[{"data":1,"prerenderedAt":93089},["ShallowReactive",2],{"post-\u002Fblog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension":3,"i-ci:tag":36456,"i-mingcute:time-line":36456},{"post":4,"nextPost":36467,"prevPost":47886},{"id":5,"title":6,"body":7,"description":13,"draft":36453,"enableCodeHighlight":36453,"enableCodeHightlight":36453,"enableComment":36454,"enableD2":36454,"enableKatex":36454,"extension":36455,"image":36419,"important":36453,"location":36456,"meta":36457,"navigation":36454,"ogImage":36456,"path":36458,"seo":36459,"stem":36460,"summary":36461,"tags":36462,"time":36465,"__hash__":36466},"blog\u002Fblog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension.md","Vapnik-Chervonenkis Dimension 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1，等于什么都没说。",[11,564,565],{},"这两个事实放在一起，说明「参数个数」和「容量」是两件不同的事。VC 维（Vapnik–Chervonenkis dimension）就是 Vapnik 与 Chervonenkis 在 1971 年为了把后者说清楚而发明的量：它不数参数，不假设分布，只看一个假设类作为「函数集合」的组合结构有多丰富。",[11,567,568],{},"本文的路线是：第一至四节把定义、Sauer–Shelah 引理、一致收敛定理这条逻辑链完整搭起来；第五至七节讨论它在深度学习里的处境——为什么界会空、空界是否意味着理论失效、今天用什么替代它；第八节清理一批常见误解。",[570,571,573],"h2",{"id":572},"一学习问题","一、学习问题",[575,576,578],"h3",{"id":577},"_11-设定","1.1 设定",[11,580,581,582,616,617,696,697,701,702,800,801,804,805,836,837,891,892,922],{},"设输入空间为 ",[15,583,585,601],{"className":584,"translate":19},[23],[15,586,588],{"className":587},[27],[29,589,590],{"xmlns":31},[34,591,592,598],{},[37,593,594],{},[43,595,597],{"mathvariant":596},"script","X",[109,599,600],{"encoding":111},"\\mathcal X",[15,602,604],{"className":603,"ariaHidden":72},[116],[15,605,607,611],{"className":606},[120],[15,608],{"className":609,"style":610},[124],"height:0.6833em;",[15,612,597],{"className":613,"style":615},[129,614],"mathcal","margin-right:0.1464em;","，标签空间为 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",[15,1286,1288,1301],{"className":1287,"translate":19},[23],[15,1289,1291],{"className":1290},[27],[29,1292,1293],{"xmlns":31},[34,1294,1295,1299],{},[37,1296,1297],{},[43,1298,906],{"mathvariant":596},[109,1300,909],{"encoding":111},[15,1302,1304],{"className":1303,"ariaHidden":72},[116],[15,1305,1307,1310],{"className":1306},[120],[15,1308],{"className":1309,"style":610},[124],[15,1311,906],{"className":1312,"style":170},[129,614],"。",[11,1315,1316,1317,1345,1346,1349,1350,1353],{},"假设 ",[15,1318,1320,1333],{"className":1319,"translate":19},[23],[15,1321,1323],{"className":1322},[27],[29,1324,1325],{"xmlns":31},[34,1326,1327,1331],{},[37,1328,1329],{},[43,1330,45],{},[109,1332,45],{"encoding":111},[15,1334,1336],{"className":1335,"ariaHidden":72},[116],[15,1337,1339,1342],{"className":1338},[120],[15,1340],{"className":1341,"style":522},[124],[15,1343,45],{"className":1344},[129,133]," 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y_i\\bigr].",[15,1499,1501,1569,1708,1790,2049],{"className":1500,"ariaHidden":72},[116],[15,1502,1504,1507,1551,1554,1557,1560,1563,1566],{"className":1503},[120],[15,1505],{"className":1506,"style":125},[124],[15,1508,1510,1513],{"className":1509},[129],[15,1511,1373],{"className":1512},[129,133],[15,1514,1516],{"className":1515},[137],[15,1517,1519,1543],{"className":1518},[141,142],[15,1520,1522,1540],{"className":1521},[146],[15,1523,1526],{"className":1524,"style":1525},[150],"height:0.3283em;",[15,1527,1528,1531],{"style":154},[15,1529],{"className":1530,"style":159},[158],[15,1532,1534],{"className":1533},[163,164,165,166],[15,1535,1537],{"className":1536},[129,166],[15,1538,906],{"className":1539,"style":170},[129,614,166],[15,1541,175],{"className":1542},[174],[15,1544,1546],{"className":1545},[146],[15,1547,1549],{"className":1548,"style":182},[150],[15,1550],{},[15,1552,53],{"className":1553},[188],[15,1555,45],{"className":1556},[129,133],[15,1558,59],{"className":1559},[195],[15,1561],{"className":1562,"style":199},[93],[15,1564,62],{"className":1565},[203],[15,1567],{"className":1568,"style":199},[93],[15,1570,1572,1576,1640,1646,1649,1652,1655,1658,1661,1705],{"className":1571},[120],[15,1573],{"className":1574,"style":1575},[124],"height:1.2052em;vertical-align:-0.3552em;",[15,1577,1579,1582],{"className":1578},[129],[15,1580,1388],{"className":1581},[129,285],[15,1583,1585],{"className":1584},[137],[15,1586,1588,1631],{"className":1587},[141,142],[15,1589,1591,1628],{"className":1590},[146],[15,1592,1595],{"className":1593,"style":1594},[150],"height:0.3448em;",[15,1596,1598,1601],{"style":1597},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[15,1599],{"className":1600,"style":159},[158],[15,1602,1604],{"className":1603},[163,164,165,166],[15,1605,1607,1610,1613,1616,1619,1622,1625],{"className":1606},[129,166],[15,1608,53],{"className":1609},[188,166],[15,1611,56],{"className":1612},[129,133,166],[15,1614,91],{"className":1615},[254,166],[15,1617,857],{"className":1618,"style":887},[129,133,166],[15,1620,59],{"className":1621},[195,166],[15,1623,991],{"className":1624},[203,166],[15,1626,906],{"className":1627,"style":170},[129,614,166],[15,1629,175],{"className":1630},[174],[15,1632,1634],{"className":1633},[146],[15,1635,1638],{"className":1636,"style":1637},[150],"height:0.3552em;",[15,1639],{},[15,1641,1643],{"className":1642},[188],[15,1644,1407],{"className":1645},[228,229],[15,1647,45],{"className":1648},[129,133],[15,1650,53],{"className":1651},[188],[15,1653,56],{"className":1654},[129,133],[15,1656,59],{"className":1657},[195],[15,1659],{"className":1660,"style":199},[93],[15,1662,1664,1698,1702],{"className":1663},[203],[15,1665,1667],{"className":1666},[203],[15,1668,1671],{"className":1669},[129,1670],"vbox",[15,1672,1675],{"className":1673},[1674],"thinbox",[15,1676,1679,1683,1694],{"className":1677},[1678],"rlap",[15,1680],{"className":1681,"style":1682},[124],"height:0.8889em;vertical-align:-0.1944em;",[15,1684,1687],{"className":1685},[1686],"inner",[15,1688,1690],{"className":1689},[129],[15,1691,1693],{"className":1692},[203],"",[15,1695],{"className":1696},[1697],"fix",[15,1699],{"className":1700},[93,1701],"nobreak",[15,1703,62],{"className":1704},[203],[15,1706],{"className":1707,"style":199},[93],[15,1709,1711,1714,1717,1723,1726,1729,1732,1772,1775,1778,1781,1784,1787],{"className":1710},[120],[15,1712],{"className":1713,"style":213},[124],[15,1715,857],{"className":1716,"style":887},[129,133],[15,1718,1720],{"className":1719},[195],[15,1721,1423],{"className":1722},[228,229],[15,1724,91],{"className":1725},[254],[15,1727],{"className":1728,"style":258},[93],[15,1730],{"className":1731,"style":262},[93],[15,1733,1735,1738],{"className":1734},[129],[15,1736,1373],{"className":1737},[129,133],[15,1739,1741],{"className":1740},[137],[15,1742,1744,1764],{"className":1743},[141,142],[15,1745,1747,1761],{"className":1746},[146],[15,1748,1750],{"className":1749,"style":1525},[150],[15,1751,1752,1755],{"style":154},[15,1753],{"className":1754,"style":159},[158],[15,1756,1758],{"className":1757},[163,164,165,166],[15,1759,940],{"className":1760,"style":1013},[129,133,166],[15,1762,175],{"className":1763},[174],[15,1765,1767],{"className":1766},[146],[15,1768,1770],{"className":1769,"style":182},[150],[15,1771],{},[15,1773,53],{"className":1774},[188],[15,1776,45],{"className":1777},[129,133],[15,1779,59],{"className":1780},[195],[15,1782],{"className":1783,"style":199},[93],[15,1785,62],{"className":1786},[203],[15,1788],{"className":1789,"style":199},[93],[15,1791,1793,1797,1868,1871,1944,1947,1951,1957,1960,1963,2004,2007,2010,2046],{"className":1792},[120],[15,1794],{"className":1795,"style":1796},[124],"height:2.9291em;vertical-align:-1.2777em;",[15,1798,1800,1804,1865],{"className":1799},[129],[15,1801],{"className":1802},[188,1803],"nulldelimiter",[15,1805,1807],{"className":1806},[1443],[15,1808,1810,1856],{"className":1809},[141,142],[15,1811,1813,1853],{"className":1812},[146],[15,1814,1817,1830,1841],{"className":1815,"style":1816},[150],"height:1.3214em;",[15,1818,1820,1824],{"style":1819},"top:-2.314em;",[15,1821],{"className":1822,"style":1823},[158],"height:3em;",[15,1825,1827],{"className":1826},[129],[15,1828,348],{"className":1829},[129,133],[15,1831,1833,1836],{"style":1832},"top:-3.23em;",[15,1834],{"className":1835,"style":1823},[158],[15,1837],{"className":1838,"style":1840},[1839],"frac-line","border-bottom-width:0.04em;",[15,1842,1844,1847],{"style":1843},"top:-3.677em;",[15,1845],{"className":1846,"style":1823},[158],[15,1848,1850],{"className":1849},[129],[15,1851,318],{"className":1852},[129],[15,1854,175],{"className":1855},[174],[15,1857,1859],{"className":1858},[146],[15,1860,1863],{"className":1861,"style":1862},[150],"height:0.686em;",[15,1864],{},[15,1866],{"className":1867},[195,1803],[15,1869],{"className":1870,"style":262},[93],[15,1872,1875],{"className":1873},[217,1874],"op-limits",[15,1876,1878,1935],{"className":1877},[141,142],[15,1879,1881,1932],{"className":1880},[146],[15,1882,1885,1907,1920],{"className":1883,"style":1884},[150],"height:1.6514em;",[15,1886,1888,1892],{"style":1887},"top:-1.8723em;margin-left:0em;",[15,1889],{"className":1890,"style":1891},[158],"height:3.05em;",[15,1893,1895],{"className":1894},[163,164,165,166],[15,1896,1898,1901,1904],{"className":1897},[129,166],[15,1899,1459],{"className":1900},[129,133,166],[15,1902,62],{"className":1903},[203,166],[15,1905,318],{"className":1906},[129,166],[15,1908,1910,1913],{"style":1909},"top:-3.05em;",[15,1911],{"className":1912,"style":1891},[158],[15,1914,1915],{},[15,1916,1454],{"className":1917},[217,1918,1919],"op-symbol","large-op",[15,1921,1923,1926],{"style":1922},"top:-4.3em;margin-left:0em;",[15,1924],{"className":1925,"style":1891},[158],[15,1927,1929],{"className":1928},[163,164,165,166],[15,1930,348],{"className":1931},[129,133,166],[15,1933,175],{"className":1934},[174],[15,1936,1938],{"className":1937},[146],[15,1939,1942],{"className":1940,"style":1941},[150],"height:1.2777em;",[15,1943],{},[15,1945],{"className":1946,"style":262},[93],[15,1948,318],{"className":1949},[129,1950],"mathbf",[15,1952,1954],{"className":1953},[188],[15,1955,1407],{"className":1956},[228,229],[15,1958,45],{"className":1959},[129,133],[15,1961,53],{"className":1962},[188],[15,1964,1966,1969],{"className":1965},[129],[15,1967,56],{"className":1968},[129,133],[15,1970,1972],{"className":1971},[137],[15,1973,1975,1996],{"className":1974},[141,142],[15,1976,1978,1993],{"className":1977},[146],[15,1979,1982],{"className":1980,"style":1981},[150],"height:0.3117em;",[15,1983,1984,1987],{"style":154},[15,1985],{"className":1986,"style":159},[158],[15,1988,1990],{"className":1989},[163,164,165,166],[15,1991,1459],{"className":1992},[129,133,166],[15,1994,175],{"className":1995},[174],[15,1997,1999],{"className":1998},[146],[15,2000,2002],{"className":2001,"style":182},[150],[15,2003],{},[15,2005,59],{"className":2006},[195],[15,2008],{"className":2009,"style":199},[93],[15,2011,2013,2040,2043],{"className":2012},[203],[15,2014,2016],{"className":2015},[203],[15,2017,2019],{"className":2018},[129,1670],[15,2020,2022],{"className":2021},[1674],[15,2023,2025,2028,2037],{"className":2024},[1678],[15,2026],{"className":2027,"style":1682},[124],[15,2029,2031],{"className":2030},[1686],[15,2032,2034],{"className":2033},[129],[15,2035,1693],{"className":2036},[203],[15,2038],{"className":2039},[1697],[15,2041],{"className":2042},[93,1701],[15,2044,62],{"className":2045},[203],[15,2047],{"className":2048,"style":199},[93],[15,2050,2052,2055,2095,2101],{"className":2051},[120],[15,2053],{"className":2054,"style":213},[124],[15,2056,2058,2061],{"className":2057},[129],[15,2059,857],{"className":2060,"style":887},[129,133],[15,2062,2064],{"className":2063},[137],[15,2065,2067,2087],{"className":2066},[141,142],[15,2068,2070,2084],{"className":2069},[146],[15,2071,2073],{"className":2072,"style":1981},[150],[15,2074,2075,2078],{"style":1100},[15,2076],{"className":2077,"style":159},[158],[15,2079,2081],{"className":2080},[163,164,165,166],[15,2082,1459],{"className":2083},[129,133,166],[15,2085,175],{"className":2086},[174],[15,2088,2090],{"className":2089},[146],[15,2091,2093],{"className":2092,"style":182},[150],[15,2094],{},[15,2096,2098],{"className":2097},[195],[15,2099,1423],{"className":2100},[228,229],[15,2102,107],{"className":2103},[129],[11,2105,2106,2107,2110],{},"前者是我们真正关心但看不见的量，后者是能算出来的量。",[698,2108,2109],{},"经验风险最小化","（ERM）只做一件事：",[15,2112,2114],{"className":2113,"translate":19},[18],[15,2115,2117,2174],{"className":2116,"translate":19},[23],[15,2118,2120],{"className":2119},[27],[29,2121,2122],{"xmlns":31,"display":32},[34,2123,2124,2171],{},[37,2125,2126,2132,2134,2137,2139,2157,2163,2165,2167,2169],{},[40,2127,2128,2130],{},[43,2129,45],{},[43,2131,940],{},[50,2133,100],{},[43,2135,2136],{},"arg",[50,2138,69],{},[2140,2141,2142,2149],"munder",{},[37,2143,2144,2147],{},[43,2145,2146],{},"min",[50,2148,69],{},[37,2150,2151,2153,2155],{},[43,2152,45],{},[50,2154,100],{},[43,2156,819],{"mathvariant":596},[40,2158,2159,2161],{},[43,2160,1373],{},[43,2162,940],{},[50,2164,53],{"stretchy":52},[43,2166,45],{},[50,2168,59],{"stretchy":52},[43,2170,107],{"mathvariant":65},[109,2172,2173],{"encoding":111},"h_S\\in\\arg\\min_{h\\in\\mathcal H}L_S(h).",[15,2175,2177,2233],{"className":2176,"ariaHidden":72},[116],[15,2178,2180,2184,2224,2227,2230],{"className":2179},[120],[15,2181],{"className":2182,"style":2183},[124],"height:0.8444em;vertical-align:-0.15em;",[15,2185,2187,2190],{"className":2186},[129],[15,2188,45],{"className":2189},[129,133],[15,2191,2193],{"className":2192},[137],[15,2194,2196,2216],{"className":2195},[141,142],[15,2197,2199,2213],{"className":2198},[146],[15,2200,2202],{"className":2201,"style":1525},[150],[15,2203,2204,2207],{"style":154},[15,2205],{"className":2206,"style":159},[158],[15,2208,2210],{"className":2209},[163,164,165,166],[15,2211,940],{"className":2212,"style":1013},[129,133,166],[15,2214,175],{"className":2215},[174],[15,2217,2219],{"className":2218},[146],[15,2220,2222],{"className":2221,"style":182},[150],[15,2223],{},[15,2225],{"className":2226,"style":199},[93],[15,2228,100],{"className":2229},[203],[15,2231],{"className":2232,"style":199},[93],[15,2234,2236,2240,2248,2251,2308,2311,2351,2354,2357,2360],{"className":2235},[120],[15,2237],{"className":2238,"style":2239},[124],"height:1.5295em;vertical-align:-0.7795em;",[15,2241,2243,2244],{"className":2242},[217],"ar",[15,2245,2247],{"style":2246},"margin-right:0.0139em;","g",[15,2249],{"className":2250,"style":262},[93],[15,2252,2254],{"className":2253},[217,1874],[15,2255,2257,2299],{"className":2256},[141,142],[15,2258,2260,2296],{"className":2259},[146],[15,2261,2264,2285],{"className":2262,"style":2263},[150],"height:0.6679em;",[15,2265,2267,2270],{"style":2266},"top:-2.3479em;margin-left:0em;",[15,2268],{"className":2269,"style":1823},[158],[15,2271,2273],{"className":2272},[163,164,165,166],[15,2274,2276,2279,2282],{"className":2275},[129,166],[15,2277,45],{"className":2278},[129,133,166],[15,2280,100],{"className":2281},[203,166],[15,2283,819],{"className":2284,"style":835},[129,614,166],[15,2286,2288,2291],{"style":2287},"top:-3em;",[15,2289],{"className":2290,"style":1823},[158],[15,2292,2293],{},[15,2294,2146],{"className":2295},[217],[15,2297,175],{"className":2298},[174],[15,2300,2302],{"className":2301},[146],[15,2303,2306],{"className":2304,"style":2305},[150],"height:0.7795em;",[15,2307],{},[15,2309],{"className":2310,"style":262},[93],[15,2312,2314,2317],{"className":2313},[129],[15,2315,1373],{"className":2316},[129,133],[15,2318,2320],{"className":2319},[137],[15,2321,2323,2343],{"className":2322},[141,142],[15,2324,2326,2340],{"className":2325},[146],[15,2327,2329],{"className":2328,"style":1525},[150],[15,2330,2331,2334],{"style":154},[15,2332],{"className":2333,"style":159},[158],[15,2335,2337],{"className":2336},[163,164,165,166],[15,2338,940],{"className":2339,"style":1013},[129,133,166],[15,2341,175],{"className":2342},[174],[15,2344,2346],{"className":2345},[146],[15,2347,2349],{"className":2348,"style":182},[150],[15,2350],{},[15,2352,53],{"className":2353},[188],[15,2355,45],{"className":2356},[129,133],[15,2358,59],{"className":2359},[195],[15,2361,107],{"className":2362},[129],[575,2364,2366],{"id":2365},"_12-误差分解","1.2 误差分解",[11,2368,2369,2370,2455,2456,2585],{},"对任意 ",[15,2371,2373,2396],{"className":2372,"translate":19},[23],[15,2374,2376],{"className":2375},[27],[29,2377,2378],{"xmlns":31},[34,2379,2380,2393],{},[37,2381,2382,2389,2391],{},[305,2383,2384,2386],{},[43,2385,45],{},[50,2387,2388],{},"∗",[50,2390,100],{},[43,2392,819],{"mathvariant":596},[109,2394,2395],{"encoding":111},"h^*\\in\\mathcal H",[15,2397,2399,2446],{"className":2398,"ariaHidden":72},[116],[15,2400,2402,2406,2437,2440,2443],{"className":2401},[120],[15,2403],{"className":2404,"style":2405},[124],"height:0.7335em;vertical-align:-0.0391em;",[15,2407,2409,2412],{"className":2408},[129],[15,2410,45],{"className":2411},[129,133],[15,2413,2415],{"className":2414},[137],[15,2416,2418],{"className":2417},[141],[15,2419,2421],{"className":2420},[146],[15,2422,2425],{"className":2423,"style":2424},[150],"height:0.6887em;",[15,2426,2427,2430],{"style":383},[15,2428],{"className":2429,"style":159},[158],[15,2431,2433],{"className":2432},[163,164,165,166],[15,2434,2388],{"className":2435},[2436,166],"mbin",[15,2438],{"className":2439,"style":199},[93],[15,2441,100],{"className":2442},[203],[15,2444],{"className":2445,"style":199},[93],[15,2447,2449,2452],{"className":2448},[120],[15,2450],{"className":2451,"style":610},[124],[15,2453,819],{"className":2454,"style":835},[129,614],"，把 ",[15,2457,2459,2487],{"className":2458,"translate":19},[23],[15,2460,2462],{"className":2461},[27],[29,2463,2464],{"xmlns":31},[34,2465,2466,2484],{},[37,2467,2468,2474,2476,2482],{},[40,2469,2470,2472],{},[43,2471,1373],{},[43,2473,906],{"mathvariant":596},[50,2475,53],{"stretchy":52},[40,2477,2478,2480],{},[43,2479,45],{},[43,2481,940],{},[50,2483,59],{"stretchy":52},[109,2485,2486],{"encoding":111},"L_{\\mathcal D}(h_S)",[15,2488,2490],{"className":2489,"ariaHidden":72},[116],[15,2491,2493,2496,2539,2542,2582],{"className":2492},[120],[15,2494],{"className":2495,"style":125},[124],[15,2497,2499,2502],{"className":2498},[129],[15,2500,1373],{"className":2501},[129,133],[15,2503,2505],{"className":2504},[137],[15,2506,2508,2531],{"className":2507},[141,142],[15,2509,2511,2528],{"className":2510},[146],[15,2512,2514],{"className":2513,"style":1525},[150],[15,2515,2516,2519],{"style":154},[15,2517],{"className":2518,"style":159},[158],[15,2520,2522],{"className":2521},[163,164,165,166],[15,2523,2525],{"className":2524},[129,166],[15,2526,906],{"className":2527,"style":170},[129,614,166],[15,2529,175],{"className":2530},[174],[15,2532,2534],{"className":2533},[146],[15,2535,2537],{"className":2536,"style":182},[150],[15,2538],{},[15,2540,53],{"className":2541},[188],[15,2543,2545,2548],{"className":2544},[129],[15,2546,45],{"className":2547},[129,133],[15,2549,2551],{"className":2550},[137],[15,2552,2554,2574],{"className":2553},[141,142],[15,2555,2557,2571],{"className":2556},[146],[15,2558,2560],{"className":2559,"style":1525},[150],[15,2561,2562,2565],{"style":154},[15,2563],{"className":2564,"style":159},[158],[15,2566,2568],{"className":2567},[163,164,165,166],[15,2569,940],{"className":2570,"style":1013},[129,133,166],[15,2572,175],{"className":2573},[174],[15,2575,2577],{"className":2576},[146],[15,2578,2580],{"className":2579,"style":182},[150],[15,2581],{},[15,2583,59],{"className":2584},[195]," 拆开：",[15,2587,2589],{"className":2588,"translate":19},[18],[15,2590,2592,2979],{"className":2591,"translate":19},[23],[15,2593,2595],{"className":2594},[27],[29,2596,2597],{"xmlns":31,"display":32},[34,2598,2599,2976],{},[2600,2601,2605,2699,2774,2888],"mtable",{"rowspacing":2602,"columnalign":2603,"columnspacing":2604},"0.25em","right left","0em",[2606,2607,2608,2632],"mtr",{},[2609,2610,2611],"mtd",{},[2612,2613,2614],"mstyle",{"scriptlevel":541,"displaystyle":72},[37,2615,2616,2622,2624,2630],{},[40,2617,2618,2620],{},[43,2619,1373],{},[43,2621,906],{"mathvariant":596},[50,2623,53],{"stretchy":52},[40,2625,2626,2628],{},[43,2627,45],{},[43,2629,940],{},[50,2631,59],{"stretchy":52},[2609,2633,2634],{},[2612,2635,2636],{"scriptlevel":541,"displaystyle":72},[37,2637,2638,2640,2642,2644,2650,2652,2658,2660,2662,2668,2670,2676,2678,2680,2683,2689,2691,2697],{},[37,2639],{},[50,2641,62],{},[50,2643,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,2645,2646,2648],{},[43,2647,1373],{},[43,2649,906],{"mathvariant":596},[50,2651,53],{"stretchy":52},[40,2653,2654,2656],{},[43,2655,45],{},[43,2657,940],{},[50,2659,59],{"stretchy":52},[50,2661,315],{},[40,2663,2664,2666],{},[43,2665,1373],{},[43,2667,940],{},[50,2669,53],{"stretchy":52},[40,2671,2672,2674],{},[43,2673,45],{},[43,2675,940],{},[50,2677,59],{"stretchy":52},[50,2679,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,2681,2682],{},"+",[40,2684,2685,2687],{},[43,2686,1373],{},[43,2688,940],{},[50,2690,53],{"stretchy":52},[40,2692,2693,2695],{},[43,2694,45],{},[43,2696,940],{},[50,2698,59],{"stretchy":52},[2606,2700,2701,2707],{},[2609,2702,2703],{},[2612,2704,2705],{"scriptlevel":541,"displaystyle":72},[37,2706],{},[2609,2708,2709],{},[2612,2710,2711],{"scriptlevel":541,"displaystyle":72},[37,2712,2713,2715,2718,2720,2726,2728,2734,2736,2738,2744,2746,2752,2754,2756,2758,2764,2766,2772],{},[37,2714],{},[50,2716,2717],{},"≤",[50,2719,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,2721,2722,2724],{},[43,2723,1373],{},[43,2725,906],{"mathvariant":596},[50,2727,53],{"stretchy":52},[40,2729,2730,2732],{},[43,2731,45],{},[43,2733,940],{},[50,2735,59],{"stretchy":52},[50,2737,315],{},[40,2739,2740,2742],{},[43,2741,1373],{},[43,2743,940],{},[50,2745,53],{"stretchy":52},[40,2747,2748,2750],{},[43,2749,45],{},[43,2751,940],{},[50,2753,59],{"stretchy":52},[50,2755,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,2757,2682],{},[40,2759,2760,2762],{},[43,2761,1373],{},[43,2763,940],{},[50,2765,53],{"stretchy":52},[305,2767,2768,2770],{},[43,2769,45],{},[50,2771,2388],{},[50,2773,59],{"stretchy":52},[2606,2775,2776,2782],{},[2609,2777,2778],{},[2612,2779,2780],{"scriptlevel":541,"displaystyle":72},[37,2781],{},[2609,2783,2784],{},[2612,2785,2786],{"scriptlevel":541,"displaystyle":72},[37,2787,2788,2790,2792,2794,2800,2802,2808,2810,2812,2818,2820,2826,2828,2830,2832,2834,2840,2842,2848,2850,2852,2858,2860,2866,2868,2870,2872,2878,2880,2886],{},[37,2789],{},[50,2791,62],{},[50,2793,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,2795,2796,2798],{},[43,2797,1373],{},[43,2799,906],{"mathvariant":596},[50,2801,53],{"stretchy":52},[40,2803,2804,2806],{},[43,2805,45],{},[43,2807,940],{},[50,2809,59],{"stretchy":52},[50,2811,315],{},[40,2813,2814,2816],{},[43,2815,1373],{},[43,2817,940],{},[50,2819,53],{"stretchy":52},[40,2821,2822,2824],{},[43,2823,45],{},[43,2825,940],{},[50,2827,59],{"stretchy":52},[50,2829,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,2831,2682],{},[50,2833,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,2835,2836,2838],{},[43,2837,1373],{},[43,2839,940],{},[50,2841,53],{"stretchy":52},[305,2843,2844,2846],{},[43,2845,45],{},[50,2847,2388],{},[50,2849,59],{"stretchy":52},[50,2851,315],{},[40,2853,2854,2856],{},[43,2855,1373],{},[43,2857,906],{"mathvariant":596},[50,2859,53],{"stretchy":52},[305,2861,2862,2864],{},[43,2863,45],{},[50,2865,2388],{},[50,2867,59],{"stretchy":52},[50,2869,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,2871,2682],{},[40,2873,2874,2876],{},[43,2875,1373],{},[43,2877,906],{"mathvariant":596},[50,2879,53],{"stretchy":52},[305,2881,2882,2884],{},[43,2883,45],{},[50,2885,2388],{},[50,2887,59],{"stretchy":52},[2606,2889,2890,2896],{},[2609,2891,2892],{},[2612,2893,2894],{"scriptlevel":541,"displaystyle":72},[37,2895],{},[2609,2897,2898],{},[2612,2899,2900],{"scriptlevel":541,"displaystyle":72},[37,2901,2902,2904,2906,2912,2914,2920,2922,2924,2926,2943,2946,2952,2954,2956,2958,2960,2966,2968,2970,2972,2974],{},[37,2903],{},[50,2905,2717],{},[40,2907,2908,2910],{},[43,2909,1373],{},[43,2911,906],{"mathvariant":596},[50,2913,53],{"stretchy":52},[305,2915,2916,2918],{},[43,2917,45],{},[50,2919,2388],{},[50,2921,59],{"stretchy":52},[50,2923,2682],{},[308,2925,310],{},[2140,2927,2928,2935],{},[37,2929,2930,2933],{},[43,2931,2932],{},"sup",[50,2934,69],{},[37,2936,2937,2939,2941],{},[43,2938,45],{},[50,2940,100],{},[43,2942,819],{"mathvariant":596},[50,2944,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},"∣",[40,2947,2948,2950],{},[43,2949,1373],{},[43,2951,906],{"mathvariant":596},[50,2953,53],{"stretchy":52},[43,2955,45],{},[50,2957,59],{"stretchy":52},[50,2959,315],{},[40,2961,2962,2964],{},[43,2963,1373],{},[43,2965,940],{},[50,2967,53],{"stretchy":52},[43,2969,45],{},[50,2971,59],{"stretchy":52},[50,2973,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,2975,107],{"mathvariant":65},[109,2977,2978],{"encoding":111},"\\begin{aligned}\nL_{\\mathcal D}(h_S)\n&=\\bigl[L_{\\mathcal D}(h_S)-L_S(h_S)\\bigr]+L_S(h_S)\\\\\n&\\le \\bigl[L_{\\mathcal D}(h_S)-L_S(h_S)\\bigr]+L_S(h^*)\\\\\n&=\\bigl[L_{\\mathcal D}(h_S)-L_S(h_S)\\bigr]+\\bigl[L_S(h^*)-L_{\\mathcal D}(h^*)\\bigr]+L_{\\mathcal D}(h^*)\\\\\n&\\le L_{\\mathcal D}(h^*)+2\\sup_{h\\in\\mathcal H}\\bigl|L_{\\mathcal 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convergence）。",[11,5370,5371,5403,5404,5432,5433,5461,5462,5532,5533,5561],{},[15,5372,5374,5390],{"className":5373,"translate":19},[23],[15,5375,5377],{"className":5376},[27],[29,5378,5379],{"xmlns":31},[34,5380,5381,5387],{},[37,5382,5383,5385],{},[43,5384,2932],{},[50,5386,69],{},[109,5388,5389],{"encoding":111},"\\sup",[15,5391,5393],{"className":5392,"ariaHidden":72},[116],[15,5394,5396,5400],{"className":5395},[120],[15,5397],{"className":5398,"style":5399},[124],"height:0.625em;vertical-align:-0.1944em;",[15,5401,2932],{"className":5402},[217]," 在概率里面，这个顺序很关键：不是「对每个 ",[15,5405,5407,5420],{"className":5406,"translate":19},[23],[15,5408,5410],{"className":5409},[27],[29,5411,5412],{"xmlns":31},[34,5413,5414,5418],{},[37,5415,5416],{},[43,5417,45],{},[109,5419,45],{"encoding":111},[15,5421,5423],{"className":5422,"ariaHidden":72},[116],[15,5424,5426,5429],{"className":5425},[120],[15,5427],{"className":5428,"style":522},[124],[15,5430,45],{"className":5431},[129,133]," 分别收敛」，而是「所有 ",[15,5434,5436,5449],{"className":5435,"translate":19},[23],[15,5437,5439],{"className":5438},[27],[29,5440,5441],{"xmlns":31},[34,5442,5443,5447],{},[37,5444,5445],{},[43,5446,45],{},[109,5448,45],{"encoding":111},[15,5450,5452],{"className":5451,"ariaHidden":72},[116],[15,5453,5455,5458],{"className":5454},[120],[15,5456],{"className":5457,"style":522},[124],[15,5459,45],{"className":5460},[129,133]," 同时收敛」。前者由大数定律自动成立，后者才是学习所需要的——因为 ",[15,5463,5465,5483],{"className":5464,"translate":19},[23],[15,5466,5468],{"className":5467},[27],[29,5469,5470],{"xmlns":31},[34,5471,5472,5480],{},[37,5473,5474],{},[40,5475,5476,5478],{},[43,5477,45],{},[43,5479,940],{},[109,5481,5482],{"encoding":111},"h_S",[15,5484,5486],{"className":5485,"ariaHidden":72},[116],[15,5487,5489,5492],{"className":5488},[120],[15,5490],{"className":5491,"style":2183},[124],[15,5493,5495,5498],{"className":5494},[129],[15,5496,45],{"className":5497},[129,133],[15,5499,5501],{"className":5500},[137],[15,5502,5504,5524],{"className":5503},[141,142],[15,5505,5507,5521],{"className":5506},[146],[15,5508,5510],{"className":5509,"style":1525},[150],[15,5511,5512,5515],{"style":154},[15,5513],{"className":5514,"style":159},[158],[15,5516,5518],{"className":5517},[163,164,165,166],[15,5519,940],{"className":5520,"style":1013},[129,133,166],[15,5522,175],{"className":5523},[174],[15,5525,5527],{"className":5526},[146],[15,5528,5530],{"className":5529,"style":182},[150],[15,5531],{}," 本身依赖数据，你不能先固定 ",[15,5534,5536,5549],{"className":5535,"translate":19},[23],[15,5537,5539],{"className":5538},[27],[29,5540,5541],{"xmlns":31},[34,5542,5543,5547],{},[37,5544,5545],{},[43,5546,45],{},[109,5548,45],{"encoding":111},[15,5550,5552],{"className":5551,"ariaHidden":72},[116],[15,5553,5555,5558],{"className":5554},[120],[15,5556],{"className":5557,"style":522},[124],[15,5559,45],{"className":5560},[129,133]," 再用大数定律。",[11,5563,5564,5565,5593],{},"于是问题变成：什么样的 ",[15,5566,5568,5581],{"className":5567,"translate":19},[23],[15,5569,5571],{"className":5570},[27],[29,5572,5573],{"xmlns":31},[34,5574,5575,5579],{},[37,5576,5577],{},[43,5578,819],{"mathvariant":596},[109,5580,822],{"encoding":111},[15,5582,5584],{"className":5583,"ariaHidden":72},[116],[15,5585,5587,5590],{"className":5586},[120],[15,5588],{"className":5589,"style":610},[124],[15,5591,819],{"className":5592,"style":835},[129,614]," 具有一致收敛性？答案由 VC 维给出。",[570,5595,5597],{"id":5596},"二打散与-vc-维","二、打散与 VC 维",[575,5599,5601],{"id":5600},"_21-二分","2.1 二分",[11,5603,5604,5605,1313,5803,5831,5832,5860,5861,5864],{},"固定一个有限的点集 ",[15,5606,5608,5653],{"className":5607,"translate":19},[23],[15,5609,5611],{"className":5610},[27],[29,5612,5613],{"xmlns":31},[34,5614,5615,5650],{},[37,5616,5617,5620,5622,5624,5631,5633,5635,5637,5643,5645,5648],{},[43,5618,5619],{},"C",[50,5621,62],{},[50,5623,636],{"stretchy":52},[40,5625,5626,5629],{},[43,5627,5628],{},"c",[308,5630,318],{},[50,5632,91],{"separator":72},[50,5634,335],{},[50,5636,91],{"separator":72},[40,5638,5639,5641],{},[43,5640,5628],{},[43,5642,348],{},[50,5644,645],{"stretchy":52},[50,5646,5647],{},"⊆",[43,5649,597],{"mathvariant":596},[109,5651,5652],{"encoding":111},"C=\\{c_1,\\dots,c_m\\}\\subseteq\\mathcal X",[15,5654,5656,5675,5794],{"className":5655,"ariaHidden":72},[116],[15,5657,5659,5662,5666,5669,5672],{"className":5658},[120],[15,5660],{"className":5661,"style":610},[124],[15,5663,5619],{"className":5664,"style":5665},[129,133],"margin-right:0.0715em;",[15,5667],{"className":5668,"style":199},[93],[15,5670,62],{"className":5671},[203],[15,5673],{"className":5674,"style":199},[93],[15,5676,5678,5681,5684,5724,5727,5730,5733,5736,5739,5742,5782,5785,5788,5791],{"className":5677},[120],[15,5679],{"className":5680,"style":125},[124],[15,5682,636],{"className":5683},[188],[15,5685,5687,5690],{"className":5686},[129],[15,5688,5628],{"className":5689},[129,133],[15,5691,5693],{"className":5692},[137],[15,5694,5696,5716],{"className":5695},[141,142],[15,5697,5699,5713],{"className":5698},[146],[15,5700,5702],{"className":5701,"style":1051},[150],[15,5703,5704,5707],{"style":154},[15,5705],{"className":5706,"style":159},[158],[15,5708,5710],{"className":5709},[163,164,165,166],[15,5711,318],{"className":5712},[129,166],[15,5714,175],{"className":5715},[174],[15,5717,5719],{"className":5718},[146],[15,5720,5722],{"className":5721,"style":182},[150],[15,5723],{},[15,5725,91],{"className":5726},[254],[15,5728],{"className":5729,"style":262},[93],[15,5731,335],{"className":5732},[449],[15,5734],{"className":5735,"style":262},[93],[15,5737,91],{"className":5738},[254],[15,5740],{"className":5741,"style":262},[93],[15,5743,5745,5748],{"className":5744},[129],[15,5746,5628],{"className":5747},[129,133],[15,5749,5751],{"className":5750},[137],[15,5752,5754,5774],{"className":5753},[141,142],[15,5755,5757,5771],{"className":5756},[146],[15,5758,5760],{"className":5759,"style":1163},[150],[15,5761,5762,5765],{"style":154},[15,5763],{"className":5764,"style":159},[158],[15,5766,5768],{"className":5767},[163,164,165,166],[15,5769,348],{"className":5770},[129,133,166],[15,5772,175],{"className":5773},[174],[15,5775,5777],{"className":5776},[146],[15,5778,5780],{"className":5779,"style":182},[150],[15,5781],{},[15,5783,645],{"className":5784},[195],[15,5786],{"className":5787,"style":199},[93],[15,5789,5647],{"className":5790},[203],[15,5792],{"className":5793,"style":199},[93],[15,5795,5797,5800],{"className":5796},[120],[15,5798],{"className":5799,"style":610},[124],[15,5801,597],{"className":5802,"style":615},[129,614],[15,5804,5806,5819],{"className":5805,"translate":19},[23],[15,5807,5809],{"className":5808},[27],[29,5810,5811],{"xmlns":31},[34,5812,5813,5817],{},[37,5814,5815],{},[43,5816,819],{"mathvariant":596},[109,5818,822],{"encoding":111},[15,5820,5822],{"className":5821,"ariaHidden":72},[116],[15,5823,5825,5828],{"className":5824},[120],[15,5826],{"className":5827,"style":610},[124],[15,5829,819],{"className":5830,"style":835},[129,614]," 在 ",[15,5833,5835,5848],{"className":5834,"translate":19},[23],[15,5836,5838],{"className":5837},[27],[29,5839,5840],{"xmlns":31},[34,5841,5842,5846],{},[37,5843,5844],{},[43,5845,5619],{},[109,5847,5619],{"encoding":111},[15,5849,5851],{"className":5850,"ariaHidden":72},[116],[15,5852,5854,5857],{"className":5853},[120],[15,5855],{"className":5856,"style":610},[124],[15,5858,5619],{"className":5859,"style":5665},[129,133]," 上的",[698,5862,5863],{},"限制","（restriction）是",[15,5866,5868],{"className":5867,"translate":19},[18],[15,5869,5871,5957],{"className":5870,"translate":19},[23],[15,5872,5874],{"className":5873},[27],[29,5875,5876],{"xmlns":31,"display":32},[34,5877,5878,5954],{},[37,5879,5880,5882,5888,5890,5892,5894,5896,5898,5904,5906,5908,5910,5912,5914,5916,5922,5924,5926,5928,5930,5932,5934,5936,5938,5940,5942,5944,5946,5952],{},[43,5881,819],{"mathvariant":596},[40,5883,5884,5886],{},[43,5885,2945],{"mathvariant":65},[43,5887,5619],{},[50,5889,62],{},[50,5891,636],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,5893,53],{"stretchy":52},[43,5895,45],{},[50,5897,53],{"stretchy":52},[40,5899,5900,5902],{},[43,5901,5628],{},[308,5903,318],{},[50,5905,59],{"stretchy":52},[50,5907,91],{"separator":72},[50,5909,335],{},[50,5911,91],{"separator":72},[43,5913,45],{},[50,5915,53],{"stretchy":52},[40,5917,5918,5920],{},[43,5919,5628],{},[43,5921,348],{},[50,5923,59],{"stretchy":52},[50,5925,59],{"stretchy":52},[50,5927,718],{},[43,5929,45],{},[50,5931,100],{},[43,5933,819],{"mathvariant":596},[50,5935,645],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,5937,5647],{},[50,5939,636],{"stretchy":52},[308,5941,541],{},[50,5943,91],{"separator":72},[308,5945,318],{},[305,5947,5948,5950],{},[50,5949,645],{"stretchy":52},[43,5951,348],{},[43,5953,107],{"mathvariant":65},[109,5955,5956],{"encoding":111},"\\mathcal H|_C=\\bigl\\{(h(c_1),\\dots,h(c_m)):h\\in\\mathcal H\\bigr\\}\\subseteq\\{0,1\\}^m.",[15,5958,5960,6018,6159,6177,6201],{"className":5959,"ariaHidden":72},[116],[15,5961,5963,5966,5969,6009,6012,6015],{"className":5962},[120],[15,5964],{"className":5965,"style":125},[124],[15,5967,819],{"className":5968,"style":835},[129,614],[15,5970,5972,5975],{"className":5971},[129],[15,5973,2945],{"className":5974},[129],[15,5976,5978],{"className":5977},[137],[15,5979,5981,6001],{"className":5980},[141,142],[15,5982,5984,5998],{"className":5983},[146],[15,5985,5987],{"className":5986,"style":1525},[150],[15,5988,5989,5992],{"style":154},[15,5990],{"className":5991,"style":159},[158],[15,5993,5995],{"className":5994},[163,164,165,166],[15,5996,5619],{"className":5997,"style":5665},[129,133,166],[15,5999,175],{"className":6000},[174],[15,6002,6004],{"className":6003},[146],[15,6005,6007],{"className":6006,"style":182},[150],[15,6008],{},[15,6010],{"className":6011,"style":199},[93],[15,6013,62],{"className":6014},[203],[15,6016],{"className":6017,"style":199},[93],[15,6019,6021,6024,6030,6033,6036,6039,6079,6082,6085,6088,6091,6094,6097,6100,6103,6106,6146,6150,6153,6156],{"className":6020},[120],[15,6022],{"className":6023,"style":213},[124],[15,6025,6027],{"className":6026},[188],[15,6028,636],{"className":6029},[228,229],[15,6031,53],{"className":6032},[188],[15,6034,45],{"className":6035},[129,133],[15,6037,53],{"className":6038},[188],[15,6040,6042,6045],{"className":6041},[129],[15,6043,5628],{"className":6044},[129,133],[15,6046,6048],{"className":6047},[137],[15,6049,6051,6071],{"className":6050},[141,142],[15,6052,6054,6068],{"className":6053},[146],[15,6055,6057],{"className":6056,"style":1051},[150],[15,6058,6059,6062],{"style":154},[15,6060],{"className":6061,"style":159},[158],[15,6063,6065],{"className":6064},[163,164,165,166],[15,6066,318],{"className":6067},[129,166],[15,6069,175],{"className":6070},[174],[15,6072,6074],{"className":6073},[146],[15,6075,6077],{"className":6076,"style":182},[150],[15,6078],{},[15,6080,59],{"className":6081},[195],[15,6083,91],{"className":6084},[254],[15,6086],{"className":6087,"style":262},[93],[15,6089,335],{"className":6090},[449],[15,6092],{"className":6093,"style":262},[93],[15,6095,91],{"className":6096},[254],[15,6098],{"className":6099,"style":262},[93],[15,6101,45],{"className":6102},[129,133],[15,6104,53],{"className":6105},[188],[15,6107,6109,6112],{"className":6108},[129],[15,6110,5628],{"className":6111},[129,133],[15,6113,6115],{"className":6114},[137],[15,6116,6118,6138],{"className":6117},[141,142],[15,6119,6121,6135],{"className":6120},[146],[15,6122,6124],{"className":6123,"style":1163},[150],[15,6125,6126,6129],{"style":154},[15,6127],{"className":6128,"style":159},[158],[15,6130,6132],{"className":6131},[163,164,165,166],[15,6133,348],{"className":6134},[129,133,166],[15,6136,175],{"className":6137},[174],[15,6139,6141],{"className":6140},[146],[15,6142,6144],{"className":6143,"style":182},[150],[15,6145],{},[15,6147,6149],{"className":6148},[195],"))",[15,6151],{"className":6152,"style":199},[93],[15,6154,718],{"className":6155},[203],[15,6157],{"className":6158,"style":199},[93],[15,6160,6162,6165,6168,6171,6174],{"className":6161},[120],[15,6163],{"className":6164,"style":2405},[124],[15,6166,45],{"className":6167},[129,133],[15,6169],{"className":6170,"style":199},[93],[15,6172,100],{"className":6173},[203],[15,6175],{"className":6176,"style":199},[93],[15,6178,6180,6183,6186,6192,6195,6198],{"className":6179},[120],[15,6181],{"className":6182,"style":213},[124],[15,6184,819],{"className":6185,"style":835},[129,614],[15,6187,6189],{"className":6188},[195],[15,6190,645],{"className":6191},[228,229],[15,6193],{"className":6194,"style":199},[93],[15,6196,5647],{"className":6197},[203],[15,6199],{"className":6200,"style":199},[93],[15,6202,6204,6207,6210,6213,6216,6219,6222,6251],{"className":6203},[120],[15,6205],{"className":6206,"style":125},[124],[15,6208,636],{"className":6209},[188],[15,6211,541],{"className":6212},[129],[15,6214,91],{"className":6215},[254],[15,6217],{"className":6218,"style":262},[93],[15,6220,318],{"className":6221},[129],[15,6223,6225,6228],{"className":6224},[195],[15,6226,645],{"className":6227},[195],[15,6229,6231],{"className":6230},[137],[15,6232,6234],{"className":6233},[141],[15,6235,6237],{"className":6236},[146],[15,6238,6240],{"className":6239,"style":1251},[150],[15,6241,6242,6245],{"style":1272},[15,6243],{"className":6244,"style":159},[158],[15,6246,6248],{"className":6247},[163,164,165,166],[15,6249,348],{"className":6250},[129,133,166],[15,6252,107],{"className":6253},[129],[11,6255,6256,6257,6260,6261,6346,6347,6407,6408,6466,6467,6495,6496,6528,6529,6557],{},"这个集合里的每个元素称为一个",[698,6258,6259],{},"二分","（dichotomy）。",[15,6262,6264,6288],{"className":6263,"translate":19},[23],[15,6265,6267],{"className":6266},[27],[29,6268,6269],{"xmlns":31},[34,6270,6271,6285],{},[37,6272,6273,6275,6277,6283],{},[43,6274,2945],{"mathvariant":65},[43,6276,819],{"mathvariant":596},[40,6278,6279,6281],{},[43,6280,2945],{"mathvariant":65},[43,6282,5619],{},[43,6284,2945],{"mathvariant":65},[109,6286,6287],{"encoding":111},"|\\mathcal H|_C|",[15,6289,6291],{"className":6290,"ariaHidden":72},[116],[15,6292,6294,6297,6300,6303,6343],{"className":6293},[120],[15,6295],{"className":6296,"style":125},[124],[15,6298,2945],{"className":6299},[129],[15,6301,819],{"className":6302,"style":835},[129,614],[15,6304,6306,6309],{"className":6305},[129],[15,6307,2945],{"className":6308},[129],[15,6310,6312],{"className":6311},[137],[15,6313,6315,6335],{"className":6314},[141,142],[15,6316,6318,6332],{"className":6317},[146],[15,6319,6321],{"className":6320,"style":1525},[150],[15,6322,6323,6326],{"style":154},[15,6324],{"className":6325,"style":159},[158],[15,6327,6329],{"className":6328},[163,164,165,166],[15,6330,5619],{"className":6331,"style":5665},[129,133,166],[15,6333,175],{"className":6334},[174],[15,6336,6338],{"className":6337},[146],[15,6339,6341],{"className":6340,"style":182},[150],[15,6342],{},[15,6344,2945],{"className":6345},[129]," 最多是 ",[15,6348,6350,6368],{"className":6349,"translate":19},[23],[15,6351,6353],{"className":6352},[27],[29,6354,6355],{"xmlns":31},[34,6356,6357,6365],{},[37,6358,6359],{},[305,6360,6361,6363],{},[308,6362,310],{},[43,6364,348],{},[109,6366,6367],{"encoding":111},"2^m",[15,6369,6371],{"className":6370,"ariaHidden":72},[116],[15,6372,6374,6378],{"className":6373},[120],[15,6375],{"className":6376,"style":6377},[124],"height:0.6644em;",[15,6379,6381,6384],{"className":6380},[129],[15,6382,310],{"className":6383},[129],[15,6385,6387],{"className":6386},[137],[15,6388,6390],{"className":6389},[141],[15,6391,6393],{"className":6392},[146],[15,6394,6396],{"className":6395,"style":6377},[150],[15,6397,6398,6401],{"style":383},[15,6399],{"className":6400,"style":159},[158],[15,6402,6404],{"className":6403},[163,164,165,166],[15,6405,348],{"className":6406},[129,133,166],"。如果取到 ",[15,6409,6411,6428],{"className":6410,"translate":19},[23],[15,6412,6414],{"className":6413},[27],[29,6415,6416],{"xmlns":31},[34,6417,6418,6426],{},[37,6419,6420],{},[305,6421,6422,6424],{},[308,6423,310],{},[43,6425,348],{},[109,6427,6367],{"encoding":111},[15,6429,6431],{"className":6430,"ariaHidden":72},[116],[15,6432,6434,6437],{"className":6433},[120],[15,6435],{"className":6436,"style":6377},[124],[15,6438,6440,6443],{"className":6439},[129],[15,6441,310],{"className":6442},[129],[15,6444,6446],{"className":6445},[137],[15,6447,6449],{"className":6448},[141],[15,6450,6452],{"className":6451},[146],[15,6453,6455],{"className":6454,"style":6377},[150],[15,6456,6457,6460],{"style":383},[15,6458],{"className":6459,"style":159},[158],[15,6461,6463],{"className":6462},[163,164,165,166],[15,6464,348],{"className":6465},[129,133,166],"，说明对这 ",[15,6468,6470,6483],{"className":6469,"translate":19},[23],[15,6471,6473],{"className":6472},[27],[29,6474,6475],{"xmlns":31},[34,6476,6477,6481],{},[37,6478,6479],{},[43,6480,348],{},[109,6482,348],{"encoding":111},[15,6484,6486],{"className":6485,"ariaHidden":72},[116],[15,6487,6489,6492],{"className":6488},[120],[15,6490],{"className":6491,"style":4383},[124],[15,6493,348],{"className":6494},[129,133]," 个点的每一种 ",[15,6497,6499,6516],{"className":6498,"translate":19},[23],[15,6500,6502],{"className":6501},[27],[29,6503,6504],{"xmlns":31},[34,6505,6506,6514],{},[37,6507,6508,6510,6512],{},[308,6509,541],{},[43,6511,544],{"mathvariant":65},[308,6513,318],{},[109,6515,549],{"encoding":111},[15,6517,6519],{"className":6518,"ariaHidden":72},[116],[15,6520,6522,6525],{"className":6521},[120],[15,6523],{"className":6524,"style":125},[124],[15,6526,549],{"className":6527},[129]," 标注，",[15,6530,6532,6545],{"className":6531,"translate":19},[23],[15,6533,6535],{"className":6534},[27],[29,6536,6537],{"xmlns":31},[34,6538,6539,6543],{},[37,6540,6541],{},[43,6542,819],{"mathvariant":596},[109,6544,822],{"encoding":111},[15,6546,6548],{"className":6547,"ariaHidden":72},[116],[15,6549,6551,6554],{"className":6550},[120],[15,6552],{"className":6553,"style":610},[124],[15,6555,819],{"className":6556,"style":835},[129,614]," 里都有一个假设恰好实现它。",[6559,6560,6562],"theorem",{"title":6561},"定义 2.1（打散）",[11,6563,6564,6565,6702,6703,804,6731,6734,6735,1313],{},"若 ",[15,6566,6568,6600],{"className":6567,"translate":19},[23],[15,6569,6571],{"className":6570},[27],[29,6572,6573],{"xmlns":31},[34,6574,6575,6597],{},[37,6576,6577,6579,6581,6587,6589,6591],{},[43,6578,2945],{"mathvariant":65},[43,6580,819],{"mathvariant":596},[40,6582,6583,6585],{},[43,6584,2945],{"mathvariant":65},[43,6586,5619],{},[43,6588,2945],{"mathvariant":65},[50,6590,62],{},[305,6592,6593,6595],{},[308,6594,310],{},[43,6596,348],{},[109,6598,6599],{"encoding":111},"|\\mathcal H|_C|=2^m",[15,6601,6603,6667],{"className":6602,"ariaHidden":72},[116],[15,6604,6606,6609,6612,6615,6655,6658,6661,6664],{"className":6605},[120],[15,6607],{"className":6608,"style":125},[124],[15,6610,2945],{"className":6611},[129],[15,6613,819],{"className":6614,"style":835},[129,614],[15,6616,6618,6621],{"className":6617},[129],[15,6619,2945],{"className":6620},[129],[15,6622,6624],{"className":6623},[137],[15,6625,6627,6647],{"className":6626},[141,142],[15,6628,6630,6644],{"className":6629},[146],[15,6631,6633],{"className":6632,"style":1525},[150],[15,6634,6635,6638],{"style":154},[15,6636],{"className":6637,"style":159},[158],[15,6639,6641],{"className":6640},[163,164,165,166],[15,6642,5619],{"className":6643,"style":5665},[129,133,166],[15,6645,175],{"className":6646},[174],[15,6648,6650],{"className":6649},[146],[15,6651,6653],{"className":6652,"style":182},[150],[15,6654],{},[15,6656,2945],{"className":6657},[129],[15,6659],{"className":6660,"style":199},[93],[15,6662,62],{"className":6663},[203],[15,6665],{"className":6666,"style":199},[93],[15,6668,6670,6673],{"className":6669},[120],[15,6671],{"className":6672,"style":6377},[124],[15,6674,6676,6679],{"className":6675},[129],[15,6677,310],{"className":6678},[129],[15,6680,6682],{"className":6681},[137],[15,6683,6685],{"className":6684},[141],[15,6686,6688],{"className":6687},[146],[15,6689,6691],{"className":6690,"style":6377},[150],[15,6692,6693,6696],{"style":383},[15,6694],{"className":6695,"style":159},[158],[15,6697,6699],{"className":6698},[163,164,165,166],[15,6700,348],{"className":6701},[129,133,166],"，称 ",[15,6704,6706,6719],{"className":6705,"translate":19},[23],[15,6707,6709],{"className":6708},[27],[29,6710,6711],{"xmlns":31},[34,6712,6713,6717],{},[37,6714,6715],{},[43,6716,819],{"mathvariant":596},[109,6718,822],{"encoding":111},[15,6720,6722],{"className":6721,"ariaHidden":72},[116],[15,6723,6725,6728],{"className":6724},[120],[15,6726],{"className":6727,"style":610},[124],[15,6729,819],{"className":6730,"style":835},[129,614],[698,6732,6733],{},"打散","（shatter）了 ",[15,6736,6738,6751],{"className":6737,"translate":19},[23],[15,6739,6741],{"className":6740},[27],[29,6742,6743],{"xmlns":31},[34,6744,6745,6749],{},[37,6746,6747],{},[43,6748,5619],{},[109,6750,5619],{"encoding":111},[15,6752,6754],{"className":6753,"ariaHidden":72},[116],[15,6755,6757,6760],{"className":6756},[120],[15,6758],{"className":6759,"style":610},[124],[15,6761,5619],{"className":6762,"style":5665},[129,133],[6559,6764,6766,6802,7134],{"title":6765},"定义 2.2（VC 维）",[11,6767,6768,6769,6797,6798,6801],{},"假设类 ",[15,6770,6772,6785],{"className":6771,"translate":19},[23],[15,6773,6775],{"className":6774},[27],[29,6776,6777],{"xmlns":31},[34,6778,6779,6783],{},[37,6780,6781],{},[43,6782,819],{"mathvariant":596},[109,6784,822],{"encoding":111},[15,6786,6788],{"className":6787,"ariaHidden":72},[116],[15,6789,6791,6794],{"className":6790},[120],[15,6792],{"className":6793,"style":610},[124],[15,6795,819],{"className":6796,"style":835},[129,614]," 的 ",[698,6799,6800],{},"VC 维","是它能打散的最大有限集的大小：",[15,6803,6805],{"className":6804,"translate":19},[18],[15,6806,6808,6896],{"className":6807,"translate":19},[23],[15,6809,6811],{"className":6810},[27],[29,6812,6813],{"xmlns":31,"display":32},[34,6814,6815,6893],{},[37,6816,6817,6820,6822,6824,6826,6828,6830,6832,6834,6836,6838,6840,6843,6845,6847,6849,6851,6855,6857,6859,6861,6863,6865,6867,6869,6871,6873,6879,6881,6883,6889,6891],{},[43,6818,6819],{"mathvariant":65},"VCdim",[50,6821,69],{},[50,6823,53],{"stretchy":52},[43,6825,819],{"mathvariant":596},[50,6827,59],{"stretchy":52},[50,6829,62],{},[43,6831,2932],{},[50,6833,69],{},[50,6835,636],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,6837,348],{},[50,6839,718],{},[43,6841,6842],{"mathvariant":65},"∃",[43,6844,5619],{},[50,6846,5647],{},[43,6848,597],{"mathvariant":596},[50,6850,91],{"separator":72},[6852,6853,6854],"mtext",{}," ",[43,6856,2945],{"mathvariant":65},[43,6858,5619],{},[43,6860,2945],{"mathvariant":65},[50,6862,62],{},[43,6864,348],{},[50,6866,91],{"separator":72},[6852,6868,6854],{},[50,6870,2945],{"stretchy":52},[43,6872,819],{"mathvariant":596},[40,6874,6875,6877],{},[43,6876,2945],{"mathvariant":65},[43,6878,5619],{},[50,6880,2945],{"stretchy":52},[50,6882,62],{},[305,6884,6885,6887],{},[308,6886,310],{},[43,6888,348],{},[50,6890,645],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,6892,107],{"mathvariant":65},[109,6894,6895],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H)=\\sup\\bigl\\{m:\\exists C\\subseteq\\mathcal X,\\ |C|=m,\\ \\lvert\\mathcal H|_C\\rvert=2^m\\bigr\\}.",[15,6897,6899,6929,6956,6978,7014,7090],{"className":6898,"ariaHidden":72},[116],[15,6900,6902,6905,6911,6914,6917,6920,6923,6926],{"className":6901},[120],[15,6903],{"className":6904,"style":125},[124],[15,6906,6908],{"className":6907},[217],[15,6909,6819],{"className":6910},[129,221],[15,6912,53],{"className":6913},[188],[15,6915,819],{"className":6916,"style":835},[129,614],[15,6918,59],{"className":6919},[195],[15,6921],{"className":6922,"style":199},[93],[15,6924,62],{"className":6925},[203],[15,6927],{"className":6928,"style":199},[93],[15,6930,6932,6935,6938,6944,6947,6950,6953],{"className":6931},[120],[15,6933],{"className":6934,"style":213},[124],[15,6936,2932],{"className":6937},[217],[15,6939,6941],{"className":6940},[188],[15,6942,636],{"className":6943},[228,229],[15,6945,348],{"className":6946},[129,133],[15,6948],{"className":6949,"style":199},[93],[15,6951,718],{"className":6952},[203],[15,6954],{"className":6955,"style":199},[93],[15,6957,6959,6963,6966,6969,6972,6975],{"className":6958},[120],[15,6960],{"className":6961,"style":6962},[124],"height:0.8304em;vertical-align:-0.136em;",[15,6964,6842],{"className":6965},[129],[15,6967,5619],{"className":6968,"style":5665},[129,133],[15,6970],{"className":6971,"style":199},[93],[15,6973,5647],{"className":6974},[203],[15,6976],{"className":6977,"style":199},[93],[15,6979,6981,6984,6987,6990,6993,6996,6999,7002,7005,7008,7011],{"className":6980},[120],[15,6982],{"className":6983,"style":125},[124],[15,6985,597],{"className":6986,"style":615},[129,614],[15,6988,91],{"className":6989},[254],[15,6991,6854],{"className":6992},[93],[15,6994],{"className":6995,"style":262},[93],[15,6997,2945],{"className":6998},[129],[15,7000,5619],{"className":7001,"style":5665},[129,133],[15,7003,2945],{"className":7004},[129],[15,7006],{"className":7007,"style":199},[93],[15,7009,62],{"className":7010},[203],[15,7012],{"className":7013,"style":199},[93],[15,7015,7017,7020,7023,7026,7029,7032,7035,7038,7078,7081,7084,7087],{"className":7016},[120],[15,7018],{"className":7019,"style":125},[124],[15,7021,348],{"className":7022},[129,133],[15,7024,91],{"className":7025},[254],[15,7027,6854],{"className":7028},[93],[15,7030],{"className":7031,"style":262},[93],[15,7033,2945],{"className":7034},[188],[15,7036,819],{"className":7037,"style":835},[129,614],[15,7039,7041,7044],{"className":7040},[129],[15,7042,2945],{"className":7043},[129],[15,7045,7047],{"className":7046},[137],[15,7048,7050,7070],{"className":7049},[141,142],[15,7051,7053,7067],{"className":7052},[146],[15,7054,7056],{"className":7055,"style":1525},[150],[15,7057,7058,7061],{"style":154},[15,7059],{"className":7060,"style":159},[158],[15,7062,7064],{"className":7063},[163,164,165,166],[15,7065,5619],{"className":7066,"style":5665},[129,133,166],[15,7068,175],{"className":7069},[174],[15,7071,7073],{"className":7072},[146],[15,7074,7076],{"className":7075,"style":182},[150],[15,7077],{},[15,7079,2945],{"className":7080},[195],[15,7082],{"className":7083,"style":199},[93],[15,7085,62],{"className":7086},[203],[15,7088],{"className":7089,"style":199},[93],[15,7091,7093,7096,7125,7131],{"className":7092},[120],[15,7094],{"className":7095,"style":213},[124],[15,7097,7099,7102],{"className":7098},[129],[15,7100,310],{"className":7101},[129],[15,7103,7105],{"className":7104},[137],[15,7106,7108],{"className":7107},[141],[15,7109,7111],{"className":7110},[146],[15,7112,7114],{"className":7113,"style":1251},[150],[15,7115,7116,7119],{"style":1272},[15,7117],{"className":7118,"style":159},[158],[15,7120,7122],{"className":7121},[163,164,165,166],[15,7123,348],{"className":7124},[129,133,166],[15,7126,7128],{"className":7127},[195],[15,7129,645],{"className":7130},[228,229],[15,7132,107],{"className":7133},[129],[11,7135,7136,7137,1313],{},"若右侧无界，则记 ",[15,7138,7140,7166],{"className":7139,"translate":19},[23],[15,7141,7143],{"className":7142},[27],[29,7144,7145],{"xmlns":31},[34,7146,7147,7163],{},[37,7148,7149,7151,7153,7155,7157,7159,7161],{},[43,7150,6819],{"mathvariant":65},[50,7152,69],{},[50,7154,53],{"stretchy":52},[43,7156,819],{"mathvariant":596},[50,7158,59],{"stretchy":52},[50,7160,62],{},[43,7162,4909],{"mathvariant":65},[109,7164,7165],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H)=\\infty",[15,7167,7169,7199],{"className":7168,"ariaHidden":72},[116],[15,7170,7172,7175,7181,7184,7187,7190,7193,7196],{"className":7171},[120],[15,7173],{"className":7174,"style":125},[124],[15,7176,7178],{"className":7177},[217],[15,7179,6819],{"className":7180},[129,221],[15,7182,53],{"className":7183},[188],[15,7185,819],{"className":7186,"style":835},[129,614],[15,7188,59],{"className":7189},[195],[15,7191],{"className":7192,"style":199},[93],[15,7194,62],{"className":7195},[203],[15,7197],{"className":7198,"style":199},[93],[15,7200,7202,7205],{"className":7201},[120],[15,7203],{"className":7204,"style":4383},[124],[15,7206,4909],{"className":7207},[129],[11,7209,7210,7211,7214,7215,7243,7244,804,7247,7305,7306,804,7308,7336,7337,7365],{},"请把定义里的量词读三遍：",[698,7212,7213],{},"存在","一个点集 ",[15,7216,7218,7231],{"className":7217,"translate":19},[23],[15,7219,7221],{"className":7220},[27],[29,7222,7223],{"xmlns":31},[34,7224,7225,7229],{},[37,7226,7227],{},[43,7228,5619],{},[109,7230,5619],{"encoding":111},[15,7232,7234],{"className":7233,"ariaHidden":72},[116],[15,7235,7237,7240],{"className":7236},[120],[15,7238],{"className":7239,"style":610},[124],[15,7241,5619],{"className":7242,"style":5665},[129,133],"，使得对",[698,7245,7246],{},"所有",[15,7248,7250,7267],{"className":7249,"translate":19},[23],[15,7251,7253],{"className":7252},[27],[29,7254,7255],{"xmlns":31},[34,7256,7257,7265],{},[37,7258,7259],{},[305,7260,7261,7263],{},[308,7262,310],{},[43,7264,348],{},[109,7266,6367],{"encoding":111},[15,7268,7270],{"className":7269,"ariaHidden":72},[116],[15,7271,7273,7276],{"className":7272},[120],[15,7274],{"className":7275,"style":6377},[124],[15,7277,7279,7282],{"className":7278},[129],[15,7280,310],{"className":7281},[129],[15,7283,7285],{"className":7284},[137],[15,7286,7288],{"className":7287},[141],[15,7289,7291],{"className":7290},[146],[15,7292,7294],{"className":7293,"style":6377},[150],[15,7295,7296,7299],{"style":383},[15,7297],{"className":7298,"style":159},[158],[15,7300,7302],{"className":7301},[163,164,165,166],[15,7303,348],{"className":7304},[129,133,166]," 种标注，都",[698,7307,7213],{},[15,7309,7311,7324],{"className":7310,"translate":19},[23],[15,7312,7314],{"className":7313},[27],[29,7315,7316],{"xmlns":31},[34,7317,7318,7322],{},[37,7319,7320],{},[43,7321,45],{},[109,7323,45],{"encoding":111},[15,7325,7327],{"className":7326,"ariaHidden":72},[116],[15,7328,7330,7333],{"className":7329},[120],[15,7331],{"className":7332,"style":522},[124],[15,7334,45],{"className":7335},[129,133],"。不是「对所有 ",[15,7338,7340,7353],{"className":7339,"translate":19},[23],[15,7341,7343],{"className":7342},[27],[29,7344,7345],{"xmlns":31},[34,7346,7347,7351],{},[37,7348,7349],{},[43,7350,5619],{},[109,7352,5619],{"encoding":111},[15,7354,7356],{"className":7355,"ariaHidden":72},[116],[15,7357,7359,7362],{"className":7358},[120],[15,7360],{"className":7361,"style":610},[124],[15,7363,5619],{"className":7364,"style":5665},[129,133],"」，也不是「对某一种标注」。",[11,7367,7368],{},"这个「存在」是 VC 维最容易被误读的地方，也是它作为最坏情况度量的根源：它问的是假设类在某个精心挑选的点集上最多能变出多少花样，而不是它在典型数据上会怎么做。",[575,7370,7372],{"id":7371},"_22-几个例子","2.2 几个例子",[11,7374,7375],{},[698,7376,7377,7378,7434],{},"例 1（半直线，",[15,7379,7381,7401],{"className":7380,"translate":19},[23],[15,7382,7384],{"className":7383},[27],[29,7385,7386],{"xmlns":31},[34,7387,7388,7398],{},[37,7389,7390,7392,7394,7396],{},[43,7391,6819],{"mathvariant":65},[50,7393,69],{},[50,7395,62],{},[308,7397,318],{},[109,7399,7400],{"encoding":111},"\\operatorname{VCdim}=1",[15,7402,7404,7425],{"className":7403,"ariaHidden":72},[116],[15,7405,7407,7410,7416,7419,7422],{"className":7406},[120],[15,7408],{"className":7409,"style":522},[124],[15,7411,7413],{"className":7412},[217],[15,7414,6819],{"className":7415},[129,221],[15,7417],{"className":7418,"style":199},[93],[15,7420,62],{"className":7421},[203],[15,7423],{"className":7424,"style":199},[93],[15,7426,7428,7431],{"className":7427},[120],[15,7429],{"className":7430,"style":4886},[124],[15,7432,318],{"className":7433},[129],"）",[11,7436,7437,7438,7606,7607,1313],{},"取 ",[15,7439,7441,7488],{"className":7440,"translate":19},[23],[15,7442,7444],{"className":7443},[27],[29,7445,7446],{"xmlns":31},[34,7447,7448,7485],{},[37,7449,7450,7452,7454,7456,7458,7461,7463,7465,7467,7470,7473,7475,7477,7479,7481,7483],{},[43,7451,819],{"mathvariant":596},[50,7453,62],{},[50,7455,636],{"stretchy":52},[43,7457,56],{},[50,7459,7460],{},"↦",[308,7462,318],{"mathvariant":1468},[50,7464,1407],{"stretchy":52},[43,7466,56],{},[50,7468,7469],{},"≥",[43,7471,7472],{},"t",[50,7474,1423],{"stretchy":52},[50,7476,718],{},[43,7478,7472],{},[50,7480,100],{},[43,7482,104],{"mathvariant":103},[50,7484,645],{"stretchy":52},[109,7486,7487],{"encoding":111},"\\mathcal H=\\{x\\mapsto\\mathbf 1[x\\ge t]:t\\in\\mathbb R\\}",[15,7489,7491,7509,7530,7554,7575,7594],{"className":7490,"ariaHidden":72},[116],[15,7492,7494,7497,7500,7503,7506],{"className":7493},[120],[15,7495],{"className":7496,"style":610},[124],[15,7498,819],{"className":7499,"style":835},[129,614],[15,7501],{"className":7502,"style":199},[93],[15,7504,62],{"className":7505},[203],[15,7507],{"className":7508,"style":199},[93],[15,7510,7512,7515,7518,7521,7524,7527],{"className":7511},[120],[15,7513],{"className":7514,"style":125},[124],[15,7516,636],{"className":7517},[188],[15,7519,56],{"className":7520},[129,133],[15,7522],{"className":7523,"style":199},[93],[15,7525,7460],{"className":7526},[203],[15,7528],{"className":7529,"style":199},[93],[15,7531,7533,7536,7539,7542,7545,7548,7551],{"className":7532},[120],[15,7534],{"className":7535,"style":125},[124],[15,7537,318],{"className":7538},[129,1950],[15,7540,1407],{"className":7541},[188],[15,7543,56],{"className":7544},[129,133],[15,7546],{"className":7547,"style":199},[93],[15,7549,7469],{"className":7550},[203],[15,7552],{"className":7553,"style":199},[93],[15,7555,7557,7560,7563,7566,7569,7572],{"className":7556},[120],[15,7558],{"className":7559,"style":125},[124],[15,7561,7472],{"className":7562},[129,133],[15,7564,1423],{"className":7565},[195],[15,7567],{"className":7568,"style":199},[93],[15,7570,718],{"className":7571},[203],[15,7573],{"className":7574,"style":199},[93],[15,7576,7578,7582,7585,7588,7591],{"className":7577},[120],[15,7579],{"className":7580,"style":7581},[124],"height:0.6542em;vertical-align:-0.0391em;",[15,7583,7472],{"className":7584},[129,133],[15,7586],{"className":7587,"style":199},[93],[15,7589,100],{"className":7590},[203],[15,7592],{"className":7593,"style":199},[93],[15,7595,7597,7600,7603],{"className":7596},[120],[15,7598],{"className":7599,"style":125},[124],[15,7601,104],{"className":7602},[129,285],[15,7604,645],{"className":7605},[195],"，",[15,7608,7610,7628],{"className":7609,"translate":19},[23],[15,7611,7613],{"className":7612},[27],[29,7614,7615],{"xmlns":31},[34,7616,7617,7625],{},[37,7618,7619,7621,7623],{},[43,7620,597],{"mathvariant":596},[50,7622,62],{},[43,7624,104],{"mathvariant":103},[109,7626,7627],{"encoding":111},"\\mathcal X=\\mathbb R",[15,7629,7631,7649],{"className":7630,"ariaHidden":72},[116],[15,7632,7634,7637,7640,7643,7646],{"className":7633},[120],[15,7635],{"className":7636,"style":610},[124],[15,7638,597],{"className":7639,"style":615},[129,614],[15,7641],{"className":7642,"style":199},[93],[15,7644,62],{"className":7645},[203],[15,7647],{"className":7648,"style":199},[93],[15,7650,7652,7655],{"className":7651},[120],[15,7653],{"className":7654,"style":281},[124],[15,7656,104],{"className":7657},[129,285],[11,7659,7660,7661,7722,7723,7775,7776,7606,7804,7775,7855,1313],{},"一个点 ",[15,7662,7664,7686],{"className":7663,"translate":19},[23],[15,7665,7667],{"className":7666},[27],[29,7668,7669],{"xmlns":31},[34,7670,7671,7683],{},[37,7672,7673,7675,7677,7679,7681],{},[43,7674,5619],{},[50,7676,62],{},[50,7678,636],{"stretchy":52},[308,7680,541],{},[50,7682,645],{"stretchy":52},[109,7684,7685],{"encoding":111},"C=\\{0\\}",[15,7687,7689,7707],{"className":7688,"ariaHidden":72},[116],[15,7690,7692,7695,7698,7701,7704],{"className":7691},[120],[15,7693],{"className":7694,"style":610},[124],[15,7696,5619],{"className":7697,"style":5665},[129,133],[15,7699],{"className":7700,"style":199},[93],[15,7702,62],{"className":7703},[203],[15,7705],{"className":7706,"style":199},[93],[15,7708,7710,7713,7716,7719],{"className":7709},[120],[15,7711],{"className":7712,"style":125},[124],[15,7714,636],{"className":7715},[188],[15,7717,541],{"className":7718},[129],[15,7720,645],{"className":7721},[195]," 显然可以被打散：",[15,7724,7726,7744],{"className":7725,"translate":19},[23],[15,7727,7729],{"className":7728},[27],[29,7730,7731],{"xmlns":31},[34,7732,7733,7741],{},[37,7734,7735,7737,7739],{},[43,7736,7472],{},[50,7738,2717],{},[308,7740,541],{},[109,7742,7743],{"encoding":111},"t\\le 0",[15,7745,7747,7766],{"className":7746,"ariaHidden":72},[116],[15,7748,7750,7754,7757,7760,7763],{"className":7749},[120],[15,7751],{"className":7752,"style":7753},[124],"height:0.7719em;vertical-align:-0.136em;",[15,7755,7472],{"className":7756},[129,133],[15,7758],{"className":7759,"style":199},[93],[15,7761,2717],{"className":7762},[203],[15,7764],{"className":7765,"style":199},[93],[15,7767,7769,7772],{"className":7768},[120],[15,7770],{"className":7771,"style":4886},[124],[15,7773,541],{"className":7774},[129]," 输出 ",[15,7777,7779,7792],{"className":7778,"translate":19},[23],[15,7780,7782],{"className":7781},[27],[29,7783,7784],{"xmlns":31},[34,7785,7786,7790],{},[37,7787,7788],{},[308,7789,318],{},[109,7791,318],{"encoding":111},[15,7793,7795],{"className":7794,"ariaHidden":72},[116],[15,7796,7798,7801],{"className":7797},[120],[15,7799],{"className":7800,"style":4886},[124],[15,7802,318],{"className":7803},[129],[15,7805,7807,7825],{"className":7806,"translate":19},[23],[15,7808,7810],{"className":7809},[27],[29,7811,7812],{"xmlns":31},[34,7813,7814,7822],{},[37,7815,7816,7818,7820],{},[43,7817,7472],{},[50,7819,4852],{},[308,7821,541],{},[109,7823,7824],{"encoding":111},"t>0",[15,7826,7828,7846],{"className":7827,"ariaHidden":72},[116],[15,7829,7831,7834,7837,7840,7843],{"className":7830},[120],[15,7832],{"className":7833,"style":7581},[124],[15,7835,7472],{"className":7836},[129,133],[15,7838],{"className":7839,"style":199},[93],[15,7841,4852],{"className":7842},[203],[15,7844],{"className":7845,"style":199},[93],[15,7847,7849,7852],{"className":7848},[120],[15,7850],{"className":7851,"style":4886},[124],[15,7853,541],{"className":7854},[129],[15,7856,7858,7871],{"className":7857,"translate":19},[23],[15,7859,7861],{"className":7860},[27],[29,7862,7863],{"xmlns":31},[34,7864,7865,7869],{},[37,7866,7867],{},[308,7868,541],{},[109,7870,541],{"encoding":111},[15,7872,7874],{"className":7873,"ariaHidden":72},[116],[15,7875,7877,7880],{"className":7876},[120],[15,7878],{"className":7879,"style":4886},[124],[15,7881,541],{"className":7882},[129],[11,7884,7885,7886,8022,8023,8075,8076,8168,8169,8261,8262,8394,8395,1313],{},"两个点 ",[15,7887,7889,7916],{"className":7888,"translate":19},[23],[15,7890,7892],{"className":7891},[27],[29,7893,7894],{"xmlns":31},[34,7895,7896,7913],{},[37,7897,7898,7904,7907],{},[40,7899,7900,7902],{},[43,7901,56],{},[308,7903,318],{},[50,7905,7906],{},"\u003C",[40,7908,7909,7911],{},[43,7910,56],{},[308,7912,310],{},[109,7914,7915],{"encoding":111},"x_1\u003Cx_2",[15,7917,7919,7975],{"className":7918,"ariaHidden":72},[116],[15,7920,7922,7926,7966,7969,7972],{"className":7921},[120],[15,7923],{"className":7924,"style":7925},[124],"height:0.6891em;vertical-align:-0.15em;",[15,7927,7929,7932],{"className":7928},[129],[15,7930,56],{"className":7931},[129,133],[15,7933,7935],{"className":7934},[137],[15,7936,7938,7958],{"className":7937},[141,142],[15,7939,7941,7955],{"className":7940},[146],[15,7942,7944],{"className":7943,"style":1051},[150],[15,7945,7946,7949],{"style":154},[15,7947],{"className":7948,"style":159},[158],[15,7950,7952],{"className":7951},[163,164,165,166],[15,7953,318],{"className":7954},[129,166],[15,7956,175],{"className":7957},[174],[15,7959,7961],{"className":7960},[146],[15,7962,7964],{"className":7963,"style":182},[150],[15,7965],{},[15,7967],{"className":7968,"style":199},[93],[15,7970,7906],{"className":7971},[203],[15,7973],{"className":7974,"style":199},[93],[15,7976,7978,7982],{"className":7977},[120],[15,7979],{"className":7980,"style":7981},[124],"height:0.5806em;vertical-align:-0.15em;",[15,7983,7985,7988],{"className":7984},[129],[15,7986,56],{"className":7987},[129,133],[15,7989,7991],{"className":7990},[137],[15,7992,7994,8014],{"className":7993},[141,142],[15,7995,7997,8011],{"className":7996},[146],[15,7998,8000],{"className":7999,"style":1051},[150],[15,8001,8002,8005],{"style":154},[15,8003],{"className":8004,"style":159},[158],[15,8006,8008],{"className":8007},[163,164,165,166],[15,8009,310],{"className":8010},[129,166],[15,8012,175],{"className":8013},[174],[15,8015,8017],{"className":8016},[146],[15,8018,8020],{"className":8019,"style":182},[150],[15,8021],{}," 则不行：标注 ",[15,8024,8026,8048],{"className":8025,"translate":19},[23],[15,8027,8029],{"className":8028},[27],[29,8030,8031],{"xmlns":31},[34,8032,8033,8045],{},[37,8034,8035,8037,8039,8041,8043],{},[50,8036,53],{"stretchy":52},[308,8038,318],{},[50,8040,91],{"separator":72},[308,8042,541],{},[50,8044,59],{"stretchy":52},[109,8046,8047],{"encoding":111},"(1,0)",[15,8049,8051],{"className":8050,"ariaHidden":72},[116],[15,8052,8054,8057,8060,8063,8066,8069,8072],{"className":8053},[120],[15,8055],{"className":8056,"style":125},[124],[15,8058,53],{"className":8059},[188],[15,8061,318],{"className":8062},[129],[15,8064,91],{"className":8065},[254],[15,8067],{"className":8068,"style":262},[93],[15,8070,541],{"className":8071},[129],[15,8073,59],{"className":8074},[195]," 要求 ",[15,8077,8079,8101],{"className":8078,"translate":19},[23],[15,8080,8082],{"className":8081},[27],[29,8083,8084],{"xmlns":31},[34,8085,8086,8098],{},[37,8087,8088,8090,8092],{},[43,8089,7472],{},[50,8091,2717],{},[40,8093,8094,8096],{},[43,8095,56],{},[308,8097,318],{},[109,8099,8100],{"encoding":111},"t\\le x_1",[15,8102,8104,8122],{"className":8103,"ariaHidden":72},[116],[15,8105,8107,8110,8113,8116,8119],{"className":8106},[120],[15,8108],{"className":8109,"style":7753},[124],[15,8111,7472],{"className":8112},[129,133],[15,8114],{"className":8115,"style":199},[93],[15,8117,2717],{"className":8118},[203],[15,8120],{"className":8121,"style":199},[93],[15,8123,8125,8128],{"className":8124},[120],[15,8126],{"className":8127,"style":7981},[124],[15,8129,8131,8134],{"className":8130},[129],[15,8132,56],{"className":8133},[129,133],[15,8135,8137],{"className":8136},[137],[15,8138,8140,8160],{"className":8139},[141,142],[15,8141,8143,8157],{"className":8142},[146],[15,8144,8146],{"className":8145,"style":1051},[150],[15,8147,8148,8151],{"style":154},[15,8149],{"className":8150,"style":159},[158],[15,8152,8154],{"className":8153},[163,164,165,166],[15,8155,318],{"className":8156},[129,166],[15,8158,175],{"className":8159},[174],[15,8161,8163],{"className":8162},[146],[15,8164,8166],{"className":8165,"style":182},[150],[15,8167],{}," 且 ",[15,8170,8172,8194],{"className":8171,"translate":19},[23],[15,8173,8175],{"className":8174},[27],[29,8176,8177],{"xmlns":31},[34,8178,8179,8191],{},[37,8180,8181,8183,8185],{},[43,8182,7472],{},[50,8184,4852],{},[40,8186,8187,8189],{},[43,8188,56],{},[308,8190,310],{},[109,8192,8193],{"encoding":111},"t>x_2",[15,8195,8197,8215],{"className":8196,"ariaHidden":72},[116],[15,8198,8200,8203,8206,8209,8212],{"className":8199},[120],[15,8201],{"className":8202,"style":7581},[124],[15,8204,7472],{"className":8205},[129,133],[15,8207],{"className":8208,"style":199},[93],[15,8210,4852],{"className":8211},[203],[15,8213],{"className":8214,"style":199},[93],[15,8216,8218,8221],{"className":8217},[120],[15,8219],{"className":8220,"style":7981},[124],[15,8222,8224,8227],{"className":8223},[129],[15,8225,56],{"className":8226},[129,133],[15,8228,8230],{"className":8229},[137],[15,8231,8233,8253],{"className":8232},[141,142],[15,8234,8236,8250],{"className":8235},[146],[15,8237,8239],{"className":8238,"style":1051},[150],[15,8240,8241,8244],{"style":154},[15,8242],{"className":8243,"style":159},[158],[15,8245,8247],{"className":8246},[163,164,165,166],[15,8248,310],{"className":8249},[129,166],[15,8251,175],{"className":8252},[174],[15,8254,8256],{"className":8255},[146],[15,8257,8259],{"className":8258,"style":182},[150],[15,8260],{},"，与 ",[15,8263,8265,8290],{"className":8264,"translate":19},[23],[15,8266,8268],{"className":8267},[27],[29,8269,8270],{"xmlns":31},[34,8271,8272,8288],{},[37,8273,8274,8280,8282],{},[40,8275,8276,8278],{},[43,8277,56],{},[308,8279,318],{},[50,8281,7906],{},[40,8283,8284,8286],{},[43,8285,56],{},[308,8287,310],{},[109,8289,7915],{"encoding":111},[15,8291,8293,8348],{"className":8292,"ariaHidden":72},[116],[15,8294,8296,8299,8339,8342,8345],{"className":8295},[120],[15,8297],{"className":8298,"style":7925},[124],[15,8300,8302,8305],{"className":8301},[129],[15,8303,56],{"className":8304},[129,133],[15,8306,8308],{"className":8307},[137],[15,8309,8311,8331],{"className":8310},[141,142],[15,8312,8314,8328],{"className":8313},[146],[15,8315,8317],{"className":8316,"style":1051},[150],[15,8318,8319,8322],{"style":154},[15,8320],{"className":8321,"style":159},[158],[15,8323,8325],{"className":8324},[163,164,165,166],[15,8326,318],{"className":8327},[129,166],[15,8329,175],{"className":8330},[174],[15,8332,8334],{"className":8333},[146],[15,8335,8337],{"className":8336,"style":182},[150],[15,8338],{},[15,8340],{"className":8341,"style":199},[93],[15,8343,7906],{"className":8344},[203],[15,8346],{"className":8347,"style":199},[93],[15,8349,8351,8354],{"className":8350},[120],[15,8352],{"className":8353,"style":7981},[124],[15,8355,8357,8360],{"className":8356},[129],[15,8358,56],{"className":8359},[129,133],[15,8361,8363],{"className":8362},[137],[15,8364,8366,8386],{"className":8365},[141,142],[15,8367,8369,8383],{"className":8368},[146],[15,8370,8372],{"className":8371,"style":1051},[150],[15,8373,8374,8377],{"style":154},[15,8375],{"className":8376,"style":159},[158],[15,8378,8380],{"className":8379},[163,164,165,166],[15,8381,310],{"className":8382},[129,166],[15,8384,175],{"className":8385},[174],[15,8387,8389],{"className":8388},[146],[15,8390,8392],{"className":8391,"style":182},[150],[15,8393],{}," 矛盾。所以 VC 维是 ",[15,8396,8398,8411],{"className":8397,"translate":19},[23],[15,8399,8401],{"className":8400},[27],[29,8402,8403],{"xmlns":31},[34,8404,8405,8409],{},[37,8406,8407],{},[308,8408,318],{},[109,8410,318],{"encoding":111},[15,8412,8414],{"className":8413,"ariaHidden":72},[116],[15,8415,8417,8420],{"className":8416},[120],[15,8418],{"className":8419,"style":4886},[124],[15,8421,318],{"className":8422},[129],[11,8424,8425,8426,8454,8455,8483,8484,8536,8537,1313],{},"这个例子说明，VC 维和「能拟合多少点」不是一回事：半直线可以把任意多个点拟合成「前面全是 ",[15,8427,8429,8442],{"className":8428,"translate":19},[23],[15,8430,8432],{"className":8431},[27],[29,8433,8434],{"xmlns":31},[34,8435,8436,8440],{},[37,8437,8438],{},[308,8439,541],{},[109,8441,541],{"encoding":111},[15,8443,8445],{"className":8444,"ariaHidden":72},[116],[15,8446,8448,8451],{"className":8447},[120],[15,8449],{"className":8450,"style":4886},[124],[15,8452,541],{"className":8453},[129],"、后面全是 ",[15,8456,8458,8471],{"className":8457,"translate":19},[23],[15,8459,8461],{"className":8460},[27],[29,8462,8463],{"xmlns":31},[34,8464,8465,8469],{},[37,8466,8467],{},[308,8468,318],{},[109,8470,318],{"encoding":111},[15,8472,8474],{"className":8473,"ariaHidden":72},[116],[15,8475,8477,8480],{"className":8476},[120],[15,8478],{"className":8479,"style":4886},[124],[15,8481,318],{"className":8482},[129],"」的形状，但它只能实现 ",[15,8485,8487,8505],{"className":8486,"translate":19},[23],[15,8488,8490],{"className":8489},[27],[29,8491,8492],{"xmlns":31},[34,8493,8494,8502],{},[37,8495,8496,8498,8500],{},[43,8497,348],{},[50,8499,2682],{},[308,8501,318],{},[109,8503,8504],{"encoding":111},"m+1",[15,8506,8508,8527],{"className":8507,"ariaHidden":72},[116],[15,8509,8511,8515,8518,8521,8524],{"className":8510},[120],[15,8512],{"className":8513,"style":8514},[124],"height:0.6667em;vertical-align:-0.0833em;",[15,8516,348],{"className":8517},[129,133],[15,8519],{"className":8520,"style":3277},[93],[15,8522,2682],{"className":8523},[2436],[15,8525],{"className":8526,"style":3277},[93],[15,8528,8530,8533],{"className":8529},[120],[15,8531],{"className":8532,"style":4886},[124],[15,8534,318],{"className":8535},[129]," 种二分，远少于 ",[15,8538,8540,8557],{"className":8539,"translate":19},[23],[15,8541,8543],{"className":8542},[27],[29,8544,8545],{"xmlns":31},[34,8546,8547,8555],{},[37,8548,8549],{},[305,8550,8551,8553],{},[308,8552,310],{},[43,8554,348],{},[109,8556,6367],{"encoding":111},[15,8558,8560],{"className":8559,"ariaHidden":72},[116],[15,8561,8563,8566],{"className":8562},[120],[15,8564],{"className":8565,"style":6377},[124],[15,8567,8569,8572],{"className":8568},[129],[15,8570,310],{"className":8571},[129],[15,8573,8575],{"className":8574},[137],[15,8576,8578],{"className":8577},[141],[15,8579,8581],{"className":8580},[146],[15,8582,8584],{"className":8583,"style":6377},[150],[15,8585,8586,8589],{"style":383},[15,8587],{"className":8588,"style":159},[158],[15,8590,8592],{"className":8591},[163,164,165,166],[15,8593,348],{"className":8594},[129,133,166],[11,8596,8597],{},[698,8598,8599,8600,7434],{},"例 2（区间，",[15,8601,8603,8623],{"className":8602,"translate":19},[23],[15,8604,8606],{"className":8605},[27],[29,8607,8608],{"xmlns":31},[34,8609,8610,8620],{},[37,8611,8612,8614,8616,8618],{},[43,8613,6819],{"mathvariant":65},[50,8615,69],{},[50,8617,62],{},[308,8619,310],{},[109,8621,8622],{"encoding":111},"\\operatorname{VCdim}=2",[15,8624,8626,8647],{"className":8625,"ariaHidden":72},[116],[15,8627,8629,8632,8638,8641,8644],{"className":8628},[120],[15,8630],{"className":8631,"style":522},[124],[15,8633,8635],{"className":8634},[217],[15,8636,6819],{"className":8637},[129,221],[15,8639],{"className":8640,"style":199},[93],[15,8642,62],{"className":8643},[203],[15,8645],{"className":8646,"style":199},[93],[15,8648,8650,8653],{"className":8649},[120],[15,8651],{"className":8652,"style":4886},[124],[15,8654,310],{"className":8655},[129],[11,8657,7437,8658,1313],{},[15,8659,8661,8711],{"className":8660,"translate":19},[23],[15,8662,8664],{"className":8663},[27],[29,8665,8666],{"xmlns":31},[34,8667,8668,8708],{},[37,8669,8670,8672,8674,8676,8678,8680,8682,8684,8687,8689,8691,8693,8696,8698,8700,8702,8704,8706],{},[43,8671,819],{"mathvariant":596},[50,8673,62],{},[50,8675,636],{"stretchy":52},[43,8677,56],{},[50,8679,7460],{},[308,8681,318],{"mathvariant":1468},[50,8683,1407],{"stretchy":52},[43,8685,8686],{},"a",[50,8688,2717],{},[43,8690,56],{},[50,8692,2717],{},[43,8694,8695],{},"b",[50,8697,1423],{"stretchy":52},[50,8699,718],{},[43,8701,8686],{},[50,8703,2717],{},[43,8705,8695],{},[50,8707,645],{"stretchy":52},[109,8709,8710],{"encoding":111},"\\mathcal H=\\{x\\mapsto\\mathbf 1[a\\le x\\le b]:a\\le 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b\\ge x_2",[15,9600,9602,9620,9739],{"className":9601,"ariaHidden":72},[116],[15,9603,9605,9608,9611,9614,9617],{"className":9604},[120],[15,9606],{"className":9607,"style":4867},[124],[15,9609,8686],{"className":9610},[129,133],[15,9612],{"className":9613,"style":199},[93],[15,9615,100],{"className":9616},[203],[15,9618],{"className":9619,"style":199},[93],[15,9621,9623,9626,9629,9669,9672,9675,9715,9718,9721,9724,9727,9730,9733,9736],{"className":9622},[120],[15,9624],{"className":9625,"style":125},[124],[15,9627,53],{"className":9628},[188],[15,9630,9632,9635],{"className":9631},[129],[15,9633,56],{"className":9634},[129,133],[15,9636,9638],{"className":9637},[137],[15,9639,9641,9661],{"className":9640},[141,142],[15,9642,9644,9658],{"className":9643},[146],[15,9645,9647],{"className":9646,"style":1051},[150],[15,9648,9649,9652],{"style":154},[15,9650],{"className":9651,"style":159},[158],[15,9653,9655],{"className":9654},[163,164,165,166],[15,9656,318],{"className":9657},[129,166],[15,9659,175],{"className":9660},[174],[15,9662,9664],{"className":9663},[146],[15,9665,9667],{"className":9666,"style":182},[150],[15,9668],{},[15,9670,91],{"className":9671},[254],[15,9673],{"className":9674,"style":262},[93],[15,9676,9678,9681],{"className":9677},[129],[15,9679,56],{"className":9680},[129,133],[15,9682,9684],{"className":9683},[137],[15,9685,9687,9707],{"className":9686},[141,142],[15,9688,9690,9704],{"className":9689},[146],[15,9691,9693],{"className":9692,"style":1051},[150],[15,9694,9695,9698],{"style":154},[15,9696],{"className":9697,"style":159},[158],[15,9699,9701],{"className":9700},[163,164,165,166],[15,9702,310],{"className":9703},[129,166],[15,9705,175],{"className":9706},[174],[15,9708,9710],{"className":9709},[146],[15,9711,9713],{"className":9712,"style":182},[150],[15,9714],{},[15,9716,1423],{"className":9717},[195],[15,9719,91],{"className":9720},[254],[15,9722,6854],{"className":9723},[93],[15,9725],{"className":9726,"style":262},[93],[15,9728,8695],{"className":9729},[129,133],[15,9731],{"className":9732,"style":199},[93],[15,9734,7469],{"className":9735},[203],[15,9737],{"className":9738,"style":199},[93],[15,9740,9742,9745],{"className":9741},[120],[15,9743],{"className":9744,"style":7981},[124],[15,9746,9748,9751],{"className":9747},[129],[15,9749,56],{"className":9750},[129,133],[15,9752,9754],{"className":9753},[137],[15,9755,9757,9777],{"className":9756},[141,142],[15,9758,9760,9774],{"className":9759},[146],[15,9761,9763],{"className":9762,"style":1051},[150],[15,9764,9765,9768],{"style":154},[15,9766],{"className":9767,"style":159},[158],[15,9769,9771],{"className":9770},[163,164,165,166],[15,9772,310],{"className":9773},[129,166],[15,9775,175],{"className":9776},[174],[15,9778,9780],{"className":9779},[146],[15,9781,9783],{"className":9782,"style":182},[150],[15,9784],{},[15,9786,9788,9810],{"className":9787,"translate":19},[23],[15,9789,9791],{"className":9790},[27],[29,9792,9793],{"xmlns":31},[34,9794,9795,9807],{},[37,9796,9797,9799,9801,9803,9805],{},[50,9798,53],{"stretchy":52},[308,9800,541],{},[50,9802,91],{"separator":72},[308,9804,541],{},[50,9806,59],{"stretchy":52},[109,9808,9809],{"encoding":111},"(0,0)",[15,9811,9813],{"className":9812,"ariaHidden":72},[116],[15,9814,9816,9819,9822,9825,9828,9831,9834],{"className":9815},[120],[15,9817],{"className":9818,"style":125},[124],[15,9820,53],{"className":9821},[188],[15,9823,541],{"className":9824},[129],[15,9826,91],{"className":9827},[254],[15,9829],{"className":9830,"style":262},[93],[15,9832,541],{"className":9833},[129],[15,9835,59],{"className":9836},[195]," 取一个避开两点的区间。",[11,9839,9840,9841,10038,10039,10104,10105,10175,10176,10246,10247,10317,10318,10346],{},"三个点 ",[15,9842,9844,9879],{"className":9843,"translate":19},[23],[15,9845,9847],{"className":9846},[27],[29,9848,9849],{"xmlns":31},[34,9850,9851,9876],{},[37,9852,9853,9859,9861,9867,9869],{},[40,9854,9855,9857],{},[43,9856,56],{},[308,9858,318],{},[50,9860,7906],{},[40,9862,9863,9865],{},[43,9864,56],{},[308,9866,310],{},[50,9868,7906],{},[40,9870,9871,9873],{},[43,9872,56],{},[308,9874,9875],{},"3",[109,9877,9878],{"encoding":111},"x_1\u003Cx_2\u003Cx_3",[15,9880,9882,9937,9992],{"className":9881,"ariaHidden":72},[116],[15,9883,9885,9888,9928,9931,9934],{"className":9884},[120],[15,9886],{"className":9887,"style":7925},[124],[15,9889,9891,9894],{"className":9890},[129],[15,9892,56],{"className":9893},[129,133],[15,9895,9897],{"className":9896},[137],[15,9898,9900,9920],{"className":9899},[141,142],[15,9901,9903,9917],{"className":9902},[146],[15,9904,9906],{"className":9905,"style":1051},[150],[15,9907,9908,9911],{"style":154},[15,9909],{"className":9910,"style":159},[158],[15,9912,9914],{"className":9913},[163,164,165,166],[15,9915,318],{"className":9916},[129,166],[15,9918,175],{"className":9919},[174],[15,9921,9923],{"className":9922},[146],[15,9924,9926],{"className":9925,"style":182},[150],[15,9927],{},[15,9929],{"className":9930,"style":199},[93],[15,9932,7906],{"className":9933},[203],[15,9935],{"className":9936,"style":199},[93],[15,9938,9940,9943,9983,9986,9989],{"className":9939},[120],[15,9941],{"className":9942,"style":7925},[124],[15,9944,9946,9949],{"className":9945},[129],[15,9947,56],{"className":9948},[129,133],[15,9950,9952],{"className":9951},[137],[15,9953,9955,9975],{"className":9954},[141,142],[15,9956,9958,9972],{"className":9957},[146],[15,9959,9961],{"className":9960,"style":1051},[150],[15,9962,9963,9966],{"style":154},[15,9964],{"className":9965,"style":159},[158],[15,9967,9969],{"className":9968},[163,164,165,166],[15,9970,310],{"className":9971},[129,166],[15,9973,175],{"className":9974},[174],[15,9976,9978],{"className":9977},[146],[15,9979,9981],{"className":9980,"style":182},[150],[15,9982],{},[15,9984],{"className":9985,"style":199},[93],[15,9987,7906],{"className":9988},[203],[15,9990],{"className":9991,"style":199},[93],[15,9993,9995,9998],{"className":9994},[120],[15,9996],{"className":9997,"style":7981},[124],[15,9999,10001,10004],{"className":10000},[129],[15,10002,56],{"className":10003},[129,133],[15,10005,10007],{"className":10006},[137],[15,10008,10010,10030],{"className":10009},[141,142],[15,10011,10013,10027],{"className":10012},[146],[15,10014,10016],{"className":10015,"style":1051},[150],[15,10017,10018,10021],{"style":154},[15,10019],{"className":10020,"style":159},[158],[15,10022,10024],{"className":10023},[163,164,165,166],[15,10025,9875],{"className":10026},[129,166],[15,10028,175],{"className":10029},[174],[15,10031,10033],{"className":10032},[146],[15,10034,10036],{"className":10035,"style":182},[150],[15,10037],{}," 不行：标注 ",[15,10040,10042,10068],{"className":10041,"translate":19},[23],[15,10043,10045],{"className":10044},[27],[29,10046,10047],{"xmlns":31},[34,10048,10049,10065],{},[37,10050,10051,10053,10055,10057,10059,10061,10063],{},[50,10052,53],{"stretchy":52},[308,10054,318],{},[50,10056,91],{"separator":72},[308,10058,541],{},[50,10060,91],{"separator":72},[308,10062,318],{},[50,10064,59],{"stretchy":52},[109,10066,10067],{"encoding":111},"(1,0,1)",[15,10069,10071],{"className":10070,"ariaHidden":72},[116],[15,10072,10074,10077,10080,10083,10086,10089,10092,10095,10098,10101],{"className":10073},[120],[15,10075],{"className":10076,"style":125},[124],[15,10078,53],{"className":10079},[188],[15,10081,318],{"className":10082},[129],[15,10084,91],{"className":10085},[254],[15,10087],{"className":10088,"style":262},[93],[15,10090,541],{"className":10091},[129],[15,10093,91],{"className":10094},[254],[15,10096],{"className":10097,"style":262},[93],[15,10099,318],{"className":10100},[129],[15,10102,59],{"className":10103},[195]," 要求区间同时包含 ",[15,10106,10108,10126],{"className":10107,"translate":19},[23],[15,10109,10111],{"className":10110},[27],[29,10112,10113],{"xmlns":31},[34,10114,10115,10123],{},[37,10116,10117],{},[40,10118,10119,10121],{},[43,10120,56],{},[308,10122,318],{},[109,10124,10125],{"encoding":111},"x_1",[15,10127,10129],{"className":10128,"ariaHidden":72},[116],[15,10130,10132,10135],{"className":10131},[120],[15,10133],{"className":10134,"style":7981},[124],[15,10136,10138,10141],{"className":10137},[129],[15,10139,56],{"className":10140},[129,133],[15,10142,10144],{"className":10143},[137],[15,10145,10147,10167],{"className":10146},[141,142],[15,10148,10150,10164],{"className":10149},[146],[15,10151,10153],{"className":10152,"style":1051},[150],[15,10154,10155,10158],{"style":154},[15,10156],{"className":10157,"style":159},[158],[15,10159,10161],{"className":10160},[163,164,165,166],[15,10162,318],{"className":10163},[129,166],[15,10165,175],{"className":10166},[174],[15,10168,10170],{"className":10169},[146],[15,10171,10173],{"className":10172,"style":182},[150],[15,10174],{}," 与 ",[15,10177,10179,10197],{"className":10178,"translate":19},[23],[15,10180,10182],{"className":10181},[27],[29,10183,10184],{"xmlns":31},[34,10185,10186,10194],{},[37,10187,10188],{},[40,10189,10190,10192],{},[43,10191,56],{},[308,10193,9875],{},[109,10195,10196],{"encoding":111},"x_3",[15,10198,10200],{"className":10199,"ariaHidden":72},[116],[15,10201,10203,10206],{"className":10202},[120],[15,10204],{"className":10205,"style":7981},[124],[15,10207,10209,10212],{"className":10208},[129],[15,10210,56],{"className":10211},[129,133],[15,10213,10215],{"className":10214},[137],[15,10216,10218,10238],{"className":10217},[141,142],[15,10219,10221,10235],{"className":10220},[146],[15,10222,10224],{"className":10223,"style":1051},[150],[15,10225,10226,10229],{"style":154},[15,10227],{"className":10228,"style":159},[158],[15,10230,10232],{"className":10231},[163,164,165,166],[15,10233,9875],{"className":10234},[129,166],[15,10236,175],{"className":10237},[174],[15,10239,10241],{"className":10240},[146],[15,10242,10244],{"className":10243,"style":182},[150],[15,10245],{},"，而区间是凸的，必然包含中间的 ",[15,10248,10250,10268],{"className":10249,"translate":19},[23],[15,10251,10253],{"className":10252},[27],[29,10254,10255],{"xmlns":31},[34,10256,10257,10265],{},[37,10258,10259],{},[40,10260,10261,10263],{},[43,10262,56],{},[308,10264,310],{},[109,10266,10267],{"encoding":111},"x_2",[15,10269,10271],{"className":10270,"ariaHidden":72},[116],[15,10272,10274,10277],{"className":10273},[120],[15,10275],{"className":10276,"style":7981},[124],[15,10278,10280,10283],{"className":10279},[129],[15,10281,56],{"className":10282},[129,133],[15,10284,10286],{"className":10285},[137],[15,10287,10289,10309],{"className":10288},[141,142],[15,10290,10292,10306],{"className":10291},[146],[15,10293,10295],{"className":10294,"style":1051},[150],[15,10296,10297,10300],{"style":154},[15,10298],{"className":10299,"style":159},[158],[15,10301,10303],{"className":10302},[163,164,165,166],[15,10304,310],{"className":10305},[129,166],[15,10307,175],{"className":10308},[174],[15,10310,10312],{"className":10311},[146],[15,10313,10315],{"className":10314,"style":182},[150],[15,10316],{},"，与中间的 ",[15,10319,10321,10334],{"className":10320,"translate":19},[23],[15,10322,10324],{"className":10323},[27],[29,10325,10326],{"xmlns":31},[34,10327,10328,10332],{},[37,10329,10330],{},[308,10331,541],{},[109,10333,541],{"encoding":111},[15,10335,10337],{"className":10336,"ariaHidden":72},[116],[15,10338,10340,10343],{"className":10339},[120],[15,10341],{"className":10342,"style":4886},[124],[15,10344,541],{"className":10345},[129]," 矛盾。",[11,10348,10349],{},[698,10350,10351,10352,7434],{},"例 3（仿射超平面，",[15,10353,10355,10380],{"className":10354,"translate":19},[23],[15,10356,10358],{"className":10357},[27],[29,10359,10360],{"xmlns":31},[34,10361,10362,10377],{},[37,10363,10364,10366,10368,10370,10373,10375],{},[43,10365,6819],{"mathvariant":65},[50,10367,69],{},[50,10369,62],{},[43,10371,10372],{},"d",[50,10374,2682],{},[308,10376,318],{},[109,10378,10379],{"encoding":111},"\\operatorname{VCdim}=d+1",[15,10381,10383,10404,10423],{"className":10382,"ariaHidden":72},[116],[15,10384,10386,10389,10395,10398,10401],{"className":10385},[120],[15,10387],{"className":10388,"style":522},[124],[15,10390,10392],{"className":10391},[217],[15,10393,6819],{"className":10394},[129,221],[15,10396],{"className":10397,"style":199},[93],[15,10399,62],{"className":10400},[203],[15,10402],{"className":10403,"style":199},[93],[15,10405,10407,10411,10414,10417,10420],{"className":10406},[120],[15,10408],{"className":10409,"style":10410},[124],"height:0.7778em;vertical-align:-0.0833em;",[15,10412,10372],{"className":10413},[129,133],[15,10415],{"className":10416,"style":3277},[93],[15,10418,2682],{"className":10419},[2436],[15,10421],{"className":10422,"style":3277},[93],[15,10424,10426,10429],{"className":10425},[120],[15,10427],{"className":10428,"style":4886},[124],[15,10430,318],{"className":10431},[129],[11,10433,7437,10434,7606,10516,1313],{},[15,10435,10437,10459],{"className":10436,"translate":19},[23],[15,10438,10440],{"className":10439},[27],[29,10441,10442],{"xmlns":31},[34,10443,10444,10456],{},[37,10445,10446,10448,10450],{},[43,10447,597],{"mathvariant":596},[50,10449,62],{},[305,10451,10452,10454],{},[43,10453,104],{"mathvariant":103},[43,10455,10372],{},[109,10457,10458],{"encoding":111},"\\mathcal X=\\mathbb R^d",[15,10460,10462,10480],{"className":10461,"ariaHidden":72},[116],[15,10463,10465,10468,10471,10474,10477],{"className":10464},[120],[15,10466],{"className":10467,"style":610},[124],[15,10469,597],{"className":10470,"style":615},[129,614],[15,10472],{"className":10473,"style":199},[93],[15,10475,62],{"className":10476},[203],[15,10478],{"className":10479,"style":199},[93],[15,10481,10483,10487],{"className":10482},[120],[15,10484],{"className":10485,"style":10486},[124],"height:0.8491em;",[15,10488,10490,10493],{"className":10489},[129],[15,10491,104],{"className":10492},[129,285],[15,10494,10496],{"className":10495},[137],[15,10497,10499],{"className":10498},[141],[15,10500,10502],{"className":10501},[146],[15,10503,10505],{"className":10504,"style":10486},[150],[15,10506,10507,10510],{"style":383},[15,10508],{"className":10509,"style":159},[158],[15,10511,10513],{"className":10512},[163,164,165,166],[15,10514,10372],{"className":10515},[129,133,166],[15,10517,10519,10563],{"className":10518,"translate":19},[23],[15,10520,10522],{"className":10521},[27],[29,10523,10524],{"xmlns":31},[34,10525,10526,10560],{},[37,10527,10528,10530,10532,10534,10536,10538,10540,10542,10544,10547,10550,10552,10554,10556,10558],{},[43,10529,819],{"mathvariant":596},[50,10531,62],{},[50,10533,636],{"stretchy":52},[43,10535,56],{},[50,10537,7460],{},[43,10539,66],{"mathvariant":65},[50,10541,69],{},[50,10543,53],{"stretchy":52},[43,10545,10546],{},"w",[50,10548,10549],{},"⋅",[43,10551,56],{},[50,10553,2682],{},[43,10555,8695],{},[50,10557,59],{"stretchy":52},[50,10559,645],{"stretchy":52},[109,10561,10562],{"encoding":111},"\\mathcal H=\\{x\\mapsto\\operatorname{sign}(w\\cdot x+b)\\}",[15,10564,10566,10584,10605,10633,10651],{"className":10565,"ariaHidden":72},[116],[15,10567,10569,10572,10575,10578,10581],{"className":10568},[120],[15,10570],{"className":10571,"style":610},[124],[15,10573,819],{"className":10574,"style":835},[129,614],[15,10576],{"className":10577,"style":199},[93],[15,10579,62],{"className":10580},[203],[15,10582],{"className":10583,"style":199},[93],[15,10585,10587,10590,10593,10596,10599,10602],{"className":10586},[120],[15,10588],{"className":10589,"style":125},[124],[15,10591,636],{"className":10592},[188],[15,10594,56],{"className":10595},[129,133],[15,10597],{"className":10598,"style":199},[93],[15,10600,7460],{"className":10601},[203],[15,10603],{"className":10604,"style":199},[93],[15,10606,10608,10611,10617,10620,10624,10627,10630],{"className":10607},[120],[15,10609],{"className":10610,"style":125},[124],[15,10612,10614],{"className":10613},[217],[15,10615,66],{"className":10616},[129,221],[15,10618,53],{"className":10619},[188],[15,10621,10546],{"className":10622,"style":10623},[129,133],"margin-right:0.0269em;",[15,10625],{"className":10626,"style":3277},[93],[15,10628,10549],{"className":10629},[2436],[15,10631],{"className":10632,"style":3277},[93],[15,10634,10636,10639,10642,10645,10648],{"className":10635},[120],[15,10637],{"className":10638,"style":8514},[124],[15,10640,56],{"className":10641},[129,133],[15,10643],{"className":10644,"style":3277},[93],[15,10646,2682],{"className":10647},[2436],[15,10649],{"className":10650,"style":3277},[93],[15,10652,10654,10657,10660],{"className":10653},[120],[15,10655],{"className":10656,"style":125},[124],[15,10658,8695],{"className":10659},[129,133],[15,10661,1235],{"className":10662},[195],[11,10664,10665,10668,10669,10855,10856,10907,10908,11022],{},[698,10666,10667],{},"下界","：取 ",[15,10670,10672,10715],{"className":10671,"translate":19},[23],[15,10673,10675],{"className":10674},[27],[29,10676,10677],{"xmlns":31},[34,10678,10679,10712],{},[37,10680,10681,10683,10685,10687,10689,10691,10698,10700,10702,10704,10710],{},[43,10682,5619],{},[50,10684,62],{},[50,10686,636],{"stretchy":52},[308,10688,541],{},[50,10690,91],{"separator":72},[40,10692,10693,10696],{},[43,10694,10695],{},"e",[308,10697,318],{},[50,10699,91],{"separator":72},[50,10701,335],{},[50,10703,91],{"separator":72},[40,10705,10706,10708],{},[43,10707,10695],{},[43,10709,10372],{},[50,10711,645],{"stretchy":52},[109,10713,10714],{"encoding":111},"C=\\{0,e_1,\\dots,e_d\\}",[15,10716,10718,10736],{"className":10717,"ariaHidden":72},[116],[15,10719,10721,10724,10727,10730,10733],{"className":10720},[120],[15,10722],{"className":10723,"style":610},[124],[15,10725,5619],{"className":10726,"style":5665},[129,133],[15,10728],{"className":10729,"style":199},[93],[15,10731,62],{"className":10732},[203],[15,10734],{"className":10735,"style":199},[93],[15,10737,10739,10742,10745,10748,10751,10754,10794,10797,10800,10803,10806,10809,10812,10852],{"className":10738},[120],[15,10740],{"className":10741,"style":125},[124],[15,10743,636],{"className":10744},[188],[15,10746,541],{"className":10747},[129],[15,10749,91],{"className":10750},[254],[15,10752],{"className":10753,"style":262},[93],[15,10755,10757,10760],{"className":10756},[129],[15,10758,10695],{"className":10759},[129,133],[15,10761,10763],{"className":10762},[137],[15,10764,10766,10786],{"className":10765},[141,142],[15,10767,10769,10783],{"className":10768},[146],[15,10770,10772],{"className":10771,"style":1051},[150],[15,10773,10774,10777],{"style":154},[15,10775],{"className":10776,"style":159},[158],[15,10778,10780],{"className":10779},[163,164,165,166],[15,10781,318],{"className":10782},[129,166],[15,10784,175],{"className":10785},[174],[15,10787,10789],{"className":10788},[146],[15,10790,10792],{"className":10791,"style":182},[150],[15,10793],{},[15,10795,91],{"className":10796},[254],[15,10798],{"className":10799,"style":262},[93],[15,10801,335],{"className":10802},[449],[15,10804],{"className":10805,"style":262},[93],[15,10807,91],{"className":10808},[254],[15,10810],{"className":10811,"style":262},[93],[15,10813,10815,10818],{"className":10814},[129],[15,10816,10695],{"className":10817},[129,133],[15,10819,10821],{"className":10820},[137],[15,10822,10824,10844],{"className":10823},[141,142],[15,10825,10827,10841],{"className":10826},[146],[15,10828,10830],{"className":10829,"style":151},[150],[15,10831,10832,10835],{"style":154},[15,10833],{"className":10834,"style":159},[158],[15,10836,10838],{"className":10837},[163,164,165,166],[15,10839,10372],{"className":10840},[129,133,166],[15,10842,175],{"className":10843},[174],[15,10845,10847],{"className":10846},[146],[15,10848,10850],{"className":10849,"style":182},[150],[15,10851],{},[15,10853,645],{"className":10854},[195],"，这是 ",[15,10857,10859,10877],{"className":10858,"translate":19},[23],[15,10860,10862],{"className":10861},[27],[29,10863,10864],{"xmlns":31},[34,10865,10866,10874],{},[37,10867,10868,10870,10872],{},[43,10869,10372],{},[50,10871,2682],{},[308,10873,318],{},[109,10875,10876],{"encoding":111},"d+1",[15,10878,10880,10898],{"className":10879,"ariaHidden":72},[116],[15,10881,10883,10886,10889,10892,10895],{"className":10882},[120],[15,10884],{"className":10885,"style":10410},[124],[15,10887,10372],{"className":10888},[129,133],[15,10890],{"className":10891,"style":3277},[93],[15,10893,2682],{"className":10894},[2436],[15,10896],{"className":10897,"style":3277},[93],[15,10899,10901,10904],{"className":10900},[120],[15,10902],{"className":10903,"style":4886},[124],[15,10905,318],{"className":10906},[129]," 个仿射无关的点。对任意标注 ",[15,10909,10911,10946],{"className":10910,"translate":19},[23],[15,10912,10914],{"className":10913},[27],[29,10915,10916],{"xmlns":31},[34,10917,10918,10943],{},[37,10919,10920,10922,10924,10926,10929,10931],{},[43,10921,857],{},[50,10923,100],{},[50,10925,636],{"stretchy":52},[50,10927,10928],{},"±",[308,10930,318],{},[305,10932,10933,10935],{},[50,10934,645],{"stretchy":52},[37,10936,10937,10939,10941],{},[43,10938,10372],{},[50,10940,2682],{},[308,10942,318],{},[109,10944,10945],{"encoding":111},"y\\in\\{\\pm1\\}^{d+1}",[15,10947,10949,10968],{"className":10948,"ariaHidden":72},[116],[15,10950,10952,10956,10959,10962,10965],{"className":10951},[120],[15,10953],{"className":10954,"style":10955},[124],"height:0.7335em;vertical-align:-0.1944em;",[15,10957,857],{"className":10958,"style":887},[129,133],[15,10960],{"className":10961,"style":199},[93],[15,10963,100],{"className":10964},[203],[15,10966],{"className":10967,"style":199},[93],[15,10969,10971,10975,10978,10981,10984],{"className":10970},[120],[15,10972],{"className":10973,"style":10974},[124],"height:1.0991em;vertical-align:-0.25em;",[15,10976,636],{"className":10977},[188],[15,10979,10928],{"className":10980},[129],[15,10982,318],{"className":10983},[129],[15,10985,10987,10990],{"className":10986},[195],[15,10988,645],{"className":10989},[195],[15,10991,10993],{"className":10992},[137],[15,10994,10996],{"className":10995},[141],[15,10997,10999],{"className":10998},[146],[15,11000,11002],{"className":11001,"style":10486},[150],[15,11003,11004,11007],{"style":383},[15,11005],{"className":11006,"style":159},[158],[15,11008,11010],{"className":11009},[163,164,165,166],[15,11011,11013,11016,11019],{"className":11012},[129,166],[15,11014,10372],{"className":11015},[129,133,166],[15,11017,2682],{"className":11018},[2436,166],[15,11020,318],{"className":11021},[129,166],"，方程组",[15,11024,11026],{"className":11025,"translate":19},[18],[15,11027,11029,11085],{"className":11028,"translate":19},[23],[15,11030,11032],{"className":11031},[27],[29,11033,11034],{"xmlns":31,"display":32},[34,11035,11036,11082],{},[37,11037,11038,11040,11042,11048,11050,11052,11054,11060,11062,11064,11066,11068,11070,11072,11074,11076,11078,11080],{},[43,11039,10546],{},[50,11041,10549],{},[40,11043,11044,11046],{},[43,11045,56],{},[43,11047,1459],{},[50,11049,2682],{},[43,11051,8695],{},[50,11053,62],{},[40,11055,11056,11058],{},[43,11057,857],{},[43,11059,1459],{},[50,11061,91],{"separator":72},[93,11063],{"width":95},[43,11065,1459],{},[50,11067,62],{},[308,11069,318],{},[50,11071,91],{"separator":72},[50,11073,335],{},[50,11075,91],{"separator":72},[43,11077,10372],{},[50,11079,2682],{},[308,11081,318],{},[109,11083,11084],{"encoding":111},"w\\cdot x_i+b=y_i,\\qquad i=1,\\dots,d+1",[15,11086,11088,11107,11163,11181,11249,11288],{"className":11087,"ariaHidden":72},[116],[15,11089,11091,11095,11098,11101,11104],{"className":11090},[120],[15,11092],{"className":11093,"style":11094},[124],"height:0.4445em;",[15,11096,10546],{"className":11097,"style":10623},[129,133],[15,11099],{"className":11100,"style":3277},[93],[15,11102,10549],{"className":11103},[2436],[15,11105],{"className":11106,"style":3277},[93],[15,11108,11110,11114,11154,11157,11160],{"className":11109},[120],[15,11111],{"className":11112,"style":11113},[124],"height:0.7333em;vertical-align:-0.15em;",[15,11115,11117,11120],{"className":11116},[129],[15,11118,56],{"className":11119},[129,133],[15,11121,11123],{"className":11122},[137],[15,11124,11126,11146],{"className":11125},[141,142],[15,11127,11129,11143],{"className":11128},[146],[15,11130,11132],{"className":11131,"style":1981},[150],[15,11133,11134,11137],{"style":154},[15,11135],{"className":11136,"style":159},[158],[15,11138,11140],{"className":11139},[163,164,165,166],[15,11141,1459],{"className":11142},[129,133,166],[15,11144,175],{"className":11145},[174],[15,11147,11149],{"className":11148},[146],[15,11150,11152],{"className":11151,"style":182},[150],[15,11153],{},[15,11155],{"className":11156,"style":3277},[93],[15,11158,2682],{"className":11159},[2436],[15,11161],{"className":11162,"style":3277},[93],[15,11164,11166,11169,11172,11175,11178],{"className":11165},[120],[15,11167],{"className":11168,"style":522},[124],[15,11170,8695],{"className":11171},[129,133],[15,11173],{"className":11174,"style":199},[93],[15,11176,62],{"className":11177},[203],[15,11179],{"className":11180,"style":199},[93],[15,11182,11184,11188,11228,11231,11234,11237,11240,11243,11246],{"className":11183},[120],[15,11185],{"className":11186,"style":11187},[124],"height:0.854em;vertical-align:-0.1944em;",[15,11189,11191,11194],{"className":11190},[129],[15,11192,857],{"className":11193,"style":887},[129,133],[15,11195,11197],{"className":11196},[137],[15,11198,11200,11220],{"className":11199},[141,142],[15,11201,11203,11217],{"className":11202},[146],[15,11204,11206],{"className":11205,"style":1981},[150],[15,11207,11208,11211],{"style":1100},[15,11209],{"className":11210,"style":159},[158],[15,11212,11214],{"className":11213},[163,164,165,166],[15,11215,1459],{"className":11216},[129,133,166],[15,11218,175],{"className":11219},[174],[15,11221,11223],{"className":11222},[146],[15,11224,11226],{"className":11225,"style":182},[150],[15,11227],{},[15,11229,91],{"className":11230},[254],[15,11232],{"className":11233,"style":258},[93],[15,11235],{"className":11236,"style":262},[93],[15,11238,1459],{"className":11239},[129,133],[15,11241],{"className":11242,"style":199},[93],[15,11244,62],{"className":11245},[203],[15,11247],{"className":11248,"style":199},[93],[15,11250,11252,11255,11258,11261,11264,11267,11270,11273,11276,11279,11282,11285],{"className":11251},[120],[15,11253],{"className":11254,"style":1682},[124],[15,11256,318],{"className":11257},[129],[15,11259,91],{"className":11260},[254],[15,11262],{"className":11263,"style":262},[93],[15,11265,335],{"className":11266},[449],[15,11268],{"className":11269,"style":262},[93],[15,11271,91],{"className":11272},[254],[15,11274],{"className":11275,"style":262},[93],[15,11277,10372],{"className":11278},[129,133],[15,11280],{"className":11281,"style":3277},[93],[15,11283,2682],{"className":11284},[2436],[15,11286],{"className":11287,"style":3277},[93],[15,11289,11291,11294],{"className":11290},[120],[15,11292],{"className":11293,"style":4886},[124],[15,11295,318],{"className":11296},[129],[11,11298,11299,11300,11393,11394,11510,11511,11539],{},"有唯一解，因为以 ",[15,11301,11303,11329],{"className":11302,"translate":19},[23],[15,11304,11306],{"className":11305},[27],[29,11307,11308],{"xmlns":31},[34,11309,11310,11326],{},[37,11311,11312,11314,11320,11322,11324],{},[50,11313,53],{"stretchy":52},[40,11315,11316,11318],{},[43,11317,56],{},[43,11319,1459],{},[50,11321,91],{"separator":72},[308,11323,318],{},[50,11325,59],{"stretchy":52},[109,11327,11328],{"encoding":111},"(x_i,1)",[15,11330,11332],{"className":11331,"ariaHidden":72},[116],[15,11333,11335,11338,11341,11381,11384,11387,11390],{"className":11334},[120],[15,11336],{"className":11337,"style":125},[124],[15,11339,53],{"className":11340},[188],[15,11342,11344,11347],{"className":11343},[129],[15,11345,56],{"className":11346},[129,133],[15,11348,11350],{"className":11349},[137],[15,11351,11353,11373],{"className":11352},[141,142],[15,11354,11356,11370],{"className":11355},[146],[15,11357,11359],{"className":11358,"style":1981},[150],[15,11360,11361,11364],{"style":154},[15,11362],{"className":11363,"style":159},[158],[15,11365,11367],{"className":11366},[163,164,165,166],[15,11368,1459],{"className":11369},[129,133,166],[15,11371,175],{"className":11372},[174],[15,11374,11376],{"className":11375},[146],[15,11377,11379],{"className":11378,"style":182},[150],[15,11380],{},[15,11382,91],{"className":11383},[254],[15,11385],{"className":11386,"style":262},[93],[15,11388,318],{"className":11389},[129],[15,11391,59],{"className":11392},[195]," 为列的 ",[15,11395,11397,11432],{"className":11396,"translate":19},[23],[15,11398,11400],{"className":11399},[27],[29,11401,11402],{"xmlns":31},[34,11403,11404,11429],{},[37,11405,11406,11408,11410,11412,11414,11416,11419,11421,11423,11425,11427],{},[50,11407,53],{"stretchy":52},[43,11409,10372],{},[50,11411,2682],{},[308,11413,318],{},[50,11415,59],{"stretchy":52},[50,11417,11418],{},"×",[50,11420,53],{"stretchy":52},[43,11422,10372],{},[50,11424,2682],{},[308,11426,318],{},[50,11428,59],{"stretchy":52},[109,11430,11431],{"encoding":111},"(d+1)\\times(d+1)",[15,11433,11435,11456,11477,11498],{"className":11434,"ariaHidden":72},[116],[15,11436,11438,11441,11444,11447,11450,11453],{"className":11437},[120],[15,11439],{"className":11440,"style":125},[124],[15,11442,53],{"className":11443},[188],[15,11445,10372],{"className":11446},[129,133],[15,11448],{"className":11449,"style":3277},[93],[15,11451,2682],{"className":11452},[2436],[15,11454],{"className":11455,"style":3277},[93],[15,11457,11459,11462,11465,11468,11471,11474],{"className":11458},[120],[15,11460],{"className":11461,"style":125},[124],[15,11463,318],{"className":11464},[129],[15,11466,59],{"className":11467},[195],[15,11469],{"className":11470,"style":3277},[93],[15,11472,11418],{"className":11473},[2436],[15,11475],{"className":11476,"style":3277},[93],[15,11478,11480,11483,11486,11489,11492,11495],{"className":11479},[120],[15,11481],{"className":11482,"style":125},[124],[15,11484,53],{"className":11485},[188],[15,11487,10372],{"className":11488},[129,133],[15,11490],{"className":11491,"style":3277},[93],[15,11493,2682],{"className":11494},[2436],[15,11496],{"className":11497,"style":3277},[93],[15,11499,11501,11504,11507],{"className":11500},[120],[15,11502],{"className":11503,"style":125},[124],[15,11505,318],{"className":11506},[129],[15,11508,59],{"className":11509},[195]," 矩阵可逆。于是每种标注都能被某个仿射函数实现，",[15,11512,11514,11527],{"className":11513,"translate":19},[23],[15,11515,11517],{"className":11516},[27],[29,11518,11519],{"xmlns":31},[34,11520,11521,11525],{},[37,11522,11523],{},[43,11524,5619],{},[109,11526,5619],{"encoding":111},[15,11528,11530],{"className":11529,"ariaHidden":72},[116],[15,11531,11533,11536],{"className":11532},[120],[15,11534],{"className":11535,"style":610},[124],[15,11537,5619],{"className":11538,"style":5665},[129,133]," 被打散。",[11,11541,11542,11545,11546,11605,11606,11657],{},[698,11543,11544],{},"上界","：",[15,11547,11549,11567],{"className":11548,"translate":19},[23],[15,11550,11552],{"className":11551},[27],[29,11553,11554],{"xmlns":31},[34,11555,11556,11564],{},[37,11557,11558],{},[305,11559,11560,11562],{},[43,11561,104],{"mathvariant":103},[43,11563,10372],{},[109,11565,11566],{"encoding":111},"\\mathbb R^d",[15,11568,11570],{"className":11569,"ariaHidden":72},[116],[15,11571,11573,11576],{"className":11572},[120],[15,11574],{"className":11575,"style":10486},[124],[15,11577,11579,11582],{"className":11578},[129],[15,11580,104],{"className":11581},[129,285],[15,11583,11585],{"className":11584},[137],[15,11586,11588],{"className":11587},[141],[15,11589,11591],{"className":11590},[146],[15,11592,11594],{"className":11593,"style":10486},[150],[15,11595,11596,11599],{"style":383},[15,11597],{"className":11598,"style":159},[158],[15,11600,11602],{"className":11601},[163,164,165,166],[15,11603,10372],{"className":11604},[129,133,166]," 中任意 ",[15,11607,11609,11627],{"className":11608,"translate":19},[23],[15,11610,11612],{"className":11611},[27],[29,11613,11614],{"xmlns":31},[34,11615,11616,11624],{},[37,11617,11618,11620,11622],{},[43,11619,10372],{},[50,11621,2682],{},[308,11623,310],{},[109,11625,11626],{"encoding":111},"d+2",[15,11628,11630,11648],{"className":11629,"ariaHidden":72},[116],[15,11631,11633,11636,11639,11642,11645],{"className":11632},[120],[15,11634],{"className":11635,"style":10410},[124],[15,11637,10372],{"className":11638},[129,133],[15,11640],{"className":11641,"style":3277},[93],[15,11643,2682],{"className":11644},[2436],[15,11646],{"className":11647,"style":3277},[93],[15,11649,11651,11654],{"className":11650},[120],[15,11652],{"className":11653,"style":4886},[124],[15,11655,310],{"className":11656},[129]," 个点都仿射相关，因此可以用 Radon 定理处理。",[6559,11659,11661],{"title":11660},"定理（Radon，1921）",[11,11662,11663,11605,11721,11771],{},[15,11664,11666,11683],{"className":11665,"translate":19},[23],[15,11667,11669],{"className":11668},[27],[29,11670,11671],{"xmlns":31},[34,11672,11673,11681],{},[37,11674,11675],{},[305,11676,11677,11679],{},[43,11678,104],{"mathvariant":103},[43,11680,10372],{},[109,11682,11566],{"encoding":111},[15,11684,11686],{"className":11685,"ariaHidden":72},[116],[15,11687,11689,11692],{"className":11688},[120],[15,11690],{"className":11691,"style":10486},[124],[15,11693,11695,11698],{"className":11694},[129],[15,11696,104],{"className":11697},[129,285],[15,11699,11701],{"className":11700},[137],[15,11702,11704],{"className":11703},[141],[15,11705,11707],{"className":11706},[146],[15,11708,11710],{"className":11709,"style":10486},[150],[15,11711,11712,11715],{"style":383},[15,11713],{"className":11714,"style":159},[158],[15,11716,11718],{"className":11717},[163,164,165,166],[15,11719,10372],{"className":11720},[129,133,166],[15,11722,11724,11741],{"className":11723,"translate":19},[23],[15,11725,11727],{"className":11726},[27],[29,11728,11729],{"xmlns":31},[34,11730,11731,11739],{},[37,11732,11733,11735,11737],{},[43,11734,10372],{},[50,11736,2682],{},[308,11738,310],{},[109,11740,11626],{"encoding":111},[15,11742,11744,11762],{"className":11743,"ariaHidden":72},[116],[15,11745,11747,11750,11753,11756,11759],{"className":11746},[120],[15,11748],{"className":11749,"style":10410},[124],[15,11751,10372],{"className":11752},[129,133],[15,11754],{"className":11755,"style":3277},[93],[15,11757,2682],{"className":11758},[2436],[15,11760],{"className":11761,"style":3277},[93],[15,11763,11765,11768],{"className":11764},[120],[15,11766],{"className":11767,"style":4886},[124],[15,11769,310],{"className":11770},[129]," 个点都可以分成两个不相交的子集，使二者的凸包相交。",[11,11773,11774,11775,7606,11935,11985,11986,12059],{},"证明很短。把点写成齐次坐标 ",[15,11776,11778,11818],{"className":11777,"translate":19},[23],[15,11779,11781],{"className":11780},[27],[29,11782,11783],{"xmlns":31},[34,11784,11785,11815],{},[37,11786,11787,11789,11795,11797,11799,11801,11803],{},[50,11788,53],{"stretchy":52},[40,11790,11791,11793],{},[43,11792,56],{},[43,11794,1459],{},[50,11796,91],{"separator":72},[308,11798,318],{},[50,11800,59],{"stretchy":52},[50,11802,100],{},[305,11804,11805,11807],{},[43,11806,104],{"mathvariant":103},[37,11808,11809,11811,11813],{},[43,11810,10372],{},[50,11812,2682],{},[308,11814,318],{},[109,11816,11817],{"encoding":111},"(x_i,1)\\in\\mathbb R^{d+1}",[15,11819,11821,11891],{"className":11820,"ariaHidden":72},[116],[15,11822,11824,11827,11830,11870,11873,11876,11879,11882,11885,11888],{"className":11823},[120],[15,11825],{"className":11826,"style":125},[124],[15,11828,53],{"className":11829},[188],[15,11831,11833,11836],{"className":11832},[129],[15,11834,56],{"className":11835},[129,133],[15,11837,11839],{"className":11838},[137],[15,11840,11842,11862],{"className":11841},[141,142],[15,11843,11845,11859],{"className":11844},[146],[15,11846,11848],{"className":11847,"style":1981},[150],[15,11849,11850,11853],{"style":154},[15,11851],{"className":11852,"style":159},[158],[15,11854,11856],{"className":11855},[163,164,165,166],[15,11857,1459],{"className":11858},[129,133,166],[15,11860,175],{"className":11861},[174],[15,11863,11865],{"className":11864},[146],[15,11866,11868],{"className":11867,"style":182},[150],[15,11869],{},[15,11871,91],{"className":11872},[254],[15,11874],{"className":11875,"style":262},[93],[15,11877,318],{"className":11878},[129],[15,11880,59],{"className":11881},[195],[15,11883],{"className":11884,"style":199},[93],[15,11886,100],{"className":11887},[203],[15,11889],{"className":11890,"style":199},[93],[15,11892,11894,11897],{"className":11893},[120],[15,11895],{"className":11896,"style":10486},[124],[15,11898,11900,11903],{"className":11899},[129],[15,11901,104],{"className":11902},[129,285],[15,11904,11906],{"className":11905},[137],[15,11907,11909],{"className":11908},[141],[15,11910,11912],{"className":11911},[146],[15,11913,11915],{"className":11914,"style":10486},[150],[15,11916,11917,11920],{"style":383},[15,11918],{"className":11919,"style":159},[158],[15,11921,11923],{"className":11922},[163,164,165,166],[15,11924,11926,11929,11932],{"className":11925},[129,166],[15,11927,10372],{"className":11928},[129,133,166],[15,11930,2682],{"className":11931},[2436,166],[15,11933,318],{"className":11934},[129,166],[15,11936,11938,11955],{"className":11937,"translate":19},[23],[15,11939,11941],{"className":11940},[27],[29,11942,11943],{"xmlns":31},[34,11944,11945,11953],{},[37,11946,11947,11949,11951],{},[43,11948,10372],{},[50,11950,2682],{},[308,11952,310],{},[109,11954,11626],{"encoding":111},[15,11956,11958,11976],{"className":11957,"ariaHidden":72},[116],[15,11959,11961,11964,11967,11970,11973],{"className":11960},[120],[15,11962],{"className":11963,"style":10410},[124],[15,11965,10372],{"className":11966},[129,133],[15,11968],{"className":11969,"style":3277},[93],[15,11971,2682],{"className":11972},[2436],[15,11974],{"className":11975,"style":3277},[93],[15,11977,11979,11982],{"className":11978},[120],[15,11980],{"className":11981,"style":4886},[124],[15,11983,310],{"className":11984},[129]," 个向量线性相关，故存在不全为零的系数 ",[15,11987,11989,12008],{"className":11988,"translate":19},[23],[15,11990,11992],{"className":11991},[27],[29,11993,11994],{"xmlns":31},[34,11995,11996,12005],{},[37,11997,11998],{},[40,11999,12000,12003],{},[43,12001,12002],{},"α",[43,12004,1459],{},[109,12006,12007],{"encoding":111},"\\alpha_i",[15,12009,12011],{"className":12010,"ariaHidden":72},[116],[15,12012,12014,12017],{"className":12013},[120],[15,12015],{"className":12016,"style":7981},[124],[15,12018,12020,12024],{"className":12019},[129],[15,12021,12002],{"className":12022,"style":12023},[129,133],"margin-right:0.0037em;",[15,12025,12027],{"className":12026},[137],[15,12028,12030,12051],{"className":12029},[141,142],[15,12031,12033,12048],{"className":12032},[146],[15,12034,12036],{"className":12035,"style":1981},[150],[15,12037,12039,12042],{"style":12038},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[15,12040],{"className":12041,"style":159},[158],[15,12043,12045],{"className":12044},[163,164,165,166],[15,12046,1459],{"className":12047},[129,133,166],[15,12049,175],{"className":12050},[174],[15,12052,12054],{"className":12053},[146],[15,12055,12057],{"className":12056,"style":182},[150],[15,12058],{},"，使",[15,12061,12063],{"className":12062,"translate":19},[18],[15,12064,12066,12121],{"className":12065,"translate":19},[23],[15,12067,12069],{"className":12068},[27],[29,12070,12071],{"xmlns":31,"display":32},[34,12072,12073,12118],{},[37,12074,12075,12081,12087,12093,12095,12097,12099,12101,12107,12113,12115],{},[2140,12076,12077,12079],{},[50,12078,1454],{},[43,12080,1459],{},[40,12082,12083,12085],{},[43,12084,12002],{},[43,12086,1459],{},[40,12088,12089,12091],{},[43,12090,56],{},[43,12092,1459],{},[50,12094,62],{},[308,12096,541],{},[50,12098,91],{"separator":72},[93,12100],{"width":95},[2140,12102,12103,12105],{},[50,12104,1454],{},[43,12106,1459],{},[40,12108,12109,12111],{},[43,12110,12002],{},[43,12112,1459],{},[50,12114,62],{},[308,12116,12117],{},"0.",[109,12119,12120],{"encoding":111},"\\sum_i \\alpha_i x_i=0,\\qquad \\sum_i\\alpha_i=0.",[15,12122,12124,12268,12382],{"className":12123,"ariaHidden":72},[116],[15,12125,12127,12131,12176,12179,12219,12259,12262,12265],{"className":12126},[120],[15,12128],{"className":12129,"style":12130},[124],"height:2.3277em;vertical-align:-1.2777em;",[15,12132,12134],{"className":12133},[217,1874],[15,12135,12137,12168],{"className":12136},[141,142],[15,12138,12140,12165],{"className":12139},[146],[15,12141,12144,12155],{"className":12142,"style":12143},[150],"height:1.05em;",[15,12145,12146,12149],{"style":1887},[15,12147],{"className":12148,"style":1891},[158],[15,12150,12152],{"className":12151},[163,164,165,166],[15,12153,1459],{"className":12154},[129,133,166],[15,12156,12157,12160],{"style":1909},[15,12158],{"className":12159,"style":1891},[158],[15,12161,12162],{},[15,12163,1454],{"className":12164},[217,1918,1919],[15,12166,175],{"className":12167},[174],[15,12169,12171],{"className":12170},[146],[15,12172,12174],{"className":12173,"style":1941},[150],[15,12175],{},[15,12177],{"className":12178,"style":262},[93],[15,12180,12182,12185],{"className":12181},[129],[15,12183,12002],{"className":12184,"style":12023},[129,133],[15,12186,12188],{"className":12187},[137],[15,12189,12191,12211],{"className":12190},[141,142],[15,12192,12194,12208],{"className":12193},[146],[15,12195,12197],{"className":12196,"style":1981},[150],[15,12198,12199,12202],{"style":12038},[15,12200],{"className":12201,"style":159},[158],[15,12203,12205],{"className":12204},[163,164,165,166],[15,12206,1459],{"className":12207},[129,133,166],[15,12209,175],{"className":12210},[174],[15,12212,12214],{"className":12213},[146],[15,12215,12217],{"className":12216,"style":182},[150],[15,12218],{},[15,12220,12222,12225],{"className":12221},[129],[15,12223,56],{"className":12224},[129,133],[15,12226,12228],{"className":12227},[137],[15,12229,12231,12251],{"className":12230},[141,142],[15,12232,12234,12248],{"className":12233},[146],[15,12235,12237],{"className":12236,"style":1981},[150],[15,12238,12239,12242],{"style":154},[15,12240],{"className":12241,"style":159},[158],[15,12243,12245],{"className":12244},[163,164,165,166],[15,12246,1459],{"className":12247},[129,133,166],[15,12249,175],{"className":12250},[174],[15,12252,12254],{"className":12253},[146],[15,12255,12257],{"className":12256,"style":182},[150],[15,12258],{},[15,12260],{"className":12261,"style":199},[93],[15,12263,62],{"className":12264},[203],[15,12266],{"className":12267,"style":199},[93],[15,12269,12271,12274,12277,12280,12283,12286,12330,12333,12373,12376,12379],{"className":12270},[120],[15,12272],{"className":12273,"style":12130},[124],[15,12275,541],{"className":12276},[129],[15,12278,91],{"className":12279},[254],[15,12281],{"className":12282,"style":258},[93],[15,12284],{"className":12285,"style":262},[93],[15,12287,12289],{"className":12288},[217,1874],[15,12290,12292,12322],{"className":12291},[141,142],[15,12293,12295,12319],{"className":12294},[146],[15,12296,12298,12309],{"className":12297,"style":12143},[150],[15,12299,12300,12303],{"style":1887},[15,12301],{"className":12302,"style":1891},[158],[15,12304,12306],{"className":12305},[163,164,165,166],[15,12307,1459],{"className":12308},[129,133,166],[15,12310,12311,12314],{"style":1909},[15,12312],{"className":12313,"style":1891},[158],[15,12315,12316],{},[15,12317,1454],{"className":12318},[217,1918,1919],[15,12320,175],{"className":12321},[174],[15,12323,12325],{"className":12324},[146],[15,12326,12328],{"className":12327,"style":1941},[150],[15,12329],{},[15,12331],{"className":12332,"style":262},[93],[15,12334,12336,12339],{"className":12335},[129],[15,12337,12002],{"className":12338,"style":12023},[129,133],[15,12340,12342],{"className":12341},[137],[15,12343,12345,12365],{"className":12344},[141,142],[15,12346,12348,12362],{"className":12347},[146],[15,12349,12351],{"className":12350,"style":1981},[150],[15,12352,12353,12356],{"style":12038},[15,12354],{"className":12355,"style":159},[158],[15,12357,12359],{"className":12358},[163,164,165,166],[15,12360,1459],{"className":12361},[129,133,166],[15,12363,175],{"className":12364},[174],[15,12366,12368],{"className":12367},[146],[15,12369,12371],{"className":12370,"style":182},[150],[15,12372],{},[15,12374],{"className":12375,"style":199},[93],[15,12377,62],{"className":12378},[203],[15,12380],{"className":12381,"style":199},[93],[15,12383,12385,12388],{"className":12384},[120],[15,12386],{"className":12387,"style":4886},[124],[15,12389,12117],{"className":12390},[129],[11,12392,12393,12394,7606,12541,12689],{},"令 ",[15,12395,12397,12431],{"className":12396,"translate":19},[23],[15,12398,12400],{"className":12399},[27],[29,12401,12402],{"xmlns":31},[34,12403,12404,12428],{},[37,12405,12406,12408,12410,12412,12414,12416,12422,12424,12426],{},[43,12407,1388],{},[50,12409,62],{},[50,12411,636],{"stretchy":52},[43,12413,1459],{},[50,12415,718],{},[40,12417,12418,12420],{},[43,12419,12002],{},[43,12421,1459],{},[50,12423,4852],{},[308,12425,541],{},[50,12427,645],{"stretchy":52},[109,12429,12430],{"encoding":111},"P=\\{i:\\alpha_i>0\\}",[15,12432,12434,12453,12474,12529],{"className":12433,"ariaHidden":72},[116],[15,12435,12437,12440,12444,12447,12450],{"className":12436},[120],[15,12438],{"className":12439,"style":610},[124],[15,12441,1388],{"className":12442,"style":12443},[129,133],"margin-right:0.1389em;",[15,12445],{"className":12446,"style":199},[93],[15,12448,62],{"className":12449},[203],[15,12451],{"className":12452,"style":199},[93],[15,12454,12456,12459,12462,12465,12468,12471],{"className":12455},[120],[15,12457],{"className":12458,"style":125},[124],[15,12460,636],{"className":12461},[188],[15,12463,1459],{"className":12464},[129,133],[15,12466],{"className":12467,"style":199},[93],[15,12469,718],{"className":12470},[203],[15,12472],{"className":12473,"style":199},[93],[15,12475,12477,12480,12520,12523,12526],{"className":12476},[120],[15,12478],{"className":12479,"style":7925},[124],[15,12481,12483,12486],{"className":12482},[129],[15,12484,12002],{"className":12485,"style":12023},[129,133],[15,12487,12489],{"className":12488},[137],[15,12490,12492,12512],{"className":12491},[141,142],[15,12493,12495,12509],{"className":12494},[146],[15,12496,12498],{"className":12497,"style":1981},[150],[15,12499,12500,12503],{"style":12038},[15,12501],{"className":12502,"style":159},[158],[15,12504,12506],{"className":12505},[163,164,165,166],[15,12507,1459],{"className":12508},[129,133,166],[15,12510,175],{"className":12511},[174],[15,12513,12515],{"className":12514},[146],[15,12516,12518],{"className":12517,"style":182},[150],[15,12519],{},[15,12521],{"className":12522,"style":199},[93],[15,12524,4852],{"className":12525},[203],[15,12527],{"className":12528,"style":199},[93],[15,12530,12532,12535,12538],{"className":12531},[120],[15,12533],{"className":12534,"style":125},[124],[15,12536,541],{"className":12537},[129],[15,12539,645],{"className":12540},[195],[15,12542,12544,12579],{"className":12543,"translate":19},[23],[15,12545,12547],{"className":12546},[27],[29,12548,12549],{"xmlns":31},[34,12550,12551,12576],{},[37,12552,12553,12556,12558,12560,12562,12564,12570,12572,12574],{},[43,12554,12555],{},"N",[50,12557,62],{},[50,12559,636],{"stretchy":52},[43,12561,1459],{},[50,12563,718],{},[40,12565,12566,12568],{},[43,12567,12002],{},[43,12569,1459],{},[50,12571,7906],{},[308,12573,541],{},[50,12575,645],{"stretchy":52},[109,12577,12578],{"encoding":111},"N=\\{i:\\alpha_i\u003C0\\}",[15,12580,12582,12601,12622,12677],{"className":12581,"ariaHidden":72},[116],[15,12583,12585,12588,12592,12595,12598],{"className":12584},[120],[15,12586],{"className":12587,"style":610},[124],[15,12589,12555],{"className":12590,"style":12591},[129,133],"margin-right:0.109em;",[15,12593],{"className":12594,"style":199},[93],[15,12596,62],{"className":12597},[203],[15,12599],{"className":12600,"style":199},[93],[15,12602,12604,12607,12610,12613,12616,12619],{"className":12603},[120],[15,12605],{"className":12606,"style":125},[124],[15,12608,636],{"className":12609},[188],[15,12611,1459],{"className":12612},[129,133],[15,12614],{"className":12615,"style":199},[93],[15,12617,718],{"className":12618},[203],[15,12620],{"className":12621,"style":199},[93],[15,12623,12625,12628,12668,12671,12674],{"className":12624},[120],[15,12626],{"className":12627,"style":7925},[124],[15,12629,12631,12634],{"className":12630},[129],[15,12632,12002],{"className":12633,"style":12023},[129,133],[15,12635,12637],{"className":12636},[137],[15,12638,12640,12660],{"className":12639},[141,142],[15,12641,12643,12657],{"className":12642},[146],[15,12644,12646],{"className":12645,"style":1981},[150],[15,12647,12648,12651],{"style":12038},[15,12649],{"className":12650,"style":159},[158],[15,12652,12654],{"className":12653},[163,164,165,166],[15,12655,1459],{"className":12656},[129,133,166],[15,12658,175],{"className":12659},[174],[15,12661,12663],{"className":12662},[146],[15,12664,12666],{"className":12665,"style":182},[150],[15,12667],{},[15,12669],{"className":12670,"style":199},[93],[15,12672,7906],{"className":12673},[203],[15,12675],{"className":12676,"style":199},[93],[15,12678,12680,12683,12686],{"className":12679},[120],[15,12681],{"className":12682,"style":125},[124],[15,12684,541],{"className":12685},[129],[15,12687,645],{"className":12688},[195],"。两者都非空，且",[15,12691,12693],{"className":12692,"translate":19},[18],[15,12694,12696,12766],{"className":12695,"translate":19},[23],[15,12697,12699],{"className":12698},[27],[29,12700,12701],{"xmlns":31,"display":32},[34,12702,12703,12763],{},[37,12704,12705,12717,12723,12729,12731,12743,12745,12747,12753,12755,12761],{},[2140,12706,12707,12709],{},[50,12708,1454],{},[37,12710,12711,12713,12715],{},[43,12712,1459],{},[50,12714,100],{},[43,12716,1388],{},[40,12718,12719,12721],{},[43,12720,12002],{},[43,12722,1459],{},[40,12724,12725,12727],{},[43,12726,56],{},[43,12728,1459],{},[50,12730,62],{},[2140,12732,12733,12735],{},[50,12734,1454],{},[37,12736,12737,12739,12741],{},[43,12738,1459],{},[50,12740,100],{},[43,12742,12555],{},[50,12744,53],{"stretchy":52},[50,12746,315],{},[40,12748,12749,12751],{},[43,12750,12002],{},[43,12752,1459],{},[50,12754,59],{"stretchy":52},[40,12756,12757,12759],{},[43,12758,56],{},[43,12760,1459],{},[43,12762,107],{"mathvariant":65},[109,12764,12765],{"encoding":111},"\\sum_{i\\in P}\\alpha_i x_i=\\sum_{i\\in N}(-\\alpha_i)x_i.",[15,12767,12769,12923],{"className":12768,"ariaHidden":72},[116],[15,12770,12772,12776,12831,12834,12874,12914,12917,12920],{"className":12771},[120],[15,12773],{"className":12774,"style":12775},[124],"height:2.3717em;vertical-align:-1.3217em;",[15,12777,12779],{"className":12778},[217,1874],[15,12780,12782,12822],{"className":12781},[141,142],[15,12783,12785,12819],{"className":12784},[146],[15,12786,12788,12809],{"className":12787,"style":12143},[150],[15,12789,12791,12794],{"style":12790},"top:-1.8557em;margin-left:0em;",[15,12792],{"className":12793,"style":1891},[158],[15,12795,12797],{"className":12796},[163,164,165,166],[15,12798,12800,12803,12806],{"className":12799},[129,166],[15,12801,1459],{"className":12802},[129,133,166],[15,12804,100],{"className":12805},[203,166],[15,12807,1388],{"className":12808,"style":12443},[129,133,166],[15,12810,12811,12814],{"style":1909},[15,12812],{"className":12813,"style":1891},[158],[15,12815,12816],{},[15,12817,1454],{"className":12818},[217,1918,1919],[15,12820,175],{"className":12821},[174],[15,12823,12825],{"className":12824},[146],[15,12826,12829],{"className":12827,"style":12828},[150],"height:1.3217em;",[15,12830],{},[15,12832],{"className":12833,"style":262},[93],[15,12835,12837,12840],{"className":12836},[129],[15,12838,12002],{"className":12839,"style":12023},[129,133],[15,12841,12843],{"className":12842},[137],[15,12844,12846,12866],{"className":12845},[141,142],[15,12847,12849,12863],{"className":12848},[146],[15,12850,12852],{"className":12851,"style":1981},[150],[15,12853,12854,12857],{"style":12038},[15,12855],{"className":12856,"style":159},[158],[15,12858,12860],{"className":12859},[163,164,165,166],[15,12861,1459],{"className":12862},[129,133,166],[15,12864,175],{"className":12865},[174],[15,12867,12869],{"className":12868},[146],[15,12870,12872],{"className":12871,"style":182},[150],[15,12873],{},[15,12875,12877,12880],{"className":12876},[129],[15,12878,56],{"className":12879},[129,133],[15,12881,12883],{"className":12882},[137],[15,12884,12886,12906],{"className":12885},[141,142],[15,12887,12889,12903],{"className":12888},[146],[15,12890,12892],{"className":12891,"style":1981},[150],[15,12893,12894,12897],{"style":154},[15,12895],{"className":12896,"style":159},[158],[15,12898,12900],{"className":12899},[163,164,165,166],[15,12901,1459],{"className":12902},[129,133,166],[15,12904,175],{"className":12905},[174],[15,12907,12909],{"className":12908},[146],[15,12910,12912],{"className":12911,"style":182},[150],[15,12913],{},[15,12915],{"className":12916,"style":199},[93],[15,12918,62],{"className":12919},[203],[15,12921],{"className":12922,"style":199},[93],[15,12924,12926,12929,12982,12985,12988,13028,13031,13071],{"className":12925},[120],[15,12927],{"className":12928,"style":12775},[124],[15,12930,12932],{"className":12931},[217,1874],[15,12933,12935,12974],{"className":12934},[141,142],[15,12936,12938,12971],{"className":12937},[146],[15,12939,12941,12961],{"className":12940,"style":12143},[150],[15,12942,12943,12946],{"style":12790},[15,12944],{"className":12945,"style":1891},[158],[15,12947,12949],{"className":12948},[163,164,165,166],[15,12950,12952,12955,12958],{"className":12951},[129,166],[15,12953,1459],{"className":12954},[129,133,166],[15,12956,100],{"className":12957},[203,166],[15,12959,12555],{"className":12960,"style":12591},[129,133,166],[15,12962,12963,12966],{"style":1909},[15,12964],{"className":12965,"style":1891},[158],[15,12967,12968],{},[15,12969,1454],{"className":12970},[217,1918,1919],[15,12972,175],{"className":12973},[174],[15,12975,12977],{"className":12976},[146],[15,12978,12980],{"className":12979,"style":12828},[150],[15,12981],{},[15,12983,53],{"className":12984},[188],[15,12986,315],{"className":12987},[129],[15,12989,12991,12994],{"className":12990},[129],[15,12992,12002],{"className":12993,"style":12023},[129,133],[15,12995,12997],{"className":12996},[137],[15,12998,13000,13020],{"className":12999},[141,142],[15,13001,13003,13017],{"className":13002},[146],[15,13004,13006],{"className":13005,"style":1981},[150],[15,13007,13008,13011],{"style":12038},[15,13009],{"className":13010,"style":159},[158],[15,13012,13014],{"className":13013},[163,164,165,166],[15,13015,1459],{"className":13016},[129,133,166],[15,13018,175],{"className":13019},[174],[15,13021,13023],{"className":13022},[146],[15,13024,13026],{"className":13025,"style":182},[150],[15,13027],{},[15,13029,59],{"className":13030},[195],[15,13032,13034,13037],{"className":13033},[129],[15,13035,56],{"className":13036},[129,133],[15,13038,13040],{"className":13039},[137],[15,13041,13043,13063],{"className":13042},[141,142],[15,13044,13046,13060],{"className":13045},[146],[15,13047,13049],{"className":13048,"style":1981},[150],[15,13050,13051,13054],{"style":154},[15,13052],{"className":13053,"style":159},[158],[15,13055,13057],{"className":13056},[163,164,165,166],[15,13058,1459],{"className":13059},[129,133,166],[15,13061,175],{"className":13062},[174],[15,13064,13066],{"className":13065},[146],[15,13067,13069],{"className":13068,"style":182},[150],[15,13070],{},[15,13072,107],{"className":13073},[129],[11,13075,13076,13077,13130,13131,13159,13160,1313],{},"两边系数之和相同，记为 ",[15,13078,13080,13099],{"className":13079,"translate":19},[23],[15,13081,13083],{"className":13082},[27],[29,13084,13085],{"xmlns":31},[34,13086,13087,13096],{},[37,13088,13089,13092,13094],{},[43,13090,13091],{},"A",[50,13093,4852],{},[308,13095,541],{},[109,13097,13098],{"encoding":111},"A>0",[15,13100,13102,13121],{"className":13101,"ariaHidden":72},[116],[15,13103,13105,13109,13112,13115,13118],{"className":13104},[120],[15,13106],{"className":13107,"style":13108},[124],"height:0.7224em;vertical-align:-0.0391em;",[15,13110,13091],{"className":13111},[129,133],[15,13113],{"className":13114,"style":199},[93],[15,13116,4852],{"className":13117},[203],[15,13119],{"className":13120,"style":199},[93],[15,13122,13124,13127],{"className":13123},[120],[15,13125],{"className":13126,"style":4886},[124],[15,13128,541],{"className":13129},[129],"。除以 ",[15,13132,13134,13147],{"className":13133,"translate":19},[23],[15,13135,13137],{"className":13136},[27],[29,13138,13139],{"xmlns":31},[34,13140,13141,13145],{},[37,13142,13143],{},[43,13144,13091],{},[109,13146,13091],{"encoding":111},[15,13148,13150],{"className":13149,"ariaHidden":72},[116],[15,13151,13153,13156],{"className":13152},[120],[15,13154],{"className":13155,"style":610},[124],[15,13157,13091],{"className":13158},[129,133],"，得到同一点的两个凸组合表示，于是 ",[15,13161,13163,13204],{"className":13162,"translate":19},[23],[15,13164,13166],{"className":13165},[27],[29,13167,13168],{"xmlns":31},[34,13169,13170,13201],{},[37,13171,13172,13175,13177,13179,13181,13183,13186,13188,13190,13192,13194,13196,13198],{},[43,13173,13174],{"mathvariant":65},"conv",[50,13176,69],{},[50,13178,53],{"stretchy":52},[43,13180,1388],{},[50,13182,59],{"stretchy":52},[50,13184,13185],{},"∩",[43,13187,13174],{"mathvariant":65},[50,13189,69],{},[50,13191,53],{"stretchy":52},[43,13193,12555],{},[50,13195,59],{"stretchy":52},[50,13197,1418],{"mathvariant":65},[43,13199,13200],{"mathvariant":65},"∅",[109,13202,13203],{"encoding":111},"\\operatorname{conv}(P)\\cap\\operatorname{conv}(N)\\neq\\varnothing",[15,13205,13207,13237,13300],{"className":13206,"ariaHidden":72},[116],[15,13208,13210,13213,13219,13222,13225,13228,13231,13234],{"className":13209},[120],[15,13211],{"className":13212,"style":125},[124],[15,13214,13216],{"className":13215},[217],[15,13217,13174],{"className":13218,"style":2246},[129,221],[15,13220,53],{"className":13221},[188],[15,13223,1388],{"className":13224,"style":12443},[129,133],[15,13226,59],{"className":13227},[195],[15,13229],{"className":13230,"style":3277},[93],[15,13232,13185],{"className":13233},[2436],[15,13235],{"className":13236,"style":3277},[93],[15,13238,13240,13243,13249,13252,13255,13258,13261,13297],{"className":13239},[120],[15,13241],{"className":13242,"style":125},[124],[15,13244,13246],{"className":13245},[217],[15,13247,13174],{"className":13248,"style":2246},[129,221],[15,13250,53],{"className":13251},[188],[15,13253,12555],{"className":13254,"style":12591},[129,133],[15,13256,59],{"className":13257},[195],[15,13259],{"className":13260,"style":199},[93],[15,13262,13264,13291,13294],{"className":13263},[203],[15,13265,13267],{"className":13266},[203],[15,13268,13270],{"className":13269},[129,1670],[15,13271,13273],{"className":13272},[1674],[15,13274,13276,13279,13288],{"className":13275},[1678],[15,13277],{"className":13278,"style":1682},[124],[15,13280,13282],{"className":13281},[1686],[15,13283,13285],{"className":13284},[129],[15,13286,1693],{"className":13287},[203],[15,13289],{"className":13290},[1697],[15,13292],{"className":13293},[93,1701],[15,13295,62],{"className":13296},[203],[15,13298],{"className":13299,"style":199},[93],[15,13301,13303,13307],{"className":13302},[120],[15,13304],{"className":13305,"style":13306},[124],"height:0.6633em;vertical-align:-0.0817em;",[15,13308,13200],{"className":13309},[129,13310],"amsrm",[11,13312,13313,13314,13342,13343,13371,13372,13400,13401,13450,13451,13479,13480,13529,13530,13580,13581,1313],{},"现在取标注「",[15,13315,13317,13330],{"className":13316,"translate":19},[23],[15,13318,13320],{"className":13319},[27],[29,13321,13322],{"xmlns":31},[34,13323,13324,13328],{},[37,13325,13326],{},[43,13327,1388],{},[109,13329,1388],{"encoding":111},[15,13331,13333],{"className":13332,"ariaHidden":72},[116],[15,13334,13336,13339],{"className":13335},[120],[15,13337],{"className":13338,"style":610},[124],[15,13340,1388],{"className":13341,"style":12443},[129,133]," 为正、",[15,13344,13346,13359],{"className":13345,"translate":19},[23],[15,13347,13349],{"className":13348},[27],[29,13350,13351],{"xmlns":31},[34,13352,13353,13357],{},[37,13354,13355],{},[43,13356,12555],{},[109,13358,12555],{"encoding":111},[15,13360,13362],{"className":13361,"ariaHidden":72},[116],[15,13363,13365,13368],{"className":13364},[120],[15,13366],{"className":13367,"style":610},[124],[15,13369,12555],{"className":13370,"style":12591},[129,133]," 为负」（系数为零的点任意归入一侧，它们不影响下面的凸组合）。任何包含 ",[15,13373,13375,13388],{"className":13374,"translate":19},[23],[15,13376,13378],{"className":13377},[27],[29,13379,13380],{"xmlns":31},[34,13381,13382,13386],{},[37,13383,13384],{},[43,13385,1388],{},[109,13387,1388],{"encoding":111},[15,13389,13391],{"className":13390,"ariaHidden":72},[116],[15,13392,13394,13397],{"className":13393},[120],[15,13395],{"className":13396,"style":610},[124],[15,13398,1388],{"className":13399,"style":12443},[129,133]," 的半空间都包含 ",[15,13402,13404,13426],{"className":13403,"translate":19},[23],[15,13405,13407],{"className":13406},[27],[29,13408,13409],{"xmlns":31},[34,13410,13411,13423],{},[37,13412,13413,13415,13417,13419,13421],{},[43,13414,13174],{"mathvariant":65},[50,13416,69],{},[50,13418,53],{"stretchy":52},[43,13420,1388],{},[50,13422,59],{"stretchy":52},[109,13424,13425],{"encoding":111},"\\operatorname{conv}(P)",[15,13427,13429],{"className":13428,"ariaHidden":72},[116],[15,13430,13432,13435,13441,13444,13447],{"className":13431},[120],[15,13433],{"className":13434,"style":125},[124],[15,13436,13438],{"className":13437},[217],[15,13439,13174],{"className":13440,"style":2246},[129,221],[15,13442,53],{"className":13443},[188],[15,13445,1388],{"className":13446,"style":12443},[129,133],[15,13448,59],{"className":13449},[195],"；任何不包含 ",[15,13452,13454,13467],{"className":13453,"translate":19},[23],[15,13455,13457],{"className":13456},[27],[29,13458,13459],{"xmlns":31},[34,13460,13461,13465],{},[37,13462,13463],{},[43,13464,12555],{},[109,13466,12555],{"encoding":111},[15,13468,13470],{"className":13469,"ariaHidden":72},[116],[15,13471,13473,13476],{"className":13472},[120],[15,13474],{"className":13475,"style":610},[124],[15,13477,12555],{"className":13478,"style":12591},[129,133]," 的半空间，其补集是凸的，因而也不包含 ",[15,13481,13483,13505],{"className":13482,"translate":19},[23],[15,13484,13486],{"className":13485},[27],[29,13487,13488],{"xmlns":31},[34,13489,13490,13502],{},[37,13491,13492,13494,13496,13498,13500],{},[43,13493,13174],{"mathvariant":65},[50,13495,69],{},[50,13497,53],{"stretchy":52},[43,13499,12555],{},[50,13501,59],{"stretchy":52},[109,13503,13504],{"encoding":111},"\\operatorname{conv}(N)",[15,13506,13508],{"className":13507,"ariaHidden":72},[116],[15,13509,13511,13514,13520,13523,13526],{"className":13510},[120],[15,13512],{"className":13513,"style":125},[124],[15,13515,13517],{"className":13516},[217],[15,13518,13174],{"className":13519,"style":2246},[129,221],[15,13521,53],{"className":13522},[188],[15,13524,12555],{"className":13525,"style":12591},[129,133],[15,13527,59],{"className":13528},[195],"。两者相交，矛盾。所以 ",[15,13531,13533,13550],{"className":13532,"translate":19},[23],[15,13534,13536],{"className":13535},[27],[29,13537,13538],{"xmlns":31},[34,13539,13540,13548],{},[37,13541,13542,13544,13546],{},[43,13543,10372],{},[50,13545,2682],{},[308,13547,310],{},[109,13549,11626],{"encoding":111},[15,13551,13553,13571],{"className":13552,"ariaHidden":72},[116],[15,13554,13556,13559,13562,13565,13568],{"className":13555},[120],[15,13557],{"className":13558,"style":10410},[124],[15,13560,10372],{"className":13561},[129,133],[15,13563],{"className":13564,"style":3277},[93],[15,13566,2682],{"className":13567},[2436],[15,13569],{"className":13570,"style":3277},[93],[15,13572,13574,13577],{"className":13573},[120],[15,13575],{"className":13576,"style":4886},[124],[15,13578,310],{"className":13579},[129]," 个点无法被打散，",[15,13582,13584,13607],{"className":13583,"translate":19},[23],[15,13585,13587],{"className":13586},[27],[29,13588,13589],{"xmlns":31},[34,13590,13591,13605],{},[37,13592,13593,13595,13597,13599,13601,13603],{},[43,13594,6819],{"mathvariant":65},[50,13596,69],{},[50,13598,62],{},[43,13600,10372],{},[50,13602,2682],{},[308,13604,318],{},[109,13606,10379],{"encoding":111},[15,13608,13610,13631,13649],{"className":13609,"ariaHidden":72},[116],[15,13611,13613,13616,13622,13625,13628],{"className":13612},[120],[15,13614],{"className":13615,"style":522},[124],[15,13617,13619],{"className":13618},[217],[15,13620,6819],{"className":13621},[129,221],[15,13623],{"className":13624,"style":199},[93],[15,13626,62],{"className":13627},[203],[15,13629],{"className":13630,"style":199},[93],[15,13632,13634,13637,13640,13643,13646],{"className":13633},[120],[15,13635],{"className":13636,"style":10410},[124],[15,13638,10372],{"className":13639},[129,133],[15,13641],{"className":13642,"style":3277},[93],[15,13644,2682],{"className":13645},[2436],[15,13647],{"className":13648,"style":3277},[93],[15,13650,13652,13655],{"className":13651},[120],[15,13653],{"className":13654,"style":4886},[124],[15,13656,318],{"className":13657},[129],[11,13659,13660],{},"这个证明值得记住：它把「线性分类器的容量」归结为「仿射无关点的最大数目」，而后者正是线性代数意义上的维数加一。VC 维里的「维」字，在这里有了具体的所指。",[11,13662,13663],{},[698,13664,13665,13666,7434],{},"例 4（轴对齐矩形，",[15,13667,13669,13691],{"className":13668,"translate":19},[23],[15,13670,13672],{"className":13671},[27],[29,13673,13674],{"xmlns":31},[34,13675,13676,13688],{},[37,13677,13678,13680,13682,13684,13686],{},[43,13679,6819],{"mathvariant":65},[50,13681,69],{},[50,13683,62],{},[308,13685,310],{},[43,13687,10372],{},[109,13689,13690],{"encoding":111},"\\operatorname{VCdim}=2d",[15,13692,13694,13715],{"className":13693,"ariaHidden":72},[116],[15,13695,13697,13700,13706,13709,13712],{"className":13696},[120],[15,13698],{"className":13699,"style":522},[124],[15,13701,13703],{"className":13702},[217],[15,13704,6819],{"className":13705},[129,221],[15,13707],{"className":13708,"style":199},[93],[15,13710,62],{"className":13711},[203],[15,13713],{"className":13714,"style":199},[93],[15,13716,13718,13721,13724],{"className":13717},[120],[15,13719],{"className":13720,"style":522},[124],[15,13722,310],{"className":13723},[129],[15,13725,10372],{"className":13726},[129,133],[11,13728,13729,13757,13758,13816,13817,13845],{},[15,13730,13732,13745],{"className":13731,"translate":19},[23],[15,13733,13735],{"className":13734},[27],[29,13736,13737],{"xmlns":31},[34,13738,13739,13743],{},[37,13740,13741],{},[43,13742,819],{"mathvariant":596},[109,13744,822],{"encoding":111},[15,13746,13748],{"className":13747,"ariaHidden":72},[116],[15,13749,13751,13754],{"className":13750},[120],[15,13752],{"className":13753,"style":610},[124],[15,13755,819],{"className":13756,"style":835},[129,614]," 是 ",[15,13759,13761,13778],{"className":13760,"translate":19},[23],[15,13762,13764],{"className":13763},[27],[29,13765,13766],{"xmlns":31},[34,13767,13768,13776],{},[37,13769,13770],{},[305,13771,13772,13774],{},[43,13773,104],{"mathvariant":103},[43,13775,10372],{},[109,13777,11566],{"encoding":111},[15,13779,13781],{"className":13780,"ariaHidden":72},[116],[15,13782,13784,13787],{"className":13783},[120],[15,13785],{"className":13786,"style":10486},[124],[15,13788,13790,13793],{"className":13789},[129],[15,13791,104],{"className":13792},[129,285],[15,13794,13796],{"className":13795},[137],[15,13797,13799],{"className":13798},[141],[15,13800,13802],{"className":13801},[146],[15,13803,13805],{"className":13804,"style":10486},[150],[15,13806,13807,13810],{"style":383},[15,13808],{"className":13809,"style":159},[158],[15,13811,13813],{"className":13812},[163,164,165,166],[15,13814,10372],{"className":13815},[129,133,166]," 中所有轴对齐矩形（即 ",[15,13818,13820,13833],{"className":13819,"translate":19},[23],[15,13821,13823],{"className":13822},[27],[29,13824,13825],{"xmlns":31},[34,13826,13827,13831],{},[37,13828,13829],{},[43,13830,10372],{},[109,13832,10372],{"encoding":111},[15,13834,13836],{"className":13835,"ariaHidden":72},[116],[15,13837,13839,13842],{"className":13838},[120],[15,13840],{"className":13841,"style":522},[124],[15,13843,10372],{"className":13844},[129,133]," 个区间的乘积）的指示函数。",[11,13847,13848,13849,13905,13906,13934,13935,13965,13966,14038,14039,14067,14068,14137,14138,1313],{},"上界：任取 ",[15,13850,13852,13872],{"className":13851,"translate":19},[23],[15,13853,13855],{"className":13854},[27],[29,13856,13857],{"xmlns":31},[34,13858,13859,13869],{},[37,13860,13861,13863,13865,13867],{},[308,13862,310],{},[43,13864,10372],{},[50,13866,2682],{},[308,13868,318],{},[109,13870,13871],{"encoding":111},"2d+1",[15,13873,13875,13896],{"className":13874,"ariaHidden":72},[116],[15,13876,13878,13881,13884,13887,13890,13893],{"className":13877},[120],[15,13879],{"className":13880,"style":10410},[124],[15,13882,310],{"className":13883},[129],[15,13885,10372],{"className":13886},[129,133],[15,13888],{"className":13889,"style":3277},[93],[15,13891,2682],{"className":13892},[2436],[15,13894],{"className":13895,"style":3277},[93],[15,13897,13899,13902],{"className":13898},[120],[15,13900],{"className":13901,"style":4886},[124],[15,13903,318],{"className":13904},[129]," 个点。若存在一点 ",[15,13907,13909,13922],{"className":13908,"translate":19},[23],[15,13910,13912],{"className":13911},[27],[29,13913,13914],{"xmlns":31},[34,13915,13916,13920],{},[37,13917,13918],{},[43,13919,11],{},[109,13921,11],{"encoding":111},[15,13923,13925],{"className":13924,"ariaHidden":72},[116],[15,13926,13928,13931],{"className":13927},[120],[15,13929],{"className":13930,"style":5399},[124],[15,13932,11],{"className":13933},[129,133],"，使它在每个坐标上都不是严格极值——即对每个坐标 ",[15,13936,13938,13952],{"className":13937,"translate":19},[23],[15,13939,13941],{"className":13940},[27],[29,13942,13943],{"xmlns":31},[34,13944,13945,13950],{},[37,13946,13947],{},[43,13948,13949],{},"j",[109,13951,13949],{"encoding":111},[15,13953,13955],{"className":13954,"ariaHidden":72},[116],[15,13956,13958,13961],{"className":13957},[120],[15,13959],{"className":13960,"style":11187},[124],[15,13962,13949],{"className":13963,"style":13964},[129,133],"margin-right:0.0572em;","，都有别的点在 ",[15,13967,13969,13987],{"className":13968,"translate":19},[23],[15,13970,13972],{"className":13971},[27],[29,13973,13974],{"xmlns":31},[34,13975,13976,13984],{},[37,13977,13978],{},[40,13979,13980,13982],{},[43,13981,11],{},[43,13983,13949],{},[109,13985,13986],{"encoding":111},"p_j",[15,13988,13990],{"className":13989,"ariaHidden":72},[116],[15,13991,13993,13997],{"className":13992},[120],[15,13994],{"className":13995,"style":13996},[124],"height:0.7167em;vertical-align:-0.2861em;",[15,13998,14000,14003],{"className":13999},[129],[15,14001,11],{"className":14002},[129,133],[15,14004,14006],{"className":14005},[137],[15,14007,14009,14029],{"className":14008},[141,142],[15,14010,14012,14026],{"className":14011},[146],[15,14013,14015],{"className":14014,"style":1981},[150],[15,14016,14017,14020],{"style":154},[15,14018],{"className":14019,"style":159},[158],[15,14021,14023],{"className":14022},[163,164,165,166],[15,14024,13949],{"className":14025,"style":13964},[129,133,166],[15,14027,175],{"className":14028},[174],[15,14030,14032],{"className":14031},[146],[15,14033,14036],{"className":14034,"style":14035},[150],"height:0.2861em;",[15,14037],{}," 的左边和右边——那么标注「",[15,14040,14042,14055],{"className":14041,"translate":19},[23],[15,14043,14045],{"className":14044},[27],[29,14046,14047],{"xmlns":31},[34,14048,14049,14053],{},[37,14050,14051],{},[43,14052,11],{},[109,14054,11],{"encoding":111},[15,14056,14058],{"className":14057,"ariaHidden":72},[116],[15,14059,14061,14064],{"className":14060},[120],[15,14062],{"className":14063,"style":5399},[124],[15,14065,11],{"className":14066},[129,133]," 为负、其余为正」无法实现：任何包含其余所有点的矩形，在每个坐标上的区间都必须覆盖 ",[15,14069,14071,14088],{"className":14070,"translate":19},[23],[15,14072,14074],{"className":14073},[27],[29,14075,14076],{"xmlns":31},[34,14077,14078,14086],{},[37,14079,14080],{},[40,14081,14082,14084],{},[43,14083,11],{},[43,14085,13949],{},[109,14087,13986],{"encoding":111},[15,14089,14091],{"className":14090,"ariaHidden":72},[116],[15,14092,14094,14097],{"className":14093},[120],[15,14095],{"className":14096,"style":13996},[124],[15,14098,14100,14103],{"className":14099},[129],[15,14101,11],{"className":14102},[129,133],[15,14104,14106],{"className":14105},[137],[15,14107,14109,14129],{"className":14108},[141,142],[15,14110,14112,14126],{"className":14111},[146],[15,14113,14115],{"className":14114,"style":1981},[150],[15,14116,14117,14120],{"style":154},[15,14118],{"className":14119,"style":159},[158],[15,14121,14123],{"className":14122},[163,164,165,166],[15,14124,13949],{"className":14125,"style":13964},[129,133,166],[15,14127,175],{"className":14128},[174],[15,14130,14132],{"className":14131},[146],[15,14133,14135],{"className":14134,"style":14035},[150],[15,14136],{}," 两侧的值，因而必然包含 ",[15,14139,14141,14154],{"className":14140,"translate":19},[23],[15,14142,14144],{"className":14143},[27],[29,14145,14146],{"xmlns":31},[34,14147,14148,14152],{},[37,14149,14150],{},[43,14151,11],{},[109,14153,11],{"encoding":111},[15,14155,14157],{"className":14156,"ariaHidden":72},[116],[15,14158,14160,14163],{"className":14159},[120],[15,14161],{"className":14162,"style":5399},[124],[15,14164,11],{"className":14165},[129,133],[11,14167,14168,14169,14197,14198,14226,14227,14255,14256,14290,14291,14346,14347,14380],{},"这样的 ",[15,14170,14172,14185],{"className":14171,"translate":19},[23],[15,14173,14175],{"className":14174},[27],[29,14176,14177],{"xmlns":31},[34,14178,14179,14183],{},[37,14180,14181],{},[43,14182,11],{},[109,14184,11],{"encoding":111},[15,14186,14188],{"className":14187,"ariaHidden":72},[116],[15,14189,14191,14194],{"className":14190},[120],[15,14192],{"className":14193,"style":5399},[124],[15,14195,11],{"className":14196},[129,133]," 一定存在。否则每个点都在某个坐标上严格小于所有其他点、或严格大于所有其他点。把「坐标 ",[15,14199,14201,14214],{"className":14200,"translate":19},[23],[15,14202,14204],{"className":14203},[27],[29,14205,14206],{"xmlns":31},[34,14207,14208,14212],{},[37,14209,14210],{},[43,14211,13949],{},[109,14213,13949],{"encoding":111},[15,14215,14217],{"className":14216,"ariaHidden":72},[116],[15,14218,14220,14223],{"className":14219},[120],[15,14221],{"className":14222,"style":11187},[124],[15,14224,13949],{"className":14225,"style":13964},[129,133]," 的最小值」和「坐标 ",[15,14228,14230,14243],{"className":14229,"translate":19},[23],[15,14231,14233],{"className":14232},[27],[29,14234,14235],{"xmlns":31},[34,14236,14237,14241],{},[37,14238,14239],{},[43,14240,13949],{},[109,14242,13949],{"encoding":111},[15,14244,14246],{"className":14245,"ariaHidden":72},[116],[15,14247,14249,14252],{"className":14248},[120],[15,14250],{"className":14251,"style":11187},[124],[15,14253,13949],{"className":14254,"style":13964},[129,133]," 的最大值」看作 ",[15,14257,14259,14275],{"className":14258,"translate":19},[23],[15,14260,14262],{"className":14261},[27],[29,14263,14264],{"xmlns":31},[34,14265,14266,14272],{},[37,14267,14268,14270],{},[308,14269,310],{},[43,14271,10372],{},[109,14273,14274],{"encoding":111},"2d",[15,14276,14278],{"className":14277,"ariaHidden":72},[116],[15,14279,14281,14284,14287],{"className":14280},[120],[15,14282],{"className":14283,"style":522},[124],[15,14285,310],{"className":14286},[129],[15,14288,10372],{"className":14289},[129,133]," 个位置，每个点占据其中一个；若两个点占据同一个位置，立刻矛盾（不可能同时严格小于对方）。",[15,14292,14294,14313],{"className":14293,"translate":19},[23],[15,14295,14297],{"className":14296},[27],[29,14298,14299],{"xmlns":31},[34,14300,14301,14311],{},[37,14302,14303,14305,14307,14309],{},[308,14304,310],{},[43,14306,10372],{},[50,14308,2682],{},[308,14310,318],{},[109,14312,13871],{"encoding":111},[15,14314,14316,14337],{"className":14315,"ariaHidden":72},[116],[15,14317,14319,14322,14325,14328,14331,14334],{"className":14318},[120],[15,14320],{"className":14321,"style":10410},[124],[15,14323,310],{"className":14324},[129],[15,14326,10372],{"className":14327},[129,133],[15,14329],{"className":14330,"style":3277},[93],[15,14332,2682],{"className":14333},[2436],[15,14335],{"className":14336,"style":3277},[93],[15,14338,14340,14343],{"className":14339},[120],[15,14341],{"className":14342,"style":4886},[124],[15,14344,318],{"className":14345},[129]," 个点放进 ",[15,14348,14350,14365],{"className":14349,"translate":19},[23],[15,14351,14353],{"className":14352},[27],[29,14354,14355],{"xmlns":31},[34,14356,14357,14363],{},[37,14358,14359,14361],{},[308,14360,310],{},[43,14362,10372],{},[109,14364,14274],{"encoding":111},[15,14366,14368],{"className":14367,"ariaHidden":72},[116],[15,14369,14371,14374,14377],{"className":14370},[120],[15,14372],{"className":14373,"style":522},[124],[15,14375,310],{"className":14376},[129],[15,14378,10372],{"className":14379},[129,133]," 个位置，必有一对冲突。所以上界成立。",[11,14382,14383,14384,14417,14418,14469,14470,14631,14632,1313],{},"下界是初等的但需要一点构造：确实存在 ",[15,14385,14387,14402],{"className":14386,"translate":19},[23],[15,14388,14390],{"className":14389},[27],[29,14391,14392],{"xmlns":31},[34,14393,14394,14400],{},[37,14395,14396,14398],{},[308,14397,310],{},[43,14399,10372],{},[109,14401,14274],{"encoding":111},[15,14403,14405],{"className":14404,"ariaHidden":72},[116],[15,14406,14408,14411,14414],{"className":14407},[120],[15,14409],{"className":14410,"style":522},[124],[15,14412,310],{"className":14413},[129],[15,14415,10372],{"className":14416},[129,133]," 个点可以被矩形打散（",[15,14419,14421,14439],{"className":14420,"translate":19},[23],[15,14422,14424],{"className":14423},[27],[29,14425,14426],{"xmlns":31},[34,14427,14428,14436],{},[37,14429,14430,14432,14434],{},[43,14431,10372],{},[50,14433,62],{},[308,14435,310],{},[109,14437,14438],{"encoding":111},"d=2",[15,14440,14442,14460],{"className":14441,"ariaHidden":72},[116],[15,14443,14445,14448,14451,14454,14457],{"className":14444},[120],[15,14446],{"className":14447,"style":522},[124],[15,14449,10372],{"className":14450},[129,133],[15,14452],{"className":14453,"style":199},[93],[15,14455,62],{"className":14456},[203],[15,14458],{"className":14459,"style":199},[93],[15,14461,14463,14466],{"className":14462},[120],[15,14464],{"className":14465,"style":4886},[124],[15,14467,310],{"className":14468},[129]," 时，",[15,14471,14473,14532],{"className":14472,"translate":19},[23],[15,14474,14476],{"className":14475},[27],[29,14477,14478],{"xmlns":31},[34,14479,14480,14529],{},[37,14481,14482,14484,14486,14488,14490,14492,14494,14496,14498,14500,14503,14505,14507,14509,14511,14513,14515,14517,14519,14521,14523,14525,14527],{},[50,14483,53],{"stretchy":52},[308,14485,318],{},[50,14487,91],{"separator":72},[308,14489,310],{},[50,14491,59],{"stretchy":52},[50,14493,91],{"separator":72},[50,14495,53],{"stretchy":52},[308,14497,310],{},[50,14499,91],{"separator":72},[308,14501,14502],{},"4",[50,14504,59],{"stretchy":52},[50,14506,91],{"separator":72},[50,14508,53],{"stretchy":52},[308,14510,9875],{},[50,14512,91],{"separator":72},[308,14514,318],{},[50,14516,59],{"stretchy":52},[50,14518,91],{"separator":72},[50,14520,53],{"stretchy":52},[308,14522,14502],{},[50,14524,91],{"separator":72},[308,14526,9875],{},[50,14528,59],{"stretchy":52},[109,14530,14531],{"encoding":111},"(1,2),(2,4),(3,1),(4,3)",[15,14533,14535],{"className":14534,"ariaHidden":72},[116],[15,14536,14538,14541,14544,14547,14550,14553,14556,14559,14562,14565,14568,14571,14574,14577,14580,14583,14586,14589,14592,14595,14598,14601,14604,14607,14610,14613,14616,14619,14622,14625,14628],{"className":14537},[120],[15,14539],{"className":14540,"style":125},[124],[15,14542,53],{"className":14543},[188],[15,14545,318],{"className":14546},[129],[15,14548,91],{"className":14549},[254],[15,14551],{"className":14552,"style":262},[93],[15,14554,310],{"className":14555},[129],[15,14557,59],{"className":14558},[195],[15,14560,91],{"className":14561},[254],[15,14563],{"className":14564,"style":262},[93],[15,14566,53],{"className":14567},[188],[15,14569,310],{"className":14570},[129],[15,14572,91],{"className":14573},[254],[15,14575],{"className":14576,"style":262},[93],[15,14578,14502],{"className":14579},[129],[15,14581,59],{"className":14582},[195],[15,14584,91],{"className":14585},[254],[15,14587],{"className":14588,"style":262},[93],[15,14590,53],{"className":14591},[188],[15,14593,9875],{"className":14594},[129],[15,14596,91],{"className":14597},[254],[15,14599],{"className":14600,"style":262},[93],[15,14602,318],{"className":14603},[129],[15,14605,59],{"className":14606},[195],[15,14608,91],{"className":14609},[254],[15,14611],{"className":14612,"style":262},[93],[15,14614,53],{"className":14615},[188],[15,14617,14502],{"className":14618},[129],[15,14620,91],{"className":14621},[254],[15,14623],{"className":14624,"style":262},[93],[15,14626,9875],{"className":14627},[129],[15,14629,59],{"className":14630},[195]," 这四个点就可以；注意不能取正方形的四个角）。于是 ",[15,14633,14635,14656],{"className":14634,"translate":19},[23],[15,14636,14638],{"className":14637},[27],[29,14639,14640],{"xmlns":31},[34,14641,14642,14654],{},[37,14643,14644,14646,14648,14650,14652],{},[43,14645,6819],{"mathvariant":65},[50,14647,69],{},[50,14649,62],{},[308,14651,310],{},[43,14653,10372],{},[109,14655,13690],{"encoding":111},[15,14657,14659,14680],{"className":14658,"ariaHidden":72},[116],[15,14660,14662,14665,14671,14674,14677],{"className":14661},[120],[15,14663],{"className":14664,"style":522},[124],[15,14666,14668],{"className":14667},[217],[15,14669,6819],{"className":14670},[129,221],[15,14672],{"className":14673,"style":199},[93],[15,14675,62],{"className":14676},[203],[15,14678],{"className":14679,"style":199},[93],[15,14681,14683,14686,14689],{"className":14682},[120],[15,14684],{"className":14685,"style":522},[124],[15,14687,310],{"className":14688},[129],[15,14690,10372],{"className":14691},[129,133],[11,14693,14694],{},[698,14695,14696,14697,7434],{},"例 5（有限类，",[15,14698,14700,14735],{"className":14699,"translate":19},[23],[15,14701,14703],{"className":14702},[27],[29,14704,14705],{"xmlns":31},[34,14706,14707,14732],{},[37,14708,14709,14711,14713,14715,14726,14728,14730],{},[43,14710,6819],{"mathvariant":65},[50,14712,69],{},[50,14714,2717],{},[40,14716,14717,14724],{},[37,14718,14719,14722],{},[43,14720,14721],{},"log",[50,14723,69],{},[308,14725,310],{},[43,14727,2945],{"mathvariant":65},[43,14729,819],{"mathvariant":596},[43,14731,2945],{"mathvariant":65},[109,14733,14734],{"encoding":111},"\\operatorname{VCdim}\\le\\log_2|\\mathcal H|",[15,14736,14738,14759],{"className":14737,"ariaHidden":72},[116],[15,14739,14741,14744,14750,14753,14756],{"className":14740},[120],[15,14742],{"className":14743,"style":6962},[124],[15,14745,14747],{"className":14746},[217],[15,14748,6819],{"className":14749},[129,221],[15,14751],{"className":14752,"style":199},[93],[15,14754,2717],{"className":14755},[203],[15,14757],{"className":14758,"style":199},[93],[15,14760,14762,14765,14811,14814,14817,14820],{"className":14761},[120],[15,14763],{"className":14764,"style":125},[124],[15,14766,14768,14774],{"className":14767},[217],[15,14769,14771,14772],{"className":14770},[217],"lo",[15,14773,2247],{"style":2246},[15,14775,14777],{"className":14776},[137],[15,14778,14780,14802],{"className":14779},[141,142],[15,14781,14783,14799],{"className":14782},[146],[15,14784,14787],{"className":14785,"style":14786},[150],"height:0.207em;",[15,14788,14790,14793],{"style":14789},"top:-2.4559em;margin-right:0.05em;",[15,14791],{"className":14792,"style":159},[158],[15,14794,14796],{"className":14795},[163,164,165,166],[15,14797,310],{"className":14798},[129,166],[15,14800,175],{"className":14801},[174],[15,14803,14805],{"className":14804},[146],[15,14806,14809],{"className":14807,"style":14808},[150],"height:0.2441em;",[15,14810],{},[15,14812],{"className":14813,"style":262},[93],[15,14815,2945],{"className":14816},[129],[15,14818,819],{"className":14819,"style":835},[129,614],[15,14821,2945],{"className":14822},[129],[11,14824,6564,14825,14853,14854,14882,14883,15068,15069,1313],{},[15,14826,14828,14841],{"className":14827,"translate":19},[23],[15,14829,14831],{"className":14830},[27],[29,14832,14833],{"xmlns":31},[34,14834,14835,14839],{},[37,14836,14837],{},[43,14838,819],{"mathvariant":596},[109,14840,822],{"encoding":111},[15,14842,14844],{"className":14843,"ariaHidden":72},[116],[15,14845,14847,14850],{"className":14846},[120],[15,14848],{"className":14849,"style":610},[124],[15,14851,819],{"className":14852,"style":835},[129,614]," 有限且能打散 ",[15,14855,14857,14870],{"className":14856,"translate":19},[23],[15,14858,14860],{"className":14859},[27],[29,14861,14862],{"xmlns":31},[34,14863,14864,14868],{},[37,14865,14866],{},[43,14867,5619],{},[109,14869,5619],{"encoding":111},[15,14871,14873],{"className":14872,"ariaHidden":72},[116],[15,14874,14876,14879],{"className":14875},[120],[15,14877],{"className":14878,"style":610},[124],[15,14880,5619],{"className":14881,"style":5665},[129,133],"，则 ",[15,14884,14886,14932],{"className":14885,"translate":19},[23],[15,14887,14889],{"className":14888},[27],[29,14890,14891],{"xmlns":31},[34,14892,14893,14929],{},[37,14894,14895,14907,14909,14911,14913,14919,14921,14923,14925,14927],{},[305,14896,14897,14899],{},[308,14898,310],{},[37,14900,14901,14903,14905],{},[43,14902,2945],{"mathvariant":65},[43,14904,5619],{},[43,14906,2945],{"mathvariant":65},[50,14908,62],{},[43,14910,2945],{"mathvariant":65},[43,14912,819],{"mathvariant":596},[40,14914,14915,14917],{},[43,14916,2945],{"mathvariant":65},[43,14918,5619],{},[43,14920,2945],{"mathvariant":65},[50,14922,2717],{},[43,14924,2945],{"mathvariant":65},[43,14926,819],{"mathvariant":596},[43,14928,2945],{"mathvariant":65},[109,14930,14931],{"encoding":111},"2^{|C|}=|\\mathcal H|_C|\\le|\\mathcal H|",[15,14933,14935,14989,15053],{"className":14934,"ariaHidden":72},[116],[15,14936,14938,14942,14980,14983,14986],{"className":14937},[120],[15,14939],{"className":14940,"style":14941},[124],"height:0.888em;",[15,14943,14945,14948],{"className":14944},[129],[15,14946,310],{"className":14947},[129],[15,14949,14951],{"className":14950},[137],[15,14952,14954],{"className":14953},[141],[15,14955,14957],{"className":14956},[146],[15,14958,14960],{"className":14959,"style":14941},[150],[15,14961,14962,14965],{"style":383},[15,14963],{"className":14964,"style":159},[158],[15,14966,14968],{"className":14967},[163,164,165,166],[15,14969,14971,14974,14977],{"className":14970},[129,166],[15,14972,2945],{"className":14973},[129,166],[15,14975,5619],{"className":14976,"style":5665},[129,133,166],[15,14978,2945],{"className":14979},[129,166],[15,14981],{"className":14982,"style":199},[93],[15,14984,62],{"className":14985},[203],[15,14987],{"className":14988,"style":199},[93],[15,14990,14992,14995,14998,15001,15041,15044,15047,15050],{"className":14991},[120],[15,14993],{"className":14994,"style":125},[124],[15,14996,2945],{"className":14997},[129],[15,14999,819],{"className":15000,"style":835},[129,614],[15,15002,15004,15007],{"className":15003},[129],[15,15005,2945],{"className":15006},[129],[15,15008,15010],{"className":15009},[137],[15,15011,15013,15033],{"className":15012},[141,142],[15,15014,15016,15030],{"className":15015},[146],[15,15017,15019],{"className":15018,"style":1525},[150],[15,15020,15021,15024],{"style":154},[15,15022],{"className":15023,"style":159},[158],[15,15025,15027],{"className":15026},[163,164,165,166],[15,15028,5619],{"className":15029,"style":5665},[129,133,166],[15,15031,175],{"className":15032},[174],[15,15034,15036],{"className":15035},[146],[15,15037,15039],{"className":15038,"style":182},[150],[15,15040],{},[15,15042,2945],{"className":15043},[129],[15,15045],{"className":15046,"style":199},[93],[15,15048,2717],{"className":15049},[203],[15,15051],{"className":15052,"style":199},[93],[15,15054,15056,15059,15062,15065],{"className":15055},[120],[15,15057],{"className":15058,"style":125},[124],[15,15060,2945],{"className":15061},[129],[15,15063,819],{"className":15064,"style":835},[129,614],[15,15066,2945],{"className":15067},[129],"，所以 ",[15,15070,15072,15108],{"className":15071,"translate":19},[23],[15,15073,15075],{"className":15074},[27],[29,15076,15077],{"xmlns":31},[34,15078,15079,15105],{},[37,15080,15081,15083,15085,15087,15089,15099,15101,15103],{},[43,15082,2945],{"mathvariant":65},[43,15084,5619],{},[43,15086,2945],{"mathvariant":65},[50,15088,2717],{},[40,15090,15091,15097],{},[37,15092,15093,15095],{},[43,15094,14721],{},[50,15096,69],{},[308,15098,310],{},[43,15100,2945],{"mathvariant":65},[43,15102,819],{"mathvariant":596},[43,15104,2945],{"mathvariant":65},[109,15106,15107],{"encoding":111},"|C|\\le\\log_2|\\mathcal H|",[15,15109,15111,15135],{"className":15110,"ariaHidden":72},[116],[15,15112,15114,15117,15120,15123,15126,15129,15132],{"className":15113},[120],[15,15115],{"className":15116,"style":125},[124],[15,15118,2945],{"className":15119},[129],[15,15121,5619],{"className":15122,"style":5665},[129,133],[15,15124,2945],{"className":15125},[129],[15,15127],{"className":15128,"style":199},[93],[15,15130,2717],{"className":15131},[203],[15,15133],{"className":15134,"style":199},[93],[15,15136,15138,15141,15183,15186,15189,15192],{"className":15137},[120],[15,15139],{"className":15140,"style":125},[124],[15,15142,15144,15149],{"className":15143},[217],[15,15145,14771,15147],{"className":15146},[217],[15,15148,2247],{"style":2246},[15,15150,15152],{"className":15151},[137],[15,15153,15155,15175],{"className":15154},[141,142],[15,15156,15158,15172],{"className":15157},[146],[15,15159,15161],{"className":15160,"style":14786},[150],[15,15162,15163,15166],{"style":14789},[15,15164],{"className":15165,"style":159},[158],[15,15167,15169],{"className":15168},[163,164,165,166],[15,15170,310],{"className":15171},[129,166],[15,15173,175],{"className":15174},[174],[15,15176,15178],{"className":15177},[146],[15,15179,15181],{"className":15180,"style":14808},[150],[15,15182],{},[15,15184],{"className":15185,"style":262},[93],[15,15187,2945],{"className":15188},[129],[15,15190,819],{"className":15191,"style":835},[129,614],[15,15193,2945],{"className":15194},[129],[11,15196,15197],{},[698,15198,15199,15200,7434],{},"例 6（正弦类，",[15,15201,15203,15223],{"className":15202,"translate":19},[23],[15,15204,15206],{"className":15205},[27],[29,15207,15208],{"xmlns":31},[34,15209,15210,15220],{},[37,15211,15212,15214,15216,15218],{},[43,15213,6819],{"mathvariant":65},[50,15215,69],{},[50,15217,62],{},[43,15219,4909],{"mathvariant":65},[109,15221,15222],{"encoding":111},"\\operatorname{VCdim}=\\infty",[15,15224,15226,15247],{"className":15225,"ariaHidden":72},[116],[15,15227,15229,15232,15238,15241,15244],{"className":15228},[120],[15,15230],{"className":15231,"style":522},[124],[15,15233,15235],{"className":15234},[217],[15,15236,6819],{"className":15237},[129,221],[15,15239],{"className":15240,"style":199},[93],[15,15242,62],{"className":15243},[203],[15,15245],{"className":15246,"style":199},[93],[15,15248,15250,15253],{"className":15249},[120],[15,15251],{"className":15252,"style":4383},[124],[15,15254,4909],{"className":15255},[129],[11,15257,7437,15258,7606,15424,15474,15475,15503,15504,15637,15638,15734],{},[15,15259,15261,15313],{"className":15260,"translate":19},[23],[15,15262,15264],{"className":15263},[27],[29,15265,15266],{"xmlns":31},[34,15267,15268,15310],{},[37,15269,15270,15272,15274,15276,15278,15280,15282,15284,15286,15288,15290,15292,15294,15296,15298,15300,15302,15304,15306,15308],{},[43,15271,819],{"mathvariant":596},[50,15273,62],{},[50,15275,636],{"stretchy":52},[43,15277,56],{},[50,15279,7460],{},[43,15281,66],{"mathvariant":65},[50,15283,69],{},[50,15285,53],{"stretchy":52},[43,15287,76],{},[50,15289,69],{},[50,15291,53],{"stretchy":52},[43,15293,48],{},[43,15295,56],{},[50,15297,59],{"stretchy":52},[50,15299,59],{"stretchy":52},[50,15301,718],{},[43,15303,48],{},[50,15305,4852],{},[308,15307,541],{},[50,15309,645],{"stretchy":52},[109,15311,15312],{"encoding":111},"\\mathcal H=\\{x\\mapsto\\operatorname{sign}(\\sin(\\theta x)):\\theta>0\\}",[15,15314,15316,15334,15355,15394,15412],{"className":15315,"ariaHidden":72},[116],[15,15317,15319,15322,15325,15328,15331],{"className":15318},[120],[15,15320],{"className":15321,"style":610},[124],[15,15323,819],{"className":15324,"style":835},[129,614],[15,15326],{"className":15327,"style":199},[93],[15,15329,62],{"className":15330},[203],[15,15332],{"className":15333,"style":199},[93],[15,15335,15337,15340,15343,15346,15349,15352],{"className":15336},[120],[15,15338],{"className":15339,"style":125},[124],[15,15341,636],{"className":15342},[188],[15,15344,56],{"className":15345},[129,133],[15,15347],{"className":15348,"style":199},[93],[15,15350,7460],{"className":15351},[203],[15,15353],{"className":15354,"style":199},[93],[15,15356,15358,15361,15367,15370,15373,15376,15379,15382,15385,15388,15391],{"className":15357},[120],[15,15359],{"className":15360,"style":125},[124],[15,15362,15364],{"className":15363},[217],[15,15365,66],{"className":15366},[129,221],[15,15368,53],{"className":15369},[188],[15,15371,76],{"className":15372},[217],[15,15374,53],{"className":15375},[188],[15,15377,48],{"className":15378,"style":170},[129,133],[15,15380,56],{"className":15381},[129,133],[15,15383,6149],{"className":15384},[195],[15,15386],{"className":15387,"style":199},[93],[15,15389,718],{"className":15390},[203],[15,15392],{"className":15393,"style":199},[93],[15,15395,15397,15400,15403,15406,15409],{"className":15396},[120],[15,15398],{"className":15399,"style":2405},[124],[15,15401,48],{"className":15402,"style":170},[129,133],[15,15404],{"className":15405,"style":199},[93],[15,15407,4852],{"className":15408},[203],[15,15410],{"className":15411,"style":199},[93],[15,15413,15415,15418,15421],{"className":15414},[120],[15,15416],{"className":15417,"style":125},[124],[15,15419,541],{"className":15420},[129],[15,15422,645],{"className":15423},[195],[15,15425,15427,15444],{"className":15426,"translate":19},[23],[15,15428,15430],{"className":15429},[27],[29,15431,15432],{"xmlns":31},[34,15433,15434,15442],{},[37,15435,15436,15438,15440],{},[43,15437,597],{"mathvariant":596},[50,15439,62],{},[43,15441,104],{"mathvariant":103},[109,15443,7627],{"encoding":111},[15,15445,15447,15465],{"className":15446,"ariaHidden":72},[116],[15,15448,15450,15453,15456,15459,15462],{"className":15449},[120],[15,15451],{"className":15452,"style":610},[124],[15,15454,597],{"className":15455,"style":615},[129,614],[15,15457],{"className":15458,"style":199},[93],[15,15460,62],{"className":15461},[203],[15,15463],{"className":15464,"style":199},[93],[15,15466,15468,15471],{"className":15467},[120],[15,15469],{"className":15470,"style":281},[124],[15,15472,104],{"className":15473},[129,285],"。对任意 ",[15,15476,15478,15491],{"className":15477,"translate":19},[23],[15,15479,15481],{"className":15480},[27],[29,15482,15483],{"xmlns":31},[34,15484,15485,15489],{},[37,15486,15487],{},[43,15488,348],{},[109,15490,348],{"encoding":111},[15,15492,15494],{"className":15493,"ariaHidden":72},[116],[15,15495,15497,15500],{"className":15496},[120],[15,15498],{"className":15499,"style":4383},[124],[15,15501,348],{"className":15502},[129,133],"，取 ",[15,15505,15507,15537],{"className":15506,"translate":19},[23],[15,15508,15510],{"className":15509},[27],[29,15511,15512],{"xmlns":31},[34,15513,15514,15534],{},[37,15515,15516,15522,15524],{},[40,15517,15518,15520],{},[43,15519,56],{},[43,15521,1459],{},[50,15523,62],{},[305,15525,15526,15528],{},[308,15527,310],{},[37,15529,15530,15532],{},[50,15531,315],{},[43,15533,1459],{},[109,15535,15536],{"encoding":111},"x_i=2^{-i}",[15,15538,15540,15595],{"className":15539,"ariaHidden":72},[116],[15,15541,15543,15546,15586,15589,15592],{"className":15542},[120],[15,15544],{"className":15545,"style":7981},[124],[15,15547,15549,15552],{"className":15548},[129],[15,15550,56],{"className":15551},[129,133],[15,15553,15555],{"className":15554},[137],[15,15556,15558,15578],{"className":15557},[141,142],[15,15559,15561,15575],{"className":15560},[146],[15,15562,15564],{"className":15563,"style":1981},[150],[15,15565,15566,15569],{"style":154},[15,15567],{"className":15568,"style":159},[158],[15,15570,15572],{"className":15571},[163,164,165,166],[15,15573,1459],{"className":15574},[129,133,166],[15,15576,175],{"className":15577},[174],[15,15579,15581],{"className":15580},[146],[15,15582,15584],{"className":15583,"style":182},[150],[15,15585],{},[15,15587],{"className":15588,"style":199},[93],[15,15590,62],{"className":15591},[203],[15,15593],{"className":15594,"style":199},[93],[15,15596,15598,15602],{"className":15597},[120],[15,15599],{"className":15600,"style":15601},[124],"height:0.8247em;",[15,15603,15605,15608],{"className":15604},[129],[15,15606,310],{"className":15607},[129],[15,15609,15611],{"className":15610},[137],[15,15612,15614],{"className":15613},[141],[15,15615,15617],{"className":15616},[146],[15,15618,15620],{"className":15619,"style":15601},[150],[15,15621,15622,15625],{"style":383},[15,15623],{"className":15624,"style":159},[158],[15,15626,15628],{"className":15627},[163,164,165,166],[15,15629,15631,15634],{"className":15630},[129,166],[15,15632,315],{"className":15633},[129,166],[15,15635,1459],{"className":15636},[129,133,166],"，给定任意标注 ",[15,15639,15641,15669],{"className":15640,"translate":19},[23],[15,15642,15644],{"className":15643},[27],[29,15645,15646],{"xmlns":31},[34,15647,15648,15666],{},[37,15649,15650,15652,15654,15656,15658,15660],{},[43,15651,857],{},[50,15653,100],{},[50,15655,636],{"stretchy":52},[50,15657,10928],{},[308,15659,318],{},[305,15661,15662,15664],{},[50,15663,645],{"stretchy":52},[43,15665,348],{},[109,15667,15668],{"encoding":111},"y\\in\\{\\pm1\\}^m",[15,15670,15672,15690],{"className":15671,"ariaHidden":72},[116],[15,15673,15675,15678,15681,15684,15687],{"className":15674},[120],[15,15676],{"className":15677,"style":10955},[124],[15,15679,857],{"className":15680,"style":887},[129,133],[15,15682],{"className":15683,"style":199},[93],[15,15685,100],{"className":15686},[203],[15,15688],{"className":15689,"style":199},[93],[15,15691,15693,15696,15699,15702,15705],{"className":15692},[120],[15,15694],{"className":15695,"style":125},[124],[15,15697,636],{"className":15698},[188],[15,15700,10928],{"className":15701},[129],[15,15703,318],{"className":15704},[129],[15,15706,15708,15711],{"className":15707},[195],[15,15709,645],{"className":15710},[195],[15,15712,15714],{"className":15713},[137],[15,15715,15717],{"className":15716},[141],[15,15718,15720],{"className":15719},[146],[15,15721,15723],{"className":15722,"style":6377},[150],[15,15724,15725,15728],{"style":383},[15,15726],{"className":15727,"style":159},[158],[15,15729,15731],{"className":15730},[163,164,165,166],[15,15732,348],{"className":15733},[129,133,166],"，令",[15,15736,15738],{"className":15737,"translate":19},[18],[15,15739,15741,15807],{"className":15740,"translate":19},[23],[15,15742,15744],{"className":15743},[27],[29,15745,15746],{"xmlns":31,"display":32},[34,15747,15748,15804],{},[37,15749,15750,15752,15754,15757,15802],{},[43,15751,48],{},[50,15753,62],{},[43,15755,15756],{},"π",[37,15758,15759,15761,15767,15769,15794,15800],{},[50,15760,53],{"fence":72},[1443,15762,15763,15765],{},[308,15764,318],{},[308,15766,310],{},[50,15768,2682],{},[2140,15770,15771,15773],{},[50,15772,1454],{},[37,15774,15775,15777,15779,15782,15788,15790,15792],{},[43,15776,1459],{},[50,15778,718],{},[6852,15780,15781],{}," ",[40,15783,15784,15786],{},[43,15785,857],{},[43,15787,1459],{},[50,15789,62],{},[50,15791,315],{},[308,15793,318],{},[305,15795,15796,15798],{},[308,15797,310],{},[43,15799,1459],{},[50,15801,59],{"fence":72},[43,15803,107],{"mathvariant":65},[109,15805,15806],{"encoding":111},"\\theta=\\pi\\left(\\frac12+\\sum_{i:\\,y_i=-1}2^i\\right).",[15,15808,15810,15828],{"className":15809,"ariaHidden":72},[116],[15,15811,15813,15816,15819,15822,15825],{"className":15812},[120],[15,15814],{"className":15815,"style":522},[124],[15,15817,48],{"className":15818,"style":170},[129,133],[15,15820],{"className":15821,"style":199},[93],[15,15823,62],{"className":15824},[203],[15,15826],{"className":15827,"style":199},[93],[15,15829,15831,15835,15838,15841,16070,16073],{"className":15830},[120],[15,15832],{"className":15833,"style":15834},[124],"height:3.1638em;vertical-align:-1.4138em;",[15,15836,15756],{"className":15837,"style":887},[129,133],[15,15839],{"className":15840,"style":262},[93],[15,15842,15844,15851,15913,15916,15919,15922,16031,16034,16064],{"className":15843},[449],[15,15845,15847],{"className":15846,"style":5052},[188,5051],[15,15848,53],{"className":15849},[228,15850],"size4",[15,15852,15854,15857,15910],{"className":15853},[129],[15,15855],{"className":15856},[188,1803],[15,15858,15860],{"className":15859},[1443],[15,15861,15863,15902],{"className":15862},[141,142],[15,15864,15866,15899],{"className":15865},[146],[15,15867,15869,15880,15888],{"className":15868,"style":1816},[150],[15,15870,15871,15874],{"style":1819},[15,15872],{"className":15873,"style":1823},[158],[15,15875,15877],{"className":15876},[129],[15,15878,310],{"className":15879},[129],[15,15881,15882,15885],{"style":1832},[15,15883],{"className":15884,"style":1823},[158],[15,15886],{"className":15887,"style":1840},[1839],[15,15889,15890,15893],{"style":1843},[15,15891],{"className":15892,"style":1823},[158],[15,15894,15896],{"className":15895},[129],[15,15897,318],{"className":15898},[129],[15,15900,175],{"className":15901},[174],[15,15903,15905],{"className":15904},[146],[15,15906,15908],{"className":15907,"style":1862},[150],[15,15909],{},[15,15911],{"className":15912},[195,1803],[15,15914],{"className":15915,"style":3277},[93],[15,15917,2682],{"className":15918},[2436],[15,15920],{"className":15921,"style":3277},[93],[15,15923,15925],{"className":15924},[217,1874],[15,15926,15928,16022],{"className":15927},[141,142],[15,15929,15931,16019],{"className":15930},[146],[15,15932,15934,16009],{"className":15933,"style":12143},[150],[15,15935,15936,15939],{"style":1887},[15,15937],{"className":15938,"style":1891},[158],[15,15940,15942],{"className":15941},[163,164,165,166],[15,15943,15945,15948,15951,15955,16000,16003,16006],{"className":15944},[129,166],[15,15946,1459],{"className":15947},[129,133,166],[15,15949,718],{"className":15950},[203,166],[15,15952],{"className":15953,"style":15954},[93,166],"margin-right:0.1952em;",[15,15956,15958,15961],{"className":15957},[129,166],[15,15959,857],{"className":15960,"style":887},[129,133,166],[15,15962,15964],{"className":15963},[137],[15,15965,15967,15991],{"className":15966},[141,142],[15,15968,15970,15988],{"className":15969},[146],[15,15971,15974],{"className":15972,"style":15973},[150],"height:0.3281em;",[15,15975,15977,15981],{"style":15976},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[15,15978],{"className":15979,"style":15980},[158],"height:2.5em;",[15,15982,15985],{"className":15983},[163,15984,229,166],"reset-size3",[15,15986,1459],{"className":15987},[129,133,166],[15,15989,175],{"className":15990},[174],[15,15992,15994],{"className":15993},[146],[15,15995,15998],{"className":15996,"style":15997},[150],"height:0.143em;",[15,15999],{},[15,16001,62],{"className":16002},[203,166],[15,16004,315],{"className":16005},[129,166],[15,16007,318],{"className":16008},[129,166],[15,16010,16011,16014],{"style":1909},[15,16012],{"className":16013,"style":1891},[158],[15,16015,16016],{},[15,16017,1454],{"className":16018},[217,1918,1919],[15,16020,175],{"className":16021},[174],[15,16023,16025],{"className":16024},[146],[15,16026,16029],{"className":16027,"style":16028},[150],"height:1.4138em;",[15,16030],{},[15,16032],{"className":16033,"style":262},[93],[15,16035,16037,16040],{"className":16036},[129],[15,16038,310],{"className":16039},[129],[15,16041,16043],{"className":16042},[137],[15,16044,16046],{"className":16045},[141],[15,16047,16049],{"className":16048},[146],[15,16050,16053],{"className":16051,"style":16052},[150],"height:0.8747em;",[15,16054,16055,16058],{"style":1272},[15,16056],{"className":16057,"style":159},[158],[15,16059,16061],{"className":16060},[163,164,165,166],[15,16062,1459],{"className":16063},[129,133,166],[15,16065,16067],{"className":16066,"style":5052},[195,5051],[15,16068,59],{"className":16069},[228,15850],[15,16071],{"className":16072,"style":262},[93],[15,16074,107],{"className":16075},[129],[11,16077,16078,16079,2455,16108,16183,16184,16218,16219,16270,16271,16304,16305,16270,16356,16385,16386,16484,16485,16536,16537,16656,16657,16825],{},"对每个 ",[15,16080,16082,16095],{"className":16081,"translate":19},[23],[15,16083,16085],{"className":16084},[27],[29,16086,16087],{"xmlns":31},[34,16088,16089,16093],{},[37,16090,16091],{},[43,16092,1459],{},[109,16094,1459],{"encoding":111},[15,16096,16098],{"className":16097,"ariaHidden":72},[116],[15,16099,16101,16105],{"className":16100},[120],[15,16102],{"className":16103,"style":16104},[124],"height:0.6595em;",[15,16106,1459],{"className":16107},[129,133],[15,16109,16111,16131],{"className":16110,"translate":19},[23],[15,16112,16114],{"className":16113},[27],[29,16115,16116],{"xmlns":31},[34,16117,16118,16128],{},[37,16119,16120,16122],{},[43,16121,48],{},[40,16123,16124,16126],{},[43,16125,56],{},[43,16127,1459],{},[109,16129,16130],{"encoding":111},"\\theta x_i",[15,16132,16134],{"className":16133,"ariaHidden":72},[116],[15,16135,16137,16140,16143],{"className":16136},[120],[15,16138],{"className":16139,"style":2183},[124],[15,16141,48],{"className":16142,"style":170},[129,133],[15,16144,16146,16149],{"className":16145},[129],[15,16147,56],{"className":16148},[129,133],[15,16150,16152],{"className":16151},[137],[15,16153,16155,16175],{"className":16154},[141,142],[15,16156,16158,16172],{"className":16157},[146],[15,16159,16161],{"className":16160,"style":1981},[150],[15,16162,16163,16166],{"style":154},[15,16164],{"className":16165,"style":159},[158],[15,16167,16169],{"className":16168},[163,164,165,166],[15,16170,1459],{"className":16171},[129,133,166],[15,16173,175],{"className":16174},[174],[15,16176,16178],{"className":16177},[146],[15,16179,16181],{"className":16180,"style":182},[150],[15,16182],{}," 对 ",[15,16185,16187,16203],{"className":16186,"translate":19},[23],[15,16188,16190],{"className":16189},[27],[29,16191,16192],{"xmlns":31},[34,16193,16194,16200],{},[37,16195,16196,16198],{},[308,16197,310],{},[43,16199,15756],{},[109,16201,16202],{"encoding":111},"2\\pi",[15,16204,16206],{"className":16205,"ariaHidden":72},[116],[15,16207,16209,16212,16215],{"className":16208},[120],[15,16210],{"className":16211,"style":4886},[124],[15,16213,310],{"className":16214},[129],[15,16216,15756],{"className":16217,"style":887},[129,133]," 取模。",[15,16220,16222,16240],{"className":16221,"translate":19},[23],[15,16223,16225],{"className":16224},[27],[29,16226,16227],{"xmlns":31},[34,16228,16229,16237],{},[37,16230,16231,16233,16235],{},[43,16232,13949],{},[50,16234,4852],{},[43,16236,1459],{},[109,16238,16239],{"encoding":111},"j>i",[15,16241,16243,16261],{"className":16242,"ariaHidden":72},[116],[15,16244,16246,16249,16252,16255,16258],{"className":16245},[120],[15,16247],{"className":16248,"style":11187},[124],[15,16250,13949],{"className":16251,"style":13964},[129,133],[15,16253],{"className":16254,"style":199},[93],[15,16256,4852],{"className":16257},[203],[15,16259],{"className":16260,"style":199},[93],[15,16262,16264,16267],{"className":16263},[120],[15,16265],{"className":16266,"style":16104},[124],[15,16268,1459],{"className":16269},[129,133]," 的项贡献 ",[15,16272,16274,16289],{"className":16273,"translate":19},[23],[15,16275,16277],{"className":16276},[27],[29,16278,16279],{"xmlns":31},[34,16280,16281,16287],{},[37,16282,16283,16285],{},[308,16284,310],{},[43,16286,15756],{},[109,16288,16202],{"encoding":111},[15,16290,16292],{"className":16291,"ariaHidden":72},[116],[15,16293,16295,16298,16301],{"className":16294},[120],[15,16296],{"className":16297,"style":4886},[124],[15,16299,310],{"className":16300},[129],[15,16302,15756],{"className":16303,"style":887},[129,133]," 的整数倍，可以丢掉；",[15,16306,16308,16326],{"className":16307,"translate":19},[23],[15,16309,16311],{"className":16310},[27],[29,16312,16313],{"xmlns":31},[34,16314,16315,16323],{},[37,16316,16317,16319,16321],{},[43,16318,13949],{},[50,16320,62],{},[43,16322,1459],{},[109,16324,16325],{"encoding":111},"j=i",[15,16327,16329,16347],{"className":16328,"ariaHidden":72},[116],[15,16330,16332,16335,16338,16341,16344],{"className":16331},[120],[15,16333],{"className":16334,"style":11187},[124],[15,16336,13949],{"className":16337,"style":13964},[129,133],[15,16339],{"className":16340,"style":199},[93],[15,16342,62],{"className":16343},[203],[15,16345],{"className":16346,"style":199},[93],[15,16348,16350,16353],{"className":16349},[120],[15,16351],{"className":16352,"style":16104},[124],[15,16354,1459],{"className":16355},[129,133],[15,16357,16359,16373],{"className":16358,"translate":19},[23],[15,16360,16362],{"className":16361},[27],[29,16363,16364],{"xmlns":31},[34,16365,16366,16370],{},[37,16367,16368],{},[43,16369,15756],{},[109,16371,16372],{"encoding":111},"\\pi",[15,16374,16376],{"className":16375,"ariaHidden":72},[116],[15,16377,16379,16382],{"className":16378},[120],[15,16380],{"className":16381,"style":4383},[124],[15,16383,15756],{"className":16384,"style":887},[129,133],"（当 ",[15,16387,16389,16413],{"className":16388,"translate":19},[23],[15,16390,16392],{"className":16391},[27],[29,16393,16394],{"xmlns":31},[34,16395,16396,16410],{},[37,16397,16398,16404,16406,16408],{},[40,16399,16400,16402],{},[43,16401,857],{},[43,16403,1459],{},[50,16405,62],{},[50,16407,315],{},[308,16409,318],{},[109,16411,16412],{"encoding":111},"y_i=-1",[15,16414,16416,16471],{"className":16415,"ariaHidden":72},[116],[15,16417,16419,16422,16462,16465,16468],{"className":16418},[120],[15,16420],{"className":16421,"style":5399},[124],[15,16423,16425,16428],{"className":16424},[129],[15,16426,857],{"className":16427,"style":887},[129,133],[15,16429,16431],{"className":16430},[137],[15,16432,16434,16454],{"className":16433},[141,142],[15,16435,16437,16451],{"className":16436},[146],[15,16438,16440],{"className":16439,"style":1981},[150],[15,16441,16442,16445],{"style":1100},[15,16443],{"className":16444,"style":159},[158],[15,16446,16448],{"className":16447},[163,164,165,166],[15,16449,1459],{"className":16450},[129,133,166],[15,16452,175],{"className":16453},[174],[15,16455,16457],{"className":16456},[146],[15,16458,16460],{"className":16459,"style":182},[150],[15,16461],{},[15,16463],{"className":16464,"style":199},[93],[15,16466,62],{"className":16467},[203],[15,16469],{"className":16470,"style":199},[93],[15,16472,16474,16478,16481],{"className":16473},[120],[15,16475],{"className":16476,"style":16477},[124],"height:0.7278em;vertical-align:-0.0833em;",[15,16479,315],{"className":16480},[129],[15,16482,318],{"className":16483},[129],"）；",[15,16486,16488,16506],{"className":16487,"translate":19},[23],[15,16489,16491],{"className":16490},[27],[29,16492,16493],{"xmlns":31},[34,16494,16495,16503],{},[37,16496,16497,16499,16501],{},[43,16498,13949],{},[50,16500,7906],{},[43,16502,1459],{},[109,16504,16505],{"encoding":111},"j\u003Ci",[15,16507,16509,16527],{"className":16508,"ariaHidden":72},[116],[15,16510,16512,16515,16518,16521,16524],{"className":16511},[120],[15,16513],{"className":16514,"style":11187},[124],[15,16516,13949],{"className":16517,"style":13964},[129,133],[15,16519],{"className":16520,"style":199},[93],[15,16522,7906],{"className":16523},[203],[15,16525],{"className":16526,"style":199},[93],[15,16528,16530,16533],{"className":16529},[120],[15,16531],{"className":16532,"style":16104},[124],[15,16534,1459],{"className":16535},[129,133]," 的项合并为 ",[15,16538,16540,16569],{"className":16539,"translate":19},[23],[15,16541,16543],{"className":16542},[27],[29,16544,16545],{"xmlns":31},[34,16546,16547,16566],{},[37,16548,16549,16551,16558,16560],{},[43,16550,15756],{},[40,16552,16553,16556],{},[43,16554,16555],{},"k",[43,16557,1459],{},[43,16559,544],{"mathvariant":65},[305,16561,16562,16564],{},[308,16563,310],{},[43,16565,1459],{},[109,16567,16568],{"encoding":111},"\\pi k_i\u002F2^i",[15,16570,16572],{"className":16571,"ariaHidden":72},[116],[15,16573,16575,16579,16582,16624,16627],{"className":16574},[120],[15,16576],{"className":16577,"style":16578},[124],"height:1.0747em;vertical-align:-0.25em;",[15,16580,15756],{"className":16581,"style":887},[129,133],[15,16583,16585,16589],{"className":16584},[129],[15,16586,16555],{"className":16587,"style":16588},[129,133],"margin-right:0.0315em;",[15,16590,16592],{"className":16591},[137],[15,16593,16595,16616],{"className":16594},[141,142],[15,16596,16598,16613],{"className":16597},[146],[15,16599,16601],{"className":16600,"style":1981},[150],[15,16602,16604,16607],{"style":16603},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[15,16605],{"className":16606,"style":159},[158],[15,16608,16610],{"className":16609},[163,164,165,166],[15,16611,1459],{"className":16612},[129,133,166],[15,16614,175],{"className":16615},[174],[15,16617,16619],{"className":16618},[146],[15,16620,16622],{"className":16621,"style":182},[150],[15,16623],{},[15,16625,544],{"className":16626},[129],[15,16628,16630,16633],{"className":16629},[129],[15,16631,310],{"className":16632},[129],[15,16634,16636],{"className":16635},[137],[15,16637,16639],{"className":16638},[141],[15,16640,16642],{"className":16641},[146],[15,16643,16645],{"className":16644,"style":15601},[150],[15,16646,16647,16650],{"style":383},[15,16648],{"className":16649,"style":159},[158],[15,16651,16653],{"className":16652},[163,164,165,166],[15,16654,1459],{"className":16655},[129,133,166],"，其中 ",[15,16658,16660,16694],{"className":16659,"translate":19},[23],[15,16661,16663],{"className":16662},[27],[29,16664,16665],{"xmlns":31},[34,16666,16667,16691],{},[37,16668,16669,16671,16673,16679,16681,16687,16689],{},[308,16670,541],{},[50,16672,2717],{},[40,16674,16675,16677],{},[43,16676,16555],{},[43,16678,1459],{},[50,16680,2717],{},[305,16682,16683,16685],{},[308,16684,310],{},[43,16686,1459],{},[50,16688,315],{},[308,16690,318],{},[109,16692,16693],{"encoding":111},"0\\le k_i\\le 2^i-1",[15,16695,16697,16716,16771,16816],{"className":16696,"ariaHidden":72},[116],[15,16698,16700,16704,16707,16710,16713],{"className":16699},[120],[15,16701],{"className":16702,"style":16703},[124],"height:0.7804em;vertical-align:-0.136em;",[15,16705,541],{"className":16706},[129],[15,16708],{"className":16709,"style":199},[93],[15,16711,2717],{"className":16712},[203],[15,16714],{"className":16715,"style":199},[93],[15,16717,16719,16722,16762,16765,16768],{"className":16718},[120],[15,16720],{"className":16721,"style":2183},[124],[15,16723,16725,16728],{"className":16724},[129],[15,16726,16555],{"className":16727,"style":16588},[129,133],[15,16729,16731],{"className":16730},[137],[15,16732,16734,16754],{"className":16733},[141,142],[15,16735,16737,16751],{"className":16736},[146],[15,16738,16740],{"className":16739,"style":1981},[150],[15,16741,16742,16745],{"style":16603},[15,16743],{"className":16744,"style":159},[158],[15,16746,16748],{"className":16747},[163,164,165,166],[15,16749,1459],{"className":16750},[129,133,166],[15,16752,175],{"className":16753},[174],[15,16755,16757],{"className":16756},[146],[15,16758,16760],{"className":16759,"style":182},[150],[15,16761],{},[15,16763],{"className":16764,"style":199},[93],[15,16766,2717],{"className":16767},[203],[15,16769],{"className":16770,"style":199},[93],[15,16772,16774,16778,16807,16810,16813],{"className":16773},[120],[15,16775],{"className":16776,"style":16777},[124],"height:0.908em;vertical-align:-0.0833em;",[15,16779,16781,16784],{"className":16780},[129],[15,16782,310],{"className":16783},[129],[15,16785,16787],{"className":16786},[137],[15,16788,16790],{"className":16789},[141],[15,16791,16793],{"className":16792},[146],[15,16794,16796],{"className":16795,"style":15601},[150],[15,16797,16798,16801],{"style":383},[15,16799],{"className":16800,"style":159},[158],[15,16802,16804],{"className":16803},[163,164,165,166],[15,16805,1459],{"className":16806},[129,133,166],[15,16808],{"className":16809,"style":3277},[93],[15,16811,315],{"className":16812},[2436],[15,16814],{"className":16815,"style":3277},[93],[15,16817,16819,16822],{"className":16818},[120],[15,16820],{"className":16821,"style":4886},[124],[15,16823,318],{"className":16824},[129],"。于是",[15,16827,16829],{"className":16828,"translate":19},[18],[15,16830,16832,16940],{"className":16831,"translate":19},[23],[15,16833,16835],{"className":16834},[27],[29,16836,16837],{"xmlns":31,"display":32},[34,16838,16839,16937],{},[37,16840,16841,16843,16849,16852,16854,16910,16912,16915,16917,16926,16929,16931,16933,16935],{},[43,16842,48],{},[40,16844,16845,16847],{},[43,16846,56],{},[43,16848,1459],{},[50,16850,16851],{},"≡",[43,16853,15756],{},[37,16855,16856,16858,16872,16874,16876,16878,16884,16886,16888,16890,16892,16894,16908],{},[50,16857,53],{"fence":72},[305,16859,16860,16862],{},[308,16861,310],{},[37,16863,16864,16866,16868,16870],{},[50,16865,315],{},[43,16867,1459],{},[50,16869,315],{},[308,16871,318],{},[50,16873,2682],{},[308,16875,318],{"mathvariant":1468},[50,16877,1407],{"stretchy":52},[40,16879,16880,16882],{},[43,16881,857],{},[43,16883,1459],{},[50,16885,62],{},[50,16887,315],{},[308,16889,318],{},[50,16891,1423],{"stretchy":52},[50,16893,2682],{},[1443,16895,16896,16902],{},[40,16897,16898,16900],{},[43,16899,16555],{},[43,16901,1459],{},[305,16903,16904,16906],{},[308,16905,310],{},[43,16907,1459],{},[50,16909,59],{"fence":72},[93,16911],{},[93,16913],{"width":16914},"1em",[50,16916,53],{"stretchy":52},[37,16918,16919,16921,16924],{},[43,16920,348],{"mathvariant":65},[43,16922,16923],{"mathvariant":65},"o",[43,16925,10372],{"mathvariant":65},[93,16927],{"width":16928},"0.3333em",[308,16930,310],{},[43,16932,15756],{},[50,16934,59],{"stretchy":52},[43,16936,107],{"mathvariant":65},[109,16938,16939],{"encoding":111},"\\theta x_i\\equiv \\pi\\left(2^{-i-1}+\\mathbf 1[y_i=-1]+\\frac{k_i}{2^i}\\right)\\pmod{2\\pi}.",[15,16941,16943,17001,17291],{"className":16942,"ariaHidden":72},[116],[15,16944,16946,16949,16952,16992,16995,16998],{"className":16945},[120],[15,16947],{"className":16948,"style":2183},[124],[15,16950,48],{"className":16951,"style":170},[129,133],[15,16953,16955,16958],{"className":16954},[129],[15,16956,56],{"className":16957},[129,133],[15,16959,16961],{"className":16960},[137],[15,16962,16964,16984],{"className":16963},[141,142],[15,16965,16967,16981],{"className":16966},[146],[15,16968,16970],{"className":16969,"style":1981},[150],[15,16971,16972,16975],{"style":154},[15,16973],{"className":16974,"style":159},[158],[15,16976,16978],{"className":16977},[163,164,165,166],[15,16979,1459],{"className":16980},[129,133,166],[15,16982,175],{"className":16983},[174],[15,16985,16987],{"className":16986},[146],[15,16988,16990],{"className":16989,"style":182},[150],[15,16991],{},[15,16993],{"className":16994,"style":199},[93],[15,16996,16851],{"className":16997},[203],[15,16999],{"className":17000,"style":199},[93],[15,17002,17004,17008,17011,17014,17280,17284,17288],{"className":17003},[120],[15,17005],{"className":17006,"style":17007},[124],"height:2.4em;vertical-align:-0.95em;",[15,17009,15756],{"className":17010,"style":887},[129,133],[15,17012],{"className":17013,"style":262},[93],[15,17015,17017,17023,17064,17067,17070,17073,17076,17079,17119,17122,17125,17128,17131,17134,17137,17140,17143,17146,17274],{"className":17016},[449],[15,17018,17020],{"className":17019,"style":5052},[188,5051],[15,17021,53],{"className":17022},[228,165],[15,17024,17026,17029],{"className":17025},[129],[15,17027,310],{"className":17028},[129],[15,17030,17032],{"className":17031},[137],[15,17033,17035],{"className":17034},[141],[15,17036,17038],{"className":17037},[146],[15,17039,17041],{"className":17040,"style":16052},[150],[15,17042,17043,17046],{"style":1272},[15,17044],{"className":17045,"style":159},[158],[15,17047,17049],{"className":17048},[163,164,165,166],[15,17050,17052,17055,17058,17061],{"className":17051},[129,166],[15,17053,315],{"className":17054},[129,166],[15,17056,1459],{"className":17057},[129,133,166],[15,17059,315],{"className":17060},[2436,166],[15,17062,318],{"className":17063},[129,166],[15,17065],{"className":17066,"style":3277},[93],[15,17068,2682],{"className":17069},[2436],[15,17071],{"className":17072,"style":3277},[93],[15,17074,318],{"className":17075},[129,1950],[15,17077,1407],{"className":17078},[188],[15,17080,17082,17085],{"className":17081},[129],[15,17083,857],{"className":17084,"style":887},[129,133],[15,17086,17088],{"className":17087},[137],[15,17089,17091,17111],{"className":17090},[141,142],[15,17092,17094,17108],{"className":17093},[146],[15,17095,17097],{"className":17096,"style":1981},[150],[15,17098,17099,17102],{"style":1100},[15,17100],{"className":17101,"style":159},[158],[15,17103,17105],{"className":17104},[163,164,165,166],[15,17106,1459],{"className":17107},[129,133,166],[15,17109,175],{"className":17110},[174],[15,17112,17114],{"className":17113},[146],[15,17115,17117],{"className":17116,"style":182},[150],[15,17118],{},[15,17120],{"className":17121,"style":199},[93],[15,17123,62],{"className":17124},[203],[15,17126],{"className":17127,"style":199},[93],[15,17129,315],{"className":17130},[129],[15,17132,318],{"className":17133},[129],[15,17135,1423],{"className":17136},[195],[15,17138],{"className":17139,"style":3277},[93],[15,17141,2682],{"className":17142},[2436],[15,17144],{"className":17145,"style":3277},[93],[15,17147,17149,17152,17271],{"className":17148},[129],[15,17150],{"className":17151},[188,1803],[15,17153,17155],{"className":17154},[1443],[15,17156,17158,17263],{"className":17157},[141,142],[15,17159,17161,17260],{"className":17160},[146],[15,17162,17165,17204,17212],{"className":17163,"style":17164},[150],"height:1.3714em;",[15,17166,17167,17170],{"style":1819},[15,17168],{"className":17169,"style":1823},[158],[15,17171,17173],{"className":17172},[129],[15,17174,17176,17179],{"className":17175},[129],[15,17177,310],{"className":17178},[129],[15,17180,17182],{"className":17181},[137],[15,17183,17185],{"className":17184},[141],[15,17186,17188],{"className":17187},[146],[15,17189,17192],{"className":17190,"style":17191},[150],"height:0.7507em;",[15,17193,17195,17198],{"style":17194},"top:-2.989em;margin-right:0.05em;",[15,17196],{"className":17197,"style":159},[158],[15,17199,17201],{"className":17200},[163,164,165,166],[15,17202,1459],{"className":17203},[129,133,166],[15,17205,17206,17209],{"style":1832},[15,17207],{"className":17208,"style":1823},[158],[15,17210],{"className":17211,"style":1840},[1839],[15,17213,17214,17217],{"style":1843},[15,17215],{"className":17216,"style":1823},[158],[15,17218,17220],{"className":17219},[129],[15,17221,17223,17226],{"className":17222},[129],[15,17224,16555],{"className":17225,"style":16588},[129,133],[15,17227,17229],{"className":17228},[137],[15,17230,17232,17252],{"className":17231},[141,142],[15,17233,17235,17249],{"className":17234},[146],[15,17236,17238],{"className":17237,"style":1981},[150],[15,17239,17240,17243],{"style":16603},[15,17241],{"className":17242,"style":159},[158],[15,17244,17246],{"className":17245},[163,164,165,166],[15,17247,1459],{"className":17248},[129,133,166],[15,17250,175],{"className":17251},[174],[15,17253,17255],{"className":17254},[146],[15,17256,17258],{"className":17257,"style":182},[150],[15,17259],{},[15,17261,175],{"className":17262},[174],[15,17264,17266],{"className":17265},[146],[15,17267,17269],{"className":17268,"style":1862},[150],[15,17270],{},[15,17272],{"className":17273},[195,1803],[15,17275,17277],{"className":17276,"style":5052},[195,5051],[15,17278,59],{"className":17279},[228,165],[15,17281],{"className":17282},[93,17283],"allowbreak",[15,17285],{"className":17286,"style":17287},[93],"margin-right:1em;",[15,17289],{"className":17290,"style":262},[93],[15,17292,17294,17297,17300,17310,17314,17317,17320,17323],{"className":17293},[120],[15,17295],{"className":17296,"style":125},[124],[15,17298,53],{"className":17299},[188],[15,17301,17303],{"className":17302},[129],[15,17304,17306],{"className":17305},[129],[15,17307,17309],{"className":17308},[129,221],"mod",[15,17311],{"className":17312,"style":17313},[93],"margin-right:0.3333em;",[15,17315,310],{"className":17316},[129],[15,17318,15756],{"className":17319,"style":887},[129,133],[15,17321,59],{"className":17322},[195],[15,17324,107],{"className":17325},[129],[11,17327,17328,17329,17426,17427,17478,17479,17575,17576,17628,17629,17801,17802,17830],{},"当 ",[15,17330,17332,17356],{"className":17331,"translate":19},[23],[15,17333,17335],{"className":17334},[27],[29,17336,17337],{"xmlns":31},[34,17338,17339,17353],{},[37,17340,17341,17347,17349,17351],{},[40,17342,17343,17345],{},[43,17344,857],{},[43,17346,1459],{},[50,17348,62],{},[50,17350,2682],{},[308,17352,318],{},[109,17354,17355],{"encoding":111},"y_i=+1",[15,17357,17359,17414],{"className":17358,"ariaHidden":72},[116],[15,17360,17362,17365,17405,17408,17411],{"className":17361},[120],[15,17363],{"className":17364,"style":5399},[124],[15,17366,17368,17371],{"className":17367},[129],[15,17369,857],{"className":17370,"style":887},[129,133],[15,17372,17374],{"className":17373},[137],[15,17375,17377,17397],{"className":17376},[141,142],[15,17378,17380,17394],{"className":17379},[146],[15,17381,17383],{"className":17382,"style":1981},[150],[15,17384,17385,17388],{"style":1100},[15,17386],{"className":17387,"style":159},[158],[15,17389,17391],{"className":17390},[163,164,165,166],[15,17392,1459],{"className":17393},[129,133,166],[15,17395,175],{"className":17396},[174],[15,17398,17400],{"className":17399},[146],[15,17401,17403],{"className":17402,"style":182},[150],[15,17404],{},[15,17406],{"className":17407,"style":199},[93],[15,17409,62],{"className":17410},[203],[15,17412],{"className":17413,"style":199},[93],[15,17415,17417,17420,17423],{"className":17416},[120],[15,17418],{"className":17419,"style":16477},[124],[15,17421,2682],{"className":17422},[129],[15,17424,318],{"className":17425},[129]," 时，括号内的值落在 ",[15,17428,17430,17451],{"className":17429,"translate":19},[23],[15,17431,17433],{"className":17432},[27],[29,17434,17435],{"xmlns":31},[34,17436,17437,17449],{},[37,17438,17439,17441,17443,17445,17447],{},[50,17440,53],{"stretchy":52},[308,17442,541],{},[50,17444,91],{"separator":72},[308,17446,318],{},[50,17448,59],{"stretchy":52},[109,17450,9520],{"encoding":111},[15,17452,17454],{"className":17453,"ariaHidden":72},[116],[15,17455,17457,17460,17463,17466,17469,17472,17475],{"className":17456},[120],[15,17458],{"className":17459,"style":125},[124],[15,17461,53],{"className":17462},[188],[15,17464,541],{"className":17465},[129],[15,17467,91],{"className":17468},[254],[15,17470],{"className":17471,"style":262},[93],[15,17473,318],{"className":17474},[129],[15,17476,59],{"className":17477},[195],"；当 ",[15,17480,17482,17505],{"className":17481,"translate":19},[23],[15,17483,17485],{"className":17484},[27],[29,17486,17487],{"xmlns":31},[34,17488,17489,17503],{},[37,17490,17491,17497,17499,17501],{},[40,17492,17493,17495],{},[43,17494,857],{},[43,17496,1459],{},[50,17498,62],{},[50,17500,315],{},[308,17502,318],{},[109,17504,16412],{"encoding":111},[15,17506,17508,17563],{"className":17507,"ariaHidden":72},[116],[15,17509,17511,17514,17554,17557,17560],{"className":17510},[120],[15,17512],{"className":17513,"style":5399},[124],[15,17515,17517,17520],{"className":17516},[129],[15,17518,857],{"className":17519,"style":887},[129,133],[15,17521,17523],{"className":17522},[137],[15,17524,17526,17546],{"className":17525},[141,142],[15,17527,17529,17543],{"className":17528},[146],[15,17530,17532],{"className":17531,"style":1981},[150],[15,17533,17534,17537],{"style":1100},[15,17535],{"className":17536,"style":159},[158],[15,17538,17540],{"className":17539},[163,164,165,166],[15,17541,1459],{"className":17542},[129,133,166],[15,17544,175],{"className":17545},[174],[15,17547,17549],{"className":17548},[146],[15,17550,17552],{"className":17551,"style":182},[150],[15,17553],{},[15,17555],{"className":17556,"style":199},[93],[15,17558,62],{"className":17559},[203],[15,17561],{"className":17562,"style":199},[93],[15,17564,17566,17569,17572],{"className":17565},[120],[15,17567],{"className":17568,"style":16477},[124],[15,17570,315],{"className":17571},[129],[15,17573,318],{"className":17574},[129]," 时落在 ",[15,17577,17579,17601],{"className":17578,"translate":19},[23],[15,17580,17582],{"className":17581},[27],[29,17583,17584],{"xmlns":31},[34,17585,17586,17598],{},[37,17587,17588,17590,17592,17594,17596],{},[50,17589,53],{"stretchy":52},[308,17591,318],{},[50,17593,91],{"separator":72},[308,17595,310],{},[50,17597,59],{"stretchy":52},[109,17599,17600],{"encoding":111},"(1,2)",[15,17602,17604],{"className":17603,"ariaHidden":72},[116],[15,17605,17607,17610,17613,17616,17619,17622,17625],{"className":17606},[120],[15,17608],{"className":17609,"style":125},[124],[15,17611,53],{"className":17612},[188],[15,17614,318],{"className":17615},[129],[15,17617,91],{"className":17618},[254],[15,17620],{"className":17621,"style":262},[93],[15,17623,310],{"className":17624},[129],[15,17626,59],{"className":17627},[195],"。所以 ",[15,17630,17632,17676],{"className":17631,"translate":19},[23],[15,17633,17635],{"className":17634},[27],[29,17636,17637],{"xmlns":31},[34,17638,17639,17673],{},[37,17640,17641,17643,17645,17647,17649,17651,17653,17655,17661,17663,17665,17667],{},[43,17642,66],{"mathvariant":65},[50,17644,69],{},[50,17646,53],{"stretchy":52},[43,17648,76],{},[50,17650,69],{},[50,17652,53],{"stretchy":52},[43,17654,48],{},[40,17656,17657,17659],{},[43,17658,56],{},[43,17660,1459],{},[50,17662,59],{"stretchy":52},[50,17664,59],{"stretchy":52},[50,17666,62],{},[40,17668,17669,17671],{},[43,17670,857],{},[43,17672,1459],{},[109,17674,17675],{"encoding":111},"\\operatorname{sign}(\\sin(\\theta x_i))=y_i",[15,17677,17679,17755],{"className":17678,"ariaHidden":72},[116],[15,17680,17682,17685,17691,17694,17697,17700,17703,17743,17746,17749,17752],{"className":17681},[120],[15,17683],{"className":17684,"style":125},[124],[15,17686,17688],{"className":17687},[217],[15,17689,66],{"className":17690},[129,221],[15,17692,53],{"className":17693},[188],[15,17695,76],{"className":17696},[217],[15,17698,53],{"className":17699},[188],[15,17701,48],{"className":17702,"style":170},[129,133],[15,17704,17706,17709],{"className":17705},[129],[15,17707,56],{"className":17708},[129,133],[15,17710,17712],{"className":17711},[137],[15,17713,17715,17735],{"className":17714},[141,142],[15,17716,17718,17732],{"className":17717},[146],[15,17719,17721],{"className":17720,"style":1981},[150],[15,17722,17723,17726],{"style":154},[15,17724],{"className":17725,"style":159},[158],[15,17727,17729],{"className":17728},[163,164,165,166],[15,17730,1459],{"className":17731},[129,133,166],[15,17733,175],{"className":17734},[174],[15,17736,17738],{"className":17737},[146],[15,17739,17741],{"className":17740,"style":182},[150],[15,17742],{},[15,17744,6149],{"className":17745},[195],[15,17747],{"className":17748,"style":199},[93],[15,17750,62],{"className":17751},[203],[15,17753],{"className":17754,"style":199},[93],[15,17756,17758,17761],{"className":17757},[120],[15,17759],{"className":17760,"style":5399},[124],[15,17762,17764,17767],{"className":17763},[129],[15,17765,857],{"className":17766,"style":887},[129,133],[15,17768,17770],{"className":17769},[137],[15,17771,17773,17793],{"className":17772},[141,142],[15,17774,17776,17790],{"className":17775},[146],[15,17777,17779],{"className":17778,"style":1981},[150],[15,17780,17781,17784],{"style":1100},[15,17782],{"className":17783,"style":159},[158],[15,17785,17787],{"className":17786},[163,164,165,166],[15,17788,1459],{"className":17789},[129,133,166],[15,17791,175],{"className":17792},[174],[15,17794,17796],{"className":17795},[146],[15,17797,17799],{"className":17798,"style":182},[150],[15,17800],{},"。任意 ",[15,17803,17805,17818],{"className":17804,"translate":19},[23],[15,17806,17808],{"className":17807},[27],[29,17809,17810],{"xmlns":31},[34,17811,17812,17816],{},[37,17813,17814],{},[43,17815,348],{},[109,17817,348],{"encoding":111},[15,17819,17821],{"className":17820,"ariaHidden":72},[116],[15,17822,17824,17827],{"className":17823},[120],[15,17825],{"className":17826,"style":4383},[124],[15,17828,348],{"className":17829},[129,133]," 个点都能被打散，VC 维无穷。",[11,17832,17833],{},"一个参数，无穷容量。这就是「参数量不等于容量」最干净的反例。",[11,17835,17836],{},[698,17837,17838,17839,7434],{},"例 7（最近邻，",[15,17840,17842,17861],{"className":17841,"translate":19},[23],[15,17843,17845],{"className":17844},[27],[29,17846,17847],{"xmlns":31},[34,17848,17849,17859],{},[37,17850,17851,17853,17855,17857],{},[43,17852,6819],{"mathvariant":65},[50,17854,69],{},[50,17856,62],{},[43,17858,4909],{"mathvariant":65},[109,17860,15222],{"encoding":111},[15,17862,17864,17885],{"className":17863,"ariaHidden":72},[116],[15,17865,17867,17870,17876,17879,17882],{"className":17866},[120],[15,17868],{"className":17869,"style":522},[124],[15,17871,17873],{"className":17872},[217],[15,17874,6819],{"className":17875},[129,221],[15,17877],{"className":17878,"style":199},[93],[15,17880,62],{"className":17881},[203],[15,17883],{"className":17884,"style":199},[93],[15,17886,17888,17891],{"className":17887},[120],[15,17889],{"className":17890,"style":4383},[124],[15,17892,4909],{"className":17893},[129],[11,17895,17896,17897,17925,17926,17954,17955,18025],{},"1-NN 没有参数：给定训练集，它把测试点判为最近训练点的标签。给定任意有限点集 ",[15,17898,17900,17913],{"className":17899,"translate":19},[23],[15,17901,17903],{"className":17902},[27],[29,17904,17905],{"xmlns":31},[34,17906,17907,17911],{},[37,17908,17909],{},[43,17910,5619],{},[109,17912,5619],{"encoding":111},[15,17914,17916],{"className":17915,"ariaHidden":72},[116],[15,17917,17919,17922],{"className":17918},[120],[15,17920],{"className":17921,"style":610},[124],[15,17923,5619],{"className":17924,"style":5665},[129,133],"（点互不相同、没有距离并列）和任意标注，直接令训练集就是 ",[15,17927,17929,17942],{"className":17928,"translate":19},[23],[15,17930,17932],{"className":17931},[27],[29,17933,17934],{"xmlns":31},[34,17935,17936,17940],{},[37,17937,17938],{},[43,17939,5619],{},[109,17941,5619],{"encoding":111},[15,17943,17945],{"className":17944,"ariaHidden":72},[116],[15,17946,17948,17951],{"className":17947},[120],[15,17949],{"className":17950,"style":610},[124],[15,17952,5619],{"className":17953,"style":5665},[129,133]," 本身（标签按标注给）；对每个 ",[15,17956,17958,17976],{"className":17957,"translate":19},[23],[15,17959,17961],{"className":17960},[27],[29,17962,17963],{"xmlns":31},[34,17964,17965,17973],{},[37,17966,17967],{},[40,17968,17969,17971],{},[43,17970,56],{},[43,17972,1459],{},[109,17974,17975],{"encoding":111},"x_i",[15,17977,17979],{"className":17978,"ariaHidden":72},[116],[15,17980,17982,17985],{"className":17981},[120],[15,17983],{"className":17984,"style":7981},[124],[15,17986,17988,17991],{"className":17987},[129],[15,17989,56],{"className":17990},[129,133],[15,17992,17994],{"className":17993},[137],[15,17995,17997,18017],{"className":17996},[141,142],[15,17998,18000,18014],{"className":17999},[146],[15,18001,18003],{"className":18002,"style":1981},[150],[15,18004,18005,18008],{"style":154},[15,18006],{"className":18007,"style":159},[158],[15,18009,18011],{"className":18010},[163,164,165,166],[15,18012,1459],{"className":18013},[129,133,166],[15,18015,175],{"className":18016},[174],[15,18018,18020],{"className":18019},[146],[15,18021,18023],{"className":18022,"style":182},[150],[15,18024],{},"，最近邻是它自己，预测即标注。所以任意有限集都被打散，VC 维无穷。",[11,18027,18028],{},"这个例子说明，VC 维无穷并不妨碍一个算法在真实数据上表现良好；它只说明不存在一个对所有分布一致的泛化保证。",[11,18030,18031],{},"把上面几个例子汇总：",[18033,18034,18035,18049],"table",{},[18036,18037,18038],"thead",{},[18039,18040,18041,18044,18047],"tr",{},[18042,18043,803],"th",{},[18042,18045,18046],{},"输入空间",[18042,18048,6800],{},[18050,18051,18052,18119,18184,18301,18401,18518,18661,18794],"tbody",{},[18039,18053,18054,18058,18089],{},[18055,18056,18057],"td",{},"半直线",[18055,18059,18060],{},[15,18061,18063,18077],{"className":18062,"translate":19},[23],[15,18064,18066],{"className":18065},[27],[29,18067,18068],{"xmlns":31},[34,18069,18070,18074],{},[37,18071,18072],{},[43,18073,104],{"mathvariant":103},[109,18075,18076],{"encoding":111},"\\mathbb R",[15,18078,18080],{"className":18079,"ariaHidden":72},[116],[15,18081,18083,18086],{"className":18082},[120],[15,18084],{"className":18085,"style":281},[124],[15,18087,104],{"className":18088},[129,285],[18055,18090,18091],{},[15,18092,18094,18107],{"className":18093,"translate":19},[23],[15,18095,18097],{"className":18096},[27],[29,18098,18099],{"xmlns":31},[34,18100,18101,18105],{},[37,18102,18103],{},[308,18104,318],{},[109,18106,318],{"encoding":111},[15,18108,18110],{"className":18109,"ariaHidden":72},[116],[15,18111,18113,18116],{"className":18112},[120],[15,18114],{"className":18115,"style":4886},[124],[15,18117,318],{"className":18118},[129],[18039,18120,18121,18124,18154],{},[18055,18122,18123],{},"区间",[18055,18125,18126],{},[15,18127,18129,18142],{"className":18128,"translate":19},[23],[15,18130,18132],{"className":18131},[27],[29,18133,18134],{"xmlns":31},[34,18135,18136,18140],{},[37,18137,18138],{},[43,18139,104],{"mathvariant":103},[109,18141,18076],{"encoding":111},[15,18143,18145],{"className":18144,"ariaHidden":72},[116],[15,18146,18148,18151],{"className":18147},[120],[15,18149],{"className":18150,"style":281},[124],[15,18152,104],{"className":18153},[129,285],[18055,18155,18156],{},[15,18157,18159,18172],{"className":18158,"translate":19},[23],[15,18160,18162],{"className":18161},[27],[29,18163,18164],{"xmlns":31},[34,18165,18166,18170],{},[37,18167,18168],{},[308,18169,310],{},[109,18171,310],{"encoding":111},[15,18173,18175],{"className":18174,"ariaHidden":72},[116],[15,18176,18178,18181],{"className":18177},[120],[15,18179],{"className":18180,"style":4886},[124],[15,18182,310],{"className":18183},[129],[18039,18185,18186,18189,18249],{},[18055,18187,18188],{},"仿射超平面",[18055,18190,18191],{},[15,18192,18194,18211],{"className":18193,"translate":19},[23],[15,18195,18197],{"className":18196},[27],[29,18198,18199],{"xmlns":31},[34,18200,18201,18209],{},[37,18202,18203],{},[305,18204,18205,18207],{},[43,18206,104],{"mathvariant":103},[43,18208,10372],{},[109,18210,11566],{"encoding":111},[15,18212,18214],{"className":18213,"ariaHidden":72},[116],[15,18215,18217,18220],{"className":18216},[120],[15,18218],{"className":18219,"style":10486},[124],[15,18221,18223,18226],{"className":18222},[129],[15,18224,104],{"className":18225},[129,285],[15,18227,18229],{"className":18228},[137],[15,18230,18232],{"className":18231},[141],[15,18233,18235],{"className":18234},[146],[15,18236,18238],{"className":18237,"style":10486},[150],[15,18239,18240,18243],{"style":383},[15,18241],{"className":18242,"style":159},[158],[15,18244,18246],{"className":18245},[163,164,165,166],[15,18247,10372],{"className":18248},[129,133,166],[18055,18250,18251],{},[15,18252,18254,18271],{"className":18253,"translate":19},[23],[15,18255,18257],{"className":18256},[27],[29,18258,18259],{"xmlns":31},[34,18260,18261,18269],{},[37,18262,18263,18265,18267],{},[43,18264,10372],{},[50,18266,2682],{},[308,18268,318],{},[109,18270,10876],{"encoding":111},[15,18272,18274,18292],{"className":18273,"ariaHidden":72},[116],[15,18275,18277,18280,18283,18286,18289],{"className":18276},[120],[15,18278],{"className":18279,"style":10410},[124],[15,18281,10372],{"className":18282},[129,133],[15,18284],{"className":18285,"style":3277},[93],[15,18287,2682],{"className":18288},[2436],[15,18290],{"className":18291,"style":3277},[93],[15,18293,18295,18298],{"className":18294},[120],[15,18296],{"className":18297,"style":4886},[124],[15,18299,318],{"className":18300},[129],[18039,18302,18303,18306,18366],{},[18055,18304,18305],{},"轴对齐矩形",[18055,18307,18308],{},[15,18309,18311,18328],{"className":18310,"translate":19},[23],[15,18312,18314],{"className":18313},[27],[29,18315,18316],{"xmlns":31},[34,18317,18318,18326],{},[37,18319,18320],{},[305,18321,18322,18324],{},[43,18323,104],{"mathvariant":103},[43,18325,10372],{},[109,18327,11566],{"encoding":111},[15,18329,18331],{"className":18330,"ariaHidden":72},[116],[15,18332,18334,18337],{"className":18333},[120],[15,18335],{"className":18336,"style":10486},[124],[15,18338,18340,18343],{"className":18339},[129],[15,18341,104],{"className":18342},[129,285],[15,18344,18346],{"className":18345},[137],[15,18347,18349],{"className":18348},[141],[15,18350,18352],{"className":18351},[146],[15,18353,18355],{"className":18354,"style":10486},[150],[15,18356,18357,18360],{"style":383},[15,18358],{"className":18359,"style":159},[158],[15,18361,18363],{"className":18362},[163,164,165,166],[15,18364,10372],{"className":18365},[129,133,166],[18055,18367,18368],{},[15,18369,18371,18386],{"className":18370,"translate":19},[23],[15,18372,18374],{"className":18373},[27],[29,18375,18376],{"xmlns":31},[34,18377,18378,18384],{},[37,18379,18380,18382],{},[308,18381,310],{},[43,18383,10372],{},[109,18385,14274],{"encoding":111},[15,18387,18389],{"className":18388,"ariaHidden":72},[116],[15,18390,18392,18395,18398],{"className":18391},[120],[15,18393],{"className":18394,"style":522},[124],[15,18396,310],{"className":18397},[129],[1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",[15,18524,18526,18539],{"className":18525,"translate":19},[23],[15,18527,18529],{"className":18528},[27],[29,18530,18531],{"xmlns":31},[34,18532,18533,18537],{},[37,18534,18535],{},[43,18536,819],{"mathvariant":596},[109,18538,822],{"encoding":111},[15,18540,18542],{"className":18541,"ariaHidden":72},[116],[15,18543,18545,18548],{"className":18544},[120],[15,18546],{"className":18547,"style":610},[124],[15,18549,819],{"className":18550,"style":835},[129,614],[18055,18552,18553],{},"任意",[18055,18555,18556],{},[15,18557,18559,18589],{"className":18558,"translate":19},[23],[15,18560,18562],{"className":18561},[27],[29,18563,18564],{"xmlns":31},[34,18565,18566,18586],{},[37,18567,18568,18570,18580,18582,18584],{},[50,18569,2717],{},[40,18571,18572,18578],{},[37,18573,18574,18576],{},[43,18575,14721],{},[50,18577,69],{},[308,18579,310],{},[50,18581,2945],{"stretchy":52},[43,18583,819],{"mathvariant":596},[50,18585,2945],{"stretchy":52},[109,18587,18588],{"encoding":111},"\\le\\log_2\\lvert\\mathcal H\\rvert",[15,18590,18592,18604],{"className":18591,"ariaHidden":72},[116],[15,18593,18595,18598,18601],{"className":18594},[120],[15,18596],{"className":18597,"style":7753},[124],[15,18599,2717],{"className":18600},[203],[15,18602],{"className":18603,"style":199},[93],[15,18605,18607,18610,18652,18655,18658],{"className":18606},[120],[15,18608],{"className":18609,"style":125},[124],[15,18611,18613,18618],{"className":18612},[217],[15,18614,14771,18616],{"className":18615},[217],[15,18617,2247],{"style":2246},[15,18619,18621],{"className":18620},[137],[15,18622,18624,18644],{"className":18623},[141,142],[15,18625,18627,18641],{"className":18626},[146],[15,18628,18630],{"className":18629,"style":14786},[150],[15,18631,18632,18635],{"style":14789},[15,18633],{"className":18634,"style":159},[158],[15,18636,18638],{"className":18637},[163,164,165,166],[15,18639,310],{"className":18640},[129,166],[15,18642,175],{"className":18643},[174],[15,18645,18647],{"className":18646},[146],[15,18648,18650],{"className":18649,"style":14808},[150],[15,18651],{},[15,18653,2945],{"className":18654},[188],[15,18656,819],{"className":18657,"style":835},[129,614],[15,18659,2945],{"className":18660},[195],[18039,18662,18663,18733,18763],{},[18055,18664,18665],{},[15,18666,18668,18700],{"className":18667,"translate":19},[23],[15,18669,18671],{"className":18670},[27],[29,18672,18673],{"xmlns":31},[34,18674,18675,18697],{},[37,18676,18677,18679,18681,18683,18685,18687,18689,18691,18693,18695],{},[43,18678,66],{"mathvariant":65},[50,18680,69],{},[50,18682,53],{"stretchy":52},[43,18684,76],{},[50,18686,69],{},[50,18688,53],{"stretchy":52},[43,18690,48],{},[43,18692,56],{},[50,18694,59],{"stretchy":52},[50,18696,59],{"stretchy":52},[109,18698,18699],{"encoding":111},"\\operatorname{sign}(\\sin(\\theta x))",[15,18701,18703],{"className":18702,"ariaHidden":72},[116],[15,18704,18706,18709,18715,18718,18721,18724,18727,18730],{"className":18705},[120],[15,18707],{"className":18708,"style":125},[124],[15,18710,18712],{"className":18711},[217],[15,18713,66],{"className":18714},[129,221],[15,18716,53],{"className":18717},[188],[15,18719,76],{"className":18720},[217],[15,18722,53],{"className":18723},[188],[15,18725,48],{"className":18726,"style":170},[129,133],[15,18728,56],{"className":18729},[129,133],[15,18731,6149],{"className":18732},[195],[18055,18734,18735],{},[15,18736,18738,18751],{"className":18737,"translate":19},[23],[15,18739,18741],{"className":18740},[27],[29,18742,18743],{"xmlns":31},[34,18744,18745,18749],{},[37,18746,18747],{},[43,18748,104],{"mathvariant":103},[109,18750,18076],{"encoding":111},[15,18752,18754],{"className":18753,"ariaHidden":72},[116],[15,18755,18757,18760],{"className":18756},[120],[15,18758],{"className":18759,"style":281},[124],[15,18761,104],{"className":18762},[129,285],[18055,18764,18765],{},[15,18766,18768,18782],{"className":18767,"translate":19},[23],[15,18769,18771],{"className":18770},[27],[29,18772,18773],{"xmlns":31},[34,18774,18775,18779],{},[37,18776,18777],{},[43,18778,4909],{"mathvariant":65},[109,18780,18781],{"encoding":111},"\\infty",[15,18783,18785],{"className":18784,"ariaHidden":72},[116],[15,18786,18788,18791],{"className":18787},[120],[15,18789],{"className":18790,"style":4383},[124],[15,18792,4909],{"className":18793},[129],[18039,18795,18796,18799,18859],{},[18055,18797,18798],{},"1-NN",[18055,18800,18801],{},[15,18802,18804,18821],{"className":18803,"translate":19},[23],[15,18805,18807],{"className":18806},[27],[29,18808,18809],{"xmlns":31},[34,18810,18811,18819],{},[37,18812,18813],{},[305,18814,18815,18817],{},[43,18816,104],{"mathvariant":103},[43,18818,10372],{},[109,18820,11566],{"encoding":111},[15,18822,18824],{"className":18823,"ariaHidden":72},[116],[15,18825,18827,18830],{"className":18826},[120],[15,18828],{"className":18829,"style":10486},[124],[15,18831,18833,18836],{"className":18832},[129],[15,18834,104],{"className":18835},[129,285],[15,18837,18839],{"className":18838},[137],[15,18840,18842],{"className":18841},[141],[15,18843,18845],{"className":18844},[146],[15,18846,18848],{"className":18847,"style":10486},[150],[15,18849,18850,18853],{"style":383},[15,18851],{"className":18852,"style":159},[158],[15,18854,18856],{"className":18855},[163,164,165,166],[15,18857,10372],{"className":18858},[129,133,166],[18055,18860,18861],{},[15,18862,18864,18877],{"className":18863,"translate":19},[23],[15,18865,18867],{"className":18866},[27],[29,18868,18869],{"xmlns":31},[34,18870,18871,18875],{},[37,18872,18873],{},[43,18874,4909],{"mathvariant":65},[109,18876,18781],{"encoding":111},[15,18878,18880],{"className":18879,"ariaHidden":72},[116],[15,18881,18883,18886],{"className":18882},[120],[15,18884],{"className":18885,"style":4383},[124],[15,18887,4909],{"className":18888},[129],[6559,18890,18892],{"title":18891},"注（怎么算）",[11,18893,18894,18895,19067,19068,19164,19165,19217,19218,19410,19411,19439,19440,19499,19500,19528,19529,19557],{},"对线性分类器，固定点集 ",[15,18896,18898,18936],{"className":18897,"translate":19},[23],[15,18899,18901],{"className":18900},[27],[29,18902,18903],{"xmlns":31},[34,18904,18905,18933],{},[37,18906,18907,18909,18911,18913,18919,18921,18923,18925,18931],{},[43,18908,5619],{},[50,18910,62],{},[50,18912,636],{"stretchy":52},[40,18914,18915,18917],{},[43,18916,56],{},[308,18918,318],{},[50,18920,91],{"separator":72},[50,18922,335],{},[50,18924,91],{"separator":72},[40,18926,18927,18929],{},[43,18928,56],{},[43,18930,16555],{},[50,18932,645],{"stretchy":52},[109,18934,18935],{"encoding":111},"C=\\{x_1,\\dots,x_k\\}",[15,18937,18939,18957],{"className":18938,"ariaHidden":72},[116],[15,18940,18942,18945,18948,18951,18954],{"className":18941},[120],[15,18943],{"className":18944,"style":610},[124],[15,18946,5619],{"className":18947,"style":5665},[129,133],[15,18949],{"className":18950,"style":199},[93],[15,18952,62],{"className":18953},[203],[15,18955],{"className":18956,"style":199},[93],[15,18958,18960,18963,18966,19006,19009,19012,19015,19018,19021,19024,19064],{"className":18959},[120],[15,18961],{"className":18962,"style":125},[124],[15,18964,636],{"className":18965},[188],[15,18967,18969,18972],{"className":18968},[129],[15,18970,56],{"className":18971},[129,133],[15,18973,18975],{"className":18974},[137],[15,18976,18978,18998],{"className":18977},[141,142],[15,18979,18981,18995],{"className":18980},[146],[15,18982,18984],{"className":18983,"style":1051},[150],[15,18985,18986,18989],{"style":154},[15,18987],{"className":18988,"style":159},[158],[15,18990,18992],{"className":18991},[163,164,165,166],[15,18993,318],{"className":18994},[129,166],[15,18996,175],{"className":18997},[174],[15,18999,19001],{"className":19000},[146],[15,19002,19004],{"className":19003,"style":182},[150],[15,19005],{},[15,19007,91],{"className":19008},[254],[15,19010],{"className":19011,"style":262},[93],[15,19013,335],{"className":19014},[449],[15,19016],{"className":19017,"style":262},[93],[15,19019,91],{"className":19020},[254],[15,19022],{"className":19023,"style":262},[93],[15,19025,19027,19030],{"className":19026},[129],[15,19028,56],{"className":19029},[129,133],[15,19031,19033],{"className":19032},[137],[15,19034,19036,19056],{"className":19035},[141,142],[15,19037,19039,19053],{"className":19038},[146],[15,19040,19042],{"className":19041,"style":151},[150],[15,19043,19044,19047],{"style":154},[15,19045],{"className":19046,"style":159},[158],[15,19048,19050],{"className":19049},[163,164,165,166],[15,19051,16555],{"className":19052,"style":16588},[129,133,166],[15,19054,175],{"className":19055},[174],[15,19057,19059],{"className":19058},[146],[15,19060,19062],{"className":19061,"style":182},[150],[15,19063],{},[15,19065,645],{"className":19066},[195],"，对每个标注 ",[15,19069,19071,19099],{"className":19070,"translate":19},[23],[15,19072,19074],{"className":19073},[27],[29,19075,19076],{"xmlns":31},[34,19077,19078,19096],{},[37,19079,19080,19082,19084,19086,19088,19090],{},[43,19081,857],{},[50,19083,100],{},[50,19085,636],{"stretchy":52},[50,19087,10928],{},[308,19089,318],{},[305,19091,19092,19094],{},[50,19093,645],{"stretchy":52},[43,19095,16555],{},[109,19097,19098],{"encoding":111},"y\\in\\{\\pm1\\}^k",[15,19100,19102,19120],{"className":19101,"ariaHidden":72},[116],[15,19103,19105,19108,19111,19114,19117],{"className":19104},[120],[15,19106],{"className":19107,"style":10955},[124],[15,19109,857],{"className":19110,"style":887},[129,133],[15,19112],{"className":19113,"style":199},[93],[15,19115,100],{"className":19116},[203],[15,19118],{"className":19119,"style":199},[93],[15,19121,19123,19126,19129,19132,19135],{"className":19122},[120],[15,19124],{"className":19125,"style":10974},[124],[15,19127,636],{"className":19128},[188],[15,19130,10928],{"className":19131},[129],[15,19133,318],{"className":19134},[129],[15,19136,19138,19141],{"className":19137},[195],[15,19139,645],{"className":19140},[195],[15,19142,19144],{"className":19143},[137],[15,19145,19147],{"className":19146},[141],[15,19148,19150],{"className":19149},[146],[15,19151,19153],{"className":19152,"style":10486},[150],[15,19154,19155,19158],{"style":383},[15,19156],{"className":19157,"style":159},[158],[15,19159,19161],{"className":19160},[163,164,165,166],[15,19162,16555],{"className":19163,"style":16588},[129,133,166]," 解可行性问题：是否存在 ",[15,19166,19168,19190],{"className":19167,"translate":19},[23],[15,19169,19171],{"className":19170},[27],[29,19172,19173],{"xmlns":31},[34,19174,19175,19187],{},[37,19176,19177,19179,19181,19183,19185],{},[50,19178,53],{"stretchy":52},[43,19180,10546],{},[50,19182,91],{"separator":72},[43,19184,8695],{},[50,19186,59],{"stretchy":52},[109,19188,19189],{"encoding":111},"(w,b)",[15,19191,19193],{"className":19192,"ariaHidden":72},[116],[15,19194,19196,19199,19202,19205,19208,19211,19214],{"className":19195},[120],[15,19197],{"className":19198,"style":125},[124],[15,19200,53],{"className":19201},[188],[15,19203,10546],{"className":19204,"style":10623},[129,133],[15,19206,91],{"className":19207},[254],[15,19209],{"className":19210,"style":262},[93],[15,19212,8695],{"className":19213},[129,133],[15,19215,59],{"className":19216},[195]," 使 ",[15,19219,19221,19261],{"className":19220,"translate":19},[23],[15,19222,19224],{"className":19223},[27],[29,19225,19226],{"xmlns":31},[34,19227,19228,19258],{},[37,19229,19230,19236,19238,19240,19242,19248,19250,19252,19254,19256],{},[40,19231,19232,19234],{},[43,19233,857],{},[43,19235,1459],{},[50,19237,53],{"stretchy":52},[43,19239,10546],{},[50,19241,10549],{},[40,19243,19244,19246],{},[43,19245,56],{},[43,19247,1459],{},[50,19249,2682],{},[43,19251,8695],{},[50,19253,59],{"stretchy":52},[50,19255,7469],{},[308,19257,318],{},[109,19259,19260],{"encoding":111},"y_i(w\\cdot x_i+b)\\ge 1",[15,19262,19264,19325,19380,19401],{"className":19263,"ariaHidden":72},[116],[15,19265,19267,19270,19310,19313,19316,19319,19322],{"className":19266},[120],[15,19268],{"className":19269,"style":125},[124],[15,19271,19273,19276],{"className":19272},[129],[15,19274,857],{"className":19275,"style":887},[129,133],[15,19277,19279],{"className":19278},[137],[15,19280,19282,19302],{"className":19281},[141,142],[15,19283,19285,19299],{"className":19284},[146],[15,19286,19288],{"className":19287,"style":1981},[150],[15,19289,19290,19293],{"style":1100},[15,19291],{"className":19292,"style":159},[158],[15,19294,19296],{"className":19295},[163,164,165,166],[15,19297,1459],{"className":19298},[129,133,166],[15,19300,175],{"className":19301},[174],[15,19303,19305],{"className":19304},[146],[15,19306,19308],{"className":19307,"style":182},[150],[15,19309],{},[15,19311,53],{"className":19312},[188],[15,19314,10546],{"className":19315,"style":10623},[129,133],[15,19317],{"className":19318,"style":3277},[93],[15,19320,10549],{"className":19321},[2436],[15,19323],{"className":19324,"style":3277},[93],[15,19326,19328,19331,19371,19374,19377],{"className":19327},[120],[15,19329],{"className":19330,"style":11113},[124],[15,19332,19334,19337],{"className":19333},[129],[15,19335,56],{"className":19336},[129,133],[15,19338,19340],{"className":19339},[137],[15,19341,19343,19363],{"className":19342},[141,142],[15,19344,19346,19360],{"className":19345},[146],[15,19347,19349],{"className":19348,"style":1981},[150],[15,19350,19351,19354],{"style":154},[15,19352],{"className":19353,"style":159},[158],[15,19355,19357],{"className":19356},[163,164,165,166],[15,19358,1459],{"className":19359},[129,133,166],[15,19361,175],{"className":19362},[174],[15,19364,19366],{"className":19365},[146],[15,19367,19369],{"className":19368,"style":182},[150],[15,19370],{},[15,19372],{"className":19373,"style":3277},[93],[15,19375,2682],{"className":19376},[2436],[15,19378],{"className":19379,"style":3277},[93],[15,19381,19383,19386,19389,19392,19395,19398],{"className":19382},[120],[15,19384],{"className":19385,"style":125},[124],[15,19387,8695],{"className":19388},[129,133],[15,19390,59],{"className":19391},[195],[15,19393],{"className":19394,"style":199},[93],[15,19396,7469],{"className":19397},[203],[15,19399],{"className":19400,"style":199},[93],[15,19402,19404,19407],{"className":19403},[120],[15,19405],{"className":19406,"style":4886},[124],[15,19408,318],{"className":19409},[129]," 对所有 ",[15,19412,19414,19427],{"className":19413,"translate":19},[23],[15,19415,19417],{"className":19416},[27],[29,19418,19419],{"xmlns":31},[34,19420,19421,19425],{},[37,19422,19423],{},[43,19424,1459],{},[109,19426,1459],{"encoding":111},[15,19428,19430],{"className":19429,"ariaHidden":72},[116],[15,19431,19433,19436],{"className":19432},[120],[15,19434],{"className":19435,"style":16104},[124],[15,19437,1459],{"className":19438},[129,133]," 成立。这是一个线性规划。若 ",[15,19441,19443,19461],{"className":19442,"translate":19},[23],[15,19444,19446],{"className":19445},[27],[29,19447,19448],{"xmlns":31},[34,19449,19450,19458],{},[37,19451,19452],{},[305,19453,19454,19456],{},[308,19455,310],{},[43,19457,16555],{},[109,19459,19460],{"encoding":111},"2^k",[15,19462,19464],{"className":19463,"ariaHidden":72},[116],[15,19465,19467,19470],{"className":19466},[120],[15,19468],{"className":19469,"style":10486},[124],[15,19471,19473,19476],{"className":19472},[129],[15,19474,310],{"className":19475},[129],[15,19477,19479],{"className":19478},[137],[15,19480,19482],{"className":19481},[141],[15,19483,19485],{"className":19484},[146],[15,19486,19488],{"className":19487,"style":10486},[150],[15,19489,19490,19493],{"style":383},[15,19491],{"className":19492,"style":159},[158],[15,19494,19496],{"className":19495},[163,164,165,166],[15,19497,16555],{"className":19498,"style":16588},[129,133,166]," 个标注全部可行，",[15,19501,19503,19516],{"className":19502,"translate":19},[23],[15,19504,19506],{"className":19505},[27],[29,19507,19508],{"xmlns":31},[34,19509,19510,19514],{},[37,19511,19512],{},[43,19513,5619],{},[109,19515,5619],{"encoding":111},[15,19517,19519],{"className":19518,"ariaHidden":72},[116],[15,19520,19522,19525],{"className":19521},[120],[15,19523],{"className":19524,"style":610},[124],[15,19526,5619],{"className":19527,"style":5665},[129,133]," 就被打散；对所有 ",[15,19530,19532,19545],{"className":19531,"translate":19},[23],[15,19533,19535],{"className":19534},[27],[29,19536,19537],{"xmlns":31},[34,19538,19539,19543],{},[37,19540,19541],{},[43,19542,5619],{},[109,19544,5619],{"encoding":111},[15,19546,19548],{"className":19547,"ariaHidden":72},[116],[15,19549,19551,19554],{"className":19550},[120],[15,19552],{"className":19553,"style":610},[124],[15,19555,5619],{"className":19556,"style":5665},[129,133]," 取最大，就得到 VC 维。对一般的假设类，精确判定通常是组合优化问题，实践中只能靠随机采样加优化或可满足性求解来估计。",[575,19559,19561],{"id":19560},"_23-三个误区","2.3 三个误区",[11,19563,19564,19567],{},[698,19565,19566],{},"第一，「存在」不是「所有」。"," 定义只要求存在一个点集被打散。所以 VC 维大，不意味着假设类在任意数据上都很灵活；VC 维小，也不意味着它在某些特殊点集上不灵活。",[11,19569,19570,19573,19574,19577],{},[698,19571,19572],{},"第二，打散不是「能拟合这组数据」。"," 给定一组带标签的数据，几乎任何足够灵活的类都能拟合。打散要求的是：把这组点的标签",[698,19575,19576],{},"换着花样","全部试一遍，每一种都能实现。前者是插值能力，后者是组合丰富度。",[11,19579,19580,19583,19584,19612,19613,19641],{},[698,19581,19582],{},"第三，VC 维与参数化方式无关。"," 同一个函数集合可以有一万种参数化，VC 维不变。反过来，同一个参数数目可以对应 VC 维为 ",[15,19585,19587,19600],{"className":19586,"translate":19},[23],[15,19588,19590],{"className":19589},[27],[29,19591,19592],{"xmlns":31},[34,19593,19594,19598],{},[37,19595,19596],{},[308,19597,318],{},[109,19599,318],{"encoding":111},[15,19601,19603],{"className":19602,"ariaHidden":72},[116],[15,19604,19606,19609],{"className":19605},[120],[15,19607],{"className":19608,"style":4886},[124],[15,19610,318],{"className":19611},[129]," 的类，也可以对应 VC 维为 ",[15,19614,19616,19629],{"className":19615,"translate":19},[23],[15,19617,19619],{"className":19618},[27],[29,19620,19621],{"xmlns":31},[34,19622,19623,19627],{},[37,19624,19625],{},[43,19626,4909],{"mathvariant":65},[109,19628,18781],{"encoding":111},[15,19630,19632],{"className":19631,"ariaHidden":72},[116],[15,19633,19635,19638],{"className":19634},[120],[15,19636],{"className":19637,"style":4383},[124],[15,19639,4909],{"className":19640},[129]," 的类。参数只是描述函数的一种方式，容量是函数集合本身的性质。",[11,19643,19644,19645,19648],{},"还有一点值得提前说明：上面定义的是",[698,19646,19647],{},"二值","函数类的 VC 维。如果假设输出实数（回归、概率、间隔），就需要 pseudo-dimension 或 fat-shattering dimension，后文会再提。",[570,19650,19652],{"id":19651},"三增长函数与-sauershelah-引理","三、增长函数与 Sauer–Shelah 引理",[575,19654,19656],{"id":19655},"_31-增长函数","3.1 增长函数",[11,19658,19659],{},"逐个点集去数二分太麻烦。我们只关心最坏情况下能产生多少种二分：",[15,19661,19663],{"className":19662,"translate":19},[18],[15,19664,19666,19738],{"className":19665,"translate":19},[23],[15,19667,19669],{"className":19668},[27],[29,19670,19671],{"xmlns":31,"display":32},[34,19672,19673,19735],{},[37,19674,19675,19682,19684,19686,19688,19690,19721,19723,19725,19731,19733],{},[40,19676,19677,19680],{},[43,19678,19679],{"mathvariant":65},"Π",[43,19681,819],{"mathvariant":596},[50,19683,53],{"stretchy":52},[43,19685,348],{},[50,19687,59],{"stretchy":52},[50,19689,62],{},[2140,19691,19692,19699],{},[37,19693,19694,19697],{},[43,19695,19696],{},"max",[50,19698,69],{},[37,19700,19701,19703,19705,19707,19709,19711,19713,19715,19717,19719],{},[43,19702,5619],{},[50,19704,5647],{},[43,19706,597],{"mathvariant":596},[50,19708,91],{"separator":72},[6852,19710,6854],{},[43,19712,2945],{"mathvariant":65},[43,19714,5619],{},[43,19716,2945],{"mathvariant":65},[50,19718,62],{},[43,19720,348],{},[50,19722,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,19724,819],{"mathvariant":596},[40,19726,19727,19729],{},[43,19728,2945],{"mathvariant":65},[43,19730,5619],{},[50,19732,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,19734,107],{"mathvariant":65},[109,19736,19737],{"encoding":111},"\\Pi_{\\mathcal H}(m)=\\max_{C\\subseteq\\mathcal X,\\ |C|=m}\\bigl|\\mathcal H|_C\\bigr|.",[15,19739,19741,19808],{"className":19740,"ariaHidden":72},[116],[15,19742,19744,19747,19790,19793,19796,19799,19802,19805],{"className":19743},[120],[15,19745],{"className":19746,"style":125},[124],[15,19748,19750,19753],{"className":19749},[129],[15,19751,19679],{"className":19752},[129],[15,19754,19756],{"className":19755},[137],[15,19757,19759,19782],{"className":19758},[141,142],[15,19760,19762,19779],{"className":19761},[146],[15,19763,19765],{"className":19764,"style":1525},[150],[15,19766,19767,19770],{"style":154},[15,19768],{"className":19769,"style":159},[158],[15,19771,19773],{"className":19772},[163,164,165,166],[15,19774,19776],{"className":19775},[129,166],[15,19777,819],{"className":19778,"style":835},[129,614,166],[15,19780,175],{"className":19781},[174],[15,19783,19785],{"className":19784},[146],[15,19786,19788],{"className":19787,"style":182},[150],[15,19789],{},[15,19791,53],{"className":19792},[188],[15,19794,348],{"className":19795},[129,133],[15,19797,59],{"className":19798},[195],[15,19800],{"className":19801,"style":199},[93],[15,19803,62],{"className":19804},[203],[15,19806],{"className":19807,"style":199},[93],[15,19809,19811,19815,19894,19931,19934,19974,20011],{"className":19810},[120],[15,19812],{"className":19813,"style":19814},[124],"height:1.816em;vertical-align:-0.966em;",[15,19816,19818],{"className":19817},[217,1874],[15,19819,19821,19885],{"className":19820},[141,142],[15,19822,19824,19882],{"className":19823},[146],[15,19825,19827,19872],{"className":19826,"style":4383},[150],[15,19828,19830,19833],{"style":19829},"top:-2.309em;margin-left:0em;",[15,19831],{"className":19832,"style":1823},[158],[15,19834,19836],{"className":19835},[163,164,165,166],[15,19837,19839,19842,19845,19848,19851,19857,19860,19863,19866,19869],{"className":19838},[129,166],[15,19840,5619],{"className":19841,"style":5665},[129,133,166],[15,19843,5647],{"className":19844},[203,166],[15,19846,597],{"className":19847,"style":615},[129,614,166],[15,19849,91],{"className":19850},[254,166],[15,19852,19854],{"className":19853},[93,166],[15,19855,6854],{"className":19856},[166],[15,19858,2945],{"className":19859},[129,166],[15,19861,5619],{"className":19862,"style":5665},[129,133,166],[15,19864,2945],{"className":19865},[129,166],[15,19867,62],{"className":19868},[203,166],[15,19870,348],{"className":19871},[129,133,166],[15,19873,19874,19877],{"style":2287},[15,19875],{"className":19876,"style":1823},[158],[15,19878,19879],{},[15,19880,19696],{"className":19881},[217],[15,19883,175],{"className":19884},[174],[15,19886,19888],{"className":19887},[146],[15,19889,19892],{"className":19890,"style":19891},[150],"height:0.966em;",[15,19893],{},[15,19895,19897],{"className":19896},[188],[15,19898,19900],{"className":19899},[228,4433],[15,19901,19903,19923],{"className":19902},[141,142],[15,19904,19906,19920],{"className":19905},[146],[15,19907,19909],{"className":19908,"style":4443},[150],[15,19910,19911,19914],{"style":4446},[15,19912],{"className":19913,"style":4450},[158],[15,19915,19916],{"style":4453},[4455,19917,19918],{"xmlns":4457,"width":4458,"height":73,"viewBox":4459},[4461,19919],{"d":4463},[15,19921,175],{"className":19922},[174],[15,19924,19926],{"className":19925},[146],[15,19927,19929],{"className":19928,"style":4473},[150],[15,19930],{},[15,19932,819],{"className":19933,"style":835},[129,614],[15,19935,19937,19940],{"className":19936},[129],[15,19938,2945],{"className":19939},[129],[15,19941,19943],{"className":19942},[137],[15,19944,19946,19966],{"className":19945},[141,142],[15,19947,19949,19963],{"className":19948},[146],[15,19950,19952],{"className":19951,"style":1525},[150],[15,19953,19954,19957],{"style":154},[15,19955],{"className":19956,"style":159},[158],[15,19958,19960],{"className":19959},[163,164,165,166],[15,19961,5619],{"className":19962,"style":5665},[129,133,166],[15,19964,175],{"className":19965},[174],[15,19967,19969],{"className":19968},[146],[15,19970,19972],{"className":19971,"style":182},[150],[15,19973],{},[15,19975,19977],{"className":19976},[195],[15,19978,19980],{"className":19979},[228,4433],[15,19981,19983,20003],{"className":19982},[141,142],[15,19984,19986,20000],{"className":19985},[146],[15,19987,19989],{"className":19988,"style":4443},[150],[15,19990,19991,19994],{"style":4446},[15,19992],{"className":19993,"style":4450},[158],[15,19995,19996],{"style":4453},[4455,19997,19998],{"xmlns":4457,"width":4458,"height":73,"viewBox":4459},[4461,19999],{"d":4463},[15,20001,175],{"className":20002},[174],[15,20004,20006],{"className":20005},[146],[15,20007,20009],{"className":20008,"style":4473},[150],[15,20010],{},[15,20012,107],{"className":20013},[129],[11,20015,20016,20017,1345,20045,20048,20049,20189,20190,20264],{},"它称为 ",[15,20018,20020,20033],{"className":20019,"translate":19},[23],[15,20021,20023],{"className":20022},[27],[29,20024,20025],{"xmlns":31},[34,20026,20027,20031],{},[37,20028,20029],{},[43,20030,819],{"mathvariant":596},[109,20032,822],{"encoding":111},[15,20034,20036],{"className":20035,"ariaHidden":72},[116],[15,20037,20039,20042],{"className":20038},[120],[15,20040],{"className":20041,"style":610},[124],[15,20043,819],{"className":20044,"style":835},[129,614],[698,20046,20047],{},"增长函数","（growth function）。显然 ",[15,20050,20052,20084],{"className":20051,"translate":19},[23],[15,20053,20055],{"className":20054},[27],[29,20056,20057],{"xmlns":31},[34,20058,20059,20081],{},[37,20060,20061,20067,20069,20071,20073,20075],{},[40,20062,20063,20065],{},[43,20064,19679],{"mathvariant":65},[43,20066,819],{"mathvariant":596},[50,20068,53],{"stretchy":52},[43,20070,348],{},[50,20072,59],{"stretchy":52},[50,20074,2717],{},[305,20076,20077,20079],{},[308,20078,310],{},[43,20080,348],{},[109,20082,20083],{"encoding":111},"\\Pi_{\\mathcal H}(m)\\le 2^m",[15,20085,20087,20154],{"className":20086,"ariaHidden":72},[116],[15,20088,20090,20093,20136,20139,20142,20145,20148,20151],{"className":20089},[120],[15,20091],{"className":20092,"style":125},[124],[15,20094,20096,20099],{"className":20095},[129],[15,20097,19679],{"className":20098},[129],[15,20100,20102],{"className":20101},[137],[15,20103,20105,20128],{"className":20104},[141,142],[15,20106,20108,20125],{"className":20107},[146],[15,20109,20111],{"className":20110,"style":1525},[150],[15,20112,20113,20116],{"style":154},[15,20114],{"className":20115,"style":159},[158],[15,20117,20119],{"className":20118},[163,164,165,166],[15,20120,20122],{"className":20121},[129,166],[15,20123,819],{"className":20124,"style":835},[129,614,166],[15,20126,175],{"className":20127},[174],[15,20129,20131],{"className":20130},[146],[15,20132,20134],{"className":20133,"style":182},[150],[15,20135],{},[15,20137,53],{"className":20138},[188],[15,20140,348],{"className":20141},[129,133],[15,20143,59],{"className":20144},[195],[15,20146],{"className":20147,"style":199},[93],[15,20149,2717],{"className":20150},[203],[15,20152],{"className":20153,"style":199},[93],[15,20155,20157,20160],{"className":20156},[120],[15,20158],{"className":20159,"style":6377},[124],[15,20161,20163,20166],{"className":20162},[129],[15,20164,310],{"className":20165},[129],[15,20167,20169],{"className":20168},[137],[15,20170,20172],{"className":20171},[141],[15,20173,20175],{"className":20174},[146],[15,20176,20178],{"className":20177,"style":6377},[150],[15,20179,20180,20183],{"style":383},[15,20181],{"className":20182,"style":159},[158],[15,20184,20186],{"className":20185},[163,164,165,166],[15,20187,348],{"className":20188},[129,133,166],"，且 ",[15,20191,20193,20211],{"className":20192,"translate":19},[23],[15,20194,20196],{"className":20195},[27],[29,20197,20198],{"xmlns":31},[34,20199,20200,20208],{},[37,20201,20202],{},[40,20203,20204,20206],{},[43,20205,19679],{"mathvariant":65},[43,20207,819],{"mathvariant":596},[109,20209,20210],{"encoding":111},"\\Pi_{\\mathcal H}",[15,20212,20214],{"className":20213,"ariaHidden":72},[116],[15,20215,20217,20221],{"className":20216},[120],[15,20218],{"className":20219,"style":20220},[124],"height:0.8333em;vertical-align:-0.15em;",[15,20222,20224,20227],{"className":20223},[129],[15,20225,19679],{"className":20226},[129],[15,20228,20230],{"className":20229},[137],[15,20231,20233,20256],{"className":20232},[141,142],[15,20234,20236,20253],{"className":20235},[146],[15,20237,20239],{"className":20238,"style":1525},[150],[15,20240,20241,20244],{"style":154},[15,20242],{"className":20243,"style":159},[158],[15,20245,20247],{"className":20246},[163,164,165,166],[15,20248,20250],{"className":20249},[129,166],[15,20251,819],{"className":20252,"style":835},[129,614,166],[15,20254,175],{"className":20255},[174],[15,20257,20259],{"className":20258},[146],[15,20260,20262],{"className":20261,"style":182},[150],[15,20263],{}," 单调不减。",[11,20266,20267,20268,20317,20318,20458,20459,20487,20488,20516,20517,20545,20546,20574,20575,20603,20604,20700,20701,20752,20753,1313],{},"由定义，",[15,20269,20271,20293],{"className":20270,"translate":19},[23],[15,20272,20274],{"className":20273},[27],[29,20275,20276],{"xmlns":31},[34,20277,20278,20290],{},[37,20279,20280,20282,20284,20286,20288],{},[43,20281,6819],{"mathvariant":65},[50,20283,69],{},[50,20285,53],{"stretchy":52},[43,20287,819],{"mathvariant":596},[50,20289,59],{"stretchy":52},[109,20291,20292],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H)",[15,20294,20296],{"className":20295,"ariaHidden":72},[116],[15,20297,20299,20302,20308,20311,20314],{"className":20298},[120],[15,20300],{"className":20301,"style":125},[124],[15,20303,20305],{"className":20304},[217],[15,20306,6819],{"className":20307},[129,221],[15,20309,53],{"className":20310},[188],[15,20312,819],{"className":20313,"style":835},[129,614],[15,20315,59],{"className":20316},[195]," 就是使 ",[15,20319,20321,20353],{"className":20320,"translate":19},[23],[15,20322,20324],{"className":20323},[27],[29,20325,20326],{"xmlns":31},[34,20327,20328,20350],{},[37,20329,20330,20336,20338,20340,20342,20344],{},[40,20331,20332,20334],{},[43,20333,19679],{"mathvariant":65},[43,20335,819],{"mathvariant":596},[50,20337,53],{"stretchy":52},[43,20339,348],{},[50,20341,59],{"stretchy":52},[50,20343,62],{},[305,20345,20346,20348],{},[308,20347,310],{},[43,20349,348],{},[109,20351,20352],{"encoding":111},"\\Pi_{\\mathcal H}(m)=2^m",[15,20354,20356,20423],{"className":20355,"ariaHidden":72},[116],[15,20357,20359,20362,20405,20408,20411,20414,20417,20420],{"className":20358},[120],[15,20360],{"className":20361,"style":125},[124],[15,20363,20365,20368],{"className":20364},[129],[15,20366,19679],{"className":20367},[129],[15,20369,20371],{"className":20370},[137],[15,20372,20374,20397],{"className":20373},[141,142],[15,20375,20377,20394],{"className":20376},[146],[15,20378,20380],{"className":20379,"style":1525},[150],[15,20381,20382,20385],{"style":154},[15,20383],{"className":20384,"style":159},[158],[15,20386,20388],{"className":20387},[163,164,165,166],[15,20389,20391],{"className":20390},[129,166],[15,20392,819],{"className":20393,"style":835},[129,614,166],[15,20395,175],{"className":20396},[174],[15,20398,20400],{"className":20399},[146],[15,20401,20403],{"className":20402,"style":182},[150],[15,20404],{},[15,20406,53],{"className":20407},[188],[15,20409,348],{"className":20410},[129,133],[15,20412,59],{"className":20413},[195],[15,20415],{"className":20416,"style":199},[93],[15,20418,62],{"className":20419},[203],[15,20421],{"className":20422,"style":199},[93],[15,20424,20426,20429],{"className":20425},[120],[15,20427],{"className":20428,"style":6377},[124],[15,20430,20432,20435],{"className":20431},[129],[15,20433,310],{"className":20434},[129],[15,20436,20438],{"className":20437},[137],[15,20439,20441],{"className":20440},[141],[15,20442,20444],{"className":20443},[146],[15,20445,20447],{"className":20446,"style":6377},[150],[15,20448,20449,20452],{"style":383},[15,20450],{"className":20451,"style":159},[158],[15,20453,20455],{"className":20454},[163,164,165,166],[15,20456,348],{"className":20457},[129,133,166]," 成立的最大的 ",[15,20460,20462,20475],{"className":20461,"translate":19},[23],[15,20463,20465],{"className":20464},[27],[29,20466,20467],{"xmlns":31},[34,20468,20469,20473],{},[37,20470,20471],{},[43,20472,348],{},[109,20474,348],{"encoding":111},[15,20476,20478],{"className":20477,"ariaHidden":72},[116],[15,20479,20481,20484],{"className":20480},[120],[15,20482],{"className":20483,"style":4383},[124],[15,20485,348],{"className":20486},[129,133],"（若对所有 ",[15,20489,20491,20504],{"className":20490,"translate":19},[23],[15,20492,20494],{"className":20493},[27],[29,20495,20496],{"xmlns":31},[34,20497,20498,20502],{},[37,20499,20500],{},[43,20501,348],{},[109,20503,348],{"encoding":111},[15,20505,20507],{"className":20506,"ariaHidden":72},[116],[15,20508,20510,20513],{"className":20509},[120],[15,20511],{"className":20512,"style":4383},[124],[15,20514,348],{"className":20515},[129,133]," 都成立，则为 ",[15,20518,20520,20533],{"className":20519,"translate":19},[23],[15,20521,20523],{"className":20522},[27],[29,20524,20525],{"xmlns":31},[34,20526,20527,20531],{},[37,20528,20529],{},[43,20530,4909],{"mathvariant":65},[109,20532,18781],{"encoding":111},[15,20534,20536],{"className":20535,"ariaHidden":72},[116],[15,20537,20539,20542],{"className":20538},[120],[15,20540],{"className":20541,"style":4383},[124],[15,20543,4909],{"className":20544},[129],"）。注意这个性质是「向下封闭」的：若 ",[15,20547,20549,20562],{"className":20548,"translate":19},[23],[15,20550,20552],{"className":20551},[27],[29,20553,20554],{"xmlns":31},[34,20555,20556,20560],{},[37,20557,20558],{},[43,20559,5619],{},[109,20561,5619],{"encoding":111},[15,20563,20565],{"className":20564,"ariaHidden":72},[116],[15,20566,20568,20571],{"className":20567},[120],[15,20569],{"className":20570,"style":610},[124],[15,20572,5619],{"className":20573,"style":5665},[129,133]," 被打散，",[15,20576,20578,20591],{"className":20577,"translate":19},[23],[15,20579,20581],{"className":20580},[27],[29,20582,20583],{"xmlns":31},[34,20584,20585,20589],{},[37,20586,20587],{},[43,20588,5619],{},[109,20590,5619],{"encoding":111},[15,20592,20594],{"className":20593,"ariaHidden":72},[116],[15,20595,20597,20600],{"className":20596},[120],[15,20598],{"className":20599,"style":610},[124],[15,20601,5619],{"className":20602,"style":5665},[129,133]," 的任何子集也被打散，所以一旦 ",[15,20605,20607,20635],{"className":20606,"translate":19},[23],[15,20608,20610],{"className":20609},[27],[29,20611,20612],{"xmlns":31},[34,20613,20614,20632],{},[37,20615,20616,20618,20620,20622,20624,20626],{},[43,20617,19679],{"mathvariant":65},[50,20619,53],{"stretchy":52},[43,20621,348],{},[50,20623,59],{"stretchy":52},[50,20625,62],{},[305,20627,20628,20630],{},[308,20629,310],{},[43,20631,348],{},[109,20633,20634],{"encoding":111},"\\Pi(m)=2^m",[15,20636,20638,20665],{"className":20637,"ariaHidden":72},[116],[15,20639,20641,20644,20647,20650,20653,20656,20659,20662],{"className":20640},[120],[15,20642],{"className":20643,"style":125},[124],[15,20645,19679],{"className":20646},[129],[15,20648,53],{"className":20649},[188],[15,20651,348],{"className":20652},[129,133],[15,20654,59],{"className":20655},[195],[15,20657],{"className":20658,"style":199},[93],[15,20660,62],{"className":20661},[203],[15,20663],{"className":20664,"style":199},[93],[15,20666,20668,20671],{"className":20667},[120],[15,20669],{"className":20670,"style":6377},[124],[15,20672,20674,20677],{"className":20673},[129],[15,20675,310],{"className":20676},[129],[15,20678,20680],{"className":20679},[137],[15,20681,20683],{"className":20682},[141],[15,20684,20686],{"className":20685},[146],[15,20687,20689],{"className":20688,"style":6377},[150],[15,20690,20691,20694],{"style":383},[15,20692],{"className":20693,"style":159},[158],[15,20695,20697],{"className":20696},[163,164,165,166],[15,20698,348],{"className":20699},[129,133,166],"，对所有 ",[15,20702,20704,20722],{"className":20703,"translate":19},[23],[15,20705,20707],{"className":20706},[27],[29,20708,20709],{"xmlns":31},[34,20710,20711,20719],{},[37,20712,20713,20715,20717],{},[43,20714,16555],{},[50,20716,2717],{},[43,20718,348],{},[109,20720,20721],{"encoding":111},"k\\le m",[15,20723,20725,20743],{"className":20724,"ariaHidden":72},[116],[15,20726,20728,20731,20734,20737,20740],{"className":20727},[120],[15,20729],{"className":20730,"style":6962},[124],[15,20732,16555],{"className":20733,"style":16588},[129,133],[15,20735],{"className":20736,"style":199},[93],[15,20738,2717],{"className":20739},[203],[15,20741],{"className":20742,"style":199},[93],[15,20744,20746,20749],{"className":20745},[120],[15,20747],{"className":20748,"style":4383},[124],[15,20750,348],{"className":20751},[129,133]," 都有 ",[15,20754,20756,20784],{"className":20755,"translate":19},[23],[15,20757,20759],{"className":20758},[27],[29,20760,20761],{"xmlns":31},[34,20762,20763,20781],{},[37,20764,20765,20767,20769,20771,20773,20775],{},[43,20766,19679],{"mathvariant":65},[50,20768,53],{"stretchy":52},[43,20770,16555],{},[50,20772,59],{"stretchy":52},[50,20774,62],{},[305,20776,20777,20779],{},[308,20778,310],{},[43,20780,16555],{},[109,20782,20783],{"encoding":111},"\\Pi(k)=2^k",[15,20785,20787,20814],{"className":20786,"ariaHidden":72},[116],[15,20788,20790,20793,20796,20799,20802,20805,20808,20811],{"className":20789},[120],[15,20791],{"className":20792,"style":125},[124],[15,20794,19679],{"className":20795},[129],[15,20797,53],{"className":20798},[188],[15,20800,16555],{"className":20801,"style":16588},[129,133],[15,20803,59],{"className":20804},[195],[15,20806],{"className":20807,"style":199},[93],[15,20809,62],{"className":20810},[203],[15,20812],{"className":20813,"style":199},[93],[15,20815,20817,20820],{"className":20816},[120],[15,20818],{"className":20819,"style":10486},[124],[15,20821,20823,20826],{"className":20822},[129],[15,20824,310],{"className":20825},[129],[15,20827,20829],{"className":20828},[137],[15,20830,20832],{"className":20831},[141],[15,20833,20835],{"className":20834},[146],[15,20836,20838],{"className":20837,"style":10486},[150],[15,20839,20840,20843],{"style":383},[15,20841],{"className":20842,"style":159},[158],[15,20844,20846],{"className":20845},[163,164,165,166],[15,20847,16555],{"className":20848,"style":16588},[129,133,166],[575,20850,20852],{"id":20851},"_32-sauershelah-引理","3.2 Sauer–Shelah 引理",[6559,20854,20856,20980],{"title":20855},"定理 3.1（Sauer–Shelah）",[11,20857,6564,20858,20951,20952,7606],{},[15,20859,20861,20891],{"className":20860,"translate":19},[23],[15,20862,20864],{"className":20863},[27],[29,20865,20866],{"xmlns":31},[34,20867,20868,20888],{},[37,20869,20870,20872,20874,20876,20878,20880,20882,20884,20886],{},[43,20871,6819],{"mathvariant":65},[50,20873,69],{},[50,20875,53],{"stretchy":52},[43,20877,819],{"mathvariant":596},[50,20879,59],{"stretchy":52},[50,20881,62],{},[43,20883,10372],{},[50,20885,7906],{},[43,20887,4909],{"mathvariant":65},[109,20889,20890],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H)=d\u003C\\infty",[15,20892,20894,20924,20942],{"className":20893,"ariaHidden":72},[116],[15,20895,20897,20900,20906,20909,20912,20915,20918,20921],{"className":20896},[120],[15,20898],{"className":20899,"style":125},[124],[15,20901,20903],{"className":20902},[217],[15,20904,6819],{"className":20905},[129,221],[15,20907,53],{"className":20908},[188],[15,20910,819],{"className":20911,"style":835},[129,614],[15,20913,59],{"className":20914},[195],[15,20916],{"className":20917,"style":199},[93],[15,20919,62],{"className":20920},[203],[15,20922],{"className":20923,"style":199},[93],[15,20925,20927,20930,20933,20936,20939],{"className":20926},[120],[15,20928],{"className":20929,"style":2405},[124],[15,20931,10372],{"className":20932},[129,133],[15,20934],{"className":20935,"style":199},[93],[15,20937,7906],{"className":20938},[203],[15,20940],{"className":20941,"style":199},[93],[15,20943,20945,20948],{"className":20944},[120],[15,20946],{"className":20947,"style":4383},[124],[15,20949,4909],{"className":20950},[129],"，则对任意 ",[15,20953,20955,20968],{"className":20954,"translate":19},[23],[15,20956,20958],{"className":20957},[27],[29,20959,20960],{"xmlns":31},[34,20961,20962,20966],{},[37,20963,20964],{},[43,20965,348],{},[109,20967,348],{"encoding":111},[15,20969,20971],{"className":20970,"ariaHidden":72},[116],[15,20972,20974,20977],{"className":20973},[120],[15,20975],{"className":20976,"style":4383},[124],[15,20978,348],{"className":20979},[129,133],[15,20981,20983],{"className":20982,"translate":19},[18],[15,20984,20986,21041],{"className":20985,"translate":19},[23],[15,20987,20989],{"className":20988},[27],[29,20990,20991],{"xmlns":31,"display":32},[34,20992,20993,21038],{},[37,20994,20995,21001,21003,21005,21007,21009,21023,21036],{},[40,20996,20997,20999],{},[43,20998,19679],{"mathvariant":65},[43,21000,819],{"mathvariant":596},[50,21002,53],{"stretchy":52},[43,21004,348],{},[50,21006,59],{"stretchy":52},[50,21008,2717],{},[1450,21010,21011,21013,21021],{},[50,21012,1454],{},[37,21014,21015,21017,21019],{},[43,21016,1459],{},[50,21018,62],{},[308,21020,541],{},[43,21022,10372],{},[37,21024,21025,21027,21034],{},[50,21026,53],{"fence":72},[1443,21028,21030,21032],{"linethickness":21029},"0px",[43,21031,348],{},[43,21033,1459],{},[50,21035,59],{"fence":72},[43,21037,107],{"mathvariant":65},[109,21039,21040],{"encoding":111},"\\Pi_{\\mathcal H}(m)\\le\\sum_{i=0}^{d}\\binom{m}{i}.",[15,21042,21044,21111],{"className":21043,"ariaHidden":72},[116],[15,21045,21047,21050,21093,21096,21099,21102,21105,21108],{"className":21046},[120],[15,21048],{"className":21049,"style":125},[124],[15,21051,21053,21056],{"className":21052},[129],[15,21054,19679],{"className":21055},[129],[15,21057,21059],{"className":21058},[137],[15,21060,21062,21085],{"className":21061},[141,142],[15,21063,21065,21082],{"className":21064},[146],[15,21066,21068],{"className":21067,"style":1525},[150],[15,21069,21070,21073],{"style":154},[15,21071],{"className":21072,"style":159},[158],[15,21074,21076],{"className":21075},[163,164,165,166],[15,21077,21079],{"className":21078},[129,166],[15,21080,819],{"className":21081,"style":835},[129,614,166],[15,21083,175],{"className":21084},[174],[15,21086,21088],{"className":21087},[146],[15,21089,21091],{"className":21090,"style":182},[150],[15,21092],{},[15,21094,53],{"className":21095},[188],[15,21097,348],{"className":21098},[129,133],[15,21100,59],{"className":21101},[195],[15,21103],{"className":21104,"style":199},[93],[15,21106,2717],{"className":21107},[203],[15,21109],{"className":21110,"style":199},[93],[15,21112,21114,21118,21186,21189,21250],{"className":21113},[120],[15,21115],{"className":21116,"style":21117},[124],"height:3.1138em;vertical-align:-1.2777em;",[15,21119,21121],{"className":21120},[217,1874],[15,21122,21124,21178],{"className":21123},[141,142],[15,21125,21127,21175],{"className":21126},[146],[15,21128,21131,21151,21161],{"className":21129,"style":21130},[150],"height:1.8361em;",[15,21132,21133,21136],{"style":1887},[15,21134],{"className":21135,"style":1891},[158],[15,21137,21139],{"className":21138},[163,164,165,166],[15,21140,21142,21145,21148],{"className":21141},[129,166],[15,21143,1459],{"className":21144},[129,133,166],[15,21146,62],{"className":21147},[203,166],[15,21149,541],{"className":21150},[129,166],[15,21152,21153,21156],{"style":1909},[15,21154],{"className":21155,"style":1891},[158],[15,21157,21158],{},[15,21159,1454],{"className":21160},[217,1918,1919],[15,21162,21163,21166],{"style":1922},[15,21164],{"className":21165,"style":1891},[158],[15,21167,21169],{"className":21168},[163,164,165,166],[15,21170,21172],{"className":21171},[129,166],[15,21173,10372],{"className":21174},[129,133,166],[15,21176,175],{"className":21177},[174],[15,21179,21181],{"className":21180},[146],[15,21182,21184],{"className":21183,"style":1941},[150],[15,21185],{},[15,21187],{"className":21188,"style":262},[93],[15,21190,21192,21198,21244],{"className":21191},[129],[15,21193,21195],{"className":21194,"style":5052},[188,5051],[15,21196,53],{"className":21197},[228,165],[15,21199,21201],{"className":21200},[1443],[15,21202,21204,21236],{"className":21203},[141,142],[15,21205,21207,21233],{"className":21206},[146],[15,21208,21211,21222],{"className":21209,"style":21210},[150],"height:1.1076em;",[15,21212,21213,21216],{"style":1819},[15,21214],{"className":21215,"style":1823},[158],[15,21217,21219],{"className":21218},[129],[15,21220,1459],{"className":21221},[129,133],[15,21223,21224,21227],{"style":1843},[15,21225],{"className":21226,"style":1823},[158],[15,21228,21230],{"className":21229},[129],[15,21231,348],{"className":21232},[129,133],[15,21234,175],{"className":21235},[174],[15,21237,21239],{"className":21238},[146],[15,21240,21242],{"className":21241,"style":1862},[150],[15,21243],{},[15,21245,21247],{"className":21246,"style":5052},[195,5051],[15,21248,59],{"className":21249},[228,165],[15,21251,107],{"className":21252},[129],[11,21254,21255,21256,21307,21308,21480,21481,21641,21642,21717],{},"证明用归纳法，对 ",[15,21257,21259,21277],{"className":21258,"translate":19},[23],[15,21260,21262],{"className":21261},[27],[29,21263,21264],{"xmlns":31},[34,21265,21266,21274],{},[37,21267,21268,21270,21272],{},[43,21269,348],{},[50,21271,2682],{},[43,21273,10372],{},[109,21275,21276],{"encoding":111},"m+d",[15,21278,21280,21298],{"className":21279,"ariaHidden":72},[116],[15,21281,21283,21286,21289,21292,21295],{"className":21282},[120],[15,21284],{"className":21285,"style":8514},[124],[15,21287,348],{"className":21288},[129,133],[15,21290],{"className":21291,"style":3277},[93],[15,21293,2682],{"className":21294},[2436],[15,21296],{"className":21297,"style":3277},[93],[15,21299,21301,21304],{"className":21300},[120],[15,21302],{"className":21303,"style":522},[124],[15,21305,10372],{"className":21306},[129,133]," 归纳。取任意 ",[15,21309,21311,21349],{"className":21310,"translate":19},[23],[15,21312,21314],{"className":21313},[27],[29,21315,21316],{"xmlns":31},[34,21317,21318,21346],{},[37,21319,21320,21322,21324,21326,21332,21334,21336,21338,21344],{},[43,21321,5619],{},[50,21323,62],{},[50,21325,636],{"stretchy":52},[40,21327,21328,21330],{},[43,21329,5628],{},[308,21331,318],{},[50,21333,91],{"separator":72},[50,21335,335],{},[50,21337,91],{"separator":72},[40,21339,21340,21342],{},[43,21341,5628],{},[43,21343,348],{},[50,21345,645],{"stretchy":52},[109,21347,21348],{"encoding":111},"C=\\{c_1,\\dots,c_m\\}",[15,21350,21352,21370],{"className":21351,"ariaHidden":72},[116],[15,21353,21355,21358,21361,21364,21367],{"className":21354},[120],[15,21356],{"className":21357,"style":610},[124],[15,21359,5619],{"className":21360,"style":5665},[129,133],[15,21362],{"className":21363,"style":199},[93],[15,21365,62],{"className":21366},[203],[15,21368],{"className":21369,"style":199},[93],[15,21371,21373,21376,21379,21419,21422,21425,21428,21431,21434,21437,21477],{"className":21372},[120],[15,21374],{"className":21375,"style":125},[124],[15,21377,636],{"className":21378},[188],[15,21380,21382,21385],{"className":21381},[129],[15,21383,5628],{"className":21384},[129,133],[15,21386,21388],{"className":21387},[137],[15,21389,21391,21411],{"className":21390},[141,142],[15,21392,21394,21408],{"className":21393},[146],[15,21395,21397],{"className":21396,"style":1051},[150],[15,21398,21399,21402],{"style":154},[15,21400],{"className":21401,"style":159},[158],[15,21403,21405],{"className":21404},[163,164,165,166],[15,21406,318],{"className":21407},[129,166],[15,21409,175],{"className":21410},[174],[15,21412,21414],{"className":21413},[146],[15,21415,21417],{"className":21416,"style":182},[150],[15,21418],{},[15,21420,91],{"className":21421},[254],[15,21423],{"className":21424,"style":262},[93],[15,21426,335],{"className":21427},[449],[15,21429],{"className":21430,"style":262},[93],[15,21432,91],{"className":21433},[254],[15,21435],{"className":21436,"style":262},[93],[15,21438,21440,21443],{"className":21439},[129],[15,21441,5628],{"className":21442},[129,133],[15,21444,21446],{"className":21445},[137],[15,21447,21449,21469],{"className":21448},[141,142],[15,21450,21452,21466],{"className":21451},[146],[15,21453,21455],{"className":21454,"style":1163},[150],[15,21456,21457,21460],{"style":154},[15,21458],{"className":21459,"style":159},[158],[15,21461,21463],{"className":21462},[163,164,165,166],[15,21464,348],{"className":21465},[129,133,166],[15,21467,175],{"className":21468},[174],[15,21470,21472],{"className":21471},[146],[15,21473,21475],{"className":21474,"style":182},[150],[15,21476],{},[15,21478,645],{"className":21479},[195],"，记 ",[15,21482,21484,21520],{"className":21483,"translate":19},[23],[15,21485,21487],{"className":21486},[27],[29,21488,21489],{"xmlns":31},[34,21490,21491,21517],{},[37,21492,21493,21500,21502,21504,21507,21509,21515],{},[305,21494,21495,21497],{},[43,21496,5619],{},[50,21498,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},"′",[50,21501,62],{},[43,21503,5619],{},[50,21505,21506],{},"∖",[50,21508,636],{"stretchy":52},[40,21510,21511,21513],{},[43,21512,5628],{},[43,21514,348],{},[50,21516,645],{"stretchy":52},[109,21518,21519],{"encoding":111},"C'=C\\setminus\\{c_m\\}",[15,21521,21523,21571,21589],{"className":21522,"ariaHidden":72},[116],[15,21524,21526,21530,21562,21565,21568],{"className":21525},[120],[15,21527],{"className":21528,"style":21529},[124],"height:0.7519em;",[15,21531,21533,21536],{"className":21532},[129],[15,21534,5619],{"className":21535,"style":5665},[129,133],[15,21537,21539],{"className":21538},[137],[15,21540,21542],{"className":21541},[141],[15,21543,21545],{"className":21544},[146],[15,21546,21548],{"className":21547,"style":21529},[150],[15,21549,21550,21553],{"style":383},[15,21551],{"className":21552,"style":159},[158],[15,21554,21556],{"className":21555},[163,164,165,166],[15,21557,21559],{"className":21558},[129,166],[15,21560,21499],{"className":21561},[129,166],[15,21563],{"className":21564,"style":199},[93],[15,21566,62],{"className":21567},[203],[15,21569],{"className":21570,"style":199},[93],[15,21572,21574,21577,21580,21583,21586],{"className":21573},[120],[15,21575],{"className":21576,"style":125},[124],[15,21578,5619],{"className":21579,"style":5665},[129,133],[15,21581],{"className":21582,"style":3277},[93],[15,21584,21506],{"className":21585},[2436],[15,21587],{"className":21588,"style":3277},[93],[15,21590,21592,21595,21598,21638],{"className":21591},[120],[15,21593],{"className":21594,"style":125},[124],[15,21596,636],{"className":21597},[188],[15,21599,21601,21604],{"className":21600},[129],[15,21602,5628],{"className":21603},[129,133],[15,21605,21607],{"className":21606},[137],[15,21608,21610,21630],{"className":21609},[141,142],[15,21611,21613,21627],{"className":21612},[146],[15,21614,21616],{"className":21615,"style":1163},[150],[15,21617,21618,21621],{"style":154},[15,21619],{"className":21620,"style":159},[158],[15,21622,21624],{"className":21623},[163,164,165,166],[15,21625,348],{"className":21626},[129,133,166],[15,21628,175],{"className":21629},[174],[15,21631,21633],{"className":21632},[146],[15,21634,21636],{"className":21635,"style":182},[150],[15,21637],{},[15,21639,645],{"className":21640},[195],"。把 ",[15,21643,21645,21665],{"className":21644,"translate":19},[23],[15,21646,21648],{"className":21647},[27],[29,21649,21650],{"xmlns":31},[34,21651,21652,21662],{},[37,21653,21654,21656],{},[43,21655,819],{"mathvariant":596},[40,21657,21658,21660],{},[43,21659,2945],{"mathvariant":65},[43,21661,5619],{},[109,21663,21664],{"encoding":111},"\\mathcal H|_C",[15,21666,21668],{"className":21667,"ariaHidden":72},[116],[15,21669,21671,21674,21677],{"className":21670},[120],[15,21672],{"className":21673,"style":125},[124],[15,21675,819],{"className":21676,"style":835},[129,614],[15,21678,21680,21683],{"className":21679},[129],[15,21681,2945],{"className":21682},[129],[15,21684,21686],{"className":21685},[137],[15,21687,21689,21709],{"className":21688},[141,142],[15,21690,21692,21706],{"className":21691},[146],[15,21693,21695],{"className":21694,"style":1525},[150],[15,21696,21697,21700],{"style":154},[15,21698],{"className":21699,"style":159},[158],[15,21701,21703],{"className":21702},[163,164,165,166],[15,21704,5619],{"className":21705,"style":5665},[129,133,166],[15,21707,175],{"className":21708},[174],[15,21710,21712],{"className":21711},[146],[15,21713,21715],{"className":21714,"style":182},[150],[15,21716],{}," 按最后一个坐标分成两类：",[21719,21720,21721,21824],"ul",{},[21722,21723,21724,21794,21795,21823],"li",{},[15,21725,21727,21745],{"className":21726,"translate":19},[23],[15,21728,21730],{"className":21729},[27],[29,21731,21732],{"xmlns":31},[34,21733,21734,21742],{},[37,21735,21736],{},[40,21737,21738,21740],{},[43,21739,13091],{},[308,21741,541],{},[109,21743,21744],{"encoding":111},"A_0",[15,21746,21748],{"className":21747,"ariaHidden":72},[116],[15,21749,21751,21754],{"className":21750},[120],[15,21752],{"className":21753,"style":20220},[124],[15,21755,21757,21760],{"className":21756},[129],[15,21758,13091],{"className":21759},[129,133],[15,21761,21763],{"className":21762},[137],[15,21764,21766,21786],{"className":21765},[141,142],[15,21767,21769,21783],{"className":21768},[146],[15,21770,21772],{"className":21771,"style":1051},[150],[15,21773,21774,21777],{"style":154},[15,21775],{"className":21776,"style":159},[158],[15,21778,21780],{"className":21779},[163,164,165,166],[15,21781,541],{"className":21782},[129,166],[15,21784,175],{"className":21785},[174],[15,21787,21789],{"className":21788},[146],[15,21790,21792],{"className":21791,"style":182},[150],[15,21793],{},"：最后一个坐标为 ",[15,21796,21798,21811],{"className":21797,"translate":19},[23],[15,21799,21801],{"className":21800},[27],[29,21802,21803],{"xmlns":31},[34,21804,21805,21809],{},[37,21806,21807],{},[308,21808,541],{},[109,21810,541],{"encoding":111},[15,21812,21814],{"className":21813,"ariaHidden":72},[116],[15,21815,21817,21820],{"className":21816},[120],[15,21818],{"className":21819,"style":4886},[124],[15,21821,541],{"className":21822},[129]," 的二分；",[21722,21825,21826,21794,21896,21924],{},[15,21827,21829,21847],{"className":21828,"translate":19},[23],[15,21830,21832],{"className":21831},[27],[29,21833,21834],{"xmlns":31},[34,21835,21836,21844],{},[37,21837,21838],{},[40,21839,21840,21842],{},[43,21841,13091],{},[308,21843,318],{},[109,21845,21846],{"encoding":111},"A_1",[15,21848,21850],{"className":21849,"ariaHidden":72},[116],[15,21851,21853,21856],{"className":21852},[120],[15,21854],{"className":21855,"style":20220},[124],[15,21857,21859,21862],{"className":21858},[129],[15,21860,13091],{"className":21861},[129,133],[15,21863,21865],{"className":21864},[137],[15,21866,21868,21888],{"className":21867},[141,142],[15,21869,21871,21885],{"className":21870},[146],[15,21872,21874],{"className":21873,"style":1051},[150],[15,21875,21876,21879],{"style":154},[15,21877],{"className":21878,"style":159},[158],[15,21880,21882],{"className":21881},[163,164,165,166],[15,21883,318],{"className":21884},[129,166],[15,21886,175],{"className":21887},[174],[15,21889,21891],{"className":21890},[146],[15,21892,21894],{"className":21893,"style":182},[150],[15,21895],{},[15,21897,21899,21912],{"className":21898,"translate":19},[23],[15,21900,21902],{"className":21901},[27],[29,21903,21904],{"xmlns":31},[34,21905,21906,21910],{},[37,21907,21908],{},[308,21909,318],{},[109,21911,318],{"encoding":111},[15,21913,21915],{"className":21914,"ariaHidden":72},[116],[15,21916,21918,21921],{"className":21917},[120],[15,21919],{"className":21920,"style":4886},[124],[15,21922,318],{"className":21923},[129]," 的二分。",[11,21926,21927,21928,10175,21997,5831,22066,22128,22129,22158],{},"设 ",[15,21929,21931,21948],{"className":21930,"translate":19},[23],[15,21932,21934],{"className":21933},[27],[29,21935,21936],{"xmlns":31},[34,21937,21938,21946],{},[37,21939,21940],{},[40,21941,21942,21944],{},[43,21943,13091],{},[308,21945,541],{},[109,21947,21744],{"encoding":111},[15,21949,21951],{"className":21950,"ariaHidden":72},[116],[15,21952,21954,21957],{"className":21953},[120],[15,21955],{"className":21956,"style":20220},[124],[15,21958,21960,21963],{"className":21959},[129],[15,21961,13091],{"className":21962},[129,133],[15,21964,21966],{"className":21965},[137],[15,21967,21969,21989],{"className":21968},[141,142],[15,21970,21972,21986],{"className":21971},[146],[15,21973,21975],{"className":21974,"style":1051},[150],[15,21976,21977,21980],{"style":154},[15,21978],{"className":21979,"style":159},[158],[15,21981,21983],{"className":21982},[163,164,165,166],[15,21984,541],{"className":21985},[129,166],[15,21987,175],{"className":21988},[174],[15,21990,21992],{"className":21991},[146],[15,21993,21995],{"className":21994,"style":182},[150],[15,21996],{},[15,21998,22000,22017],{"className":21999,"translate":19},[23],[15,22001,22003],{"className":22002},[27],[29,22004,22005],{"xmlns":31},[34,22006,22007,22015],{},[37,22008,22009],{},[40,22010,22011,22013],{},[43,22012,13091],{},[308,22014,318],{},[109,22016,21846],{"encoding":111},[15,22018,22020],{"className":22019,"ariaHidden":72},[116],[15,22021,22023,22026],{"className":22022},[120],[15,22024],{"className":22025,"style":20220},[124],[15,22027,22029,22032],{"className":22028},[129],[15,22030,13091],{"className":22031},[129,133],[15,22033,22035],{"className":22034},[137],[15,22036,22038,22058],{"className":22037},[141,142],[15,22039,22041,22055],{"className":22040},[146],[15,22042,22044],{"className":22043,"style":1051},[150],[15,22045,22046,22049],{"style":154},[15,22047],{"className":22048,"style":159},[158],[15,22050,22052],{"className":22051},[163,164,165,166],[15,22053,318],{"className":22054},[129,166],[15,22056,175],{"className":22057},[174],[15,22059,22061],{"className":22060},[146],[15,22062,22064],{"className":22063,"style":182},[150],[15,22065],{},[15,22067,22069,22087],{"className":22068,"translate":19},[23],[15,22070,22072],{"className":22071},[27],[29,22073,22074],{"xmlns":31},[34,22075,22076,22084],{},[37,22077,22078],{},[305,22079,22080,22082],{},[43,22081,5619],{},[50,22083,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[109,22085,22086],{"encoding":111},"C'",[15,22088,22090],{"className":22089,"ariaHidden":72},[116],[15,22091,22093,22096],{"className":22092},[120],[15,22094],{"className":22095,"style":21529},[124],[15,22097,22099,22102],{"className":22098},[129],[15,22100,5619],{"className":22101,"style":5665},[129,133],[15,22103,22105],{"className":22104},[137],[15,22106,22108],{"className":22107},[141],[15,22109,22111],{"className":22110},[146],[15,22112,22114],{"className":22113,"style":21529},[150],[15,22115,22116,22119],{"style":383},[15,22117],{"className":22118,"style":159},[158],[15,22120,22122],{"className":22121},[163,164,165,166],[15,22123,22125],{"className":22124},[129,166],[15,22126,21499],{"className":22127},[129,166]," 上的投影的交集大小为 ",[15,22130,22132,22146],{"className":22131,"translate":19},[23],[15,22133,22135],{"className":22134},[27],[29,22136,22137],{"xmlns":31},[34,22138,22139,22144],{},[37,22140,22141],{},[43,22142,22143],{},"s",[109,22145,22143],{"encoding":111},[15,22147,22149],{"className":22148,"ariaHidden":72},[116],[15,22150,22152,22155],{"className":22151},[120],[15,22153],{"className":22154,"style":4383},[124],[15,22156,22143],{"className":22157},[129,133],"，则",[15,22160,22162],{"className":22161,"translate":19},[18],[15,22163,22165,22237],{"className":22164,"translate":19},[23],[15,22166,22168],{"className":22167},[27],[29,22169,22170],{"xmlns":31,"display":32},[34,22171,22172,22234],{},[37,22173,22174,22176,22178,22184,22186,22188,22190,22196,22198,22200,22202,22208,22210,22212,22218,22220,22222,22224,22226,22228,22230,22232],{},[43,22175,2945],{"mathvariant":65},[43,22177,819],{"mathvariant":596},[40,22179,22180,22182],{},[43,22181,2945],{"mathvariant":65},[43,22183,5619],{},[43,22185,2945],{"mathvariant":65},[50,22187,62],{},[43,22189,2945],{"mathvariant":65},[40,22191,22192,22194],{},[43,22193,13091],{},[308,22195,541],{},[43,22197,2945],{"mathvariant":65},[50,22199,2682],{},[43,22201,2945],{"mathvariant":65},[40,22203,22204,22206],{},[43,22205,13091],{},[308,22207,318],{},[43,22209,2945],{"mathvariant":65},[50,22211,2717],{},[40,22213,22214,22216],{},[43,22215,19679],{"mathvariant":65},[43,22217,819],{"mathvariant":596},[50,22219,53],{"stretchy":52},[43,22221,348],{},[50,22223,315],{},[308,22225,318],{},[50,22227,59],{"stretchy":52},[50,22229,2682],{},[43,22231,22143],{},[43,22233,107],{"mathvariant":65},[109,22235,22236],{"encoding":111},"|\\mathcal H|_C|=|A_0|+|A_1|\\le \\Pi_{\\mathcal H}(m-1)+s.",[15,22238,22240,22304,22365,22426,22490,22511],{"className":22239,"ariaHidden":72},[116],[15,22241,22243,22246,22249,22252,22292,22295,22298,22301],{"className":22242},[120],[15,22244],{"className":22245,"style":125},[124],[15,22247,2945],{"className":22248},[129],[15,22250,819],{"className":22251,"style":835},[129,614],[15,22253,22255,22258],{"className":22254},[129],[15,22256,2945],{"className":22257},[129],[15,22259,22261],{"className":22260},[137],[15,22262,22264,22284],{"className":22263},[141,142],[15,22265,22267,22281],{"className":22266},[146],[15,22268,22270],{"className":22269,"style":1525},[150],[15,22271,22272,22275],{"style":154},[15,22273],{"className":22274,"style":159},[158],[15,22276,22278],{"className":22277},[163,164,165,166],[15,22279,5619],{"className":22280,"style":5665},[129,133,166],[15,22282,175],{"className":22283},[174],[15,22285,22287],{"className":22286},[146],[15,22288,22290],{"className":22289,"style":182},[150],[15,22291],{},[15,22293,2945],{"className":22294},[129],[15,22296],{"className":22297,"style":199},[93],[15,22299,62],{"className":22300},[203],[15,22302],{"className":22303,"style":199},[93],[15,22305,22307,22310,22313,22353,22356,22359,22362],{"className":22306},[120],[15,22308],{"className":22309,"style":125},[124],[15,22311,2945],{"className":22312},[129],[15,22314,22316,22319],{"className":22315},[129],[15,22317,13091],{"className":22318},[129,133],[15,22320,22322],{"className":22321},[137],[15,22323,22325,22345],{"className":22324},[141,142],[15,22326,22328,22342],{"className":22327},[146],[15,22329,22331],{"className":22330,"style":1051},[150],[15,22332,22333,22336],{"style":154},[15,22334],{"className":22335,"style":159},[158],[15,22337,22339],{"className":22338},[163,164,165,166],[15,22340,541],{"className":22341},[129,166],[15,22343,175],{"className":22344},[174],[15,22346,22348],{"className":22347},[146],[15,22349,22351],{"className":22350,"style":182},[150],[15,22352],{},[15,22354,2945],{"className":22355},[129],[15,22357],{"className":22358,"style":3277},[93],[15,22360,2682],{"className":22361},[2436],[15,22363],{"className":22364,"style":3277},[93],[15,22366,22368,22371,22374,22414,22417,22420,22423],{"className":22367},[120],[15,22369],{"className":22370,"style":125},[124],[15,22372,2945],{"className":22373},[129],[15,22375,22377,22380],{"className":22376},[129],[15,22378,13091],{"className":22379},[129,133],[15,22381,22383],{"className":22382},[137],[15,22384,22386,22406],{"className":22385},[141,142],[15,22387,22389,22403],{"className":22388},[146],[15,22390,22392],{"className":22391,"style":1051},[150],[15,22393,22394,22397],{"style":154},[15,22395],{"className":22396,"style":159},[158],[15,22398,22400],{"className":22399},[163,164,165,166],[15,22401,318],{"className":22402},[129,166],[15,22404,175],{"className":22405},[174],[15,22407,22409],{"className":22408},[146],[15,22410,22412],{"className":22411,"style":182},[150],[15,22413],{},[15,22415,2945],{"className":22416},[129],[15,22418],{"className":22419,"style":199},[93],[15,22421,2717],{"className":22422},[203],[15,22424],{"className":22425,"style":199},[93],[15,22427,22429,22432,22475,22478,22481,22484,22487],{"className":22428},[120],[15,22430],{"className":22431,"style":125},[124],[15,22433,22435,22438],{"className":22434},[129],[15,22436,19679],{"className":22437},[129],[15,22439,22441],{"className":22440},[137],[15,22442,22444,22467],{"className":22443},[141,142],[15,22445,22447,22464],{"className":22446},[146],[15,22448,22450],{"className":22449,"style":1525},[150],[15,22451,22452,22455],{"style":154},[15,22453],{"className":22454,"style":159},[158],[15,22456,22458],{"className":22457},[163,164,165,166],[15,22459,22461],{"className":22460},[129,166],[15,22462,819],{"className":22463,"style":835},[129,614,166],[15,22465,175],{"className":22466},[174],[15,22468,22470],{"className":22469},[146],[15,22471,22473],{"className":22472,"style":182},[150],[15,22474],{},[15,22476,53],{"className":22477},[188],[15,22479,348],{"className":22480},[129,133],[15,22482],{"className":22483,"style":3277},[93],[15,22485,315],{"className":22486},[2436],[15,22488],{"className":22489,"style":3277},[93],[15,22491,22493,22496,22499,22502,22505,22508],{"className":22492},[120],[15,22494],{"className":22495,"style":125},[124],[15,22497,318],{"className":22498},[129],[15,22500,59],{"className":22501},[195],[15,22503],{"className":22504,"style":3277},[93],[15,22506,2682],{"className":22507},[2436],[15,22509],{"className":22510,"style":3277},[93],[15,22512,22514,22517,22520],{"className":22513},[120],[15,22515],{"className":22516,"style":4383},[124],[15,22518,22143],{"className":22519},[129,133],[15,22521,107],{"className":22522},[129],[11,22524,22525,22526,22587,22588,22655,22656,22721,22722,22783,22784,22812,22813,22841,22842,22922,22923,22973,22974,23105,23106,1313],{},"交集里的每个二分，都对应 ",[15,22527,22529,22546],{"className":22528,"translate":19},[23],[15,22530,22532],{"className":22531},[27],[29,22533,22534],{"xmlns":31},[34,22535,22536,22544],{},[37,22537,22538],{},[305,22539,22540,22542],{},[43,22541,5619],{},[50,22543,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[109,22545,22086],{"encoding":111},[15,22547,22549],{"className":22548,"ariaHidden":72},[116],[15,22550,22552,22555],{"className":22551},[120],[15,22553],{"className":22554,"style":21529},[124],[15,22556,22558,22561],{"className":22557},[129],[15,22559,5619],{"className":22560,"style":5665},[129,133],[15,22562,22564],{"className":22563},[137],[15,22565,22567],{"className":22566},[141],[15,22568,22570],{"className":22569},[146],[15,22571,22573],{"className":22572,"style":21529},[150],[15,22574,22575,22578],{"style":383},[15,22576],{"className":22577,"style":159},[158],[15,22579,22581],{"className":22580},[163,164,165,166],[15,22582,22584],{"className":22583},[129,166],[15,22585,21499],{"className":22586},[129,166]," 上一种「最后一位可以任意翻转」的模式。把这些模式组成的集合记为 ",[15,22589,22591,22613],{"className":22590,"translate":19},[23],[15,22592,22594],{"className":22593},[27],[29,22595,22596],{"xmlns":31},[34,22597,22598,22610],{},[37,22599,22600],{},[305,22601,22602,22604],{},[43,22603,819],{"mathvariant":596},[37,22605,22606,22608],{},[50,22607,21499],{"mathvariant":65},[50,22609,21499],{"mathvariant":65},[109,22611,22612],{"encoding":111},"\\mathcal H''",[15,22614,22616],{"className":22615,"ariaHidden":72},[116],[15,22617,22619,22622],{"className":22618},[120],[15,22620],{"className":22621,"style":21529},[124],[15,22623,22625,22628],{"className":22624},[129],[15,22626,819],{"className":22627,"style":835},[129,614],[15,22629,22631],{"className":22630},[137],[15,22632,22634],{"className":22633},[141],[15,22635,22637],{"className":22636},[146],[15,22638,22640],{"className":22639,"style":21529},[150],[15,22641,22642,22645],{"style":383},[15,22643],{"className":22644,"style":159},[158],[15,22646,22648],{"className":22647},[163,164,165,166],[15,22649,22651],{"className":22650},[129,166],[15,22652,22654],{"className":22653},[129,166],"′′","。若 ",[15,22657,22659,22680],{"className":22658,"translate":19},[23],[15,22660,22662],{"className":22661},[27],[29,22663,22664],{"xmlns":31},[34,22665,22666,22678],{},[37,22667,22668],{},[305,22669,22670,22672],{},[43,22671,819],{"mathvariant":596},[37,22673,22674,22676],{},[50,22675,21499],{"mathvariant":65},[50,22677,21499],{"mathvariant":65},[109,22679,22612],{"encoding":111},[15,22681,22683],{"className":22682,"ariaHidden":72},[116],[15,22684,22686,22689],{"className":22685},[120],[15,22687],{"className":22688,"style":21529},[124],[15,22690,22692,22695],{"className":22691},[129],[15,22693,819],{"className":22694,"style":835},[129,614],[15,22696,22698],{"className":22697},[137],[15,22699,22701],{"className":22700},[141],[15,22702,22704],{"className":22703},[146],[15,22705,22707],{"className":22706,"style":21529},[150],[15,22708,22709,22712],{"style":383},[15,22710],{"className":22711,"style":159},[158],[15,22713,22715],{"className":22714},[163,164,165,166],[15,22716,22718],{"className":22717},[129,166],[15,22719,22654],{"className":22720},[129,166]," 打散了 ",[15,22723,22725,22742],{"className":22724,"translate":19},[23],[15,22726,22728],{"className":22727},[27],[29,22729,22730],{"xmlns":31},[34,22731,22732,22740],{},[37,22733,22734],{},[305,22735,22736,22738],{},[43,22737,5619],{},[50,22739,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[109,22741,22086],{"encoding":111},[15,22743,22745],{"className":22744,"ariaHidden":72},[116],[15,22746,22748,22751],{"className":22747},[120],[15,22749],{"className":22750,"style":21529},[124],[15,22752,22754,22757],{"className":22753},[129],[15,22755,5619],{"className":22756,"style":5665},[129,133],[15,22758,22760],{"className":22759},[137],[15,22761,22763],{"className":22762},[141],[15,22764,22766],{"className":22765},[146],[15,22767,22769],{"className":22768,"style":21529},[150],[15,22770,22771,22774],{"style":383},[15,22772],{"className":22773,"style":159},[158],[15,22775,22777],{"className":22776},[163,164,165,166],[15,22778,22780],{"className":22779},[129,166],[15,22781,21499],{"className":22782},[129,166]," 的某个 ",[15,22785,22787,22800],{"className":22786,"translate":19},[23],[15,22788,22790],{"className":22789},[27],[29,22791,22792],{"xmlns":31},[34,22793,22794,22798],{},[37,22795,22796],{},[43,22797,10372],{},[109,22799,10372],{"encoding":111},[15,22801,22803],{"className":22802,"ariaHidden":72},[116],[15,22804,22806,22809],{"className":22805},[120],[15,22807],{"className":22808,"style":522},[124],[15,22810,10372],{"className":22811},[129,133]," 元子集，那么 ",[15,22814,22816,22829],{"className":22815,"translate":19},[23],[15,22817,22819],{"className":22818},[27],[29,22820,22821],{"xmlns":31},[34,22822,22823,22827],{},[37,22824,22825],{},[43,22826,819],{"mathvariant":596},[109,22828,822],{"encoding":111},[15,22830,22832],{"className":22831,"ariaHidden":72},[116],[15,22833,22835,22838],{"className":22834},[120],[15,22836],{"className":22837,"style":610},[124],[15,22839,819],{"className":22840,"style":835},[129,614]," 就打散了该子集并上 ",[15,22843,22845,22867],{"className":22844,"translate":19},[23],[15,22846,22848],{"className":22847},[27],[29,22849,22850],{"xmlns":31},[34,22851,22852,22864],{},[37,22853,22854,22856,22862],{},[50,22855,636],{"stretchy":52},[40,22857,22858,22860],{},[43,22859,5628],{},[43,22861,348],{},[50,22863,645],{"stretchy":52},[109,22865,22866],{"encoding":111},"\\{c_m\\}",[15,22868,22870],{"className":22869,"ariaHidden":72},[116],[15,22871,22873,22876,22879,22919],{"className":22872},[120],[15,22874],{"className":22875,"style":125},[124],[15,22877,636],{"className":22878},[188],[15,22880,22882,22885],{"className":22881},[129],[15,22883,5628],{"className":22884},[129,133],[15,22886,22888],{"className":22887},[137],[15,22889,22891,22911],{"className":22890},[141,142],[15,22892,22894,22908],{"className":22893},[146],[15,22895,22897],{"className":22896,"style":1163},[150],[15,22898,22899,22902],{"style":154},[15,22900],{"className":22901,"style":159},[158],[15,22903,22905],{"className":22904},[163,164,165,166],[15,22906,348],{"className":22907},[129,133,166],[15,22909,175],{"className":22910},[174],[15,22912,22914],{"className":22913},[146],[15,22915,22917],{"className":22916,"style":182},[150],[15,22918],{},[15,22920,645],{"className":22921},[195],"，共 ",[15,22924,22926,22943],{"className":22925,"translate":19},[23],[15,22927,22929],{"className":22928},[27],[29,22930,22931],{"xmlns":31},[34,22932,22933,22941],{},[37,22934,22935,22937,22939],{},[43,22936,10372],{},[50,22938,2682],{},[308,22940,318],{},[109,22942,10876],{"encoding":111},[15,22944,22946,22964],{"className":22945,"ariaHidden":72},[116],[15,22947,22949,22952,22955,22958,22961],{"className":22948},[120],[15,22950],{"className":22951,"style":10410},[124],[15,22953,10372],{"className":22954},[129,133],[15,22956],{"className":22957,"style":3277},[93],[15,22959,2682],{"className":22960},[2436],[15,22962],{"className":22963,"style":3277},[93],[15,22965,22967,22970],{"className":22966},[120],[15,22968],{"className":22969,"style":4886},[124],[15,22971,318],{"className":22972},[129]," 个点，矛盾。所以 ",[15,22975,22977,23015],{"className":22976,"translate":19},[23],[15,22978,22980],{"className":22979},[27],[29,22981,22982],{"xmlns":31},[34,22983,22984,23012],{},[37,22985,22986,22988,22990,22992,23002,23004,23006,23008,23010],{},[43,22987,6819],{"mathvariant":65},[50,22989,69],{},[50,22991,53],{"stretchy":52},[305,22993,22994,22996],{},[43,22995,819],{"mathvariant":596},[37,22997,22998,23000],{},[50,22999,21499],{"mathvariant":65},[50,23001,21499],{"mathvariant":65},[50,23003,59],{"stretchy":52},[50,23005,2717],{},[43,23007,10372],{},[50,23009,315],{},[308,23011,318],{},[109,23013,23014],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H'')\\le d-1",[15,23016,23018,23078,23096],{"className":23017,"ariaHidden":72},[116],[15,23019,23021,23025,23031,23034,23066,23069,23072,23075],{"className":23020},[120],[15,23022],{"className":23023,"style":23024},[124],"height:1.0019em;vertical-align:-0.25em;",[15,23026,23028],{"className":23027},[217],[15,23029,6819],{"className":23030},[129,221],[15,23032,53],{"className":23033},[188],[15,23035,23037,23040],{"className":23036},[129],[15,23038,819],{"className":23039,"style":835},[129,614],[15,23041,23043],{"className":23042},[137],[15,23044,23046],{"className":23045},[141],[15,23047,23049],{"className":23048},[146],[15,23050,23052],{"className":23051,"style":21529},[150],[15,23053,23054,23057],{"style":383},[15,23055],{"className":23056,"style":159},[158],[15,23058,23060],{"className":23059},[163,164,165,166],[15,23061,23063],{"className":23062},[129,166],[15,23064,22654],{"className":23065},[129,166],[15,23067,59],{"className":23068},[195],[15,23070],{"className":23071,"style":199},[93],[15,23073,2717],{"className":23074},[203],[15,23076],{"className":23077,"style":199},[93],[15,23079,23081,23084,23087,23090,23093],{"className":23080},[120],[15,23082],{"className":23083,"style":10410},[124],[15,23085,10372],{"className":23086},[129,133],[15,23088],{"className":23089,"style":3277},[93],[15,23091,315],{"className":23092},[2436],[15,23094],{"className":23095,"style":3277},[93],[15,23097,23099,23102],{"className":23098},[120],[15,23100],{"className":23101,"style":4886},[124],[15,23103,318],{"className":23104},[129],"，从而 ",[15,23107,23109,23190],{"className":23108,"translate":19},[23],[15,23110,23112],{"className":23111},[27],[29,23113,23114],{"xmlns":31},[34,23115,23116,23187],{},[37,23117,23118,23120,23122,23136,23138,23140,23142,23144,23146,23148,23169],{},[43,23119,22143],{},[50,23121,2717],{},[40,23123,23124,23126],{},[43,23125,19679],{"mathvariant":65},[305,23127,23128,23130],{},[43,23129,819],{"mathvariant":596},[37,23131,23132,23134],{},[50,23133,21499],{"mathvariant":65},[50,23135,21499],{"mathvariant":65},[50,23137,53],{"stretchy":52},[43,23139,348],{},[50,23141,315],{},[308,23143,318],{},[50,23145,59],{"stretchy":52},[50,23147,2717],{},[23149,23150,23151,23153,23161],"msubsup",{},[50,23152,1454],{},[37,23154,23155,23157,23159],{},[43,23156,1459],{},[50,23158,62],{},[308,23160,541],{},[37,23162,23163,23165,23167],{},[43,23164,10372],{},[50,23166,315],{},[308,23168,318],{},[37,23170,23171,23173,23185],{},[50,23172,53],{"fence":72},[1443,23174,23175,23183],{"linethickness":21029},[37,23176,23177,23179,23181],{},[43,23178,348],{},[50,23180,315],{},[308,23182,318],{},[43,23184,1459],{},[50,23186,59],{"fence":72},[109,23188,23189],{"encoding":111},"s\\le\\Pi_{\\mathcal H''}(m-1)\\le\\sum_{i=0}^{d-1}\\binom{m-1}{i}",[15,23191,23193,23211,23306,23327],{"className":23192,"ariaHidden":72},[116],[15,23194,23196,23199,23202,23205,23208],{"className":23195},[120],[15,23197],{"className":23198,"style":7753},[124],[15,23200,22143],{"className":23201},[129,133],[15,23203],{"className":23204,"style":199},[93],[15,23206,2717],{"className":23207},[203],[15,23209],{"className":23210,"style":199},[93],[15,23212,23214,23217,23291,23294,23297,23300,23303],{"className":23213},[120],[15,23215],{"className":23216,"style":125},[124],[15,23218,23220,23223],{"className":23219},[129],[15,23221,19679],{"className":23222},[129],[15,23224,23226],{"className":23225},[137],[15,23227,23229,23283],{"className":23228},[141,142],[15,23230,23232,23280],{"className":23231},[146],[15,23233,23235],{"className":23234,"style":1525},[150],[15,23236,23237,23240],{"style":154},[15,23238],{"className":23239,"style":159},[158],[15,23241,23243],{"className":23242},[163,164,165,166],[15,23244,23246],{"className":23245},[129,166],[15,23247,23249,23252],{"className":23248},[129,166],[15,23250,819],{"className":23251,"style":835},[129,614,166],[15,23253,23255],{"className":23254},[137],[15,23256,23258],{"className":23257},[141],[15,23259,23261],{"className":23260},[146],[15,23262,23265],{"className":23263,"style":23264},[150],"height:0.6828em;",[15,23266,23268,23271],{"style":23267},"top:-2.786em;margin-right:0.0714em;",[15,23269],{"className":23270,"style":15980},[158],[15,23272,23274],{"className":23273},[163,15984,229,166],[15,23275,23277],{"className":23276},[129,166],[15,23278,22654],{"className":23279},[129,166],[15,23281,175],{"className":23282},[174],[15,23284,23286],{"className":23285},[146],[15,23287,23289],{"className":23288,"style":182},[150],[15,23290],{},[15,23292,53],{"className":23293},[188],[15,23295,348],{"className":23296},[129,133],[15,23298],{"className":23299,"style":3277},[93],[15,23301,315],{"className":23302},[2436],[15,23304],{"className":23305,"style":3277},[93],[15,23307,23309,23312,23315,23318,23321,23324],{"className":23308},[120],[15,23310],{"className":23311,"style":125},[124],[15,23313,318],{"className":23314},[129],[15,23316,59],{"className":23317},[195],[15,23319],{"className":23320,"style":199},[93],[15,23322,2717],{"className":23323},[203],[15,23325],{"className":23326,"style":199},[93],[15,23328,23330,23334,23409,23412],{"className":23329},[120],[15,23331],{"className":23332,"style":23333},[124],"height:1.339em;vertical-align:-0.35em;",[15,23335,23337,23342],{"className":23336},[217],[15,23338,1454],{"className":23339,"style":23341},[217,1918,23340],"small-op","position:relative;top:0em;",[15,23343,23345],{"className":23344},[137],[15,23346,23348,23400],{"className":23347},[141,142],[15,23349,23351,23397],{"className":23350},[146],[15,23352,23355,23376],{"className":23353,"style":23354},[150],"height:0.989em;",[15,23356,23358,23361],{"style":23357},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[15,23359],{"className":23360,"style":159},[158],[15,23362,23364],{"className":23363},[163,164,165,166],[15,23365,23367,23370,23373],{"className":23366},[129,166],[15,23368,1459],{"className":23369},[129,133,166],[15,23371,62],{"className":23372},[203,166],[15,23374,541],{"className":23375},[129,166],[15,23377,23379,23382],{"style":23378},"top:-3.2029em;margin-right:0.05em;",[15,23380],{"className":23381,"style":159},[158],[15,23383,23385],{"className":23384},[163,164,165,166],[15,23386,23388,23391,23394],{"className":23387},[129,166],[15,23389,10372],{"className":23390},[129,133,166],[15,23392,315],{"className":23393},[2436,166],[15,23395,318],{"className":23396},[129,166],[15,23398,175],{"className":23399},[174],[15,23401,23403],{"className":23402},[146],[15,23404,23407],{"className":23405,"style":23406},[150],"height:0.2997em;",[15,23408],{},[15,23410],{"className":23411,"style":262},[93],[15,23413,23415,23421,23482],{"className":23414},[129],[15,23416,23418],{"className":23417,"style":5052},[188,5051],[15,23419,53],{"className":23420},[228,229],[15,23422,23424],{"className":23423},[1443],[15,23425,23427,23473],{"className":23426},[141,142],[15,23428,23430,23470],{"className":23429},[146],[15,23431,23434,23449],{"className":23432,"style":23433},[150],"height:0.8951em;",[15,23435,23437,23440],{"style":23436},"top:-2.355em;",[15,23438],{"className":23439,"style":159},[158],[15,23441,23443],{"className":23442},[163,164,165,166],[15,23444,23446],{"className":23445},[129,166],[15,23447,1459],{"className":23448},[129,133,166],[15,23450,23452,23455],{"style":23451},"top:-3.144em;",[15,23453],{"className":23454,"style":159},[158],[15,23456,23458],{"className":23457},[163,164,165,166],[15,23459,23461,23464,23467],{"className":23460},[129,166],[15,23462,348],{"className":23463},[129,133,166],[15,23465,315],{"className":23466},[2436,166],[15,23468,318],{"className":23469},[129,166],[15,23471,175],{"className":23472},[174],[15,23474,23476],{"className":23475},[146],[15,23477,23480],{"className":23478,"style":23479},[150],"height:0.345em;",[15,23481],{},[15,23483,23485],{"className":23484,"style":5052},[195,5051],[15,23486,59],{"className":23487},[228,229],[11,23489,23490],{},"代回并用 Pascal 公式：",[15,23492,23494],{"className":23493,"translate":19},[18],[15,23495,23497,23625],{"className":23496,"translate":19},[23],[15,23498,23500],{"className":23499},[27],[29,23501,23502],{"xmlns":31,"display":32},[34,23503,23504,23622],{},[37,23505,23506,23512,23514,23516,23518,23520,23534,23552,23554,23574,23592,23594,23608,23620],{},[40,23507,23508,23510],{},[43,23509,19679],{"mathvariant":65},[43,23511,819],{"mathvariant":596},[50,23513,53],{"stretchy":52},[43,23515,348],{},[50,23517,59],{"stretchy":52},[50,23519,2717],{},[1450,23521,23522,23524,23532],{},[50,23523,1454],{},[37,23525,23526,23528,23530],{},[43,23527,1459],{},[50,23529,62],{},[308,23531,541],{},[43,23533,10372],{},[37,23535,23536,23538,23550],{},[50,23537,53],{"fence":72},[1443,23539,23540,23548],{"linethickness":21029},[37,23541,23542,23544,23546],{},[43,23543,348],{},[50,23545,315],{},[308,23547,318],{},[43,23549,1459],{},[50,23551,59],{"fence":72},[50,23553,2682],{},[1450,23555,23556,23558,23566],{},[50,23557,1454],{},[37,23559,23560,23562,23564],{},[43,23561,1459],{},[50,23563,62],{},[308,23565,541],{},[37,23567,23568,23570,23572],{},[43,23569,10372],{},[50,23571,315],{},[308,23573,318],{},[37,23575,23576,23578,23590],{},[50,23577,53],{"fence":72},[1443,23579,23580,23588],{"linethickness":21029},[37,23581,23582,23584,23586],{},[43,23583,348],{},[50,23585,315],{},[308,23587,318],{},[43,23589,1459],{},[50,23591,59],{"fence":72},[50,23593,62],{},[1450,23595,23596,23598,23606],{},[50,23597,1454],{},[37,23599,23600,23602,23604],{},[43,23601,1459],{},[50,23603,62],{},[308,23605,541],{},[43,23607,10372],{},[37,23609,23610,23612,23618],{},[50,23611,53],{"fence":72},[1443,23613,23614,23616],{"linethickness":21029},[43,23615,348],{},[43,23617,1459],{},[50,23619,59],{"fence":72},[43,23621,107],{"mathvariant":65},[109,23623,23624],{"encoding":111},"\\Pi_{\\mathcal H}(m)\\le\\sum_{i=0}^{d}\\binom{m-1}{i}+\\sum_{i=0}^{d-1}\\binom{m-1}{i}\n=\\sum_{i=0}^{d}\\binom{m}{i}.",[15,23626,23628,23695,23852,24015],{"className":23627,"ariaHidden":72},[116],[15,23629,23631,23634,23677,23680,23683,23686,23689,23692],{"className":23630},[120],[15,23632],{"className":23633,"style":125},[124],[15,23635,23637,23640],{"className":23636},[129],[15,23638,19679],{"className":23639},[129],[15,23641,23643],{"className":23642},[137],[15,23644,23646,23669],{"className":23645},[141,142],[15,23647,23649,23666],{"className":23648},[146],[15,23650,23652],{"className":23651,"style":1525},[150],[15,23653,23654,23657],{"style":154},[15,23655],{"className":23656,"style":159},[158],[15,23658,23660],{"className":23659},[163,164,165,166],[15,23661,23663],{"className":23662},[129,166],[15,23664,819],{"className":23665,"style":835},[129,614,166],[15,23667,175],{"className":23668},[174],[15,23670,23672],{"className":23671},[146],[15,23673,23675],{"className":23674,"style":182},[150],[15,23676],{},[15,23678,53],{"className":23679},[188],[15,23681,348],{"className":23682},[129,133],[15,23684,59],{"className":23685},[195],[15,23687],{"className":23688,"style":199},[93],[15,23690,2717],{"className":23691},[203],[15,23693],{"className":23694,"style":199},[93],[15,23696,23698,23701,23768,23771,23843,23846,23849],{"className":23697},[120],[15,23699],{"className":23700,"style":21117},[124],[15,23702,23704],{"className":23703},[217,1874],[15,23705,23707,23760],{"className":23706},[141,142],[15,23708,23710,23757],{"className":23709},[146],[15,23711,23713,23733,23743],{"className":23712,"style":21130},[150],[15,23714,23715,23718],{"style":1887},[15,23716],{"className":23717,"style":1891},[158],[15,23719,23721],{"className":23720},[163,164,165,166],[15,23722,23724,23727,23730],{"className":23723},[129,166],[15,23725,1459],{"className":23726},[129,133,166],[15,23728,62],{"className":23729},[203,166],[15,23731,541],{"className":23732},[129,166],[15,23734,23735,23738],{"style":1909},[15,23736],{"className":23737,"style":1891},[158],[15,23739,23740],{},[15,23741,1454],{"className":23742},[217,1918,1919],[15,23744,23745,23748],{"style":1922},[15,23746],{"className":23747,"style":1891},[158],[15,23749,23751],{"className":23750},[163,164,165,166],[15,23752,23754],{"className":23753},[129,166],[15,23755,10372],{"className":23756},[129,133,166],[15,23758,175],{"className":23759},[174],[15,23761,23763],{"className":23762},[146],[15,23764,23766],{"className":23765,"style":1941},[150],[15,23767],{},[15,23769],{"className":23770,"style":262},[93],[15,23772,23774,23780,23837],{"className":23773},[129],[15,23775,23777],{"className":23776,"style":5052},[188,5051],[15,23778,53],{"className":23779},[228,165],[15,23781,23783],{"className":23782},[1443],[15,23784,23786,23829],{"className":23785},[141,142],[15,23787,23789,23826],{"className":23788},[146],[15,23790,23792,23803],{"className":23791,"style":1816},[150],[15,23793,23794,23797],{"style":1819},[15,23795],{"className":23796,"style":1823},[158],[15,23798,23800],{"className":23799},[129],[15,23801,1459],{"className":23802},[129,133],[15,23804,23805,23808],{"style":1843},[15,23806],{"className":23807,"style":1823},[158],[15,23809,23811,23814,23817,23820,23823],{"className":23810},[129],[15,23812,348],{"className":23813},[129,133],[15,23815],{"className":23816,"style":3277},[93],[15,23818,315],{"className":23819},[2436],[15,23821],{"className":23822,"style":3277},[93],[15,23824,318],{"className":23825},[129],[15,23827,175],{"className":23828},[174],[15,23830,23832],{"className":23831},[146],[15,23833,23835],{"className":23834,"style":1862},[150],[15,23836],{},[15,23838,23840],{"className":23839,"style":5052},[195,5051],[15,23841,59],{"className":23842},[228,165],[15,23844],{"className":23845,"style":3277},[93],[15,23847,2682],{"className":23848},[2436],[15,23850],{"className":23851,"style":3277},[93],[15,23853,23855,23858,23931,23934,24006,24009,24012],{"className":23854},[120],[15,23856],{"className":23857,"style":21117},[124],[15,23859,23861],{"className":23860},[217,1874],[15,23862,23864,23923],{"className":23863},[141,142],[15,23865,23867,23920],{"className":23866},[146],[15,23868,23870,23890,23900],{"className":23869,"style":21130},[150],[15,23871,23872,23875],{"style":1887},[15,23873],{"className":23874,"style":1891},[158],[15,23876,23878],{"className":23877},[163,164,165,166],[15,23879,23881,23884,23887],{"className":23880},[129,166],[15,23882,1459],{"className":23883},[129,133,166],[15,23885,62],{"className":23886},[203,166],[15,23888,541],{"className":23889},[129,166],[15,23891,23892,23895],{"style":1909},[15,23893],{"className":23894,"style":1891},[158],[15,23896,23897],{},[15,23898,1454],{"className":23899},[217,1918,1919],[15,23901,23902,23905],{"style":1922},[15,23903],{"className":23904,"style":1891},[158],[15,23906,23908],{"className":23907},[163,164,165,166],[15,23909,23911,23914,23917],{"className":23910},[129,166],[15,23912,10372],{"className":23913},[129,133,166],[15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多项式增长",[11,24162,17328,24163,24214],{},[15,24164,24166,24184],{"className":24165,"translate":19},[23],[15,24167,24169],{"className":24168},[27],[29,24170,24171],{"xmlns":31},[34,24172,24173,24181],{},[37,24174,24175,24177,24179],{},[43,24176,348],{},[50,24178,7469],{},[43,24180,10372],{},[109,24182,24183],{"encoding":111},"m\\ge d",[15,24185,24187,24205],{"className":24186,"ariaHidden":72},[116],[15,24188,24190,24193,24196,24199,24202],{"className":24189},[120],[15,24191],{"className":24192,"style":7753},[124],[15,24194,348],{"className":24195},[129,133],[15,24197],{"className":24198,"style":199},[93],[15,24200,7469],{"className":24201},[203],[15,24203],{"className":24204,"style":199},[93],[15,24206,24208,24211],{"className":24207},[120],[15,24209],{"className":24210,"style":522},[124],[15,24212,10372],{"className":24213},[129,133]," 时，上述和式可以被一个更粗但更好用的界控制：",[15,24216,24218],{"className":24217,"translate":19},[18],[15,24219,24221,24283],{"className":24220,"translate":19},[23],[15,24222,24224],{"className":24223},[27],[29,24225,24226],{"xmlns":31,"display":32},[34,24227,24228,24280],{},[37,24229,24230,24244,24256,24258,24278],{},[1450,24231,24232,24234,24242],{},[50,24233,1454],{},[37,24235,24236,24238,24240],{},[43,24237,1459],{},[50,24239,62],{},[308,24241,541],{},[43,24243,10372],{},[37,24245,24246,24248,24254],{},[50,24247,53],{"fence":72},[1443,24249,24250,24252],{"linethickness":21029},[43,24251,348],{},[43,24253,1459],{},[50,24255,59],{"fence":72},[50,24257,2717],{},[305,24259,24260,24276],{},[37,24261,24262,24264,24274],{},[50,24263,53],{"fence":72},[1443,24265,24266,24272],{},[37,24267,24268,24270],{},[43,24269,10695],{},[43,24271,348],{},[43,24273,10372],{},[50,24275,59],{"fence":72},[43,24277,10372],{},[43,24279,107],{"mathvariant":65},[109,24281,24282],{"encoding":111},"\\sum_{i=0}^{d}\\binom{m}{i}\\le\\left(\\frac{em}{d}\\right)^d.",[15,24284,24286,24431],{"className":24285,"ariaHidden":72},[116],[15,24287,24289,24292,24359,24362,24422,24425,24428],{"className":24288},[120],[15,24290],{"className":24291,"style":21117},[124],[15,24293,24295],{"className":24294},[217,1874],[15,24296,24298,24351],{"className":24297},[141,142],[15,24299,24301,24348],{"className":24300},[146],[15,24302,24304,24324,24334],{"className":24303,"style":21130},[150],[15,24305,24306,24309],{"style":1887},[15,24307],{"className":24308,"style":1891},[158],[15,24310,24312],{"className":24311},[163,164,165,166],[15,24313,24315,24318,24321],{"className":24314},[129,166],[15,24316,1459],{"className":24317},[129,133,166],[15,24319,62],{"className":24320},[203,166],[15,24322,541],{"className":24323},[129,166],[15,24325,24326,24329],{"style":1909},[15,24327],{"className":24328,"style":1891},[158],[15,24330,24331],{},[15,24332,1454],{"className":24333},[217,1918,1919],[15,24335,24336,24339],{"style":1922},[15,24337],{"className":24338,"style":1891},[158],[15,24340,24342],{"className":24341},[163,164,165,166],[15,24343,24345],{"className":24344},[129,166],[15,24346,10372],{"className":24347},[129,133,166],[15,24349,175],{"className":24350},[174],[15,24352,24354],{"className":24353},[146],[15,24355,24357],{"className":24356,"style":1941},[150],[15,24358],{},[15,24360],{"className":24361,"style":262},[93],[15,24363,24365,24371,24416],{"className":24364},[129],[15,24366,24368],{"className":24367,"style":5052},[188,5051],[15,24369,53],{"className":24370},[228,165],[15,24372,24374],{"className":24373},[1443],[15,24375,24377,24408],{"className":24376},[141,142],[15,24378,24380,24405],{"className":24379},[146],[15,24381,24383,24394],{"className":24382,"style":21210},[150],[15,24384,24385,24388],{"style":1819},[15,24386],{"className":24387,"style":1823},[158],[15,24389,24391],{"className":24390},[129],[15,24392,1459],{"className":24393},[129,133],[15,24395,24396,24399],{"style":1843},[15,24397],{"className":24398,"style":1823},[158],[15,24400,24402],{"className":24401},[129],[15,24403,348],{"className":24404},[129,133],[15,24406,175],{"className":24407},[174],[15,24409,24411],{"className":24410},[146],[15,24412,24414],{"className":24413,"style":1862},[150],[15,24415],{},[15,24417,24419],{"className":24418,"style":5052},[195,5051],[15,24420,59],{"className":24421},[228,165],[15,24423],{"className":24424,"style":199},[93],[15,24426,2717],{"className":24427},[203],[15,24429],{"className":24430,"style":199},[93],[15,24432,24434,24438,24547,24550],{"className":24433},[120],[15,24435],{"className":24436,"style":24437},[124],"height:2.075em;vertical-align:-0.686em;",[15,24439,24441,24522],{"className":24440},[449],[15,24442,24444,24451,24516],{"className":24443},[449],[15,24445,24447],{"className":24446,"style":5052},[188,5051],[15,24448,53],{"className":24449},[228,24450],"size2",[15,24452,24454,24457,24513],{"className":24453},[129],[15,24455],{"className":24456},[188,1803],[15,24458,24460],{"className":24459},[1443],[15,24461,24463,24505],{"className":24462},[141,142],[15,24464,24466,24502],{"className":24465},[146],[15,24467,24469,24480,24488],{"className":24468,"style":21210},[150],[15,24470,24471,24474],{"style":1819},[15,24472],{"className":24473,"style":1823},[158],[15,24475,24477],{"className":24476},[129],[15,24478,10372],{"className":24479},[129,133],[15,24481,24482,24485],{"style":1832},[15,24483],{"className":24484,"style":1823},[158],[15,24486],{"className":24487,"style":1840},[1839],[15,24489,24490,24493],{"style":1843},[15,24491],{"className":24492,"style":1823},[158],[15,24494,24496,24499],{"className":24495},[129],[15,24497,10695],{"className":24498},[129,133],[15,24500,348],{"className":24501},[129,133],[15,24503,175],{"className":24504},[174],[15,24506,24508],{"className":24507},[146],[15,24509,24511],{"className":24510,"style":1862},[150],[15,24512],{},[15,24514],{"className":24515},[195,1803],[15,24517,24519],{"className":24518,"style":5052},[195,5051],[15,2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维有限的假设类，在 ",[15,25973,25975,25988],{"className":25974,"translate":19},[23],[15,25976,25978],{"className":25977},[27],[29,25979,25980],{"xmlns":31},[34,25981,25982,25986],{},[37,25983,25984],{},[43,25985,348],{},[109,25987,348],{"encoding":111},[15,25989,25991],{"className":25990,"ariaHidden":72},[116],[15,25992,25994,25997],{"className":25993},[120],[15,25995],{"className":25996,"style":4383},[124],[15,25998,348],{"className":25999},[129,133]," 个点上最多只有 ",[15,26002,26004,26022],{"className":26003,"translate":19},[23],[15,26005,26007],{"className":26006},[27],[29,26008,26009],{"xmlns":31},[34,26010,26011,26019],{},[37,26012,26013],{},[305,26014,26015,26017],{},[43,26016,348],{},[43,26018,10372],{},[109,26020,26021],{"encoding":111},"m^d",[15,26023,26025],{"className":26024,"ariaHidden":72},[116],[15,26026,26028,26031],{"className":26027},[120],[15,26029],{"className":26030,"style":10486},[124],[15,26032,26034,26037],{"className":26033},[129],[15,26035,348],{"className":26036},[129,133],[15,26038,26040],{"className":26039},[137],[15,26041,26043],{"className":26042},[141],[15,26044,26046],{"className":26045},[146],[15,26047,26049],{"className":26048,"style":10486},[150],[15,26050,26051,26054],{"style":383},[15,26052],{"className":26053,"style":159},[158],[15,26055,26057],{"className":26056},[163,164,165,166],[15,26058,10372],{"className":26059},[129,133,166]," 种行为，而不是 ",[15,26062,26064,26081],{"className":26063,"translate":19},[23],[15,26065,26067],{"className":26066},[27],[29,26068,26069],{"xmlns":31},[34,26070,26071,26079],{},[37,26072,26073],{},[305,26074,26075,26077],{},[308,26076,310],{},[43,26078,348],{},[109,26080,6367],{"encoding":111},[15,26082,26084],{"className":26083,"ariaHidden":72},[116],[15,26085,26087,26090],{"className":26086},[120],[15,26088],{"className":26089,"style":6377},[124],[15,26091,26093,26096],{"className":26092},[129],[15,26094,310],{"className":26095},[129],[15,26097,26099],{"className":26098},[137],[15,26100,26102],{"className":26101},[141],[15,26103,26105],{"className":26104},[146],[15,26106,26108],{"className":26107,"style":6377},[150],[15,26109,26110,26113],{"style":383},[15,26111],{"className":26112,"style":159},[158],[15,26114,26116],{"className":26115},[163,164,165,166],[15,26117,348],{"className":26118},[129,133,166]," 种。当 ",[15,26121,26123,26136],{"className":26122,"translate":19},[23],[15,26124,26126],{"className":26125},[27],[29,26127,26128],{"xmlns":31},[34,26129,26130,26134],{},[37,26131,26132],{},[43,26133,348],{},[109,26135,348],{"encoding":111},[15,26137,26139],{"className":26138,"ariaHidden":72},[116],[15,26140,26142,26145],{"className":26141},[120],[15,26143],{"className":26144,"style":4383},[124],[15,26146,348],{"className":26147},[129,133]," 增大时，这两种增长速度的差距是压倒性的——",[15,26150,26152,26169],{"className":26151,"translate":19},[23],[15,26153,26155],{"className":26154},[27],[29,26156,26157],{"xmlns":31},[34,26158,26159,26167],{},[37,26160,26161],{},[305,26162,26163,26165],{},[43,26164,348],{},[43,26166,10372],{},[109,26168,26021],{"encoding":111},[15,26170,26172],{"className":26171,"ariaHidden":72},[116],[15,26173,26175,26178],{"className":26174},[120],[15,26176],{"className":26177,"style":10486},[124],[15,26179,26181,26184],{"className":26180},[129],[15,26182,348],{"className":26183},[129,133],[15,26185,26187],{"className":26186},[137],[15,26188,26190],{"className":26189},[141],[15,26191,26193],{"className":26192},[146],[15,26194,26196],{"className":26195,"style":10486},[150],[15,26197,26198,26201],{"style":383},[15,26199],{"className":26200,"style":159},[158],[15,26202,26204],{"className":26203},[163,164,165,166],[15,26205,10372],{"className":26206},[129,133,166]," 取对数后只有 ",[15,26209,26211,26231],{"className":26210,"translate":19},[23],[15,26212,26214],{"className":26213},[27],[29,26215,26216],{"xmlns":31},[34,26217,26218,26228],{},[37,26219,26220,26222,26224,26226],{},[43,26221,10372],{},[43,26223,14721],{},[50,26225,69],{},[43,26227,348],{},[109,26229,26230],{"encoding":111},"d\\log m",[15,26232,26234],{"className":26233,"ariaHidden":72},[116],[15,26235,26237,26240,26243,26246,26251,26254],{"className":26236},[120],[15,26238],{"className":26239,"style":1682},[124],[15,26241,10372],{"className":26242},[129,133],[15,26244],{"className":26245,"style":262},[93],[15,26247,14771,26249],{"className":26248},[217],[15,26250,2247],{"style":2246},[15,26252],{"className":26253,"style":262},[93],[15,26255,348],{"className":26256},[129,133],"，而 ",[15,26259,26261,26278],{"className":26260,"translate":19},[23],[15,26262,26264],{"className":26263},[27],[29,26265,26266],{"xmlns":31},[34,26267,26268,26276],{},[37,26269,26270],{},[305,26271,26272,26274],{},[308,26273,310],{},[43,26275,348],{},[109,26277,6367],{"encoding":111},[15,26279,26281],{"className":26280,"ariaHidden":72},[116],[15,26282,26284,26287],{"className":26283},[120],[15,26285],{"className":26286,"style":6377},[124],[15,26288,26290,26293],{"className":26289},[129],[15,26291,310],{"className":26292},[129],[15,26294,26296],{"className":26295},[137],[15,26297,26299],{"className":26298},[141],[15,26300,26302],{"className":26301},[146],[15,26303,26305],{"className":26304,"style":6377},[150],[15,26306,26307,26310],{"style":383},[15,26308],{"className":26309,"style":159},[158],[15,26311,26313],{"className":26312},[163,164,165,166],[15,26314,348],{"className":26315},[129,133,166]," 取对数后是 ",[15,26318,26320,26333],{"className":26319,"translate":19},[23],[15,26321,26323],{"className":26322},[27],[29,26324,26325],{"xmlns":31},[34,26326,26327,26331],{},[37,26328,26329],{},[43,26330,348],{},[109,26332,348],{"encoding":111},[15,26334,26336],{"className":26335,"ariaHidden":72},[116],[15,26337,26339,26342],{"className":26338},[120],[15,26340],{"className":26341,"style":4383},[124],[15,26343,348],{"className":26344},[129,133],[11,26346,26347,26348,26351],{},"换句话说，",[698,26349,26350],{},"有限 VC 维等价于增长函数是多项式级的","。这就是「容量」一词的数学来源：容量不是参数个数，而是假设类能产生的二分数量随样本量的增长速度。",[570,26353,26355],{"id":26354},"四一致收敛定理","四、一致收敛定理",[575,26357,26359],{"id":26358},"_41-对称化","4.1 对称化",[11,26361,26362,26363,26366],{},"增长函数是组合量，要把它变成泛化界，需要一步概率论证，称为",[698,26364,26365],{},"对称化","（symmetrization）。",[11,26368,21927,26369,26431,26432,26460,26461,26489],{},[15,26370,26372,26390],{"className":26371,"translate":19},[23],[15,26373,26375],{"className":26374},[27],[29,26376,26377],{"xmlns":31},[34,26378,26379,26387],{},[37,26380,26381],{},[305,26382,26383,26385],{},[43,26384,940],{},[50,26386,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[109,26388,26389],{"encoding":111},"S'",[15,26391,26393],{"className":26392,"ariaHidden":72},[116],[15,26394,26396,26399],{"className":26395},[120],[15,26397],{"className":26398,"style":21529},[124],[15,26400,26402,26405],{"className":26401},[129],[15,26403,940],{"className":26404,"style":1013},[129,133],[15,26406,26408],{"className":26407},[137],[15,26409,26411],{"className":26410},[141],[15,26412,26414],{"className":26413},[146],[15,26415,26417],{"className":26416,"style":21529},[150],[15,26418,26419,26422],{"style":383},[15,26420],{"className":26421,"style":159},[158],[15,26423,26425],{"className":26424},[163,164,165,166],[15,26426,26428],{"className":26427},[129,166],[15,26429,21499],{"className":26430},[129,166]," 是与 ",[15,26433,26435,26448],{"className":26434,"translate":19},[23],[15,26436,26438],{"className":26437},[27],[29,26439,26440],{"xmlns":31},[34,26441,26442,26446],{},[37,26443,26444],{},[43,26445,940],{},[109,26447,940],{"encoding":111},[15,26449,26451],{"className":26450,"ariaHidden":72},[116],[15,26452,26454,26457],{"className":26453},[120],[15,26455],{"className":26456,"style":610},[124],[15,26458,940],{"className":26459,"style":1013},[129,133]," 独立同分布的另一组 ",[15,26462,26464,26477],{"className":26463,"translate":19},[23],[15,26465,26467],{"className":26466},[27],[29,26468,26469],{"xmlns":31},[34,26470,26471,26475],{},[37,26472,26473],{},[43,26474,348],{},[109,26476,348],{"encoding":111},[15,26478,26480],{"className":26479,"ariaHidden":72},[116],[15,26481,26483,26486],{"className":26482},[120],[15,26484],{"className":26485,"style":4383},[124],[15,26487,348],{"className":26488},[129,133]," 个样本。核心不等式是",[15,26491,26493],{"className":26492,"translate":19},[18],[15,26494,26496,26631],{"className":26495,"translate":19},[23],[15,26497,26499],{"className":26498},[27],[29,26500,26501],{"xmlns":31,"display":32},[34,26502,26503,26628],{},[37,26504,26505,26521,26523,26529,26531,26533,26535,26537,26543,26545,26547,26549,26551,26554,26557,26559,26576,26592,26594,26600,26602,26604,26606,26608,26618,26620,26622,26624,26626],{},[2140,26506,26507,26513],{},[37,26508,26509,26511],{},[43,26510,2932],{},[50,26512,69],{},[37,26514,26515,26517,26519],{},[43,26516,45],{},[50,26518,100],{},[43,26520,819],{"mathvariant":596},[50,26522,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,26524,26525,26527],{},[43,26526,1373],{},[43,26528,906],{"mathvariant":596},[50,26530,53],{"stretchy":52},[43,26532,45],{},[50,26534,59],{"stretchy":52},[50,26536,315],{},[40,26538,26539,26541],{},[43,26540,1373],{},[43,26542,940],{},[50,26544,53],{"stretchy":52},[43,26546,45],{},[50,26548,59],{"stretchy":52},[50,26550,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[6852,26552,26553],{},"  ",[50,26555,26556],{},"≲",[6852,26558,26553],{},[40,26560,26561,26564],{},[43,26562,26563],{"mathvariant":103},"E",[37,26565,26566,26568,26570],{},[43,26567,940],{},[50,26569,91],{"separator":72},[305,26571,26572,26574],{},[43,26573,940],{},[50,26575,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[2140,26577,26578,26584],{},[37,26579,26580,26582],{},[43,26581,2932],{},[50,26583,69],{},[37,26585,26586,26588,26590],{},[43,26587,45],{},[50,26589,100],{},[43,26591,819],{"mathvariant":596},[50,26593,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,26595,26596,26598],{},[43,26597,1373],{},[43,26599,940],{},[50,26601,53],{"stretchy":52},[43,26603,45],{},[50,26605,59],{"stretchy":52},[50,26607,315],{},[40,26609,26610,26612],{},[43,26611,1373],{},[305,26613,26614,26616],{},[43,26615,940],{},[50,26617,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[50,26619,53],{"stretchy":52},[43,26621,45],{},[50,26623,59],{"stretchy":52},[50,26625,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,26627,91],{"separator":72},[109,26629,26630],{"encoding":111},"\\sup_{h\\in\\mathcal H}\\bigl|L_{\\mathcal D}(h)-L_S(h)\\bigr|\n\\;\\lesssim\\;\n\\mathbb E_{S,S'}\\sup_{h\\in\\mathcal H}\\bigl|L_S(h)-L_{S'}(h)\\bigr|,",[15,26632,26634,26792,26899,27134],{"className":26633,"ariaHidden":72},[116],[15,26635,26637,26641,26694,26731,26774,26777,26780,26783,26786,26789],{"className":26636},[120],[15,26638],{"className":26639,"style":26640},[124],"height:1.8239em;vertical-align:-0.9739em;",[15,26642,26644],{"className":26643},[217,1874],[15,26645,26647,26686],{"className":26646},[141,142],[15,26648,26650,26683],{"className":26649},[146],[15,26651,26653,26673],{"className":26652,"style":4383},[150],[15,26654,26655,26658],{"style":4386},[15,26656],{"className":26657,"style":1823},[158],[15,26659,26661],{"className":26660},[163,164,165,166],[15,26662,26664,26667,26670],{"className":26663},[129,166],[15,26665,45],{"className":26666},[129,133,166],[15,26668,100],{"className":26669},[203,166],[15,26671,819],{"className":26672,"style":835},[129,614,166],[15,26674,26675,26678],{"style":2287},[15,26676],{"className":26677,"style":1823},[158],[15,26679,26680],{},[15,26681,2932],{"className":26682},[217],[15,26684,175],{"className":26685},[174],[15,26687,26689],{"className":26688},[146],[15,26690,26692],{"className":26691,"style":4424},[150],[15,26693],{},[15,26695,26697],{"className":26696},[188],[15,26698,26700],{"className":26699},[228,4433],[15,26701,26703,26723],{"className":26702},[141,142],[15,26704,26706,26720],{"className":26705},[146],[15,26707,26709],{"className":26708,"style":4443},[150],[15,26710,26711,26714],{"style":4446},[15,26712],{"className":26713,"style":4450},[158],[15,26715,26716],{"style":4453},[4455,26717,26718],{"xmlns":4457,"width":4458,"height":73,"viewBox":4459},[4461,26719],{"d":4463},[15,26721,175],{"className":26722},[174],[15,26724,26726],{"className":26725},[146],[15,26727,26729],{"className":26728,"style":4473},[150],[15,26730],{},[15,26732,26734,26737],{"className":26733},[129],[15,26735,1373],{"className":26736},[129,133],[15,26738,26740],{"className":26739},[137],[15,26741,26743,26766],{"className":26742},[141,142],[15,26744,26746,26763],{"className":26745},[146],[15,26747,26749],{"className":26748,"style":1525},[150],[15,26750,26751,26754],{"style":154},[15,26752],{"className":26753,"style":159},[158],[15,26755,26757],{"className":26756},[163,164,165,166],[15,26758,26760],{"className":26759},[129,166],[15,26761,906],{"className":26762,"style":170},[129,614,166],[15,26764,175],{"className":26765},[174],[15,26767,26769],{"className":26768},[146],[15,26770,26772],{"className":26771,"style":182},[150],[15,26773],{},[15,26775,53],{"className":26776},[188],[15,26778,45],{"className":26779},[129,133],[15,26781,59],{"className":26782},[195],[15,26784],{"className":26785,"style":3277},[93],[15,26787,315],{"className":26788},[2436],[15,26790],{"className":26791,"style":3277},[93],[15,26793,26795,26798,26838,26841,26844,26847,26884,26887,26890,26893,26896],{"className":26794},[120],[15,26796],{"className":26797,"style":213},[124],[15,26799,26801,26804],{"className":26800},[129],[15,26802,1373],{"className":26803},[129,133],[15,26805,26807],{"className":26806},[137],[15,26808,26810,26830],{"className":26809},[141,142],[15,26811,26813,26827],{"className":26812},[146],[15,26814,26816],{"className":26815,"style":1525},[150],[15,26817,26818,26821],{"style":154},[15,26819],{"className":26820,"style":159},[158],[15,26822,26824],{"className":26823},[163,164,165,166],[15,26825,940],{"className":26826,"style":1013},[129,133,166],[15,26828,175],{"className":26829},[174],[15,26831,26833],{"className":26832},[146],[15,26834,26836],{"className":26835,"style":182},[150],[15,26837],{},[15,26839,53],{"className":26840},[188],[15,26842,45],{"className":26843},[129,133],[15,26845,59],{"className":26846},[195],[15,26848,26850],{"className":26849},[195],[15,26851,26853],{"className":26852},[228,4433],[15,26854,26856,26876],{"className":26855},[141,142],[15,26857,26859,26873],{"className":26858},[146],[15,26860,26862],{"className":26861,"style":4443},[150],[15,26863,26864,26867],{"style":4446},[15,26865],{"className":26866,"style":4450},[158],[15,26868,26869],{"style":4453},[4455,26870,26871],{"xmlns":4457,"width":4458,"height":73,"viewBox":4459},[4461,26872],{"d":4463},[15,26874,175],{"className":26875},[174],[15,26877,26879],{"className":26878},[146],[15,26880,26882],{"className":26881,"style":4473},[150],[15,26883],{},[15,26885],{"className":26886,"style":199},[93],[15,26888],{"className":26889,"style":199},[93],[15,26891,26556],{"className":26892},[203,13310],[15,26894],{"className":26895,"style":199},[93],[15,26897],{"className":26898,"style":199},[93],[15,26900,26902,26905,26983,26986,27039,27076,27116,27119,27122,27125,27128,27131],{"className":26901},[120],[15,26903],{"className":26904,"style":26640},[124],[15,26906,26908,26911],{"className":26907},[129],[15,26909,26563],{"className":26910},[129,285],[15,26912,26914],{"className":26913},[137],[15,26915,26917,26975],{"className":26916},[141,142],[15,26918,26920,26972],{"className":26919},[146],[15,26921,26923],{"className":26922,"style":1525},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不等式），代价是概率意义上的常数因子。",[11,27265,27266,27267,27375,27376,27446],{},"接着引入独立同分布的 Rademacher 变量 ",[15,27268,27270,27299],{"className":27269,"translate":19},[23],[15,27271,27273],{"className":27272},[27],[29,27274,27275],{"xmlns":31},[34,27276,27277,27296],{},[37,27278,27279,27286,27288,27290,27292,27294],{},[40,27280,27281,27284],{},[43,27282,27283],{},"σ",[43,27285,1459],{},[50,27287,100],{},[50,27289,636],{"stretchy":52},[50,27291,10928],{},[308,27293,318],{},[50,27295,645],{"stretchy":52},[109,27297,27298],{"encoding":111},"\\sigma_i\\in\\{\\pm1\\}",[15,27300,27302,27357],{"className":27301,"ariaHidden":72},[116],[15,27303,27305,27308,27348,27351,27354],{"className":27304},[120],[15,27306],{"className":27307,"style":7925},[124],[15,27309,27311,27314],{"className":27310},[129],[15,27312,27283],{"className":27313,"style":887},[129,133],[15,27315,27317],{"className":27316},[137],[15,27318,27320,27340],{"className":27319},[141,142],[15,27321,27323,27337],{"className":27322},[146],[15,27324,27326],{"className":27325,"style":1981},[150],[15,27327,27328,27331],{"style":1100},[15,27329],{"className":27330,"style":159},[158],[15,27332,27334],{"className":27333},[163,164,165,166],[15,27335,1459],{"className":27336},[129,133,166],[15,27338,175],{"className":27339},[174],[15,27341,27343],{"className":27342},[146],[15,27344,27346],{"className":27345,"style":182},[150],[15,27347],{},[15,27349],{"className":27350,"style":199},[93],[15,27352,100],{"className":27353},[203],[15,27355],{"className":27356,"style":199},[93],[15,27358,27360,27363,27366,27369,27372],{"className":27359},[120],[15,27361],{"className":27362,"style":125},[124],[15,27364,636],{"className":27365},[188],[15,27367,10928],{"className":27368},[129],[15,27370,318],{"className":27371},[129],[15,27373,645],{"className":27374},[195],"。由于 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",[15,28688,28690,28734],{"className":28689,"translate":19},[23],[15,28691,28693],{"className":28692},[27],[29,28694,28695],{"xmlns":31},[34,28696,28697,28731],{},[37,28698,28699,28705,28707,28709,28711,28713,28715,28717,28719,28721,28723,28725,28727,28729],{},[40,28700,28701,28703],{},[43,28702,27518],{"mathvariant":65},[43,28704,45],{},[50,28706,53],{"stretchy":52},[43,28708,27527],{},[50,28710,59],{"stretchy":52},[50,28712,62],{},[308,28714,318],{"mathvariant":1468},[50,28716,1407],{"stretchy":52},[43,28718,45],{},[50,28720,53],{"stretchy":52},[43,28722,56],{},[50,28724,59],{"stretchy":52},[50,28726,1418],{"mathvariant":65},[43,28728,857],{},[50,28730,1423],{"stretchy":52},[109,28732,28733],{"encoding":111},"\\ell_h(z)=\\mathbf 1[h(x)\\neq y]",[15,28735,28737,28801,28867],{"className":28736,"ariaHidden":72},[116],[15,28738,28740,28743,28783,28786,28789,28792,28795,28798],{"className":28739},[120],[15,28741],{"className":28742,"style":125},[124],[15,28744,28746,28749],{"className":28745},[129],[15,28747,27518],{"className":28748},[129],[15,28750,28752],{"className":28751},[137],[15,28753,28755,28775],{"className":28754},[141,142],[15,28756,28758,28772],{"className":28757},[146],[15,28759,28761],{"className":28760,"style":151},[150],[15,28762,28763,28766],{"style":154},[15,28764],{"className":28765,"style":159},[158],[15,28767,28769],{"className":28768},[163,164,165,166],[15,28770,45],{"className":28771},[129,133,166],[15,28773,175],{"className":28774},[174],[15,28776,28778],{"className":28777},[146],[15,28779,28781],{"className":28780,"style":182},[150],[15,28782],{},[15,28784,53],{"className":28785},[188],[15,28787,27527],{"className":28788,"style":28017},[129,133],[15,28790,59],{"className":28791},[195],[15,28793],{"className":28794,"style":199},[93],[15,28796,62],{"className":28797},[203],[15,28799],{"className":28800,"style":199},[93],[15,28802,28804,28807,28810,28813,28816,28819,28822,28825,28828,28864],{"className":28803},[120],[15,28805],{"className":28806,"style":125},[124],[15,28808,318],{"className":28809},[129,1950],[15,28811,1407],{"className":28812},[188],[15,28814,45],{"className":28815},[129,133],[15,28817,53],{"className":28818},[188],[15,28820,56],{"className":28821},[129,133],[15,28823,59],{"className":28824},[195],[15,28826],{"className":28827,"style":199},[93],[15,28829,28831,28858,28861],{"className":28830},[203],[15,28832,28834],{"className":28833},[203],[15,28835,28837],{"className":28836},[129,1670],[15,28838,28840],{"className":28839},[1674],[15,28841,28843,28846,28855],{"className":28842},[1678],[15,28844],{"className":28845,"style":1682},[124],[15,28847,28849],{"className":28848},[1686],[15,28850,28852],{"className":28851},[129],[15,28853,1693],{"className":28854},[203],[15,28856],{"className":28857},[1697],[15,28859],{"className":28860},[93,1701],[15,28862,62],{"className":28863},[203],[15,28865],{"className":28866,"style":199},[93],[15,28868,28870,28873,28876],{"className":28869},[120],[15,28871],{"className":28872,"style":125},[124],[15,28874,857],{"className":28875,"style":887},[129,133],[15,28877,1423],{"className":28878},[195],"。右侧就是 ",[698,28881,28882],{},"Rademacher 复杂度",[15,28884,28886,28911],{"className":28885,"translate":19},[23],[15,28887,28889],{"className":28888},[27],[29,28890,28891],{"xmlns":31},[34,28892,28893,28908],{},[37,28894,28895,28902,28904,28906],{},[40,28896,28897,28900],{},[43,28898,104],{"mathvariant":28899},"fraktur",[43,28901,348],{},[50,28903,53],{"stretchy":52},[43,28905,819],{"mathvariant":596},[50,28907,59],{"stretchy":52},[109,28909,28910],{"encoding":111},"\\mathfrak R_m(\\mathcal H)",[15,28912,28914],{"className":28913,"ariaHidden":72},[116],[15,28915,28917,28920,28961,28964,28967],{"className":28916},[120],[15,28918],{"className":28919,"style":125},[124],[15,28921,28923,28927],{"className":28922},[129],[15,28924,104],{"className":28925},[129,28926],"mathfrak",[15,28928,28930],{"className":28929},[137],[15,28931,28933,28953],{"className":28932},[141,142],[15,28934,28936,28950],{"className":28935},[146],[15,28937,28939],{"className":28938,"style":1163},[150],[15,28940,28941,28944],{"style":154},[15,28942],{"className":28943,"style":159},[158],[15,28945,28947],{"className":28946},[163,164,165,166],[15,28948,348],{"className":28949},[129,133,166],[15,28951,175],{"className":28952},[174],[15,28954,28956],{"className":28955},[146],[15,28957,28959],{"className":28958,"style":182},[150],[15,28960],{},[15,28962,53],{"className":28963},[188],[15,28965,819],{"className":28966,"style":835},[129,614],[15,28968,59],{"className":28969},[195],"，它度量的是：给每个样本随机翻一个符号，假设类还能拟合得多好。随机符号是纯噪声，能拟合噪声就说明容量大。",[11,28972,28973],{},"对二值函数类，Rademacher 复杂度可以被增长函数控制：",[15,28975,28977],{"className":28976,"translate":19},[18],[15,28978,28980,29035],{"className":28979,"translate":19},[23],[15,28981,28983],{"className":28982},[27],[29,28984,28985],{"xmlns":31,"display":32},[34,28986,28987,29032],{},[37,28988,28989,28995,28997,28999,29001,29003,29030],{},[40,28990,28991,28993],{},[43,28992,104],{"mathvariant":28899},[43,28994,348],{},[50,28996,53],{"stretchy":52},[43,28998,819],{"mathvariant":596},[50,29000,59],{"stretchy":52},[50,29002,2717],{},[29004,29005,29006],"msqrt",{},[1443,29007,29008,29028],{},[37,29009,29010,29012,29014,29016,29022,29024,29026],{},[308,29011,310],{},[43,29013,14721],{},[50,29015,69],{},[40,29017,29018,29020],{},[43,29019,19679],{"mathvariant":65},[43,29021,819],{"mathvariant":596},[50,29023,53],{"stretchy":52},[43,29025,348],{},[50,29027,59],{"stretchy":52},[43,29029,348],{},[43,29031,107],{"mathvariant":65},[109,29033,29034],{"encoding":111},"\\mathfrak R_m(\\mathcal H)\\le\\sqrt{\\frac{2\\log\\Pi_{\\mathcal H}(m)}{m}}.",[15,29036,29038,29102],{"className":29037,"ariaHidden":72},[116],[15,29039,29041,29044,29084,29087,29090,29093,29096,29099],{"className":29040},[120],[15,29042],{"className":29043,"style":125},[124],[15,29045,29047,29050],{"className":29046},[129],[15,29048,104],{"className":29049},[129,28926],[15,29051,29053],{"className":29052},[137],[15,29054,29056,29076],{"className":29055},[141,142],[15,29057,29059,29073],{"className":29058},[146],[15,29060,29062],{"className":29061,"style":1163},[150],[15,29063,29064,29067],{"style":154},[15,29065],{"className":29066,"style":159},[158],[15,29068,29070],{"className":29069},[163,164,165,166],[15,29071,348],{"className":29072},[129,133,166],[15,29074,175],{"className":29075},[174],[15,29077,29079],{"className":29078},[146],[15,29080,29082],{"className":29081,"style":182},[150],[15,29083],{},[15,29085,53],{"className":29086},[188],[15,29088,819],{"className":29089,"style":835},[129,614],[15,29091,59],{"className":29092},[195],[15,29094],{"className":29095,"style":199},[93],[15,29097,2717],{"className":29098},[203],[15,29100],{"className":29101,"style":199},[93],[15,29103,29105,29109,29294],{"className":29104},[120],[15,29106],{"className":29107,"style":29108},[124],"height:2.44em;vertical-align:-0.7356em;",[15,29110,29113],{"className":29111},[129,29112],"sqrt",[15,29114,29116,29285],{"className":29115},[141,142],[15,29117,29119,29282],{"className":29118},[146],[15,29120,29123,29262],{"className":29121,"style":29122},[150],"height:1.7044em;",[15,29124,29128,29132],{"className":29125,"style":29127},[29126],"svg-align","top:-4.4em;",[15,29129],{"className":29130,"style":29131},[158],"height:4.4em;",[15,29133,29136],{"className":29134,"style":29135},[129],"padding-left:1em;",[15,29137,29139,29142,29259],{"className":29138},[129],[15,29140],{"className":29141},[188,1803],[15,29143,29145],{"className":29144},[1443],[15,29146,29148,29251],{"className":29147},[141,142],[15,29149,29151,29248],{"className":29150},[146],[15,29152,29155,29166,29174],{"className":29153,"style":29154},[150],"height:1.427em;",[15,29156,29157,29160],{"style":1819},[15,29158],{"className":29159,"style":1823},[158],[15,29161,29163],{"className":29162},[129],[15,29164,348],{"className":29165},[129,133],[15,29167,29168,29171],{"style":1832},[15,29169],{"className":29170,"style":1823},[158],[15,29172],{"className":29173,"style":1840},[1839],[15,29175,29176,29179],{"style":1843},[15,29177],{"className":29178,"style":1823},[158],[15,29180,29182,29185,29188,29193,29196,29239,29242,29245],{"className":29181},[129],[15,29183,310],{"className":29184},[129],[15,29186],{"className":29187,"style":262},[93],[15,29189,14771,29191],{"className":29190},[217],[15,29192,2247],{"style":2246},[15,29194],{"className":29195,"style":262},[93],[15,29197,29199,29202],{"className":29198},[129],[15,29200,19679],{"className":29201},[129],[15,29203,29205],{"className":29204},[137],[15,29206,29208,29231],{"className":29207},[141,142],[15,29209,29211,29228],{"className":29210},[146],[15,29212,29214],{"className":29213,"style":1525},[150],[15,29215,29216,29219],{"style":154},[15,29217],{"className":29218,"style":159},[158],[15,29220,29222],{"className":29221},[163,164,165,166],[15,29223,29225],{"className":29224},[129,166],[15,29226,819],{"className":29227,"style":835},[129,614,166],[15,29229,175],{"className":29230},[174],[15,29232,29234],{"className":29233},[146],[15,29235,29237],{"className":29236,"style":182},[150],[15,29238],{},[15,29240,53],{"className":29241},[188],[15,29243,348],{"className":29244},[129,133],[15,29246,59],{"className":29247},[195],[15,29249,175],{"className":29250},[174],[15,29252,29254],{"className":29253},[146],[15,29255,29257],{"className":29256,"style":1862},[150],[15,29258],{},[15,29260],{"className":29261},[195,1803],[15,29263,29265,29268],{"style":29264},"top:-3.6644em;",[15,29266],{"className":29267,"style":29131},[158],[15,29269,29273],{"className":29270,"style":29272},[29271],"hide-tail","min-width:1.02em;height:2.48em;",[4455,29274,29279],{"xmlns":4457,"width":29275,"height":29276,"viewBox":29277,"preserveAspectRatio":29278},"400em","2.48em","0 0 400000 2592","xMinYMin slice",[4461,29280],{"d":29281},"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z",[15,29283,175],{"className":29284},[174],[15,29286,29288],{"className":29287},[146],[15,29289,29292],{"className":29290,"style":29291},[150],"height:0.7356em;",[15,29293],{},[15,29295,107],{"className":29296},[129],[11,29298,29299,29300,29328,29329,29417],{},"这一步用的是有限类的 Massart 引理：把 ",[15,29301,29303,29316],{"className":29302,"translate":19},[23],[15,29304,29306],{"className":29305},[27],[29,29307,29308],{"xmlns":31},[34,29309,29310,29314],{},[37,29311,29312],{},[43,29313,819],{"mathvariant":596},[109,29315,822],{"encoding":111},[15,29317,29319],{"className":29318,"ariaHidden":72},[116],[15,29320,29322,29325],{"className":29321},[120],[15,29323],{"className":29324,"style":610},[124],[15,29326,819],{"className":29327,"style":835},[129,614]," 限制到样本上，最多只有 ",[15,29330,29332,29356],{"className":29331,"translate":19},[23],[15,29333,29335],{"className":29334},[27],[29,29336,29337],{"xmlns":31},[34,29338,29339,29353],{},[37,29340,29341,29347,29349,29351],{},[40,29342,29343,29345],{},[43,29344,19679],{"mathvariant":65},[43,29346,819],{"mathvariant":596},[50,29348,53],{"stretchy":52},[43,29350,348],{},[50,29352,59],{"stretchy":52},[109,29354,29355],{"encoding":111},"\\Pi_{\\mathcal H}(m)",[15,29357,29359],{"className":29358,"ariaHidden":72},[116],[15,29360,29362,29365,29408,29411,29414],{"className":29361},[120],[15,29363],{"className":29364,"style":125},[124],[15,29366,29368,29371],{"className":29367},[129],[15,29369,19679],{"className":29370},[129],[15,29372,29374],{"className":29373},[137],[15,29375,29377,29400],{"className":29376},[141,142],[15,29378,29380,29397],{"className":29379},[146],[15,29381,29383],{"className":29382,"style":1525},[150],[15,29384,29385,29388],{"style":154},[15,29386],{"className":29387,"style":159},[158],[15,29389,29391],{"className":29390},[163,164,165,166],[15,29392,29394],{"className":29393},[129,166],[15,29395,819],{"className":29396,"style":835},[129,614,166],[15,29398,175],{"className":29399},[174],[15,29401,29403],{"className":29402},[146],[15,29404,29406],{"className":29405,"style":182},[150],[15,29407],{},[15,29409,53],{"className":29410},[188],[15,29412,348],{"className":29413},[129,133],[15,29415,59],{"className":29416},[195]," 种不同的二分，再用 union bound 或鞅不等式。",[575,29419,29421],{"id":29420},"_42-泛化界","4.2 泛化界",[11,29423,29424,29425,14469],{},"把上面的估计串起来，得到本文唯一需要记住的公式。当 ",[15,29426,29428,29445],{"className":29427,"translate":19},[23],[15,29429,29431],{"className":29430},[27],[29,29432,29433],{"xmlns":31},[34,29434,29435,29443],{},[37,29436,29437,29439,29441],{},[43,29438,348],{},[50,29440,7469],{},[43,29442,10372],{},[109,29444,24183],{"encoding":111},[15,29446,29448,29466],{"className":29447,"ariaHidden":72},[116],[15,29449,29451,29454,29457,29460,29463],{"className":29450},[120],[15,29452],{"className":29453,"style":7753},[124],[15,29455,348],{"className":29456},[129,133],[15,29458],{"className":29459,"style":199},[93],[15,29461,7469],{"className":29462},[203],[15,29464],{"className":29465,"style":199},[93],[15,29467,29469,29472],{"className":29468},[120],[15,29470],{"className":29471,"style":522},[124],[15,29473,10372],{"className":29474},[129,133],[15,29476,29478],{"className":29477,"translate":19},[18],[15,29479,29481,29574],{"className":29480,"translate":19},[23],[15,29482,29484],{"className":29483},[27],[29,29485,29486],{"xmlns":31,"display":32},[34,29487,29488,29571],{},[37,29489,29490,29496,29498,29500,29502,29504,29506,29508,29514,29516,29518,29520,29522,29524,29526,29528,29569],{},[40,29491,29492,29494],{},[43,29493,1373],{},[43,29495,906],{"mathvariant":596},[50,29497,53],{"stretchy":52},[43,29499,45],{},[50,29501,59],{"stretchy":52},[6852,29503,26553],{},[50,29505,2717],{},[6852,29507,26553],{},[40,29509,29510,29512],{},[43,29511,1373],{},[43,29513,940],{},[50,29515,53],{"stretchy":52},[43,29517,45],{},[50,29519,59],{"stretchy":52},[6852,29521,26553],{},[50,29523,2682],{},[6852,29525,26553],{},[43,29527,5619],{},[29004,29529,29530],{},[1443,29531,29532,29567],{},[37,29533,29534,29536,29538,29540,29542,29544,29546,29548,29550,29552,29554,29556,29558,29560,29562,29565],{},[43,29535,10372],{},[43,29537,14721],{},[50,29539,69],{},[50,29541,53],{"stretchy":52},[43,29543,348],{},[43,29545,544],{"mathvariant":65},[43,29547,10372],{},[50,29549,59],{"stretchy":52},[50,29551,2682],{},[43,29553,14721],{},[50,29555,69],{},[50,29557,53],{"stretchy":52},[308,29559,318],{},[43,29561,544],{"mathvariant":65},[43,29563,29564],{},"δ",[50,29566,59],{"stretchy":52},[43,29568,348],{},[50,29570,91],{"separator":72},[109,29572,29573],{"encoding":111},"L_{\\mathcal D}(h)\\;\\le\\;L_S(h)\\;+\\;C\\sqrt{\\frac{d\\log(m\u002Fd)+\\log(1\u002F\\delta)}{m}},",[15,29575,29577,29650,29720],{"className":29576,"ariaHidden":72},[116],[15,29578,29580,29583,29626,29629,29632,29635,29638,29641,29644,29647],{"className":29579},[120],[15,29581],{"className":29582,"style":125},[124],[15,29584,29586,29589],{"className":29585},[129],[15,29587,1373],{"className":29588},[129,133],[15,29590,29592],{"className":29591},[137],[15,29593,29595,29618],{"className":29594},[141,142],[15,29596,29598,29615],{"className":29597},[146],[15,29599,29601],{"className":29600,"style":1525},[150],[15,29602,29603,29606],{"style":154},[15,29604],{"className":29605,"style":159},[158],[15,29607,29609],{"className":29608},[163,164,165,166],[15,29610,29612],{"className":29611},[129,166],[15,29613,906],{"className":29614,"style":170},[129,614,166],[15,29616,175],{"className":29617},[174],[15,29619,29621],{"className":29620},[146],[15,29622,29624],{"className":29623,"style":182},[150],[15,29625],{},[15,29627,53],{"className":29628},[188],[15,29630,45],{"className":29631},[129,133],[15,29633,59],{"className":29634},[195],[15,29636],{"className":29637,"style":199},[93],[15,29639],{"className":29640,"style":199},[93],[15,29642,2717],{"className":29643},[203],[15,29645],{"className":29646,"style":199},[93],[15,29648],{"className":29649,"style":199},[93],[15,29651,29653,29656,29696,29699,29702,29705,29708,29711,29714,29717],{"className":29652},[120],[15,29654],{"className":29655,"style":125},[124],[15,29657,29659,29662],{"className":29658},[129],[15,29660,1373],{"className":29661},[129,133],[15,29663,29665],{"className":29664},[137],[15,29666,29668,29688],{"className":29667},[141,142],[15,29669,29671,29685],{"className":29670},[146],[15,29672,29674],{"className":29673,"style":1525},[150],[15,29675,29676,29679],{"style":154},[15,29677],{"className":29678,"style":159},[158],[15,29680,29682],{"className":29681},[163,164,165,166],[15,29683,940],{"className":29684,"style":1013},[129,133,166],[15,29686,175],{"className":29687},[174],[15,29689,29691],{"className":29690},[146],[15,29692,29694],{"className":29693,"style":182},[150],[15,29695],{},[15,29697,53],{"className":29698},[188],[15,29700,45],{"className":29701},[129,133],[15,29703,59],{"className":29704},[195],[15,29706],{"className":29707,"style":199},[93],[15,29709],{"className":29710,"style":3277},[93],[15,29712,2682],{"className":29713},[2436],[15,29715],{"className":29716,"style":199},[93],[15,29718],{"className":29719,"style":3277},[93],[15,29721,29723,29726,29729,29886],{"className":29722},[120],[15,29724],{"className":29725,"style":29108},[124],[15,29727,5619],{"className":29728,"style":5665},[129,133],[15,29730,29732],{"className":29731},[129,29112],[15,29733,29735,29878],{"className":29734},[141,142],[15,29736,29738,29875],{"className":29737},[146],[15,29739,29741,29863],{"className":29740,"style":29122},[150],[15,29742,29744,29747],{"className":29743,"style":29127},[29126],[15,29745],{"className":29746,"style":29131},[158],[15,29748,29750],{"className":29749,"style":29135},[129],[15,29751,29753,29756,29860],{"className":29752},[129],[15,29754],{"className":29755},[188,1803],[15,29757,29759],{"className":29758},[1443],[15,29760,29762,29852],{"className":29761},[141,142],[15,29763,29765,29849],{"className":29764},[146],[15,29766,29768,29779,29787],{"className":29767,"style":29154},[150],[15,29769,29770,29773],{"style":1819},[15,29771],{"className":29772,"style":1823},[158],[15,29774,29776],{"className":29775},[129],[15,29777,348],{"className":29778},[129,133],[15,29780,29781,29784],{"style":1832},[15,29782],{"className":29783,"style":1823},[158],[15,29785],{"className":29786,"style":1840},[1839],[15,29788,29789,29792],{"style":1843},[15,29790],{"className":29791,"style":1823},[158],[15,29793,29795,29798,29801,29806,29809,29812,29815,29818,29821,29824,29827,29830,29835,29838,29842,29846],{"className":29794},[129],[15,29796,10372],{"className":29797},[129,133],[15,29799],{"className":29800,"style":262},[93],[15,29802,14771,29804],{"className":29803},[217],[15,29805,2247],{"style":2246},[15,29807,53],{"className":29808},[188],[15,29810,348],{"className":29811},[129,133],[15,29813,544],{"className":29814},[129],[15,29816,10372],{"className":29817},[129,133],[15,29819,59],{"className":29820},[195],[15,29822],{"className":29823,"style":3277},[93],[15,29825,2682],{"className":29826},[2436],[15,29828],{"className":29829,"style":3277},[93],[15,29831,14771,29833],{"className":29832},[217],[15,29834,2247],{"style":2246},[15,29836,53],{"className":29837},[188],[15,29839,29841],{"className":29840},[129],"1\u002F",[15,29843,29564],{"className":29844,"style":29845},[129,133],"margin-right:0.0379em;",[15,29847,59],{"className":29848},[195],[15,29850,175],{"className":29851},[174],[15,29853,29855],{"className":29854},[146],[15,29856,29858],{"className":29857,"style":1862},[150],[15,29859],{},[15,29861],{"className":29862},[195,1803],[15,29864,29865,29868],{"style":29264},[15,29866],{"className":29867,"style":29131},[158],[15,29869,29871],{"className":29870,"style":29272},[29271],[4455,29872,29873],{"xmlns":4457,"width":29275,"height":29276,"viewBox":29277,"preserveAspectRatio":29278},[4461,29874],{"d":29281},[15,29876,175],{"className":29877},[174],[15,29879,29881],{"className":29880},[146],[15,29882,29884],{"className":29883,"style":29291},[150],[15,29885],{},[15,29887,91],{"className":29888},[254],[11,29890,29891,29892,29943,29944,804,29946,29997,29998,7606,30069,30097],{},"以至少 ",[15,29893,29895,29913],{"className":29894,"translate":19},[23],[15,29896,29898],{"className":29897},[27],[29,29899,29900],{"xmlns":31},[34,29901,29902,29910],{},[37,29903,29904,29906,29908],{},[308,29905,318],{},[50,29907,315],{},[43,29909,29564],{},[109,29911,29912],{"encoding":111},"1-\\delta",[15,29914,29916,29934],{"className":29915,"ariaHidden":72},[116],[15,29917,29919,29922,29925,29928,29931],{"className":29918},[120],[15,29920],{"className":29921,"style":16477},[124],[15,29923,318],{"className":29924},[129],[15,29926],{"className":29927,"style":3277},[93],[15,29929,315],{"className":29930},[2436],[15,29932],{"className":29933,"style":3277},[93],[15,29935,29937,29940],{"className":29936},[120],[15,29938],{"className":29939,"style":522},[124],[15,29941,29564],{"className":29942,"style":29845},[129,133]," 的概率对",[698,29945,7246],{},[15,29947,29949,29967],{"className":29948,"translate":19},[23],[15,29950,29952],{"className":29951},[27],[29,29953,29954],{"xmlns":31},[34,29955,29956,29964],{},[37,29957,29958,29960,29962],{},[43,29959,45],{},[50,29961,100],{},[43,29963,819],{"mathvariant":596},[109,29965,29966],{"encoding":111},"h\\in\\mathcal H",[15,29968,29970,29988],{"className":29969,"ariaHidden":72},[116],[15,29971,29973,29976,29979,29982,29985],{"className":29972},[120],[15,29974],{"className":29975,"style":2405},[124],[15,29977,45],{"className":29978},[129,133],[15,29980],{"className":29981,"style":199},[93],[15,29983,100],{"className":29984},[203],[15,29986],{"className":29987,"style":199},[93],[15,29989,29991,29994],{"className":29990},[120],[15,29992],{"className":29993,"style":610},[124],[15,29995,819],{"className":29996,"style":835},[129,614]," 同时成立，其中 ",[15,29999,30001,30027],{"className":30000,"translate":19},[23],[15,30002,30004],{"className":30003},[27],[29,30005,30006],{"xmlns":31},[34,30007,30008,30024],{},[37,30009,30010,30012,30014,30016,30018,30020,30022],{},[43,30011,10372],{},[50,30013,62],{},[43,30015,6819],{"mathvariant":65},[50,30017,69],{},[50,30019,53],{"stretchy":52},[43,30021,819],{"mathvariant":596},[50,30023,59],{"stretchy":52},[109,30025,30026],{"encoding":111},"d=\\operatorname{VCdim}(\\mathcal H)",[15,30028,30030,30048],{"className":30029,"ariaHidden":72},[116],[15,30031,30033,30036,30039,30042,30045],{"className":30032},[120],[15,30034],{"className":30035,"style":522},[124],[15,30037,10372],{"className":30038},[129,133],[15,30040],{"className":30041,"style":199},[93],[15,30043,62],{"className":30044},[203],[15,30046],{"className":30047,"style":199},[93],[15,30049,30051,30054,30060,30063,30066],{"className":30050},[120],[15,30052],{"className":30053,"style":125},[124],[15,30055,30057],{"className":30056},[217],[15,30058,6819],{"className":30059},[129,221],[15,30061,53],{"className":30062},[188],[15,30064,819],{"className":30065,"style":835},[129,614],[15,30067,59],{"className":30068},[195],[15,30070,30072,30085],{"className":30071,"translate":19},[23],[15,30073,30075],{"className":30074},[27],[29,30076,30077],{"xmlns":31},[34,30078,30079,30083],{},[37,30080,30081],{},[43,30082,5619],{},[109,30084,5619],{"encoding":111},[15,30086,30088],{"className":30087,"ariaHidden":72},[116],[15,30089,30091,30094],{"className":30090},[120],[15,30092],{"className":30093,"style":610},[124],[15,30095,5619],{"className":30096,"style":5665},[129,133]," 是某个绝对常数。",[11,30099,30100,30101,30134,30135,30201,30202,30278],{},"不同教材的常数和 ",[15,30102,30104,30120],{"className":30103,"translate":19},[23],[15,30105,30107],{"className":30106},[27],[29,30108,30109],{"xmlns":31},[34,30110,30111,30117],{},[37,30112,30113,30115],{},[43,30114,14721],{},[50,30116,69],{},[109,30118,30119],{"encoding":111},"\\log",[15,30121,30123],{"className":30122,"ariaHidden":72},[116],[15,30124,30126,30129],{"className":30125},[120],[15,30127],{"className":30128,"style":1682},[124],[15,30130,14771,30132],{"className":30131},[217],[15,30133,2247],{"style":2246}," 里的具体形式略有出入（常见的一种写法是把 ",[15,30136,30138,30166],{"className":30137,"translate":19},[23],[15,30139,30141],{"className":30140},[27],[29,30142,30143],{"xmlns":31},[34,30144,30145,30163],{},[37,30146,30147,30149,30151,30153,30155,30157,30159,30161],{},[43,30148,10372],{},[43,30150,14721],{},[50,30152,69],{},[50,30154,53],{"stretchy":52},[43,30156,348],{},[43,30158,544],{"mathvariant":65},[43,30160,10372],{},[50,30162,59],{"stretchy":52},[109,30164,30165],{"encoding":111},"d\\log(m\u002Fd)",[15,30167,30169],{"className":30168,"ariaHidden":72},[116],[15,30170,30172,30175,30178,30181,30186,30189,30192,30195,30198],{"className":30171},[120],[15,30173],{"className":30174,"style":125},[124],[15,30176,10372],{"className":30177},[129,133],[15,30179],{"className":30180,"style":262},[93],[15,30182,14771,30184],{"className":30183},[217],[15,30185,2247],{"style":2246},[15,30187,53],{"className":30188},[188],[15,30190,348],{"className":30191},[129,133],[15,30193,544],{"className":30194},[129],[15,30196,10372],{"className":30197},[129,133],[15,30199,59],{"className":30200},[195]," 换成 ",[15,30203,30205,30237],{"className":30204,"translate":19},[23],[15,30206,30208],{"className":30207},[27],[29,30209,30210],{"xmlns":31},[34,30211,30212,30234],{},[37,30213,30214,30216,30218,30220,30222,30224,30226,30228,30230,30232],{},[43,30215,10372],{},[43,30217,14721],{},[50,30219,69],{},[50,30221,53],{"stretchy":52},[308,30223,310],{},[43,30225,10695],{},[43,30227,348],{},[43,30229,544],{"mathvariant":65},[43,30231,10372],{},[50,30233,59],{"stretchy":52},[109,30235,30236],{"encoding":111},"d\\log(2em\u002Fd)",[15,30238,30240],{"className":30239,"ariaHidden":72},[116],[15,30241,30243,30246,30249,30252,30257,30260,30263,30266,30269,30272,30275],{"className":30242},[120],[15,30244],{"className":30245,"style":125},[124],[15,30247,10372],{"className":30248},[129,133],[15,30250],{"className":30251,"style":262},[93],[15,30253,14771,30255],{"className":30254},[217],[15,30256,2247],{"style":2246},[15,30258,53],{"className":30259},[188],[15,30261,310],{"className":30262},[129],[15,30264,10695],{"className":30265},[129,133],[15,30267,348],{"className":30268},[129,133],[15,30270,544],{"className":30271},[129],[15,30273,10372],{"className":30274},[129,133],[15,30276,59],{"className":30277},[195],"），但量级是一致的。重要的是它的结构：",[21719,30280,30281,30369,30401],{},[21722,30282,30283,30284,30312,30313,30368],{},"分子里的 ",[15,30285,30287,30300],{"className":30286,"translate":19},[23],[15,30288,30290],{"className":30289},[27],[29,30291,30292],{"xmlns":31},[34,30293,30294,30298],{},[37,30295,30296],{},[43,30297,10372],{},[109,30299,10372],{"encoding":111},[15,30301,30303],{"className":30302,"ariaHidden":72},[116],[15,30304,30306,30309],{"className":30305},[120],[15,30307],{"className":30308,"style":522},[124],[15,30310,10372],{"className":30311},[129,133]," 是容量，",[15,30314,30316,30342],{"className":30315,"translate":19},[23],[15,30317,30319],{"className":30318},[27],[29,30320,30321],{"xmlns":31},[34,30322,30323,30339],{},[37,30324,30325,30327,30329,30331,30333,30335,30337],{},[43,30326,14721],{},[50,30328,69],{},[50,30330,53],{"stretchy":52},[308,30332,318],{},[43,30334,544],{"mathvariant":65},[43,30336,29564],{},[50,30338,59],{"stretchy":52},[109,30340,30341],{"encoding":111},"\\log(1\u002F\\delta)",[15,30343,30345],{"className":30344,"ariaHidden":72},[116],[15,30346,30348,30351,30356,30359,30362,30365],{"className":30347},[120],[15,30349],{"className":30350,"style":125},[124],[15,30352,14771,30354],{"className":30353},[217],[15,30355,2247],{"style":2246},[15,30357,53],{"className":30358},[188],[15,30360,29841],{"className":30361},[129],[15,30363,29564],{"className":30364,"style":29845},[129,133],[15,30366,59],{"className":30367},[195]," 是置信度；",[21722,30370,30371,30372,30400],{},"分母里的 ",[15,30373,30375,30388],{"className":30374,"translate":19},[23],[15,30376,30378],{"className":30377},[27],[29,30379,30380],{"xmlns":31},[34,30381,30382,30386],{},[37,30383,30384],{},[43,30385,348],{},[109,30387,348],{"encoding":111},[15,30389,30391],{"className":30390,"ariaHidden":72},[116],[15,30392,30394,30397],{"className":30393},[120],[15,30395],{"className":30396,"style":4383},[124],[15,30398,348],{"className":30399},[129,133]," 是样本量；",[21722,30402,30403,30404,30492],{},"整个界以 ",[15,30405,30407,30427],{"className":30406,"translate":19},[23],[15,30408,30410],{"className":30409},[27],[29,30411,30412],{"xmlns":31},[34,30413,30414,30424],{},[37,30415,30416,30418,30420],{},[308,30417,318],{},[43,30419,544],{"mathvariant":65},[29004,30421,30422],{},[43,30423,348],{},[109,30425,30426],{"encoding":111},"1\u002F\\sqrt m",[15,30428,30430],{"className":30429,"ariaHidden":72},[116],[15,30431,30433,30437,30440],{"className":30432},[120],[15,30434],{"className":30435,"style":30436},[124],"height:1.0503em;vertical-align:-0.25em;",[15,30438,29841],{"className":30439},[129],[15,30441,30443],{"className":30442},[129,29112],[15,30444,30446,30483],{"className":30445},[141,142],[15,30447,30449,30480],{"className":30448},[146],[15,30450,30453,30463],{"className":30451,"style":30452},[150],"height:0.8003em;",[15,30454,30456,30459],{"className":30455,"style":2287},[29126],[15,30457],{"className":30458,"style":1823},[158],[15,30460,348],{"className":30461,"style":30462},[129,133],"padding-left:0.833em;",[15,30464,30466,30469],{"style":30465},"top:-2.7603em;",[15,30467],{"className":30468,"style":1823},[158],[15,30470,30473],{"className":30471,"style":30472},[29271],"min-width:0.853em;height:1.08em;",[4455,30474,30477],{"xmlns":4457,"width":29275,"height":30475,"viewBox":30476,"preserveAspectRatio":29278},"1.08em","0 0 400000 1080",[4461,30478],{"d":30479},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[15,30481,175],{"className":30482},[174],[15,30484,30486],{"className":30485},[146],[15,30487,30490],{"className":30488,"style":30489},[150],"height:0.2397em;",[15,30491],{}," 的速度下降。在 agnostic 设定下，这个速度是最优的：存在匹配的下界，样本量再少就不可能保证同样的一致收敛。",[11,30494,30495,30496,30589],{},"把 ",[15,30497,30499,30525],{"className":30498,"translate":19},[23],[15,30500,30502],{"className":30501},[27],[29,30503,30504],{"xmlns":31},[34,30505,30506,30522],{},[37,30507,30508,30514,30516,30518,30520],{},[40,30509,30510,30512],{},[43,30511,19679],{"mathvariant":65},[43,30513,819],{"mathvariant":596},[50,30515,53],{"stretchy":52},[308,30517,310],{},[43,30519,348],{},[50,30521,59],{"stretchy":52},[109,30523,30524],{"encoding":111},"\\Pi_{\\mathcal H}(2m)",[15,30526,30528],{"className":30527,"ariaHidden":72},[116],[15,30529,30531,30534,30577,30580,30583,30586],{"className":30530},[120],[15,30532],{"className":30533,"style":125},[124],[15,30535,30537,30540],{"className":30536},[129],[15,30538,19679],{"className":30539},[129],[15,30541,30543],{"className":30542},[137],[15,30544,30546,30569],{"className":30545},[141,142],[15,30547,30549,30566],{"className":30548},[146],[15,30550,30552],{"className":30551,"style":1525},[150],[15,30553,30554,30557],{"style":154},[15,30555],{"className":30556,"style":159},[158],[15,30558,30560],{"className":30559},[163,164,165,166],[15,30561,30563],{"className":30562},[129,166],[15,30564,819],{"className":30565,"style":835},[129,614,166],[15,30567,175],{"className":30568},[174],[15,30570,30572],{"className":30571},[146],[15,30573,30575],{"className":30574,"style":182},[150],[15,30576],{},[15,30578,53],{"className":30579},[188],[15,30581,310],{"className":30582},[129],[15,30584,348],{"className":30585},[129,133],[15,30587,59],{"className":30588},[195]," 用 Sauer 界代入，就得到只依赖 VC 维的形式。再把它代回误差分解，就得到 ERM 的泛化保证。",[575,30591,30593],{"id":30592},"_43-基本定理","4.3 基本定理",[6559,30595,30597,30658],{"title":30596},"定理 4.1（统计学习基本定理）",[11,30598,30599,30600,30628,30629,30657],{},"对二分类、",[15,30601,30603,30616],{"className":30602,"translate":19},[23],[15,30604,30606],{"className":30605},[27],[29,30607,30608],{"xmlns":31},[34,30609,30610,30614],{},[37,30611,30612],{},[308,30613,541],{},[109,30615,541],{"encoding":111},[15,30617,30619],{"className":30618,"ariaHidden":72},[116],[15,30620,30622,30625],{"className":30621},[120],[15,30623],{"className":30624,"style":4886},[124],[15,30626,541],{"className":30627},[129],"-",[15,30630,30632,30645],{"className":30631,"translate":19},[23],[15,30633,30635],{"className":30634},[27],[29,30636,30637],{"xmlns":31},[34,30638,30639,30643],{},[37,30640,30641],{},[308,30642,318],{},[109,30644,318],{"encoding":111},[15,30646,30648],{"className":30647,"ariaHidden":72},[116],[15,30649,30651,30654],{"className":30650},[120],[15,30652],{"className":30653,"style":4886},[124],[15,30655,318],{"className":30656},[129]," 损失、分布无关（distribution-free）的设定，以下四条等价：",[30659,30660,30661,30734,30765,30768],"ol",{},[21722,30662,30663,9207],{},[15,30664,30666,30692],{"className":30665,"translate":19},[23],[15,30667,30669],{"className":30668},[27],[29,30670,30671],{"xmlns":31},[34,30672,30673,30689],{},[37,30674,30675,30677,30679,30681,30683,30685,30687],{},[43,30676,6819],{"mathvariant":65},[50,30678,69],{},[50,30680,53],{"stretchy":52},[43,30682,819],{"mathvariant":596},[50,30684,59],{"stretchy":52},[50,30686,7906],{},[43,30688,4909],{"mathvariant":65},[109,30690,30691],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H)\u003C\\infty",[15,30693,30695,30725],{"className":30694,"ariaHidden":72},[116],[15,30696,30698,30701,30707,30710,30713,30716,30719,30722],{"className":30697},[120],[15,30699],{"className":30700,"style":125},[124],[15,30702,30704],{"className":30703},[217],[15,30705,6819],{"className":30706},[129,221],[15,30708,53],{"className":30709},[188],[15,30711,819],{"className":30712,"style":835},[129,614],[15,30714,59],{"className":30715},[195],[15,30717],{"className":30718,"style":199},[93],[15,30720,7906],{"className":30721},[203],[15,30723],{"className":30724,"style":199},[93],[15,30726,30728,30731],{"className":30727},[120],[15,30729],{"className":30730,"style":4383},[124],[15,30732,4909],{"className":30733},[129],[21722,30735,30736,30764],{},[15,30737,30739,30752],{"className":30738,"translate":19},[23],[15,30740,30742],{"className":30741},[27],[29,30743,30744],{"xmlns":31},[34,30745,30746,30750],{},[37,30747,30748],{},[43,30749,819],{"mathvariant":596},[109,30751,822],{"encoding":111},[15,30753,30755],{"className":30754,"ariaHidden":72},[116],[15,30756,30758,30761],{"className":30757},[120],[15,30759],{"className":30760,"style":610},[124],[15,30762,819],{"className":30763,"style":835},[129,614]," 具有一致收敛性；",[21722,30766,30767],{},"ERM 是 agnostic PAC 一致的；",[21722,30769,30770,30798],{},[15,30771,30773,30786],{"className":30772,"translate":19},[23],[15,30774,30776],{"className":30775},[27],[29,30777,30778],{"xmlns":31},[34,30779,30780,30784],{},[37,30781,30782],{},[43,30783,819],{"mathvariant":596},[109,30785,822],{"encoding":111},[15,30787,30789],{"className":30788,"ariaHidden":72},[116],[15,30790,30792,30795],{"className":30791},[120],[15,30793],{"className":30794,"style":610},[124],[15,30796,819],{"className":30797,"style":835},[129,614]," 是 agnostic PAC 可学习的。",[11,30800,30801,30802],{},"在可实现（realizable）情形下，有限 VC 维同样等价于 PAC 可学习。这个定理把「组合上的有限性」和「统计上的可学习性」严格地焊在了一起：",[698,30803,30804],{},"在分布无关的框架里，有限 VC 维不只是充分条件，而是充要条件。",[575,30806,30808],{"id":30807},"_44-样本复杂度","4.4 样本复杂度",[11,30810,30811,30812,30840],{},"由上面的界解出 ",[15,30813,30815,30828],{"className":30814,"translate":19},[23],[15,30816,30818],{"className":30817},[27],[29,30819,30820],{"xmlns":31},[34,30821,30822,30826],{},[37,30823,30824],{},[43,30825,348],{},[109,30827,348],{"encoding":111},[15,30829,30831],{"className":30830,"ariaHidden":72},[116],[15,30832,30834,30837],{"className":30833},[120],[15,30835],{"className":30836,"style":4383},[124],[15,30838,348],{"className":30839},[129,133],"，得到 agnostic 情形下的样本复杂度：",[15,30842,30844],{"className":30843,"translate":19},[18],[15,30845,30847,30902],{"className":30846,"translate":19},[23],[15,30848,30850],{"className":30849},[27],[29,30851,30852],{"xmlns":31,"display":32},[34,30853,30854,30899],{},[37,30855,30856,30858,30860,30863,30897],{},[43,30857,348],{},[50,30859,62],{},[43,30861,30862],{"mathvariant":65},"Θ",[37,30864,30865,30867,30895],{},[50,30866,53],{"fence":72},[1443,30868,30869,30889],{},[37,30870,30871,30873,30875,30877,30879,30881,30883,30885,30887],{},[43,30872,10372],{},[50,30874,2682],{},[43,30876,14721],{},[50,30878,69],{},[50,30880,53],{"stretchy":52},[308,30882,318],{},[43,30884,544],{"mathvariant":65},[43,30886,29564],{},[50,30888,59],{"stretchy":52},[305,30890,30891,30893],{},[43,30892,4849],{},[308,30894,310],{},[50,30896,59],{"fence":72},[50,30898,91],{"separator":72},[109,30900,30901],{"encoding":111},"m=\\Theta\\left(\\frac{d+\\log(1\u002F\\delta)}{\\varepsilon^2}\\right),",[15,30903,30905,30923],{"className":30904,"ariaHidden":72},[116],[15,30906,30908,30911,30914,30917,30920],{"className":30907},[120],[15,30909],{"className":30910,"style":4383},[124],[15,30912,348],{"className":30913},[129,133],[15,30915],{"className":30916,"style":199},[93],[15,30918,62],{"className":30919},[203],[15,30921],{"className":30922,"style":199},[93],[15,30924,30926,30929,30932,30935,31065,31068],{"className":30925},[120],[15,30927],{"className":30928,"style":17007},[124],[15,30930,30862],{"className":30931},[129],[15,30933],{"className":30934,"style":262},[93],[15,30936,30938,30944,31059],{"className":30937},[449],[15,30939,30941],{"className":30940,"style":5052},[188,5051],[15,30942,53],{"className":30943},[228,165],[15,30945,30947,30950,31056],{"className":30946},[129],[15,30948],{"className":30949},[188,1803],[15,30951,30953],{"className":30952},[1443],[15,30954,30956,31048],{"className":30955},[141,142],[15,30957,30959,31045],{"className":30958},[146],[15,30960,30962,31000,31008],{"className":30961,"style":29154},[150],[15,30963,30964,30967],{"style":1819},[15,30965],{"className":30966,"style":1823},[158],[15,30968,30970],{"className":30969},[129],[15,30971,30973,30976],{"className":30972},[129],[15,30974,4849],{"className":30975},[129,133],[15,30977,30979],{"className":30978},[137],[15,30980,30982],{"className":30981},[141],[15,30983,30985],{"className":30984},[146],[15,30986,30989],{"className":30987,"style":30988},[150],"height:0.7401em;",[15,30990,30991,30994],{"style":17194},[15,30992],{"className":30993,"style":159},[158],[15,30995,30997],{"className":30996},[163,164,165,166],[15,30998,310],{"className":30999},[129,166],[15,31001,31002,31005],{"style":1832},[15,31003],{"className":31004,"style":1823},[158],[15,31006],{"className":31007,"style":1840},[1839],[15,31009,31010,31013],{"style":1843},[15,31011],{"className":31012,"style":1823},[158],[15,31014,31016,31019,31022,31025,31028,31033,31036,31039,31042],{"className":31015},[129],[15,31017,10372],{"className":31018},[129,133],[15,31020],{"className":31021,"style":3277},[93],[15,31023,2682],{"className":31024},[2436],[15,31026],{"className":31027,"style":3277},[93],[15,31029,14771,31031],{"className":31030},[217],[15,31032,2247],{"style":2246},[15,31034,53],{"className":31035},[188],[15,31037,29841],{"className":31038},[129],[15,31040,29564],{"className":31041,"style":29845},[129,133],[15,31043,59],{"className":31044},[195],[15,31046,175],{"className":31047},[174],[15,31049,31051],{"className":31050},[146],[15,31052,31054],{"className":31053,"style":1862},[150],[15,31055],{},[15,31057],{"className":31058},[195,1803],[15,31060,31062],{"className":31061,"style":5052},[195,5051],[15,31063,59],{"className":31064},[228,165],[15,31066],{"className":31067,"style":262},[93],[15,31069,91],{"className":31070},[254],[11,31072,31073],{},"可实现情形下则是",[15,31075,31077],{"className":31076,"translate":19},[18],[15,31078,31080,31130],{"className":31079,"translate":19},[23],[15,31081,31083],{"className":31082},[27],[29,31084,31085],{"xmlns":31,"display":32},[34,31086,31087,31127],{},[37,31088,31089,31091,31093,31095,31125],{},[43,31090,348],{},[50,31092,62],{},[43,31094,30862],{"mathvariant":65},[37,31096,31097,31099,31123],{},[50,31098,53],{"fence":72},[1443,31100,31101,31121],{},[37,31102,31103,31105,31107,31109,31111,31113,31115,31117,31119],{},[43,31104,10372],{},[50,31106,2682],{},[43,31108,14721],{},[50,31110,69],{},[50,31112,53],{"stretchy":52},[308,31114,318],{},[43,31116,544],{"mathvariant":65},[43,31118,29564],{},[50,31120,59],{"stretchy":52},[43,31122,4849],{},[50,31124,59],{"fence":72},[43,31126,107],{"mathvariant":65},[109,31128,31129],{"encoding":111},"m=\\Theta\\left(\\frac{d+\\log(1\u002F\\delta)}{\\varepsilon}\\right).",[15,31131,31133,31151],{"className":31132,"ariaHidden":72},[116],[15,31134,31136,31139,31142,31145,31148],{"className":31135},[120],[15,31137],{"className":31138,"style":4383},[124],[15,31140,348],{"className":31141},[129,133],[15,31143],{"className":31144,"style":199},[93],[15,31146,62],{"className":31147},[203],[15,31149],{"className":31150,"style":199},[93],[15,31152,31154,31157,31160,31163,31266,31269],{"className":31153},[120],[15,31155],{"className":31156,"style":17007},[124],[15,31158,30862],{"className":31159},[129],[15,31161],{"className":31162,"style":262},[93],[15,31164,31166,31172,31260],{"className":31165},[449],[15,31167,31169],{"className":31168,"style":5052},[188,5051],[15,31170,53],{"className":31171},[228,165],[15,31173,31175,31178,31257],{"className":31174},[129],[15,31176],{"className":31177},[188,1803],[15,31179,31181],{"className":31180},[1443],[15,31182,31184,31249],{"className":31183},[141,142],[15,31185,31187,31246],{"className":31186},[146],[15,31188,31190,31201,31209],{"className":31189,"style":29154},[150],[15,31191,31192,31195],{"style":1819},[15,31193],{"className":31194,"style":1823},[158],[15,31196,31198],{"className":31197},[129],[15,31199,4849],{"className":31200},[129,133],[15,31202,31203,31206],{"style":1832},[15,31204],{"className":31205,"style":1823},[158],[15,31207],{"className":31208,"style":1840},[1839],[15,31210,31211,31214],{"style":1843},[15,31212],{"className":31213,"style":1823},[158],[15,31215,31217,31220,31223,31226,31229,31234,31237,31240,31243],{"className":31216},[129],[15,31218,10372],{"className":31219},[129,133],[15,31221],{"className":31222,"style":3277},[93],[15,31224,2682],{"className":31225},[2436],[15,31227],{"className":31228,"style":3277},[93],[15,31230,14771,31232],{"className":31231},[217],[15,31233,2247],{"style":2246},[15,31235,53],{"className":31236},[188],[15,31238,29841],{"className":31239},[129],[15,31241,29564],{"className":31242,"style":29845},[129,133],[15,31244,59],{"className":31245},[195],[15,31247,175],{"className":31248},[174],[15,31250,31252],{"className":31251},[146],[15,31253,31255],{"className":31254,"style":1862},[150],[15,31256],{},[15,31258],{"className":31259},[195,1803],[15,31261,31263],{"className":31262,"style":5052},[195,5051],[15,31264,59],{"className":31265},[228,165],[15,31267],{"className":31268,"style":262},[93],[15,31270,107],{"className":31271},[129],[11,31273,31274,31275,31304,31305,31308],{},"这两个 ",[15,31276,31278,31292],{"className":31277,"translate":19},[23],[15,31279,31281],{"className":31280},[27],[29,31282,31283],{"xmlns":31},[34,31284,31285,31289],{},[37,31286,31287],{},[43,31288,30862],{"mathvariant":65},[109,31290,31291],{"encoding":111},"\\Theta",[15,31293,31295],{"className":31294,"ariaHidden":72},[116],[15,31296,31298,31301],{"className":31297},[120],[15,31299],{"className":31300,"style":610},[124],[15,31302,30862],{"className":31303},[129]," 是双向的：上界由 ERM 达到，下界则是信息论下界——存在一个分布，使任何算法都需要这么多样本。所以 VC 维不仅给出了一个可用的界，还在 minimax 意义下给出了",[698,31306,31307],{},"最优","的样本复杂度量级。",[11,31310,31311],{},"这一点很重要。很多人批评 VC 界「太松」，但在分布无关、最坏情况的设定里，它并不松：你不可能用更少的样本保证同样的一致收敛。问题不在于界本身，而在于这个设定是否匹配你关心的场景。",[570,31313,31315],{"id":31314},"五vc-维的含义","五、VC 维的含义",[11,31317,31318],{},"到这里，数学已经完整了。但公式正确不等于理解。这一节把 VC 维的语义拆成五句话。",[11,31320,31321,31324,31325,31401,31402,31479,31480,1313],{},[698,31322,31323],{},"第一，它是 min-max 的。"," 定义里的 ",[15,31326,31328,31350],{"className":31327,"translate":19},[23],[15,31329,31331],{"className":31330},[27],[29,31332,31333],{"xmlns":31},[34,31334,31335,31347],{},[37,31336,31337],{},[40,31338,31339,31345],{},[37,31340,31341,31343],{},[43,31342,2932],{},[50,31344,69],{},[43,31346,5619],{},[109,31348,31349],{"encoding":111},"\\sup_C",[15,31351,31353],{"className":31352,"ariaHidden":72},[116],[15,31354,31356,31360],{"className":31355},[120],[15,31357],{"className":31358,"style":31359},[124],"height:0.6747em;vertical-align:-0.2441em;",[15,31361,31363,31366],{"className":31362},[217],[15,31364,2932],{"className":31365},[217],[15,31367,31369],{"className":31368},[137],[15,31370,31372,31393],{"className":31371},[141,142],[15,31373,31375,31390],{"className":31374},[146],[15,31376,31379],{"className":31377,"style":31378},[150],"height:0.2342em;",[15,31380,31381,31384],{"style":14789},[15,31382],{"className":31383,"style":159},[158],[15,31385,31387],{"className":31386},[163,164,165,166],[15,31388,5619],{"className":31389,"style":5665},[129,133,166],[15,31391,175],{"className":31392},[174],[15,31394,31396],{"className":31395},[146],[15,31397,31399],{"className":31398,"style":14808},[150],[15,31400],{}," 对应界里的 ",[15,31403,31405,31427],{"className":31404,"translate":19},[23],[15,31406,31408],{"className":31407},[27],[29,31409,31410],{"xmlns":31},[34,31411,31412,31424],{},[37,31413,31414],{},[40,31415,31416,31422],{},[37,31417,31418,31420],{},[43,31419,2932],{},[50,31421,69],{},[43,31423,906],{"mathvariant":596},[109,31425,31426],{"encoding":111},"\\sup_{\\mathcal D}",[15,31428,31430],{"className":31429,"ariaHidden":72},[116],[15,31431,31433,31436],{"className":31432},[120],[15,31434],{"className":31435,"style":31359},[124],[15,31437,31439,31442],{"className":31438},[217],[15,31440,2932],{"className":31441},[217],[15,31443,31445],{"className":31444},[137],[15,31446,31448,31471],{"className":31447},[141,142],[15,31449,31451,31468],{"className":31450},[146],[15,31452,31454],{"className":31453,"style":31378},[150],[15,31455,31456,31459],{"style":14789},[15,31457],{"className":31458,"style":159},[158],[15,31460,31462],{"className":31461},[163,164,165,166],[15,31463,31465],{"className":31464},[129,166],[15,31466,906],{"className":31467,"style":170},[129,614,166],[15,31469,175],{"className":31470},[174],[15,31472,31474],{"className":31473},[146],[15,31475,31477],{"className":31476,"style":14808},[150],[15,31478],{},"。VC 理论先让对手选一个最坏的分布，再让对手选一个最坏的点集，然后问：你还剩什么保证？所以它天生悲观。它给出的不是一个典型情形下的估计，而是一个",[698,31481,31482],{},"任何情形下都不被打破的承诺",[11,31484,31485,5831,31488,30628,31516,31544],{},[698,31486,31487],{},"第二，它是充要的。",[15,31489,31491,31504],{"className":31490,"translate":19},[23],[15,31492,31494],{"className":31493},[27],[29,31495,31496],{"xmlns":31},[34,31497,31498,31502],{},[37,31499,31500],{},[308,31501,541],{},[109,31503,541],{"encoding":111},[15,31505,31507],{"className":31506,"ariaHidden":72},[116],[15,31508,31510,31513],{"className":31509},[120],[15,31511],{"className":31512,"style":4886},[124],[15,31514,541],{"className":31515},[129],[15,31517,31519,31532],{"className":31518,"translate":19},[23],[15,31520,31522],{"className":31521},[27],[29,31523,31524],{"xmlns":31},[34,31525,31526,31530],{},[37,31527,31528],{},[308,31529,318],{},[109,31531,318],{"encoding":111},[15,31533,31535],{"className":31534,"ariaHidden":72},[116],[15,31536,31538,31541],{"className":31537},[120],[15,31539],{"className":31540,"style":4886},[124],[15,31542,318],{"className":31543},[129]," 损失、二分类、要求对所有分布一致的框架里，有限 VC 维精确刻画了可学习性。它不是「一个充分但过强的条件」；它就是那个条件。问题不在定理，而在框架。",[11,31546,31547,31550],{},[698,31548,31549],{},"第三，它是组合量，概率只出现在最后一步。"," 打散、增长函数、Sauer 引理都是纯组合的，与分布、与概率无关；对称化把组合量变成概率界。这也解释了为什么 VC 维对数据长什么样一无所知：它的输入只有假设类的集合结构。",[11,31552,31553,31556,31557,31621],{},[698,31554,31555],{},"第四，它衡量的是「能实现多少种二分」，不是「能拟合多复杂的函数」。"," 一个类可以 VC 维无穷却在某个分布上表现很好（1-NN），也可以 VC 维很小却拟合得很差（区间无法实现 ",[15,31558,31560,31585],{"className":31559,"translate":19},[23],[15,31561,31563],{"className":31562},[27],[29,31564,31565],{"xmlns":31},[34,31566,31567,31583],{},[37,31568,31569,31571,31573,31575,31577,31579,31581],{},[50,31570,53],{"stretchy":52},[308,31572,318],{},[50,31574,91],{"separator":72},[308,31576,541],{},[50,31578,91],{"separator":72},[308,31580,318],{},[50,31582,59],{"stretchy":52},[109,31584,10067],{"encoding":111},[15,31586,31588],{"className":31587,"ariaHidden":72},[116],[15,31589,31591,31594,31597,31600,31603,31606,31609,31612,31615,31618],{"className":31590},[120],[15,31592],{"className":31593,"style":125},[124],[15,31595,53],{"className":31596},[188],[15,31598,318],{"className":31599},[129],[15,31601,91],{"className":31602},[254],[15,31604],{"className":31605,"style":262},[93],[15,31607,541],{"className":31608},[129],[15,31610,91],{"className":31611},[254],[15,31613],{"className":31614,"style":262},[93],[15,31616,318],{"className":31617},[129],[15,31619,59],{"className":31620},[195]," 这种标注）。VC 维是容量，不是质量。",[11,31623,31624,804,31627,31787,31788,31816],{},[698,31625,31626],{},"第五，它是上界，不是预测。",[15,31628,31630,31661],{"className":31629,"translate":19},[23],[15,31631,31633],{"className":31632},[27],[29,31634,31635],{"xmlns":31},[34,31636,31637,31658],{},[37,31638,31639,31645,31647,31653,31655],{},[40,31640,31641,31643],{},[43,31642,1373],{},[43,31644,906],{"mathvariant":596},[50,31646,2717],{},[40,31648,31649,31651],{},[43,31650,1373],{},[43,31652,940],{},[50,31654,2682],{},[50,31656,31657],{},"⋯",[109,31659,31660],{"encoding":111},"L_{\\mathcal D}\\le L_S+\\cdots",[15,31662,31664,31722,31777],{"className":31663,"ariaHidden":72},[116],[15,31665,31667,31670,31713,31716,31719],{"className":31666},[120],[15,31668],{"className":31669,"style":20220},[124],[15,31671,31673,31676],{"className":31672},[129],[15,31674,1373],{"className":31675},[129,133],[15,31677,31679],{"className":31678},[137],[15,31680,31682,31705],{"className":31681},[141,142],[15,31683,31685,31702],{"className":31684},[146],[15,31686,31688],{"className":31687,"style":1525},[150],[15,31689,31690,31693],{"style":154},[15,31691],{"className":31692,"style":159},[158],[15,31694,31696],{"className":31695},[163,164,165,166],[15,31697,31699],{"className":31698},[129,166],[15,31700,906],{"className":31701,"style":170},[129,614,166],[15,31703,175],{"className":31704},[174],[15,31706,31708],{"className":31707},[146],[15,31709,31711],{"className":31710,"style":182},[150],[15,31712],{},[15,31714],{"className":31715,"style":199},[93],[15,31717,2717],{"className":31718},[203],[15,31720],{"className":31721,"style":199},[93],[15,31723,31725,31728,31768,31771,31774],{"className":31724},[120],[15,31726],{"className":31727,"style":20220},[124],[15,31729,31731,31734],{"className":31730},[129],[15,31732,1373],{"className":31733},[129,133],[15,31735,31737],{"className":31736},[137],[15,31738,31740,31760],{"className":31739},[141,142],[15,31741,31743,31757],{"className":31742},[146],[15,31744,31746],{"className":31745,"style":1525},[150],[15,31747,31748,31751],{"style":154},[15,31749],{"className":31750,"style":159},[158],[15,31752,31754],{"className":31753},[163,164,165,166],[15,31755,940],{"className":31756,"style":1013},[129,133,166],[15,31758,175],{"className":31759},[174],[15,31761,31763],{"className":31762},[146],[15,31764,31766],{"className":31765,"style":182},[150],[15,31767],{},[15,31769],{"className":31770,"style":3277},[93],[15,31772,2682],{"className":31773},[2436],[15,31775],{"className":31776,"style":3277},[93],[15,31778,31780,31784],{"className":31779},[120],[15,31781],{"className":31782,"style":31783},[124],"height:0.313em;",[15,31785,31657],{"className":31786},[449]," 是一个保证（guarantee），不是一个估计（estimate）。当界大于 ",[15,31789,31791,31804],{"className":31790,"translate":19},[23],[15,31792,31794],{"className":31793},[27],[29,31795,31796],{"xmlns":31},[34,31797,31798,31802],{},[37,31799,31800],{},[308,31801,318],{},[109,31803,318],{"encoding":111},[15,31805,31807],{"className":31806,"ariaHidden":72},[116],[15,31808,31810,31813],{"className":31809},[120],[15,31811],{"className":31812,"style":4886},[124],[15,31814,318],{"className":31815},[129]," 时，它没有说「误差很大」，它只是没有信息。把泛化界当成测试误差的预测器，是误用的常见来源。",[11,31818,31819,31820,31823],{},"这五句话合起来，解释了为什么 VC 理论在深度学习时代显得「失灵」：它回答的是一个比深度学习实际面对的问题",[698,31821,31822],{},"更难","的问题。深度网络不是在一个任意的分布上学习，而是在图像、文本、语音这些有强结构的分布上学习；不是用任意算法，而是用 SGD 及其变体；不是在整个假设类里搜索，而是在一个由初始化、架构和训练动力学共同决定的子集里搜索。VC 维对这三件事都不敏感。",[31825,31826,31827],"div",{},[31828,31829],"img",{"height":31830,"width":31831,"style":31832,"data-d2-theme":31833,"alt":31834,"src":31835},68,760,"max-width:100%;height:auto","default","VC 理论的逻辑链：从打散到泛化界","data:image\u002Fsvg+xml;base64,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\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",[570,31837,31839],{"id":31838},"六深度学习中的-vc-维","六、深度学习中的 VC 维",[575,31841,31843],{"id":31842},"_61-参数量估算","6.1 参数量估算",[11,31845,31846,31847,31943,31944,31972,31973,32002,32003,32035],{},"把上一节的界套到实际模型上，只需估算 ",[15,31848,31850,31872],{"className":31849,"translate":19},[23],[15,31851,31853],{"className":31852},[27],[29,31854,31855],{"xmlns":31},[34,31856,31857,31869],{},[37,31858,31859],{},[29004,31860,31861],{},[37,31862,31863,31865,31867],{},[43,31864,10372],{},[43,31866,544],{"mathvariant":65},[43,31868,348],{},[109,31870,31871],{"encoding":111},"\\sqrt{d\u002Fm}",[15,31873,31875],{"className":31874,"ariaHidden":72},[116],[15,31876,31878,31882],{"className":31877},[120],[15,31879],{"className":31880,"style":31881},[124],"height:1.24em;vertical-align:-0.305em;",[15,31883,31885],{"className":31884},[129,29112],[15,31886,31888,31934],{"className":31887},[141,142],[15,31889,31891,31931],{"className":31890},[146],[15,31892,31895,31914],{"className":31893,"style":31894},[150],"height:0.935em;",[15,31896,31899,31902],{"className":31897,"style":31898},[29126],"top:-3.2em;",[15,31900],{"className":31901,"style":4450},[158],[15,31903,31905,31908,31911],{"className":31904,"style":29135},[129],[15,31906,10372],{"className":31907},[129,133],[15,31909,544],{"className":31910},[129],[15,31912,348],{"className":31913},[129,133],[15,31915,31917,31920],{"style":31916},"top:-2.895em;",[15,31918],{"className":31919,"style":4450},[158],[15,31921,31924],{"className":31922,"style":31923},[29271],"min-width:1.02em;height:1.28em;",[4455,31925,31928],{"xmlns":4457,"width":29275,"height":31926,"viewBox":31927,"preserveAspectRatio":29278},"1.28em","0 0 400000 1296",[4461,31929],{"d":31930},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[15,31932,175],{"className":31933},[174],[15,31935,31937],{"className":31936},[146],[15,31938,31941],{"className":31939,"style":31940},[150],"height:0.305em;",[15,31942],{}," 的量级（暂时把 ",[15,31945,31947,31960],{"className":31946,"translate":19},[23],[15,31948,31950],{"className":31949},[27],[29,31951,31952],{"xmlns":31},[34,31953,31954,31958],{},[37,31955,31956],{},[43,31957,10372],{},[109,31959,10372],{"encoding":111},[15,31961,31963],{"className":31962,"ariaHidden":72},[116],[15,31964,31966,31969],{"className":31965},[120],[15,31967],{"className":31968,"style":522},[124],[15,31970,10372],{"className":31971},[129,133]," 低估成参数量 ",[15,31974,31976,31990],{"className":31975,"translate":19},[23],[15,31977,31979],{"className":31978},[27],[29,31980,31981],{"xmlns":31},[34,31982,31983,31988],{},[37,31984,31985],{},[43,31986,31987],{},"W",[109,31989,31987],{"encoding":111},[15,31991,31993],{"className":31992,"ariaHidden":72},[116],[15,31994,31996,31999],{"className":31995},[120],[15,31997],{"className":31998,"style":610},[124],[15,32000,31987],{"className":32001,"style":12443},[129,133],"，忽略 ",[15,32004,32006,32021],{"className":32005,"translate":19},[23],[15,32007,32009],{"className":32008},[27],[29,32010,32011],{"xmlns":31},[34,32012,32013,32019],{},[37,32014,32015,32017],{},[43,32016,14721],{},[50,32018,69],{},[109,32020,30119],{"encoding":111},[15,32022,32024],{"className":32023,"ariaHidden":72},[116],[15,32025,32027,32030],{"className":32026},[120],[15,32028],{"className":32029,"style":1682},[124],[15,32031,14771,32033],{"className":32032},[217],[15,32034,2247],{"style":2246}," 因子）：",[18033,32037,32038,32196],{},[18036,32039,32040],{},[18039,32041,32042,32045,32076,32107],{},[18042,32043,32044],{},"模型",[18042,32046,32047,32048],{},"参数量 ",[15,32049,32051,32064],{"className":32050,"translate":19},[23],[15,32052,32054],{"className":32053},[27],[29,32055,32056],{"xmlns":31},[34,32057,32058,32062],{},[37,32059,32060],{},[43,32061,31987],{},[109,32063,31987],{"encoding":111},[15,32065,32067],{"className":32066,"ariaHidden":72},[116],[15,32068,32070,32073],{"className":32069},[120],[15,32071],{"className":32072,"style":610},[124],[15,32074,31987],{"className":32075,"style":12443},[129,133],[18042,32077,32078,32079],{},"训练样本 ",[15,32080,32082,32095],{"className":32081,"translate":19},[23],[15,32083,32085],{"className":32084},[27],[29,32086,32087],{"xmlns":31},[34,32088,32089,32093],{},[37,32090,32091],{},[43,32092,348],{},[109,32094,348],{"encoding":111},[15,32096,32098],{"className":32097,"ariaHidden":72},[116],[15,32099,32101,32104],{"className":32100},[120],[15,32102],{"className":32103,"style":4383},[124],[15,32105,348],{"className":32106},[129,133],[18042,32108,32109],{},[15,32110,32112,32134],{"className":32111,"translate":19},[23],[15,32113,32115],{"className":32114},[27],[29,32116,32117],{"xmlns":31},[34,32118,32119,32131],{},[37,32120,32121],{},[29004,32122,32123],{},[37,32124,32125,32127,32129],{},[43,32126,31987],{},[43,32128,544],{"mathvariant":65},[43,32130,348],{},[109,32132,32133],{"encoding":111},"\\sqrt{W\u002Fm}",[15,32135,32137],{"className":32136,"ariaHidden":72},[116],[15,32138,32140,32143],{"className":32139},[120],[15,32141],{"className":32142,"style":31881},[124],[15,32144,32146],{"className":32145},[129,29112],[15,32147,32149,32188],{"className":32148},[141,142],[15,32150,32152,32185],{"className":32151},[146],[15,32153,32155,32173],{"className":32154,"style":31894},[150],[15,32156,32158,32161],{"className":32157,"style":31898},[29126],[15,32159],{"className":32160,"style":4450},[158],[15,32162,32164,32167,32170],{"className":32163,"style":29135},[129],[15,32165,31987],{"className":32166,"style":12443},[129,133],[15,32168,544],{"className":32169},[129],[15,32171,348],{"className":32172},[129,133],[15,32174,32175,32178],{"style":31916},[15,32176],{"className":32177,"style":4450},[158],[15,32179,32181],{"className":32180,"style":31923},[29271],[4455,32182,32183],{"xmlns":4457,"width":29275,"height":31926,"viewBox":31927,"preserveAspectRatio":29278},[4461,32184],{"d":31930},[15,32186,175],{"className":32187},[174],[15,32189,32191],{"className":32190},[146],[15,32192,32194],{"className":32193,"style":31940},[150],[15,32195],{},[18050,32197,32198,32423,32649,32872],{},[18039,32199,32200,32203,32291,32376],{},[18055,32201,32202],{},"LeNet-5（MNIST）",[18055,32204,32205],{},[15,32206,32208,32232],{"className":32207,"translate":19},[23],[15,32209,32211],{"className":32210},[27],[29,32212,32213],{"xmlns":31},[34,32214,32215,32229],{},[37,32216,32217,32220,32222],{},[308,32218,32219],{},"6",[50,32221,11418],{},[305,32223,32224,32227],{},[308,32225,32226],{},"10",[308,32228,14502],{},[109,32230,32231],{"encoding":111},"6\\times10^4",[15,32233,32235,32253],{"className":32234,"ariaHidden":72},[116],[15,32236,32238,32241,32244,32247,32250],{"className":32237},[120],[15,32239],{"className":32240,"style":16477},[124],[15,32242,32219],{"className":32243},[129],[15,32245],{"className":32246,"style":3277},[93],[15,32248,11418],{"className":32249},[2436],[15,32251],{"className":32252,"style":3277},[93],[15,32254,32256,32259,32262],{"className":32255},[120],[15,32257],{"className":32258,"style":380},[124],[15,32260,318],{"className":32261},[129],[15,32263,32265,32268],{"className":32264},[129],[15,32266,541],{"className":32267},[129],[15,32269,32271],{"className":32270},[137],[15,32272,32274],{"className":32273},[141],[15,32275,32277],{"className":32276},[146],[15,32278,32280],{"className":32279,"style":380},[150],[15,32281,32282,32285],{"style":383},[15,32283],{"className":32284,"style":159},[158],[15,32286,32288],{"className":32287},[163,164,165,166],[15,32289,14502],{"className":32290},[129,166],[18055,32292,32293],{},[15,32294,32296,32317],{"className":32295,"translate":19},[23],[15,32297,32299],{"className":32298},[27],[29,32300,32301],{"xmlns":31},[34,32302,32303,32315],{},[37,32304,32305,32307,32309],{},[308,32306,32219],{},[50,32308,11418],{},[305,32310,32311,32313],{},[308,32312,32226],{},[308,32314,14502],{},[109,32316,32231],{"encoding":111},[15,32318,32320,32338],{"className":32319,"ariaHidden":72},[116],[15,32321,32323,32326,32329,32332,32335],{"className":32322},[120],[15,32324],{"className":32325,"style":16477},[124],[15,32327,32219],{"className":32328},[129],[15,32330],{"className":32331,"style":3277},[93],[15,32333,11418],{"className":32334},[2436],[15,32336],{"className":32337,"style":3277},[93],[15,32339,32341,32344,32347],{"className":32340},[120],[15,32342],{"className":32343,"style":380},[124],[15,32345,318],{"className":32346},[129],[15,32348,32350,32353],{"className":32349},[129],[15,32351,541],{"className":32352},[129],[15,32354,32356],{"className":32355},[137],[15,32357,32359],{"className":32358},[141],[15,32360,32362],{"className":32361},[146],[15,32363,32365],{"className":32364,"style":380},[150],[15,32366,32367,32370],{"style":383},[15,32368],{"className":32369,"style":159},[158],[15,32371,32373],{"className":32372},[163,164,165,166],[15,32374,14502],{"className":32375},[129,166],[18055,32377,32378],{},[15,32379,32381,32398],{"className":32380,"translate":19},[23],[15,32382,32384],{"className":32383},[27],[29,32385,32386],{"xmlns":31},[34,32387,32388,32395],{},[37,32389,32390,32393],{},[50,32391,32392],{},"≈",[308,32394,318],{},[109,32396,32397],{"encoding":111},"\\approx 1",[15,32399,32401,32414],{"className":32400,"ariaHidden":72},[116],[15,32402,32404,32408,32411],{"className":32403},[120],[15,32405],{"className":32406,"style":32407},[124],"height:0.4831em;",[15,32409,32392],{"className":32410},[203],[15,32412],{"className":32413,"style":199},[93],[15,32415,32417,32420],{"className":32416},[120],[15,32418],{"className":32419,"style":4886},[124],[15,32421,318],{"className":32422},[129],[18039,32424,32425,32428,32516,32603],{},[18055,32426,32427],{},"ResNet-18（ImageNet）",[18055,32429,32430],{},[15,32431,32433,32457],{"className":32432,"translate":19},[23],[15,32434,32436],{"className":32435},[27],[29,32437,32438],{"xmlns":31},[34,32439,32440,32454],{},[37,32441,32442,32445,32447],{},[308,32443,32444],{},"1.17",[50,32446,11418],{},[305,32448,32449,32451],{},[308,32450,32226],{},[308,32452,32453],{},"7",[109,32455,32456],{"encoding":111},"1.17\\times10^7",[15,32458,32460,32478],{"className":32459,"ariaHidden":72},[116],[15,32461,32463,32466,32469,32472,32475],{"className":32462},[120],[15,32464],{"className":32465,"style":16477},[124],[15,32467,32444],{"className":32468},[129],[15,32470],{"className":32471,"style":3277},[93],[15,32473,11418],{"className":32474},[2436],[15,32476],{"className":32477,"style":3277},[93],[15,32479,32481,32484,32487],{"className":32480},[120],[15,32482],{"className":32483,"style":380},[124],[15,32485,318],{"className":32486},[129],[15,32488,32490,32493],{"className":32489},[129],[15,32491,541],{"className":32492},[129],[15,32494,32496],{"className":32495},[137],[15,32497,32499],{"className":32498},[141],[15,32500,32502],{"className":32501},[146],[15,32503,32505],{"className":32504,"style":380},[150],[15,32506,32507,32510],{"style":383},[15,32508],{"className":32509,"style":159},[158],[15,32511,32513],{"className":32512},[163,164,165,166],[15,32514,32453],{"className":32515},[129,166],[18055,32517,32518],{},[15,32519,32521,32544],{"className":32520,"translate":19},[23],[15,32522,32524],{"className":32523},[27],[29,32525,32526],{"xmlns":31},[34,32527,32528,32541],{},[37,32529,32530,32533,32535],{},[308,32531,32532],{},"1.28",[50,32534,11418],{},[305,32536,32537,32539],{},[308,32538,32226],{},[308,32540,32219],{},[109,32542,32543],{"encoding":111},"1.28\\times10^6",[15,32545,32547,32565],{"className":32546,"ariaHidden":72},[116],[15,32548,32550,32553,32556,32559,32562],{"className":32549},[120],[15,32551],{"className":32552,"style":16477},[124],[15,32554,32532],{"className":32555},[129],[15,32557],{"className":32558,"style":3277},[93],[15,32560,11418],{"className":32561},[2436],[15,32563],{"className":32564,"style":3277},[93],[15,32566,32568,32571,32574],{"className":32567},[120],[15,32569],{"className":32570,"style":380},[124],[15,32572,318],{"className":32573},[129],[15,32575,32577,32580],{"className":32576},[129],[15,32578,541],{"className":32579},[129],[15,32581,32583],{"className":32582},[137],[15,32584,32586],{"className":32585},[141],[15,32587,32589],{"className":32588},[146],[15,32590,32592],{"className":32591,"style":380},[150],[15,32593,32594,32597],{"style":383},[15,32595],{"className":32596,"style":159},[158],[15,32598,32600],{"className":32599},[163,164,165,166],[15,32601,32219],{"className":32602},[129,166],[18055,32604,32605],{},[15,32606,32608,32625],{"className":32607,"translate":19},[23],[15,32609,32611],{"className":32610},[27],[29,32612,32613],{"xmlns":31},[34,32614,32615,32622],{},[37,32616,32617,32619],{},[50,32618,32392],{},[308,32620,32621],{},"3.0",[109,32623,32624],{"encoding":111},"\\approx 3.0",[15,32626,32628,32640],{"className":32627,"ariaHidden":72},[116],[15,32629,32631,32634,32637],{"className":32630},[120],[15,32632],{"className":32633,"style":32407},[124],[15,32635,32392],{"className":32636},[203],[15,32638],{"className":32639,"style":199},[93],[15,32641,32643,32646],{"className":32642},[120],[15,32644],{"className":32645,"style":4886},[124],[15,32647,32621],{"className":32648},[129],[18039,32650,32651,32654,32741,32826],{},[18055,32652,32653],{},"ResNet-50（ImageNet）",[18055,32655,32656],{},[15,32657,32659,32682],{"className":32658,"translate":19},[23],[15,32660,32662],{"className":32661},[27],[29,32663,32664],{"xmlns":31},[34,32665,32666,32679],{},[37,32667,32668,32671,32673],{},[308,32669,32670],{},"2.56",[50,32672,11418],{},[305,32674,32675,32677],{},[308,32676,32226],{},[308,32678,32453],{},[109,32680,32681],{"encoding":111},"2.56\\times10^7",[15,32683,32685,32703],{"className":32684,"ariaHidden":72},[116],[15,32686,32688,32691,32694,32697,32700],{"className":32687},[120],[15,32689],{"className":32690,"style":16477},[124],[15,32692,32670],{"className":32693},[129],[15,32695],{"className":32696,"style":3277},[93],[15,32698,11418],{"className":32699},[2436],[15,32701],{"className":32702,"style":3277},[93],[15,32704,32706,32709,32712],{"className":32705},[120],[15,32707],{"className":32708,"style":380},[124],[15,32710,318],{"className":32711},[129],[15,32713,32715,32718],{"className":32714},[129],[15,32716,541],{"className":32717},[129],[15,32719,32721],{"className":32720},[137],[15,32722,32724],{"className":32723},[141],[15,32725,32727],{"className":32726},[146],[15,32728,32730],{"className":32729,"style":380},[150],[15,32731,32732,32735],{"style":383},[15,32733],{"className":32734,"style":159},[158],[15,32736,32738],{"className":32737},[163,164,165,166],[15,32739,32453],{"className":32740},[129,166],[18055,32742,32743],{},[15,32744,32746,32767],{"className":32745,"translate":19},[23],[15,32747,32749],{"className":32748},[27],[29,32750,32751],{"xmlns":31},[34,32752,32753,32765],{},[37,32754,32755,32757,32759],{},[308,32756,32532],{},[50,32758,11418],{},[305,32760,32761,32763],{},[308,32762,32226],{},[308,32764,32219],{},[109,32766,32543],{"encoding":111},[15,32768,32770,32788],{"className":32769,"ariaHidden":72},[116],[15,32771,32773,32776,32779,32782,32785],{"className":32772},[120],[15,32774],{"className":32775,"style":16477},[124],[15,32777,32532],{"className":32778},[129],[15,32780],{"className":32781,"style":3277},[93],[15,32783,11418],{"className":32784},[2436],[15,32786],{"className":32787,"style":3277},[93],[15,32789,32791,32794,32797],{"className":32790},[120],[15,32792],{"className":32793,"style":380},[124],[15,32795,318],{"className":32796},[129],[15,32798,32800,32803],{"className":32799},[129],[15,32801,541],{"className":32802},[129],[15,32804,32806],{"className":32805},[137],[15,32807,32809],{"className":32808},[141],[15,32810,32812],{"className":32811},[146],[15,32813,32815],{"className":32814,"style":380},[150],[15,32816,32817,32820],{"style":383},[15,32818],{"className":32819,"style":159},[158],[15,32821,32823],{"className":32822},[163,164,165,166],[15,32824,32219],{"className":32825},[129,166],[18055,32827,32828],{},[15,32829,32831,32848],{"className":32830,"translate":19},[23],[15,32832,32834],{"className":32833},[27],[29,32835,32836],{"xmlns":31},[34,32837,32838,32845],{},[37,32839,32840,32842],{},[50,32841,32392],{},[308,32843,32844],{},"4.5",[109,32846,32847],{"encoding":111},"\\approx 4.5",[15,32849,32851,32863],{"className":32850,"ariaHidden":72},[116],[15,32852,32854,32857,32860],{"className":32853},[120],[15,32855],{"className":32856,"style":32407},[124],[15,32858,32392],{"className":32859},[203],[15,32861],{"className":32862,"style":199},[93],[15,32864,32866,32869],{"className":32865},[120],[15,32867],{"className":32868,"style":4886},[124],[15,32870,32844],{"className":32871},[129],[18039,32873,32874,32877,32965,33050],{},[18055,32875,32876],{},"ViT-L\u002F16（ImageNet）",[18055,32878,32879],{},[15,32880,32882,32906],{"className":32881,"translate":19},[23],[15,32883,32885],{"className":32884},[27],[29,32886,32887],{"xmlns":31},[34,32888,32889,32903],{},[37,32890,32891,32894,32896],{},[308,32892,32893],{},"3.07",[50,32895,11418],{},[305,32897,32898,32900],{},[308,32899,32226],{},[308,32901,32902],{},"8",[109,32904,32905],{"encoding":111},"3.07\\times10^8",[15,32907,32909,32927],{"className":32908,"ariaHidden":72},[116],[15,32910,32912,32915,32918,32921,32924],{"className":32911},[120],[15,32913],{"className":32914,"style":16477},[124],[15,32916,32893],{"className":32917},[129],[15,32919],{"className":32920,"style":3277},[93],[15,32922,11418],{"className":32923},[2436],[15,32925],{"className":32926,"style":3277},[93],[15,32928,32930,32933,32936],{"className":32929},[120],[15,32931],{"className":32932,"style":380},[124],[15,32934,318],{"className":32935},[129],[15,32937,32939,32942],{"className":32938},[129],[15,32940,541],{"className":32941},[129],[15,32943,32945],{"className":32944},[137],[15,32946,32948],{"className":32947},[141],[15,32949,32951],{"className":32950},[146],[15,32952,32954],{"className":32953,"style":380},[150],[15,32955,32956,32959],{"style":383},[15,32957],{"className":32958,"style":159},[158],[15,32960,32962],{"className":32961},[163,164,165,166],[15,32963,32902],{"className":32964},[129,166],[18055,32966,32967],{},[15,32968,32970,32991],{"className":32969,"translate":19},[23],[15,32971,32973],{"className":32972},[27],[29,32974,32975],{"xmlns":31},[34,32976,32977,32989],{},[37,32978,32979,32981,32983],{},[308,32980,32532],{},[50,32982,11418],{},[305,32984,32985,32987],{},[308,32986,32226],{},[308,32988,32219],{},[109,32990,32543],{"encoding":111},[15,32992,32994,33012],{"className":32993,"ariaHidden":72},[116],[15,32995,32997,33000,33003,33006,33009],{"className":32996},[120],[15,32998],{"className":32999,"style":16477},[124],[15,33001,32532],{"className":33002},[129],[15,33004],{"className":33005,"style":3277},[93],[15,33007,11418],{"className":33008},[2436],[15,33010],{"className":33011,"style":3277},[93],[15,33013,33015,33018,33021],{"className":33014},[120],[15,33016],{"className":33017,"style":380},[124],[15,33019,318],{"className":33020},[129],[15,33022,33024,33027],{"className":33023},[129],[15,33025,541],{"className":33026},[129],[15,33028,33030],{"className":33029},[137],[15,33031,33033],{"className":33032},[141],[15,33034,33036],{"className":33035},[146],[15,33037,33039],{"className":33038,"style":380},[150],[15,33040,33041,33044],{"style":383},[15,33042],{"className":33043,"style":159},[158],[15,33045,33047],{"className":33046},[163,164,165,166],[15,33048,32219],{"className":33049},[129,166],[18055,33051,33052],{},[15,33053,33055,33072],{"className":33054,"translate":19},[23],[15,33056,33058],{"className":33057},[27],[29,33059,33060],{"xmlns":31},[34,33061,33062,33069],{},[37,33063,33064,33066],{},[50,33065,32392],{},[308,33067,33068],{},"15.5",[109,33070,33071],{"encoding":111},"\\approx 15.5",[15,33073,33075,33087],{"className":33074,"ariaHidden":72},[116],[15,33076,33078,33081,33084],{"className":33077},[120],[15,33079],{"className":33080,"style":32407},[124],[15,33082,32392],{"className":33083},[203],[15,33085],{"className":33086,"style":199},[93],[15,33088,33090,33093],{"className":33089},[120],[15,33091],{"className":33092,"style":4886},[124],[15,33094,33068],{"className":33095},[129],[11,33097,33098,33099,33127,33128,33156,33157,33185,33186,33218],{},"每一个都在 ",[15,33100,33102,33115],{"className":33101,"translate":19},[23],[15,33103,33105],{"className":33104},[27],[29,33106,33107],{"xmlns":31},[34,33108,33109,33113],{},[37,33110,33111],{},[308,33112,318],{},[109,33114,318],{"encoding":111},[15,33116,33118],{"className":33117,"ariaHidden":72},[116],[15,33119,33121,33124],{"className":33120},[120],[15,33122],{"className":33123,"style":4886},[124],[15,33125,318],{"className":33126},[129]," 附近或远大于 ",[15,33129,33131,33144],{"className":33130,"translate":19},[23],[15,33132,33134],{"className":33133},[27],[29,33135,33136],{"xmlns":31},[34,33137,33138,33142],{},[37,33139,33140],{},[308,33141,318],{},[109,33143,318],{"encoding":111},[15,33145,33147],{"className":33146,"ariaHidden":72},[116],[15,33148,33150,33153],{"className":33149},[120],[15,33151],{"className":33152,"style":4886},[124],[15,33154,318],{"className":33155},[129],"。一个「错误率上界」一旦达到或超过 ",[15,33158,33160,33173],{"className":33159,"translate":19},[23],[15,33161,33163],{"className":33162},[27],[29,33164,33165],{"xmlns":31},[34,33166,33167,33171],{},[37,33168,33169],{},[308,33170,318],{},[109,33172,318],{"encoding":111},[15,33174,33176],{"className":33175,"ariaHidden":72},[116],[15,33177,33179,33182],{"className":33178},[120],[15,33180],{"className":33181,"style":4886},[124],[15,33183,318],{"className":33184},[129],"，就等于什么都没说——它连「模型会犯错」这种同义反复都不如，因为它没有排除任何东西。而且这还是把 VC 维低估成参数量的结果；下一小节会看到，深度网络的 VC 维还要乘上 ",[15,33187,33189,33204],{"className":33188,"translate":19},[23],[15,33190,33192],{"className":33191},[27],[29,33193,33194],{"xmlns":31},[34,33195,33196,33202],{},[37,33197,33198,33200],{},[43,33199,14721],{},[50,33201,69],{},[109,33203,30119],{"encoding":111},[15,33205,33207],{"className":33206,"ariaHidden":72},[116],[15,33208,33210,33213],{"className":33209},[120],[15,33211],{"className":33212,"style":1682},[124],[15,33214,14771,33216],{"className":33215},[217],[15,33217,2247],{"style":2246}," 因子。",[11,33220,33221,33222,33250,33251,33279,33280,33341],{},"有人会说：那就把界改紧一点。但问题不是常数，是量级。ImageNet 有 128 万张训练图，ResNet-50 有 2560 万参数，",[15,33223,33225,33238],{"className":33224,"translate":19},[23],[15,33226,33228],{"className":33227},[27],[29,33229,33230],{"xmlns":31},[34,33231,33232,33236],{},[37,33233,33234],{},[43,33235,348],{},[109,33237,348],{"encoding":111},[15,33239,33241],{"className":33240,"ariaHidden":72},[116],[15,33242,33244,33247],{"className":33243},[120],[15,33245],{"className":33246,"style":4383},[124],[15,33248,348],{"className":33249},[129,133]," 比 ",[15,33252,33254,33267],{"className":33253,"translate":19},[23],[15,33255,33257],{"className":33256},[27],[29,33258,33259],{"xmlns":31},[34,33260,33261,33265],{},[37,33262,33263],{},[43,33264,10372],{},[109,33266,10372],{"encoding":111},[15,33268,33270],{"className":33269,"ariaHidden":72},[116],[15,33271,33273,33276],{"className":33272},[120],[15,33274],{"className":33275,"style":522},[124],[15,33277,10372],{"className":33278},[129,133]," 小二十倍。",[698,33281,33282,33283,33311,33312,33340],{},"任何以 ",[15,33284,33286,33299],{"className":33285,"translate":19},[23],[15,33287,33289],{"className":33288},[27],[29,33290,33291],{"xmlns":31},[34,33292,33293,33297],{},[37,33294,33295],{},[43,33296,10372],{},[109,33298,10372],{"encoding":111},[15,33300,33302],{"className":33301,"ariaHidden":72},[116],[15,33303,33305,33308],{"className":33304},[120],[15,33306],{"className":33307,"style":522},[124],[15,33309,10372],{"className":33310},[129,133]," 为主项的界，在这个区间里都只能给出与 ",[15,33313,33315,33328],{"className":33314,"translate":19},[23],[15,33316,33318],{"className":33317},[27],[29,33319,33320],{"xmlns":31},[34,33321,33322,33326],{},[37,33323,33324],{},[308,33325,318],{},[109,33327,318],{"encoding":111},[15,33329,33331],{"className":33330,"ariaHidden":72},[116],[15,33332,33334,33337],{"className":33333},[120],[15,33335],{"className":33336,"style":4886},[124],[15,33338,318],{"className":33339},[129]," 同阶或更大的量级。"," 这是样本量与容量之间的量级失配，不是分析技巧的问题。",[575,33343,33345],{"id":33344},"_62-深度网络的-vc-维","6.2 深度网络的 VC 维",[11,33347,33348],{},"先确认一件事：深度网络的 VC 维是有限的。只要参数个数有限、激活函数是分段多项式（ReLU、Leaky ReLU、MaxOut 都是），VC 维就有限。深度网络并不是一个无限容量的对象。",[11,33350,33351],{},"那么它有多大？",[21719,33353,33354,33516],{},[21722,33355,33356,33359,33360,16656,33425,33453,33454,33486,33487,33515],{},[698,33357,33358],{},"阈值网络。"," Baum 与 Haussler（1989）证明，由阈值门组成的多层网络，VC 维是 ",[15,33361,33363,33390],{"className":33362,"translate":19},[23],[15,33364,33366],{"className":33365},[27],[29,33367,33368],{"xmlns":31},[34,33369,33370,33387],{},[37,33371,33372,33375,33377,33379,33381,33383,33385],{},[43,33373,33374],{},"O",[50,33376,53],{"stretchy":52},[43,33378,31987],{},[43,33380,14721],{},[50,33382,69],{},[43,33384,31987],{},[50,33386,59],{"stretchy":52},[109,33388,33389],{"encoding":111},"O(W\\log W)",[15,33391,33393],{"className":33392,"ariaHidden":72},[116],[15,33394,33396,33399,33402,33405,33408,33411,33416,33419,33422],{"className":33395},[120],[15,33397],{"className":33398,"style":125},[124],[15,33400,33374],{"className":33401,"style":170},[129,133],[15,33403,53],{"className":33404},[188],[15,33406,31987],{"className":33407,"style":12443},[129,133],[15,33409],{"className":33410,"style":262},[93],[15,33412,14771,33414],{"className":33413},[217],[15,33415,2247],{"style":2246},[15,33417],{"className":33418,"style":262},[93],[15,33420,31987],{"className":33421,"style":12443},[129,133],[15,33423,59],{"className":33424},[195],[15,33426,33428,33441],{"className":33427,"translate":19},[23],[15,33429,33431],{"className":33430},[27],[29,33432,33433],{"xmlns":31},[34,33434,33435,33439],{},[37,33436,33437],{},[43,33438,31987],{},[109,33440,31987],{"encoding":111},[15,33442,33444],{"className":33443,"ariaHidden":72},[116],[15,33445,33447,33450],{"className":33446},[120],[15,33448],{"className":33449,"style":610},[124],[15,33451,31987],{"className":33452,"style":12443},[129,133]," 是权重个数。",[15,33455,33457,33472],{"className":33456,"translate":19},[23],[15,33458,33460],{"className":33459},[27],[29,33461,33462],{"xmlns":31},[34,33463,33464,33470],{},[37,33465,33466,33468],{},[43,33467,14721],{},[50,33469,69],{},[109,33471,30119],{"encoding":111},[15,33473,33475],{"className":33474,"ariaHidden":72},[116],[15,33476,33478,33481],{"className":33477},[120],[15,33479],{"className":33480,"style":1682},[124],[15,33482,14771,33484],{"className":33483},[217],[15,33485,2247],{"style":2246}," 因子来自网络内部的分段线性结构：不同的权重符号组合对应不同的线性区域，区域数量随 ",[15,33488,33490,33503],{"className":33489,"translate":19},[23],[15,33491,33493],{"className":33492},[27],[29,33494,33495],{"xmlns":31},[34,33496,33497,33501],{},[37,33498,33499],{},[43,33500,31987],{},[109,33502,31987],{"encoding":111},[15,33504,33506],{"className":33505,"ariaHidden":72},[116],[15,33507,33509,33512],{"className":33508},[120],[15,33510],{"className":33511,"style":610},[124],[15,33513,31987],{"className":33514,"style":12443},[129,133]," 组合增长。",[21722,33517,33518,33521,33522,33591,33592,33620,33621,33705],{},[698,33519,33520],{},"ReLU 网络。"," Bartlett、Harvey、Liaw 与 Mehrabian（2019）给出 ",[15,33523,33525,33553],{"className":33524,"translate":19},[23],[15,33526,33528],{"className":33527},[27],[29,33529,33530],{"xmlns":31},[34,33531,33532,33550],{},[37,33533,33534,33536,33538,33540,33542,33544,33546,33548],{},[43,33535,33374],{},[50,33537,53],{"stretchy":52},[43,33539,31987],{},[43,33541,1373],{},[43,33543,14721],{},[50,33545,69],{},[43,33547,31987],{},[50,33549,59],{"stretchy":52},[109,33551,33552],{"encoding":111},"O(WL\\log W)",[15,33554,33556],{"className":33555,"ariaHidden":72},[116],[15,33557,33559,33562,33565,33568,33571,33574,33577,33582,33585,33588],{"className":33558},[120],[15,33560],{"className":33561,"style":125},[124],[15,33563,33374],{"className":33564,"style":170},[129,133],[15,33566,53],{"className":33567},[188],[15,33569,31987],{"className":33570,"style":12443},[129,133],[15,33572,1373],{"className":33573},[129,133],[15,33575],{"className":33576,"style":262},[93],[15,33578,14771,33580],{"className":33579},[217],[15,33581,2247],{"style":2246},[15,33583],{"className":33584,"style":262},[93],[15,33586,31987],{"className":33587,"style":12443},[129,133],[15,33589,59],{"className":33590},[195]," 的上界（",[15,33593,33595,33608],{"className":33594,"translate":19},[23],[15,33596,33598],{"className":33597},[27],[29,33599,33600],{"xmlns":31},[34,33601,33602,33606],{},[37,33603,33604],{},[43,33605,1373],{},[109,33607,1373],{"encoding":111},[15,33609,33611],{"className":33610,"ariaHidden":72},[116],[15,33612,33614,33617],{"className":33613},[120],[15,33615],{"className":33616,"style":610},[124],[15,33618,1373],{"className":33619},[129,133]," 为层数），并给出几乎匹配的下界 ",[15,33622,33624,33661],{"className":33623,"translate":19},[23],[15,33625,33627],{"className":33626},[27],[29,33628,33629],{"xmlns":31},[34,33630,33631,33658],{},[37,33632,33633,33636,33638,33640,33642,33644,33646,33648,33650,33652,33654,33656],{},[43,33634,33635],{"mathvariant":65},"Ω",[50,33637,53],{"stretchy":52},[43,33639,31987],{},[43,33641,1373],{},[43,33643,14721],{},[50,33645,69],{},[50,33647,53],{"stretchy":52},[43,33649,31987],{},[43,33651,544],{"mathvariant":65},[43,33653,1373],{},[50,33655,59],{"stretchy":52},[50,33657,59],{"stretchy":52},[109,33659,33660],{"encoding":111},"\\Omega(WL\\log(W\u002FL))",[15,33662,33664],{"className":33663,"ariaHidden":72},[116],[15,33665,33667,33670,33673,33676,33679,33682,33685,33690,33693,33696,33699,33702],{"className":33666},[120],[15,33668],{"className":33669,"style":125},[124],[15,33671,33635],{"className":33672},[129],[15,33674,53],{"className":33675},[188],[15,33677,31987],{"className":33678,"style":12443},[129,133],[15,33680,1373],{"className":33681},[129,133],[15,33683],{"className":33684,"style":262},[93],[15,33686,14771,33688],{"className":33687},[217],[15,33689,2247],{"style":2246},[15,33691,53],{"className":33692},[188],[15,33694,31987],{"className":33695,"style":12443},[129,133],[15,33697,544],{"className":33698},[129],[15,33700,1373],{"className":33701},[129,133],[15,33703,6149],{"className":33704},[195],"；伪维（pseudo-dimension）同阶。",[11,33707,33708,33709,33745],{},"结论很清楚：",[698,33710,33711,33712,33744],{},"深度网络的 VC 维随参数线性增长（乘一个 ",[15,33713,33715,33730],{"className":33714,"translate":19},[23],[15,33716,33718],{"className":33717},[27],[29,33719,33720],{"xmlns":31},[34,33721,33722,33728],{},[37,33723,33724,33726],{},[43,33725,14721],{},[50,33727,69],{},[109,33729,30119],{"encoding":111},[15,33731,33733],{"className":33732,"ariaHidden":72},[116],[15,33734,33736,33739],{"className":33735},[120],[15,33737],{"className":33738,"style":1682},[124],[15,33740,14771,33742],{"className":33741},[217],[15,33743,2247],{"style":2246},"）。"," 「深度学习之所以泛化，是因为它的 VC 维很小」这个流行说法是错的。深度网络和线性模型一样，容量随参数增长；区别在于，深度网络的实际泛化误差可以远小于这个容量所暗示的上界。",[11,33747,33748],{},"另一个方向的反例是 1-NN：VC 维无穷，但在很多分布上表现并不差。这说明 VC 维无穷并不等于「不能用」；它只意味着不存在一个对所有分布一致的保证。",[575,33750,33752],{"id":33751},"_63-三条出路","6.3 三条出路",[11,33754,33755],{},"既然「整个假设类的 VC 维」给不出有用的话，理论就往三个方向走。",[11,33757,33758],{},[698,33759,33760],{},"（a）把容量做得更细：从假设类到算法输出",[11,33762,33763,33764,33792],{},"与其用整个 ",[15,33765,33767,33780],{"className":33766,"translate":19},[23],[15,33768,33770],{"className":33769},[27],[29,33771,33772],{"xmlns":31},[34,33773,33774,33778],{},[37,33775,33776],{},[43,33777,819],{"mathvariant":596},[109,33779,822],{"encoding":111},[15,33781,33783],{"className":33782,"ariaHidden":72},[116],[15,33784,33786,33789],{"className":33785},[120],[15,33787],{"className":33788,"style":610},[124],[15,33790,819],{"className":33791,"style":835},[129,614]," 的 VC 维，不如用「训练解所在的那个小子类」的容量。",[11,33794,33795,33796,33859,33860,33923,33924,33955,33956,34041,34042,34070],{},"线性情形是最好的例子。若 ",[15,33797,33799,33822],{"className":33798,"translate":19},[23],[15,33800,33802],{"className":33801},[27],[29,33803,33804],{"xmlns":31},[34,33805,33806,33819],{},[37,33807,33808,33811,33813,33815,33817],{},[50,33809,33810],{"stretchy":52},"∥",[43,33812,56],{},[50,33814,33810],{"stretchy":52},[50,33816,2717],{},[43,33818,104],{},[109,33820,33821],{"encoding":111},"\\lVert x\\rVert\\le R",[15,33823,33825,33849],{"className":33824,"ariaHidden":72},[116],[15,33826,33828,33831,33834,33837,33840,33843,33846],{"className":33827},[120],[15,33829],{"className":33830,"style":125},[124],[15,33832,33810],{"className":33833},[188],[15,33835,56],{"className":33836},[129,133],[15,33838,33810],{"className":33839},[195],[15,33841],{"className":33842,"style":199},[93],[15,33844,2717],{"className":33845},[203],[15,33847],{"className":33848,"style":199},[93],[15,33850,33852,33855],{"className":33851},[120],[15,33853],{"className":33854,"style":610},[124],[15,33856,104],{"className":33857,"style":33858},[129,133],"margin-right:0.0077em;","、",[15,33861,33863,33886],{"className":33862,"translate":19},[23],[15,33864,33866],{"className":33865},[27],[29,33867,33868],{"xmlns":31},[34,33869,33870,33883],{},[37,33871,33872,33874,33876,33878,33880],{},[50,33873,33810],{"stretchy":52},[43,33875,10546],{},[50,33877,33810],{"stretchy":52},[50,33879,2717],{},[43,33881,33882],{},"B",[109,33884,33885],{"encoding":111},"\\lVert w\\rVert\\le B",[15,33887,33889,33913],{"className":33888,"ariaHidden":72},[116],[15,33890,33892,33895,33898,33901,33904,33907,33910],{"className":33891},[120],[15,33893],{"className":33894,"style":125},[124],[15,33896,33810],{"className":33897},[188],[15,33899,10546],{"className":33900,"style":10623},[129,133],[15,33902,33810],{"className":33903},[195],[15,33905],{"className":33906,"style":199},[93],[15,33908,2717],{"className":33909},[203],[15,33911],{"className":33912,"style":199},[93],[15,33914,33916,33919],{"className":33915},[120],[15,33917],{"className":33918,"style":610},[124],[15,33920,33882],{"className":33921,"style":33922},[129,133],"margin-right:0.0502em;","，且分类间隔至少为 ",[15,33925,33927,33942],{"className":33926,"translate":19},[23],[15,33928,33930],{"className":33929},[27],[29,33931,33932],{"xmlns":31},[34,33933,33934,33939],{},[37,33935,33936],{},[43,33937,33938],{},"γ",[109,33940,33941],{"encoding":111},"\\gamma",[15,33943,33945],{"className":33944,"ariaHidden":72},[116],[15,33946,33948,33951],{"className":33947},[120],[15,33949],{"className":33950,"style":5399},[124],[15,33952,33938],{"className":33953,"style":33954},[129,133],"margin-right:0.0556em;","，则有效容量约为 ",[15,33957,33959,33987],{"className":33958,"translate":19},[23],[15,33960,33962],{"className":33961},[27],[29,33963,33964],{"xmlns":31},[34,33965,33966,33984],{},[37,33967,33968,33970,33972,33974,33976,33978],{},[50,33969,53],{"stretchy":52},[43,33971,104],{},[43,33973,33882],{},[43,33975,544],{"mathvariant":65},[43,33977,33938],{},[305,33979,33980,33982],{},[50,33981,59],{"stretchy":52},[308,33983,310],{},[109,33985,33986],{"encoding":111},"(RB\u002F\\gamma)^2",[15,33988,33990],{"className":33989,"ariaHidden":72},[116],[15,33991,33993,33997,34000,34003,34006,34009,34012],{"className":33992},[120],[15,33994],{"className":33995,"style":33996},[124],"height:1.0641em;vertical-align:-0.25em;",[15,33998,53],{"className":33999},[188],[15,34001,104],{"className":34002,"style":33858},[129,133],[15,34004,33882],{"className":34005,"style":33922},[129,133],[15,34007,544],{"className":34008},[129],[15,34010,33938],{"className":34011,"style":33954},[129,133],[15,34013,34015,34018],{"className":34014},[195],[15,34016,59],{"className":34017},[195],[15,34019,34021],{"className":34020},[137],[15,34022,34024],{"className":34023},[141],[15,34025,34027],{"className":34026},[146],[15,34028,34030],{"className":34029,"style":380},[150],[15,34031,34032,34035],{"style":383},[15,34033],{"className":34034,"style":159},[158],[15,34036,34038],{"className":34037},[163,164,165,166],[15,34039,310],{"className":34040},[129,166],"，与输入维度 ",[15,34043,34045,34058],{"className":34044,"translate":19},[23],[15,34046,34048],{"className":34047},[27],[29,34049,34050],{"xmlns":31},[34,34051,34052,34056],{},[37,34053,34054],{},[43,34055,10372],{},[109,34057,10372],{"encoding":111},[15,34059,34061],{"className":34060,"ariaHidden":72},[116],[15,34062,34064,34067],{"className":34063},[120],[15,34065],{"className":34066,"style":522},[124],[15,34068,10372],{"className":34069},[129,133]," 无关。这正是 SVM 的理论基础：在高维甚至无限维的特征空间里，只要间隔大，容量就小。",[11,34072,34073,34074,34113,34114,34287],{},"深度网络有类似的谱范数版本。把 ",[15,34075,34077,34095],{"className":34076,"translate":19},[23],[15,34078,34080],{"className":34079},[27],[29,34081,34082],{"xmlns":31},[34,34083,34084,34092],{},[37,34085,34086,34088,34090],{},[50,34087,33810],{"stretchy":52},[43,34089,10546],{},[50,34091,33810],{"stretchy":52},[109,34093,34094],{"encoding":111},"\\lVert w\\rVert",[15,34096,34098],{"className":34097,"ariaHidden":72},[116],[15,34099,34101,34104,34107,34110],{"className":34100},[120],[15,34102],{"className":34103,"style":125},[124],[15,34105,33810],{"className":34106},[188],[15,34108,10546],{"className":34109,"style":10623},[129,133],[15,34111,33810],{"className":34112},[195]," 换成各层谱范数的乘积 ",[15,34115,34117,34151],{"className":34116,"translate":19},[23],[15,34118,34120],{"className":34119},[27],[29,34121,34122],{"xmlns":31},[34,34123,34124,34148],{},[37,34125,34126,34134,34136,34142],{},[40,34127,34128,34131],{},[50,34129,34130],{},"∏",[43,34132,34133],{},"l",[50,34135,33810],{"stretchy":52},[40,34137,34138,34140],{},[43,34139,31987],{},[43,34141,34133],{},[40,34143,34144,34146],{},[50,34145,33810],{"stretchy":52},[43,34147,27283],{},[109,34149,34150],{"encoding":111},"\\prod_l\\lVert W_l\\rVert_\\sigma",[15,34152,34154],{"className":34153,"ariaHidden":72},[116],[15,34155,34157,34161,34203,34206,34247],{"className":34156},[120],[15,34158],{"className":34159,"style":34160},[124],"height:1.0497em;vertical-align:-0.2997em;",[15,34162,34164,34167],{"className":34163},[217],[15,34165,34130],{"className":34166,"style":23341},[217,1918,23340],[15,34168,34170],{"className":34169},[137],[15,34171,34173,34195],{"className":34172},[141,142],[15,34174,34176,34192],{"className":34175},[146],[15,34177,34180],{"className":34178,"style":34179},[150],"height:0.1864em;",[15,34181,34182,34185],{"style":23357},[15,34183],{"className":34184,"style":159},[158],[15,34186,34188],{"className":34187},[163,164,165,166],[15,34189,34133],{"className":34190,"style":34191},[129,133,166],"margin-right:0.0197em;",[15,34193,175],{"className":34194},[174],[15,34196,34198],{"className":34197},[146],[15,34199,34201],{"className":34200,"style":23406},[150],[15,34202],{},[15,34204,33810],{"className":34205},[188],[15,34207,34209,34212],{"className":34208},[129],[15,34210,31987],{"className":34211,"style":12443},[129,133],[15,34213,34215],{"className":34214},[137],[15,34216,34218,34239],{"className":34217},[141,142],[15,34219,34221,34236],{"className":34220},[146],[15,34222,34224],{"className":34223,"style":151},[150],[15,34225,34227,34230],{"style":34226},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[15,34228],{"className":34229,"style":159},[158],[15,34231,34233],{"className":34232},[163,164,165,166],[15,34234,34133],{"className":34235,"style":34191},[129,133,166],[15,34237,175],{"className":34238},[174],[15,34240,34242],{"className":34241},[146],[15,34243,34245],{"className":34244,"style":182},[150],[15,34246],{},[15,34248,34250,34253],{"className":34249},[195],[15,34251,33810],{"className":34252},[195],[15,34254,34256],{"className":34255},[137],[15,34257,34259,34279],{"className":34258},[141,142],[15,34260,34262,34276],{"className":34261},[146],[15,34263,34265],{"className":34264,"style":1163},[150],[15,34266,34267,34270],{"style":154},[15,34268],{"className":34269,"style":159},[158],[15,34271,34273],{"className":34272},[163,164,165,166],[15,34274,27283],{"className":34275,"style":887},[129,133,166],[15,34277,175],{"className":34278},[174],[15,34280,34282],{"className":34281},[146],[15,34283,34285],{"className":34284,"style":182},[150],[15,34286],{},"，再乘上深度和宽度带来的多项式因子、除以间隔，就得到 Bartlett–Foster–Telgarsky（2017）与 Neyshabur 等人（2018）的界。这类界的形式大致是「分子为谱范数乘积的平方，分母为间隔与样本量」；它们在经过特殊正则的小网络上可以非空，但在标准训练的 ImageNet 网络上仍然空泛。",[11,34289,34290,34291,34320],{},"覆盖数（covering number）是同一思想的度量几何版本：VC 维回答「能打散多少点」，覆盖数回答「用多少个半径为 ",[15,34292,34294,34308],{"className":34293,"translate":19},[23],[15,34295,34297],{"className":34296},[27],[29,34298,34299],{"xmlns":31},[34,34300,34301,34305],{},[37,34302,34303],{},[43,34304,4849],{},[109,34306,34307],{"encoding":111},"\\varepsilon",[15,34309,34311],{"className":34310,"ariaHidden":72},[116],[15,34312,34314,34317],{"className":34313},[120],[15,34315],{"className":34316,"style":4383},[124],[15,34318,4849],{"className":34319},[129,133]," 的球能把假设类盖住」。二者通过 Dudley 熵积分相连。",[11,34322,34323,34326,34327,34357,34358,34408],{},[698,34324,34325],{},"PAC-Bayes"," 则把容量变成了「后验与先验的距离」。对任意后验分布 ",[15,34328,34330,34344],{"className":34329,"translate":19},[23],[15,34331,34333],{"className":34332},[27],[29,34334,34335],{"xmlns":31},[34,34336,34337,34342],{},[37,34338,34339],{},[43,34340,34341],{},"Q",[109,34343,34341],{"encoding":111},[15,34345,34347],{"className":34346,"ariaHidden":72},[116],[15,34348,34350,34354],{"className":34349},[120],[15,34351],{"className":34352,"style":34353},[124],"height:0.8778em;vertical-align:-0.1944em;",[15,34355,34341],{"className":34356},[129,133],"（通常是训练后权重的分布），以 ",[15,34359,34361,34378],{"className":34360,"translate":19},[23],[15,34362,34364],{"className":34363},[27],[29,34365,34366],{"xmlns":31},[34,34367,34368,34376],{},[37,34369,34370,34372,34374],{},[308,34371,318],{},[50,34373,315],{},[43,34375,29564],{},[109,34377,29912],{"encoding":111},[15,34379,34381,34399],{"className":34380,"ariaHidden":72},[116],[15,34382,34384,34387,34390,34393,34396],{"className":34383},[120],[15,34385],{"className":34386,"style":16477},[124],[15,34388,318],{"className":34389},[129],[15,34391],{"className":34392,"style":3277},[93],[15,34394,315],{"className":34395},[2436],[15,34397],{"className":34398,"style":3277},[93],[15,34400,34402,34405],{"className":34401},[120],[15,34403],{"className":34404,"style":522},[124],[15,34406,29564],{"className":34407,"style":29845},[129,133]," 的概率，",[15,34410,34412],{"className":34411,"translate":19},[18],[15,34413,34415,34549],{"className":34414,"translate":19},[23],[15,34416,34418],{"className":34417},[27],[29,34419,34420],{"xmlns":31,"display":32},[34,34421,34422,34546],{},[37,34423,34424,34436,34438,34444,34446,34448,34450,34452,34454,34456,34458,34470,34472,34478,34480,34482,34484,34486,34488,34490,34492,34494,34497,34544],{},[40,34425,34426,34428],{},[43,34427,26563],{"mathvariant":103},[37,34429,34430,34432,34434],{},[43,34431,45],{},[50,34433,991],{},[43,34435,34341],{},[50,34437,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,34439,34440,34442],{},[43,34441,1373],{},[43,34443,906],{"mathvariant":596},[50,34445,53],{"stretchy":52},[43,34447,45],{},[50,34449,59],{"stretchy":52},[50,34451,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[6852,34453,26553],{},[50,34455,2717],{},[6852,34457,26553],{},[40,34459,34460,34462],{},[43,34461,26563],{"mathvariant":103},[37,34463,34464,34466,34468],{},[43,34465,45],{},[50,34467,991],{},[43,34469,34341],{},[50,34471,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,34473,34474,34476],{},[43,34475,1373],{},[43,34477,940],{},[50,34479,53],{"stretchy":52},[43,34481,45],{},[50,34483,59],{"stretchy":52},[50,34485,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[6852,34487,26553],{},[50,34489,2682],{},[6852,34491,26553],{},[43,34493,33374],{},[6852,34495,34496],{}," ⁣",[37,34498,34499,34501,34542],{},[50,34500,53],{"fence":72},[29004,34502,34503],{},[1443,34504,34505,34540],{},[37,34506,34507,34514,34516,34518,34520,34522,34524,34526,34528,34530,34532,34534,34536,34538],{},[37,34508,34509,34512],{},[43,34510,34511],{"mathvariant":65},"K",[43,34513,1373],{"mathvariant":65},[50,34515,53],{"stretchy":52},[43,34517,34341],{},[43,34519,33810],{"mathvariant":65},[43,34521,1388],{},[50,34523,59],{"stretchy":52},[50,34525,2682],{},[43,34527,14721],{},[50,34529,69],{},[50,34531,53],{"stretchy":52},[308,34533,318],{},[43,34535,544],{"mathvariant":65},[43,34537,29564],{},[50,34539,59],{"stretchy":52},[43,34541,348],{},[50,34543,59],{"fence":72},[43,34545,107],{"mathvariant":65},[109,34547,34548],{"encoding":111},"\\mathbb E_{h\\sim Q}\\bigl[L_{\\mathcal D}(h)\\bigr]\n\\;\\le\\;\n\\mathbb E_{h\\sim Q}\\bigl[L_S(h)\\bigr]\n\\;+\\;O\\!\\left(\\sqrt{\\frac{\\mathrm{KL}(Q\\Vert P)+\\log(1\u002F\\delta)}{m}}\\right).",[15,34550,34552,34686,34817],{"className":34551,"ariaHidden":72},[116],[15,34553,34555,34558,34607,34613,34656,34659,34662,34665,34671,34674,34677,34680,34683],{"className":34554},[120],[15,34556],{"className":34557,"style":213},[124],[15,34559,34561,34564],{"className":34560},[129],[15,34562,26563],{"className":34563},[129,285],[15,34565,34567],{"className":34566},[137],[15,34568,34570,34599],{"className":34569},[141,142],[15,34571,34573,34596],{"className":34572},[146],[15,34574,34576],{"className":34575,"style":151},[150],[15,34577,34578,34581],{"style":154},[15,34579],{"className":34580,"style":159},[158],[15,34582,34584],{"className":34583},[163,164,165,166],[15,34585,34587,34590,34593],{"className":34586},[129,166],[15,34588,45],{"className":34589},[129,133,166],[15,34591,991],{"className":34592},[203,166],[15,34594,34341],{"className":34595},[129,133,166],[15,34597,175],{"className":34598},[174],[15,34600,34602],{"className":34601},[146],[15,34603,34605],{"className":34604,"style":14035},[150],[15,34606],{},[15,34608,34610],{"className":34609},[188],[15,34611,1407],{"className":34612},[228,229],[15,34614,34616,34619],{"className":34615},[129],[15,34617,1373],{"className":34618},[129,133],[15,34620,34622],{"className":34621},[137],[15,34623,34625,34648],{"className":34624},[141,142],[15,34626,34628,34645],{"className":34627},[146],[15,34629,34631],{"className":34630,"style":1525},[150],[15,34632,34633,34636],{"style":154},[15,34634],{"className":34635,"style":159},[158],[15,34637,34639],{"className":34638},[163,164,165,166],[15,34640,34642],{"className":34641},[129,166],[15,34643,906],{"className":34644,"style":170},[129,614,166],[15,34646,175],{"className":34647},[174],[15,34649,34651],{"className":34650},[146],[15,34652,34654],{"className":34653,"style":182},[150],[15,34655],{},[15,34657,53],{"className":34658},[188],[15,34660,45],{"className":34661},[129,133],[15,34663,59],{"className":34664},[195],[15,34666,34668],{"className":34667},[195],[15,34669,1423],{"className":34670},[228,229],[15,34672],{"className":34673,"style":199},[93],[15,34675],{"className":34676,"style":199},[93],[15,34678,2717],{"className":34679},[203],[15,34681],{"className":34682,"style":199},[93],[15,34684],{"className":34685,"style":199},[93],[15,34687,34689,34692,34741,34747,34787,34790,34793,34796,34802,34805,34808,34811,34814],{"className":34688},[120],[15,34690],{"className":34691,"style":213},[124],[15,34693,34695,34698],{"className":34694},[129],[15,34696,26563],{"className":34697},[129,285],[15,34699,34701],{"className":34700},[137],[15,34702,34704,34733],{"className":34703},[141,142],[15,34705,34707,34730],{"className":34706},[146],[15,34708,34710],{"className":34709,"style":151},[150],[15,34711,34712,34715],{"style":154},[15,34713],{"className":34714,"style":159},[158],[15,34716,34718],{"className":34717},[163,164,165,166],[15,34719,34721,34724,34727],{"className":34720},[129,166],[15,34722,45],{"className":34723},[129,133,166],[15,34725,991],{"className":34726},[203,166],[15,34728,34341],{"className":34729},[129,133,166],[15,34731,175],{"className":34732},[174],[15,34734,34736],{"className":34735},[146],[15,34737,34739],{"className":34738,"style":14035},[150],[15,34740],{},[15,34742,34744],{"className":34743},[188],[15,34745,1407],{"className":34746},[228,229],[15,34748,34750,34753],{"className":34749},[129],[15,34751,1373],{"className":34752},[129,133],[15,34754,34756],{"className":34755},[137],[15,34757,34759,34779],{"className":34758},[141,142],[15,34760,34762,34776],{"className":34761},[146],[15,34763,34765],{"className":34764,"style":1525},[150],[15,34766,34767,34770],{"style":154},[15,34768],{"className":34769,"style":159},[158],[15,34771,34773],{"className":34772},[163,164,165,166],[15,34774,940],{"className":34775,"style":1013},[129,133,166],[15,34777,175],{"className":34778},[174],[15,34780,34782],{"className":34781},[146],[15,34783,34785],{"className":34784,"style":182},[150],[15,34786],{},[15,34788,53],{"className":34789},[188],[15,34791,45],{"className":34792},[129,133],[15,34794,59],{"className":34795},[195],[15,34797,34799],{"className":34798},[195],[15,34800,1423],{"className":34801},[228,229],[15,34803],{"className":34804,"style":199},[93],[15,34806],{"className":34807,"style":3277},[93],[15,34809,2682],{"className":34810},[2436],[15,34812],{"className":34813,"style":199},[93],[15,34815],{"className":34816,"style":3277},[93],[15,34818,34820,34824,34827,34831,34834,35000,35003],{"className":34819},[120],[15,34821],{"className":34822,"style":34823},[124],"height:3em;vertical-align:-1.25em;",[15,34825,33374],{"className":34826,"style":170},[129,133],[15,34828],{"className":34829,"style":34830},[93],"margin-right:-0.1667em;",[15,34832],{"className":34833,"style":262},[93],[15,34835,34837,34843,34994],{"className":34836},[449],[15,34838,34840],{"className":34839,"style":5052},[188,5051],[15,34841,53],{"className":34842},[228,15850],[15,34844,34846],{"className":3484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VC 维，而是「训练后的分布 ",[15,35010,35012,35025],{"className":35011,"translate":19},[23],[15,35013,35015],{"className":35014},[27],[29,35016,35017],{"xmlns":31},[34,35018,35019,35023],{},[37,35020,35021],{},[43,35022,34341],{},[109,35024,34341],{"encoding":111},[15,35026,35028],{"className":35027,"ariaHidden":72},[116],[15,35029,35031,35034],{"className":35030},[120],[15,35032],{"className":35033,"style":34353},[124],[15,35035,34341],{"className":35036},[129,133]," 离先验 ",[15,35039,35041,35054],{"className":35040,"translate":19},[23],[15,35042,35044],{"className":35043},[27],[29,35045,35046],{"xmlns":31},[34,35047,35048,35052],{},[37,35049,35050],{},[43,35051,1388],{},[109,35053,1388],{"encoding":111},[15,35055,35057],{"className":35056,"ariaHidden":72},[116],[15,35058,35060,35063],{"className":35059},[120],[15,35061],{"className":35062,"style":610},[124],[15,35064,1388],{"className":35065,"style":12443},[129,133]," 有多远」。Dziugaite 与 Roy（2017）用它在 MNIST 规模的网络上算出了非空的界，这是 PAC-Bayes 在深度网络上最早显示出实际内容的结果之一。",[11,35068,35069,35072,35073,35101,35102,35189],{},[698,35070,35071],{},"压缩"," 是另一条路：若训练好的网络可以被压缩成 ",[15,35074,35076,35089],{"className":35075,"translate":19},[23],[15,35077,35079],{"className":35078},[27],[29,35080,35081],{"xmlns":31},[34,35082,35083,35087],{},[37,35084,35085],{},[43,35086,16555],{},[109,35088,16555],{"encoding":111},[15,35090,35092],{"className":35091,"ariaHidden":72},[116],[15,35093,35095,35098],{"className":35094},[120],[15,35096],{"className":35097,"style":522},[124],[15,35099,16555],{"className":35100,"style":16588},[129,133]," 比特而不改变其在训练集上的预测，则泛化误差约为 ",[15,35103,35105,35127],{"className":35104,"translate":19},[23],[15,35106,35108],{"className":35107},[27],[29,35109,35110],{"xmlns":31},[34,35111,35112,35124],{},[37,35113,35114],{},[29004,35115,35116],{},[37,35117,35118,35120,35122],{},[43,35119,16555],{},[43,35121,544],{"mathvariant":65},[43,35123,348],{},[109,35125,35126],{"encoding":111},"\\sqrt{k\u002Fm}",[15,35128,35130],{"className":35129,"ariaHidden":72},[116],[15,35131,35133,35136],{"className":35132},[120],[15,35134],{"className":35135,"style":31881},[124],[15,35137,35139],{"className":35138},[129,29112],[15,35140,35142,35181],{"className":35141},[141,142],[15,35143,35145,35178],{"className":35144},[146],[15,35146,35148,35166],{"className":35147,"style":31894},[150],[15,35149,35151,35154],{"className":35150,"style":31898},[29126],[15,35152],{"className":35153,"style":4450},[158],[15,35155,35157,35160,35163],{"className":35156,"style":29135},[129],[15,35158,16555],{"className":35159,"style":16588},[129,133],[15,35161,544],{"className":35162},[129],[15,35164,348],{"className":35165},[129,133],[15,35167,35168,35171],{"style":31916},[15,35169],{"className":35170,"style":4450},[158],[15,35172,35174],{"className":35173,"style":31923},[29271],[4455,35175,35176],{"xmlns":4457,"width":29275,"height":31926,"viewBox":31927,"preserveAspectRatio":29278},[4461,35177],{"d":31930},[15,35179,175],{"className":35180},[174],[15,35182,35184],{"className":35183},[146],[15,35185,35187],{"className":35186,"style":31940},[150],[15,35188],{},"。Zhou 等人（2019）用剪枝加量化的压缩方案，在 ImageNet 规模上给出了非空（但仍然很松）的界。「非空」是里程碑，但离「解释」还很远。",[11,35191,35192,35195],{},[698,35193,35194],{},"稳定性"," 则把容量挂在算法上：若算法对替换一个样本不敏感，泛化误差就可以被稳定性系数控制（Bousquet–Elisseeff，2002）。Hardt、Recht 与 Singer（2016）把这一框架用到了 SGD 上。",[11,35197,35198,35199,35202,35203,35231],{},"这些界的共同点很重要：",[698,35200,35201],{},"它们中的大多数不再是纯粹「假设类」的属性，而是「算法输出」或「数据」的属性。"," 容量从单一的 ",[15,35204,35206,35219],{"className":35205,"translate":19},[23],[15,35207,35209],{"className":35208},[27],[29,35210,35211],{"xmlns":31},[34,35212,35213,35217],{},[37,35214,35215],{},[43,35216,819],{"mathvariant":596},[109,35218,822],{"encoding":111},[15,35220,35222],{"className":35221,"ariaHidden":72},[116],[15,35223,35225,35228],{"className":35224},[120],[15,35226],{"className":35227,"style":610},[124],[15,35229,819],{"className":35230,"style":835},[129,614]," 转移到了假设类、算法、数据三者共同决定的整体上。",[11,35233,35234],{},[698,35235,35236],{},"（b）算法偏置：SGD 只在一个很小的子集里搜索",[11,35238,35239,35267,35268,35296,35297,35325],{},[15,35240,35242,35255],{"className":35241,"translate":19},[23],[15,35243,35245],{"className":35244},[27],[29,35246,35247],{"xmlns":31},[34,35248,35249,35253],{},[37,35250,35251],{},[43,35252,819],{"mathvariant":596},[109,35254,822],{"encoding":111},[15,35256,35258],{"className":35257,"ariaHidden":72},[116],[15,35259,35261,35264],{"className":35260},[120],[15,35262],{"className":35263,"style":610},[124],[15,35265,819],{"className":35266,"style":835},[129,614]," 很大，但 SGD 不是均匀地在 ",[15,35269,35271,35284],{"className":35270,"translate":19},[23],[15,35272,35274],{"className":35273},[27],[29,35275,35276],{"xmlns":31},[34,35277,35278,35282],{},[37,35279,35280],{},[43,35281,819],{"mathvariant":596},[109,35283,822],{"encoding":111},[15,35285,35287],{"className":35286,"ariaHidden":72},[116],[15,35288,35290,35293],{"className":35289},[120],[15,35291],{"className":35292,"style":610},[124],[15,35294,819],{"className":35295,"style":835},[129,614]," 里搜索。从初始化出发、沿着梯度的方向、在有限步内，它只能到达 ",[15,35298,35300,35313],{"className":35299,"translate":19},[23],[15,35301,35303],{"className":35302},[27],[29,35304,35305],{"xmlns":31},[34,35306,35307,35311],{},[37,35308,35309],{},[43,35310,819],{"mathvariant":596},[109,35312,822],{"encoding":111},[15,35314,35316],{"className":35315,"ariaHidden":72},[116],[15,35317,35319,35322],{"className":35318},[120],[15,35320],{"className":35321,"style":610},[124],[15,35323,819],{"className":35324,"style":835},[129,614]," 的一个很小的子集。",[11,35327,35328],{},"线性可分的情形已经被严格刻画：梯度下降收敛到最大间隔解（Soudry 等，2018），即使假设类包含所有线性分类器。深度网络里也有大量关于隐式正则的工作，试图说明梯度方法偏向于某种「最小范数」或「最大间隔」的解。",[11,35330,35331],{},"神经正切核（NTK）提供了一个更清晰的图景：当网络宽度趋于无穷时，训练动力学等价于核回归，泛化由核的谱和 RKHS 范数控制。这把容量从参数空间搬到了函数空间：不是问「有多少个参数」，而是问「在核的度量下，解有多大的范数」。",[11,35333,35334],{},"但要小心：NTK 只描述「懒惰训练」（lazy regime），即特征几乎不变的情形。真实网络会学特征，这时 NTK 不再适用，隐式偏置的理论也远未完成。这是当前深度学习理论最核心的开放问题之一。",[11,35336,35337],{},[698,35338,35339],{},"（c）数据分布：最坏情况太坏",[11,35341,35342],{},"VC 界对每个分布都成立，因此必须按最坏的分布来定价。真实数据不是最坏的：图像有低维结构，类别之间有间隔，长尾分布有层次。",[11,35344,35345,35348,35349,35377,35378,35406],{},[698,35346,35347],{},"双下降","（double descent）是一个典型现象：当模型容量、训练时间或样本量越过插值阈值时，测试误差可以出现第二次下降，而不是经典 U 形曲线预言的那样一路变差（Belkin 等，2019；Nakkiran 等，2020）。需要注意，经典理论从来没有证明过 U 形曲线——它只是小模型时代的经验总结；双下降也不「推翻」VC 理论，它只是说明在 ",[15,35350,35352,35365],{"className":35351,"translate":19},[23],[15,35353,35355],{"className":35354},[27],[29,35356,35357],{"xmlns":31},[34,35358,35359,35363],{},[37,35360,35361],{},[43,35362,10372],{},[109,35364,10372],{"encoding":111},[15,35366,35368],{"className":35367,"ariaHidden":72},[116],[15,35369,35371,35374],{"className":35370},[120],[15,35372],{"className":35373,"style":522},[124],[15,35375,10372],{"className":35376},[129,133]," 随 ",[15,35379,35381,35394],{"className":35380,"translate":19},[23],[15,35382,35384],{"className":35383},[27],[29,35385,35386],{"xmlns":31},[34,35387,35388,35392],{},[37,35389,35390],{},[43,35391,348],{},[109,35393,348],{"encoding":111},[15,35395,35397],{"className":35396,"ariaHidden":72},[116],[15,35398,35400,35403],{"className":35399},[120],[15,35401],{"className":35402,"style":4383},[124],[15,35404,348],{"className":35405},[129,133]," 变化的区域，偏差 - 方差分解的直觉不再适用。",[11,35408,35409,35412],{},[698,35410,35411],{},"良性过拟合","（benign overfitting）则更反直觉：在高维线性回归中，最小范数插值器可以是一致的，只要数据协方差的谱衰减得足够快（Bartlett、Long、Lugosi、Tsigler，2020）。「插值」不等于「过拟合」——这是一个已经被严格证明的现象，而它的成立强烈依赖数据分布的具体性质。",[11,35414,35415],{},"这些结果都指向同一件事：泛化不是假设类单独的性质，而是假设类、算法、分布三者共同的性质。VC 维只看到了第一个。",[575,35417,35419],{"id":35418},"_64-一致收敛的局限","6.4 一致收敛的局限",[11,35421,35422,35423,35426],{},"还有一条更根本的批评。Nagarajan 与 Kolter（2019）构造了一类学习问题，使得真实泛化误差可以很小，但",[698,35424,35425],{},"任何","基于一致收敛的界都必然很大。换句话说，「把 VC 界改紧」这条路存在原理性的上限。",[11,35428,35429],{},"这不是说他们的构造代表了真实网络，而是说：要解释深度学习，必须放弃「对所有假设一致」的要求，转而利用算法或分布的结构。一致收敛是一个充分条件，不是必要条件。",[11,35431,35432,35433,30628,35461,35489,35490],{},"这里有一个容易被忽略的逻辑细节：对二分类 ",[15,35434,35436,35449],{"className":35435,"translate":19},[23],[15,35437,35439],{"className":35438},[27],[29,35440,35441],{"xmlns":31},[34,35442,35443,35447],{},[37,35444,35445],{},[308,35446,541],{},[109,35448,541],{"encoding":111},[15,35450,35452],{"className":35451,"ariaHidden":72},[116],[15,35453,35455,35458],{"className":35454},[120],[15,35456],{"className":35457,"style":4886},[124],[15,35459,541],{"className":35460},[129],[15,35462,35464,35477],{"className":35463,"translate":19},[23],[15,35465,35467],{"className":35466},[27],[29,35468,35469],{"xmlns":31},[34,35470,35471,35475],{},[37,35472,35473],{},[308,35474,318],{},[109,35476,318],{"encoding":111},[15,35478,35480],{"className":35479,"ariaHidden":72},[116],[15,35481,35483,35486],{"className":35482},[120],[15,35484],{"className":35485,"style":4886},[124],[15,35487,318],{"className":35488},[129]," 损失，有限 VC 维确实是 PAC 可学习的充要条件。所以如果深度网络在分布无关的意义下可学，它的 VC 维必须有限——而它确实有限。逻辑上没有矛盾；矛盾只在「界」与「实际」的巨大差距。",[698,35491,35492],{},"VC 理论没有被证伪，它只是没有提供我们想要的那种解释。",[570,35494,35496],{"id":35495},"七容量度量谱系","七、容量度量谱系",[11,35498,35499,35500,35503,35504,35532,35533,35536,35537,35565],{},"在前面的讨论里，VC 维一直默认假设输出是二值的。对实值函数类（回归、概率输出、间隔），它不能直接使用。标准的替代有两种：把实值函数按阈值二值化后取 VC 维，得到 ",[698,35501,35502],{},"pseudo-dimension","；或者要求在某个尺度 ",[15,35505,35507,35520],{"className":35506,"translate":19},[23],[15,35508,35510],{"className":35509},[27],[29,35511,35512],{"xmlns":31},[34,35513,35514,35518],{},[37,35515,35516],{},[43,35517,33938],{},[109,35519,33941],{"encoding":111},[15,35521,35523],{"className":35522,"ariaHidden":72},[116],[15,35524,35526,35529],{"className":35525},[120],[15,35527],{"className":35528,"style":5399},[124],[15,35530,33938],{"className":35531,"style":33954},[129,133]," 下仍能打散，得到 ",[698,35534,35535],{},"fat-shattering dimension","。后者是间隔理论的正式语言：一个类完全可以在 VC 维意义下无穷，却在每个固定的 ",[15,35538,35540,35553],{"className":35539,"translate":19},[23],[15,35541,35543],{"className":35542},[27],[29,35544,35545],{"xmlns":31},[34,35546,35547,35551],{},[37,35548,35549],{},[43,35550,33938],{},[109,35552,33941],{"encoding":111},[15,35554,35556],{"className":35555,"ariaHidden":72},[116],[15,35557,35559,35562],{"className":35558},[120],[15,35560],{"className":35561,"style":5399},[124],[15,35563,33938],{"className":35564,"style":33954},[129,133]," 下只有有限的 fat-shattering 维数——这正是「间隔大则容量小」的精确表述。",[11,35567,35568],{},"把上面提到的各种度量放在一张表里，可以看到它们各自回答的是不同的问题。",[18033,35570,35571,35590],{},[18036,35572,35573],{},[18039,35574,35575,35578,35581,35584,35587],{},[18042,35576,35577],{},"度量",[18042,35579,35580],{},"度量对象",[18042,35582,35583],{},"核心思想",[18042,35585,35586],{},"典型界",[18042,35588,35589],{},"主要局限",[18050,35591,35592,35800,35815,35832,35877,35923,36079,36237,36284,36385],{},[18039,35593,35594,35596,35599,35602,35740],{},[18055,35595,6800],{},[18055,35597,35598],{},"二值函数类",[18055,35600,35601],{},"最大打散集",[18055,35603,35604],{},[15,35605,35607,35649],{"className":35606,"translate":19},[23],[15,35608,35610],{"className":35609},[27],[29,35611,35612],{"xmlns":31},[34,35613,35614,35646],{},[37,35615,35616],{},[29004,35617,35618],{},[37,35619,35620,35622,35624,35626,35628,35630,35632,35634,35636,35638,35640,35642,35644],{},[50,35621,53],{"stretchy":52},[43,35623,10372],{},[50,35625,2682],{},[43,35627,14721],{},[50,35629,69],{},[50,35631,53],{"stretchy":52},[308,35633,318],{},[43,35635,544],{"mathvariant":65},[43,35637,29564],{},[50,35639,59],{"stretchy":52},[50,35641,59],{"stretchy":52},[43,35643,544],{"mathvariant":65},[43,35645,348],{},[109,35647,35648],{"encoding":111},"\\sqrt{(d+\\log(1\u002F\\delta))\u002Fm}",[15,35650,35652],{"className":35651,"ariaHidden":72},[116],[15,35653,35655,35658],{"className":35654},[120],[15,35656],{"className":35657,"style":31881},[124],[15,35659,35661],{"className":35660},[129,29112],[15,35662,35664,35732],{"className":35663},[141,142],[15,35665,35667,35729],{"className":35666},[146],[15,35668,35670,35717],{"className":35669,"style":31894},[150],[15,35671,35673,35676],{"className":35672,"style":31898},[29126],[15,35674],{"className":35675,"style":4450},[158],[15,35677,35679,35682,35685,35688,35691,35694,35699,35702,35705,35708,35711,35714],{"className":35678,"style":29135},[129],[15,35680,53],{"className":35681},[188],[15,35683,10372],{"className":35684},[129,133],[15,35686],{"className":35687,"style":3277},[93],[15,35689,2682],{"className":35690},[2436],[15,35692],{"className":35693,"style":3277},[93],[15,35695,14771,35697],{"className":35696},[217],[15,35698,2247],{"style":2246},[15,35700,53],{"className":35701},[188],[15,35703,29841],{"className":35704},[129],[15,35706,29564],{"className":35707,"style":29845},[129,133],[15,35709,6149],{"className":35710},[195],[15,35712,544],{"className":35713},[129],[15,35715,348],{"className":35716},[129,133],[15,35718,35719,35722],{"style":31916},[15,35720],{"className":35721,"style":4450},[158],[15,35723,35725],{"className":35724,"style":31923},[29271],[4455,35726,35727],{"xmlns":4457,"width":29275,"height":31926,"viewBox":31927,"preserveAspectRatio":29278},[4461,35728],{"d":31930},[15,35730,175],{"className":35731},[174],[15,35733,35735],{"className":35734},[146],[15,35736,35738],{"className":35737,"style":31940},[150],[15,35739],{},[18055,35741,35742,35743,30628,35771,35799],{},"分布无关、最坏情况、只适用 ",[15,35744,35746,35759],{"className":35745,"translate":19},[23],[15,35747,35749],{"className":35748},[27],[29,35750,35751],{"xmlns":31},[34,35752,35753,35757],{},[37,35754,35755],{},[308,35756,541],{},[109,35758,541],{"encoding":111},[15,35760,35762],{"className":35761,"ariaHidden":72},[116],[15,35763,35765,35768],{"className":35764},[120],[15,35766],{"className":35767,"style":4886},[124],[15,35769,541],{"className":35770},[129],[15,35772,35774,35787],{"className":35773,"translate":19},[23],[15,35775,35777],{"className":35776},[27],[29,35778,35779],{"xmlns":31},[34,35780,35781,35785],{},[37,35782,35783],{},[308,35784,318],{},[109,35786,318],{"encoding":111},[15,35788,35790],{"className":35789,"ariaHidden":72},[116],[15,35791,35793,35796],{"className":35792},[120],[15,35794],{"className":35795,"style":4886},[124],[15,35797,318],{"className":35798},[129]," 损失",[18039,35801,35802,35804,35806,35809,35812],{},[18055,35803,20047],{},[18055,35805,35598],{},[18055,35807,35808],{},"限制类的最大规模",[18055,35810,35811],{},"Sauer 多项式界",[18055,35813,35814],{},"组合量，需要再转概率",[18039,35816,35817,35820,35823,35826,35829],{},[18055,35818,35819],{},"Pseudo-dimension",[18055,35821,35822],{},"实值函数类",[18055,35824,35825],{},"阈值化后的 VC 维",[18055,35827,35828],{},"与 VC 同阶",[18055,35830,35831],{},"仍是最坏情况",[18039,35833,35834,35837,35839,35871,35874],{},[18055,35835,35836],{},"Fat-shattering",[18055,35838,35822],{},[18055,35840,35841,35842,35870],{},"尺度 ",[15,35843,35845,35858],{"className":35844,"translate":19},[23],[15,35846,35848],{"className":35847},[27],[29,35849,35850],{"xmlns":31},[34,35851,35852,35856],{},[37,35853,35854],{},[43,35855,33938],{},[109,35857,33941],{"encoding":111},[15,35859,35861],{"className":35860,"ariaHidden":72},[116],[15,35862,35864,35867],{"className":35863},[120],[15,35865],{"className":35866,"style":5399},[124],[15,35868,33938],{"className":35869,"style":33954},[129,133]," 下的打散",[18055,35872,35873],{},"间隔界",[18055,35875,35876],{},"依赖尺度选择",[18039,35878,35879,35882,35885,35917,35920],{},[18055,35880,35881],{},"覆盖数",[18055,35883,35884],{},"度量空间",[18055,35886,35887,35888,35916],{},"用 ",[15,35889,35891,35904],{"className":35890,"translate":19},[23],[15,35892,35894],{"className":35893},[27],[29,35895,35896],{"xmlns":31},[34,35897,35898,35902],{},[37,35899,35900],{},[43,35901,4849],{},[109,35903,34307],{"encoding":111},[15,35905,35907],{"className":35906,"ariaHidden":72},[116],[15,35908,35910,35913],{"className":35909},[120],[15,35911],{"className":35912,"style":4383},[124],[15,35914,4849],{"className":35915},[129,133],"-球覆盖",[18055,35918,35919],{},"Dudley 熵积分",[18055,35921,35922],{},"计算困难",[18039,35924,35925,35927,35930,35933,36076],{},[18055,35926,28882],{},[18055,35928,35929],{},"数据相关",[18055,35931,35932],{},"随机符号下的拟合能力",[18055,35934,35935],{},[15,35936,35938,35978],{"className":35937,"translate":19},[23],[15,35939,35941],{"className":35940},[27],[29,35942,35943],{"xmlns":31},[34,35944,35945,35975],{},[37,35946,35947,35949,35951,35953],{},[308,35948,310],{},[43,35950,104],{"mathvariant":28899},[50,35952,2682],{},[29004,35954,35955],{},[37,35956,35957,35959,35961,35963,35965,35967,35969,35971,35973],{},[43,35958,14721],{},[50,35960,69],{},[50,35962,53],{"stretchy":52},[308,35964,318],{},[43,35966,544],{"mathvariant":65},[43,35968,29564],{},[50,35970,59],{"stretchy":52},[43,35972,544],{"mathvariant":65},[43,35974,348],{},[109,35976,35977],{"encoding":111},"2\\mathfrak R+\\sqrt{\\log(1\u002F\\delta)\u002Fm}",[15,35979,35981,36003],{"className":35980,"ariaHidden":72},[116],[15,35982,35984,35988,35991,35994,35997,36000],{"className":35983},[120],[15,35985],{"className":35986,"style":35987},[124],"height:0.7747em;vertical-align:-0.0833em;",[15,35989,310],{"className":35990},[129],[15,35992,104],{"className":35993},[129,28926],[15,35995],{"className":35996,"style":3277},[93],[15,35998,2682],{"className":35999},[2436],[15,36001],{"className":36002,"style":3277},[93],[15,36004,36006,36009],{"className":36005},[120],[15,36007],{"className":36008,"style":31881},[124],[15,36010,36012],{"className":36011},[129,29112],[15,36013,36015,36068],{"className":36014},[141,142],[15,36016,36018,36065],{"className":36017},[146],[15,36019,36021,36053],{"className":36020,"style":31894},[150],[15,36022,36024,36027],{"className":36023,"style":31898},[29126],[15,36025],{"className":36026,"style":4450},[158],[15,36028,36030,36035,36038,36041,36044,36047,36050],{"className":36029,"style":29135},[129],[15,36031,14771,36033],{"className":36032},[217],[15,36034,2247],{"style":2246},[15,36036,53],{"className":36037},[188],[15,36039,29841],{"className":36040},[129],[15,36042,29564],{"className":36043,"style":29845},[129,133],[15,36045,59],{"className":36046},[195],[15,36048,544],{"className":36049},[129],[15,36051,348],{"className":36052},[129,133],[15,36054,36055,36058],{"style":31916},[15,36056],{"className":36057,"style":4450},[158],[15,36059,36061],{"className":36060,"style":31923},[29271],[4455,36062,36063],{"xmlns":4457,"width":29275,"height":31926,"viewBox":31927,"preserveAspectRatio":29278},[4461,36064],{"d":31930},[15,36066,175],{"className":36067},[174],[15,36069,36071],{"className":36070},[146],[15,36072,36074],{"className":36073,"style":31940},[150],[15,36075],{},[18055,36077,36078],{},"需要估计，通常偏松",[18039,36080,36081,36083,36086,36089,36234],{},[18055,36082,34325],{},[18055,36084,36085],{},"后验与先验",[18055,36087,36088],{},"KL 复杂度",[18055,36090,36091],{},[15,36092,36094,36140],{"className":36093,"translate":19},[23],[15,36095,36097],{"className":36096},[27],[29,36098,36099],{"xmlns":31},[34,36100,36101,36137],{},[37,36102,36103],{},[29004,36104,36105],{},[37,36106,36107,36109,36115,36117,36119,36121,36123,36125,36127,36129,36131,36133,36135],{},[50,36108,53],{"stretchy":52},[37,36110,36111,36113],{},[43,36112,34511],{"mathvariant":65},[43,36114,1373],{"mathvariant":65},[50,36116,2682],{},[43,36118,14721],{},[50,36120,69],{},[50,36122,53],{"stretchy":52},[308,36124,318],{},[43,36126,544],{"mathvariant":65},[43,36128,29564],{},[50,36130,59],{"stretchy":52},[50,36132,59],{"stretchy":52},[43,36134,544],{"mathvariant":65},[43,36136,348],{},[109,36138,36139],{"encoding":111},"\\sqrt{(\\mathrm{KL}+\\log(1\u002F\\delta))\u002Fm}",[15,36141,36143],{"className":36142,"ariaHidden":72},[116],[15,36144,36146,36149],{"className":36145},[120],[15,36147],{"className":36148,"style":31881},[124],[15,36150,36152],{"className":36151},[129,29112],[15,36153,36155,36226],{"className":36154},[141,142],[15,36156,36158,36223],{"className":36157},[146],[15,36159,36161,36211],{"className":36160,"style":31894},[150],[15,36162,36164,36167],{"className":36163,"style":31898},[29126],[15,36165],{"className":36166,"style":4450},[158],[15,36168,36170,36173,36179,36182,36185,36188,36193,36196,36199,36202,36205,36208],{"className":36169,"style":29135},[129],[15,36171,53],{"className":36172},[188],[15,36174,36176],{"className":36175},[129],[15,36177,34915],{"className":36178},[129,221],[15,36180],{"className":36181,"style":3277},[93],[15,36183,2682],{"className":36184},[2436],[15,36186],{"className":36187,"style":3277},[93],[15,36189,14771,36191],{"className":36190},[217],[15,36192,2247],{"style":2246},[15,36194,53],{"className":36195},[188],[15,36197,29841],{"className":36198},[129],[15,36200,29564],{"className":36201,"style":29845},[129,133],[15,36203,6149],{"className":36204},[195],[15,36206,544],{"className":36207},[129],[15,36209,348],{"className":36210},[129,133],[15,36212,36213,36216],{"style":31916},[15,36214],{"className":36215,"style":4450},[158],[15,36217,36219],{"className":36218,"style":31923},[29271],[4455,36220,36221],{"xmlns":4457,"width":29275,"height":31926,"viewBox":31927,"preserveAspectRatio":29278},[4461,36222],{"d":31930},[15,36224,175],{"className":36225},[174],[15,36227,36229],{"className":36228},[146],[15,36230,36232],{"className":36231,"style":31940},[150],[15,36233],{},[18055,36235,36236],{},"先验的选择",[18039,36238,36239,36241,36244,36247,36281],{},[18055,36240,35194],{},[18055,36242,36243],{},"算法",[18055,36245,36246],{},"替换一个样本的影响",[18055,36248,36249,36250],{},"稳定性系数 ",[15,36251,36253,36268],{"className":36252,"translate":19},[23],[15,36254,36256],{"className":36255},[27],[29,36257,36258],{"xmlns":31},[34,36259,36260,36265],{},[37,36261,36262],{},[43,36263,36264],{},"β",[109,36266,36267],{"encoding":111},"\\beta",[15,36269,36271],{"className":36270,"ariaHidden":72},[116],[15,36272,36274,36277],{"className":36273},[120],[15,36275],{"className":36276,"style":1682},[124],[15,36278,36264],{"className":36279,"style":36280},[129,133],"margin-right:0.0528em;",[18055,36282,36283],{},"对 SGD 难以验证",[18039,36285,36286,36288,36291,36294,36382],{},[18055,36287,35071],{},[18055,36289,36290],{},"算法输出",[18055,36292,36293],{},"描述长度",[18055,36295,36296],{},[15,36297,36299,36320],{"className":36298,"translate":19},[23],[15,36300,36302],{"className":36301},[27],[29,36303,36304],{"xmlns":31},[34,36305,36306,36318],{},[37,36307,36308],{},[29004,36309,36310],{},[37,36311,36312,36314,36316],{},[43,36313,16555],{},[43,36315,544],{"mathvariant":65},[43,36317,348],{},[109,36319,35126],{"encoding":111},[15,36321,36323],{"className":36322,"ariaHidden":72},[116],[15,36324,36326,36329],{"className":36325},[120],[15,36327],{"className":36328,"style":31881},[124],[15,36330,36332],{"className":36331},[129,29112],[15,36333,36335,36374],{"className":36334},[141,142],[15,36336,36338,36371],{"className":36337},[146],[15,36339,36341,36359],{"className":36340,"style":31894},[150],[15,36342,36344,36347],{"className":36343,"style":31898},[29126],[15,36345],{"className":36346,"style":4450},[158],[15,36348,36350,36353,36356],{"className":36349,"style":29135},[129],[15,36351,16555],{"className":36352,"style":16588},[129,133],[15,36354,544],{"className":36355},[129],[15,36357,348],{"className":36358},[129,133],[15,36360,36361,36364],{"style":31916},[15,36362],{"className":36363,"style":4450},[158],[15,36365,36367],{"className":36366,"style":31923},[29271],[4455,36368,36369],{"xmlns":4457,"width":29275,"height":31926,"viewBox":31927,"preserveAspectRatio":29278},[4461,36370],{"d":31930},[15,36372,175],{"className":36373},[174],[15,36375,36377],{"className":36376},[146],[15,36378,36380],{"className":36379,"style":31940},[150],[15,36381],{},[18055,36383,36384],{},"需要构造压缩方案",[18039,36386,36387,36390,36393,36396,36399],{},[18055,36388,36389],{},"NTK \u002F RKHS 范数",[18055,36391,36392],{},"核函数",[18055,36394,36395],{},"函数空间中的范数",[18055,36397,36398],{},"谱条件",[18055,36400,36401],{},"只适用于懒惰训练",[11,36403,36404,36405,36408],{},"这张表想说明一件事：",[698,36406,36407],{},"「容量」不是一个数，而是一个依设定而变的函数。"," 你选择哪种度量，等于选择了对「什么算简单」的回答。VC 维的回答是「组合上简单」，间隔界的回答是「几何上简单」，PAC-Bayes 的回答是「相对先验简单」，压缩的回答是「描述上简单」。它们不是互相替代的关系，而是从不同角度逼近同一个问题。",[570,36410,36411],{"id":36411},"结语",[11,36413,36414,36415],{},"回到开头那两个数字。一个只有 1 个参数的正弦分类器可以有无限 VC 维；一个 2500 万参数的 ResNet 却只能在 ImageNet 上得到一个大于 1 的界。这两件事并不矛盾，它们从两个方向说明了同一件事：",[698,36416,36417],{},"容量不是参数的函数，而是假设类结构的函数；泛化不是容量的函数，而是容量、算法与分布三者共同的函数。",{"title":36419,"searchDepth":36420,"depth":36420,"links":36421},"",2,[36422,36428,36433,36438,36444,36445,36451,36452],{"id":572,"depth":36420,"text":573,"children":36423},[36424,36426,36427],{"id":577,"depth":36425,"text":578},3,{"id":2365,"depth":36425,"text":2366},{"id":4801,"depth":36425,"text":4802},{"id":5596,"depth":36420,"text":5597,"children":36429},[36430,36431,36432],{"id":5600,"depth":36425,"text":5601},{"id":7371,"depth":36425,"text":7372},{"id":19560,"depth":36425,"text":19561},{"id":19651,"depth":36420,"text":19652,"children":36434},[36435,36436,36437],{"id":19655,"depth":36425,"text":19656},{"id":20851,"depth":36425,"text":20852},{"id":24159,"depth":36425,"text":24160},{"id":26354,"depth":36420,"text":26355,"children":36439},[36440,36441,36442,36443],{"id":26358,"depth":36425,"text":26359},{"id":29420,"depth":36425,"text":29421},{"id":30592,"depth":36425,"text":30593},{"id":30807,"depth":36425,"text":30808},{"id":31314,"depth":36420,"text":31315},{"id":31838,"depth":36420,"text":31839,"children":36446},[36447,36448,36449,36450],{"id":31842,"depth":36425,"text":31843},{"id":33344,"depth":36425,"text":33345},{"id":33751,"depth":36425,"text":33752},{"id":35418,"depth":36425,"text":35419},{"id":35495,"depth":36420,"text":35496},{"id":36411,"depth":36420,"text":36411},false,true,"md",null,{},"\u002Fblog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension",{"title":6,"description":13},"blog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension","Vapnik-Chervonenkis Dimension 是统计学习理论中刻画假设类容量的核心概念。本文从打散与增长函数出发，完整推导 Sauer–Shelah 引理与一致收敛泛化界，再讨论它为何在深度网络上失效，以及间隔界、PAC-Bayes、稳定性等替代方案如何回应这一困境。",[36463,36464],"mathematics","machine-learning","2025-10-05T18:09:22+08:00","6HatO2q1pkHp4S2UlrkM8KhGJ3NYOLL6EHDVErNnkYM",{"id":36468,"title":36469,"body":36470,"description":36419,"draft":36453,"enableCodeHighlight":36453,"enableCodeHightlight":36453,"enableComment":36454,"enableD2":36453,"enableKatex":36454,"extension":36455,"image":36419,"important":36453,"location":36456,"meta":47878,"navigation":36454,"ogImage":36456,"path":47879,"seo":47880,"stem":47881,"summary":47882,"tags":47883,"time":47884,"__hash__":47885},"blog\u002Fblog\u002F2026\u002F2026-04-06-nonlinearity-with-activations.md","非线性与激活函数",{"type":8,"value":36471,"toc":47864},[36472,36475,36478,36481,36572,36575,36578,36721,37518,37521,37524,37683,38450,38453,38456,38516,38749,38752,38789,39283,39286,39289,39427,39524,39584,39587,39840,40059,40062,40786,40789,40792,41125,41391,41451,41454,41457,41460,41638,41641,41801,41804,42038,42041,42044,42262,42265,42413,42416,42419,42515,42683,42869,42901,42904,43143,43217,43250,43287,43290,43293,43296,43299,43302,43307,43310,43312,43532,43621,43935,43938,43941,44638,44697,44700,44907,44910,44913,45435,45438,45657,45660,46302,46305,46308,46508,46511,46570,46573,46811,46814,46817,47556,47559,47633,47636,47639,47642,47645,47648,47651,47843,47846,47849,47852,47855,47858,47861],[570,36473,36474],{"id":36474},"线性组合",[11,36476,36477],{},"「线性」（linearity）这一概念贯穿数学、物理、工程学以及许多其他领域。简单来说，线性描述了两种基本的运算性质：可加性与齐次性。",[11,36479,36480],{},"如果一个关系、函数或系统同时满足以下两个条件，则称其为线性的。",[11,36482,36483,36484,36512,36513,36541,36542,1313],{},"考虑作用在输入 ",[15,36485,36487,36500],{"className":36486,"translate":19},[23],[15,36488,36490],{"className":36489},[27],[29,36491,36492],{"xmlns":31},[34,36493,36494,36498],{},[37,36495,36496],{},[43,36497,56],{},[109,36499,56],{"encoding":111},[15,36501,36503],{"className":36502,"ariaHidden":72},[116],[15,36504,36506,36509],{"className":36505},[120],[15,36507],{"className":36508,"style":4383},[124],[15,36510,56],{"className":36511},[129,133]," 和 ",[15,36514,36516,36529],{"className":36515,"translate":19},[23],[15,36517,36519],{"className":36518},[27],[29,36520,36521],{"xmlns":31},[34,36522,36523,36527],{},[37,36524,36525],{},[43,36526,857],{},[109,36528,857],{"encoding":111},[15,36530,36532],{"className":36531,"ariaHidden":72},[116],[15,36533,36535,36538],{"className":36534},[120],[15,36536],{"className":36537,"style":5399},[124],[15,36539,857],{"className":36540,"style":887},[129,133]," 上的变换或函数 ",[15,36543,36545,36559],{"className":36544,"translate":19},[23],[15,36546,36548],{"className":36547},[27],[29,36549,36550],{"xmlns":31},[34,36551,36552,36557],{},[37,36553,36554],{},[43,36555,36556],{},"f",[109,36558,36556],{"encoding":111},[15,36560,36562],{"className":36561,"ariaHidden":72},[116],[15,36563,36565,36568],{"className":36564},[120],[15,36566],{"className":36567,"style":1682},[124],[15,36569,36556],{"className":36570,"style":36571},[129,133],"margin-right:0.1076em;",[575,36573,36574],{"id":36574},"可加性",[11,36576,36577],{},"「部分之和等于整体」。也就是说，分别处理两个输入再相加结果，与先将输入相加再处理其和，是等价的。",[15,36579,36581],{"className":36580,"translate":19},[18],[15,36582,36584,36628],{"className":36583,"translate":19},[23],[15,36585,36587],{"className":36586},[27],[29,36588,36589],{"xmlns":31,"display":32},[34,36590,36591,36625],{},[37,36592,36593,36595,36597,36599,36601,36603,36605,36607,36609,36611,36613,36615,36617,36619,36621,36623],{},[43,36594,36556],{},[50,36596,53],{"stretchy":52},[43,36598,56],{},[50,36600,2682],{},[43,36602,857],{},[50,36604,59],{"stretchy":52},[50,36606,62],{},[43,36608,36556],{},[50,36610,53],{"stretchy":52},[43,36612,56],{},[50,36614,59],{"stretchy":52},[50,36616,2682],{},[43,36618,36556],{},[50,36620,53],{"stretchy":52},[43,36622,857],{},[50,36624,59],{"stretchy":52},[109,36626,36627],{"encoding":111},"f(x + y) = f(x) + f(y)",[15,36629,36631,36655,36676,36703],{"className":36630,"ariaHidden":72},[116],[15,36632,36634,36637,36640,36643,36646,36649,36652],{"className":36633},[120],[15,36635],{"className":36636,"style":125},[124],[15,36638,36556],{"className":36639,"style":36571},[129,133],[15,36641,53],{"className":36642},[188],[15,36644,56],{"className":36645},[129,133],[15,36647],{"className":36648,"style":3277},[93],[15,36650,2682],{"className":36651},[2436],[15,36653],{"className":36654,"style":3277},[93],[15,36656,36658,36661,36664,36667,36670,36673],{"className":36657},[120],[15,36659],{"className":36660,"style":125},[124],[15,36662,857],{"className":36663,"style":887},[129,133],[15,36665,59],{"className":36666},[195],[15,36668],{"className":36669,"style":199},[93],[15,36671,62],{"className":36672},[203],[15,36674],{"className":36675,"style":199},[93],[15,36677,36679,36682,36685,36688,36691,36694,36697,36700],{"className":36678},[120],[15,36680],{"className":36681,"style":125},[124],[15,36683,36556],{"className":36684,"style":36571},[129,133],[15,36686,53],{"className":36687},[188],[15,36689,56],{"className":36690},[129,133],[15,36692,59],{"className":36693},[195],[15,36695],{"className":36696,"style":3277},[93],[15,36698,2682],{"className":36699},[2436],[15,36701],{"className":36702,"style":3277},[93],[15,36704,36706,36709,36712,36715,36718],{"className":36705},[120],[15,36707],{"className":36708,"style":125},[124],[15,36710,36556],{"className":36711,"style":36571},[129,133],[15,36713,53],{"className":36714},[188],[15,36716,857],{"className":36717,"style":887},[129,133],[15,36719,59],{"className":36720},[195],[36722,36723,36724,36800,37120,37220],"blockquote",{},[11,36725,36726,36727,36799],{},"示例：乘法 ",[15,36728,36730,36757],{"className":36729,"translate":19},[23],[15,36731,36733],{"className":36732},[27],[29,36734,36735],{"xmlns":31},[34,36736,36737,36754],{},[37,36738,36739,36741,36743,36745,36747,36749,36752],{},[43,36740,36556],{},[50,36742,53],{"stretchy":52},[43,36744,56],{},[50,36746,59],{"stretchy":52},[50,36748,62],{},[308,36750,36751],{},"5",[43,36753,56],{},[109,36755,36756],{"encoding":111},"f(x) = 5x",[15,36758,36760,36787],{"className":36759,"ariaHidden":72},[116],[15,36761,36763,36766,36769,36772,36775,36778,36781,36784],{"className":36762},[120],[15,36764],{"className":36765,"style":125},[124],[15,36767,36556],{"className":36768,"style":36571},[129,133],[15,36770,53],{"className":36771},[188],[15,36773,56],{"className":36774},[129,133],[15,36776,59],{"className":36777},[195],[15,36779],{"className":36780,"style":199},[93],[15,36782,62],{"className":36783},[203],[15,36785],{"className":36786,"style":199},[93],[15,36788,36790,36793,36796],{"className":36789},[120],[15,36791],{"className":36792,"style":4886},[124],[15,36794,36751],{"className":36795},[129],[15,36797,56],{"className":36798},[129,133]," 满足可加性。",[21719,36801,36802],{},[21722,36803,36804,36805,36970,36971,37119],{},"左边：",[15,36806,36808,36853],{"className":36807,"translate":19},[23],[15,36809,36811],{"className":36810},[27],[29,36812,36813],{"xmlns":31},[34,36814,36815,36850],{},[37,36816,36817,36819,36821,36823,36825,36827,36829,36831,36833,36835,36837,36839,36841,36843,36845,36847],{},[43,36818,36556],{},[50,36820,53],{"stretchy":52},[308,36822,310],{},[50,36824,2682],{},[308,36826,9875],{},[50,36828,59],{"stretchy":52},[50,36830,62],{},[308,36832,36751],{},[50,36834,11418],{},[50,36836,53],{"stretchy":52},[308,36838,310],{},[50,36840,2682],{},[308,36842,9875],{},[50,36844,59],{"stretchy":52},[50,36846,62],{},[308,36848,36849],{},"25",[109,36851,36852],{"encoding":111},"f(2+3) = 5 \\times (2+3) = 25",[15,36854,36856,36880,36901,36919,36940,36961],{"className":36855,"ariaHidden":72},[116],[15,36857,36859,36862,36865,36868,36871,36874,36877],{"className":36858},[120],[15,36860],{"className":36861,"style":125},[124],[15,36863,36556],{"className":36864,"style":36571},[129,133],[15,36866,53],{"className":36867},[188],[15,36869,310],{"className":36870},[129],[15,36872],{"className":36873,"style":3277},[93],[15,36875,2682],{"className":36876},[2436],[15,36878],{"className":36879,"style":3277},[93],[15,36881,36883,36886,36889,36892,36895,36898],{"className":36882},[120],[15,36884],{"className":36885,"style":125},[124],[15,36887,9875],{"className":36888},[129],[15,36890,59],{"className":36891},[195],[15,36893],{"className":36894,"style":199},[93],[15,36896,62],{"className":36897},[203],[15,36899],{"className":36900,"style":199},[93],[15,36902,36904,36907,36910,36913,36916],{"className":36903},[120],[15,36905],{"className":36906,"style":16477},[124],[15,36908,36751],{"className":36909},[129],[15,36911],{"className":36912,"style":3277},[93],[15,36914,11418],{"className":36915},[2436],[15,36917],{"className":36918,"style":3277},[93],[15,36920,36922,36925,36928,36931,36934,36937],{"className":36921},[120],[15,36923],{"className":36924,"style":125},[124],[15,36926,53],{"className":36927},[188],[15,36929,310],{"className":36930},[129],[15,36932],{"className":36933,"style":3277},[93],[15,36935,2682],{"className":36936},[2436],[15,36938],{"className":36939,"style":3277},[93],[15,36941,36943,36946,36949,36952,36955,36958],{"className":36942},[120],[15,36944],{"className":36945,"style":125},[124],[15,36947,9875],{"className":36948},[129],[15,36950,59],{"className":36951},[195],[15,36953],{"className":36954,"style":199},[93],[15,36956,62],{"className":36957},[203],[15,36959],{"className":36960,"style":199},[93],[15,36962,36964,36967],{"className":36963},[120],[15,36965],{"className":36966,"style":4886},[124],[15,36968,36849],{"className":36969},[129],"，右边：",[15,36972,36974,37017],{"className":36973,"translate":19},[23],[15,36975,36977],{"className":36976},[27],[29,36978,36979],{"xmlns":31},[34,36980,36981,37014],{},[37,36982,36983,36985,36987,36989,36991,36993,36995,36997,36999,37001,37003,37005,37007,37010,37012],{},[43,36984,36556],{},[50,36986,53],{"stretchy":52},[308,36988,310],{},[50,36990,59],{"stretchy":52},[50,36992,2682],{},[43,36994,36556],{},[50,36996,53],{"stretchy":52},[308,36998,9875],{},[50,37000,59],{"stretchy":52},[50,37002,62],{},[308,37004,32226],{},[50,37006,2682],{},[308,37008,37009],{},"15",[50,37011,62],{},[308,37013,36849],{},[109,37015,37016],{"encoding":111},"f(2) + f(3) = 10 + 15 = 25",[15,37018,37020,37047,37074,37092,37110],{"className":37019,"ariaHidden":72},[116],[15,37021,37023,37026,37029,37032,37035,37038,37041,37044],{"className":37022},[120],[15,37024],{"className":37025,"style":125},[124],[15,37027,36556],{"className":37028,"style":36571},[129,133],[15,37030,53],{"className":37031},[188],[15,37033,310],{"className":37034},[129],[15,37036,59],{"className":37037},[195],[15,37039],{"className":37040,"style":3277},[93],[15,37042,2682],{"className":37043},[2436],[15,37045],{"className":37046,"style":3277},[93],[15,37048,37050,37053,37056,37059,37062,37065,37068,37071],{"className":37049},[120],[15,37051],{"className":37052,"style":125},[124],[15,37054,36556],{"className":37055,"style":36571},[129,133],[15,37057,53],{"className":37058},[188],[15,37060,9875],{"className":37061},[129],[15,37063,59],{"className":37064},[195],[15,37066],{"className":37067,"style":199},[93],[15,37069,62],{"className":37070},[203],[15,37072],{"className":37073,"style":199},[93],[15,37075,37077,37080,37083,37086,37089],{"className":37076},[120],[15,37078],{"className":37079,"style":16477},[124],[15,37081,32226],{"className":37082},[129],[15,37084],{"className":37085,"style":3277},[93],[15,37087,2682],{"className":37088},[2436],[15,37090],{"className":37091,"style":3277},[93],[15,37093,37095,37098,37101,37104,37107],{"className":37094},[120],[15,37096],{"className":37097,"style":4886},[124],[15,37099,37009],{"className":37100},[129],[15,37102],{"className":37103,"style":199},[93],[15,37105,62],{"className":37106},[203],[15,37108],{"className":37109,"style":199},[93],[15,37111,37113,37116],{"className":37112},[120],[15,37114],{"className":37115,"style":4886},[124],[15,37117,36849],{"className":37118},[129],"，相等。",[11,37121,37122,37123,37219],{},"反例：平方函数 ",[15,37124,37126,37154],{"className":37125,"translate":19},[23],[15,37127,37129],{"className":37128},[27],[29,37130,37131],{"xmlns":31},[34,37132,37133,37151],{},[37,37134,37135,37137,37139,37141,37143,37145],{},[43,37136,36556],{},[50,37138,53],{"stretchy":52},[43,37140,56],{},[50,37142,59],{"stretchy":52},[50,37144,62],{},[305,37146,37147,37149],{},[43,37148,56],{},[308,37150,310],{},[109,37152,37153],{"encoding":111},"f(x) = x^2",[15,37155,37157,37184],{"className":37156,"ariaHidden":72},[116],[15,37158,37160,37163,37166,37169,37172,37175,37178,37181],{"className":37159},[120],[15,37161],{"className":37162,"style":125},[124],[15,37164,36556],{"className":37165,"style":36571},[129,133],[15,37167,53],{"className":37168},[188],[15,37170,56],{"className":37171},[129,133],[15,37173,59],{"className":37174},[195],[15,37176],{"className":37177,"style":199},[93],[15,37179,62],{"className":37180},[203],[15,37182],{"className":37183,"style":199},[93],[15,37185,37187,37190],{"className":37186},[120],[15,37188],{"className":37189,"style":380},[124],[15,37191,37193,37196],{"className":37192},[129],[15,37194,56],{"className":37195},[129,133],[15,37197,37199],{"className":37198},[137],[15,37200,37202],{"className":37201},[141],[15,37203,37205],{"className":37204},[146],[15,37206,37208],{"className":37207,"style":380},[150],[15,37209,37210,37213],{"style":383},[15,37211],{"className":37212,"style":159},[158],[15,37214,37216],{"className":37215},[163,164,165,166],[15,37217,310],{"className":37218},[129,166]," 不满足可加性。",[21719,37221,37222],{},[21722,37223,36804,37224,36970,37337,37517],{},[15,37225,37227,37257],{"className":37226,"translate":19},[23],[15,37228,37230],{"className":37229},[27],[29,37231,37232],{"xmlns":31},[34,37233,37234,37254],{},[37,37235,37236,37238,37240,37242,37244,37250,37252],{},[50,37237,53],{"stretchy":52},[308,37239,310],{},[50,37241,2682],{},[308,37243,9875],{},[305,37245,37246,37248],{},[50,37247,59],{"stretchy":52},[308,37249,310],{},[50,37251,62],{},[308,37253,36849],{},[109,37255,37256],{"encoding":111},"(2+3)^2 = 25",[15,37258,37260,37281,37328],{"className":37259,"ariaHidden":72},[116],[15,37261,37263,37266,37269,37272,37275,37278],{"className":37262},[120],[15,37264],{"className":37265,"style":125},[124],[15,37267,53],{"className":37268},[188],[15,37270,310],{"className":37271},[129],[15,37273],{"className":37274,"style":3277},[93],[15,37276,2682],{"className":37277},[2436],[15,37279],{"className":37280,"style":3277},[93],[15,37282,37284,37287,37290,37319,37322,37325],{"className":37283},[120],[15,37285],{"className":37286,"style":33996},[124],[15,37288,9875],{"className":37289},[129],[15,37291,37293,37296],{"className":37292},[195],[15,37294,59],{"className":37295},[195],[15,37297,37299],{"className":37298},[137],[15,37300,37302],{"className":37301},[141],[15,37303,37305],{"className":37304},[146],[15,37306,37308],{"className":37307,"style":380},[150],[15,37309,37310,37313],{"style":383},[15,37311],{"className":37312,"style":159},[158],[15,37314,37316],{"className":37315},[163,164,165,166],[15,37317,310],{"className":37318},[129,166],[15,37320],{"className":37321,"style":199},[93],[15,37323,62],{"className":37324},[203],[15,37326],{"className":37327,"style":199},[93],[15,37329,37331,37334],{"className":37330},[120],[15,37332],{"className":37333,"style":4886},[124],[15,37335,36849],{"className":37336},[129],[15,37338,37340,37380],{"className":37339,"translate":19},[23],[15,37341,37343],{"className":37342},[27],[29,37344,37345],{"xmlns":31},[34,37346,37347,37377],{},[37,37348,37349,37355,37357,37363,37365,37367,37369,37372,37374],{},[305,37350,37351,37353],{},[308,37352,310],{},[308,37354,310],{},[50,37356,2682],{},[305,37358,37359,37361],{},[308,37360,9875],{},[308,37362,310],{},[50,37364,62],{},[308,37366,14502],{},[50,37368,2682],{},[308,37370,37371],{},"9",[50,37373,62],{},[308,37375,37376],{},"13",[109,37378,37379],{"encoding":111},"2^2 + 3^2 = 4+9=13",[15,37381,37383,37428,37472,37490,37508],{"className":37382,"ariaHidden":72},[116],[15,37384,37386,37390,37419,37422,37425],{"className":37385},[120],[15,37387],{"className":37388,"style":37389},[124],"height:0.8974em;vertical-align:-0.0833em;",[15,37391,37393,37396],{"className":37392},[129],[15,37394,310],{"className":37395},[129],[15,37397,37399],{"className":37398},[137],[15,37400,37402],{"className":37401},[141],[15,37403,37405],{"className":37404},[146],[15,37406,37408],{"className":37407,"style":380},[150],[15,37409,37410,37413],{"style":383},[15,37411],{"className":37412,"style":159},[158],[15,37414,37416],{"className":37415},[163,164,165,166],[15,37417,310],{"className":37418},[129,166],[15,37420],{"className":37421,"style":3277},[93],[15,37423,2682],{"className":37424},[2436],[15,37426],{"className":37427,"style":3277},[93],[15,37429,37431,37434,37463,37466,37469],{"className":37430},[120],[15,37432],{"className":37433,"style":380},[124],[15,37435,37437,37440],{"className":37436},[129],[15,37438,9875],{"className":37439},[129],[15,37441,37443],{"className":37442},[137],[15,37444,37446],{"className":37445},[141],[15,37447,37449],{"className":37448},[146],[15,37450,37452],{"className":37451,"style":380},[150],[15,37453,37454,37457],{"style":383},[15,37455],{"className":37456,"style":159},[158],[15,37458,37460],{"className":37459},[163,164,165,166],[15,37461,310],{"className":37462},[129,166],[15,37464],{"className":37465,"style":199},[93],[15,37467,62],{"className":37468},[203],[15,37470],{"className":37471,"style":199},[93],[15,37473,37475,37478,37481,37484,37487],{"className":37474},[120],[15,37476],{"className":37477,"style":16477},[124],[15,37479,14502],{"className":37480},[129],[15,37482],{"className":37483,"style":3277},[93],[15,37485,2682],{"className":37486},[2436],[15,37488],{"className":37489,"style":3277},[93],[15,37491,37493,37496,37499,37502,37505],{"className":37492},[120],[15,37494],{"className":37495,"style":4886},[124],[15,37497,37371],{"className":37498},[129],[15,37500],{"className":37501,"style":199},[93],[15,37503,62],{"className":37504},[203],[15,37506],{"className":37507,"style":199},[93],[15,37509,37511,37514],{"className":37510},[120],[15,37512],{"className":37513,"style":4886},[124],[15,37515,37376],{"className":37516},[129],"，不相等。",[575,37519,37520],{"id":37520},"齐次性",[11,37522,37523],{},"「输入的缩放会使输出按相同比例缩放」。也就是说，将输入乘以一个常数（标量），得到的结果等于原输出乘以同一个常数。",[15,37525,37527],{"className":37526,"translate":19},[18],[15,37528,37530,37577],{"className":37529,"translate":19},[23],[15,37531,37533],{"className":37532},[27],[29,37534,37535],{"xmlns":31,"display":32},[34,37536,37537,37574],{},[37,37538,37539,37541,37543,37545,37547,37549,37551,37553,37555,37557,37559,37561,37563,37565,37567,37569,37572],{},[43,37540,36556],{},[50,37542,53],{"stretchy":52},[43,37544,5628],{},[50,37546,10549],{},[43,37548,56],{},[50,37550,59],{"stretchy":52},[50,37552,62],{},[43,37554,5628],{},[50,37556,10549],{},[43,37558,36556],{},[50,37560,53],{"stretchy":52},[43,37562,56],{},[50,37564,59],{"stretchy":52},[93,37566],{"width":16914},[50,37568,53],{"stretchy":52},[6852,37570,37571],{},"c 为任意常数",[50,37573,59],{"stretchy":52},[109,37575,37576],{"encoding":111},"f(c \\cdot x) = c \\cdot f(x) \\quad (\\text{c 为任意常数})",[15,37578,37580,37604,37625,37643],{"className":37579,"ariaHidden":72},[116],[15,37581,37583,37586,37589,37592,37595,37598,37601],{"className":37582},[120],[15,37584],{"className":37585,"style":125},[124],[15,37587,36556],{"className":37588,"style":36571},[129,133],[15,37590,53],{"className":37591},[188],[15,37593,5628],{"className":37594},[129,133],[15,37596],{"className":37597,"style":3277},[93],[15,37599,10549],{"className":37600},[2436],[15,37602],{"className":37603,"style":3277},[93],[15,37605,37607,37610,37613,37616,37619,37622],{"className":37606},[120],[15,37608],{"className":37609,"style":125},[124],[15,37611,56],{"className":37612},[129,133],[15,37614,59],{"className":37615},[195],[15,37617],{"className":37618,"style":199},[93],[15,37620,62],{"className":37621},[203],[15,37623],{"className":37624,"style":199},[93],[15,37626,37628,37631,37634,37637,37640],{"className":37627},[120],[15,37629],{"className":37630,"style":11094},[124],[15,37632,5628],{"className":37633},[129,133],[15,37635],{"className":37636,"style":3277},[93],[15,37638,10549],{"className":37639},[2436],[15,37641],{"className":37642,"style":3277},[93],[15,37644,37646,37649,37652,37655,37658,37661,37664,37667,37680],{"className":37645},[120],[15,37647],{"className":37648,"style":125},[124],[15,37650,36556],{"className":37651,"style":36571},[129,133],[15,37653,53],{"className":37654},[188],[15,37656,56],{"className":37657},[129,133],[15,37659,59],{"className":37660},[195],[15,37662],{"className":37663,"style":17287},[93],[15,37665,53],{"className":37666},[188],[15,37668,37671,37675],{"className":37669},[129,37670],"text",[15,37672,37674],{"className":37673},[129],"c ",[15,37676,37679],{"className":37677},[129,37678],"cjk_fallback","为任意常数",[15,37681,59],{"className":37682},[195],[36722,37684,37685,37759,38029,38126],{},[11,37686,37687,37688,37758],{},"示例：",[15,37689,37691,37716],{"className":37690,"translate":19},[23],[15,37692,37694],{"className":37693},[27],[29,37695,37696],{"xmlns":31},[34,37697,37698,37714],{},[37,37699,37700,37702,37704,37706,37708,37710,37712],{},[43,37701,36556],{},[50,37703,53],{"stretchy":52},[43,37705,56],{},[50,37707,59],{"stretchy":52},[50,37709,62],{},[308,37711,36751],{},[43,37713,56],{},[109,37715,36756],{"encoding":111},[15,37717,37719,37746],{"className":37718,"ariaHidden":72},[116],[15,37720,37722,37725,37728,37731,37734,37737,37740,37743],{"className":37721},[120],[15,37723],{"className":37724,"style":125},[124],[15,37726,36556],{"className":37727,"style":36571},[129,133],[15,37729,53],{"className":37730},[188],[15,37732,56],{"className":37733},[129,133],[15,37735,59],{"className":37736},[195],[15,37738],{"className":37739,"style":199},[93],[15,37741,62],{"className":37742},[203],[15,37744],{"className":37745,"style":199},[93],[15,37747,37749,37752,37755],{"className":37748},[120],[15,37750],{"className":37751,"style":4886},[124],[15,37753,36751],{"className":37754},[129],[15,37756,56],{"className":37757},[129,133]," 满足齐次性。",[21719,37760,37761],{},[21722,37762,36804,37763,36970,37897,37119],{},[15,37764,37766,37804],{"className":37765,"translate":19},[23],[15,37767,37769],{"className":37768},[27],[29,37770,37771],{"xmlns":31},[34,37772,37773,37801],{},[37,37774,37775,37777,37779,37781,37783,37785,37787,37789,37791,37793,37796,37798],{},[43,37776,36556],{},[50,37778,53],{"stretchy":52},[308,37780,14502],{},[50,37782,11418],{},[308,37784,9875],{},[50,37786,59],{"stretchy":52},[50,37788,62],{},[308,37790,36751],{},[50,37792,11418],{},[308,37794,37795],{},"12",[50,37797,62],{},[308,37799,37800],{},"60",[109,37802,37803],{"encoding":111},"f(4 \\times 3) = 5 \\times 12 = 60",[15,37805,37807,37831,37852,37870,37888],{"className":37806,"ariaHidden":72},[116],[15,37808,37810,37813,37816,37819,37822,37825,37828],{"className":37809},[120],[15,37811],{"className":37812,"style":125},[124],[15,37814,36556],{"className":37815,"style":36571},[129,133],[15,37817,53],{"className":37818},[188],[15,37820,14502],{"className":37821},[129],[15,37823],{"className":37824,"style":3277},[93],[15,37826,11418],{"className":37827},[2436],[15,37829],{"className":37830,"style":3277},[93],[15,37832,37834,37837,37840,37843,37846,37849],{"className":37833},[120],[15,37835],{"className":37836,"style":125},[124],[15,37838,9875],{"className":37839},[129],[15,37841,59],{"className":37842},[195],[15,37844],{"className":37845,"style":199},[93],[15,37847,62],{"className":37848},[203],[15,37850],{"className":37851,"style":199},[93],[15,37853,37855,37858,37861,37864,37867],{"className":37854},[120],[15,37856],{"className":37857,"style":16477},[124],[15,37859,36751],{"className":37860},[129],[15,37862],{"className":37863,"style":3277},[93],[15,37865,11418],{"className":37866},[2436],[15,37868],{"className":37869,"style":3277},[93],[15,37871,37873,37876,37879,37882,37885],{"className":37872},[120],[15,37874],{"className":37875,"style":4886},[124],[15,37877,37795],{"className":37878},[129],[15,37880],{"className":37881,"style":199},[93],[15,37883,62],{"className":37884},[203],[15,37886],{"className":37887,"style":199},[93],[15,37889,37891,37894],{"className":37890},[120],[15,37892],{"className":37893,"style":4886},[124],[15,37895,37800],{"className":37896},[129],[15,37898,37900,37936],{"className":37899,"translate":19},[23],[15,37901,37903],{"className":37902},[27],[29,37904,37905],{"xmlns":31},[34,37906,37907,37933],{},[37,37908,37909,37911,37913,37915,37917,37919,37921,37923,37925,37927,37929,37931],{},[308,37910,14502],{},[50,37912,11418],{},[43,37914,36556],{},[50,37916,53],{"stretchy":52},[308,37918,9875],{},[50,37920,59],{"stretchy":52},[50,37922,62],{},[308,37924,14502],{},[50,37926,11418],{},[308,37928,37009],{},[50,37930,62],{},[308,37932,37800],{},[109,37934,37935],{"encoding":111},"4 \\times f(3) = 4 \\times 15 = 60",[15,37937,37939,37957,37984,38002,38020],{"className":37938,"ariaHidden":72},[116],[15,37940,37942,37945,37948,37951,37954],{"className":37941},[120],[15,37943],{"className":37944,"style":16477},[124],[15,37946,14502],{"className":37947},[129],[15,37949],{"className":37950,"style":3277},[93],[15,37952,11418],{"className":37953},[2436],[15,37955],{"className":37956,"style":3277},[93],[15,37958,37960,37963,37966,37969,37972,37975,37978,37981],{"className":37959},[120],[15,37961],{"className":37962,"style":125},[124],[15,37964,36556],{"className":37965,"style":36571},[129,133],[15,37967,53],{"className":37968},[188],[15,37970,9875],{"className":37971},[129],[15,37973,59],{"className":37974},[195],[15,37976],{"className":37977,"style":199},[93],[15,37979,62],{"className":37980},[203],[15,37982],{"className":37983,"style":199},[93],[15,37985,37987,37990,37993,37996,37999],{"className":37986},[120],[15,37988],{"className":37989,"style":16477},[124],[15,37991,14502],{"className":37992},[129],[15,37994],{"className":37995,"style":3277},[93],[15,37997,11418],{"className":37998},[2436],[15,38000],{"className":38001,"style":3277},[93],[15,38003,38005,38008,38011,38014,38017],{"className":38004},[120],[15,38006],{"className":38007,"style":4886},[124],[15,38009,37009],{"className":38010},[129],[15,38012],{"className":38013,"style":199},[93],[15,38015,62],{"className":38016},[203],[15,38018],{"className":38019,"style":199},[93],[15,38021,38023,38026],{"className":38022},[120],[15,38024],{"className":38025,"style":4886},[124],[15,38027,37800],{"className":38028},[129],[11,38030,38031,38032,38125],{},"反例：加上常数项 ",[15,38033,38035,38065],{"className":38034,"translate":19},[23],[15,38036,38038],{"className":38037},[27],[29,38039,38040],{"xmlns":31},[34,38041,38042,38062],{},[37,38043,38044,38046,38048,38050,38052,38054,38056,38058,38060],{},[43,38045,36556],{},[50,38047,53],{"stretchy":52},[43,38049,56],{},[50,38051,59],{"stretchy":52},[50,38053,62],{},[308,38055,36751],{},[43,38057,56],{},[50,38059,2682],{},[308,38061,310],{},[109,38063,38064],{"encoding":111},"f(x) = 5x + 2",[15,38066,38068,38095,38116],{"className":38067,"ariaHidden":72},[116],[15,38069,38071,38074,38077,38080,38083,38086,38089,38092],{"className":38070},[120],[15,38072],{"className":38073,"style":125},[124],[15,38075,36556],{"className":38076,"style":36571},[129,133],[15,38078,53],{"className":38079},[188],[15,38081,56],{"className":38082},[129,133],[15,38084,59],{"className":38085},[195],[15,38087],{"className":38088,"style":199},[93],[15,38090,62],{"className":38091},[203],[15,38093],{"className":38094,"style":199},[93],[15,38096,38098,38101,38104,38107,38110,38113],{"className":38097},[120],[15,38099],{"className":38100,"style":16477},[124],[15,38102,36751],{"className":38103},[129],[15,38105,56],{"className":38106},[129,133],[15,38108],{"className":38109,"style":3277},[93],[15,38111,2682],{"className":38112},[2436],[15,38114],{"className":38115,"style":3277},[93],[15,38117,38119,38122],{"className":38118},[120],[15,38120],{"className":38121,"style":4886},[124],[15,38123,310],{"className":38124},[129]," 不满足齐次性。",[21719,38127,38128],{},[21722,38129,36804,38130,36970,38285,37517],{},[15,38131,38133,38174],{"className":38132,"translate":19},[23],[15,38134,38136],{"className":38135},[27],[29,38137,38138],{"xmlns":31},[34,38139,38140,38171],{},[37,38141,38142,38144,38146,38148,38150,38152,38154,38156,38158,38160,38162,38164,38166,38168],{},[43,38143,36556],{},[50,38145,53],{"stretchy":52},[308,38147,14502],{},[50,38149,11418],{},[308,38151,9875],{},[50,38153,59],{"stretchy":52},[50,38155,62],{},[308,38157,36751],{},[50,38159,11418],{},[308,38161,37795],{},[50,38163,2682],{},[308,38165,310],{},[50,38167,62],{},[308,38169,38170],{},"62",[109,38172,38173],{"encoding":111},"f(4 \\times 3) = 5 \\times 12 + 2 = 62",[15,38175,38177,38201,38222,38240,38258,38276],{"className":38176,"ariaHidden":72},[116],[15,38178,38180,38183,38186,38189,38192,38195,38198],{"className":38179},[120],[15,38181],{"className":38182,"style":125},[124],[15,38184,36556],{"className":38185,"style":36571},[129,133],[15,38187,53],{"className":38188},[188],[15,38190,14502],{"className":38191},[129],[15,38193],{"className":38194,"style":3277},[93],[15,38196,11418],{"className":38197},[2436],[15,38199],{"className":38200,"style":3277},[93],[15,38202,38204,38207,38210,38213,38216,38219],{"className":38203},[120],[15,38205],{"className":38206,"style":125},[124],[15,38208,9875],{"className":38209},[129],[15,38211,59],{"className":38212},[195],[15,38214],{"className":38215,"style":199},[93],[15,38217,62],{"className":38218},[203],[15,38220],{"className":38221,"style":199},[93],[15,38223,38225,38228,38231,38234,38237],{"className":38224},[120],[15,38226],{"className":38227,"style":16477},[124],[15,38229,36751],{"className":38230},[129],[15,38232],{"className":38233,"style":3277},[93],[15,38235,11418],{"className":38236},[2436],[15,38238],{"className":38239,"style":3277},[93],[15,38241,38243,38246,38249,38252,38255],{"className":38242},[120],[15,38244],{"className":38245,"style":16477},[124],[15,38247,37795],{"className":38248},[129],[15,38250],{"className":38251,"style":3277},[93],[15,38253,2682],{"className":38254},[2436],[15,38256],{"className":38257,"style":3277},[93],[15,38259,38261,38264,38267,38270,38273],{"className":38260},[120],[15,38262],{"className":38263,"style":4886},[124],[15,38265,310],{"className":38266},[129],[15,38268],{"className":38269,"style":199},[93],[15,38271,62],{"className":38272},[203],[15,38274],{"className":38275,"style":199},[93],[15,38277,38279,38282],{"className":38278},[120],[15,38280],{"className":38281,"style":4886},[124],[15,38283,38170],{"className":38284},[129],[15,38286,38288,38333],{"className":38287,"translate":19},[23],[15,38289,38291],{"className":38290},[27],[29,38292,38293],{"xmlns":31},[34,38294,38295,38330],{},[37,38296,38297,38299,38301,38303,38305,38307,38309,38311,38313,38315,38317,38319,38321,38323,38325,38327],{},[308,38298,14502],{},[50,38300,11418],{},[43,38302,36556],{},[50,38304,53],{"stretchy":52},[308,38306,9875],{},[50,38308,59],{"stretchy":52},[50,38310,62],{},[308,38312,14502],{},[50,38314,11418],{},[50,38316,53],{"stretchy":52},[308,38318,37009],{},[50,38320,2682],{},[308,38322,310],{},[50,38324,59],{"stretchy":52},[50,38326,62],{},[308,38328,38329],{},"68",[109,38331,38332],{"encoding":111},"4 \\times f(3) = 4 \\times (15+2) = 68",[15,38334,38336,38354,38381,38399,38420,38441],{"className":38335,"ariaHidden":72},[116],[15,38337,38339,38342,38345,38348,38351],{"className":38338},[120],[15,38340],{"className":38341,"style":16477},[124],[15,38343,14502],{"className":38344},[129],[15,38346],{"className":38347,"style":3277},[93],[15,38349,11418],{"className":38350},[2436],[15,38352],{"className":38353,"style":3277},[93],[15,38355,38357,38360,38363,38366,38369,38372,38375,38378],{"className":38356},[120],[15,38358],{"className":38359,"style":125},[124],[15,38361,36556],{"className":38362,"style":36571},[129,133],[15,38364,53],{"className":38365},[188],[15,38367,9875],{"className":38368},[129],[15,38370,59],{"className":38371},[195],[15,38373],{"className":38374,"style":199},[93],[15,38376,62],{"className":38377},[203],[15,38379],{"className":38380,"style":199},[93],[15,38382,38384,38387,38390,38393,38396],{"className":38383},[120],[15,38385],{"className":38386,"style":16477},[124],[15,38388,14502],{"className":38389},[129],[15,38391],{"className":38392,"style":3277},[93],[15,38394,11418],{"className":38395},[2436],[15,38397],{"className":38398,"style":3277},[93],[15,38400,38402,38405,38408,38411,38414,38417],{"className":38401},[120],[15,38403],{"className":38404,"style":125},[124],[15,38406,53],{"className":38407},[188],[15,38409,37009],{"className":38410},[129],[15,38412],{"className":38413,"style":3277},[93],[15,38415,2682],{"className":38416},[2436],[15,38418],{"className":38419,"style":3277},[93],[15,38421,38423,38426,38429,38432,38435,38438],{"className":38422},[120],[15,38424],{"className":38425,"style":125},[124],[15,38427,310],{"className":38428},[129],[15,38430,59],{"className":38431},[195],[15,38433],{"className":38434,"style":199},[93],[15,38436,62],{"className":38437},[203],[15,38439],{"className":38440,"style":199},[93],[15,38442,38444,38447],{"className":38443},[120],[15,38445],{"className":38446,"style":4886},[124],[15,38448,38329],{"className":38449},[129],[11,38451,38452],{},"在上述定义中，同时满足可加性与齐次性是线性成立的充分必要条件。",[11,38454,38455],{},"许多人错误地认为「线性」就是「图像是一条直线」。这种理解并不准确。",[11,38457,38458,38459,38515],{},"「真正的线性」：一条必须过原点的直线。例如 ",[15,38460,38462,38482],{"className":38461,"translate":19},[23],[15,38463,38465],{"className":38464},[27],[29,38466,38467],{"xmlns":31},[34,38468,38469,38479],{},[37,38470,38471,38473,38475,38477],{},[43,38472,857],{},[50,38474,62],{},[308,38476,310],{},[43,38478,56],{},[109,38480,38481],{"encoding":111},"y = 2x",[15,38483,38485,38503],{"className":38484,"ariaHidden":72},[116],[15,38486,38488,38491,38494,38497,38500],{"className":38487},[120],[15,38489],{"className":38490,"style":5399},[124],[15,38492,857],{"className":38493,"style":887},[129,133],[15,38495],{"className":38496,"style":199},[93],[15,38498,62],{"className":38499},[203],[15,38501],{"className":38502,"style":199},[93],[15,38504,38506,38509,38512],{"className":38505},[120],[15,38507],{"className":38508,"style":4886},[124],[15,38510,310],{"className":38511},[129],[15,38513,56],{"className":38514},[129,133],"。它同时满足可加性与齐次性。如果输入为 0，输出也必为 0。",[11,38517,38518,38519,38547,38548,38626,38627,38748],{},"严格来说，如果加入偏置项 ",[15,38520,38522,38535],{"className":38521,"translate":19},[23],[15,38523,38525],{"className":38524},[27],[29,38526,38527],{"xmlns":31},[34,38528,38529,38533],{},[37,38530,38531],{},[43,38532,8695],{},[109,38534,8695],{"encoding":111},[15,38536,38538],{"className":38537,"ariaHidden":72},[116],[15,38539,38541,38544],{"className":38540},[120],[15,38542],{"className":38543,"style":522},[124],[15,38545,8695],{"className":38546},[129,133],"，它就变成了仿射（affine）。它仍是一条直线，但未必经过原点。例如 ",[15,38549,38551,38575],{"className":38550,"translate":19},[23],[15,38552,38554],{"className":38553},[27],[29,38555,38556],{"xmlns":31},[34,38557,38558,38572],{},[37,38559,38560,38562,38564,38566,38568,38570],{},[43,38561,857],{},[50,38563,62],{},[308,38565,310],{},[43,38567,56],{},[50,38569,2682],{},[308,38571,318],{},[109,38573,38574],{"encoding":111},"y = 2x + 1",[15,38576,38578,38596,38617],{"className":38577,"ariaHidden":72},[116],[15,38579,38581,38584,38587,38590,38593],{"className":38580},[120],[15,38582],{"className":38583,"style":5399},[124],[15,38585,857],{"className":38586,"style":887},[129,133],[15,38588],{"className":38589,"style":199},[93],[15,38591,62],{"className":38592},[203],[15,38594],{"className":38595,"style":199},[93],[15,38597,38599,38602,38605,38608,38611,38614],{"className":38598},[120],[15,38600],{"className":38601,"style":16477},[124],[15,38603,310],{"className":38604},[129],[15,38606,56],{"className":38607},[129,133],[15,38609],{"className":38610,"style":3277},[93],[15,38612,2682],{"className":38613},[2436],[15,38615],{"className":38616,"style":3277},[93],[15,38618,38620,38623],{"className":38619},[120],[15,38621],{"className":38622,"style":4886},[124],[15,38624,318],{"className":38625},[129],"。它不满足齐次性（因为 ",[15,38628,38630,38658],{"className":38629,"translate":19},[23],[15,38631,38633],{"className":38632},[27],[29,38634,38635],{"xmlns":31},[34,38636,38637,38655],{},[37,38638,38639,38641,38643,38645,38647,38649,38651,38653],{},[43,38640,36556],{},[50,38642,53],{"stretchy":52},[308,38644,541],{},[50,38646,59],{"stretchy":52},[50,38648,62],{},[308,38650,318],{},[50,38652,1418],{"mathvariant":65},[308,38654,541],{},[109,38656,38657],{"encoding":111},"f(0)=1 \\neq 0",[15,38659,38661,38688,38739],{"className":38660,"ariaHidden":72},[116],[15,38662,38664,38667,38670,38673,38676,38679,38682,38685],{"className":38663},[120],[15,38665],{"className":38666,"style":125},[124],[15,38668,36556],{"className":38669,"style":36571},[129,133],[15,38671,53],{"className":38672},[188],[15,38674,541],{"className":38675},[129],[15,38677,59],{"className":38678},[195],[15,38680],{"className":38681,"style":199},[93],[15,38683,62],{"className":38684},[203],[15,38686],{"className":38687,"style":199},[93],[15,38689,38691,38694,38697,38700,38736],{"className":38690},[120],[15,38692],{"className":38693,"style":1682},[124],[15,38695,318],{"className":38696},[129],[15,38698],{"className":38699,"style":199},[93],[15,38701,38703,38730,38733],{"className":38702},[203],[15,38704,38706],{"className":38705},[203],[15,38707,38709],{"className":38708},[129,1670],[15,38710,38712],{"className":38711},[1674],[15,38713,38715,38718,38727],{"className":38714},[1678],[15,38716],{"className":38717,"style":1682},[124],[15,38719,38721],{"className":38720},[1686],[15,38722,38724],{"className":38723},[129],[15,38725,1693],{"className":38726},[203],[15,38728],{"className":38729},[1697],[15,38731],{"className":38732},[93,1701],[15,38734,62],{"className":38735},[203],[15,38737],{"className":38738,"style":199},[93],[15,38740,38742,38745],{"className":38741},[120],[15,38743],{"className":38744,"style":4886},[124],[15,38746,541],{"className":38747},[129],"）。尽管它常被称为「线性函数」，但在严格定义下它并非线性，而是「线性 + 平移」，即仿射。",[11,38750,38751],{},"现在我们就能理解为什么称为「线性」组合了。这种组合只允许两类「线性」运算：",[30659,38753,38754,38786],{},[21722,38755,38756,38757,38785],{},"数乘（对应齐次性）：将向量按因子 ",[15,38758,38760,38773],{"className":38759,"translate":19},[23],[15,38761,38763],{"className":38762},[27],[29,38764,38765],{"xmlns":31},[34,38766,38767,38771],{},[37,38768,38769],{},[43,38770,5628],{},[109,38772,5628],{"encoding":111},[15,38774,38776],{"className":38775,"ariaHidden":72},[116],[15,38777,38779,38782],{"className":38778},[120],[15,38780],{"className":38781,"style":4383},[124],[15,38783,5628],{"className":38784},[129,133]," 拉伸或压缩。",[21722,38787,38788],{},"加法（对应可加性）：将处理后的向量相加。",[11,38790,38791,38792,39034,39035,39282],{},"像平方、取绝对值或对数这类非线性运算是不被允许的。因此，形如 ",[15,38793,38795,38836],{"className":38794,"translate":19},[23],[15,38796,38798],{"className":38797},[27],[29,38799,38800],{"xmlns":31},[34,38801,38802,38833],{},[37,38803,38804,38810,38819,38821,38827],{},[40,38805,38806,38808],{},[43,38807,5628],{},[308,38809,318],{},[23149,38811,38812,38815,38817],{},[43,38813,38814],{},"v",[308,38816,318],{},[308,38818,310],{},[50,38820,2682],{},[40,38822,38823,38825],{},[43,38824,5628],{},[308,38826,310],{},[40,38828,38829,38831],{},[43,38830,38814],{},[308,38832,310],{},[109,38834,38835],{"encoding":111},"c_1 v_1^2 + c_2 v_2",[15,38837,38839,38948],{"className":38838,"ariaHidden":72},[116],[15,38840,38842,38846,38886,38939,38942,38945],{"className":38841},[120],[15,38843],{"className":38844,"style":38845},[124],"height:1.0622em;vertical-align:-0.2481em;",[15,38847,38849,38852],{"className":38848},[129],[15,38850,5628],{"className":38851},[129,133],[15,38853,38855],{"className":38854},[137],[15,38856,38858,38878],{"className":38857},[141,142],[15,38859,38861,38875],{"className":38860},[146],[15,38862,38864],{"className":38863,"style":1051},[150],[15,38865,38866,38869],{"style":154},[15,38867],{"className":38868,"style":159},[158],[15,38870,38872],{"className":38871},[163,164,165,166],[15,38873,318],{"className":38874},[129,166],[15,38876,175],{"className":38877},[174],[15,38879,38881],{"className":38880},[146],[15,38882,38884],{"className":38883,"style":182},[150],[15,38885],{},[15,38887,38889,38892],{"className":38888},[129],[15,38890,38814],{"className":38891,"style":887},[129,133],[15,38893,38895],{"className":38894},[137],[15,38896,38898,38930],{"className":38897},[141,142],[15,38899,38901,38927],{"className":38900},[146],[15,38902,38904,38916],{"className":38903,"style":380},[150],[15,38905,38907,38910],{"style":38906},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[15,38908],{"className":38909,"style":159},[158],[15,38911,38913],{"className":38912},[163,164,165,166],[15,38914,318],{"className":38915},[129,166],[15,38917,38918,38921],{"style":383},[15,38919],{"className":38920,"style":159},[158],[15,38922,38924],{"className":38923},[163,164,165,166],[15,38925,310],{"className":38926},[129,166],[15,38928,175],{"className":38929},[174],[15,38931,38933],{"className":38932},[146],[15,38934,38937],{"className":38935,"style":38936},[150],"height:0.2481em;",[15,38938],{},[15,38940],{"className":38941,"style":3277},[93],[15,38943,2682],{"className":38944},[2436],[15,38946],{"className":38947,"style":3277},[93],[15,38949,38951,38954,38994],{"className":38950},[120],[15,38952],{"className":38953,"style":7981},[124],[15,38955,38957,38960],{"className":38956},[129],[15,38958,5628],{"className":38959},[129,133],[15,38961,38963],{"className":38962},[137],[15,38964,38966,38986],{"className":38965},[141,142],[15,38967,38969,38983],{"className":38968},[146],[15,38970,38972],{"className":38971,"style":1051},[150],[15,38973,38974,38977],{"style":154},[15,38975],{"className":38976,"style":159},[158],[15,38978,38980],{"className":38979},[163,164,165,166],[15,38981,310],{"className":38982},[129,166],[15,38984,175],{"className":38985},[174],[15,38987,38989],{"className":38988},[146],[15,38990,38992],{"className":38991,"style":182},[150],[15,38993],{},[15,38995,38997,39000],{"className":38996},[129],[15,38998,38814],{"className":38999,"style":887},[129,133],[15,39001,39003],{"className":39002},[137],[15,39004,39006,39026],{"className":39005},[141,142],[15,39007,39009,39023],{"className":39008},[146],[15,39010,39012],{"className":39011,"style":1051},[150],[15,39013,39014,39017],{"style":1100},[15,39015],{"className":39016,"style":159},[158],[15,39018,39020],{"className":39019},[163,164,165,166],[15,39021,310],{"className":39022},[129,166],[15,39024,175],{"className":39025},[174],[15,39027,39029],{"className":39028},[146],[15,39030,39032],{"className":39031,"style":182},[150],[15,39033],{}," 或 ",[15,39036,39038,39080],{"className":39037,"translate":19},[23],[15,39039,39041],{"className":39040},[27],[29,39042,39043],{"xmlns":31},[34,39044,39045,39077],{},[37,39046,39047,39053,39059,39061,39067,39073,39075],{},[40,39048,39049,39051],{},[43,39050,5628],{},[308,39052,318],{},[40,39054,39055,39057],{},[43,39056,38814],{},[308,39058,318],{},[50,39060,2682],{},[40,39062,39063,39065],{},[43,39064,5628],{},[308,39066,310],{},[40,39068,39069,39071],{},[43,39070,38814],{},[308,39072,310],{},[50,39074,2682],{},[308,39076,9875],{},[109,39078,39079],{"encoding":111},"c_1 v_1 + c_2 v_2 + 3",[15,39081,39083,39178,39273],{"className":39082,"ariaHidden":72},[116],[15,39084,39086,39089,39129,39169,39172,39175],{"className":39085},[120],[15,39087],{"className":39088,"style":11113},[124],[15,39090,39092,39095],{"className":39091},[129],[15,39093,5628],{"className":39094},[129,133],[15,39096,39098],{"className":39097},[137],[15,39099,39101,39121],{"className":39100},[141,142],[15,39102,39104,39118],{"className":39103},[146],[15,39105,39107],{"className":39106,"style":1051},[150],[15,39108,39109,39112],{"style":154},[15,39110],{"className":39111,"style":159},[158],[15,39113,39115],{"className":39114},[163,164,165,166],[15,39116,318],{"className":39117},[129,166],[15,39119,175],{"className":39120},[174],[15,39122,39124],{"className":39123},[146],[15,39125,39127],{"className":39126,"style":182},[150],[15,39128],{},[15,39130,39132,39135],{"className":39131},[129],[15,39133,38814],{"className":39134,"style":887},[129,133],[15,39136,39138],{"className":39137},[137],[15,39139,39141,39161],{"className":39140},[141,142],[15,39142,39144,39158],{"className":39143},[146],[15,39145,39147],{"className":39146,"style":1051},[150],[15,39148,39149,39152],{"style":1100},[15,39150],{"className":39151,"style":159},[158],[15,39153,39155],{"className":39154},[163,164,165,166],[15,39156,318],{"className":39157},[129,166],[15,39159,175],{"className":39160},[174],[15,39162,39164],{"className":39163},[146],[15,39165,39167],{"className":39166,"style":182},[150],[15,39168],{},[15,39170],{"className":39171,"style":3277},[93],[15,39173,2682],{"className":39174},[2436],[15,39176],{"className":39177,"style":3277},[93],[15,39179,39181,39184,39224,39264,39267,39270],{"className":39180},[120],[15,39182],{"className":39183,"style":11113},[124],[15,39185,39187,39190],{"className":39186},[129],[15,39188,5628],{"className":39189},[129,133],[15,39191,39193],{"className":39192},[137],[15,39194,39196,39216],{"className":39195},[141,142],[15,39197,39199,39213],{"className":39198},[146],[15,39200,39202],{"className":39201,"style":1051},[150],[15,39203,39204,39207],{"style":154},[15,39205],{"className":39206,"style":159},[158],[15,39208,39210],{"className":39209},[163,164,165,166],[15,39211,310],{"className":39212},[129,166],[15,39214,175],{"className":39215},[174],[15,39217,39219],{"className":39218},[146],[15,39220,39222],{"className":39221,"style":182},[150],[15,39223],{},[15,39225,39227,39230],{"className":39226},[129],[15,39228,38814],{"className":39229,"style":887},[129,133],[15,39231,39233],{"className":39232},[137],[15,39234,39236,39256],{"className":39235},[141,142],[15,39237,39239,39253],{"className":39238},[146],[15,39240,39242],{"className":39241,"style":1051},[150],[15,39243,39244,39247],{"style":1100},[15,39245],{"className":39246,"style":159},[158],[15,39248,39250],{"className":39249},[163,164,165,166],[15,39251,310],{"className":39252},[129,166],[15,39254,175],{"className":39255},[174],[15,39257,39259],{"className":39258},[146],[15,39260,39262],{"className":39261,"style":182},[150],[15,39263],{},[15,39265],{"className":39266,"style":3277},[93],[15,39268,2682],{"className":39269},[2436],[15,39271],{"className":39272,"style":3277},[93],[15,39274,39276,39279],{"className":39275},[120],[15,39277],{"className":39278,"style":4886},[124],[15,39280,9875],{"className":39281},[129]," 的表达式不能称为线性组合。",[11,39284,39285],{},"可以说「线性」是数学与科学中最简单、最核心的关系，因为它允许我们把复杂问题分解为简单部分的叠加。",[570,39287,39288],{"id":39288},"仿射",[11,39290,39291,39292,39426],{},"仿射函数 ",[15,39293,39295,39326],{"className":39294,"translate":19},[23],[15,39296,39298],{"className":39297},[27],[29,39299,39300],{"xmlns":31},[34,39301,39302,39323],{},[37,39303,39304,39306,39308,39315,39317],{},[43,39305,36556],{},[50,39307,718],{},[305,39309,39310,39312],{},[43,39311,104],{"mathvariant":103},[43,39313,39314],{},"n",[50,39316,723],{},[305,39318,39319,39321],{},[43,39320,104],{"mathvariant":103},[43,39322,348],{},[109,39324,39325],{"encoding":111},"f: \\mathbb{R}^n \\to \\mathbb{R}^m",[15,39327,39329,39347,39391],{"className":39328,"ariaHidden":72},[116],[15,39330,39332,39335,39338,39341,39344],{"className":39331},[120],[15,39333],{"className":39334,"style":1682},[124],[15,39336,36556],{"className":39337,"style":36571},[129,133],[15,39339],{"className":39340,"style":199},[93],[15,39342,718],{"className":39343},[203],[15,39345],{"className":39346,"style":199},[93],[15,39348,39350,39353,39382,39385,39388],{"className":39349},[120],[15,39351],{"className":39352,"style":281},[124],[15,39354,39356,39359],{"className":39355},[129],[15,39357,104],{"className":39358},[129,285],[15,39360,39362],{"className":39361},[137],[15,39363,39365],{"className":39364},[141],[15,39366,39368],{"className":39367},[146],[15,39369,39371],{"className":39370,"style":6377},[150],[15,39372,39373,39376],{"style":383},[15,39374],{"className":39375,"style":159},[158],[15,39377,39379],{"className":39378},[163,164,165,166],[15,39380,39314],{"className":39381},[129,133,166],[15,39383],{"className":39384,"style":199},[93],[15,39386,723],{"className":39387},[203],[15,39389],{"className":39390,"style":199},[93],[15,39392,39394,39397],{"className":39393},[120],[15,39395],{"className":39396,"style":281},[124],[15,39398,39400,39403],{"className":39399},[129],[15,39401,104],{"className":39402},[129,285],[15,39404,39406],{"className":39405},[137],[15,39407,39409],{"className":39408},[141],[15,39410,39412],{"className":39411},[146],[15,39413,39415],{"className":39414,"style":6377},[150],[15,39416,39417,39420],{"style":383},[15,39418],{"className":39419,"style":159},[158],[15,39421,39423],{"className":39422},[163,164,165,166],[15,39424,348],{"className":39425},[129,133,166]," 的定义如下：",[15,39428,39430],{"className":39429,"translate":19},[18],[15,39431,39433,39463],{"className":39432,"translate":19},[23],[15,39434,39436],{"className":39435},[27],[29,39437,39438],{"xmlns":31,"display":32},[34,39439,39440,39460],{},[37,39441,39442,39444,39446,39448,39450,39452,39454,39456,39458],{},[43,39443,36556],{},[50,39445,53],{"stretchy":52},[43,39447,56],{},[50,39449,59],{"stretchy":52},[50,39451,62],{},[43,39453,13091],{},[43,39455,56],{},[50,39457,2682],{},[43,39459,8695],{},[109,39461,39462],{"encoding":111},"f(x) = Ax + b",[15,39464,39466,39493,39515],{"className":39465,"ariaHidden":72},[116],[15,39467,39469,39472,39475,39478,39481,39484,39487,39490],{"className":39468},[120],[15,39470],{"className":39471,"style":125},[124],[15,39473,36556],{"className":39474,"style":36571},[129,133],[15,39476,53],{"className":39477},[188],[15,39479,56],{"className":39480},[129,133],[15,39482,59],{"className":39483},[195],[15,39485],{"className":39486,"style":199},[93],[15,39488,62],{"className":39489},[203],[15,39491],{"className":39492,"style":199},[93],[15,39494,39496,39500,39503,39506,39509,39512],{"className":39495},[120],[15,39497],{"className":39498,"style":39499},[124],"height:0.7667em;vertical-align:-0.0833em;",[15,39501,13091],{"className":39502},[129,133],[15,39504,56],{"className":39505},[129,133],[15,39507],{"className":39508,"style":3277},[93],[15,39510,2682],{"className":39511},[2436],[15,39513],{"className":39514,"style":3277},[93],[15,39516,39518,39521],{"className":39517},[120],[15,39519],{"className":39520,"style":522},[124],[15,39522,8695],{"className":39523},[129,133],[11,39525,28686,39526,39554,39555,39583],{},[15,39527,39529,39542],{"className":39528,"translate":19},[23],[15,39530,39532],{"className":39531},[27],[29,39533,39534],{"xmlns":31},[34,39535,39536,39540],{},[37,39537,39538],{},[43,39539,13091],{},[109,39541,13091],{"encoding":111},[15,39543,39545],{"className":39544,"ariaHidden":72},[116],[15,39546,39548,39551],{"className":39547},[120],[15,39549],{"className":39550,"style":610},[124],[15,39552,13091],{"className":39553},[129,133]," 是矩阵，",[15,39556,39558,39571],{"className":39557,"translate":19},[23],[15,39559,39561],{"className":39560},[27],[29,39562,39563],{"xmlns":31},[34,39564,39565,39569],{},[37,39566,39567],{},[43,39568,8695],{},[109,39570,8695],{"encoding":111},[15,39572,39574],{"className":39573,"ariaHidden":72},[116],[15,39575,39577,39580],{"className":39576},[120],[15,39578],{"className":39579,"style":522},[124],[15,39581,8695],{"className":39582},[129,133]," 是常向量。它就是线性函数加上偏置。",[11,39585,39586],{},"如果取仿射函数的加权线性组合（系数可以是标量）：",[15,39588,39590],{"className":39589,"translate":19},[18],[15,39591,39593,39647],{"className":39592,"translate":19},[23],[15,39594,39596],{"className":39595},[27],[29,39597,39598],{"xmlns":31,"display":32},[34,39599,39600,39644],{},[37,39601,39602,39604,39606,39608,39610,39612,39626,39632,39638,39640,39642],{},[43,39603,2247],{},[50,39605,53],{"stretchy":52},[43,39607,56],{},[50,39609,59],{"stretchy":52},[50,39611,62],{},[1450,39613,39614,39616,39624],{},[50,39615,1454],{},[37,39617,39618,39620,39622],{},[43,39619,1459],{},[50,39621,62],{},[308,39623,318],{},[43,39625,16555],{},[40,39627,39628,39630],{},[43,39629,12002],{},[43,39631,1459],{},[40,39633,39634,39636],{},[43,39635,36556],{},[43,39637,1459],{},[50,39639,53],{"stretchy":52},[43,39641,56],{},[50,39643,59],{"stretchy":52},[109,39645,39646],{"encoding":111},"g(x) = \\sum_{i=1}^k \\alpha_i f_i(x)",[15,39648,39650,39677],{"className":39649,"ariaHidden":72},[116],[15,39651,39653,39656,39659,39662,39665,39668,39671,39674],{"className":39652},[120],[15,39654],{"className":39655,"style":125},[124],[15,39657,2247],{"className":39658,"style":887},[129,133],[15,39660,53],{"className":39661},[188],[15,39663,56],{"className":39664},[129,133],[15,39666,59],{"className":39667},[195],[15,39669],{"className":39670,"style":199},[93],[15,39672,62],{"className":39673},[203],[15,39675],{"className":39676,"style":199},[93],[15,39678,39680,39683,39747,39750,39790,39831,39834,39837],{"className":39679},[120],[15,39681],{"className":39682,"style":21117},[124],[15,39684,39686],{"className":39685},[217,1874],[15,39687,39689,39739],{"className":39688},[141,142],[15,39690,39692,39736],{"className":39691},[146],[15,39693,39695,39715,39725],{"className":39694,"style":21130},[150],[15,39696,39697,39700],{"style":1887},[15,39698],{"className":39699,"style":1891},[158],[15,39701,39703],{"className":39702},[163,164,165,166],[15,39704,39706,39709,39712],{"className":39705},[129,166],[15,39707,1459],{"className":39708},[129,133,166],[15,39710,62],{"className":39711},[203,166],[15,39713,318],{"className":39714},[129,166],[15,39716,39717,39720],{"style":1909},[15,39718],{"className":39719,"style":1891},[158],[15,39721,39722],{},[15,39723,1454],{"className":39724},[217,1918,1919],[15,39726,39727,39730],{"style":1922},[15,39728],{"className":39729,"style":1891},[158],[15,39731,39733],{"className":39732},[163,164,165,166],[15,39734,16555],{"className":39735,"style":16588},[129,133,166],[15,39737,175],{"className":39738},[174],[15,39740,39742],{"className":39741},[146],[15,39743,39745],{"className":39744,"style":1941},[150],[15,39746],{},[15,39748],{"className":39749,"style":262},[93],[15,39751,39753,39756],{"className":39752},[129],[15,39754,12002],{"className":39755,"style":12023},[129,133],[15,39757,39759],{"className":39758},[137],[15,39760,39762,39782],{"className":39761},[141,142],[15,39763,39765,39779],{"className":39764},[146],[15,39766,39768],{"className":39767,"style":1981},[150],[15,39769,39770,39773],{"style":12038},[15,39771],{"className":39772,"style":159},[158],[15,39774,39776],{"className":39775},[163,164,165,166],[15,39777,1459],{"className":39778},[129,133,166],[15,39780,175],{"className":39781},[174],[15,39783,39785],{"className":39784},[146],[15,39786,39788],{"className":39787,"style":182},[150],[15,39789],{},[15,39791,39793,39796],{"className":39792},[129],[15,39794,36556],{"className":39795,"style":36571},[129,133],[15,39797,39799],{"className":39798},[137],[15,39800,39802,39823],{"className":39801},[141,142],[15,39803,39805,39820],{"className":39804},[146],[15,39806,39808],{"className":39807,"style":1981},[150],[15,39809,39811,39814],{"style":39810},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[15,39812],{"className":39813,"style":159},[158],[15,39815,39817],{"className":39816},[163,164,165,166],[15,39818,1459],{"className":39819},[129,133,166],[15,39821,175],{"className":39822},[174],[15,39824,39826],{"className":39825},[146],[15,39827,39829],{"className":39828,"style":182},[150],[15,39830],{},[15,39832,53],{"className":39833},[188],[15,39835,56],{"className":39836},[129,133],[15,39838,59],{"className":39839},[195],[11,39841,39842,39843,1313],{},"其中每个 ",[15,39844,39846,39888],{"className":39845,"translate":19},[23],[15,39847,39849],{"className":39848},[27],[29,39850,39851],{"xmlns":31},[34,39852,39853,39885],{},[37,39854,39855,39861,39863,39865,39867,39869,39875,39877,39879],{},[40,39856,39857,39859],{},[43,39858,36556],{},[43,39860,1459],{},[50,39862,53],{"stretchy":52},[43,39864,56],{},[50,39866,59],{"stretchy":52},[50,39868,62],{},[40,39870,39871,39873],{},[43,39872,13091],{},[43,39874,1459],{},[43,39876,56],{},[50,39878,2682],{},[40,39880,39881,39883],{},[43,39882,8695],{},[43,39884,1459],{},[109,39886,39887],{"encoding":111},"f_i(x) = A_i x + b_i",[15,39889,39891,39955,40013],{"className":39890,"ariaHidden":72},[116],[15,39892,39894,39897,39937,39940,39943,39946,39949,39952],{"className":39893},[120],[15,39895],{"className":39896,"style":125},[124],[15,39898,39900,39903],{"className":39899},[129],[15,39901,36556],{"className":39902,"style":36571},[129,133],[15,39904,39906],{"className":39905},[137],[15,39907,39909,39929],{"className":39908},[141,142],[15,39910,39912,39926],{"className":39911},[146],[15,39913,39915],{"className":39914,"style":1981},[150],[15,39916,39917,39920],{"style":39810},[15,39918],{"className":39919,"style":159},[158],[15,39921,39923],{"className":39922},[163,164,165,166],[15,39924,1459],{"className":39925},[129,133,166],[15,39927,175],{"className":39928},[174],[15,39930,39932],{"className":39931},[146],[15,39933,39935],{"className":39934,"style":182},[150],[15,39936],{},[15,39938,53],{"className":39939},[188],[15,39941,56],{"className":39942},[129,133],[15,39944,59],{"className":39945},[195],[15,39947],{"className":39948,"style":199},[93],[15,39950,62],{"className":39951},[203],[15,39953],{"className":39954,"style":199},[93],[15,39956,39958,39961,40001,40004,40007,40010],{"className":39957},[120],[15,39959],{"className":39960,"style":20220},[124],[15,39962,39964,39967],{"className":39963},[129],[15,39965,13091],{"className":39966},[129,133],[15,39968,39970],{"className":39969},[137],[15,39971,39973,39993],{"className":39972},[141,142],[15,39974,39976,39990],{"className":39975},[146],[15,39977,39979],{"className":39978,"style":1981},[150],[15,39980,39981,39984],{"style":154},[15,39982],{"className":39983,"style":159},[158],[15,39985,39987],{"className":39986},[163,164,165,166],[15,39988,1459],{"className":39989},[129,133,166],[15,39991,175],{"className":39992},[174],[15,39994,39996],{"className":39995},[146],[15,39997,39999],{"className":39998,"style":182},[150],[15,40000],{},[15,40002,56],{"className":40003},[129,133],[15,40005],{"className":40006,"style":3277},[93],[15,40008,2682],{"className":40009},[2436],[15,40011],{"className":40012,"style":3277},[93],[15,40014,40016,40019],{"className":40015},[120],[15,40017],{"className":40018,"style":2183},[124],[15,40020,40022,40025],{"className":40021},[129],[15,40023,8695],{"className":40024},[129,133],[15,40026,40028],{"className":40027},[137],[15,40029,40031,40051],{"className":40030},[141,142],[15,40032,40034,40048],{"className":40033},[146],[15,40035,40037],{"className":40036,"style":1981},[150],[15,40038,40039,40042],{"style":154},[15,40040],{"className":40041,"style":159},[158],[15,40043,40045],{"className":40044},[163,164,165,166],[15,40046,1459],{"className":40047},[129,133,166],[15,40049,175],{"className":40050},[174],[15,40052,40054],{"className":40053},[146],[15,40055,40057],{"className":40056,"style":182},[150],[15,40058],{},[11,40060,40061],{},"展开后：",[15,40063,40065],{"className":40064,"translate":19},[18],[15,40066,40068,40194],{"className":40067,"translate":19},[23],[15,40069,40071],{"className":40070},[27],[29,40072,40073],{"xmlns":31,"display":32},[34,40074,40075,40191],{},[37,40076,40077,40079,40081,40083,40085,40087,40101,40107,40109,40115,40117,40119,40125,40127,40129,40161,40163,40165,40179,40185],{},[43,40078,2247],{},[50,40080,53],{"stretchy":52},[43,40082,56],{},[50,40084,59],{"stretchy":52},[50,40086,62],{},[1450,40088,40089,40091,40099],{},[50,40090,1454],{},[37,40092,40093,40095,40097],{},[43,40094,1459],{},[50,40096,62],{},[308,40098,318],{},[43,40100,16555],{},[40,40102,40103,40105],{},[43,40104,12002],{},[43,40106,1459],{},[50,40108,53],{"stretchy":52},[40,40110,40111,40113],{},[43,40112,13091],{},[43,40114,1459],{},[43,40116,56],{},[50,40118,2682],{},[40,40120,40121,40123],{},[43,40122,8695],{},[43,40124,1459],{},[50,40126,59],{"stretchy":52},[50,40128,62],{},[37,40130,40131,40133,40147,40153,40159],{},[50,40132,53],{"fence":72},[1450,40134,40135,40137,40145],{},[50,40136,1454],{},[37,40138,40139,40141,40143],{},[43,40140,1459],{},[50,40142,62],{},[308,40144,318],{},[43,40146,16555],{},[40,40148,40149,40151],{},[43,40150,12002],{},[43,40152,1459],{},[40,40154,40155,40157],{},[43,40156,13091],{},[43,40158,1459],{},[50,40160,59],{"fence":72},[43,40162,56],{},[50,40164,2682],{},[1450,40166,40167,40169,40177],{},[50,40168,1454],{},[37,40170,40171,40173,40175],{},[43,40172,1459],{},[50,40174,62],{},[308,40176,318],{},[43,40178,16555],{},[40,40180,40181,40183],{},[43,40182,12002],{},[43,40184,1459],{},[40,40186,40187,40189],{},[43,40188,8695],{},[43,40190,1459],{},[109,40192,40193],{"encoding":111},"g(x) = \\sum_{i=1}^k \\alpha_i (A_i x + b_i) = \\left(\\sum_{i=1}^k \\alpha_i A_i \\right) x + \\sum_{i=1}^k \\alpha_i b_i",[15,40195,40197,40224,40392,40450,40633],{"className":40196,"ariaHidden":72},[116],[15,40198,40200,40203,40206,40209,40212,40215,40218,40221],{"className":40199},[120],[15,40201],{"className":40202,"style":125},[124],[15,40204,2247],{"className":40205,"style":887},[129,133],[15,40207,53],{"className":40208},[188],[15,40210,56],{"className":40211},[129,133],[15,40213,59],{"className":40214},[195],[15,40216],{"className":40217,"style":199},[93],[15,40219,62],{"className":40220},[203],[15,40222],{"className":40223,"style":199},[93],[15,40225,40227,40230,40294,40297,40337,40340,40380,40383,40386,40389],{"className":40226},[120],[15,40228],{"className":40229,"style":21117},[124],[15,40231,40233],{"className":40232},[217,1874],[15,40234,40236,40286],{"className":40235},[141,142],[15,40237,40239,40283],{"className":40238},[146],[15,40240,40242,40262,40272],{"className":40241,"style":21130},[150],[15,40243,40244,40247],{"style":1887},[15,40245],{"className":40246,"style":1891},[158],[15,40248,40250],{"className":40249},[163,164,165,166],[15,40251,40253,40256,40259],{"className":40252},[129,166],[15,40254,1459],{"className":40255},[129,133,166],[15,40257,62],{"className":40258},[203,166],[15,40260,318],{"className":40261},[129,166],[15,40263,40264,40267],{"style":1909},[15,40265],{"className":40266,"style":1891},[158],[15,40268,40269],{},[15,40270,1454],{"className":40271},[217,1918,1919],[15,40273,40274,40277],{"style":1922},[15,40275],{"className":40276,"style":1891},[158],[15,40278,40280],{"className":40279},[163,164,165,166],[15,40281,16555],{"className":40282,"style":16588},[129,133,166],[15,40284,175],{"className":40285},[174],[15,40287,40289],{"className":40288},[146],[15,40290,40292],{"className":40291,"style":1941},[150],[15,40293],{},[15,40295],{"className":40296,"style":262},[93],[15,40298,40300,40303],{"className":40299},[129],[15,40301,12002],{"className":40302,"style":12023},[129,133],[15,40304,40306],{"className":40305},[137],[15,40307,40309,40329],{"className":40308},[141,142],[15,40310,40312,40326],{"className":40311},[146],[15,40313,40315],{"className":40314,"style":1981},[150],[15,40316,40317,40320],{"style":12038},[15,40318],{"className":40319,"style":159},[158],[15,40321,40323],{"className":40322},[163,164,165,166],[15,40324,1459],{"className":40325},[129,133,166],[15,40327,175],{"className":40328},[174],[15,40330,40332],{"className":40331},[146],[15,40333,40335],{"className":40334,"style":182},[150],[15,40336],{},[15,40338,53],{"className":40339},[188],[15,40341,40343,40346],{"className":40342},[129],[15,40344,13091],{"className":40345},[129,133],[15,40347,40349],{"className":40348},[137],[15,40350,40352,40372],{"className":40351},[141,142],[15,40353,40355,40369],{"className":40354},[146],[15,40356,40358],{"className":40357,"style":1981},[150],[15,40359,40360,40363],{"style":154},[15,40361],{"className":40362,"style":159},[158],[15,40364,40366],{"className":40365},[163,164,165,166],[15,40367,1459],{"className":40368},[129,133,166],[15,40370,175],{"className":40371},[174],[15,40373,40375],{"className":40374},[146],[15,40376,40378],{"className":40377,"style":182},[150],[15,40379],{},[15,40381,56],{"className":40382},[129,133],[15,40384],{"className":40385,"style":3277},[93],[15,40387,2682],{"className":40388},[2436],[15,40390],{"className":40391,"style":3277},[93],[15,40393,40395,40398,40438,40441,40444,40447],{"className":40394},[120],[15,40396],{"className":40397,"style":125},[124],[15,40399,40401,40404],{"className":40400},[129],[15,40402,8695],{"className":40403},[129,133],[15,40405,40407],{"className":40406},[137],[15,40408,40410,40430],{"className":40409},[141,142],[15,40411,40413,40427],{"className":40412},[146],[15,40414,40416],{"className":40415,"style":1981},[150],[15,40417,40418,40421],{"style":154},[15,40419],{"className":40420,"style":159},[158],[15,40422,40424],{"className":40423},[163,164,165,166],[15,40425,1459],{"className":40426},[129,133,166],[15,40428,175],{"className":40429},[174],[15,40431,40433],{"className":40432},[146],[15,40434,40436],{"className":40435,"style":182},[150],[15,40437],{},[15,40439,59],{"className":40440},[195],[15,40442],{"className":40443,"style":199},[93],[15,40445,62],{"className":40446},[203],[15,40448],{"className":40449,"style":199},[93],[15,40451,40453,40456,40618,40621,40624,40627,40630],{"className":40452},[120],[15,40454],{"className":40455,"style":21117},[124],[15,40457,40459,40465,40529,40532,40572,40612],{"className":40458},[449],[15,40460,40462],{"className":40461,"style":5052},[188,5051],[15,40463,53],{"className":40464},[228,15850],[15,40466,40468],{"className":40467},[217,1874],[15,40469,40471,40521],{"className":40470},[141,142],[15,40472,40474,40518],{"className":40473},[146],[15,40475,40477,40497,40507],{"className":40476,"style":21130},[150],[15,40478,40479,40482],{"style":1887},[15,40480],{"className":40481,"style":1891},[158],[15,40483,40485],{"className":40484},[163,164,165,166],[15,40486,40488,40491,40494],{"className":40487},[129,166],[15,40489,1459],{"className":40490},[129,133,166],[15,40492,62],{"className":40493},[203,166],[15,40495,318],{"className":40496},[129,166],[15,40498,40499,40502],{"style":1909},[15,40500],{"className":40501,"style":1891},[158],[15,40503,40504],{},[15,40505,1454],{"className":40506},[217,1918,1919],[15,40508,40509,40512],{"style":1922},[15,40510],{"className":40511,"style":1891},[158],[15,40513,40515],{"className":40514},[163,164,165,166],[15,40516,16555],{"className":40517,"style":16588},[129,133,166],[15,40519,175],{"className":40520},[174],[15,40522,40524],{"className":40523},[146],[15,40525,40527],{"className":40526,"style":1941},[150],[15,40528],{},[15,40530],{"className":40531,"style":262},[93],[15,40533,40535,40538],{"className":40534},[129],[15,40536,12002],{"className":40537,"style":12023},[129,133],[15,40539,40541],{"className":40540},[137],[15,40542,40544,40564],{"className":40543},[141,142],[15,40545,40547,40561],{"className":40546},[146],[15,40548,40550],{"className":40549,"style":1981},[150],[15,40551,40552,40555],{"style":12038},[15,40553],{"className":40554,"style":159},[158],[15,40556,40558],{"className":40557},[163,164,165,166],[15,40559,1459],{"className":40560},[129,133,166],[15,40562,175],{"className":40563},[174],[15,40565,40567],{"className":40566},[146],[15,40568,40570],{"className":40569,"style":182},[150],[15,40571],{},[15,40573,40575,40578],{"className":40574},[129],[15,40576,13091],{"className":40577},[129,133],[15,40579,40581],{"className":40580},[137],[15,40582,40584,40604],{"className":40583},[141,142],[15,40585,40587,40601],{"className":40586},[146],[15,40588,40590],{"className":40589,"style":1981},[150],[15,40591,40592,40595],{"style":154},[15,40593],{"className":40594,"style":159},[158],[15,40596,40598],{"className":40597},[163,164,165,166],[15,40599,1459],{"className":40600},[129,133,166],[15,40602,175],{"className":40603},[174],[15,40605,40607],{"className":40606},[146],[15,40608,40610],{"className":40609,"style":182},[150],[15,40611],{},[15,40613,40615],{"className":40614,"style":5052},[195,5051],[15,40616,59],{"className":40617},[228,15850],[15,40619],{"className":40620,"style":262},[93],[15,40622,56],{"className":40623},[129,133],[15,40625],{"className":40626,"style":3277},[93],[15,40628,2682],{"className":40629},[2436],[15,40631],{"className":40632,"style":3277},[93],[15,40634,40636,40639,40703,40706,40746],{"className":40635},[120],[15,40637],{"className":40638,"style":21117},[124],[15,40640,40642],{"className":40641},[217,1874],[15,40643,40645,40695],{"className":40644},[141,142],[15,40646,40648,40692],{"className":40647},[146],[15,40649,40651,40671,40681],{"className":40650,"style":21130},[150],[15,40652,40653,40656],{"style":1887},[15,40654],{"className":40655,"style":1891},[158],[15,40657,40659],{"className":40658},[163,164,165,166],[15,40660,40662,40665,40668],{"className":40661},[129,166],[15,40663,1459],{"className":40664},[129,133,166],[15,40666,62],{"className":40667},[203,166],[15,40669,318],{"className":40670},[129,166],[15,40672,40673,40676],{"style":1909},[15,40674],{"className":40675,"style":1891},[158],[15,40677,40678],{},[15,40679,1454],{"className":40680},[217,1918,1919],[15,40682,40683,40686],{"style":1922},[15,40684],{"className":40685,"style":1891},[158],[15,40687,40689],{"className":40688},[163,164,165,166],[15,40690,16555],{"className":40691,"style":16588},[129,133,166],[15,40693,175],{"className":40694},[174],[15,40696,40698],{"className":40697},[146],[15,40699,40701],{"className":40700,"style":1941},[150],[15,40702],{},[15,40704],{"className":40705,"style":262},[93],[15,40707,40709,40712],{"className":40708},[129],[15,40710,12002],{"className":40711,"style":12023},[129,133],[15,40713,40715],{"className":40714},[137],[15,40716,40718,40738],{"className":40717},[141,142],[15,40719,40721,40735],{"className":40720},[146],[15,40722,40724],{"className":40723,"style":1981},[150],[15,40725,40726,40729],{"style":12038},[15,40727],{"className":40728,"style":159},[158],[15,40730,40732],{"className":40731},[163,164,165,166],[15,40733,1459],{"className":40734},[129,133,166],[15,40736,175],{"className":40737},[174],[15,40739,40741],{"className":40740},[146],[15,40742,40744],{"className":40743,"style":182},[150],[15,40745],{},[15,40747,40749,40752],{"className":40748},[129],[15,40750,8695],{"className":40751},[129,133],[15,40753,40755],{"className":40754},[137],[15,40756,40758,40778],{"className":40757},[141,142],[15,40759,40761,40775],{"className":40760},[146],[15,40762,40764],{"className":40763,"style":1981},[150],[15,40765,40766,40769],{"style":154},[15,40767],{"className":40768,"style":159},[158],[15,40770,40772],{"className":40771},[163,164,165,166],[15,40773,1459],{"className":40774},[129,133,166],[15,40776,175],{"className":40777},[174],[15,40779,40781],{"className":40780},[146],[15,40782,40784],{"className":40783,"style":182},[150],[15,40785],{},[11,40787,40788],{},"这仍然是仿射函数。因此，加权组合不会破坏仿射性。",[11,40790,40791],{},"如果用非线性的多项式运算来组合仿射函数，例如平方或更高次幂：",[15,40793,40795],{"className":40794,"translate":19},[18],[15,40796,40798,40872],{"className":40797,"translate":19},[23],[15,40799,40801],{"className":40800},[27],[29,40802,40803],{"xmlns":31,"display":32},[34,40804,40805,40869],{},[37,40806,40807,40809,40811,40813,40815,40817,40819,40825,40827,40829,40831,40837,40839,40841,40847,40849,40851,40853,40859,40861,40863,40865,40867],{},[43,40808,45],{},[50,40810,53],{"stretchy":52},[43,40812,56],{},[50,40814,59],{"stretchy":52},[50,40816,62],{},[50,40818,53],{"stretchy":52},[40,40820,40821,40823],{},[43,40822,36556],{},[308,40824,318],{},[50,40826,53],{"stretchy":52},[43,40828,56],{},[50,40830,59],{"stretchy":52},[305,40832,40833,40835],{},[50,40834,59],{"stretchy":52},[308,40836,310],{},[50,40838,2682],{},[308,40840,9875],{},[40,40842,40843,40845],{},[43,40844,36556],{},[308,40846,310],{},[50,40848,53],{"stretchy":52},[43,40850,56],{},[50,40852,59],{"stretchy":52},[40,40854,40855,40857],{},[43,40856,36556],{},[308,40858,9875],{},[50,40860,53],{"stretchy":52},[43,40862,56],{},[50,40864,59],{"stretchy":52},[50,40866,2682],{},[308,40868,36751],{},[109,40870,40871],{"encoding":111},"h(x) = (f_1(x))^2 + 3 f_2(x) f_3(x) + 5",[15,40873,40875,40902,41000,41116],{"className":40874,"ariaHidden":72},[116],[15,40876,40878,40881,40884,40887,40890,40893,40896,40899],{"className":40877},[120],[15,40879],{"className":40880,"style":125},[124],[15,40882,45],{"className":40883},[129,133],[15,40885,53],{"className":40886},[188],[15,40888,56],{"className":40889},[129,133],[15,40891,59],{"className":40892},[195],[15,40894],{"className":40895,"style":199},[93],[15,40897,62],{"className":40898},[203],[15,40900],{"className":40901,"style":199},[93],[15,40903,40905,40909,40912,40952,40955,40958,40961,40991,40994,40997],{"className":40904},[120],[15,40906],{"className":40907,"style":40908},[124],"height:1.1141em;vertical-align:-0.25em;",[15,40910,53],{"className":40911},[188],[15,40913,40915,40918],{"className":40914},[129],[15,40916,36556],{"className":40917,"style":36571},[129,133],[15,40919,40921],{"className":40920},[137],[15,40922,40924,40944],{"className":40923},[141,142],[15,40925,40927,40941],{"className":40926},[146],[15,40928,40930],{"className":40929,"style":1051},[150],[15,40931,40932,40935],{"style":39810},[15,40933],{"className":40934,"style":159},[158],[15,40936,40938],{"className":40937},[163,164,165,166],[15,40939,318],{"className":40940},[129,166],[15,40942,175],{"className":40943},[174],[15,40945,40947],{"className":40946},[146],[15,40948,40950],{"className":40949,"style":182},[150],[15,40951],{},[15,40953,53],{"className":40954},[188],[15,40956,56],{"className":40957},[129,133],[15,40959,59],{"className":40960},[195],[15,40962,40964,40967],{"className":40963},[195],[15,40965,59],{"className":40966},[195],[15,40968,40970],{"className":40969},[137],[15,40971,40973],{"className":40972},[141],[15,40974,40976],{"className":40975},[146],[15,40977,40980],{"className":40978,"style":40979},[150],"height:0.8641em;",[15,40981,40982,40985],{"style":1272},[15,40983],{"className":40984,"style":159},[158],[15,40986,40988],{"className":40987},[163,164,165,166],[15,40989,310],{"className":40990},[129,166],[15,40992],{"className":40993,"style":3277},[93],[15,40995,2682],{"className":40996},[2436],[15,40998],{"className":40999,"style":3277},[93],[15,41001,41003,41006,41009,41049,41052,41055,41058,41098,41101,41104,41107,41110,41113],{"className":41002},[120],[15,41004],{"className":41005,"style":125},[124],[15,41007,9875],{"className":41008},[129],[15,41010,41012,41015],{"className":41011},[129],[15,41013,36556],{"className":41014,"style":36571},[129,133],[15,41016,41018],{"className":41017},[137],[15,41019,41021,41041],{"className":41020},[141,142],[15,41022,41024,41038],{"className":41023},[146],[15,41025,41027],{"className":41026,"style":1051},[150],[15,41028,41029,41032],{"style":39810},[15,41030],{"className":41031,"style":159},[158],[15,41033,41035],{"className":41034},[163,164,165,166],[15,41036,310],{"className":41037},[129,166],[15,41039,175],{"className":41040},[174],[15,41042,41044],{"className":41043},[146],[15,41045,41047],{"className":41046,"style":182},[150],[15,41048],{},[15,41050,53],{"className":41051},[188],[15,41053,56],{"className":41054},[129,133],[15,41056,59],{"className":41057},[195],[15,41059,41061,41064],{"className":41060},[129],[15,41062,36556],{"className":41063,"style":36571},[129,133],[15,41065,41067],{"className":41066},[137],[15,41068,41070,41090],{"className":41069},[141,142],[15,41071,41073,41087],{"className":41072},[146],[15,41074,41076],{"className":41075,"style":1051},[150],[15,41077,41078,41081],{"style":39810},[15,41079],{"className":41080,"style":159},[158],[15,41082,41084],{"className":41083},[163,164,165,166],[15,41085,9875],{"className":41086},[129,166],[15,41088,175],{"className":41089},[174],[15,41091,41093],{"className":41092},[146],[15,41094,41096],{"className":41095,"style":182},[150],[15,41097],{},[15,41099,53],{"className":41100},[188],[15,41102,56],{"className":41103},[129,133],[15,41105,59],{"className":41106},[195],[15,41108],{"className":41109,"style":3277},[93],[15,41111,2682],{"className":41112},[2436],[15,41114],{"className":41115,"style":3277},[93],[15,41117,41119,41122],{"className":41118},[120],[15,41120],{"className":41121,"style":4886},[124],[15,41123,36751],{"className":41124},[129],[11,41126,41127,41128,41243,41244,41390],{},"注意：",[15,41129,41131,41159],{"className":41130,"translate":19},[23],[15,41132,41134],{"className":41133},[27],[29,41135,41136],{"xmlns":31},[34,41137,41138,41156],{},[37,41139,41140,41146,41148,41150],{},[40,41141,41142,41144],{},[43,41143,36556],{},[308,41145,318],{},[50,41147,53],{"stretchy":52},[43,41149,56],{},[305,41151,41152,41154],{},[50,41153,59],{"stretchy":52},[308,41155,310],{},[109,41157,41158],{"encoding":111},"f_1(x)^2",[15,41160,41162],{"className":41161,"ariaHidden":72},[116],[15,41163,41165,41168,41208,41211,41214],{"className":41164},[120],[15,41166],{"className":41167,"style":33996},[124],[15,41169,41171,41174],{"className":41170},[129],[15,41172,36556],{"className":41173,"style":36571},[129,133],[15,41175,41177],{"className":41176},[137],[15,41178,41180,41200],{"className":41179},[141,142],[15,41181,41183,41197],{"className":41182},[146],[15,41184,41186],{"className":41185,"style":1051},[150],[15,41187,41188,41191],{"style":39810},[15,41189],{"className":41190,"style":159},[158],[15,41192,41194],{"className":41193},[163,164,165,166],[15,41195,318],{"className":41196},[129,166],[15,41198,175],{"className":41199},[174],[15,41201,41203],{"className":41202},[146],[15,41204,41206],{"className":41205,"style":182},[150],[15,41207],{},[15,41209,53],{"className":41210},[188],[15,41212,56],{"className":41213},[129,133],[15,41215,41217,41220],{"className":41216},[195],[15,41218,59],{"className":41219},[195],[15,41221,41223],{"className":41222},[137],[15,41224,41226],{"className":41225},[141],[15,41227,41229],{"className":41228},[146],[15,41230,41232],{"className":41231,"style":380},[150],[15,41233,41234,41237],{"style":383},[15,41235],{"className":41236,"style":159},[158],[15,41238,41240],{"className":41239},[163,164,165,166],[15,41241,310],{"className":41242},[129,166]," 是逐元素平方，",[15,41245,41247,41283],{"className":41246,"translate":19},[23],[15,41248,41250],{"className":41249},[27],[29,41251,41252],{"xmlns":31},[34,41253,41254,41280],{},[37,41255,41256,41262,41264,41266,41268,41274,41276,41278],{},[40,41257,41258,41260],{},[43,41259,36556],{},[308,41261,310],{},[50,41263,53],{"stretchy":52},[43,41265,56],{},[50,41267,59],{"stretchy":52},[40,41269,41270,41272],{},[43,41271,36556],{},[308,41273,9875],{},[50,41275,53],{"stretchy":52},[43,41277,56],{},[50,41279,59],{"stretchy":52},[109,41281,41282],{"encoding":111},"f_2(x) f_3(x)",[15,41284,41286],{"className":41285,"ariaHidden":72},[116],[15,41287,41289,41292,41332,41335,41338,41341,41381,41384,41387],{"className":41288},[120],[15,41290],{"className":41291,"style":125},[124],[15,41293,41295,41298],{"className":41294},[129],[15,41296,36556],{"className":41297,"style":36571},[129,133],[15,41299,41301],{"className":41300},[137],[15,41302,41304,41324],{"className":41303},[141,142],[15,41305,41307,41321],{"className":41306},[146],[15,41308,41310],{"className":41309,"style":1051},[150],[15,41311,41312,41315],{"style":39810},[15,41313],{"className":41314,"style":159},[158],[15,41316,41318],{"className":41317},[163,164,165,166],[15,41319,310],{"className":41320},[129,166],[15,41322,175],{"className":41323},[174],[15,41325,41327],{"className":41326},[146],[15,41328,41330],{"className":41329,"style":182},[150],[15,41331],{},[15,41333,53],{"className":41334},[188],[15,41336,56],{"className":41337},[129,133],[15,41339,59],{"className":41340},[195],[15,41342,41344,41347],{"className":41343},[129],[15,41345,36556],{"className":41346,"style":36571},[129,133],[15,41348,41350],{"className":41349},[137],[15,41351,41353,41373],{"className":41352},[141,142],[15,41354,41356,41370],{"className":41355},[146],[15,41357,41359],{"className":41358,"style":1051},[150],[15,41360,41361,41364],{"style":39810},[15,41362],{"className":41363,"style":159},[158],[15,41365,41367],{"className":41366},[163,164,165,166],[15,41368,9875],{"className":41369},[129,166],[15,41371,175],{"className":41372},[174],[15,41374,41376],{"className":41375},[146],[15,41377,41379],{"className":41378,"style":182},[150],[15,41380],{},[15,41382,53],{"className":41383},[188],[15,41385,56],{"className":41386},[129,133],[15,41388,59],{"className":41389},[195]," 是逐元素相乘或点积。",[11,41392,41393,41394,41450],{},"这些运算会引入二次项或更高阶项，不再保持 ",[15,41395,41397,41417],{"className":41396,"translate":19},[23],[15,41398,41400],{"className":41399},[27],[29,41401,41402],{"xmlns":31},[34,41403,41404,41414],{},[37,41405,41406,41408,41410,41412],{},[43,41407,13091],{},[43,41409,56],{},[50,41411,2682],{},[43,41413,8695],{},[109,41415,41416],{"encoding":111},"Ax+b",[15,41418,41420,41441],{"className":41419,"ariaHidden":72},[116],[15,41421,41423,41426,41429,41432,41435,41438],{"className":41422},[120],[15,41424],{"className":41425,"style":39499},[124],[15,41427,13091],{"className":41428},[129,133],[15,41430,56],{"className":41431},[129,133],[15,41433],{"className":41434,"style":3277},[93],[15,41436,2682],{"className":41437},[2436],[15,41439],{"className":41440,"style":3277},[93],[15,41442,41444,41447],{"className":41443},[120],[15,41445],{"className":41446,"style":522},[124],[15,41448,8695],{"className":41449},[129,133]," 的形式。因此，非线性多项式组合通常不是仿射函数。",[570,41452,41453],{"id":41453},"仿射复合",[11,41455,41456],{},"这个问题非常重要，是理解神经网络本质的关键。让我们仔细分析仿射函数的复合是否仍然是仿射的。",[11,41458,41459],{},"假设有两个仿射函数：",[15,41461,41463],{"className":41462,"translate":19},[18],[15,41464,41466,41518],{"className":41465,"translate":19},[23],[15,41467,41469],{"className":41468},[27],[29,41470,41471],{"xmlns":31,"display":32},[34,41472,41473,41515],{},[37,41474,41475,41477,41479,41481,41483,41485,41487,41489,41491,41493,41495,41497,41499,41501,41503,41505,41507,41509,41511,41513],{},[43,41476,36556],{},[50,41478,53],{"stretchy":52},[43,41480,56],{},[50,41482,59],{"stretchy":52},[50,41484,62],{},[43,41486,13091],{},[43,41488,56],{},[50,41490,2682],{},[43,41492,8695],{},[50,41494,91],{"separator":72},[93,41496],{"width":16914},[43,41498,2247],{},[50,41500,53],{"stretchy":52},[43,41502,56],{},[50,41504,59],{"stretchy":52},[50,41506,62],{},[43,41508,5619],{},[43,41510,56],{},[50,41512,2682],{},[43,41514,10372],{},[109,41516,41517],{"encoding":111},"f(x) = A x + b, \\quad g(x) = C x + d",[15,41519,41521,41548,41569,41608,41629],{"className":41520,"ariaHidden":72},[116],[15,41522,41524,41527,41530,41533,41536,41539,41542,41545],{"className":41523},[120],[15,41525],{"className":41526,"style":125},[124],[15,41528,36556],{"className":41529,"style":36571},[129,133],[15,41531,53],{"className":41532},[188],[15,41534,56],{"className":41535},[129,133],[15,41537,59],{"className":41538},[195],[15,41540],{"className":41541,"style":199},[93],[15,41543,62],{"className":41544},[203],[15,41546],{"className":41547,"style":199},[93],[15,41549,41551,41554,41557,41560,41563,41566],{"className":41550},[120],[15,41552],{"className":41553,"style":39499},[124],[15,41555,13091],{"className":41556},[129,133],[15,41558,56],{"className":41559},[129,133],[15,41561],{"className":41562,"style":3277},[93],[15,41564,2682],{"className":41565},[2436],[15,41567],{"className":41568,"style":3277},[93],[15,41570,41572,41575,41578,41581,41584,41587,41590,41593,41596,41599,41602,41605],{"className":41571},[120],[15,41573],{"className":41574,"style":125},[124],[15,41576,8695],{"className":41577},[129,133],[15,41579,91],{"className":41580},[254],[15,41582],{"className":41583,"style":17287},[93],[15,41585],{"className":41586,"style":262},[93],[15,41588,2247],{"className":41589,"style":887},[129,133],[15,41591,53],{"className":41592},[188],[15,41594,56],{"className":41595},[129,133],[15,41597,59],{"className":41598},[195],[15,41600],{"className":41601,"style":199},[93],[15,41603,62],{"className":41604},[203],[15,41606],{"className":41607,"style":199},[93],[15,41609,41611,41614,41617,41620,41623,41626],{"className":41610},[120],[15,41612],{"className":41613,"style":39499},[124],[15,41615,5619],{"className":41616,"style":5665},[129,133],[15,41618,56],{"className":41619},[129,133],[15,41621],{"className":41622,"style":3277},[93],[15,41624,2682],{"className":41625},[2436],[15,41627],{"className":41628,"style":3277},[93],[15,41630,41632,41635],{"className":41631},[120],[15,41633],{"className":41634,"style":522},[124],[15,41636,10372],{"className":41637},[129,133],[11,41639,41640],{},"其复合为：",[15,41642,41644],{"className":41643,"translate":19},[18],[15,41645,41647,41699],{"className":41646,"translate":19},[23],[15,41648,41650],{"className":41649},[27],[29,41651,41652],{"xmlns":31,"display":32},[34,41653,41654,41696],{},[37,41655,41656,41658,41660,41662,41664,41666,41668,41670,41672,41674,41676,41678,41680,41682,41684,41686,41688,41690,41692,41694],{},[43,41657,45],{},[50,41659,53],{"stretchy":52},[43,41661,56],{},[50,41663,59],{"stretchy":52},[50,41665,62],{},[43,41667,36556],{},[50,41669,53],{"stretchy":52},[43,41671,2247],{},[50,41673,53],{"stretchy":52},[43,41675,56],{},[50,41677,59],{"stretchy":52},[50,41679,59],{"stretchy":52},[50,41681,62],{},[43,41683,36556],{},[50,41685,53],{"stretchy":52},[43,41687,5619],{},[43,41689,56],{},[50,41691,2682],{},[43,41693,10372],{},[50,41695,59],{"stretchy":52},[109,41697,41698],{"encoding":111},"h(x) = f(g(x)) = f(Cx + d)",[15,41700,41702,41729,41762,41789],{"className":41701,"ariaHidden":72},[116],[15,41703,41705,41708,41711,41714,41717,41720,41723,41726],{"className":41704},[120],[15,41706],{"className":41707,"style":125},[124],[15,41709,45],{"className":41710},[129,133],[15,41712,53],{"className":41713},[188],[15,41715,56],{"className":41716},[129,133],[15,41718,59],{"className":41719},[195],[15,41721],{"className":41722,"style":199},[93],[15,41724,62],{"className":41725},[203],[15,41727],{"className":41728,"style":199},[93],[15,41730,41732,41735,41738,41741,41744,41747,41750,41753,41756,41759],{"className":41731},[120],[15,41733],{"className":41734,"style":125},[124],[15,41736,36556],{"className":41737,"style":36571},[129,133],[15,41739,53],{"className":41740},[188],[15,41742,2247],{"className":41743,"style":887},[129,133],[15,41745,53],{"className":41746},[188],[15,41748,56],{"className":41749},[129,133],[15,41751,6149],{"className":41752},[195],[15,41754],{"className":41755,"style":199},[93],[15,41757,62],{"className":41758},[203],[15,41760],{"className":41761,"style":199},[93],[15,41763,41765,41768,41771,41774,41777,41780,41783,41786],{"className":41764},[120],[15,41766],{"className":41767,"style":125},[124],[15,41769,36556],{"className":41770,"style":36571},[129,133],[15,41772,53],{"className":41773},[188],[15,41775,5619],{"className":41776,"style":5665},[129,133],[15,41778,56],{"className":41779},[129,133],[15,41781],{"className":41782,"style":3277},[93],[15,41784,2682],{"className":41785},[2436],[15,41787],{"className":41788,"style":3277},[93],[15,41790,41792,41795,41798],{"className":41791},[120],[15,41793],{"className":41794,"style":125},[124],[15,41796,10372],{"className":41797},[129,133],[15,41799,59],{"className":41800},[195],[11,41802,41803],{},"展开来看其形式：",[15,41805,41807],{"className":41806,"translate":19},[18],[15,41808,41810,41876],{"className":41809,"translate":19},[23],[15,41811,41813],{"className":41812},[27],[29,41814,41815],{"xmlns":31,"display":32},[34,41816,41817,41873],{},[37,41818,41819,41821,41823,41825,41827,41829,41831,41833,41835,41837,41839,41841,41843,41845,41847,41849,41851,41853,41855,41857,41859,41861,41863,41865,41867,41869,41871],{},[43,41820,45],{},[50,41822,53],{"stretchy":52},[43,41824,56],{},[50,41826,59],{"stretchy":52},[50,41828,62],{},[43,41830,13091],{},[50,41832,53],{"stretchy":52},[43,41834,5619],{},[43,41836,56],{},[50,41838,2682],{},[43,41840,10372],{},[50,41842,59],{"stretchy":52},[50,41844,2682],{},[43,41846,8695],{},[50,41848,62],{},[50,41850,53],{"stretchy":52},[43,41852,13091],{},[43,41854,5619],{},[50,41856,59],{"stretchy":52},[43,41858,56],{},[50,41860,2682],{},[50,41862,53],{"stretchy":52},[43,41864,13091],{},[43,41866,10372],{},[50,41868,2682],{},[43,41870,8695],{},[50,41872,59],{"stretchy":52},[109,41874,41875],{"encoding":111},"h(x) = A(Cx + d) + b = (AC)x + (Ad + b)",[15,41877,41879,41906,41933,41954,41972,42002,42026],{"className":41878,"ariaHidden":72},[116],[15,41880,41882,41885,41888,41891,41894,41897,41900,41903],{"className":41881},[120],[15,41883],{"className":41884,"style":125},[124],[15,41886,45],{"className":41887},[129,133],[15,41889,53],{"className":41890},[188],[15,41892,56],{"className":41893},[129,133],[15,41895,59],{"className":41896},[195],[15,41898],{"className":41899,"style":199},[93],[15,41901,62],{"className":41902},[203],[15,41904],{"className":41905,"style":199},[93],[15,41907,41909,41912,41915,41918,41921,41924,41927,41930],{"className":41908},[120],[15,41910],{"className":41911,"style":125},[124],[15,41913,13091],{"className":41914},[129,133],[15,41916,53],{"className":41917},[188],[15,41919,5619],{"className":41920,"style":5665},[129,133],[15,41922,56],{"className":41923},[129,133],[15,41925],{"className":41926,"style":3277},[93],[15,41928,2682],{"className":41929},[2436],[15,41931],{"className":41932,"style":3277},[93],[15,41934,41936,41939,41942,41945,41948,41951],{"className":41935},[120],[15,41937],{"className":41938,"style":125},[124],[15,41940,10372],{"className":41941},[129,133],[15,41943,59],{"className":41944},[195],[15,41946],{"className":41947,"style":3277},[93],[15,41949,2682],{"className":41950},[2436],[15,41952],{"className":41953,"style":3277},[93],[15,41955,41957,41960,41963,41966,41969],{"className":41956},[120],[15,41958],{"className":41959,"style":522},[124],[15,41961,8695],{"className":41962},[129,133],[15,41964],{"className":41965,"style":199},[93],[15,41967,62],{"className":41968},[203],[15,41970],{"className":41971,"style":199},[93],[15,41973,41975,41978,41981,41984,41987,41990,41993,41996,41999],{"className":41974},[120],[15,41976],{"className":41977,"style":125},[124],[15,41979,53],{"className":41980},[188],[15,41982,13091],{"className":41983},[129,133],[15,41985,5619],{"className":41986,"style":5665},[129,133],[15,41988,59],{"className":41989},[195],[15,41991,56],{"className":41992},[129,133],[15,41994],{"className":41995,"style":3277},[93],[15,41997,2682],{"className":41998},[2436],[15,42000],{"className":42001,"style":3277},[93],[15,42003,42005,42008,42011,42014,42017,42020,42023],{"className":42004},[120],[15,42006],{"className":42007,"style":125},[124],[15,42009,53],{"className":42010},[188],[15,42012,13091],{"className":42013},[129,133],[15,42015,10372],{"className":42016},[129,133],[15,42018],{"className":42019,"style":3277},[93],[15,42021,2682],{"className":42022},[2436],[15,42024],{"className":42025,"style":3277},[93],[15,42027,42029,42032,42035],{"className":42028},[120],[15,42030],{"className":42031,"style":125},[124],[15,42033,8695],{"className":42034},[129,133],[15,42036,59],{"className":42037},[195],[11,42039,42040],{},"结果仍然是仿射函数。",[11,42042,42043],{},"由此可见，仿射函数的复合仍然是仿射的。这一点可以推广：复合任意多个仿射函数：",[15,42045,42047],{"className":42046,"translate":19},[18],[15,42048,42050,42100],{"className":42049,"translate":19},[23],[15,42051,42053],{"className":42052},[27],[29,42054,42055],{"xmlns":31,"display":32},[34,42056,42057,42097],{},[37,42058,42059,42065,42067,42069,42075,42077,42083,42085,42087,42089,42091,42093,42095],{},[40,42060,42061,42063],{},[43,42062,36556],{},[43,42064,39314],{},[50,42066,53],{"stretchy":52},[50,42068,335],{},[40,42070,42071,42073],{},[43,42072,36556],{},[308,42074,310],{},[50,42076,53],{"stretchy":52},[40,42078,42079,42081],{},[43,42080,36556],{},[308,42082,318],{},[50,42084,53],{"stretchy":52},[43,42086,56],{},[50,42088,59],{"stretchy":52},[50,42090,59],{"stretchy":52},[50,42092,335],{},[6852,42094,15781],{},[50,42096,59],{"stretchy":52},[109,42098,42099],{"encoding":111},"f_n(\\dots f_2(f_1(x))\\dots)",[15,42101,42103],{"className":42102,"ariaHidden":72},[116],[15,42104,42106,42109,42149,42152,42155,42158,42198,42201,42241,42244,42247,42250,42253,42256,42259],{"className":42105},[120],[15,42107],{"className":42108,"style":125},[124],[15,42110,42112,42115],{"className":42111},[129],[15,42113,36556],{"className":42114,"style":36571},[129,133],[15,42116,42118],{"className":42117},[137],[15,42119,42121,42141],{"className":42120},[141,142],[15,42122,42124,42138],{"className":42123},[146],[15,42125,42127],{"className":42126,"style":1163},[150],[15,42128,42129,42132],{"style":39810},[15,42130],{"className":42131,"style":159},[158],[15,42133,42135],{"className":42134},[163,164,165,166],[15,42136,39314],{"className":42137},[129,133,166],[15,42139,175],{"className":42140},[174],[15,42142,42144],{"className":42143},[146],[15,42145,42147],{"className":42146,"style":182},[150],[15,42148],{},[15,42150,53],{"className":42151},[188],[15,42153,335],{"className":42154},[449],[15,42156],{"className":42157,"style":262},[93],[15,42159,42161,42164],{"className":42160},[129],[15,42162,36556],{"className":42163,"style":36571},[129,133],[15,42165,42167],{"className":42166},[137],[15,42168,42170,42190],{"className":42169},[141,142],[15,42171,42173,42187],{"className":42172},[146],[15,42174,42176],{"className":42175,"style":1051},[150],[15,42177,42178,42181],{"style":39810},[15,42179],{"className":42180,"style":159},[158],[15,42182,42184],{"className":42183},[163,164,165,166],[15,42185,310],{"className":42186},[129,166],[15,42188,175],{"className":42189},[174],[15,42191,42193],{"className":42192},[146],[15,42194,42196],{"className":42195,"style":182},[150],[15,42197],{},[15,42199,53],{"className":42200},[188],[15,42202,42204,42207],{"className":42203},[129],[15,42205,36556],{"className":42206,"style":36571},[129,133],[15,42208,42210],{"className":42209},[137],[15,42211,42213,42233],{"className":42212},[141,142],[15,42214,42216,42230],{"className":42215},[146],[15,42217,42219],{"className":42218,"style":1051},[150],[15,42220,42221,42224],{"style":39810},[15,42222],{"className":42223,"style":159},[158],[15,42225,42227],{"className":42226},[163,164,165,166],[15,42228,318],{"className":42229},[129,166],[15,42231,175],{"className":42232},[174],[15,42234,42236],{"className":42235},[146],[15,42237,42239],{"className":42238,"style":182},[150],[15,42240],{},[15,42242,53],{"className":42243},[188],[15,42245,56],{"className":42246},[129,133],[15,42248,6149],{"className":42249},[195],[15,42251],{"className":42252,"style":262},[93],[15,42254,335],{"className":42255},[449],[15,42257],{"className":42258,"style":262},[93],[15,42260,59],{"className":42261},[195],[11,42263,42264],{},"总能写成：",[15,42266,42268],{"className":42267,"translate":19},[18],[15,42269,42271,42300],{"className":42270,"translate":19},[23],[15,42272,42274],{"className":42273},[27],[29,42275,42276],{"xmlns":31,"display":32},[34,42277,42278,42297],{},[37,42279,42280,42287,42289,42291],{},[40,42281,42282,42284],{},[43,42283,13091],{},[6852,42285,42286],{},"总",[43,42288,56],{},[50,42290,2682],{},[40,42292,42293,42295],{},[43,42294,8695],{},[6852,42296,42286],{},[109,42298,42299],{"encoding":111},"A_\\text{总} x + b_\\text{总}",[15,42301,42303,42364],{"className":42302,"ariaHidden":72},[116],[15,42304,42306,42309,42352,42355,42358,42361],{"className":42305},[120],[15,42307],{"className":42308,"style":20220},[124],[15,42310,42312,42315],{"className":42311},[129],[15,42313,13091],{"className":42314},[129,133],[15,42316,42318],{"className":42317},[137],[15,42319,42321,42344],{"className":42320},[141,142],[15,42322,42324,42341],{"className":42323},[146],[15,42325,42327],{"className":42326,"style":1525},[150],[15,42328,42329,42332],{"style":154},[15,42330],{"className":42331,"style":159},[158],[15,42333,42335],{"className":42334},[163,164,165,166],[15,42336,42338],{"className":42337},[129,37670,166],[15,42339,42286],{"className":42340},[129,37678,166],[15,42342,175],{"className":42343},[174],[15,42345,42347],{"className":42346},[146],[15,42348,42350],{"className":42349,"style":182},[150],[15,42351],{},[15,42353,56],{"className":42354},[129,133],[15,42356],{"className":42357,"style":3277},[93],[15,42359,2682],{"className":42360},[2436],[15,42362],{"className":42363,"style":3277},[93],[15,42365,42367,42370],{"className":42366},[120],[15,42368],{"className":42369,"style":2183},[124],[15,42371,42373,42376],{"className":42372},[129],[15,42374,8695],{"className":42375},[129,133],[15,42377,42379],{"className":42378},[137],[15,42380,42382,42405],{"className":42381},[141,142],[15,42383,42385,42402],{"className":42384},[146],[15,42386,42388],{"className":42387,"style":1525},[150],[15,42389,42390,42393],{"style":154},[15,42391],{"className":42392,"style":159},[158],[15,42394,42396],{"className":42395},[163,164,165,166],[15,42397,42399],{"className":42398},[129,37670,166],[15,42400,42286],{"className":42401},[129,37678,166],[15,42403,175],{"className":42404},[174],[15,42406,42408],{"className":42407},[146],[15,42409,42411],{"className":42410,"style":182},[150],[15,42412],{},[11,42414,42415],{},"这正是为什么：如果神经网络的每一层都是仿射的而不使用激活函数，那么无论堆叠多少层，网络的表达能力都等同于单层。我们可以把仿射层看作拉伸、旋转或平移的运算，或这些运算的组合。复合多个仿射变换等价于连续的拉伸 + 旋转 + 平移，最终仍归结为一次整体的拉伸 + 旋转 + 平移。因此，复合仿射变换并不会增加非线性的自由度，也无法表示曲线或复杂函数。",[11,42417,42418],{},"神经网络的每一层都是一个仿射映射：",[15,42420,42422],{"className":42421,"translate":19},[18],[15,42423,42425,42455],{"className":42424,"translate":19},[23],[15,42426,42428],{"className":42427},[27],[29,42429,42430],{"xmlns":31,"display":32},[34,42431,42432,42452],{},[37,42433,42434,42436,42438,42440,42442,42444,42446,42448,42450],{},[43,42435,36556],{},[50,42437,53],{"stretchy":52},[43,42439,56],{},[50,42441,59],{"stretchy":52},[50,42443,62],{},[43,42445,31987],{},[43,42447,56],{},[50,42449,2682],{},[43,42451,8695],{},[109,42453,42454],{"encoding":111},"f(x) = Wx + b",[15,42456,42458,42485,42506],{"className":42457,"ariaHidden":72},[116],[15,42459,42461,42464,42467,42470,42473,42476,42479,42482],{"className":42460},[120],[15,42462],{"className":42463,"style":125},[124],[15,42465,36556],{"className":42466,"style":36571},[129,133],[15,42468,53],{"className":42469},[188],[15,42471,56],{"className":42472},[129,133],[15,42474,59],{"className":42475},[195],[15,42477],{"className":42478,"style":199},[93],[15,42480,62],{"className":42481},[203],[15,42483],{"className":42484,"style":199},[93],[15,42486,42488,42491,42494,42497,42500,42503],{"className":42487},[120],[15,42489],{"className":42490,"style":39499},[124],[15,42492,31987],{"className":42493,"style":12443},[129,133],[15,42495,56],{"className":42496},[129,133],[15,42498],{"className":42499,"style":3277},[93],[15,42501,2682],{"className":42502},[2436],[15,42504],{"className":42505,"style":3277},[93],[15,42507,42509,42512],{"className":42508},[120],[15,42510],{"className":42511,"style":522},[124],[15,42513,8695],{"className":42514},[129,133],[21719,42516,42517,42680],{},[21722,42518,6564,42519,42547,42548,42650,42651,42679],{},[15,42520,42522,42535],{"className":42521,"translate":19},[23],[15,42523,42525],{"className":42524},[27],[29,42526,42527],{"xmlns":31},[34,42528,42529,42533],{},[37,42530,42531],{},[43,42532,31987],{},[109,42534,31987],{"encoding":111},[15,42536,42538],{"className":42537,"ariaHidden":72},[116],[15,42539,42541,42544],{"className":42540},[120],[15,42542],{"className":42543,"style":610},[124],[15,42545,31987],{"className":42546,"style":12443},[129,133]," 可逆（即 ",[15,42549,42551,42578],{"className":42550,"translate":19},[23],[15,42552,42554],{"className":42553},[27],[29,42555,42556],{"xmlns":31},[34,42557,42558,42575],{},[37,42559,42560,42563,42565,42567,42569,42571,42573],{},[43,42561,42562],{},"det",[50,42564,69],{},[50,42566,53],{"stretchy":52},[43,42568,31987],{},[50,42570,59],{"stretchy":52},[50,42572,1418],{"mathvariant":65},[308,42574,541],{},[109,42576,42577],{"encoding":111},"\\det(W) \\neq 0",[15,42579,42581,42641],{"className":42580,"ariaHidden":72},[116],[15,42582,42584,42587,42590,42593,42596,42599,42602,42638],{"className":42583},[120],[15,42585],{"className":42586,"style":125},[124],[15,42588,42562],{"className":42589},[217],[15,42591,53],{"className":42592},[188],[15,42594,31987],{"className":42595,"style":12443},[129,133],[15,42597,59],{"className":42598},[195],[15,42600],{"className":42601,"style":199},[93],[15,42603,42605,42632,42635],{"className":42604},[203],[15,42606,42608],{"className":42607},[203],[15,42609,42611],{"className":42610},[129,1670],[15,42612,42614],{"className":42613},[1674],[15,42615,42617,42620,42629],{"className":42616},[1678],[15,42618],{"className":42619,"style":1682},[124],[15,42621,42623],{"className":42622},[1686],[15,42624,42626],{"className":42625},[129],[15,42627,1693],{"className":42628},[203],[15,42630],{"className":42631},[1697],[15,42633],{"className":42634},[93,1701],[15,42636,62],{"className":42637},[203],[15,42639],{"className":42640,"style":199},[93],[15,42642,42644,42647],{"className":42643},[120],[15,42645],{"className":42646,"style":4886},[124],[15,42648,541],{"className":42649},[129],"），则 ",[15,42652,42654,42667],{"className":42653,"translate":19},[23],[15,42655,42657],{"className":42656},[27],[29,42658,42659],{"xmlns":31},[34,42660,42661,42665],{},[37,42662,42663],{},[43,42664,36556],{},[109,42666,36556],{"encoding":111},[15,42668,42670],{"className":42669,"ariaHidden":72},[116],[15,42671,42673,42676],{"className":42672},[120],[15,42674],{"className":42675,"style":1682},[124],[15,42677,36556],{"className":42678,"style":36571},[129,133]," 是双射且连续的。",[21722,42681,42682],{},"其逆映射为：",[15,42684,42686],{"className":42685,"translate":19},[18],[15,42687,42689,42739],{"className":42688,"translate":19},[23],[15,42690,42692],{"className":42691},[27],[29,42693,42694],{"xmlns":31,"display":32},[34,42695,42696,42736],{},[37,42697,42698,42708,42710,42712,42714,42716,42726,42728,42730,42732,42734],{},[305,42699,42700,42702],{},[43,42701,36556],{},[37,42703,42704,42706],{},[50,42705,315],{},[308,42707,318],{},[50,42709,53],{"stretchy":52},[43,42711,857],{},[50,42713,59],{"stretchy":52},[50,42715,62],{},[305,42717,42718,42720],{},[43,42719,13091],{},[37,42721,42722,42724],{},[50,42723,315],{},[308,42725,318],{},[50,42727,53],{"stretchy":52},[43,42729,857],{},[50,42731,315],{},[43,42733,8695],{},[50,42735,59],{"stretchy":52},[109,42737,42738],{"encoding":111},"f^{-1}(y) = A^{-1}(y - b)",[15,42740,42742,42801,42857],{"className":42741,"ariaHidden":72},[116],[15,42743,42745,42748,42783,42786,42789,42792,42795,42798],{"className":42744},[120],[15,42746],{"className":42747,"style":40908},[124],[15,42749,42751,42754],{"className":42750},[129],[15,42752,36556],{"className":42753,"style":36571},[129,133],[15,42755,42757],{"className":42756},[137],[15,42758,42760],{"className":42759},[141],[15,42761,42763],{"className":42762},[146],[15,42764,42766],{"className":42765,"style":40979},[150],[15,42767,42768,42771],{"style":1272},[15,42769],{"className":42770,"style":159},[158],[15,42772,42774],{"className":42773},[163,164,165,166],[15,42775,42777,42780],{"className":42776},[129,166],[15,42778,315],{"className":42779},[129,166],[15,42781,318],{"className":42782},[129,166],[15,42784,53],{"className":42785},[188],[15,42787,857],{"className":42788,"style":887},[129,133],[15,42790,59],{"className":42791},[195],[15,42793],{"className":42794,"style":199},[93],[15,42796,62],{"className":42797},[203],[15,42799],{"className":42800,"style":199},[93],[15,42802,42804,42807,42842,42845,42848,42851,42854],{"className":42803},[120],[15,42805],{"className":42806,"style":40908},[124],[15,42808,42810,42813],{"className":42809},[129],[15,42811,13091],{"className":42812},[129,133],[15,42814,42816],{"className":42815},[137],[15,42817,42819],{"className":42818},[141],[15,42820,42822],{"className":42821},[146],[15,42823,42825],{"className":42824,"style":40979},[150],[15,42826,42827,42830],{"style":1272},[15,42828],{"className":42829,"style":159},[158],[15,42831,42833],{"className":42832},[163,164,165,166],[15,42834,42836,42839],{"className":42835},[129,166],[15,42837,315],{"className":42838},[129,166],[15,42840,318],{"className":42841},[129,166],[15,42843,53],{"className":42844},[188],[15,42846,857],{"className":42847,"style":887},[129,133],[15,42849],{"className":42850,"style":3277},[93],[15,42852,315],{"className":42853},[2436],[15,42855],{"className":42856,"style":3277},[93],[15,42858,42860,42863,42866],{"className":42859},[120],[15,42861],{"className":42862,"style":125},[124],[15,42864,8695],{"className":42865},[129,133],[15,42867,59],{"className":42868},[195],[11,42870,42871,42872,42900],{},"它同样是仿射且连续的，因此是一个同胚（homeomorphism）。但这里有一个核心条件：",[15,42873,42875,42888],{"className":42874,"translate":19},[23],[15,42876,42878],{"className":42877},[27],[29,42879,42880],{"xmlns":31},[34,42881,42882,42886],{},[37,42883,42884],{},[43,42885,31987],{},[109,42887,31987],{"encoding":111},[15,42889,42891],{"className":42890,"ariaHidden":72},[116],[15,42892,42894,42897],{"className":42893},[120],[15,42895],{"className":42896,"style":610},[124],[15,42898,31987],{"className":42899,"style":12443},[129,133]," 必须可逆。",[11,42902,42903],{},"复合多个仿射变换等价于复合多个同胚。假设有多个连续的可逆仿射映射：",[15,42905,42907],{"className":42906,"translate":19},[18],[15,42908,42910,42955],{"className":42909,"translate":19},[23],[15,42911,42913],{"className":42912},[27],[29,42914,42915],{"xmlns":31,"display":32},[34,42916,42917,42952],{},[37,42918,42919,42925,42928,42940,42942,42944,42946],{},[40,42920,42921,42923],{},[43,42922,36556],{},[43,42924,39314],{},[50,42926,42927],{},"∘",[40,42929,42930,42932],{},[43,42931,36556],{},[37,42933,42934,42936,42938],{},[43,42935,39314],{},[50,42937,315],{},[308,42939,318],{},[50,42941,42927],{},[50,42943,31657],{},[50,42945,42927],{},[40,42947,42948,42950],{},[43,42949,36556],{},[308,42951,318],{},[109,42953,42954],{"encoding":111},"f_n \\circ f_{n-1} \\circ \\cdots \\circ f_1",[15,42956,42958,43013,43079,43097],{"className":42957,"ariaHidden":72},[116],[15,42959,42961,42964,43004,43007,43010],{"className":42960},[120],[15,42962],{"className":42963,"style":1682},[124],[15,42965,42967,42970],{"className":42966},[129],[15,42968,36556],{"className":42969,"style":36571},[129,133],[15,42971,42973],{"className":42972},[137],[15,42974,42976,42996],{"className":42975},[141,142],[15,42977,42979,42993],{"className":42978},[146],[15,42980,42982],{"className":42981,"style":1163},[150],[15,42983,42984,42987],{"style":39810},[15,42985],{"className":42986,"style":159},[158],[15,42988,42990],{"className":42989},[163,164,165,166],[15,42991,39314],{"className":42992},[129,133,166],[15,42994,175],{"className":42995},[174],[15,42997,42999],{"className":42998},[146],[15,43000,43002],{"className":43001,"style":182},[150],[15,43003],{},[15,43005],{"className":43006,"style":3277},[93],[15,43008,42927],{"className":43009},[2436],[15,43011],{"className":43012,"style":3277},[93],[15,43014,43016,43020,43070,43073,43076],{"className":43015},[120],[15,43017],{"className":43018,"style":43019},[124],"height:0.9028em;vertical-align:-0.2083em;",[15,43021,43023,43026],{"className":43022},[129],[15,43024,36556],{"className":43025,"style":36571},[129,133],[15,43027,43029],{"className":43028},[137],[15,43030,43032,43061],{"className":43031},[141,142],[15,43033,43035,43058],{"className":43034},[146],[15,43036,43038],{"className":43037,"style":1051},[150],[15,43039,43040,43043],{"style":39810},[15,43041],{"className":43042,"style":159},[158],[15,43044,43046],{"className":43045},[163,164,165,166],[15,43047,43049,43052,43055],{"className":43048},[129,166],[15,43050,39314],{"className":43051},[129,133,166],[15,43053,315],{"className":43054},[2436,166],[15,43056,318],{"className":43057},[129,166],[15,43059,175],{"className":43060},[174],[15,43062,43064],{"className":43063},[146],[15,43065,43068],{"className":43066,"style":43067},[150],"height:0.2083em;",[15,43069],{},[15,43071],{"className":43072,"style":3277},[93],[15,43074,42927],{"className":43075},[2436],[15,43077],{"className":43078,"style":3277},[93],[15,43080,43082,43085,43088,43091,43094],{"className":43081},[120],[15,43083],{"className":43084,"style":11094},[124],[15,43086,31657],{"className":43087},[449],[15,43089],{"className":43090,"style":3277},[93],[15,43092,42927],{"className":43093},[2436],[15,43095],{"className":43096,"style":3277},[93],[15,43098,43100,43103],{"className":43099},[120],[15,43101],{"className":43102,"style":1682},[124],[15,43104,43106,43109],{"className":43105},[129],[15,43107,36556],{"className":43108,"style":36571},[129,133],[15,43110,43112],{"className":43111},[137],[15,43113,43115,43135],{"className":43114},[141,142],[15,43116,43118,43132],{"className":43117},[146],[15,43119,43121],{"className":43120,"style":1051},[150],[15,43122,43123,43126],{"style":39810},[15,43124],{"className":43125,"style":159},[158],[15,43127,43129],{"className":43128},[163,164,165,166],[15,43130,318],{"className":43131},[129,166],[15,43133,175],{"className":43134},[174],[15,43136,43138],{"className":43137},[146],[15,43139,43141],{"className":43140,"style":182},[150],[15,43142],{},[11,43144,43145,43146,43216],{},"每个 ",[15,43147,43149,43167],{"className":43148,"translate":19},[23],[15,43150,43152],{"className":43151},[27],[29,43153,43154],{"xmlns":31},[34,43155,43156,43164],{},[37,43157,43158],{},[40,43159,43160,43162],{},[43,43161,36556],{},[43,43163,1459],{},[109,43165,43166],{"encoding":111},"f_i",[15,43168,43170],{"className":43169,"ariaHidden":72},[116],[15,43171,43173,43176],{"className":43172},[120],[15,43174],{"className":43175,"style":1682},[124],[15,43177,43179,43182],{"className":43178},[129],[15,43180,36556],{"className":43181,"style":36571},[129,133],[15,43183,43185],{"className":43184},[137],[15,43186,43188,43208],{"className":43187},[141,142],[15,43189,43191,43205],{"className":43190},[146],[15,43192,43194],{"className":43193,"style":1981},[150],[15,43195,43196,43199],{"style":39810},[15,43197],{"className":43198,"style":159},[158],[15,43200,43202],{"className":43201},[163,164,165,166],[15,43203,1459],{"className":43204},[129,133,166],[15,43206,175],{"className":43207},[174],[15,43209,43211],{"className":43210},[146],[15,43212,43214],{"className":43213,"style":182},[150],[15,43215],{}," 都是同胚，而同胚的复合仍然是同胚。因此，最终的映射也是同胚。",[11,43218,43219,43220,43249],{},"于是我们得出如下结论：设 ",[15,43221,43223,43237],{"className":43222,"translate":19},[23],[15,43224,43226],{"className":43225},[27],[29,43227,43228],{"xmlns":31},[34,43229,43230,43235],{},[37,43231,43232],{},[43,43233,43234],{},"M",[109,43236,43234],{"encoding":111},[15,43238,43240],{"className":43239,"ariaHidden":72},[116],[15,43241,43243,43246],{"className":43242},[120],[15,43244],{"className":43245,"style":610},[124],[15,43247,43234],{"className":43248,"style":12591},[129,133]," 为原始流形：",[21719,43251,43252,43255],{},[21722,43253,43254],{},"连续且双射的仿射映射，保持流形结构（同胚）。",[21722,43256,43257,43258,43286],{},"连续复合多个这样的仿射映射，得到的流形与 ",[15,43259,43261,43274],{"className":43260,"translate":19},[23],[15,43262,43264],{"className":43263},[27],[29,43265,43266],{"xmlns":31},[34,43267,43268,43272],{},[37,43269,43270],{},[43,43271,43234],{},[109,43273,43234],{"encoding":111},[15,43275,43277],{"className":43276,"ariaHidden":72},[116],[15,43278,43280,43283],{"className":43279},[120],[15,43281],{"className":43282,"style":610},[124],[15,43284,43234],{"className":43285,"style":12591},[129,133]," 同胚。",[11,43288,43289],{},"换句话说：「连续线性 + 偏置的变换不会改变流形的拓扑，只会对其进行拉伸、旋转、平移或剪切。」",[11,43291,43292],{},"这就是为什么仅使用线性层无法「弯曲」数据流形：在嵌套的线性层下，数据始终保持其「原始形状」（拓扑不变）。如果决策边界需要切开一个复杂流形，就必须引入非线性（激活函数）。",[11,43294,43295],{},"这也是深度学习的目标。如果原始流形的结构对于给定任务而言不可分，而学习目标又要求可分性，那么原始流形与目标流形必然不是同胚的。",[11,43297,43298],{},"在从原始流形到目标流形的变换中，要失去同胚性就必须摆脱线性、仿射的变换。因为仅靠仿射变换（更一般地说是同胚变换）通常不足以解开类别。深度网络往往需要通过非同胚的非线性操作来重构数据流形，使其最终线性可分。",[11,43300,43301],{},"我们需要考虑如何让仿射变换的复合变得非线性。这一洞见正是神经网络几何解释的核心：",[36722,43303,43304],{},[11,43305,43306],{},"线性层只能执行保持拓扑的仿射变换；非线性层则允许拓扑改变，从而使神经网络能够表示复杂函数。",[570,43308,43309],{"id":43309},"仿射层的激活",[11,43311,42418],{},[15,43313,43315],{"className":43314,"translate":19},[18],[15,43316,43318,43360],{"className":43317,"translate":19},[23],[15,43319,43321],{"className":43320},[27],[29,43322,43323],{"xmlns":31,"display":32},[34,43324,43325,43357],{},[37,43326,43327,43333,43335,43337,43339,43341,43347,43349,43351],{},[40,43328,43329,43331],{},[43,43330,34133],{},[43,43332,1459],{},[50,43334,53],{"stretchy":52},[43,43336,56],{},[50,43338,59],{"stretchy":52},[50,43340,62],{},[40,43342,43343,43345],{},[43,43344,31987],{},[43,43346,1459],{},[43,43348,56],{},[50,43350,2682],{},[40,43352,43353,43355],{},[43,43354,8695],{},[43,43356,1459],{},[109,43358,43359],{"encoding":111},"l_i(x) = W_i x + b_i",[15,43361,43363,43428,43486],{"className":43362,"ariaHidden":72},[116],[15,43364,43366,43369,43410,43413,43416,43419,43422,43425],{"className":43365},[120],[15,43367],{"className":43368,"style":125},[124],[15,43370,43372,43375],{"className":43371},[129],[15,43373,34133],{"className":43374,"style":34191},[129,133],[15,43376,43378],{"className":43377},[137],[15,43379,43381,43402],{"className":43380},[141,142],[15,43382,43384,43399],{"className":43383},[146],[15,43385,43387],{"className":43386,"style":1981},[150],[15,43388,43390,43393],{"style":43389},"top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;",[15,43391],{"className":43392,"style":159},[158],[15,43394,43396],{"className":43395},[163,164,165,166],[15,43397,1459],{"className":43398},[129,133,166],[15,43400,175],{"className":43401},[174],[15,43403,43405],{"className":43404},[146],[15,43406,43408],{"className":43407,"style":182},[150],[15,43409],{},[15,43411,53],{"className":43412},[188],[15,43414,56],{"className":43415},[129,133],[15,43417,59],{"className":43418},[195],[15,43420],{"className":43421,"style":199},[93],[15,43423,62],{"className":43424},[203],[15,43426],{"className":43427,"style":199},[93],[15,43429,43431,43434,43474,43477,43480,43483],{"className":43430},[120],[15,43432],{"className":43433,"style":20220},[124],[15,43435,43437,43440],{"className":43436},[129],[15,43438,31987],{"className":43439,"style":12443},[129,133],[15,43441,43443],{"className":43442},[137],[15,43444,43446,43466],{"className":43445},[141,142],[15,43447,43449,43463],{"className":43448},[146],[15,43450,43452],{"className":43451,"style":1981},[150],[15,43453,43454,43457],{"style":34226},[15,43455],{"className":43456,"style":159},[158],[15,43458,43460],{"className":43459},[163,164,165,166],[15,43461,1459],{"className":43462},[129,133,166],[15,43464,175],{"className":43465},[174],[15,43467,43469],{"className":43468},[146],[15,43470,43472],{"className":43471,"style":182},[150],[15,43473],{},[15,43475,56],{"className":43476},[129,133],[15,43478],{"className":43479,"style":3277},[93],[15,43481,2682],{"className":43482},[2436],[15,43484],{"className":43485,"style":3277},[93],[15,43487,43489,43492],{"className":43488},[120],[15,43490],{"className":43491,"style":2183},[124],[15,43493,43495,43498],{"className":43494},[129],[15,43496,8695],{"className":43497},[129,133],[15,43499,43501],{"className":43500},[137],[15,43502,43504,43524],{"className":43503},[141,142],[15,43505,43507,43521],{"className":43506},[146],[15,43508,43510],{"className":43509,"style":1981},[150],[15,43511,43512,43515],{"style":154},[15,43513],{"className":43514,"style":159},[158],[15,43516,43518],{"className":43517},[163,164,165,166],[15,43519,1459],{"className":43520},[129,133,166],[15,43522,175],{"className":43523},[174],[15,43525,43527],{"className":43526},[146],[15,43528,43530],{"className":43529,"style":182},[150],[15,43531],{},[11,43533,28686,43534,43562,43563,43591,43592,43620],{},[15,43535,43537,43550],{"className":43536,"translate":19},[23],[15,43538,43540],{"className":43539},[27],[29,43541,43542],{"xmlns":31},[34,43543,43544,43548],{},[37,43545,43546],{},[43,43547,1459],{},[109,43549,1459],{"encoding":111},[15,43551,43553],{"className":43552,"ariaHidden":72},[116],[15,43554,43556,43559],{"className":43555},[120],[15,43557],{"className":43558,"style":16104},[124],[15,43560,1459],{"className":43561},[129,133]," 表示第 ",[15,43564,43566,43579],{"className":43565,"translate":19},[23],[15,43567,43569],{"className":43568},[27],[29,43570,43571],{"xmlns":31},[34,43572,43573,43577],{},[37,43574,43575],{},[43,43576,1459],{},[109,43578,1459],{"encoding":111},[15,43580,43582],{"className":43581,"ariaHidden":72},[116],[15,43583,43585,43588],{"className":43584},[120],[15,43586],{"className":43587,"style":16104},[124],[15,43589,1459],{"className":43590},[129,133]," 层。",[15,43593,43595,43608],{"className":43594,"translate":19},[23],[15,43596,43598],{"className":43597},[27],[29,43599,43600],{"xmlns":31},[34,43601,43602,43606],{},[37,43603,43604],{},[43,43605,39314],{},[109,43607,39314],{"encoding":111},[15,43609,43611],{"className":43610,"ariaHidden":72},[116],[15,43612,43614,43617],{"className":43613},[120],[15,43615],{"className":43616,"style":4383},[124],[15,43618,39314],{"className":43619},[129,133]," 层仿射映射的嵌套复合为：",[15,43622,43624],{"className":43623,"translate":19},[18],[15,43625,43627,43685],{"className":43626,"translate":19},[23],[15,43628,43630],{"className":43629},[27],[29,43631,43632],{"xmlns":31,"display":32},[34,43633,43634,43682],{},[37,43635,43636,43642,43644,43656,43658,43670,43672,43674,43676],{},[40,43637,43638,43640],{},[43,43639,34133],{},[43,43641,39314],{},[50,43643,42927],{},[40,43645,43646,43648],{},[43,43647,34133],{},[37,43649,43650,43652,43654],{},[43,43651,39314],{},[50,43653,315],{},[308,43655,318],{},[50,43657,42927],{},[40,43659,43660,43662],{},[43,43661,34133],{},[37,43663,43664,43666,43668],{},[43,43665,39314],{},[50,43667,315],{},[308,43669,310],{},[50,43671,42927],{},[50,43673,31657],{},[50,43675,42927],{},[40,43677,43678,43680],{},[43,43679,34133],{},[308,43681,541],{},[109,43683,43684],{"encoding":111},"l_n \\circ l_{n-1} \\circ l_{n-2} \\circ \\dots \\circ l_0",[15,43686,43688,43743,43807,43871,43889],{"className":43687,"ariaHidden":72},[116],[15,43689,43691,43694,43734,43737,43740],{"className":43690},[120],[15,43692],{"className":43693,"style":2183},[124],[15,43695,43697,43700],{"className":43696},[129],[15,43698,34133],{"className":43699,"style":34191},[129,133],[15,43701,43703],{"className":43702},[137],[15,43704,43706,43726],{"className":43705},[141,142],[15,43707,43709,43723],{"className":43708},[146],[15,43710,43712],{"className":43711,"style":1163},[150],[15,43713,43714,43717],{"style":43389},[15,43715],{"className":43716,"style":159},[158],[15,43718,43720],{"className":43719},[163,164,165,166],[15,43721,39314],{"className":43722},[129,133,166],[15,43724,175],{"className":43725},[174],[15,43727,43729],{"className":43728},[146],[15,43730,43732],{"className":43731,"style":182},[150],[15,43733],{},[15,43735],{"className":43736,"style":3277},[93],[15,43738,42927],{"className":43739},[2436],[15,43741],{"className":43742,"style":3277},[93],[15,43744,43746,43749,43798,43801,43804],{"className":43745},[120],[15,43747],{"className":43748,"style":43019},[124],[15,43750,43752,43755],{"className":43751},[129],[15,43753,34133],{"className":43754,"style":34191},[129,133],[15,43756,43758],{"className":43757},[137],[15,43759,43761,43790],{"className":43760},[141,142],[15,43762,43764,43787],{"className":43763},[146],[15,43765,43767],{"className":43766,"style":1051},[150],[15,43768,43769,43772],{"style":43389},[15,43770],{"className":43771,"style":159},[158],[15,43773,43775],{"className":43774},[163,164,165,166],[15,43776,43778,43781,43784],{"className":43777},[129,166],[15,43779,39314],{"className":43780},[129,133,166],[15,43782,315],{"className":43783},[2436,166],[15,43785,318],{"className":43786},[129,166],[15,43788,175],{"className":43789},[174],[15,43791,43793],{"className":43792},[146],[15,43794,43796],{"className":43795,"style":43067},[150],[15,43797],{},[15,43799],{"className":43800,"style":3277},[93],[15,43802,42927],{"className":43803},[2436],[15,43805],{"className":43806,"style":3277},[93],[15,43808,43810,43813,43862,43865,43868],{"className":43809},[120],[15,43811],{"className":43812,"style":43019},[124],[15,43814,43816,43819],{"className":43815},[129],[15,43817,34133],{"className":43818,"style":34191},[129,133],[15,43820,43822],{"className":43821},[137],[15,43823,43825,43854],{"className":43824},[141,142],[15,43826,43828,43851],{"className":43827},[146],[15,43829,43831],{"className":43830,"style":1051},[150],[15,43832,43833,43836],{"style":43389},[15,43834],{"className":43835,"style":159},[158],[15,43837,43839],{"className":43838},[163,164,165,166],[15,43840,43842,43845,43848],{"className":43841},[129,166],[15,43843,39314],{"className":43844},[129,133,166],[15,43846,315],{"className":43847},[2436,166],[15,43849,310],{"className":43850},[129,166],[15,43852,175],{"className":43853},[174],[15,43855,43857],{"className":43856},[146],[15,43858,43860],{"className":43859,"style":43067},[150],[15,43861],{},[15,43863],{"className":43864,"style":3277},[93],[15,43866,42927],{"className":43867},[2436],[15,43869],{"className":43870,"style":3277},[93],[15,43872,43874,43877,43880,43883,43886],{"className":43873},[120],[15,43875],{"className":43876,"style":11094},[124],[15,43878,31657],{"className":43879},[449],[15,43881],{"className":43882,"style":3277},[93],[15,43884,42927],{"className":43885},[2436],[15,43887],{"className":43888,"style":3277},[93],[15,43890,43892,43895],{"className":43891},[120],[15,43893],{"className":43894,"style":2183},[124],[15,43896,43898,43901],{"className":43897},[129],[15,43899,34133],{"className":43900,"style":34191},[129,133],[15,43902,43904],{"className":43903},[137],[15,43905,43907,43927],{"className":43906},[141,142],[15,43908,43910,43924],{"className":43909},[146],[15,43911,43913],{"className":43912,"style":1051},[150],[15,43914,43915,43918],{"style":43389},[15,43916],{"className":43917,"style":159},[158],[15,43919,43921],{"className":43920},[163,164,165,166],[15,43922,541],{"className":43923},[129,166],[15,43925,175],{"className":43926},[174],[15,43928,43930],{"className":43929},[146],[15,43931,43933],{"className":43932,"style":182},[150],[15,43934],{},[11,43936,43937],{},"如果不引入激活函数，上述结果仍然是仿射函数。",[11,43939,43940],{},"证明很简单：",[15,43942,43944],{"className":43943,"translate":19},[18],[15,43945,43947,44063],{"className":43946,"translate":19},[23],[15,43948,43950],{"className":43949},[27],[29,43951,43952],{"xmlns":31,"display":32},[34,43953,43954,44060],{},[37,43955,43956,43962,43964,43970,43972,43974,43976,43978,43980,43986,43988,43994,43996,43998,44004,44006,44008,44014,44016,44018,44024,44030,44032,44034,44036,44038,44044,44050,44052,44058],{},[40,43957,43958,43960],{},[43,43959,34133],{},[308,43961,310],{},[50,43963,53],{"stretchy":52},[40,43965,43966,43968],{},[43,43967,34133],{},[308,43969,318],{},[50,43971,53],{"stretchy":52},[43,43973,56],{},[50,43975,59],{"stretchy":52},[50,43977,59],{"stretchy":52},[50,43979,62],{},[40,43981,43982,43984],{},[43,43983,31987],{},[308,43985,310],{},[50,43987,53],{"stretchy":52},[40,43989,43990,43992],{},[43,43991,31987],{},[308,43993,318],{},[43,43995,56],{},[50,43997,2682],{},[40,43999,44000,44002],{},[43,44001,8695],{},[308,44003,318],{},[50,44005,59],{"stretchy":52},[50,44007,2682],{},[40,44009,44010,44012],{},[43,44011,8695],{},[308,44013,310],{},[50,44015,62],{},[50,44017,53],{"stretchy":52},[40,44019,44020,44022],{},[43,44021,31987],{},[308,44023,310],{},[40,44025,44026,44028],{},[43,44027,31987],{},[308,44029,318],{},[50,44031,59],{"stretchy":52},[43,44033,56],{},[50,44035,2682],{},[50,44037,53],{"stretchy":52},[40,44039,44040,44042],{},[43,44041,31987],{},[308,44043,310],{},[40,44045,44046,44048],{},[43,44047,8695],{},[308,44049,318],{},[50,44051,2682],{},[40,44053,44054,44056],{},[43,44055,8695],{},[308,44057,310],{},[50,44059,59],{"stretchy":52},[109,44061,44062],{"encoding":111},"l_2(l_1(x)) = W_2(W_1x+b_1)+b_2 = (W_2W_1)x+(W_2b_1+b_2)",[15,44064,44066,44173,44274,44332,44387,44491,44589],{"className":44065,"ariaHidden":72},[116],[15,44067,44069,44072,44112,44115,44155,44158,44161,44164,44167,44170],{"className":44068},[120],[15,44070],{"className":44071,"style":125},[124],[15,44073,44075,44078],{"className":44074},[129],[15,44076,34133],{"className":44077,"style":34191},[129,133],[15,44079,44081],{"className":44080},[137],[15,44082,44084,44104],{"className":44083},[141,142],[15,44085,44087,44101],{"className":44086},[146],[15,44088,44090],{"className":44089,"style":1051},[150],[15,44091,44092,44095],{"style":43389},[15,44093],{"className":44094,"style":159},[158],[15,44096,44098],{"className":44097},[163,164,165,166],[15,44099,310],{"className":44100},[129,166],[15,44102,175],{"className":44103},[174],[15,44105,44107],{"className":44106},[146],[15,44108,44110],{"className":44109,"style":182},[150],[15,44111],{},[15,44113,53],{"className":44114},[188],[15,44116,44118,44121],{"className":44117},[129],[15,44119,34133],{"className":44120,"style":34191},[129,133],[15,44122,44124],{"className":44123},[137],[15,44125,44127,44147],{"className":44126},[141,142],[15,44128,44130,44144],{"className":44129},[146],[15,44131,44133],{"className":44132,"style":1051},[150],[15,44134,44135,44138],{"style":43389},[15,44136],{"className":44137,"style":159},[158],[15,44139,44141],{"className":44140},[163,164,165,166],[15,44142,318],{"className":44143},[129,166],[15,44145,175],{"className":44146},[174],[15,44148,44150],{"className":44149},[146],[15,44151,44153],{"className":44152,"style":182},[150],[15,44154],{},[15,44156,53],{"className":44157},[188],[15,44159,56],{"className":44160},[129,133],[15,44162,6149],{"className":44163},[195],[15,44165],{"className":44166,"style":199},[93],[15,44168,62],{"className":44169},[203],[15,44171],{"className":44172,"style":199},[93],[15,44174,44176,44179,44219,44222,44262,44265,44268,44271],{"className":44175},[120],[15,44177],{"className":44178,"style":125},[124],[15,44180,44182,44185],{"className":44181},[129],[15,44183,31987],{"className":44184,"style":12443},[129,133],[15,44186,44188],{"className":44187},[137],[15,44189,44191,44211],{"className":44190},[141,142],[15,44192,44194,44208],{"className":44193},[146],[15,44195,44197],{"className":44196,"style":1051},[150],[15,44198,44199,44202],{"style":34226},[15,44200],{"className":44201,"style":159},[158],[15,44203,44205],{"className":44204},[163,164,165,166],[15,44206,310],{"className":44207},[129,166],[15,44209,175],{"className":44210},[174],[15,44212,44214],{"className":44213},[146],[15,44215,44217],{"className":44216,"style":182},[150],[15,44218],{},[15,44220,53],{"className":44221},[188],[15,44223,44225,44228],{"className":44224},[129],[15,44226,31987],{"className":44227,"style":12443},[129,133],[15,44229,44231],{"className":44230},[137],[15,44232,44234,44254],{"className":44233},[141,142],[15,44235,44237,44251],{"className":44236},[146],[15,44238,44240],{"className":44239,"style":1051},[150],[15,44241,44242,44245],{"style":34226},[15,44243],{"className":44244,"style":159},[158],[15,44246,44248],{"className":44247},[163,164,165,166],[15,44249,318],{"className":44250},[129,166],[15,44252,175],{"className":44253},[174],[15,44255,44257],{"className":44256},[146],[15,44258,44260],{"className":44259,"style":182},[150],[15,44261],{},[15,44263,56],{"className":44264},[129,133],[15,44266],{"className":44267,"style":3277},[93],[15,44269,2682],{"className":44270},[2436],[15,44272],{"className":44273,"style":3277},[93],[15,44275,44277,44280,44320,44323,44326,44329],{"className":44276},[120],[15,44278],{"className":44279,"style":125},[124],[15,44281,44283,44286],{"className":44282},[129],[15,44284,8695],{"className":44285},[129,133],[15,44287,44289],{"className":44288},[137],[15,44290,44292,44312],{"className":44291},[141,142],[15,44293,44295,44309],{"className":44294},[146],[15,44296,44298],{"className":44297,"style":1051},[150],[15,44299,44300,44303],{"style":154},[15,44301],{"className":44302,"style":159},[158],[15,44304,44306],{"className":44305},[163,164,165,166],[15,44307,318],{"className":44308},[129,166],[15,44310,175],{"className":44311},[174],[15,44313,44315],{"className":44314},[146],[15,44316,44318],{"className":44317,"style":182},[150],[15,44319],{},[15,44321,59],{"className":44322},[195],[15,44324],{"className":44325,"style":3277},[93],[15,44327,2682],{"className":44328},[2436],[15,44330],{"className":44331,"style":3277},[93],[15,44333,44335,44338,44378,44381,44384],{"className":44334},[120],[15,44336],{"className":44337,"style":2183},[124],[15,44339,44341,44344],{"className":44340},[129],[15,44342,8695],{"className":44343},[129,133],[15,44345,44347],{"className":44346},[137],[15,44348,44350,44370],{"className":44349},[141,142],[15,44351,44353,44367],{"className":44352},[146],[15,44354,44356],{"className":44355,"style":1051},[150],[15,44357,44358,44361],{"style":154},[15,44359],{"className":44360,"style":159},[158],[15,44362,44364],{"className":44363},[163,164,165,166],[15,44365,310],{"className":44366},[129,166],[15,44368,175],{"className":44369},[174],[15,44371,44373],{"className":44372},[146],[15,44374,44376],{"className":44375,"style":182},[150],[15,44377],{},[15,44379],{"className":44380,"style":199},[93],[15,44382,62],{"className":44383},[203],[15,44385],{"className":44386,"style":199},[93],[15,44388,44390,44393,44396,44436,44476,44479,44482,44485,44488],{"className":44389},[120],[15,44391],{"className":44392,"style":125},[124],[15,44394,53],{"className":44395},[188],[15,44397,44399,44402],{"className":44398},[129],[15,44400,31987],{"className":44401,"style":12443},[129,133],[15,44403,44405],{"className":44404},[137],[15,44406,44408,44428],{"className":44407},[141,142],[15,44409,44411,44425],{"className":44410},[146],[15,44412,44414],{"className":44413,"style":1051},[150],[15,44415,44416,44419],{"style":34226},[15,44417],{"className":44418,"style":159},[158],[15,44420,44422],{"className":44421},[163,164,165,166],[15,44423,310],{"className":44424},[129,166],[15,44426,175],{"className":44427},[174],[15,44429,44431],{"className":44430},[146],[15,44432,44434],{"className":44433,"style":182},[150],[15,44435],{},[15,44437,44439,44442],{"className":44438},[129],[15,44440,31987],{"className":44441,"style":12443},[129,133],[15,44443,44445],{"className":44444},[137],[15,44446,44448,44468],{"className":44447},[141,142],[15,44449,44451,44465],{"className":44450},[146],[15,44452,44454],{"className":44453,"style":1051},[150],[15,44455,44456,44459],{"style":34226},[15,44457],{"className":44458,"style":159},[158],[15,44460,44462],{"className":44461},[163,164,165,166],[15,44463,318],{"className":44464},[129,166],[15,44466,175],{"className":44467},[174],[15,44469,44471],{"className":44470},[146],[15,44472,44474],{"className":44473,"style":182},[150],[15,44475],{},[15,44477,59],{"className":44478},[195],[15,44480,56],{"className":44481},[129,133],[15,44483],{"className":44484,"style":3277},[93],[15,44486,2682],{"className":44487},[2436],[15,44489],{"className":44490,"style":3277},[93],[15,44492,44494,44497,44500,44540,44580,44583,44586],{"className":44493},[120],[15,44495],{"className":44496,"style":125},[124],[15,44498,53],{"className":44499},[188],[15,44501,44503,44506],{"className":44502},[129],[15,44504,31987],{"className":44505,"style":12443},[129,133],[15,44507,44509],{"className":44508},[137],[15,44510,44512,44532],{"className":44511},[141,142],[15,44513,44515,44529],{"className":44514},[146],[15,44516,44518],{"className":44517,"style":1051},[150],[15,44519,44520,44523],{"style":34226},[15,44521],{"className":44522,"style":159},[158],[15,44524,44526],{"className":44525},[163,164,165,166],[15,44527,310],{"className":44528},[129,166],[15,44530,175],{"className":44531},[174],[15,44533,44535],{"className":44534},[146],[15,44536,44538],{"className":44537,"style":182},[150],[15,44539],{},[15,44541,44543,44546],{"className":44542},[129],[15,44544,8695],{"className":44545},[129,133],[15,44547,44549],{"className":44548},[137],[15,44550,44552,44572],{"className":44551},[141,142],[15,44553,44555,44569],{"className":44554},[146],[15,44556,44558],{"className":44557,"style":1051},[150],[15,44559,44560,44563],{"style":154},[15,44561],{"className":44562,"style":159},[158],[15,44564,44566],{"className":44565},[163,164,165,166],[15,44567,318],{"className":44568},[129,166],[15,44570,175],{"className":44571},[174],[15,44573,44575],{"className":44574},[146],[15,44576,44578],{"className":44577,"style":182},[150],[15,44579],{},[15,44581],{"className":44582,"style":3277},[93],[15,44584,2682],{"className":44585},[2436],[15,44587],{"className":44588,"style":3277},[93],[15,44590,44592,44595,44635],{"className":44591},[120],[15,44593],{"className":44594,"style":125},[124],[15,44596,44598,44601],{"className":44597},[129],[15,44599,8695],{"className":44600},[129,133],[15,44602,44604],{"className":44603},[137],[15,44605,44607,44627],{"className":44606},[141,142],[15,44608,44610,44624],{"className":44609},[146],[15,44611,44613],{"className":44612,"style":1051},[150],[15,44614,44615,44618],{"style":154},[15,44616],{"className":44617,"style":159},[158],[15,44619,44621],{"className":44620},[163,164,165,166],[15,44622,310],{"className":44623},[129,166],[15,44625,175],{"className":44626},[174],[15,44628,44630],{"className":44629},[146],[15,44631,44633],{"className":44632,"style":182},[150],[15,44634],{},[15,44636,59],{"className":44637},[195],[11,44639,44640,44641,44696],{},"它仍然保持 ",[15,44642,44644,44663],{"className":44643,"translate":19},[23],[15,44645,44647],{"className":44646},[27],[29,44648,44649],{"xmlns":31},[34,44650,44651,44661],{},[37,44652,44653,44655,44657,44659],{},[43,44654,13091],{},[43,44656,56],{},[50,44658,2682],{},[43,44660,8695],{},[109,44662,41416],{"encoding":111},[15,44664,44666,44687],{"className":44665,"ariaHidden":72},[116],[15,44667,44669,44672,44675,44678,44681,44684],{"className":44668},[120],[15,44670],{"className":44671,"style":39499},[124],[15,44673,13091],{"className":44674},[129,133],[15,44676,56],{"className":44677},[129,133],[15,44679],{"className":44680,"style":3277},[93],[15,44682,2682],{"className":44683},[2436],[15,44685],{"className":44686,"style":3277},[93],[15,44688,44690,44693],{"className":44689},[120],[15,44691],{"className":44692,"style":522},[124],[15,44694,8695],{"className":44695},[129,133]," 的形式。",[11,44698,44699],{},"由归纳法可得：",[15,44701,44703],{"className":44702,"translate":19},[18],[15,44704,44706,44746],{"className":44705,"translate":19},[23],[15,44707,44709],{"className":44708},[27],[29,44710,44711],{"xmlns":31,"display":32},[34,44712,44713,44743],{},[37,44714,44715,44721,44723,44725,44727,44733,44735,44737,44739,44741],{},[40,44716,44717,44719],{},[43,44718,34133],{},[43,44720,39314],{},[50,44722,42927],{},[50,44724,31657],{},[50,44726,42927],{},[40,44728,44729,44731],{},[43,44730,34133],{},[308,44732,541],{},[50,44734,62],{},[43,44736,31987],{},[43,44738,56],{},[50,44740,2682],{},[43,44742,8695],{},[109,44744,44745],{"encoding":111},"l_n \\circ \\cdots \\circ l_0 = Wx + b",[15,44747,44749,44804,44822,44877,44898],{"className":44748,"ariaHidden":72},[116],[15,44750,44752,44755,44795,44798,44801],{"className":44751},[120],[15,44753],{"className":44754,"style":2183},[124],[15,44756,44758,44761],{"className":44757},[129],[15,44759,34133],{"className":44760,"style":34191},[129,133],[15,44762,44764],{"className":44763},[137],[15,44765,44767,44787],{"className":44766},[141,142],[15,44768,44770,44784],{"className":44769},[146],[15,44771,44773],{"className":44772,"style":1163},[150],[15,44774,44775,44778],{"style":43389},[15,44776],{"className":44777,"style":159},[158],[15,44779,44781],{"className":44780},[163,164,165,166],[15,44782,39314],{"className":44783},[129,133,166],[15,44785,175],{"className":44786},[174],[15,44788,44790],{"className":44789},[146],[15,44791,44793],{"className":44792,"style":182},[150],[15,44794],{},[15,44796],{"className":44797,"style":3277},[93],[15,44799,42927],{"className":44800},[2436],[15,44802],{"className":44803,"style":3277},[93],[15,44805,44807,44810,44813,44816,44819],{"className":44806},[120],[15,44808],{"className":44809,"style":11094},[124],[15,44811,31657],{"className":44812},[449],[15,44814],{"className":44815,"style":3277},[93],[15,44817,42927],{"className":44818},[2436],[15,44820],{"className":44821,"style":3277},[93],[15,44823,44825,44828,44868,44871,44874],{"className":44824},[120],[15,44826],{"className":44827,"style":2183},[124],[15,44829,44831,44834],{"className":44830},[129],[15,44832,34133],{"className":44833,"style":34191},[129,133],[15,44835,44837],{"className":44836},[137],[15,44838,44840,44860],{"className":44839},[141,142],[15,44841,44843,44857],{"className":44842},[146],[15,44844,44846],{"className":44845,"style":1051},[150],[15,44847,44848,44851],{"style":43389},[15,44849],{"className":44850,"style":159},[158],[15,44852,44854],{"className":44853},[163,164,165,166],[15,44855,541],{"className":44856},[129,166],[15,44858,175],{"className":44859},[174],[15,44861,44863],{"className":44862},[146],[15,44864,44866],{"className":44865,"style":182},[150],[15,44867],{},[15,44869],{"className":44870,"style":199},[93],[15,44872,62],{"className":44873},[203],[15,44875],{"className":44876,"style":199},[93],[15,44878,44880,44883,44886,44889,44892,44895],{"className":44879},[120],[15,44881],{"className":44882,"style":39499},[124],[15,44884,31987],{"className":44885,"style":12443},[129,133],[15,44887,56],{"className":44888},[129,133],[15,44890],{"className":44891,"style":3277},[93],[15,44893,2682],{"className":44894},[2436],[15,44896],{"className":44897,"style":3277},[93],[15,44899,44901,44904],{"className":44900},[120],[15,44902],{"className":44903,"style":522},[124],[15,44905,8695],{"className":44906},[129,133],[11,44908,44909],{},"因此，网络的深度完全失去了意义。",[11,44911,44912],{},"为了引入非线性，我们可以在每一层上添加激活函数：",[15,44914,44916],{"className":44915,"translate":19},[18],[15,44917,44919,45021],{"className":44918,"translate":19},[23],[15,44920,44922],{"className":44921},[27],[29,44923,44924],{"xmlns":31,"display":32},[34,44925,44926,45018],{},[37,44927,44928,44930,44932,44934,44936,44938,44940,44946,44948,44954,44956,44958,44960,44972,44974,44986,44988,44990,44992,44994,44996,45002,45004,45010,45012,45014,45016],{},[43,44929,36556],{},[50,44931,53],{"stretchy":52},[43,44933,56],{},[50,44935,59],{"stretchy":52},[50,44937,62],{},[50,44939,53],{"stretchy":52},[40,44941,44942,44944],{},[43,44943,27283],{},[43,44945,39314],{},[50,44947,42927],{},[40,44949,44950,44952],{},[43,44951,34133],{},[43,44953,39314],{},[50,44955,59],{"stretchy":52},[50,44957,42927],{},[50,44959,53],{"stretchy":52},[40,44961,44962,44964],{},[43,44963,27283],{},[37,44965,44966,44968,44970],{},[43,44967,39314],{},[50,44969,315],{},[308,44971,318],{},[50,44973,42927],{},[40,44975,44976,44978],{},[43,44977,34133],{},[37,44979,44980,44982,44984],{},[43,44981,39314],{},[50,44983,315],{},[308,44985,318],{},[50,44987,59],{"stretchy":52},[50,44989,42927],{},[50,44991,31657],{},[50,44993,42927],{},[50,44995,53],{"stretchy":52},[40,44997,44998,45000],{},[43,44999,27283],{},[308,45001,541],{},[50,45003,42927],{},[40,45005,45006,45008],{},[43,45007,34133],{},[308,45009,541],{},[50,45011,59],{"stretchy":52},[50,45013,53],{"stretchy":52},[43,45015,56],{},[50,45017,59],{"stretchy":52},[109,45019,45020],{"encoding":111},"f(x) = (\\sigma_n \\circ l_n) \\circ (\\sigma_{n-1} \\circ l_{n-1}) \\circ \\cdots \\circ (\\sigma_0 \\circ l_0)(x)",[15,45022,45024,45051,45109,45167,45234,45301,45319,45377],{"className":45023,"ariaHidden":72},[116],[15,45025,45027,45030,45033,45036,45039,45042,45045,45048],{"className":45026},[120],[15,45028],{"className":45029,"style":125},[124],[15,45031,36556],{"className":45032,"style":36571},[129,133],[15,45034,53],{"className":45035},[188],[15,45037,56],{"className":45038},[129,133],[15,45040,59],{"className":45041},[195],[15,45043],{"className":45044,"style":199},[93],[15,45046,62],{"className":45047},[203],[15,45049],{"className":45050,"style":199},[93],[15,45052,45054,45057,45060,45100,45103,45106],{"className":45053},[120],[15,45055],{"className":45056,"style":125},[124],[15,45058,53],{"className":45059},[188],[15,45061,45063,45066],{"className":45062},[129],[15,45064,27283],{"className":45065,"style":887},[129,133],[15,45067,45069],{"className":45068},[137],[15,45070,45072,45092],{"className":45071},[141,142],[15,45073,45075,45089],{"className":45074},[146],[15,45076,45078],{"className":45077,"style":1163},[150],[15,45079,45080,45083],{"style":1100},[15,45081],{"className":45082,"style":159},[158],[15,45084,45086],{"className":45085},[163,164,165,166],[15,45087,39314],{"className":45088},[129,133,166],[15,45090,175],{"className":45091},[174],[15,45093,45095],{"className":45094},[146],[15,45096,45098],{"className":45097,"style":182},[150],[15,45099],{},[15,45101],{"className":45102,"style":3277},[93],[15,45104,42927],{"className":45105},[2436],[15,45107],{"className":45108,"style":3277},[93],[15,45110,45112,45115,45155,45158,45161,45164],{"className":45111},[120],[15,45113],{"className":45114,"style":125},[124],[15,45116,45118,45121],{"className":45117},[129],[15,45119,34133],{"className":45120,"style":34191},[129,133],[15,45122,45124],{"className":45123},[137],[15,45125,45127,45147],{"className":45126},[141,142],[15,45128,45130,45144],{"className":45129},[146],[15,45131,45133],{"className":45132,"style":1163},[150],[15,45134,45135,45138],{"style":43389},[15,45136],{"className":45137,"style":159},[158],[15,45139,45141],{"className":45140},[163,164,165,166],[15,45142,39314],{"className":45143},[129,133,166],[15,45145,175],{"className":45146},[174],[15,45148,45150],{"className":45149},[146],[15,45151,45153],{"className":45152,"style":182},[150],[15,45154],{},[15,45156,59],{"className":45157},[195],[15,45159],{"className":45160,"style":3277},[93],[15,45162,42927],{"className":45163},[2436],[15,45165],{"className":45166,"style":3277},[93],[15,45168,45170,45173,45176,45225,45228,45231],{"className":45169},[120],[15,45171],{"className":45172,"style":125},[124],[15,45174,53],{"className":45175},[188],[15,45177,45179,45182],{"className":45178},[129],[15,45180,27283],{"className":45181,"style":887},[129,133],[15,45183,45185],{"className":45184},[137],[15,45186,45188,45217],{"className":45187},[141,142],[15,45189,45191,45214],{"className":45190},[146],[15,45192,45194],{"className":45193,"style":1051},[150],[15,45195,45196,45199],{"style":1100},[15,45197],{"className":45198,"style":159},[158],[15,45200,45202],{"className":45201},[163,164,165,166],[15,45203,45205,45208,45211],{"className":45204},[129,166],[15,45206,39314],{"className":45207},[129,133,166],[15,45209,315],{"className":45210},[2436,166],[15,45212,318],{"className":45213},[129,166],[15,45215,175],{"className":45216},[174],[15,45218,45220],{"className":45219},[146],[15,45221,45223],{"className":45222,"style":43067},[150],[15,45224],{},[15,45226],{"className":45227,"style":3277},[93],[15,45229,42927],{"className":45230},[2436],[15,45232],{"className":45233,"style":3277},[93],[15,45235,45237,45240,45289,45292,45295,45298],{"className":45236},[120],[15,45238],{"className":45239,"style":125},[124],[15,45241,45243,45246],{"className":45242},[129],[15,45244,34133],{"className":45245,"style":34191},[129,133],[15,45247,45249],{"className":45248},[137],[15,45250,45252,45281],{"className":45251},[141,142],[15,45253,45255,45278],{"className":45254},[146],[15,45256,45258],{"className":45257,"style":1051},[150],[15,45259,45260,45263],{"style":43389},[15,45261],{"className":45262,"style":159},[158],[15,45264,45266],{"className":45265},[163,164,165,166],[15,45267,45269,45272,45275],{"className":45268},[129,166],[15,45270,39314],{"className":45271},[129,133,166],[15,45273,315],{"className":45274},[2436,166],[15,45276,318],{"className":45277},[129,166],[15,45279,175],{"className":45280},[174],[15,45282,45284],{"className":45283},[146],[15,45285,45287],{"className":45286,"style":43067},[150],[15,45288],{},[15,45290,59],{"className":45291},[195],[15,45293],{"className":45294,"style":3277},[93],[15,45296,42927],{"className":45297},[2436],[15,45299],{"className":45300,"style":3277},[93],[15,45302,45304,45307,45310,45313,45316],{"className":45303},[120],[15,45305],{"className":45306,"style":11094},[124],[15,45308,31657],{"className":45309},[449],[15,45311],{"className":45312,"style":3277},[93],[15,45314,42927],{"className":45315},[2436],[15,45317],{"className":45318,"style":3277},[93],[15,45320,45322,45325,45328,45368,45371,45374],{"className":45321},[120],[15,45323],{"className":45324,"style":125},[124],[15,45326,53],{"className":45327},[188],[15,45329,45331,45334],{"className":45330},[129],[15,45332,27283],{"className":45333,"style":887},[129,133],[15,45335,45337],{"className":45336},[137],[15,45338,45340,45360],{"className":45339},[141,142],[15,45341,45343,45357],{"className":45342},[146],[15,45344,45346],{"className":45345,"style":1051},[150],[15,45347,45348,45351],{"style":1100},[15,45349],{"className":45350,"style":159},[158],[15,45352,45354],{"className":45353},[163,164,165,166],[15,45355,541],{"className":45356},[129,166],[15,45358,175],{"className":45359},[174],[15,45361,45363],{"className":45362},[146],[15,45364,45366],{"className":45365,"style":182},[150],[15,45367],{},[15,45369],{"className":45370,"style":3277},[93],[15,45372,42927],{"className":45373},[2436],[15,45375],{"className":45376,"style":3277},[93],[15,45378,45380,45383,45423,45426,45429,45432],{"className":45379},[120],[15,45381],{"className":45382,"style":125},[124],[15,45384,45386,45389],{"className":45385},[129],[15,45387,34133],{"className":45388,"style":34191},[129,133],[15,45390,45392],{"className":45391},[137],[15,45393,45395,45415],{"className":45394},[141,142],[15,45396,45398,45412],{"className":45397},[146],[15,45399,45401],{"className":45400,"style":1051},[150],[15,45402,45403,45406],{"style":43389},[15,45404],{"className":45405,"style":159},[158],[15,45407,45409],{"className":45408},[163,164,165,166],[15,45410,541],{"className":45411},[129,166],[15,45413,175],{"className":45414},[174],[15,45416,45418],{"className":45417},[146],[15,45419,45421],{"className":45420,"style":182},[150],[15,45422],{},[15,45424,59],{"className":45425},[195],[15,45427,53],{"className":45428},[188],[15,45430,56],{"className":45431},[129,133],[15,45433,59],{"className":45434},[195],[11,45436,45437],{},"其中",[15,45439,45441],{"className":45440,"translate":19},[18],[15,45442,45444,45486],{"className":45443,"translate":19},[23],[15,45445,45447],{"className":45446},[27],[29,45448,45449],{"xmlns":31,"display":32},[34,45450,45451,45483],{},[37,45452,45453,45459,45461,45463,45465,45467,45473,45475,45477],{},[40,45454,45455,45457],{},[43,45456,34133],{},[43,45458,1459],{},[50,45460,53],{"stretchy":52},[43,45462,56],{},[50,45464,59],{"stretchy":52},[50,45466,62],{},[40,45468,45469,45471],{},[43,45470,31987],{},[43,45472,1459],{},[43,45474,56],{},[50,45476,2682],{},[40,45478,45479,45481],{},[43,45480,8695],{},[43,45482,1459],{},[109,45484,45485],{"encoding":111},"l_i(x)=W_i x+b_i",[15,45487,45489,45553,45611],{"className":45488,"ariaHidden":72},[116],[15,45490,45492,45495,45535,45538,45541,45544,45547,45550],{"className":45491},[120],[15,45493],{"className":45494,"style":125},[124],[15,45496,45498,45501],{"className":45497},[129],[15,45499,34133],{"className":45500,"style":34191},[129,133],[15,45502,45504],{"className":45503},[137],[15,45505,45507,45527],{"className":45506},[141,142],[15,45508,45510,45524],{"className":45509},[146],[15,45511,45513],{"className":45512,"style":1981},[150],[15,45514,45515,45518],{"style":43389},[15,45516],{"className":45517,"style":159},[158],[15,45519,45521],{"className":45520},[163,164,165,166],[15,45522,1459],{"className":45523},[129,133,166],[15,45525,175],{"className":45526},[174],[15,45528,45530],{"className":45529},[146],[15,45531,45533],{"className":45532,"style":182},[150],[15,45534],{},[15,45536,53],{"className":45537},[188],[15,45539,56],{"className":45540},[129,133],[15,45542,59],{"className":45543},[195],[15,45545],{"className":45546,"style":199},[93],[15,45548,62],{"className":45549},[203],[15,45551],{"className":45552,"style":199},[93],[15,45554,45556,45559,45599,45602,45605,45608],{"className":45555},[120],[15,45557],{"className":45558,"style":20220},[124],[15,45560,45562,45565],{"className":45561},[129],[15,45563,31987],{"className":45564,"style":12443},[129,133],[15,45566,45568],{"className":45567},[137],[15,45569,45571,45591],{"className":45570},[141,142],[15,45572,45574,45588],{"className":45573},[146],[15,45575,45577],{"className":45576,"style":1981},[150],[15,45578,45579,45582],{"style":34226},[15,45580],{"className":45581,"style":159},[158],[15,45583,45585],{"className":45584},[163,164,165,166],[15,45586,1459],{"className":45587},[129,133,166],[15,45589,175],{"className":45590},[174],[15,45592,45594],{"className":45593},[146],[15,45595,45597],{"className":45596,"style":182},[150],[15,45598],{},[15,45600,56],{"className":45601},[129,133],[15,45603],{"className":45604,"style":3277},[93],[15,45606,2682],{"className":45607},[2436],[15,45609],{"className":45610,"style":3277},[93],[15,45612,45614,45617],{"className":45613},[120],[15,45615],{"className":45616,"style":2183},[124],[15,45618,45620,45623],{"className":45619},[129],[15,45621,8695],{"className":45622},[129,133],[15,45624,45626],{"className":45625},[137],[15,45627,45629,45649],{"className":45628},[141,142],[15,45630,45632,45646],{"className":45631},[146],[15,45633,45635],{"className":45634,"style":1981},[150],[15,45636,45637,45640],{"style":154},[15,45638],{"className":45639,"style":159},[158],[15,45641,45643],{"className":45642},[163,164,165,166],[15,45644,1459],{"className":45645},[129,133,166],[15,45647,175],{"className":45648},[174],[15,45650,45652],{"className":45651},[146],[15,45653,45655],{"className":45654,"style":182},[150],[15,45656],{},[11,45658,45659],{},"展开后即为：",[15,45661,45663],{"className":45662,"translate":19},[18],[15,45664,45666,45785],{"className":45665,"translate":19},[23],[15,45667,45669],{"className":45668},[27],[29,45670,45671],{"xmlns":31,"display":32},[34,45672,45673,45782],{},[37,45674,45675,45677,45679,45681,45683,45685,45691,45694,45700,45712,45714,45726,45728,45734,45736,45742,45744,45746,45752,45754,45756,45758,45770,45772,45774,45780],{},[43,45676,36556],{},[50,45678,53],{"stretchy":52},[43,45680,56],{},[50,45682,59],{"stretchy":52},[50,45684,62],{},[40,45686,45687,45689],{},[43,45688,27283],{},[43,45690,39314],{},[50,45692,53],{"fence":72,"stretchy":72,"minsize":45693,"maxsize":45693},"1.8em",[40,45695,45696,45698],{},[43,45697,31987],{},[43,45699,39314],{},[40,45701,45702,45704],{},[43,45703,27283],{},[37,45705,45706,45708,45710],{},[43,45707,39314],{},[50,45709,315],{},[308,45711,318],{},[50,45713,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,45715,45716,45718],{},[43,45717,31987],{},[37,45719,45720,45722,45724],{},[43,45721,39314],{},[50,45723,315],{},[308,45725,318],{},[50,45727,31657],{},[40,45729,45730,45732],{},[43,45731,27283],{},[308,45733,541],{},[50,45735,53],{"stretchy":52},[40,45737,45738,45740],{},[43,45739,31987],{},[308,45741,541],{},[43,45743,56],{},[50,45745,2682],{},[40,45747,45748,45750],{},[43,45749,8695],{},[308,45751,541],{},[50,45753,59],{"stretchy":52},[50,45755,31657],{},[50,45757,2682],{},[40,45759,45760,45762],{},[43,45761,8695],{},[37,45763,45764,45766,45768],{},[43,45765,39314],{},[50,45767,315],{},[308,45769,318],{},[50,45771,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,45773,2682],{},[40,45775,45776,45778],{},[43,45777,8695],{},[43,45779,39314],{},[50,45781,59],{"fence":72,"stretchy":72,"minsize":45693,"maxsize":45693},[109,45783,45784],{"encoding":111},"f(x) = \\sigma_n \\Bigl( W_n \\sigma_{n-1} \\bigl( W_{n-1} \\cdots \\sigma_0(W_0x+b_0) \\cdots +b_{n-1} \\bigr) + b_n \\Bigr)",[15,45786,45788,45815,46116,46180,46250],{"className":45787,"ariaHidden":72},[116],[15,45789,45791,45794,45797,45800,45803,45806,45809,45812],{"className":45790},[120],[15,45792],{"className":45793,"style":125},[124],[15,45795,36556],{"className":45796,"style":36571},[129,133],[15,45798,53],{"className":45799},[188],[15,45801,56],{"className":45802},[129,133],[15,45804,59],{"className":45805},[195],[15,45807],{"className":45808,"style":199},[93],[15,45810,62],{"className":45811},[203],[15,45813],{"className":45814,"style":199},[93],[15,45816,45818,45822,45862,45868,45908,45957,45963,46012,46015,46018,46021,46061,46064,46104,46107,46110,46113],{"className":45817},[120],[15,45819],{"className":45820,"style":45821},[124],"height:1.8em;vertical-align:-0.65em;",[15,45823,45825,45828],{"className":45824},[129],[15,45826,27283],{"className":45827,"style":887},[129,133],[15,45829,45831],{"className":45830},[137],[15,45832,45834,45854],{"className":45833},[141,142],[15,45835,45837,45851],{"className":45836},[146],[15,45838,45840],{"className":45839,"style":1163},[150],[15,45841,45842,45845],{"style":1100},[15,45843],{"className":45844,"style":159},[158],[15,45846,45848],{"className":45847},[163,164,165,166],[15,45849,39314],{"className":45850},[129,133,166],[15,45852,175],{"className":45853},[174],[15,45855,45857],{"className":45856},[146],[15,45858,45860],{"className":45859,"style":182},[150],[15,45861],{},[15,45863,45865],{"className":45864},[188],[15,45866,53],{"className":45867},[228,24450],[15,45869,45871,45874],{"className":45870},[129],[15,45872,31987],{"className":45873,"style":12443},[129,133],[15,45875,45877],{"className":45876},[137],[15,45878,45880,45900],{"className":45879},[141,142],[15,45881,45883,45897],{"className":45882},[146],[15,45884,45886],{"className":45885,"style":1163},[150],[15,45887,45888,45891],{"style":34226},[15,45889],{"className":45890,"style":159},[158],[15,45892,45894],{"className":45893},[163,164,165,166],[15,45895,39314],{"className":45896},[129,133,166],[15,45898,175],{"className":45899},[174],[15,45901,45903],{"className":45902},[146],[15,45904,45906],{"className":45905,"style":182},[150],[15,45907],{},[15,45909,45911,45914],{"className":45910},[129],[15,45912,27283],{"className":45913,"style":887},[129,133],[15,45915,45917],{"className":45916},[137],[15,45918,45920,45949],{"className":45919},[141,142],[15,45921,45923,45946],{"className":45922},[146],[15,45924,45926],{"className":45925,"style":1051},[150],[15,45927,45928,45931],{"style":1100},[15,45929],{"className":45930,"style":159},[158],[15,45932,45934],{"className":45933},[163,164,165,166],[15,45935,45937,45940,45943],{"className":45936},[129,166],[15,45938,39314],{"className":45939},[129,133,166],[15,45941,315],{"className":45942},[2436,166],[15,45944,318],{"className":45945},[129,166],[15,45947,175],{"className":45948},[174],[15,45950,45952],{"className":45951},[146],[15,45953,45955],{"className":45954,"style":43067},[150],[15,45956],{},[15,45958,45960],{"className":45959},[188],[15,45961,53],{"className":45962},[228,229],[15,45964,45966,45969],{"className":45965},[129],[15,45967,31987],{"className":45968,"style":12443},[129,133],[15,45970,45972],{"className":45971},[137],[15,45973,45975,46004],{"className":45974},[141,142],[15,45976,45978,46001],{"className":45977},[146],[15,45979,45981],{"className":45980,"style":1051},[150],[15,45982,45983,45986],{"style":34226},[15,45984],{"className":45985,"style":159},[158],[15,45987,45989],{"className":45988},[163,164,165,166],[15,45990,45992,45995,45998],{"className":45991},[129,166],[15,45993,39314],{"className":45994},[129,133,166],[15,45996,315],{"className":45997},[2436,166],[15,45999,318],{"className":46000},[129,166],[15,46002,175],{"className":46003},[174],[15,46005,46007],{"className":46006},[146],[15,46008,46010],{"className":46009,"style":43067},[150],[15,46011],{},[15,46013],{"className":46014,"style":262},[93],[15,46016,31657],{"className":46017},[449],[15,46019],{"className":46020,"style":262},[93],[15,46022,46024,46027],{"className":46023},[129],[15,46025,27283],{"className":46026,"style":887},[129,133],[15,46028,46030],{"className":46029},[137],[15,46031,46033,46053],{"className":46032},[141,142],[15,46034,46036,46050],{"className":46035},[146],[15,46037,46039],{"className":46038,"style":1051},[150],[15,46040,46041,46044],{"style":1100},[15,46042],{"className":46043,"style":159},[158],[15,46045,46047],{"className":46046},[163,164,165,166],[15,46048,541],{"className":46049},[129,166],[15,46051,175],{"className":46052},[174],[15,46054,46056],{"className":46055},[146],[15,46057,46059],{"className":46058,"style":182},[150],[15,46060],{},[15,46062,53],{"className":46063},[188],[15,46065,46067,46070],{"className":46066},[129],[15,46068,31987],{"className":46069,"style":12443},[129,133],[15,46071,46073],{"className":46072},[137],[15,46074,46076,46096],{"className":46075},[141,142],[15,46077,46079,46093],{"className":46078},[146],[15,46080,46082],{"className":46081,"style":1051},[150],[15,46083,46084,46087],{"style":34226},[15,46085],{"className":46086,"style":159},[158],[15,46088,46090],{"className":46089},[163,164,165,166],[15,46091,541],{"className":46092},[129,166],[15,46094,175],{"className":46095},[174],[15,46097,46099],{"className":46098},[146],[15,46100,46102],{"className":46101,"style":182},[150],[15,46103],{},[15,46105,56],{"className":46106},[129,133],[15,46108],{"className":46109,"style":3277},[93],[15,46111,2682],{"className":46112},[2436],[15,46114],{"className":46115,"style":3277},[93],[15,46117,46119,46122,46162,46165,46168,46171,46174,46177],{"className":46118},[120],[15,46120],{"className":46121,"style":125},[124],[15,46123,46125,46128],{"className":46124},[129],[15,46126,8695],{"className":46127},[129,133],[15,46129,46131],{"className":46130},[137],[15,46132,46134,46154],{"className":46133},[141,142],[15,46135,46137,46151],{"className":46136},[146],[15,46138,46140],{"className":46139,"style":1051},[150],[15,46141,46142,46145],{"style":154},[15,46143],{"className":46144,"style":159},[158],[15,46146,46148],{"className":46147},[163,164,165,166],[15,46149,541],{"className":46150},[129,166],[15,46152,175],{"className":46153},[174],[15,46155,46157],{"className":46156},[146],[15,46158,46160],{"className":46159,"style":182},[150],[15,46161],{},[15,46163,59],{"className":46164},[195],[15,46166],{"className":46167,"style":262},[93],[15,46169,31657],{"className":46170},[449],[15,46172],{"className":46173,"style":3277},[93],[15,46175,2682],{"className":46176},[2436],[15,46178],{"className":46179,"style":3277},[93],[15,46181,46183,46186,46235,46241,46244,46247],{"className":46182},[120],[15,46184],{"className":46185,"style":213},[124],[15,46187,46189,46192],{"className":46188},[129],[15,46190,8695],{"className":46191},[129,133],[15,46193,46195],{"className":46194},[137],[15,46196,46198,46227],{"className":46197},[141,142],[15,46199,46201,46224],{"className":46200},[146],[15,46202,46204],{"className":46203,"style":1051},[150],[15,46205,46206,46209],{"style":154},[15,46207],{"className":46208,"style":159},[158],[15,46210,46212],{"className":46211},[163,164,165,166],[15,46213,46215,46218,46221],{"className":46214},[129,166],[15,46216,39314],{"className":46217},[129,133,166],[15,46219,315],{"className":46220},[2436,166],[15,46222,318],{"className":46223},[129,166],[15,46225,175],{"className":46226},[174],[15,46228,46230],{"className":46229},[146],[15,46231,46233],{"className":46232,"style":43067},[150],[15,46234],{},[15,46236,46238],{"className":46237},[195],[15,46239,59],{"className":46240},[228,229],[15,46242],{"className":46243,"style":3277},[93],[15,46245,2682],{"className":46246},[2436],[15,46248],{"className":46249,"style":3277},[93],[15,46251,46253,46256,46296],{"className":46252},[120],[15,46254],{"className":46255,"style":45821},[124],[15,46257,46259,46262],{"className":46258},[129],[15,46260,8695],{"className":46261},[129,133],[15,46263,46265],{"className":46264},[137],[15,46266,46268,46288],{"className":46267},[141,142],[15,46269,46271,46285],{"className":46270},[146],[15,46272,46274],{"className":46273,"style":1163},[150],[15,46275,46276,46279],{"style":154},[15,46277],{"className":46278,"style":159},[158],[15,46280,46282],{"className":46281},[163,164,165,166],[15,46283,39314],{"className":46284},[129,133,166],[15,46286,175],{"className":46287},[174],[15,46289,46291],{"className":46290},[146],[15,46292,46294],{"className":46293,"style":182},[150],[15,46295],{},[15,46297,46299],{"className":46298},[195],[15,46300,59],{"className":46301},[228,24450],[11,46303,46304],{},"这就是神经网络的标准形式。",[11,46306,46307],{},"在加入激活函数后：",[15,46309,46311],{"className":46310,"translate":19},[18],[15,46312,46314,46356],{"className":46313,"translate":19},[23],[15,46315,46317],{"className":46316},[27],[29,46318,46319],{"xmlns":31,"display":32},[34,46320,46321,46353],{},[37,46322,46323,46325,46327,46329,46335,46337,46339,46341,46343,46345,46351],{},[50,46324,53],{"stretchy":52},[43,46326,27283],{},[50,46328,42927],{},[40,46330,46331,46333],{},[43,46332,34133],{},[308,46334,310],{},[50,46336,59],{"stretchy":52},[50,46338,42927],{},[50,46340,53],{"stretchy":52},[43,46342,27283],{},[50,46344,42927],{},[40,46346,46347,46349],{},[43,46348,34133],{},[308,46350,318],{},[50,46352,59],{"stretchy":52},[109,46354,46355],{"encoding":111},"(\\sigma \\circ l_2) \\circ (\\sigma \\circ l_1)",[15,46357,46359,46380,46438,46459],{"className":46358,"ariaHidden":72},[116],[15,46360,46362,46365,46368,46371,46374,46377],{"className":46361},[120],[15,46363],{"className":46364,"style":125},[124],[15,46366,53],{"className":46367},[188],[15,46369,27283],{"className":46370,"style":887},[129,133],[15,46372],{"className":46373,"style":3277},[93],[15,46375,42927],{"className":46376},[2436],[15,46378],{"className":46379,"style":3277},[93],[15,46381,46383,46386,46426,46429,46432,46435],{"className":46382},[120],[15,46384],{"className":46385,"style":125},[124],[15,46387,46389,46392],{"className":46388},[129],[15,46390,34133],{"className":46391,"style":34191},[129,133],[15,46393,46395],{"className":46394},[137],[15,46396,46398,46418],{"className":46397},[141,142],[15,46399,46401,46415],{"className":46400},[146],[15,46402,46404],{"className":46403,"style":1051},[150],[15,46405,46406,46409],{"style":43389},[15,46407],{"className":46408,"style":159},[158],[15,46410,46412],{"className":46411},[163,164,165,166],[15,46413,310],{"className":46414},[129,166],[15,46416,175],{"className":46417},[174],[15,46419,46421],{"className":46420},[146],[15,46422,46424],{"className":46423,"style":182},[150],[15,46425],{},[15,46427,59],{"className":46428},[195],[15,46430],{"className":46431,"style":3277},[93],[15,46433,42927],{"className":46434},[2436],[15,46436],{"className":46437,"style":3277},[93],[15,46439,46441,46444,46447,46450,46453,46456],{"className":46440},[120],[15,46442],{"className":46443,"style":125},[124],[15,46445,53],{"className":46446},[188],[15,46448,27283],{"className":46449,"style":887},[129,133],[15,46451],{"className":46452,"style":3277},[93],[15,46454,42927],{"className":46455},[2436],[15,46457],{"className":46458,"style":3277},[93],[15,46460,46462,46465,46505],{"className":46461},[120],[15,46463],{"className":46464,"style":125},[124],[15,46466,46468,46471],{"className":46467},[129],[15,46469,34133],{"className":46470,"style":34191},[129,133],[15,46472,46474],{"className":46473},[137],[15,46475,46477,46497],{"className":46476},[141,142],[15,46478,46480,46494],{"className":46479},[146],[15,46481,46483],{"className":46482,"style":1051},[150],[15,46484,46485,46488],{"style":43389},[15,46486],{"className":46487,"style":159},[158],[15,46489,46491],{"className":46490},[163,164,165,166],[15,46492,318],{"className":46493},[129,166],[15,46495,175],{"className":46496},[174],[15,46498,46500],{"className":46499},[146],[15,46501,46503],{"className":46502,"style":182},[150],[15,46504],{},[15,46506,59],{"className":46507},[195],[11,46509,46510],{},"无法化简为",[15,46512,46514],{"className":46513,"translate":19},[18],[15,46515,46517,46537],{"className":46516,"translate":19},[23],[15,46518,46520],{"className":46519},[27],[29,46521,46522],{"xmlns":31,"display":32},[34,46523,46524,46534],{},[37,46525,46526,46528,46530,46532],{},[43,46527,31987],{},[43,46529,56],{},[50,46531,2682],{},[43,46533,8695],{},[109,46535,46536],{"encoding":111},"Wx+b",[15,46538,46540,46561],{"className":46539,"ariaHidden":72},[116],[15,46541,46543,46546,46549,46552,46555,46558],{"className":46542},[120],[15,46544],{"className":46545,"style":39499},[124],[15,46547,31987],{"className":46548,"style":12443},[129,133],[15,46550,56],{"className":46551},[129,133],[15,46553],{"className":46554,"style":3277},[93],[15,46556,2682],{"className":46557},[2436],[15,46559],{"className":46560,"style":3277},[93],[15,46562,46564,46567],{"className":46563},[120],[15,46565],{"className":46566,"style":522},[124],[15,46568,8695],{"className":46569},[129,133],[11,46571,46572],{},"因为一般来说：",[15,46574,46576],{"className":46575,"translate":19},[18],[15,46577,46579,46629],{"className":46578,"translate":19},[23],[15,46580,46582],{"className":46581},[27],[29,46583,46584],{"xmlns":31,"display":32},[34,46585,46586,46626],{},[37,46587,46588,46590,46592,46594,46596,46598,46600,46602,46604,46610,46612,46614,46616,46618,46620],{},[43,46589,27283],{},[50,46591,53],{"stretchy":52},[43,46593,13091],{},[43,46595,56],{},[50,46597,2682],{},[43,46599,8695],{},[50,46601,59],{"stretchy":52},[50,46603,1418],{"mathvariant":65},[305,46605,46606,46608],{},[43,46607,13091],{},[50,46609,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[43,46611,27283],{},[50,46613,53],{"stretchy":52},[43,46615,56],{},[50,46617,59],{"stretchy":52},[50,46619,2682],{},[305,46621,46622,46624],{},[43,46623,8695],{},[50,46625,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[109,46627,46628],{"encoding":111},"\\sigma(Ax+b) \\neq A'\\sigma(x)+b'",[15,46630,46632,46659,46713,46773],{"className":46631,"ariaHidden":72},[116],[15,46633,46635,46638,46641,46644,46647,46650,46653,46656],{"className":46634},[120],[15,46636],{"className":46637,"style":125},[124],[15,46639,27283],{"className":46640,"style":887},[129,133],[15,46642,53],{"className":46643},[188],[15,46645,13091],{"className":46646},[129,133],[15,46648,56],{"className":46649},[129,133],[15,46651],{"className":46652,"style":3277},[93],[15,46654,2682],{"className":46655},[2436],[15,46657],{"className":46658,"style":3277},[93],[15,46660,46662,46665,46668,46671,46674,46710],{"className":46661},[120],[15,46663],{"className":46664,"style":125},[124],[15,46666,8695],{"className":46667},[129,133],[15,46669,59],{"className":46670},[195],[15,46672],{"className":46673,"style":199},[93],[15,46675,46677,46704,46707],{"className":46676},[203],[15,46678,46680],{"className":46679},[203],[15,46681,46683],{"className":46682},[129,1670],[15,46684,46686],{"className":46685},[1674],[15,46687,46689,46692,46701],{"className":46688},[1678],[15,46690],{"className":46691,"style":1682},[124],[15,46693,46695],{"className":46694},[1686],[15,46696,46698],{"className":46697},[129],[15,46699,1693],{"className":46700},[203],[15,46702],{"className":46703},[1697],[15,46705],{"className":46706},[93,1701],[15,46708,62],{"className":46709},[203],[15,46711],{"className":46712,"style":199},[93],[15,46714,46716,46720,46752,46755,46758,46761,46764,46767,46770],{"className":46715},[120],[15,46717],{"className":46718,"style":46719},[124],"height:1.0519em;vertical-align:-0.25em;",[15,46721,46723,46726],{"className":46722},[129],[15,46724,13091],{"className":46725},[129,133],[15,46727,46729],{"className":46728},[137],[15,46730,46732],{"className":46731},[141],[15,46733,46735],{"className":46734},[146],[15,46736,46738],{"className":46737,"style":28126},[150],[15,46739,46740,46743],{"style":1272},[15,46741],{"className":46742,"style":159},[158],[15,46744,46746],{"className":46745},[163,164,165,166],[15,46747,46749],{"className":46748},[129,166],[15,46750,21499],{"className":46751},[129,166],[15,46753,27283],{"className":46754,"style":887},[129,133],[15,46756,53],{"className":46757},[188],[15,46759,56],{"className":46760},[129,133],[15,46762,59],{"className":46763},[195],[15,46765],{"className":46766,"style":3277},[93],[15,46768,2682],{"className":46769},[2436],[15,46771],{"className":46772,"style":3277},[93],[15,46774,46776,46779],{"className":46775},[120],[15,46777],{"className":46778,"style":28126},[124],[15,46780,46782,46785],{"className":46781},[129],[15,46783,8695],{"className":46784},[129,133],[15,46786,46788],{"className":46787},[137],[15,46789,46791],{"className":46790},[141],[15,46792,46794],{"className":46793},[146],[15,46795,46797],{"className":46796,"style":28126},[150],[15,46798,46799,46802],{"style":1272},[15,46800],{"className":46801,"style":159},[158],[15,46803,46805],{"className":46804},[163,164,165,166],[15,46806,46808],{"className":46807},[129,166],[15,46809,21499],{"className":46810},[129,166],[11,46812,46813],{},"激活函数破坏了仿射映射在复合下的封闭性。换言之，它打破了可加性与齐次性这两个关键特性。这本质上就是神经网络非线性表达能力的来源。",[11,46815,46816],{},"从前面讨论的流形视角来看，这可以进一步写成：",[15,46818,46820],{"className":46819,"translate":19},[18],[15,46821,46823,46922],{"className":46822,"translate":19},[23],[15,46824,46826],{"className":46825},[27],[29,46827,46828],{"xmlns":31,"display":32},[34,46829,46830,46919],{},[37,46831,46832,46834,46851,46857,46869,46877,46889,46895,46907,46915,46917],{},[43,46833,43234],{},[46835,46836,46837,46840],"mover",{},[50,46838,723],{"stretchy":72,"minsize":46839},"3.0em",[46841,46842,46845],"mpadded",{"width":46843,"lspace":46844},"+0.6em","0.3em",[40,46846,46847,46849],{},[43,46848,34133],{},[308,46850,541],{},[40,46852,46853,46855],{},[43,46854,43234],{},[308,46856,318],{},[46835,46858,46859,46861],{},[50,46860,723],{"stretchy":72,"minsize":46839},[46841,46862,46863],{"width":46843,"lspace":46844},[40,46864,46865,46867],{},[43,46866,27283],{},[308,46868,541],{},[23149,46870,46871,46873,46875],{},[43,46872,43234],{},[308,46874,318],{},[50,46876,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[46835,46878,46879,46881],{},[50,46880,723],{"stretchy":72,"minsize":46839},[46841,46882,46883],{"width":46843,"lspace":46844},[40,46884,46885,46887],{},[43,46886,34133],{},[308,46888,318],{},[40,46890,46891,46893],{},[43,46892,43234],{},[308,46894,310],{},[46835,46896,46897,46899],{},[50,46898,723],{"stretchy":72,"minsize":46839},[46841,46900,46901],{"width":46843,"lspace":46844},[40,46902,46903,46905],{},[43,46904,27283],{},[308,46906,318],{},[23149,46908,46909,46911,46913],{},[43,46910,43234],{},[308,46912,310],{},[50,46914,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[50,46916,723],{},[50,46918,31657],{},[109,46920,46921],{"encoding":111},"M \\xrightarrow{l_0} M_1 \\xrightarrow{\\sigma_0} M_1' \\xrightarrow{l_1} M_2 \\xrightarrow{\\sigma_1} M_2' \\rightarrow \\cdots",[15,46923,46925,47041,47183,47338,47477,47547],{"className":46924,"ariaHidden":72},[116],[15,46926,46928,46932,46935,46938,47038],{"className":46927},[120],[15,46929],{"className":46930,"style":46931},[124],"height:1.1191em;vertical-align:-0.011em;",[15,46933,43234],{"className":46934,"style":12591},[129,133],[15,46936],{"className":46937,"style":199},[93],[15,46939,46942],{"className":46940},[203,46941],"x-arrow",[15,46943,46945,47029],{"className":46944},[141,142],[15,46946,46948,47026],{"className":46947},[146],[15,46949,46952,47007],{"className":46950,"style":46951},[150],"height:1.1081em;",[15,46953,46955,46958],{"style":46954},"top:-3.322em;",[15,46956],{"className":46957,"style":159},[158],[15,46959,46962],{"className":46960},[163,164,165,166,46961],"x-arrow-pad",[15,46963,46965],{"className":46964},[129,166],[15,46966,46968,46971],{"className":46967},[129,166],[15,46969,34133],{"className":46970,"style":34191},[129,133,166],[15,46972,46974],{"className":46973},[137],[15,46975,46977,46999],{"className":46976},[141,142],[15,46978,46980,46996],{"className":46979},[146],[15,46981,46984],{"className":46982,"style":46983},[150],"height:0.3173em;",[15,46985,46987,46990],{"style":46986},"top:-2.357em;margin-left:-0.0197em;margin-right:0.0714em;",[15,46988],{"className":46989,"style":15980},[158],[15,46991,46993],{"className":46992},[163,15984,229,166],[15,46994,541],{"className":46995},[129,166],[15,46997,175],{"className":46998},[174],[15,47000,47002],{"className":47001},[146],[15,47003,47005],{"className":47004,"style":15997},[150],[15,47006],{},[15,47008,47011,47014],{"className":47009,"style":47010},[29126],"top:-2.689em;",[15,47012],{"className":47013,"style":159},[158],[15,47015,47018],{"className":47016,"style":47017},[29271],"height:0.522em;min-width:1.469em;",[4455,47019,47023],{"xmlns":4457,"width":29275,"height":47020,"viewBox":47021,"preserveAspectRatio":47022},"0.522em","0 0 400000 522","xMaxYMin slice",[4461,47024],{"d":47025},"M0 241v40h399891c-47.3 35.3-84 78-110 128\n-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20\n 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7\n 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85\n-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n 151.7 139 205zm0 0v40h399900v-40z",[15,47027,175],{"className":47028},[174],[15,47030,47032],{"className":47031},[146],[15,47033,47036],{"className":47034,"style":47035},[150],"height:0.011em;",[15,47037],{},[15,47039],{"className":47040,"style":199},[93],[15,47042,47044,47048,47089,47092,47180],{"className":47043},[120],[15,47045],{"className":47046,"style":47047},[124],"height:1.0734em;vertical-align:-0.15em;",[15,47049,47051,47054],{"className":47050},[129],[15,47052,43234],{"className":47053,"style":12591},[129,133],[15,47055,47057],{"className":47056},[137],[15,47058,47060,47081],{"className":47059},[141,142],[15,47061,47063,47078],{"className":47062},[146],[15,47064,47066],{"className":47065,"style":1051},[150],[15,47067,47069,47072],{"style":47068},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[15,47070],{"className":47071,"style":159},[158],[15,47073,47075],{"className":47074},[163,164,165,166],[15,47076,318],{"className":47077},[129,166],[15,47079,175],{"className":47080},[174],[15,47082,47084],{"className":47083},[146],[15,47085,47087],{"className":47086,"style":182},[150],[15,47088],{},[15,47090],{"className":47091,"style":199},[93],[15,47093,47095],{"className":47094},[203,46941],[15,47096,47098,47172],{"className":47097},[141,142],[15,47099,47101,47169],{"className":47100},[146],[15,47102,47105,47156],{"className":47103,"style":47104},[150],"height:0.9234em;",[15,47106,47107,47110],{"style":46954},[15,47108],{"className":47109,"style":159},[158],[15,47111,47113],{"className":47112},[163,164,165,166,46961],[15,47114,47116],{"className":47115},[129,166],[15,47117,47119,47122],{"className":47118},[129,166],[15,47120,27283],{"className":47121,"style":887},[129,133,166],[15,47123,47125],{"className":47124},[137],[15,47126,47128,47148],{"className":47127},[141,142],[15,47129,47131,47145],{"className":47130},[146],[15,47132,47134],{"className":47133,"style":46983},[150],[15,47135,47136,47139],{"style":15976},[15,47137],{"className":47138,"style":15980},[158],[15,47140,47142],{"className":47141},[163,15984,229,166],[15,47143,541],{"className":47144},[129,166],[15,47146,175],{"className":47147},[174],[15,47149,47151],{"className":47150},[146],[15,47152,47154],{"className":47153,"style":15997},[150],[15,47155],{},[15,47157,47159,47162],{"className":47158,"style":47010},[29126],[15,47160],{"className":47161,"style":159},[158],[15,47163,47165],{"className":47164,"style":47017},[29271],[4455,47166,47167],{"xmlns":4457,"width":29275,"height":47020,"viewBox":47021,"preserveAspectRatio":47022},[4461,47168],{"d":47025},[15,47170,175],{"className":47171},[174],[15,47173,47175],{"className":47174},[146],[15,47176,47178],{"className":47177,"style":47035},[150],[15,47179],{},[15,47181],{"className":47182,"style":199},[93],[15,47184,47186,47190,47245,47248,47335],{"className":47185},[120],[15,47187],{"className":47188,"style":47189},[124],"height:1.3551em;vertical-align:-0.247em;",[15,47191,47193,47196],{"className":47192},[129],[15,47194,43234],{"className":47195,"style":12591},[129,133],[15,47197,47199],{"className":47198},[137],[15,47200,47202,47237],{"className":47201},[141,142],[15,47203,47205,47234],{"className":47204},[146],[15,47206,47208,47220],{"className":47207,"style":28126},[150],[15,47209,47211,47214],{"style":47210},"top:-2.453em;margin-left:-0.109em;margin-right:0.05em;",[15,47212],{"className":47213,"style":159},[158],[15,47215,47217],{"className":47216},[163,164,165,166],[15,47218,318],{"className":47219},[129,166],[15,47221,47222,47225],{"style":1272},[15,47223],{"className":47224,"style":159},[158],[15,47226,47228],{"className":47227},[163,164,165,166],[15,47229,47231],{"className":47230},[129,166],[15,47232,21499],{"className":47233},[129,166],[15,47235,175],{"className":47236},[174],[15,47238,47240],{"className":47239},[146],[15,47241,47243],{"className":47242,"style":28162},[150],[15,47244],{},[15,47246],{"className":47247,"style":199},[93],[15,47249,47251],{"className":47250},[203,46941],[15,47252,47254,47327],{"className":47253},[141,142],[15,47255,47257,47324],{"className":47256},[146],[15,47258,47260,47311],{"className":47259,"style":46951},[150],[15,47261,47262,47265],{"style":46954},[15,47263],{"className":47264,"style":159},[158],[15,47266,47268],{"className":47267},[163,164,165,166,46961],[15,47269,47271],{"className":47270},[129,166],[15,47272,47274,47277],{"className":47273},[129,166],[15,47275,34133],{"className":47276,"style":34191},[129,133,166],[15,47278,47280],{"className":47279},[137],[15,47281,47283,47303],{"className":47282},[141,142],[15,47284,47286,47300],{"className":47285},[146],[15,47287,47289],{"className":47288,"style":46983},[150],[15,47290,47291,47294],{"style":46986},[15,47292],{"className":47293,"style":15980},[158],[15,47295,47297],{"className":47296},[163,15984,229,166],[15,47298,318],{"className":47299},[129,166],[15,47301,175],{"className":47302},[174],[15,47304,47306],{"className":47305},[146],[15,47307,47309],{"className":47308,"style":15997},[150],[15,47310],{},[15,47312,47314,47317],{"className":47313,"style":47010},[29126],[15,47315],{"className":47316,"style":159},[158],[15,47318,47320],{"className":47319,"style":47017},[29271],[4455,47321,47322],{"xmlns":4457,"width":29275,"height":47020,"viewBox":47021,"preserveAspectRatio":47022},[4461,47323],{"d":47025},[15,47325,175],{"className":47326},[174],[15,47328,47330],{"className":47329},[146],[15,47331,47333],{"className":47332,"style":47035},[150],[15,47334],{},[15,47336],{"className":47337,"style":199},[93],[15,47339,47341,47344,47384,47387,47474],{"className":47340},[120],[15,47342],{"className":47343,"style":47047},[124],[15,47345,47347,47350],{"className":47346},[129],[15,47348,43234],{"className":47349,"style":12591},[129,133],[15,47351,47353],{"className":47352},[137],[15,47354,47356,47376],{"className":47355},[141,142],[15,47357,47359,47373],{"className":47358},[146],[15,47360,47362],{"className":47361,"style":1051},[150],[15,47363,47364,47367],{"style":47068},[15,47365],{"className":47366,"style":159},[158],[15,47368,47370],{"className":47369},[163,164,165,166],[15,47371,310],{"className":47372},[129,166],[15,47374,175],{"className":47375},[174],[15,47377,47379],{"className":47378},[146],[15,47380,47382],{"className":47381,"style":182},[150],[15,47383],{},[15,47385],{"className":47386,"style":199},[93],[15,47388,47390],{"className":47389},[203,46941],[15,47391,47393,47466],{"className":47392},[141,142],[15,47394,47396,47463],{"className":47395},[146],[15,47397,47399,47450],{"className":47398,"style":47104},[150],[15,47400,47401,47404],{"style":46954},[15,47402],{"className":47403,"style":159},[158],[15,47405,47407],{"className":47406},[163,164,165,166,46961],[15,47408,47410],{"className":47409},[129,166],[15,47411,47413,47416],{"className":47412},[129,166],[15,47414,27283],{"className":47415,"style":887},[129,133,166],[15,47417,47419],{"className":47418},[137],[15,47420,47422,47442],{"className":47421},[141,142],[15,47423,47425,47439],{"className":47424},[146],[15,47426,47428],{"className":47427,"style":46983},[150],[15,47429,47430,47433],{"style":15976},[15,47431],{"className":47432,"style":15980},[158],[15,47434,47436],{"className":47435},[163,15984,229,166],[15,47437,318],{"className":47438},[129,166],[15,47440,175],{"className":47441},[174],[15,47443,47445],{"className":47444},[146],[15,47446,47448],{"className":47447,"style":15997},[150],[15,47449],{},[15,47451,47453,47456],{"className":47452,"style":47010},[29126],[15,47454],{"className":47455,"style":159},[158],[15,47457,47459],{"className":47458,"style":47017},[29271],[4455,47460,47461],{"xmlns":4457,"width":29275,"height":47020,"viewBox":47021,"preserveAspectRatio":47022},[4461,47462],{"d":47025},[15,47464,175],{"className":47465},[174],[15,47467,47469],{"className":47468},[146],[15,47470,47472],{"className":47471,"style":47035},[150],[15,47473],{},[15,47475],{"className":47476,"style":199},[93],[15,47478,47480,47484,47538,47541,47544],{"className":47479},[120],[15,47481],{"className":47482,"style":47483},[124],"height:1.0489em;vertical-align:-0.247em;",[15,47485,47487,47490],{"className":47486},[129],[15,47488,43234],{"className":47489,"style":12591},[129,133],[15,47491,47493],{"className":47492},[137],[15,47494,47496,47530],{"className":47495},[141,142],[15,47497,47499,47527],{"className":47498},[146],[15,47500,47502,47513],{"className":47501,"style":28126},[150],[15,47503,47504,47507],{"style":47210},[15,47505],{"className":47506,"style":159},[158],[15,47508,47510],{"className":47509},[163,164,165,166],[15,47511,310],{"className":47512},[129,166],[15,47514,47515,47518],{"style":1272},[15,47516],{"className":47517,"style":159},[158],[15,47519,47521],{"className":47520},[163,164,165,166],[15,47522,47524],{"className":47523},[129,166],[15,47525,21499],{"className":47526},[129,166],[15,47528,175],{"className":47529},[174],[15,47531,47533],{"className":47532},[146],[15,47534,47536],{"className":47535,"style":28162},[150],[15,47537],{},[15,47539],{"className":47540,"style":199},[93],[15,47542,723],{"className":47543},[203],[15,47545],{"className":47546,"style":199},[93],[15,47548,47550,47553],{"className":47549},[120],[15,47551],{"className":47552,"style":31783},[124],[15,47554,31657],{"className":47555},[449],[11,47557,47558],{},"其中：仿射层负责拉伸、旋转、平移、剪切；激活层负责对流形进行折叠、裁剪、压缩与重构。",[11,47560,47561,47562,47632],{},"如果所有 ",[15,47563,47565,47583],{"className":47564,"translate":19},[23],[15,47566,47568],{"className":47567},[27],[29,47569,47570],{"xmlns":31},[34,47571,47572,47580],{},[37,47573,47574],{},[40,47575,47576,47578],{},[43,47577,31987],{},[43,47579,1459],{},[109,47581,47582],{"encoding":111},"W_i",[15,47584,47586],{"className":47585,"ariaHidden":72},[116],[15,47587,47589,47592],{"className":47588},[120],[15,47590],{"className":47591,"style":20220},[124],[15,47593,47595,47598],{"className":47594},[129],[15,47596,31987],{"className":47597,"style":12443},[129,133],[15,47599,47601],{"className":47600},[137],[15,47602,47604,47624],{"className":47603},[141,142],[15,47605,47607,47621],{"className":47606},[146],[15,47608,47610],{"className":47609,"style":1981},[150],[15,47611,47612,47615],{"style":34226},[15,47613],{"className":47614,"style":159},[158],[15,47616,47618],{"className":47617},[163,164,165,166],[15,47619,1459],{"className":47620},[129,133,166],[15,47622,175],{"className":47623},[174],[15,47625,47627],{"className":47626},[146],[15,47628,47630],{"className":47629,"style":182},[150],[15,47631],{}," 都可逆，那么仿射层本身只是同胚变换，不会改变流形的拓扑。真正让流形逐步变得可分的，往往正是激活函数引入的非线性形变。",[11,47634,47635],{},"因此，从几何角度看，神经网络可以理解为作用在数据流形上的仿射变换与非线性流形操作的交替组合。这一视角与许多现代流形学习和几何深度学习的解释相一致。",[570,47637,47638],{"id":47638},"激活函数",[11,47640,47641],{},"激活函数并不存在「唯一最佳」的选择。不过，从神经网络的可训练性与表达能力来看，一个好的激活函数通常应满足几个核心特征：",[575,47643,47644],{"id":47644},"非线性",[11,47646,47647],{},"如前所述，需要非线性来打破堆叠仿射层的线性封闭性，使多层网络能够逼近任意函数。",[11,47649,47650],{},"数学要求：不能满足线性组合的封闭性：",[15,47652,47654],{"className":47653,"translate":19},[18],[15,47655,47657,47709],{"className":47656,"translate":19},[23],[15,47658,47660],{"className":47659},[27],[29,47661,47662],{"xmlns":31,"display":32},[34,47663,47664,47706],{},[37,47665,47666,47668,47670,47672,47674,47676,47678,47680,47682,47684,47686,47688,47690,47692,47694,47696,47698,47700,47702,47704],{},[43,47667,27283],{},[50,47669,53],{"stretchy":52},[43,47671,8686],{},[43,47673,56],{},[50,47675,2682],{},[43,47677,8695],{},[43,47679,857],{},[50,47681,59],{"stretchy":52},[50,47683,1418],{"mathvariant":65},[43,47685,8686],{},[43,47687,27283],{},[50,47689,53],{"stretchy":52},[43,47691,56],{},[50,47693,59],{"stretchy":52},[50,47695,2682],{},[43,47697,8695],{},[43,47699,27283],{},[50,47701,53],{"stretchy":52},[43,47703,857],{},[50,47705,59],{"stretchy":52},[109,47707,47708],{"encoding":111},"\\sigma(a x + b y) \\neq a \\sigma(x) + b \\sigma(y)",[15,47710,47712,47739,47796,47824],{"className":47711,"ariaHidden":72},[116],[15,47713,47715,47718,47721,47724,47727,47730,47733,47736],{"className":47714},[120],[15,47716],{"className":47717,"style":125},[124],[15,47719,27283],{"className":47720,"style":887},[129,133],[15,47722,53],{"className":47723},[188],[15,47725,8686],{"className":47726},[129,133],[15,47728,56],{"className":47729},[129,133],[15,47731],{"className":47732,"style":3277},[93],[15,47734,2682],{"className":47735},[2436],[15,47737],{"className":47738,"style":3277},[93],[15,47740,47742,47745,47748,47751,47754,47757,47793],{"className":47741},[120],[15,47743],{"className":47744,"style":125},[124],[15,47746,8695],{"className":47747},[129,133],[15,47749,857],{"className":47750,"style":887},[129,133],[15,47752,59],{"className":47753},[195],[15,47755],{"className":47756,"style":199},[93],[15,47758,47760,47787,47790],{"className":47759},[203],[15,47761,47763],{"className":47762},[203],[15,47764,47766],{"className":47765},[129,1670],[15,47767,47769],{"className":47768},[1674],[15,47770,47772,47775,47784],{"className":47771},[1678],[15,47773],{"className":47774,"style":1682},[124],[15,47776,47778],{"className":47777},[1686],[15,47779,47781],{"className":47780},[129],[15,47782,1693],{"className":47783},[203],[15,47785],{"className":47786},[1697],[15,47788],{"className":47789},[93,1701],[15,47791,62],{"className":47792},[203],[15,47794],{"className":47795,"style":199},[93],[15,47797,47799,47802,47806,47809,47812,47815,47818,47821],{"className":47798},[120],[15,47800],{"className":47801,"style":125},[124],[15,47803,47805],{"className":47804,"style":887},[129,133],"aσ",[15,47807,53],{"className":47808},[188],[15,47810,56],{"className":47811},[129,133],[15,47813,59],{"className":47814},[195],[15,47816],{"className":47817,"style":3277},[93],[15,47819,2682],{"className":47820},[2436],[15,47822],{"className":47823,"style":3277},[93],[15,47825,47827,47830,47834,47837,47840],{"className":47826},[120],[15,47828],{"className":47829,"style":125},[124],[15,47831,47833],{"className":47832,"style":887},[129,133],"bσ",[15,47835,53],{"className":47836},[188],[15,47838,857],{"className":47839,"style":887},[129,133],[15,47841,59],{"className":47842},[195],[11,47844,47845],{},"例如：ReLU、Tanh、Sigmoid、GELU。",[11,47847,47848],{},"线性函数并不适用，因为多个线性层的复合仍然是线性的，这会退化回仿射复合，从而限制表达能力。",[575,47850,47851],{"id":47851},"可微性",[11,47853,47854],{},"由于神经网络的核心依赖反向传播来确定权重，激活函数必须在其定义域的大部分区域可微，以便在反向传播过程中计算梯度。对于不可微的点（例如 ReLU 在 0 处），通常使用次梯度（subgradient）。此外，导数的性质会影响梯度流动：梯度过小会导致梯度消失（如 Sigmoid），梯度过大则会导致梯度爆炸。",[575,47856,47857],{"id":47857},"连续性",[11,47859,47860],{},"确保网络输出随输入连续变化，否则训练会变得不稳定。存在少数可控的不连续点通常是可以接受的（例如 ReLU 只有一个不连续点，实际影响很小）。",[11,47862,47863],{},"在上述性质中，满足条件只是「入场券」，而不是「成功的保证」。",{"title":36419,"searchDepth":36420,"depth":36420,"links":47865},[47866,47870,47871,47872,47873],{"id":36474,"depth":36420,"text":36474,"children":47867},[47868,47869],{"id":36574,"depth":36425,"text":36574},{"id":37520,"depth":36425,"text":37520},{"id":39288,"depth":36420,"text":39288},{"id":41453,"depth":36420,"text":41453},{"id":43309,"depth":36420,"text":43309},{"id":47638,"depth":36420,"text":47638,"children":47874},[47875,47876,47877],{"id":47644,"depth":36425,"text":47644},{"id":47851,"depth":36425,"text":47851},{"id":47857,"depth":36425,"text":47857},{},"\u002Fblog\u002F2026\u002F2026-04-06-nonlinearity-with-activations",{"title":36469,"description":36419},"blog\u002F2026\u002F2026-04-06-nonlinearity-with-activations","线性要求满足可加性与齐次性，但神经网络需要非线性才能摆脱简单的仿射变换。仿射层只能对数据进行拉伸和旋转，而不改变其拓扑结构；激活函数则能实现复杂的形变，使网络能够解开数据流形，有效地学习复杂模式。",[36463,36464],"2026-04-06T00:00:00+08:00","tJd-MURjidybpGcywsQAgaKC5QX9X65sQjlBIAqVFPg",{"id":47887,"title":47888,"body":47889,"description":47893,"draft":36453,"enableCodeHighlight":36453,"enableCodeHightlight":36453,"enableComment":36454,"enableD2":36453,"enableKatex":36454,"extension":36455,"image":36419,"important":36453,"location":36456,"meta":93081,"navigation":36454,"ogImage":36456,"path":93082,"seo":93083,"stem":93084,"summary":93085,"tags":93086,"time":93087,"__hash__":93088},"blog\u002Fblog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa.md","Fano 不等式与 Le Cam 方法",{"type":8,"value":47890,"toc":93049},[47891,47894,48443,48446,48450,48672,48675,49060,49067,49073,49076,49080,49084,49298,49599,49953,50680,50775,51161,51226,51608,51612,51673,52011,52014,52198,52205,52209,52447,52685,52846,52849,52853,52857,52860,54047,54050,54254,54472,54936,54939,55599,55602,56520,56605,56722,57564,57770,58021,58756,58759,59389,60166,60170,60173,60511,60514,60951,60954,61698,62002,62040,62044,62269,62941,63272,63854,63932,63936,63968,64314,64981,65071,65075,65079,65889,65892,66306,66428,66764,66767,67140,67144,67147,69097,69100,69364,69602,70190,70306,70562,70797,71287,71319,71860,72412,72787,72790,73451,73454,74144,74350,74354,74361,74861,74955,75347,75650,75654,75658,75690,77473,77682,78166,78514,78733,78736,79244,79432,79887,79890,79894,80042,80378,80822,80825,81616,81652,81656,81733,81736,82081,82231,82299,82583,82962,83346,83349,83725,83865,84500,84606,84638,84750,84752,85273,85588,85593,85767,85773,86084,86344,86538,86837,87227,87555,88261,88590,88972,89113,89116,90069,90181,90512,90638,90934,90937,91217,91221,92259,92262,92266,92451,92454,92591,92594,92787,92789,92857,92947,92950],[11,47892,47893],{},"统计学的下界证明有一个几乎统一的套路：先把估计问题换成一个检验问题，再证明这个检验问题很难。",[11,47895,21927,47896,48088,48089,48117,48118,48154,48155,48190,48191,48226,48227,10175,48320,48413,48414,48442],{},[15,47897,47899,47943],{"className":47898,"translate":19},[23],[15,47900,47902],{"className":47901},[27],[29,47903,47904],{"xmlns":31},[34,47905,47906,47940],{},[37,47907,47908,47914,47916,47918,47920,47926,47928,47930,47932,47934,47936,47938],{},[40,47909,47910,47912],{},[43,47911,597],{},[308,47913,318],{},[50,47915,91],{"separator":72},[50,47917,335],{},[50,47919,91],{"separator":72},[40,47921,47922,47924],{},[43,47923,597],{},[43,47925,39314],{},[50,47927,991],{},[43,47929,12555],{},[50,47931,53],{"stretchy":52},[43,47933,48],{},[50,47935,91],{"separator":72},[308,47937,318],{},[50,47939,59],{"stretchy":52},[109,47941,47942],{"encoding":111},"X_1,\\dots,X_n\\sim N(\\theta,1)",[15,47944,47946,48061],{"className":47945,"ariaHidden":72},[116],[15,47947,47949,47952,47994,47997,48000,48003,48006,48009,48012,48052,48055,48058],{"className":47948},[120],[15,47950],{"className":47951,"style":34353},[124],[15,47953,47955,47959],{"className":47954},[129],[15,47956,597],{"className":47957,"style":47958},[129,133],"margin-right:0.0785em;",[15,47960,47962],{"className":47961},[137],[15,47963,47965,47986],{"className":47964},[141,142],[15,47966,47968,47983],{"className":47967},[146],[15,47969,47971],{"className":47970,"style":1051},[150],[15,47972,47974,47977],{"style":47973},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[15,47975],{"className":47976,"style":159},[158],[15,47978,47980],{"className":47979},[163,164,165,166],[15,47981,318],{"className":47982},[129,166],[15,47984,175],{"className":47985},[174],[15,47987,47989],{"className":47988},[146],[15,47990,47992],{"className":47991,"style":182},[150],[15,47993],{},[15,47995,91],{"className":47996},[254],[15,47998],{"className":47999,"style":262},[93],[15,48001,335],{"className":48002},[449],[15,48004],{"className":48005,"style":262},[93],[15,48007,91],{"className":48008},[254],[15,48010],{"className":48011,"style":262},[93],[15,48013,48015,48018],{"className":48014},[129],[15,48016,597],{"className":48017,"style":47958},[129,133],[15,48019,48021],{"className":48020},[137],[15,48022,48024,48044],{"className":48023},[141,142],[15,48025,48027,48041],{"className":48026},[146],[15,48028,48030],{"className":48029,"style":1163},[150],[15,48031,48032,48035],{"style":47973},[15,48033],{"className":48034,"style":159},[158],[15,48036,48038],{"className":48037},[163,164,165,166],[15,48039,39314],{"className":48040},[129,133,166],[15,48042,175],{"className":48043},[174],[15,48045,48047],{"className":48046},[146],[15,48048,48050],{"className":48049,"style":182},[150],[15,48051],{},[15,48053],{"className":48054,"style":199},[93],[15,48056,991],{"className":48057},[203],[15,48059],{"className":48060,"style":199},[93],[15,48062,48064,48067,48070,48073,48076,48079,48082,48085],{"className":48063},[120],[15,48065],{"className":48066,"style":125},[124],[15,48068,12555],{"className":48069,"style":12591},[129,133],[15,48071,53],{"className":48072},[188],[15,48074,48],{"className":48075,"style":170},[129,133],[15,48077,91],{"className":48078},[254],[15,48080],{"className":48081,"style":262},[93],[15,48083,318],{"className":48084},[129],[15,48086,59],{"className":48087},[195],"，要估计 ",[15,48090,48092,48105],{"className":48091,"translate":19},[23],[15,48093,48095],{"className":48094},[27],[29,48096,48097],{"xmlns":31},[34,48098,48099,48103],{},[37,48100,48101],{},[43,48102,48],{},[109,48104,512],{"encoding":111},[15,48106,48108],{"className":48107,"ariaHidden":72},[116],[15,48109,48111,48114],{"className":48110},[120],[15,48112],{"className":48113,"style":522},[124],[15,48115,48],{"className":48116,"style":170},[129,133],"。样本均值给出风险 ",[15,48119,48121,48139],{"className":48120,"translate":19},[23],[15,48122,48124],{"className":48123},[27],[29,48125,48126],{"xmlns":31},[34,48127,48128,48136],{},[37,48129,48130,48132,48134],{},[308,48131,318],{},[43,48133,544],{"mathvariant":65},[43,48135,39314],{},[109,48137,48138],{"encoding":111},"1\u002Fn",[15,48140,48142],{"className":48141,"ariaHidden":72},[116],[15,48143,48145,48148,48151],{"className":48144},[120],[15,48146],{"className":48147,"style":125},[124],[15,48149,29841],{"className":48150},[129],[15,48152,39314],{"className":48153},[129,133],"。这个 ",[15,48156,48158,48175],{"className":48157,"translate":19},[23],[15,48159,48161],{"className":48160},[27],[29,48162,48163],{"xmlns":31},[34,48164,48165,48173],{},[37,48166,48167,48169,48171],{},[308,48168,318],{},[43,48170,544],{"mathvariant":65},[43,48172,39314],{},[109,48174,48138],{"encoding":111},[15,48176,48178],{"className":48177,"ariaHidden":72},[116],[15,48179,48181,48184,48187],{"className":48180},[120],[15,48182],{"className":48183,"style":125},[124],[15,48185,29841],{"className":48186},[129],[15,48188,39314],{"className":48189},[129,133]," 是不是最优的？也就是说，是否存在某个估计量，能让最坏情况下的风险比 ",[15,48192,48194,48211],{"className":48193,"translate":19},[23],[15,48195,48197],{"className":48196},[27],[29,48198,48199],{"xmlns":31},[34,48200,48201,48209],{},[37,48202,48203,48205,48207],{},[308,48204,318],{},[43,48206,544],{"mathvariant":65},[43,48208,39314],{},[109,48210,48138],{"encoding":111},[15,48212,48214],{"className":48213,"ariaHidden":72},[116],[15,48215,48217,48220,48223],{"className":48216},[120],[15,48218],{"className":48219,"style":125},[124],[15,48221,29841],{"className":48222},[129],[15,48224,39314],{"className":48225},[129,133]," 更小？Le Cam 的两点方法给出的回答是：不存在。它的论证只用了两个候选参数 ",[15,48228,48230,48252],{"className":48229,"translate":19},[23],[15,48231,48233],{"className":48232},[27],[29,48234,48235],{"xmlns":31},[34,48236,48237,48249],{},[37,48238,48239,48245,48247],{},[40,48240,48241,48243],{},[43,48242,48],{},[308,48244,541],{},[50,48246,62],{},[308,48248,541],{},[109,48250,48251],{"encoding":111},"\\theta_0=0",[15,48253,48255,48311],{"className":48254,"ariaHidden":72},[116],[15,48256,48258,48261,48302,48305,48308],{"className":48257},[120],[15,48259],{"className":48260,"style":2183},[124],[15,48262,48264,48267],{"className":48263},[129],[15,48265,48],{"className":48266,"style":170},[129,133],[15,48268,48270],{"className":48269},[137],[15,48271,48273,48294],{"className":48272},[141,142],[15,48274,48276,48291],{"className":48275},[146],[15,48277,48279],{"className":48278,"style":1051},[150],[15,48280,48282,48285],{"style":48281},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[15,48283],{"className":48284,"style":159},[158],[15,48286,48288],{"className":48287},[163,164,165,166],[15,48289,541],{"className":48290},[129,166],[15,48292,175],{"className":48293},[174],[15,48295,48297],{"className":48296},[146],[15,48298,48300],{"className":48299,"style":182},[150],[15,48301],{},[15,48303],{"className":48304,"style":199},[93],[15,48306,62],{"className":48307},[203],[15,48309],{"className":48310,"style":199},[93],[15,48312,48314,48317],{"className":48313},[120],[15,48315],{"className":48316,"style":4886},[124],[15,48318,541],{"className":48319},[129],[15,48321,48323,48346],{"className":48322,"translate":19},[23],[15,48324,48326],{"className":48325},[27],[29,48327,48328],{"xmlns":31},[34,48329,48330,48343],{},[37,48331,48332,48338,48340],{},[40,48333,48334,48336],{},[43,48335,48],{},[308,48337,318],{},[50,48339,62],{},[43,48341,48342],{"mathvariant":65},"Δ",[109,48344,48345],{"encoding":111},"\\theta_1=\\Delta",[15,48347,48349,48404],{"className":48348,"ariaHidden":72},[116],[15,48350,48352,48355,48395,48398,48401],{"className":48351},[120],[15,48353],{"className":48354,"style":2183},[124],[15,48356,48358,48361],{"className":48357},[129],[15,48359,48],{"className":48360,"style":170},[129,133],[15,48362,48364],{"className":48363},[137],[15,48365,48367,48387],{"className":48366},[141,142],[15,48368,48370,48384],{"className":48369},[146],[15,48371,48373],{"className":48372,"style":1051},[150],[15,48374,48375,48378],{"style":48281},[15,48376],{"className":48377,"style":159},[158],[15,48379,48381],{"className":48380},[163,164,165,166],[15,48382,318],{"className":48383},[129,166],[15,48385,175],{"className":48386},[174],[15,48388,48390],{"className":48389},[146],[15,48391,48393],{"className":48392,"style":182},[150],[15,48394],{},[15,48396],{"className":48397,"style":199},[93],[15,48399,62],{"className":48400},[203],[15,48402],{"className":48403,"style":199},[93],[15,48405,48407,48410],{"className":48406},[120],[15,48408],{"className":48409,"style":610},[124],[15,48411,48342],{"className":48412},[129],"——如果连这两个点都分不清，那就更不可能把 ",[15,48415,48417,48430],{"className":48416,"translate":19},[23],[15,48418,48420],{"className":48419},[27],[29,48421,48422],{"xmlns":31},[34,48423,48424,48428],{},[37,48425,48426],{},[43,48427,48],{},[109,48429,512],{"encoding":111},[15,48431,48433],{"className":48432,"ariaHidden":72},[116],[15,48434,48436,48439],{"className":48435},[120],[15,48437],{"className":48438,"style":522},[124],[15,48440,48],{"className":48441,"style":170},[129,133]," 估准。",[11,48444,48445],{},"本文介绍这套语言里最常用的三个工具：Le Cam 的两点方法、Fano 不等式，以及 Assouad 的超立方体引理。三者的共同点是：先选一个有限的「困难子问题」，把估计下界转化为检验下界，再用总变差、KL 散度或互信息来度量「有多难分」。",[570,48447,48449],{"id":48448},"一极小极大下界在说什么","一、极小极大：下界在说什么",[11,48451,48452,48453,48520,48521,48524,48525,14469,48553,48613,48614,48642,48643,48671],{},"一个估计量 ",[15,48454,48456,48475],{"className":48455,"translate":19},[23],[15,48457,48459],{"className":48458},[27],[29,48460,48461],{"xmlns":31},[34,48462,48463,48472],{},[37,48464,48465],{},[46835,48466,48467,48469],{"accent":72},[43,48468,48],{},[50,48470,48471],{},"^",[109,48473,48474],{"encoding":111},"\\hat\\theta",[15,48476,48478],{"className":48477,"ariaHidden":72},[116],[15,48479,48481,48485],{"className":48480},[120],[15,48482],{"className":48483,"style":48484},[124],"height:0.9579em;",[15,48486,48489],{"className":48487},[129,48488],"accent",[15,48490,48492],{"className":48491},[141],[15,48493,48495],{"className":48494},[146],[15,48496,48498,48506],{"className":48497,"style":48484},[150],[15,48499,48500,48503],{"style":2287},[15,48501],{"className":48502,"style":1823},[158],[15,48504,48],{"className":48505,"style":170},[129,133],[15,48507,48509,48512],{"style":48508},"top:-3.2634em;",[15,48510],{"className":48511,"style":1823},[158],[15,48513,48517],{"className":48514,"style":48516},[48515],"accent-body","left:-0.1667em;",[15,48518,48471],{"className":48519},[129]," 的优劣，通常用它的",[698,48522,48523],{},"风险","衡量：在真实参数为 ",[15,48526,48528,48541],{"className":48527,"translate":19},[23],[15,48529,48531],{"className":48530},[27],[29,48532,48533],{"xmlns":31},[34,48534,48535,48539],{},[37,48536,48537],{},[43,48538,48],{},[109,48540,512],{"encoding":111},[15,48542,48544],{"className":48543,"ariaHidden":72},[116],[15,48545,48547,48550],{"className":48546},[120],[15,48548],{"className":48549,"style":522},[124],[15,48551,48],{"className":48552,"style":170},[129,133],[15,48554,48556,48573],{"className":48555,"translate":19},[23],[15,48557,48559],{"className":48558},[27],[29,48560,48561],{"xmlns":31},[34,48562,48563,48571],{},[37,48564,48565],{},[46835,48566,48567,48569],{"accent":72},[43,48568,48],{},[50,48570,48471],{},[109,48572,48474],{"encoding":111},[15,48574,48576],{"className":48575,"ariaHidden":72},[116],[15,48577,48579,48582],{"className":48578},[120],[15,48580],{"className":48581,"style":48484},[124],[15,48583,48585],{"className":48584},[129,48488],[15,48586,48588],{"className":48587},[141],[15,48589,48591],{"className":48590},[146],[15,48592,48594,48602],{"className":48593,"style":48484},[150],[15,48595,48596,48599],{"style":2287},[15,48597],{"className":48598,"style":1823},[158],[15,48600,48],{"className":48601,"style":170},[129,133],[15,48603,48604,48607],{"style":48508},[15,48605],{"className":48606,"style":1823},[158],[15,48608,48610],{"className":48609,"style":48516},[48515],[15,48611,48471],{"className":48612},[129]," 的平均损失。但真实参数是未知的，所以我们不能只关心某一个 ",[15,48615,48617,48630],{"className":48616,"translate":19},[23],[15,48618,48620],{"className":48619},[27],[29,48621,48622],{"xmlns":31},[34,48623,48624,48628],{},[37,48625,48626],{},[43,48627,48],{},[109,48629,512],{"encoding":111},[15,48631,48633],{"className":48632,"ariaHidden":72},[116],[15,48634,48636,48639],{"className":48635},[120],[15,48637],{"className":48638,"style":522},[124],[15,48640,48],{"className":48641,"style":170},[129,133]," 下的风险。极小极大（minimax）的立场是：先让大自然挑一个最不利的 ",[15,48644,48646,48659],{"className":48645,"translate":19},[23],[15,48647,48649],{"className":48648},[27],[29,48650,48651],{"xmlns":31},[34,48652,48653,48657],{},[37,48654,48655],{},[43,48656,48],{},[109,48658,512],{"encoding":111},[15,48660,48662],{"className":48661,"ariaHidden":72},[116],[15,48663,48665,48668],{"className":48664},[120],[15,48666],{"className":48667,"style":522},[124],[15,48669,48],{"className":48670,"style":170},[129,133],"，再让统计学家挑估计量，看统计学家能保证什么。",[11,48673,48674],{},"用博弈的话说，极小极大风险是",[15,48676,48678],{"className":48677,"translate":19},[18],[15,48679,48681,48758],{"className":48680,"translate":19},[23],[15,48682,48684],{"className":48683},[27],[29,48685,48686],{"xmlns":31,"display":32},[34,48687,48688,48755],{},[37,48689,48690,48705,48721,48723,48733,48735,48737,48739,48741,48743,48749,48751,48753],{},[2140,48691,48692,48699],{},[37,48693,48694,48697],{},[43,48695,48696],{},"inf",[50,48698,69],{},[46835,48700,48701,48703],{"accent":72},[43,48702,48],{},[50,48704,48471],{},[2140,48706,48707,48713],{},[37,48708,48709,48711],{},[43,48710,2932],{},[50,48712,69],{},[37,48714,48715,48717,48719],{},[43,48716,48],{},[50,48718,100],{},[43,48720,30862],{"mathvariant":65},[6852,48722,6854],{},[40,48724,48725,48727],{},[43,48726,26563],{"mathvariant":103},[40,48728,48729,48731],{},[43,48730,1388],{},[43,48732,48],{},[50,48734,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,48736,27518],{"mathvariant":65},[50,48738,53],{"stretchy":52},[43,48740,48],{},[50,48742,91],{"separator":72},[46835,48744,48745,48747],{"accent":72},[43,48746,48],{},[50,48748,48471],{},[50,48750,59],{"stretchy":52},[50,48752,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,48754,107],{"mathvariant":65},[109,48756,48757],{"encoding":111},"\\inf_{\\hat\\theta}\\sup_{\\theta\\in\\Theta}\\ \\mathbb E_{P_\\theta}\\bigl[\\ell(\\theta,\\hat\\theta)\\bigr].",[15,48759,48761],{"className":48760,"ariaHidden":72},[116],[15,48762,48764,48768,48851,48854,48907,48910,48913,48996,49002,49005,49008,49011,49014,49017,49048,49051,49057],{"className":48763},[120],[15,48765],{"className":48766,"style":48767},[124],"height:1.9318em;vertical-align:-0.9739em;",[15,48769,48771],{"className":48770},[217,1874],[15,48772,48774,48842],{"className":48773},[141,142],[15,48775,48777,48839],{"className":48776},[146],[15,48778,48780,48825],{"className":48779,"style":522},[150],[15,48781,48783,48786],{"style":48782},"top:-2.1635em;margin-left:0em;",[15,48784],{"className":48785,"style":1823},[158],[15,48787,48789],{"className":48788},[163,164,165,166],[15,48790,48792],{"className":48791},[129,166],[15,48793,48795],{"className":48794},[129,48488,166],[15,48796,48798],{"className":48797},[141],[15,48799,48801],{"className":48800},[146],[15,48802,48804,48813],{"className":48803,"style":48484},[150],[15,48805,48807,48810],{"style":48806},"top:-2.7em;",[15,48808],{"className":48809,"style":159},[158],[15,48811,48],{"className":48812,"style":170},[129,133,166],[15,48814,48816,48819],{"style":48815},"top:-2.9634em;",[15,48817],{"className":48818,"style":159},[158],[15,48820,48822],{"className":48821,"style":48516},[48515],[15,48823,48471],{"className":48824},[129,166],[15,48826,48827,48830],{"style":2287},[15,48828],{"className":48829,"style":1823},[158],[15,48831,48832],{},[15,48833,48835,48836],{"className":48834},[217],"in",[15,48837,36556],{"style":48838},"margin-right:0.0778em;",[15,48840,175],{"className":48841},[174],[15,48843,48845],{"className":48844},[146],[15,48846,48849],{"className":48847,"style":48848},[150],"height:0.9365em;",[15,48850],{},[15,48852],{"className":48853,"style":262},[93],[15,48855,48857],{"className":48856},[217,1874],[15,48858,48860,48899],{"className":48859},[141,142],[15,48861,48863,48896],{"className":48862},[146],[15,48864,48866,48886],{"className":48865,"style":4383},[150],[15,48867,48868,48871],{"style":4386},[15,48869],{"className":48870,"style":1823},[158],[15,48872,48874],{"className":48873},[163,164,165,166],[15,48875,48877,48880,48883],{"className":48876},[129,166],[15,48878,48],{"className":48879,"style":170},[129,133,166],[15,48881,100],{"className":48882},[203,166],[15,48884,30862],{"className":48885},[129,166],[15,48887,48888,48891],{"style":2287},[15,48889],{"className":48890,"style":1823},[158],[15,48892,48893],{},[15,48894,2932],{"className":48895},[217],[15,48897,175],{"className":48898},[174],[15,48900,48902],{"className":48901},[146],[15,48903,48905],{"className":48904,"style":4424},[150],[15,48906],{},[15,48908,6854],{"className":48909},[93],[15,48911],{"className":48912,"style":262},[93],[15,48914,48916,48919],{"className":48915},[129],[15,48917,26563],{"className":48918},[129,285],[15,48920,48922],{"className":48921},[137],[15,48923,48925,48987],{"className":48924},[141,142],[15,48926,48928,48984],{"className":48927},[146],[15,48929,48931],{"className":48930,"style":1525},[150],[15,48932,48933,48936],{"style":154},[15,48934],{"className":48935,"style":159},[158],[15,48937,48939],{"className":48938},[163,164,165,166],[15,48940,48942],{"className":48941},[129,166],[15,48943,48945,48948],{"className":48944},[129,166],[15,48946,1388],{"className":48947,"style":12443},[129,133,166],[15,48949,48951],{"className":48950},[137],[15,48952,48954,48975],{"className":48953},[141,142],[15,48955,48957,48972],{"className":48956},[146],[15,48958,48960],{"className":48959,"style":1594},[150],[15,48961,48963,48966],{"style":48962},"top:-2.3488em;margin-left:-0.1389em;margin-right:0.0714em;",[15,48964],{"className":48965,"style":15980},[158],[15,48967,48969],{"className":48968},[163,15984,229,166],[15,48970,48],{"className":48971,"style":170},[129,133,166],[15,48973,175],{"className":48974},[174],[15,48976,48978],{"className":48977},[146],[15,48979,48982],{"className":48980,"style":48981},[150],"height:0.1512em;",[15,48983],{},[15,48985,175],{"className":48986},[174],[15,48988,48990],{"className":48989},[146],[15,48991,48994],{"className":48992,"style":48993},[150],"height:0.2559em;",[15,48995],{},[15,48997,48999],{"className":48998},[188],[15,49000,1407],{"className":49001},[228,229],[15,49003,27518],{"className":49004},[129],[15,49006,53],{"className":49007},[188],[15,49009,48],{"className":49010,"style":170},[129,133],[15,49012,91],{"className":49013},[254],[15,49015],{"className":49016,"style":262},[93],[15,49018,49020],{"className":49019},[129,48488],[15,49021,49023],{"className":49022},[141],[15,49024,49026],{"className":49025},[146],[15,49027,49029,49037],{"className":49028,"style":48484},[150],[15,49030,49031,49034],{"style":2287},[15,49032],{"className":49033,"style":1823},[158],[15,49035,48],{"className":49036,"style":170},[129,133],[15,49038,49039,49042],{"style":48508},[15,49040],{"className":49041,"style":1823},[158],[15,49043,49045],{"className":49044,"style":48516},[48515],[15,49046,48471],{"className":49047},[129],[15,49049,59],{"className":49050},[195],[15,49052,49054],{"className":49053},[195],[15,49055,1423],{"className":49056},[228,229],[15,49058,107],{"className":49059},[129],[11,49061,49062,49063,49066],{},"这个量回答的是：",[698,49064,49065],{},"在所有可能的参数中，最坏情况下的风险最小能压到多少。"," 一个上界对应一个具体的估计量；一个下界则对应一句否定：无论你多聪明，都不可能比这个数更好。",[11,49068,49069,49070,49072],{},"下界为什么难？因为你要同时对",[698,49071,7246],{},"估计量说话。逐一检查是不可能的。所以下界证明总是走同一条路：把参数空间缩小到一个有限的、精心挑选的子集，使得在这个子集上「区分参数」本身就已经很难；如果区分都做不到，估计就更做不到。",[11,49074,49075],{},"这就是「归约到检验」。接下来的三种方法，区别只在于子集有多大、用什么距离度量「难分」。",[570,49077,49079],{"id":49078},"二决策论框架","二、决策论框架",[575,49081,49083],{"id":49082},"_21-模型损失与风险","2.1 模型、损失与风险",[11,49085,21927,49086,49210,49211,49239,49240,49268,49269,49297],{},[15,49087,49089,49119],{"className":49088,"translate":19},[23],[15,49090,49092],{"className":49091},[27],[29,49093,49094],{"xmlns":31},[34,49095,49096,49116],{},[37,49097,49098,49100,49106,49108,49110,49112,49114],{},[50,49099,636],{"stretchy":52},[40,49101,49102,49104],{},[43,49103,1388],{},[43,49105,48],{},[50,49107,718],{},[43,49109,48],{},[50,49111,100],{},[43,49113,30862],{"mathvariant":65},[50,49115,645],{"stretchy":52},[109,49117,49118],{"encoding":111},"\\{P_\\theta:\\theta\\in\\Theta\\}",[15,49120,49122,49180,49198],{"className":49121,"ariaHidden":72},[116],[15,49123,49125,49128,49131,49171,49174,49177],{"className":49124},[120],[15,49126],{"className":49127,"style":125},[124],[15,49129,636],{"className":49130},[188],[15,49132,49134,49137],{"className":49133},[129],[15,49135,1388],{"className":49136,"style":12443},[129,133],[15,49138,49140],{"className":49139},[137],[15,49141,49143,49163],{"className":49142},[141,142],[15,49144,49146,49160],{"className":49145},[146],[15,49147,49149],{"className":49148,"style":151},[150],[15,49150,49151,49154],{"style":34226},[15,49152],{"className":49153,"style":159},[158],[15,49155,49157],{"className":49156},[163,164,165,166],[15,49158,48],{"className":49159,"style":170},[129,133,166],[15,49161,175],{"className":49162},[174],[15,49164,49166],{"className":49165},[146],[15,49167,49169],{"className":49168,"style":182},[150],[15,49170],{},[15,49172],{"className":49173,"style":199},[93],[15,49175,718],{"className":49176},[203],[15,49178],{"className":49179,"style":199},[93],[15,49181,49183,49186,49189,49192,49195],{"className":49182},[120],[15,49184],{"className":49185,"style":2405},[124],[15,49187,48],{"className":49188,"style":170},[129,133],[15,49190],{"className":49191,"style":199},[93],[15,49193,100],{"className":49194},[203],[15,49196],{"className":49197,"style":199},[93],[15,49199,49201,49204,49207],{"className":49200},[120],[15,49202],{"className":49203,"style":125},[124],[15,49205,30862],{"className":49206},[129],[15,49208,645],{"className":49209},[195]," 是样本空间 ",[15,49212,49214,49227],{"className":49213,"translate":19},[23],[15,49215,49217],{"className":49216},[27],[29,49218,49219],{"xmlns":31},[34,49220,49221,49225],{},[37,49222,49223],{},[43,49224,597],{"mathvariant":596},[109,49226,600],{"encoding":111},[15,49228,49230],{"className":49229,"ariaHidden":72},[116],[15,49231,49233,49236],{"className":49232},[120],[15,49234],{"className":49235,"style":610},[124],[15,49237,597],{"className":49238,"style":615},[129,614]," 上的一族分布，参数空间 ",[15,49241,49243,49256],{"className":49242,"translate":19},[23],[15,49244,49246],{"className":49245},[27],[29,49247,49248],{"xmlns":31},[34,49249,49250,49254],{},[37,49251,49252],{},[43,49253,30862],{"mathvariant":65},[109,49255,31291],{"encoding":111},[15,49257,49259],{"className":49258,"ariaHidden":72},[116],[15,49260,49262,49265],{"className":49261},[120],[15,49263],{"className":49264,"style":610},[124],[15,49266,30862],{"className":49267},[129]," 可以是任意集合。我们观测 ",[15,49270,49272,49285],{"className":49271,"translate":19},[23],[15,49273,49275],{"className":49274},[27],[29,49276,49277],{"xmlns":31},[34,49278,49279,49283],{},[37,49280,49281],{},[43,49282,39314],{},[109,49284,39314],{"encoding":111},[15,49286,49288],{"className":49287,"ariaHidden":72},[116],[15,49289,49291,49294],{"className":49290},[120],[15,49292],{"className":49293,"style":4383},[124],[15,49295,39314],{"className":49296},[129,133]," 个独立同分布的样本",[15,49299,49301],{"className":49300,"translate":19},[18],[15,49302,49304,49363],{"className":49303,"translate":19},[23],[15,49305,49307],{"className":49306},[27],[29,49308,49309],{"xmlns":31,"display":32},[34,49310,49311,49360],{},[37,49312,49313,49319,49321,49323,49329,49331,49333,49335,49341,49343,49345,49358],{},[305,49314,49315,49317],{},[43,49316,597],{},[43,49318,39314],{},[50,49320,62],{},[50,49322,53],{"stretchy":52},[40,49324,49325,49327],{},[43,49326,597],{},[308,49328,318],{},[50,49330,91],{"separator":72},[50,49332,335],{},[50,49334,91],{"separator":72},[40,49336,49337,49339],{},[43,49338,597],{},[43,49340,39314],{},[50,49342,59],{"stretchy":52},[50,49344,991],{},[23149,49346,49347,49349,49351],{},[43,49348,1388],{},[43,49350,48],{},[37,49352,49353,49356],{},[50,49354,49355],{},"⊗",[43,49357,39314],{},[50,49359,91],{"separator":72},[109,49361,49362],{"encoding":111},"X^n=(X_1,\\dots,X_n)\\sim P_\\theta^{\\otimes n},",[15,49364,49366,49410,49529],{"className":49365,"ariaHidden":72},[116],[15,49367,49369,49372,49401,49404,49407],{"className":49368},[120],[15,49370],{"className":49371,"style":1251},[124],[15,49373,49375,49378],{"className":49374},[129],[15,49376,597],{"className":49377,"style":47958},[129,133],[15,49379,49381],{"className":49380},[137],[15,49382,49384],{"className":49383},[141],[15,49385,49387],{"className":49386},[146],[15,49388,49390],{"className":49389,"style":1251},[150],[15,49391,49392,49395],{"style":1272},[15,49393],{"className":49394,"style":159},[158],[15,49396,49398],{"className":49397},[163,164,165,166],[15,49399,39314],{"className":49400},[129,133,166],[15,49402],{"className":49403,"style":199},[93],[15,49405,62],{"className":49406},[203],[15,49408],{"className":49409,"style":199},[93],[15,49411,49413,49416,49419,49459,49462,49465,49468,49471,49474,49477,49517,49520,49523,49526],{"className":49412},[120],[15,49414],{"className":49415,"style":125},[124],[15,49417,53],{"className":49418},[188],[15,49420,49422,49425],{"className":49421},[129],[15,49423,597],{"className":49424,"style":47958},[129,133],[15,49426,49428],{"className":49427},[137],[15,49429,49431,49451],{"className":49430},[141,142],[15,49432,49434,49448],{"className":49433},[146],[15,49435,49437],{"className":49436,"style":1051},[150],[15,49438,49439,49442],{"style":47973},[15,49440],{"className":49441,"style":159},[158],[15,49443,49445],{"className":49444},[163,164,165,166],[15,49446,318],{"className":49447},[129,166],[15,49449,175],{"className":49450},[174],[15,49452,49454],{"className":49453},[146],[15,49455,49457],{"className":49456,"style":182},[150],[15,49458],{},[15,49460,91],{"className":49461},[254],[15,49463],{"className":49464,"style":262},[93],[15,49466,335],{"className":49467},[449],[15,49469],{"className":49470,"style":262},[93],[15,49472,91],{"className":49473},[254],[15,49475],{"className":49476,"style":262},[93],[15,49478,49480,49483],{"className":49479},[129],[15,49481,597],{"className":49482,"style":47958},[129,133],[15,49484,49486],{"className":49485},[137],[15,49487,49489,49509],{"className":49488},[141,142],[15,49490,49492,49506],{"className":49491},[146],[15,49493,49495],{"className":49494,"style":1163},[150],[15,49496,49497,49500],{"style":47973},[15,49498],{"className":49499,"style":159},[158],[15,49501,49503],{"className":49502},[163,164,165,166],[15,49504,39314],{"className":49505},[129,133,166],[15,49507,175],{"className":49508},[174],[15,49510,49512],{"className":49511},[146],[15,49513,49515],{"className":49514,"style":182},[150],[15,49516],{},[15,49518,59],{"className":49519},[195],[15,49521],{"className":49522,"style":199},[93],[15,49524,991],{"className":49525},[203],[15,49527],{"className":49528,"style":199},[93],[15,49530,49532,49536,49596],{"className":49531},[120],[15,49533],{"className":49534,"style":49535},[124],"height:1.1128em;vertical-align:-0.2914em;",[15,49537,49539,49542],{"className":49538},[129],[15,49540,1388],{"className":49541,"style":12443},[129,133],[15,49543,49545],{"className":49544},[137],[15,49546,49548,49587],{"className":49547},[141,142],[15,49549,49551,49584],{"className":49550},[146],[15,49552,49555,49567],{"className":49553,"style":49554},[150],"height:0.8213em;",[15,49556,49558,49561],{"style":49557},"top:-2.4086em;margin-left:-0.1389em;margin-right:0.05em;",[15,49559],{"className":49560,"style":159},[158],[15,49562,49564],{"className":49563},[163,164,165,166],[15,49565,48],{"className":49566,"style":170},[129,133,166],[15,49568,49569,49572],{"style":1272},[15,49570],{"className":49571,"style":159},[158],[15,49573,49575],{"className":49574},[163,164,165,166],[15,49576,49578,49581],{"className":49577},[129,166],[15,49579,49355],{"className":49580},[129,166],[15,49582,39314],{"className":49583},[129,133,166],[15,49585,175],{"className":49586},[174],[15,49588,49590],{"className":49589},[146],[15,49591,49594],{"className":49592,"style":49593},[150],"height:0.2914em;",[15,49595],{},[15,49597,91],{"className":49598},[254],[11,49600,49601,49602,49605,49606,1313,49741,804,49744,49862,49863,49923,49924,49952],{},"一个",[698,49603,49604],{},"估计量","是任意可测映射 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度量估计值 ",[15,49864,49866,49883],{"className":49865,"translate":19},[23],[15,49867,49869],{"className":49868},[27],[29,49870,49871],{"xmlns":31},[34,49872,49873,49881],{},[37,49874,49875],{},[46835,49876,49877,49879],{"accent":72},[43,49878,48],{},[50,49880,48471],{},[109,49882,48474],{"encoding":111},[15,49884,49886],{"className":49885,"ariaHidden":72},[116],[15,49887,49889,49892],{"className":49888},[120],[15,49890],{"className":49891,"style":48484},[124],[15,49893,49895],{"className":49894},[129,48488],[15,49896,49898],{"className":49897},[141],[15,49899,49901],{"className":49900},[146],[15,49902,49904,49912],{"className":49903,"style":48484},[150],[15,49905,49906,49909],{"style":2287},[15,49907],{"className":49908,"style":1823},[158],[15,49910,48],{"className":49911,"style":170},[129,133],[15,49913,49914,49917],{"style":48508},[15,49915],{"className":49916,"style":1823},[158],[15,49918,49920],{"className":49919,"style":48516},[48515],[15,49921,48471],{"className":49922},[129]," 与真值 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损失 ",[15,50468,50470,50516],{"className":50469,"translate":19},[23],[15,50471,50473],{"className":50472},[27],[29,50474,50475],{"xmlns":31},[34,50476,50477,50513],{},[37,50478,50479,50481,50483,50485,50487,50493,50495,50497,50499,50501,50503,50505,50511],{},[43,50480,27518],{"mathvariant":65},[50,50482,53],{"stretchy":52},[43,50484,48],{},[50,50486,91],{"separator":72},[46835,50488,50489,50491],{"accent":72},[43,50490,48],{},[50,50492,48471],{},[50,50494,59],{"stretchy":52},[50,50496,62],{},[308,50498,318],{"mathvariant":1468},[50,50500,1407],{"stretchy":52},[43,50502,48],{},[50,50504,1418],{"mathvariant":65},[46835,50506,50507,50509],{"accent":72},[43,50508,48],{},[50,50510,48471],{},[50,50512,1423],{"stretchy":52},[109,50514,50515],{"encoding":111},"\\ell(\\theta,\\hat\\theta)=\\mathbf 1[\\theta\\neq\\hat\\theta]",[15,50517,50519,50583,50640],{"className":50518,"ariaHidden":72},[116],[15,50520,50522,50525,50528,50531,50534,50537,50540,50571,50574,50577,50580],{"className":50521},[120],[15,50523],{"className":50524,"style":50020},[124],[15,50526,27518],{"className":50527},[129],[15,50529,53],{"className":50530},[188],[15,50532,48],{"className":50533,"style":170},[129,133],[15,50535,91],{"className":50536},[254],[15,50538],{"className":50539,"style":262},[93],[15,50541,50543],{"className":50542},[129,48488],[15,50544,50546],{"className":50545},[141],[15,50547,50549],{"className":50548},[146],[15,50550,50552,50560],{"className":50551,"style":48484},[150],[15,50553,50554,50557],{"style":2287},[15,50555],{"className":50556,"style":1823},[158],[15,50558,48],{"className":50559,"style":170},[129,133],[15,50561,50562,50565],{"style":48508},[15,50563],{"className":50564,"style":1823},[158],[15,50566,50568],{"className":50567,"style":48516},[48515],[15,50569,48471],{"className":50570},[129],[15,50572,59],{"className":50573},[195],[15,50575],{"className":50576,"style":199},[93],[15,50578,62],{"className":50579},[203],[15,50581],{"className":50582,"style":199},[93],[15,50584,50586,50589,50592,50595,50598,50601,50637],{"className":50585},[120],[15,50587],{"className":50588,"style":125},[124],[15,50590,318],{"className":50591},[129,1950],[15,50593,1407],{"className":50594},[188],[15,50596,48],{"className":50597,"style":170},[129,133],[15,50599],{"className":50600,"style":199},[93],[15,50602,50604,50631,50634],{"className":50603},[203],[15,50605,50607],{"className":50606},[203],[15,50608,50610],{"className":50609},[129,1670],[15,50611,50613],{"className":50612},[1674],[15,50614,50616,50619,50628],{"className":50615},[1678],[15,50617],{"className":50618,"style":1682},[124],[15,50620,50622],{"className":50621},[1686],[15,50623,50625],{"className":50624},[129],[15,50626,1693],{"className":50627},[203],[15,50629],{"className":50630},[1697],[15,50632],{"className":50633},[93,1701],[15,50635,62],{"className":50636},[203],[15,50638],{"className":50639,"style":199},[93],[15,50641,50643,50646,50677],{"className":50642},[120],[15,50644],{"className":50645,"style":50020},[124],[15,50647,50649],{"className":50648},[129,48488],[15,50650,50652],{"className":50651},[141],[15,50653,50655],{"className":50654},[146],[15,50656,50658,50666],{"className":50657,"style":48484},[150],[15,50659,50660,50663],{"style":2287},[15,50661],{"className":50662,"style":1823},[158],[15,50664,48],{"className":50665,"style":170},[129,133],[15,50667,50668,50671],{"style":48508},[15,50669],{"className":50670,"style":1823},[158],[15,50672,50674],{"className":50673,"style":48516},[48515],[15,50675,48471],{"className":50676},[129],[15,50678,1423],{"className":50679},[195],[11,50681,50682,50683,5831,50743,50771,50772,50774],{},"估计量 ",[15,50684,50686,50703],{"className":50685,"translate":19},[23],[15,50687,50689],{"className":50688},[27],[29,50690,50691],{"xmlns":31},[34,50692,50693,50701],{},[37,50694,50695],{},[46835,50696,50697,50699],{"accent":72},[43,50698,48],{},[50,50700,48471],{},[109,50702,48474],{"encoding":111},[15,50704,50706],{"className":50705,"ariaHidden":72},[116],[15,50707,50709,50712],{"className":50708},[120],[15,50710],{"className":50711,"style":48484},[124],[15,50713,50715],{"className":50714},[129,48488],[15,50716,50718],{"className":50717},[141],[15,50719,50721],{"className":50720},[146],[15,50722,50724,50732],{"className":50723,"style":48484},[150],[15,50725,50726,50729],{"style":2287},[15,50727],{"className":50728,"style":1823},[158],[15,50730,48],{"className":50731,"style":170},[129,133],[15,50733,50734,50737],{"style":48508},[15,50735],{"className":50736,"style":1823},[158],[15,50738,50740],{"className":50739,"style":48516},[48515],[15,50741,48471],{"className":50742},[129],[15,50744,50746,50759],{"className":50745,"translate":19},[23],[15,50747,50749],{"className":50748},[27],[29,50750,50751],{"xmlns":31},[34,50752,50753,50757],{},[37,50754,50755],{},[43,50756,48],{},[109,50758,512],{"encoding":111},[15,50760,50762],{"className":50761,"ariaHidden":72},[116],[15,50763,50765,50768],{"className":50764},[120],[15,50766],{"className":50767,"style":522},[124],[15,50769,48],{"className":50770,"style":170},[129,133]," 处的",[698,50773,48523],{},"是",[15,50776,50778],{"className":50777,"translate":19},[18],[15,50779,50781,50859],{"className":50780,"translate":19},[23],[15,50782,50784],{"className":50783},[27],[29,50785,50786],{"xmlns":31,"display":32},[34,50787,50788,50856],{},[37,50789,50790,50796,50798,50804,50806,50808,50824,50826,50828,50830,50832,50834,50840,50842,50848,50850,50852,50854],{},[40,50791,50792,50794],{},[43,50793,104],{},[43,50795,48],{},[50,50797,53],{"stretchy":52},[46835,50799,50800,50802],{"accent":72},[43,50801,48],{},[50,50803,48471],{},[50,50805,59],{"stretchy":52},[50,50807,62],{},[40,50809,50810,50812],{},[43,50811,26563],{"mathvariant":103},[23149,50813,50814,50816,50818],{},[43,50815,1388],{},[43,50817,48],{},[37,50819,50820,50822],{},[50,50821,49355],{},[43,50823,39314],{},[50,50825,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,50827,27518],{"mathvariant":65},[50,50829,53],{"stretchy":52},[43,50831,48],{},[50,50833,91],{"separator":72},[46835,50835,50836,50838],{"accent":72},[43,50837,48],{},[50,50839,48471],{},[50,50841,53],{"stretchy":52},[305,50843,50844,50846],{},[43,50845,597],{},[43,50847,39314],{},[50,50849,59],{"stretchy":52},[50,50851,59],{"stretchy":52},[50,50853,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,50855,107],{"mathvariant":65},[109,50857,50858],{"encoding":111},"R_\\theta(\\hat\\theta)=\\mathbb E_{P_\\theta^{\\otimes n}}\\bigl[\\ell(\\theta,\\hat\\theta(X^n))\\bigr].",[15,50860,50862,50955],{"className":50861,"ariaHidden":72},[116],[15,50863,50865,50868,50909,50912,50943,50946,50949,50952],{"className":50864},[120],[15,50866],{"className":50867,"style":50020},[124],[15,50869,50871,50874],{"className":50870},[129],[15,50872,104],{"className":50873,"style":33858},[129,133],[15,50875,50877],{"className":50876},[137],[15,50878,50880,50901],{"className":50879},[141,142],[15,50881,50883,50898],{"className":50882},[146],[15,50884,50886],{"className":50885,"style":151},[150],[15,50887,50889,50892],{"style":50888},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[15,50890],{"className":50891,"style":159},[158],[15,50893,50895],{"className":50894},[163,164,165,166],[15,50896,48],{"className":50897,"style":170},[129,133,166],[15,50899,175],{"className":50900},[174],[15,50902,50904],{"className":50903},[146],[15,50905,50907],{"className":50906,"style":182},[150],[15,50908],{},[15,50910,53],{"className":50911},[188],[15,50913,50915],{"className":50914},[129,48488],[15,50916,50918],{"className":50917},[141],[15,50919,50921],{"className":50920},[146],[15,50922,50924,50932],{"className":50923,"style":48484},[150],[15,50925,50926,50929],{"style":2287},[15,50927],{"className":50928,"style":1823},[158],[15,50930,48],{"className":50931,"style":170},[129,133],[15,50933,50934,50937],{"style":48508},[15,50935],{"className":50936,"style":1823},[158],[15,50938,50940],{"className":50939,"style":48516},[48515],[15,50941,48471],{"className":50942},[129],[15,50944,59],{"className":50945},[195],[15,50947],{"className":50948,"style":199},[93],[15,50950,62],{"className":50951},[203],[15,50953],{"className":50954,"style":199},[93],[15,50956,50958,50962,51065,51071,51074,51077,51080,51083,51086,51117,51120,51149,51152,51158],{"className":50957},[120],[15,50959],{"className":50960,"style":50961},[124],"height:1.4308em;vertical-align:-0.4729em;",[15,50963,50965,50968],{"className":50964},[129],[15,50966,26563],{"className":50967},[129,285],[15,50969,50971],{"className":50970},[137],[15,50972,50974,51056],{"className":50973},[141,142],[15,50975,50977,51053],{"className":50976},[146],[15,50978,50980],{"className":50979,"style":1594},[150],[15,50981,50983,50986],{"style":50982},"top:-2.4701em;margin-left:0em;margin-right:0.05em;",[15,50984],{"className":50985,"style":159},[158],[15,50987,50989],{"className":50988},[163,164,165,166],[15,50990,50992],{"className":50991},[129,166],[15,50993,50995,50998],{"className":50994},[129,166],[15,50996,1388],{"className":50997,"style":12443},[129,133,166],[15,50999,51001],{"className":51000},[137],[15,51002,51004,51044],{"className":51003},[141,142],[15,51005,51007,51041],{"className":51006},[146],[15,51008,51011,51023],{"className":51009,"style":51010},[150],"height:0.821em;",[15,51012,51014,51017],{"style":51013},"top:-2.1528em;margin-left:-0.1389em;margin-right:0.0714em;",[15,51015],{"className":51016,"style":15980},[158],[15,51018,51020],{"className":51019},[163,15984,229,166],[15,51021,48],{"className":51022,"style":170},[129,133,166],[15,51024,51026,51029],{"style":51025},"top:-2.9043em;margin-right:0.0714em;",[15,51027],{"className":51028,"style":15980},[158],[15,51030,51032],{"className":51031},[163,15984,229,166],[15,51033,51035,51038],{"className":51034},[129,166],[15,51036,49355],{"className":51037},[129,166],[15,51039,39314],{"className":51040},[129,133,166],[15,51042,175],{"className":51043},[174],[15,51045,51047],{"className":51046},[146],[15,51048,51051],{"className":51049,"style":51050},[150],"height:0.3472em;",[15,51052],{},[15,51054,175],{"className":51055},[174],[15,51057,51059],{"className":51058},[146],[15,51060,51063],{"className":51061,"style":51062},[150],"height:0.4729em;",[15,51064],{},[15,51066,51068],{"className":51067},[188],[15,51069,1407],{"className":51070},[228,229],[15,51072,27518],{"className":51073},[129],[15,51075,53],{"className":51076},[188],[15,51078,48],{"className":51079,"style":170},[129,133],[15,51081,91],{"className":51082},[254],[15,51084],{"className":51085,"style":262},[93],[15,51087,51089],{"className":51088},[129,48488],[15,51090,51092],{"className":51091},[141],[15,51093,51095],{"className":51094},[146],[15,51096,51098,51106],{"className":51097,"style":48484},[150],[15,51099,51100,51103],{"style":2287},[15,51101],{"className":51102,"style":1823},[158],[15,51104,48],{"className":51105,"style":170},[129,133],[15,51107,51108,51111],{"style":48508},[15,51109],{"className":51110,"style":1823},[158],[15,51112,51114],{"className":51113,"style":48516},[48515],[15,51115,48471],{"className":51116},[129],[15,51118,53],{"className":51119},[188],[15,51121,51123,51126],{"className":51122},[129],[15,51124,597],{"className":51125,"style":47958},[129,133],[15,51127,51129],{"className":51128},[137],[15,51130,51132],{"className":51131},[141],[15,51133,51135],{"className":51134},[146],[15,51136,51138],{"className":51137,"style":1251},[150],[15,51139,51140,51143],{"style":1272},[15,51141],{"className":51142,"style":159},[158],[15,51144,51146],{"className":51145},[163,164,165,166],[15,51147,39314],{"className":51148},[129,133,166],[15,51150,6149],{"className":51151},[195],[15,51153,51155],{"className":51154},[195],[15,51156,1423],{"className":51157},[228,229],[15,51159,107],{"className":51160},[129],[11,51162,51163,51164,51167,51168,51196,51197,51225],{},"注意这里的期望是对",[698,51165,51166],{},"数据","取的，",[15,51169,51171,51184],{"className":51170,"translate":19},[23],[15,51172,51174],{"className":51173},[27],[29,51175,51176],{"xmlns":31},[34,51177,51178,51182],{},[37,51179,51180],{},[43,51181,48],{},[109,51183,512],{"encoding":111},[15,51185,51187],{"className":51186,"ariaHidden":72},[116],[15,51188,51190,51193],{"className":51189},[120],[15,51191],{"className":51192,"style":522},[124],[15,51194,48],{"className":51195,"style":170},[129,133]," 固定。极小极大风险则再对 ",[15,51198,51200,51213],{"className":51199,"translate":19},[23],[15,51201,51203],{"className":51202},[27],[29,51204,51205],{"xmlns":31},[34,51206,51207,51211],{},[37,51208,51209],{},[43,51210,48],{},[109,51212,512],{"encoding":111},[15,51214,51216],{"className":51215,"ariaHidden":72},[116],[15,51217,51219,51222],{"className":51218},[120],[15,51220],{"className":51221,"style":522},[124],[15,51223,48],{"className":51224,"style":170},[129,133]," 取最坏情况：",[15,51227,51229],{"className":51228,"translate":19},[18],[15,51230,51232,51310],{"className":51231,"translate":19},[23],[15,51233,51235],{"className":51234},[27],[29,51236,51237],{"xmlns":31,"display":32},[34,51238,51239,51307],{},[37,51240,51241,51247,51249,51251,51253,51255,51257,51259,51273,51289,51295,51297,51303,51305],{},[40,51242,51243,51245],{},[43,51244,43234],{"mathvariant":28899},[43,51246,39314],{},[50,51248,53],{"stretchy":52},[43,51250,30862],{"mathvariant":65},[50,51252,91],{"separator":72},[43,51254,27518],{"mathvariant":65},[50,51256,59],{"stretchy":52},[50,51258,62],{},[2140,51260,51261,51267],{},[37,51262,51263,51265],{},[43,51264,48696],{},[50,51266,69],{},[46835,51268,51269,51271],{"accent":72},[43,51270,48],{},[50,51272,48471],{},[2140,51274,51275,51281],{},[37,51276,51277,51279],{},[43,51278,2932],{},[50,51280,69],{},[37,51282,51283,51285,51287],{},[43,51284,48],{},[50,51286,100],{},[43,51288,30862],{"mathvariant":65},[40,51290,51291,51293],{},[43,51292,104],{},[43,51294,48],{},[50,51296,53],{"stretchy":52},[46835,51298,51299,51301],{"accent":72},[43,51300,48],{},[50,51302,48471],{},[50,51304,59],{"stretchy":52},[43,51306,107],{"mathvariant":65},[109,51308,51309],{"encoding":111},"\\mathfrak M_n(\\Theta,\\ell)=\\inf_{\\hat\\theta}\\sup_{\\theta\\in\\Theta}R_\\theta(\\hat\\theta).",[15,51311,51313,51386],{"className":51312,"ariaHidden":72},[116],[15,51314,51316,51319,51359,51362,51365,51368,51371,51374,51377,51380,51383],{"className":51315},[120],[15,51317],{"className":51318,"style":125},[124],[15,51320,51322,51325],{"className":51321},[129],[15,51323,43234],{"className":51324},[129,28926],[15,51326,51328],{"className":51327},[137],[15,51329,51331,51351],{"className":51330},[141,142],[15,51332,51334,51348],{"className":51333},[146],[15,51335,51337],{"className":51336,"style":1163},[150],[15,51338,51339,51342],{"style":154},[15,51340],{"className":51341,"style":159},[158],[15,51343,51345],{"className":51344},[163,164,165,166],[15,51346,39314],{"className":51347},[129,133,166],[15,51349,175],{"className":51350},[174],[15,51352,51354],{"className":51353},[146],[15,51355,51357],{"className":51356,"style":182},[150],[15,51358],{},[15,51360,53],{"className":51361},[188],[15,51363,30862],{"className":51364},[129],[15,51366,91],{"className":51367},[254],[15,51369],{"className":51370,"style":262},[93],[15,51372,27518],{"className":51373},[129],[15,51375,59],{"className":51376},[195],[15,51378],{"className":51379,"style":199},[93],[15,51381,62],{"className":51382},[203],[15,51384],{"className":51385,"style":199},[93],[15,51387,51389,51392,51469,51472,51525,51528,51568,51571,51602,51605],{"className":51388},[120],[15,51390],{"className":51391,"style":48767},[124],[15,51393,51395],{"className":51394},[217,1874],[15,51396,51398,51461],{"className":51397},[141,142],[15,51399,51401,51458],{"className":51400},[146],[15,51402,51404,51446],{"className":51403,"style":522},[150],[15,51405,51406,51409],{"style":48782},[15,51407],{"className":51408,"style":1823},[158],[15,51410,51412],{"className":51411},[163,164,165,166],[15,51413,51415],{"className":51414},[129,166],[15,51416,51418],{"className":51417},[129,48488,166],[15,51419,51421],{"className":51420},[141],[15,51422,51424],{"className":51423},[146],[15,51425,51427,51435],{"className":51426,"style":48484},[150],[15,51428,51429,51432],{"style":48806},[15,51430],{"className":51431,"style":159},[158],[15,51433,48],{"className":51434,"style":170},[129,133,166],[15,51436,51437,51440],{"style":48815},[15,51438],{"className":51439,"style":159},[158],[15,51441,51443],{"className":51442,"style":48516},[48515],[15,51444,48471],{"className":51445},[129,166],[15,51447,51448,51451],{"style":2287},[15,51449],{"className":51450,"style":1823},[158],[15,51452,51453],{},[15,51454,48835,51456],{"className":51455},[217],[15,51457,36556],{"style":48838},[15,51459,175],{"className":51460},[174],[15,51462,51464],{"className":51463},[146],[15,51465,51467],{"className":51466,"style":48848},[150],[15,51468],{},[15,51470],{"className":51471,"style":262},[93],[15,51473,51475],{"className":51474},[217,1874],[15,51476,51478,51517],{"className":51477},[141,142],[15,51479,51481,51514],{"className":51480},[146],[15,51482,51484,51504],{"className":51483,"style":4383},[150],[15,51485,51486,51489],{"style":4386},[15,51487],{"className":51488,"style":1823},[158],[15,51490,51492],{"className":51491},[163,164,165,166],[15,51493,51495,51498,51501],{"className":51494},[129,166],[15,51496,48],{"className":51497,"style":170},[129,133,166],[15,51499,100],{"className":51500},[203,166],[15,51502,30862],{"className":51503},[129,166],[15,51505,51506,51509],{"style":2287},[15,51507],{"className":51508,"style":1823},[158],[15,51510,51511],{},[15,51512,2932],{"className":51513},[217],[15,51515,175],{"className":51516},[174],[15,51518,51520],{"className":51519},[146],[15,51521,51523],{"className":51522,"style":4424},[150],[15,51524],{},[15,51526],{"className":51527,"style":262},[93],[15,51529,51531,51534],{"className":51530},[129],[15,51532,104],{"className":51533,"style":33858},[129,133],[15,51535,51537],{"className":51536},[137],[15,51538,51540,51560],{"className":51539},[141,142],[15,51541,51543,51557],{"className":51542},[146],[15,51544,51546],{"className":51545,"style":151},[150],[15,51547,51548,51551],{"style":50888},[15,51549],{"className":51550,"style":159},[158],[15,51552,51554],{"className":51553},[163,164,165,166],[15,51555,48],{"className":51556,"style":170},[129,133,166],[15,51558,175],{"className":51559},[174],[15,51561,51563],{"className":51562},[146],[15,51564,51566],{"className":51565,"style":182},[150],[15,51567],{},[15,51569,53],{"className":51570},[188],[15,51572,51574],{"className":51573},[129,48488],[15,51575,51577],{"className":51576},[141],[15,51578,51580],{"className":51579},[146],[15,51581,51583,51591],{"className":51582,"style":48484},[150],[15,51584,51585,51588],{"style":2287},[15,51586],{"className":51587,"style":1823},[158],[15,51589,48],{"className":51590,"style":170},[129,133],[15,51592,51593,51596],{"style":48508},[15,51594],{"className":51595,"style":1823},[158],[15,51597,51599],{"className":51598,"style":48516},[48515],[15,51600,48471],{"className":51601},[129],[15,51603,59],{"className":51604},[195],[15,51606,107],{"className":51607},[129],[575,51609,51611],{"id":51610},"_22-一个先验视角","2.2 一个先验视角",[11,51613,51614,51615,51643,51644,51672],{},"极小极大风险有一个贝叶斯下界。对任意 ",[15,51616,51618,51631],{"className":51617,"translate":19},[23],[15,51619,51621],{"className":51620},[27],[29,51622,51623],{"xmlns":31},[34,51624,51625,51629],{},[37,51626,51627],{},[43,51628,30862],{"mathvariant":65},[109,51630,31291],{"encoding":111},[15,51632,51634],{"className":51633,"ariaHidden":72},[116],[15,51635,51637,51640],{"className":51636},[120],[15,51638],{"className":51639,"style":610},[124],[15,51641,30862],{"className":51642},[129]," 上的先验 ",[15,51645,51647,51660],{"className":51646,"translate":19},[23],[15,51648,51650],{"className":51649},[27],[29,51651,51652],{"xmlns":31},[34,51653,51654,51658],{},[37,51655,51656],{},[43,51657,15756],{},[109,51659,16372],{"encoding":111},[15,51661,51663],{"className":51662,"ariaHidden":72},[116],[15,51664,51666,51669],{"className":51665},[120],[15,51667],{"className":51668,"style":4383},[124],[15,51670,15756],{"className":51671,"style":887},[129,133],"，记贝叶斯风险为",[15,51674,51676],{"className":51675,"translate":19},[18],[15,51677,51679,51752],{"className":51678,"translate":19},[23],[15,51680,51682],{"className":51681},[27],[29,51683,51684],{"xmlns":31,"display":32},[34,51685,51686,51749],{},[37,51687,51688,51694,51696,51698,51700,51702,51716,51719,51725,51727,51733,51735,51737,51739,51741,51743,51745,51747],{},[40,51689,51690,51692],{},[43,51691,33882],{"mathvariant":28899},[43,51693,39314],{},[50,51695,53],{"stretchy":52},[43,51697,15756],{},[50,51699,59],{"stretchy":52},[50,51701,62],{},[2140,51703,51704,51710],{},[37,51705,51706,51708],{},[43,51707,48696],{},[50,51709,69],{},[46835,51711,51712,51714],{"accent":72},[43,51713,48],{},[50,51715,48471],{},[50,51717,51718],{},"∫",[40,51720,51721,51723],{},[43,51722,104],{},[43,51724,48],{},[50,51726,53],{"stretchy":52},[46835,51728,51729,51731],{"accent":72},[43,51730,48],{},[50,51732,48471],{},[50,51734,59],{"stretchy":52},[6852,51736,15781],{},[43,51738,15756],{},[50,51740,53],{"stretchy":52},[43,51742,10372],{},[43,51744,48],{},[50,51746,59],{"stretchy":52},[43,51748,107],{"mathvariant":65},[109,51750,51751],{"encoding":111},"\\mathfrak B_n(\\pi)=\\inf_{\\hat\\theta}\\int R_\\theta(\\hat\\theta)\\,\\pi(d\\theta).",[15,51753,51755,51819],{"className":51754,"ariaHidden":72},[116],[15,51756,51758,51761,51801,51804,51807,51810,51813,51816],{"className":51757},[120],[15,51759],{"className":51760,"style":125},[124],[15,51762,51764,51767],{"className":51763},[129],[15,51765,33882],{"className":51766},[129,28926],[15,51768,51770],{"className":51769},[137],[15,51771,51773,51793],{"className":51772},[141,142],[15,51774,51776,51790],{"className":51775},[146],[15,51777,51779],{"className":51778,"style":1163},[150],[15,51780,51781,51784],{"style":154},[15,51782],{"className":51783,"style":159},[158],[15,51785,51787],{"className":51786},[163,164,165,166],[15,51788,39314],{"className":51789},[129,133,166],[15,51791,175],{"className":51792},[174],[15,51794,51796],{"className":51795},[146],[15,51797,51799],{"className":51798,"style":182},[150],[15,51800],{},[15,51802,53],{"className":51803},[188],[15,51805,15756],{"className":51806,"style":887},[129,133],[15,51808,59],{"className":51809},[195],[15,51811],{"className":51812,"style":199},[93],[15,51814,62],{"className":51815},[203],[15,51817],{"className":51818,"style":199},[93],[15,51820,51822,51826,51903,51906,51910,51913,51953,51956,51987,51990,51993,51996,51999,52002,52005,52008],{"className":51821},[120],[15,51823],{"className":51824,"style":51825},[124],"height:2.2965em;vertical-align:-0.9365em;",[15,51827,51829],{"className":51828},[217,1874],[15,51830,51832,51895],{"className":51831},[141,142],[15,51833,51835,51892],{"className":51834},[146],[15,51836,51838,51880],{"className":51837,"style":522},[150],[15,51839,51840,51843],{"style":48782},[15,51841],{"className":51842,"style":1823},[158],[15,51844,51846],{"className":51845},[163,164,165,166],[15,51847,51849],{"className":51848},[129,166],[15,51850,51852],{"className":51851},[129,48488,166],[15,51853,51855],{"className":51854},[141],[15,51856,51858],{"className":51857},[146],[15,51859,51861,51869],{"className":51860,"style":48484},[150],[15,51862,51863,51866],{"style":48806},[15,51864],{"className":51865,"style":159},[158],[15,51867,48],{"className":51868,"style":170},[129,133,166],[15,51870,51871,51874],{"style":48815},[15,51872],{"className":51873,"style":159},[158],[15,51875,51877],{"className":51876,"style":48516},[48515],[15,51878,48471],{"className":51879},[129,166],[15,51881,51882,51885],{"style":2287},[15,51883],{"className":51884,"style":1823},[158],[15,51886,51887],{},[15,51888,48835,51890],{"className":51889},[217],[15,51891,36556],{"style":48838},[15,51893,175],{"className":51894},[174],[15,51896,51898],{"className":51897},[146],[15,51899,51901],{"className":51900,"style":48848},[150],[15,51902],{},[15,51904],{"className":51905,"style":262},[93],[15,51907,51718],{"className":51908,"style":51909},[217,1918,1919],"margin-right:0.4445em;position:relative;top:-0.0011em;",[15,51911],{"className":51912,"style":262},[93],[15,51914,51916,51919],{"className":51915},[129],[15,51917,104],{"className":51918,"style":33858},[129,133],[15,51920,51922],{"className":51921},[137],[15,51923,51925,51945],{"className":51924},[141,142],[15,51926,51928,51942],{"className":51927},[146],[15,51929,51931],{"className":51930,"style":151},[150],[15,51932,51933,51936],{"style":50888},[15,51934],{"className":51935,"style":159},[158],[15,51937,51939],{"className":51938},[163,164,165,166],[15,51940,48],{"className":51941,"style":170},[129,133,166],[15,51943,175],{"className":51944},[174],[15,51946,51948],{"className":51947},[146],[15,51949,51951],{"className":51950,"style":182},[150],[15,51952],{},[15,51954,53],{"className":51955},[188],[15,51957,51959],{"className":51958},[129,48488],[15,51960,51962],{"className":51961},[141],[15,51963,51965],{"className":51964},[146],[15,51966,51968,51976],{"className":51967,"style":48484},[150],[15,51969,51970,51973],{"style":2287},[15,51971],{"className":51972,"style":1823},[158],[15,51974,48],{"className":51975,"style":170},[129,133],[15,51977,51978,51981],{"style":48508},[15,51979],{"className":51980,"style":1823},[158],[15,51982,51984],{"className":51983,"style":48516},[48515],[15,51985,48471],{"className":51986},[129],[15,51988,59],{"className":51989},[195],[15,51991],{"className":51992,"style":262},[93],[15,51994,15756],{"className":51995,"style":887},[129,133],[15,51997,53],{"className":51998},[188],[15,52000,10372],{"className":52001},[129,133],[15,52003,48],{"className":52004,"style":170},[129,133],[15,52006,59],{"className":52007},[195],[15,52009,107],{"className":52010},[129],[11,52012,52013],{},"由于「积分 ≤ 最大值」，",[15,52015,52017],{"className":52016,"translate":19},[18],[15,52018,52020,52064],{"className":52019,"translate":19},[23],[15,52021,52023],{"className":52022},[27],[29,52024,52025],{"xmlns":31,"display":32},[34,52026,52027,52061],{},[37,52028,52029,52035,52037,52039,52041,52043,52049,52051,52053,52055,52057,52059],{},[40,52030,52031,52033],{},[43,52032,33882],{"mathvariant":28899},[43,52034,39314],{},[50,52036,53],{"stretchy":52},[43,52038,15756],{},[50,52040,59],{"stretchy":52},[50,52042,2717],{},[40,52044,52045,52047],{},[43,52046,43234],{"mathvariant":28899},[43,52048,39314],{},[50,52050,53],{"stretchy":52},[43,52052,30862],{"mathvariant":65},[50,52054,91],{"separator":72},[43,52056,27518],{"mathvariant":65},[50,52058,59],{"stretchy":52},[43,52060,107],{"mathvariant":65},[109,52062,52063],{"encoding":111},"\\mathfrak B_n(\\pi)\\le \\mathfrak M_n(\\Theta,\\ell).",[15,52065,52067,52131],{"className":52066,"ariaHidden":72},[116],[15,52068,52070,52073,52113,52116,52119,52122,52125,52128],{"className":52069},[120],[15,52071],{"className":52072,"style":125},[124],[15,52074,52076,52079],{"className":52075},[129],[15,52077,33882],{"className":52078},[129,28926],[15,52080,52082],{"className":52081},[137],[15,52083,52085,52105],{"className":52084},[141,142],[15,52086,52088,52102],{"className":52087},[146],[15,52089,52091],{"className":52090,"style":1163},[150],[15,52092,52093,52096],{"style":154},[15,52094],{"className":52095,"style":159},[158],[15,52097,52099],{"className":52098},[163,164,165,166],[15,52100,39314],{"className":52101},[129,133,166],[15,52103,175],{"className":52104},[174],[15,52106,52108],{"className":52107},[146],[15,52109,52111],{"className":52110,"style":182},[150],[15,52112],{},[15,52114,53],{"className":52115},[188],[15,52117,15756],{"className":52118,"style":887},[129,133],[15,52120,59],{"className":52121},[195],[15,52123],{"className":52124,"style":199},[93],[15,52126,2717],{"className":52127},[203],[15,52129],{"className":52130,"style":199},[93],[15,52132,52134,52137,52177,52180,52183,52186,52189,52192,52195],{"className":52133},[120],[15,52135],{"className":52136,"style":125},[124],[15,52138,52140,52143],{"className":52139},[129],[15,52141,43234],{"className":52142},[129,28926],[15,52144,52146],{"className":52145},[137],[15,52147,52149,52169],{"className":52148},[141,142],[15,52150,52152,52166],{"className":52151},[146],[15,52153,52155],{"className":52154,"style":1163},[150],[15,52156,52157,52160],{"style":154},[15,52158],{"className":52159,"style":159},[158],[15,52161,52163],{"className":52162},[163,164,165,166],[15,52164,39314],{"className":52165},[129,133,166],[15,52167,175],{"className":52168},[174],[15,52170,52172],{"className":52171},[146],[15,52173,52175],{"className":52174,"style":182},[150],[15,52176],{},[15,52178,53],{"className":52179},[188],[15,52181,30862],{"className":52182},[129],[15,52184,91],{"className":52185},[254],[15,52187],{"className":52188,"style":262},[93],[15,52190,27518],{"className":52191},[129],[15,52193,59],{"className":52194},[195],[15,52196,107],{"className":52197},[129],[11,52199,52200,52201,52204],{},"所以，",[698,52202,52203],{},"任何一个先验下的贝叶斯风险，都是极小极大风险的下界","。这个观察非常有用：下界证明往往先选一个先验（通常是有限集上的均匀分布），再算贝叶斯风险。Le Cam 与 Fano 本质上都是在做这件事，只是把贝叶斯风险进一步转化成了检验误差。",[575,52206,52208],{"id":52207},"_23-归约到有限子问题","2.3 归约到有限子问题",[11,52210,52211,52212,7606],{},"对任意有限子集 ",[15,52213,52215,52261],{"className":52214,"translate":19},[23],[15,52216,52218],{"className":52217},[27],[29,52219,52220],{"xmlns":31},[34,52221,52222,52258],{},[37,52223,52224,52230,52232,52234,52240,52242,52244,52246,52252,52254,52256],{},[40,52225,52226,52228],{},[43,52227,30862],{"mathvariant":65},[308,52229,541],{},[50,52231,62],{},[50,52233,636],{"stretchy":52},[40,52235,52236,52238],{},[43,52237,48],{},[308,52239,318],{},[50,52241,91],{"separator":72},[50,52243,335],{},[50,52245,91],{"separator":72},[40,52247,52248,52250],{},[43,52249,48],{},[43,52251,43234],{},[50,52253,645],{"stretchy":52},[50,52255,5647],{},[43,52257,30862],{"mathvariant":65},[109,52259,52260],{"encoding":111},"\\Theta_0=\\{\\theta_1,\\dots,\\theta_M\\}\\subseteq\\Theta",[15,52262,52264,52319,52438],{"className":52263,"ariaHidden":72},[116],[15,52265,52267,52270,52310,52313,52316],{"className":52266},[120],[15,52268],{"className":52269,"style":20220},[124],[15,52271,52273,52276],{"className":52272},[129],[15,52274,30862],{"className":52275},[129],[15,52277,52279],{"className":52278},[137],[15,52280,52282,52302],{"className":52281},[141,142],[15,52283,52285,52299],{"className":52284},[146],[15,52286,52288],{"className":52287,"style":1051},[150],[15,52289,52290,52293],{"style":154},[15,52291],{"className":52292,"style":159},[158],[15,52294,52296],{"className":52295},[163,164,165,166],[15,52297,541],{"className":52298},[129,166],[15,52300,175],{"className":52301},[174],[15,52303,52305],{"className":52304},[146],[15,52306,52308],{"className":52307,"style":182},[150],[15,52309],{},[15,52311],{"className":52312,"style":199},[93],[15,52314,62],{"className":52315},[203],[15,52317],{"className":52318,"style":199},[93],[15,52320,52322,52325,52328,52368,52371,52374,52377,52380,52383,52386,52426,52429,52432,52435],{"className":52321},[120],[15,52323],{"className":52324,"style":125},[124],[15,52326,636],{"className":52327},[188],[15,52329,52331,52334],{"className":52330},[129],[15,52332,48],{"className":52333,"style":170},[129,133],[15,52335,52337],{"className":52336},[137],[15,52338,52340,52360],{"className":52339},[141,142],[15,52341,52343,52357],{"className":52342},[146],[15,52344,52346],{"className":52345,"style":1051},[150],[15,52347,52348,52351],{"style":48281},[15,52349],{"className":52350,"style":159},[158],[15,52352,52354],{"className":52353},[163,164,165,166],[15,52355,318],{"className":52356},[129,166],[15,52358,175],{"className":52359},[174],[15,52361,52363],{"className":52362},[146],[15,52364,52366],{"className":52365,"style":182},[150],[15,52367],{},[15,52369,91],{"className":52370},[254],[15,52372],{"className":52373,"style":262},[93],[15,52375,335],{"className":52376},[449],[15,52378],{"className":52379,"style":262},[93],[15,52381,91],{"className":52382},[254],[15,52384],{"className":52385,"style":262},[93],[15,52387,52389,52392],{"className":52388},[129],[15,52390,48],{"className":52391,"style":170},[129,133],[15,52393,52395],{"className":52394},[137],[15,52396,52398,52418],{"className":52397},[141,142],[15,52399,52401,52415],{"className":52400},[146],[15,52402,52404],{"className":52403,"style":1525},[150],[15,52405,52406,52409],{"style":48281},[15,52407],{"className":52408,"style":159},[158],[15,52410,52412],{"className":52411},[163,164,165,166],[15,52413,43234],{"className":52414,"style":12591},[129,133,166],[15,52416,175],{"className":52417},[174],[15,52419,52421],{"className":52420},[146],[15,52422,52424],{"className":52423,"style":182},[150],[15,52425],{},[15,52427,645],{"className":52428},[195],[15,52430],{"className":52431,"style":199},[93],[15,52433,5647],{"className":52434},[203],[15,52436],{"className":52437,"style":199},[93],[15,52439,52441,52444],{"className":52440},[120],[15,52442],{"className":52443,"style":610},[124],[15,52445,30862],{"className":52446},[129],[15,52448,52450],{"className":52449,"translate":19},[18],[15,52451,52453,52505],{"className":52452,"translate":19},[23],[15,52454,52456],{"className":52455},[27],[29,52457,52458],{"xmlns":31,"display":32},[34,52459,52460,52502],{},[37,52461,52462,52468,52470,52472,52474,52476,52478,52480,52486,52488,52494,52496,52498,52500],{},[40,52463,52464,52466],{},[43,52465,43234],{"mathvariant":28899},[43,52467,39314],{},[50,52469,53],{"stretchy":52},[43,52471,30862],{"mathvariant":65},[50,52473,91],{"separator":72},[43,52475,27518],{"mathvariant":65},[50,52477,59],{"stretchy":52},[50,52479,7469],{},[40,52481,52482,52484],{},[43,52483,43234],{"mathvariant":28899},[43,52485,39314],{},[50,52487,53],{"stretchy":52},[40,52489,52490,52492],{},[43,52491,30862],{"mathvariant":65},[308,52493,541],{},[50,52495,91],{"separator":72},[43,52497,27518],{"mathvariant":65},[50,52499,59],{"stretchy":52},[43,52501,107],{"mathvariant":65},[109,52503,52504],{"encoding":111},"\\mathfrak M_n(\\Theta,\\ell)\\ge \\mathfrak M_n(\\Theta_0,\\ell).",[15,52506,52508,52581],{"className":52507,"ariaHidden":72},[116],[15,52509,52511,52514,52554,52557,52560,52563,52566,52569,52572,52575,52578],{"className":52510},[120],[15,52512],{"className":52513,"style":125},[124],[15,52515,52517,52520],{"className":52516},[129],[15,52518,43234],{"className":52519},[129,28926],[15,52521,52523],{"className":52522},[137],[15,52524,52526,52546],{"className":52525},[141,142],[15,52527,52529,52543],{"className":52528},[146],[15,52530,52532],{"className":52531,"style":1163},[150],[15,52533,52534,52537],{"style":154},[15,52535],{"className":52536,"style":159},[158],[15,52538,52540],{"className":52539},[163,164,165,166],[15,52541,39314],{"className":52542},[129,133,166],[15,52544,175],{"className":52545},[174],[15,52547,52549],{"className":52548},[146],[15,52550,52552],{"className":52551,"style":182},[150],[15,52553],{},[15,52555,53],{"className":52556},[188],[15,52558,30862],{"className":52559},[129],[15,52561,91],{"className":52562},[254],[15,52564],{"className":52565,"style":262},[93],[15,52567,27518],{"className":52568},[129],[15,52570,59],{"className":52571},[195],[15,52573],{"className":52574,"style":199},[93],[15,52576,7469],{"className":52577},[203],[15,52579],{"className":52580,"style":199},[93],[15,52582,52584,52587,52627,52630,52670,52673,52676,52679,52682],{"className":52583},[120],[15,52585],{"className":52586,"style":125},[124],[15,52588,52590,52593],{"className":52589},[129],[15,52591,43234],{"className":52592},[129,28926],[15,52594,52596],{"className":52595},[137],[15,52597,52599,52619],{"className":52598},[141,142],[15,52600,52602,52616],{"className":52601},[146],[15,52603,52605],{"className":52604,"style":1163},[150],[15,52606,52607,52610],{"style":154},[15,52608],{"className":52609,"style":159},[158],[15,52611,52613],{"className":52612},[163,164,165,166],[15,52614,39314],{"className":52615},[129,133,166],[15,52617,175],{"className":52618},[174],[15,52620,52622],{"className":52621},[146],[15,52623,52625],{"className":52624,"style":182},[150],[15,52626],{},[15,52628,53],{"className":52629},[188],[15,52631,52633,52636],{"className":52632},[129],[15,52634,30862],{"className":52635},[129],[15,52637,52639],{"className":52638},[137],[15,52640,52642,52662],{"className":52641},[141,142],[15,52643,52645,52659],{"className":52644},[146],[15,52646,52648],{"className":52647,"style":1051},[150],[15,52649,52650,52653],{"style":154},[15,52651],{"className":52652,"style":159},[158],[15,52654,52656],{"className":52655},[163,164,165,166],[15,52657,541],{"className":52658},[129,166],[15,52660,175],{"className":52661},[174],[15,52663,52665],{"className":52664},[146],[15,52666,52668],{"className":52667,"style":182},[150],[15,52669],{},[15,52671,91],{"className":52672},[254],[15,52674],{"className":52675,"style":262},[93],[15,52677,27518],{"className":52678},[129],[15,52680,59],{"className":52681},[195],[15,52683,107],{"className":52684},[129],[11,52686,52687,52688,52716,52717,52787,52788,52816,52817,52845],{},"所以我们只需要在一个有限问题上证明下界。有限问题可以看成一场 ",[15,52689,52691,52704],{"className":52690,"translate":19},[23],[15,52692,52694],{"className":52693},[27],[29,52695,52696],{"xmlns":31},[34,52697,52698,52702],{},[37,52699,52700],{},[43,52701,43234],{},[109,52703,43234],{"encoding":111},[15,52705,52707],{"className":52706,"ariaHidden":72},[116],[15,52708,52710,52713],{"className":52709},[120],[15,52711],{"className":52712,"style":610},[124],[15,52714,43234],{"className":52715,"style":12591},[129,133]," 元检验：自然以某个先验选出 ",[15,52718,52720,52738],{"className":52719,"translate":19},[23],[15,52721,52723],{"className":52722},[27],[29,52724,52725],{"xmlns":31},[34,52726,52727,52735],{},[37,52728,52729],{},[40,52730,52731,52733],{},[43,52732,48],{},[43,52734,34511],{},[109,52736,52737],{"encoding":111},"\\theta_K",[15,52739,52741],{"className":52740,"ariaHidden":72},[116],[15,52742,52744,52747],{"className":52743},[120],[15,52745],{"className":52746,"style":2183},[124],[15,52748,52750,52753],{"className":52749},[129],[15,52751,48],{"className":52752,"style":170},[129,133],[15,52754,52756],{"className":52755},[137],[15,52757,52759,52779],{"className":52758},[141,142],[15,52760,52762,52776],{"className":52761},[146],[15,52763,52765],{"className":52764,"style":1525},[150],[15,52766,52767,52770],{"style":48281},[15,52768],{"className":52769,"style":159},[158],[15,52771,52773],{"className":52772},[163,164,165,166],[15,52774,34511],{"className":52775,"style":5665},[129,133,166],[15,52777,175],{"className":52778},[174],[15,52780,52782],{"className":52781},[146],[15,52783,52785],{"className":52784,"style":182},[150],[15,52786],{},"，统计学家看到数据后要猜出 ",[15,52789,52791,52804],{"className":52790,"translate":19},[23],[15,52792,52794],{"className":52793},[27],[29,52795,52796],{"xmlns":31},[34,52797,52798,52802],{},[37,52799,52800],{},[43,52801,34511],{},[109,52803,34511],{"encoding":111},[15,52805,52807],{"className":52806,"ariaHidden":72},[116],[15,52808,52810,52813],{"className":52809},[120],[15,52811],{"className":52812,"style":610},[124],[15,52814,34511],{"className":52815,"style":5665},[129,133],"。如果连「猜出是哪一个」都做不到，那么「把 ",[15,52818,52820,52833],{"className":52819,"translate":19},[23],[15,52821,52823],{"className":52822},[27],[29,52824,52825],{"xmlns":31},[34,52826,52827,52831],{},[37,52828,52829],{},[43,52830,48],{},[109,52832,512],{"encoding":111},[15,52834,52836],{"className":52835,"ariaHidden":72},[116],[15,52837,52839,52842],{"className":52838},[120],[15,52840],{"className":52841,"style":522},[124],[15,52843,48],{"className":52844,"style":170},[129,133]," 估到很小的损失」当然也做不到。",[11,52847,52848],{},"把这句话写成不等式，就是 Le Cam 与 Fano 的工具。",[570,52850,52852],{"id":52851},"三le-cam-两点方法","三、Le Cam 两点方法",[575,52854,52856],{"id":52855},"_31-定理与证明","3.1 定理与证明",[11,52858,52859],{},"先看最简单的情形：只有两个候选参数。",[6559,52861,52863,53267],{"title":52862},"定理 3.1（Le Cam 两点方法）",[11,52864,21927,52865,7606,53011,20189,53177,53206,53207,7606],{},[15,52866,52868,52898],{"className":52867,"translate":19},[23],[15,52869,52871],{"className":52870},[27],[29,52872,52873],{"xmlns":31},[34,52874,52875,52895],{},[37,52876,52877,52883,52885,52891,52893],{},[40,52878,52879,52881],{},[43,52880,48],{},[308,52882,541],{},[50,52884,91],{"separator":72},[40,52886,52887,52889],{},[43,52888,48],{},[308,52890,318],{},[50,52892,100],{},[43,52894,30862],{"mathvariant":65},[109,52896,52897],{"encoding":111},"\\theta_0,\\theta_1\\in\\Theta",[15,52899,52901,53002],{"className":52900,"ariaHidden":72},[116],[15,52902,52904,52907,52947,52950,52953,52993,52996,52999],{"className":52903},[120],[15,52905],{"className":52906,"style":1682},[124],[15,52908,52910,52913],{"className":52909},[129],[15,52911,48],{"className":52912,"style":170},[129,133],[15,52914,52916],{"className":52915},[137],[15,52917,52919,52939],{"className":52918},[141,142],[15,52920,52922,52936],{"className":52921},[146],[15,52923,52925],{"className":52924,"style":1051},[150],[15,52926,52927,52930],{"style":48281},[15,52928],{"className":52929,"style":159},[158],[15,52931,52933],{"className":52932},[163,164,165,166],[15,52934,541],{"className":52935},[129,166],[15,52937,175],{"className":52938},[174],[15,52940,52942],{"className":52941},[146],[15,52943,52945],{"className":52944,"style":182},[150],[15,52946],{},[15,52948,91],{"className":52949},[254],[15,52951],{"className":52952,"style":262},[93],[15,52954,52956,52959],{"className":52955},[129],[15,52957,48],{"className":52958,"style":170},[129,133],[15,52960,52962],{"className":52961},[137],[15,52963,52965,52985],{"className":52964},[141,142],[15,52966,52968,52982],{"className":52967},[146],[15,52969,52971],{"className":52970,"style":1051},[150],[15,52972,52973,52976],{"style":48281},[15,52974],{"className":52975,"style":159},[158],[15,52977,52979],{"className":52978},[163,164,165,166],[15,52980,318],{"className":52981},[129,166],[15,52983,175],{"className":52984},[174],[15,52986,52988],{"className":52987},[146],[15,52989,52991],{"className":52990,"style":182},[150],[15,52992],{},[15,52994],{"className":52995,"style":199},[93],[15,52997,100],{"className":52998},[203],[15,53000],{"className":53001,"style":199},[93],[15,53003,53005,53008],{"className":53004},[120],[15,53006],{"className":53007,"style":610},[124],[15,53009,30862],{"className":53010},[129],[15,53012,53014,53052],{"className":53013,"translate":19},[23],[15,53015,53017],{"className":53016},[27],[29,53018,53019],{"xmlns":31},[34,53020,53021,53049],{},[37,53022,53023,53025,53027,53033,53035,53041,53043,53045,53047],{},[43,53024,27518],{"mathvariant":65},[50,53026,53],{"stretchy":52},[40,53028,53029,53031],{},[43,53030,48],{},[308,53032,541],{},[50,53034,91],{"separator":72},[40,53036,53037,53039],{},[43,53038,48],{},[308,53040,318],{},[50,53042,59],{"stretchy":52},[50,53044,7469],{},[308,53046,310],{},[43,53048,29564],{},[109,53050,53051],{"encoding":111},"\\ell(\\theta_0,\\theta_1)\\ge 2\\delta",[15,53053,53055,53165],{"className":53054,"ariaHidden":72},[116],[15,53056,53058,53061,53064,53067,53107,53110,53113,53153,53156,53159,53162],{"className":53057},[120],[15,53059],{"className":53060,"style":125},[124],[15,53062,27518],{"className":53063},[129],[15,53065,53],{"className":53066},[188],[15,53068,53070,53073],{"className":53069},[129],[15,53071,48],{"className":53072,"style":170},[129,133],[15,53074,53076],{"className":53075},[137],[15,53077,53079,53099],{"className":53078},[141,142],[15,53080,53082,53096],{"className":53081},[146],[15,53083,53085],{"className":53084,"style":1051},[150],[15,53086,53087,53090],{"style":48281},[15,53088],{"className":53089,"style":159},[158],[15,53091,53093],{"className":53092},[163,164,165,166],[15,53094,541],{"className":53095},[129,166],[15,53097,175],{"className":53098},[174],[15,53100,53102],{"className":53101},[146],[15,53103,53105],{"className":53104,"style":182},[150],[15,53106],{},[15,53108,91],{"className":53109},[254],[15,53111],{"className":53112,"style":262},[93],[15,53114,53116,53119],{"className":53115},[129],[15,53117,48],{"className":53118,"style":170},[129,133],[15,53120,53122],{"className":53121},[137],[15,53123,53125,53145],{"className":53124},[141,142],[15,53126,53128,53142],{"className":53127},[146],[15,53129,53131],{"className":53130,"style":1051},[150],[15,53132,53133,53136],{"style":48281},[15,53134],{"className":53135,"style":159},[158],[15,53137,53139],{"className":53138},[163,164,165,166],[15,53140,318],{"className":53141},[129,166],[15,53143,175],{"className":53144},[174],[15,53146,53148],{"className":53147},[146],[15,53149,53151],{"className":53150,"style":182},[150],[15,53152],{},[15,53154,59],{"className":53155},[195],[15,53157],{"className":53158,"style":199},[93],[15,53160,7469],{"className":53161},[203],[15,53163],{"className":53164,"style":199},[93],[15,53166,53168,53171,53174],{"className":53167},[120],[15,53169],{"className":53170,"style":522},[124],[15,53172,310],{"className":53173},[129],[15,53175,29564],{"className":53176,"style":29845},[129,133],[15,53178,53180,53194],{"className":53179,"translate":19},[23],[15,53181,53183],{"className":53182},[27],[29,53184,53185],{"xmlns":31},[34,53186,53187,53191],{},[37,53188,53189],{},[43,53190,27518],{"mathvariant":65},[109,53192,53193],{"encoding":111},"\\ell",[15,53195,53197],{"className":53196,"ariaHidden":72},[116],[15,53198,53200,53203],{"className":53199},[120],[15,53201],{"className":53202,"style":522},[124],[15,53204,27518],{"className":53205},[129]," 满足三角不等式。则对任意估计量 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E_{P_0}\\ell(\\theta_0,\\hat\\theta)+\\mathbb E_{P_1}\\ell(\\theta_1,\\hat\\theta)\n\\ge \\\\ \\delta\\bigl[1+P_0(\\psi=1)-P_1(\\psi=1)\\bigr]\n\\ge \\\\ \\delta\\bigl(1-\\lVert P_0-P_1\\rVert_{\\mathrm{TV}}\\bigr).",[15,55773,55775,55956,56134,56137,56164,56225,56246,56307,56331,56334,56361,56419],{"className":55774,"ariaHidden":72},[116],[15,55776,55778,55781,55861,55864,55867,55907,55910,55913,55944,55947,55950,55953],{"className":55777},[120],[15,55779],{"className":55780,"style":55076},[124],[15,55782,55784,55787],{"className":55783},[129],[15,55785,26563],{"className":55786},[129,285],[15,55788,55790],{"className":55789},[137],[15,55791,55793,55853],{"className":55792},[141,142],[15,55794,55796,55850],{"className":55795},[146],[15,55797,55799],{"className":55798,"style":1525},[150],[15,55800,55801,55804],{"style":154},[15,55802],{"className":55803,"style":159},[158],[15,55805,55807],{"className":55806},[163,164,165,166],[15,55808,55810],{"className":55809},[129,166],[15,55811,55813,55816],{"className":55812},[129,166],[15,55814,1388],{"className":55815,"style":12443},[129,133,166],[15,55817,55819],{"className":55818},[137],[15,55820,55822,55842],{"className":55821},[141,142],[15,55823,55825,55839],{"className":55824},[146],[15,55826,55828],{"className":55827,"style":46983},[150],[15,55829,55830,55833],{"style":55126},[15,55831],{"className":55832,"style":15980},[158],[15,55834,55836],{"className":55835},[163,15984,229,166],[15,55837,541],{"className":55838},[129,166],[15,55840,175],{"className":55841},[174],[15,55843,55845],{"className":55844},[146],[15,55846,55848],{"className":55847,"style":15997},[150],[15,55849],{},[15,55851,175],{"className":55852},[174],[15,55854,55856],{"className":55855},[146],[15,55857,55859],{"className":55858,"style":55156},[150],[15,55860],{},[15,55862,27518],{"className":55863},[129],[15,55865,53],{"className":55866},[188],[15,55868,55870,55873],{"className":55869},[129],[15,55871,48],{"className":55872,"style":170},[129,133],[15,55874,55876],{"className":55875},[137],[15,55877,55879,55899],{"className":55878},[141,142],[15,55880,55882,55896],{"className":55881},[146],[15,55883,55885],{"className":55884,"style":1051},[150],[15,55886,55887,55890],{"style":48281},[15,55888],{"className":55889,"style":159},[158],[15,55891,55893],{"className":55892},[163,164,165,166],[15,55894,541],{"className":55895},[129,166],[15,55897,175],{"className":55898},[174],[15,55900,55902],{"className":55901},[146],[15,55903,55905],{"className":55904,"style":182},[150],[15,55906],{},[15,55908,91],{"className":55909},[254],[15,55911],{"className":55912,"style":262},[93],[15,55914,55916],{"className":55915},[129,48488],[15,55917,55919],{"className":55918},[141],[15,55920,55922],{"className":55921},[146],[15,55923,55925,55933],{"className":55924,"style":48484},[150],[15,55926,55927,55930],{"style":2287},[15,55928],{"className":55929,"style":1823},[158],[15,55931,48],{"className":55932,"style":170},[129,133],[15,55934,55935,55938],{"style":48508},[15,55936],{"className":55937,"style":1823},[158],[15,55939,55941],{"className":55940,"style":48516},[48515],[15,55942,48471],{"className":55943},[129],[15,55945,59],{"className":55946},[195],[15,55948],{"className":55949,"style":3277},[93],[15,55951,2682],{"className":55952},[2436],[15,55954],{"className":55955,"style":3277},[93],[15,55957,55959,55962,56042,56045,56048,56088,56091,56094,56125,56128,56131],{"className":55958},[120],[15,55960],{"className":55961,"style":55076},[124],[15,55963,55965,55968],{"className":55964},[129],[15,55966,26563],{"className":55967},[129,285],[15,55969,55971],{"className":55970},[137],[15,55972,55974,56034],{"className":55973},[141,142],[15,55975,55977,56031],{"className":55976},[146],[15,55978,55980],{"className":55979,"style":1525},[150],[15,55981,55982,55985],{"style":154},[15,55983],{"className":55984,"style":159},[158],[15,55986,55988],{"className":55987},[163,164,165,166],[15,55989,55991],{"className":55990},[129,166],[15,55992,55994,55997],{"className":55993},[129,166],[15,55995,1388],{"className":55996,"style":12443},[129,133,166],[15,55998,56000],{"className":55999},[137],[15,56001,56003,56023],{"className":56002},[141,142],[15,56004,56006,56020],{"className":56005},[146],[15,56007,56009],{"className":56008,"style":46983},[150],[15,56010,56011,56014],{"style":55126},[15,56012],{"className":56013,"style":15980},[158],[15,56015,56017],{"className":56016},[163,15984,229,166],[15,56018,318],{"className":56019},[129,166],[15,56021,175],{"className":56022},[174],[15,56024,56026],{"className":56025},[146],[15,56027,56029],{"className":56028,"style":15997},[150],[15,56030],{},[15,56032,175],{"className":56033},[174],[15,56035,56037],{"className":56036},[146],[15,56038,56040],{"className":56039,"style":55156},[150],[15,56041],{},[15,56043,27518],{"className":56044},[129],[15,56046,53],{"className":56047},[188],[15,56049,56051,56054],{"className":56050},[129],[15,56052,48],{"className":56053,"style":170},[129,133],[15,56055,56057],{"className":56056},[137],[15,56058,56060,56080],{"className":56059},[141,142],[15,56061,56063,56077],{"className":56062},[146],[15,56064,56066],{"className":56065,"style":1051},[150],[15,56067,56068,56071],{"style":48281},[15,56069],{"className":56070,"style":159},[158],[15,56072,56074],{"className":56073},[163,164,165,166],[15,56075,318],{"className":56076},[129,166],[15,56078,175],{"className":56079},[174],[15,56081,56083],{"className":56082},[146],[15,56084,56086],{"className":56085,"style":182},[150],[15,56087],{},[15,56089,91],{"className":56090},[254],[15,56092],{"className":56093,"style":262},[93],[15,56095,56097],{"className":56096},[129,48488],[15,56098,56100],{"className":56099},[141],[15,56101,56103],{"className":56102},[146],[15,56104,56106,56114],{"className":56105,"style":48484},[150],[15,56107,56108,56111],{"style":2287},[15,56109],{"className":56110,"style":1823},[158],[15,56112,48],{"className":56113,"style":170},[129,133],[15,56115,56116,56119],{"style":48508},[15,56117],{"className":56118,"style":1823},[158],[15,56120,56122],{"className":56121,"style":48516},[48515],[15,56123,48471],{"className":56124},[129],[15,56126,59],{"className":56127},[195],[15,56129],{"className":56130,"style":199},[93],[15,56132,7469],{"className":56133},[203],[15,56135],{"className":56136},[93,24710],[15,56138,56140,56143,56146,56152,56155,56158,56161],{"className":56139},[120],[15,56141],{"className":56142,"style":213},[124],[15,56144,29564],{"className":56145,"style":29845},[129,133],[15,56147,56149],{"className":56148},[188],[15,56150,1407],{"className":56151},[228,229],[15,56153,318],{"className":56154},[129],[15,56156],{"className":56157,"style":3277},[93],[15,56159,2682],{"className":56160},[2436],[15,56162],{"className":56163,"style":3277},[93],[15,56165,56167,56170,56210,56213,56216,56219,56222],{"className":56166},[120],[15,56168],{"className":56169,"style":125},[124],[15,56171,56173,56176],{"className":56172},[129],[15,56174,1388],{"className":56175,"style":12443},[129,133],[15,56177,56179],{"className":56178},[137],[15,56180,56182,56202],{"className":56181},[141,142],[15,56183,56185,56199],{"className":56184},[146],[15,56186,56188],{"className":56187,"style":1051},[150],[15,56189,56190,56193],{"style":34226},[15,56191],{"className":56192,"style":159},[158],[15,56194,56196],{"className":56195},[163,164,165,166],[15,56197,541],{"className":56198},[129,166],[15,56200,175],{"className":56201},[174],[15,56203,56205],{"className":56204},[146],[15,56206,56208],{"className":56207,"style":182},[150],[15,56209],{},[15,56211,53],{"className":56212},[188],[15,56214,54067],{"className":56215,"style":887},[129,133],[15,56217],{"className":56218,"style":199},[93],[15,56220,62],{"className":56221},[203],[15,56223],{"className":56224,"style":199},[93],[15,56226,56228,56231,56234,56237,56240,56243],{"className":56227},[120],[15,56229],{"className":56230,"style":125},[124],[15,56232,318],{"className":56233},[129],[15,56235,59],{"className":56236},[195],[15,56238],{"className":56239,"style":3277},[93],[15,56241,315],{"className":56242},[2436],[15,56244],{"className":56245,"style":3277},[93],[15,56247,56249,56252,56292,56295,56298,56301,56304],{"className":56248},[120],[15,56250],{"className":56251,"style":125},[124],[15,56253,56255,56258],{"className":56254},[129],[15,56256,1388],{"className":56257,"style":12443},[129,133],[15,56259,56261],{"className":56260},[137],[15,56262,56264,56284],{"className":56263},[141,142],[15,56265,56267,56281],{"className":56266},[146],[15,56268,56270],{"className":56269,"style":1051},[150],[15,56271,56272,56275],{"style":34226},[15,56273],{"className":56274,"style":159},[158],[15,56276,56278],{"className":56277},[163,164,165,166],[15,56279,318],{"className":56280},[129,166],[15,56282,175],{"className":56283},[174],[15,56285,56287],{"className":56286},[146],[15,56288,56290],{"className":56289,"style":182},[150],[15,56291],{},[15,56293,53],{"className":56294},[188],[15,56296,54067],{"className":56297,"style":887},[129,133],[15,56299],{"className":56300,"style":199},[93],[15,56302,62],{"className":56303},[203],[15,56305],{"className":56306,"style":199},[93],[15,56308,56310,56313,56316,56319,56325,56328],{"className":56309},[120],[15,56311],{"className":56312,"style":213},[124],[15,56314,318],{"className":56315},[129],[15,56317,59],{"className":56318},[195],[15,56320,56322],{"className":56321},[195],[15,56323,1423],{"className":56324},[228,229],[15,56326],{"className":56327,"style":199},[93],[15,56329,7469],{"className":56330},[203],[15,56332],{"className":56333},[93,24710],[15,56335,56337,56340,56343,56349,56352,56355,56358],{"className":56336},[120],[15,56338],{"className":56339,"style":213},[124],[15,56341,29564],{"className":56342,"style":29845},[129,133],[15,56344,56346],{"className":56345},[188],[15,56347,53],{"className":56348},[228,229],[15,56350,318],{"className":56351},[129],[15,56353],{"className":56354,"style":3277},[93],[15,56356,315],{"className":56357},[2436],[15,56359],{"className":56360,"style":3277},[93],[15,56362,56364,56367,56370,56410,56413,56416],{"className":56363},[120],[15,56365],{"className":56366,"style":125},[124],[15,56368,33810],{"className":56369},[188],[15,56371,56373,56376],{"className":56372},[129],[15,56374,1388],{"className":56375,"style":12443},[129,133],[15,56377,56379],{"className":56378},[137],[15,56380,56382,56402],{"className":56381},[141,142],[15,56383,56385,56399],{"className":56384},[146],[15,56386,56388],{"className":56387,"style":1051},[150],[15,56389,56390,56393],{"style":34226},[15,56391],{"className":56392,"style":159},[158],[15,56394,56396],{"className":56395},[163,164,165,166],[15,56397,541],{"className":56398},[129,166],[15,56400,175],{"className":56401},[174],[15,56403,56405],{"className":56404},[146],[15,56406,56408],{"className":56407,"style":182},[150],[15,56409],{},[15,56411],{"className":56412,"style":3277},[93],[15,56414,315],{"className":56415},[2436],[15,56417],{"className":56418,"style":3277},[93],[15,56420,56422,56425,56465,56511,56517],{"className":56421},[120],[15,56423],{"className":56424,"style":213},[124],[15,56426,56428,56431],{"className":56427},[129],[15,56429,1388],{"className":56430,"style":12443},[129,133],[15,56432,56434],{"className":56433},[137],[15,56435,56437,56457],{"className":56436},[141,142],[15,56438,56440,56454],{"className":56439},[146],[15,56441,56443],{"className":56442,"style":1051},[150],[15,56444,56445,56448],{"style":34226},[15,56446],{"className":56447,"style":159},[158],[15,56449,56451],{"className":56450},[163,164,165,166],[15,56452,318],{"className":56453},[129,166],[15,56455,175],{"className":56456},[174],[15,56458,56460],{"className":56459},[146],[15,56461,56463],{"className":56462,"style":182},[150],[15,56464],{},[15,56466,56468,56471],{"className":56467},[195],[15,56469,33810],{"className":56470},[195],[15,56472,56474],{"className":56473},[137],[15,56475,56477,56503],{"className":56476},[141,142],[15,56478,56480,56500],{"className":56479},[146],[15,56481,56483],{"className":56482,"style":1525},[150],[15,56484,56485,56488],{"style":154},[15,56486],{"className":56487,"style":159},[158],[15,56489,56491],{"className":56490},[163,164,165,166],[15,56492,56494],{"className":56493},[129,166],[15,56495,56497],{"className":56496},[129,166],[15,56498,54025],{"className":56499,"style":2246},[129,221,166],[15,56501,175],{"className":56502},[174],[15,56504,56506],{"className":56505},[146],[15,56507,56509],{"className":56508,"style":182},[150],[15,56510],{},[15,56512,56514],{"className":56513},[195],[15,56515,59],{"className":56516},[228,229],[15,56518,107],{"className":56519},[129],[11,56521,56522,56523,56604],{},"最后，最大值不小于平均值，得到 ",[15,56524,56526,56558],{"className":56525,"translate":19},[23],[15,56527,56529],{"className":56528},[27],[29,56530,56531],{"xmlns":31},[34,56532,56533,56555],{},[37,56534,56535,56537,56539,56541,56543,56549,56551,56553],{},[43,56536,29564],{},[50,56538,53],{"stretchy":52},[308,56540,318],{},[50,56542,315],{},[37,56544,56545,56547],{},[43,56546,53394],{"mathvariant":65},[43,56548,53397],{"mathvariant":65},[50,56550,59],{"stretchy":52},[43,56552,544],{"mathvariant":65},[308,56554,310],{},[109,56556,56557],{"encoding":111},"\\delta(1-\\mathrm{TV})\u002F2",[15,56559,56561,56585],{"className":56560,"ariaHidden":72},[116],[15,56562,56564,56567,56570,56573,56576,56579,56582],{"className":56563},[120],[15,56565],{"className":56566,"style":125},[124],[15,56568,29564],{"className":56569,"style":29845},[129,133],[15,56571,53],{"className":56572},[188],[15,56574,318],{"className":56575},[129],[15,56577],{"className":56578,"style":3277},[93],[15,56580,315],{"className":56581},[2436],[15,56583],{"className":56584,"style":3277},[93],[15,56586,56588,56591,56597,56600],{"className":56587},[120],[15,56589],{"className":56590,"style":125},[124],[15,56592,56594],{"className":56593},[129],[15,56595,54025],{"className":56596,"style":2246},[129,221],[15,56598,59],{"className":56599},[195],[15,56601,56603],{"className":56602},[129],"\u002F2","。证毕。",[11,56606,56607,56608,30628,56636,56664,56665,56721],{},"如果取 ",[15,56609,56611,56624],{"className":56610,"translate":19},[23],[15,56612,56614],{"className":56613},[27],[29,56615,56616],{"xmlns":31},[34,56617,56618,56622],{},[37,56619,56620],{},[308,56621,541],{},[109,56623,541],{"encoding":111},[15,56625,56627],{"className":56626,"ariaHidden":72},[116],[15,56628,56630,56633],{"className":56629},[120],[15,56631],{"className":56632,"style":4886},[124],[15,56634,541],{"className":56635},[129],[15,56637,56639,56652],{"className":56638,"translate":19},[23],[15,56640,56642],{"className":56641},[27],[29,56643,56644],{"xmlns":31},[34,56645,56646,56650],{},[37,56647,56648],{},[308,56649,318],{},[109,56651,318],{"encoding":111},[15,56653,56655],{"className":56654,"ariaHidden":72},[116],[15,56656,56658,56661],{"className":56657},[120],[15,56659],{"className":56660,"style":4886},[124],[15,56662,318],{"className":56663},[129]," 损失、",[15,56666,56668,56690],{"className":56667,"translate":19},[23],[15,56669,56671],{"className":56670},[27],[29,56672,56673],{"xmlns":31},[34,56674,56675,56687],{},[37,56676,56677,56679,56681,56683,56685],{},[43,56678,29564],{},[50,56680,62],{},[308,56682,318],{},[43,56684,544],{"mathvariant":65},[308,56686,310],{},[109,56688,56689],{"encoding":111},"\\delta=1\u002F2",[15,56691,56693,56711],{"className":56692,"ariaHidden":72},[116],[15,56694,56696,56699,56702,56705,56708],{"className":56695},[120],[15,56697],{"className":56698,"style":522},[124],[15,56700,29564],{"className":56701,"style":29845},[129,133],[15,56703],{"className":56704,"style":199},[93],[15,56706,62],{"className":56707},[203],[15,56709],{"className":56710,"style":199},[93],[15,56712,56714,56717],{"className":56713},[120],[15,56715],{"className":56716,"style":125},[124],[15,56718,56720],{"className":56719},[129],"1\u002F2","，就得到最常见的检验形式：",[15,56723,56725],{"className":56724,"translate":19},[18],[15,56726,56728,56862],{"className":56727,"translate":19},[23],[15,56729,56731],{"className":56730},[27],[29,56732,56733],{"xmlns":31,"display":32},[34,56734,56735,56859],{},[37,56736,56737,56751,56771,56773,56785,56787,56793,56795,56801,56803,56805,56811,56813,56815,56817,56819,56831,56833,56845,56855,56857],{},[2140,56738,56739,56745],{},[37,56740,56741,56743],{},[43,56742,48696],{},[50,56744,69],{},[46835,56746,56747,56749],{"accent":72},[43,56748,48],{},[50,56750,48471],{},[2140,56752,56753,56759],{},[37,56754,56755,56757],{},[43,56756,19696],{},[50,56758,69],{},[37,56760,56761,56763,56765,56767,56769],{},[43,56762,1459],{},[50,56764,62],{},[308,56766,541],{},[50,56768,91],{"separator":72},[308,56770,318],{},[6852,56772,6854],{},[23149,56774,56775,56777,56779],{},[43,56776,1388],{},[43,56778,1459],{},[37,56780,56781,56783],{},[50,56782,49355],{},[43,56784,39314],{},[50,56786,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[46835,56788,56789,56791],{"accent":72},[43,56790,48],{},[50,56792,48471],{},[50,56794,1418],{"mathvariant":65},[40,56796,56797,56799],{},[43,56798,48],{},[43,56800,1459],{},[50,56802,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,56804,7469],{},[1443,56806,56807,56809],{},[308,56808,318],{},[308,56810,310],{},[50,56812,53],{"fence":72,"stretchy":72,"minsize":45693,"maxsize":45693},[308,56814,318],{},[50,56816,315],{},[50,56818,33810],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[23149,56820,56821,56823,56825],{},[43,56822,1388],{},[308,56824,541],{},[37,56826,56827,56829],{},[50,56828,49355],{},[43,56830,39314],{},[50,56832,315],{},[23149,56834,56835,56837,56839],{},[43,56836,1388],{},[308,56838,318],{},[37,56840,56841,56843],{},[50,56842,49355],{},[43,56844,39314],{},[40,56846,56847,56849],{},[50,56848,33810],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[37,56850,56851,56853],{},[43,56852,53394],{"mathvariant":65},[43,56854,53397],{"mathvariant":65},[50,56856,59],{"fence":72,"stretchy":72,"minsize":45693,"maxsize":45693},[43,56858,107],{"mathvariant":65},[109,56860,56861],{"encoding":111},"\\inf_{\\hat\\theta}\\max_{i=0,1}\\ P_i^{\\otimes n}\\bigl(\\hat\\theta\\neq\\theta_i\\bigr)\n\\ge\n\\frac12\\Bigl(1-\\bigl\\lVert P_0^{\\otimes n}-P_1^{\\otimes n}\\bigr\\rVert_{\\mathrm{TV}}\\Bigr).",[15,56863,56865,57155,57216,57303,57412],{"className":56864,"ariaHidden":72},[116],[15,56866,56868,56872,56949,56952,57011,57014,57017,57076,57082,57113,57116,57152],{"className":56867},[120],[15,56869],{"className":56870,"style":56871},[124],"height:1.8944em;vertical-align:-0.9365em;",[15,56873,56875],{"className":56874},[217,1874],[15,56876,56878,56941],{"className":56877},[141,142],[15,56879,56881,56938],{"className":56880},[146],[15,56882,56884,56926],{"className":56883,"style":522},[150],[15,56885,56886,56889],{"style":48782},[15,56887],{"className":56888,"style":1823},[158],[15,56890,56892],{"className":56891},[163,164,165,166],[15,56893,56895],{"className":56894},[129,166],[15,56896,56898],{"className":56897},[129,48488,166],[15,56899,56901],{"className":56900},[141],[15,56902,56904],{"className":56903},[146],[15,56905,56907,56915],{"className":56906,"style":48484},[150],[15,56908,56909,56912],{"style":48806},[15,56910],{"className":56911,"style":159},[158],[15,56913,48],{"className":56914,"style":170},[129,133,166],[15,56916,56917,56920],{"style":48815},[15,56918],{"className":56919,"style":159},[158],[15,56921,56923],{"className":56922,"style":48516},[48515],[15,56924,48471],{"className":56925},[129,166],[15,56927,56928,56931],{"style":2287},[15,56929],{"className":56930,"style":1823},[158],[15,56932,56933],{},[15,56934,48835,56936],{"className":56935},[217],[15,56937,36556],{"style":48838},[15,56939,175],{"className":56940},[174],[15,56942,56944],{"className":56943},[146],[15,56945,56947],{"className":56946,"style":48848},[150],[15,56948],{},[15,56950],{"className":56951,"style":262},[93],[15,56953,56955],{"className":56954},[217,1874],[15,56956,56958,57003],{"className":56957},[141,142],[15,56959,56961,57000],{"className":56960},[146],[15,56962,56964,56990],{"className":56963,"style":4383},[150],[15,56965,56966,56969],{"style":53429},[15,56967],{"className":56968,"style":1823},[158],[15,56970,56972],{"className":56971},[163,164,165,166],[15,56973,56975,56978,56981,56984,56987],{"className":56974},[129,166],[15,56976,1459],{"className":56977},[129,133,166],[15,56979,62],{"className":56980},[203,166],[15,56982,541],{"className":56983},[129,166],[15,56985,91],{"className":56986},[254,166],[15,56988,318],{"className":56989},[129,166],[15,56991,56992,56995],{"style":2287},[15,56993],{"className":56994,"style":1823},[158],[15,56996,56997],{},[15,56998,19696],{"className":56999},[217],[15,57001,175],{"className":57002},[174],[15,57004,57006],{"className":57005},[146],[15,57007,57009],{"className":57008,"style":53473},[150],[15,57010],{},[15,57012,6854],{"className":57013},[93],[15,57015],{"className":57016,"style":262},[93],[15,57018,57020,57023],{"className":57019},[129],[15,57021,1388],{"className":57022,"style":12443},[129,133],[15,57024,57026],{"className":57025},[137],[15,57027,57029,57067],{"className":57028},[141,142],[15,57030,57032,57064],{"className":57031},[146],[15,57033,57035,57047],{"className":57034,"style":49554},[150],[15,57036,57038,57041],{"style":57037},"top:-2.433em;margin-left:-0.1389em;margin-right:0.05em;",[15,57039],{"className":57040,"style":159},[158],[15,57042,57044],{"className":57043},[163,164,165,166],[15,57045,1459],{"className":57046},[129,133,166],[15,57048,57049,57052],{"style":1272},[15,57050],{"className":57051,"style":159},[158],[15,57053,57055],{"className":57054},[163,164,165,166],[15,57056,57058,57061],{"className":57057},[129,166],[15,57059,49355],{"className":57060},[129,166],[15,57062,39314],{"className":57063},[129,133,166],[15,57065,175],{"className":57066},[174],[15,57068,57070],{"className":57069},[146],[15,57071,57074],{"className":57072,"style":57073},[150],"height:0.267em;",[15,57075],{},[15,57077,57079],{"className":57078},[188],[15,57080,53],{"className":57081},[228,229],[15,57083,57085],{"className":57084},[129,48488],[15,57086,57088],{"className":57087},[141],[15,57089,57091],{"className":57090},[146],[15,57092,57094,57102],{"className":57093,"style":48484},[150],[15,57095,57096,57099],{"style":2287},[15,57097],{"className":57098,"style":1823},[158],[15,57100,48],{"className":57101,"style":170},[129,133],[15,57103,57104,57107],{"style":48508},[15,57105],{"className":57106,"style":1823},[158],[15,57108,57110],{"className":57109,"style":48516},[48515],[15,57111,48471],{"className":57112},[129],[15,57114],{"className":57115,"style":199},[93],[15,57117,57119,57146,57149],{"className":57118},[203],[15,57120,57122],{"className":57121},[203],[15,57123,57125],{"className":57124},[129,1670],[15,57126,57128],{"className":57127},[1674],[15,57129,57131,57134,57143],{"className":57130},[1678],[15,57132],{"className":57133,"style":1682},[124],[15,57135,57137],{"className":57136},[1686],[15,57138,57140],{"className":57139},[129],[15,57141,1693],{"className":57142},[203],[15,57144],{"className":57145},[1697],[15,57147],{"className":57148},[93,1701],[15,57150,62],{"className":57151},[203],[15,57153],{"className":57154,"style":199},[93],[15,57156,57158,57161,57201,57207,57210,57213],{"className":57157},[120],[15,57159],{"className":57160,"style":213},[124],[15,57162,57164,57167],{"className":57163},[129],[15,57165,48],{"className":57166,"style":170},[129,133],[15,57168,57170],{"className":57169},[137],[15,57171,57173,57193],{"className":57172},[141,142],[15,57174,57176,57190],{"className":57175},[146],[15,57177,57179],{"className":57178,"style":1981},[150],[15,57180,57181,57184],{"style":48281},[15,57182],{"className":57183,"style":159},[158],[15,57185,57187],{"className":57186},[163,164,165,166],[15,57188,1459],{"className":57189},[129,133,166],[15,57191,175],{"className":57192},[174],[15,57194,57196],{"className":57195},[146],[15,57197,57199],{"className":57198,"style":182},[150],[15,57200],{},[15,57202,57204],{"className":57203},[195],[15,57205,59],{"className":57206},[228,229],[15,57208],{"className":57209,"style":199},[93],[15,57211,7469],{"className":57212},[203],[15,57214],{"className":57215,"style":199},[93],[15,57217,57219,57223,57285,57291,57294,57297,57300],{"className":57218},[120],[15,57220],{"className":57221,"style":57222},[124],"height:2.0074em;vertical-align:-0.686em;",[15,57224,57226,57229,57282],{"className":57225},[129],[15,57227],{"className":57228},[188,1803],[15,57230,57232],{"className":57231},[1443],[15,57233,57235,57274],{"className":57234},[141,142],[15,57236,57238,57271],{"className":57237},[146],[15,57239,57241,57252,57260],{"className":57240,"style":1816},[150],[15,57242,57243,57246],{"style":1819},[15,57244],{"className":57245,"style":1823},[158],[15,57247,57249],{"className":57248},[129],[15,57250,310],{"className":57251},[129],[15,57253,57254,57257],{"style":1832},[15,57255],{"className":57256,"style":1823},[158],[15,57258],{"className":57259,"style":1840},[1839],[15,57261,57262,57265],{"style":1843},[15,57263],{"className":57264,"style":1823},[158],[15,57266,57268],{"className":57267},[129],[15,57269,318],{"className":57270},[129],[15,57272,175],{"className":57273},[174],[15,57275,57277],{"className":57276},[146],[15,57278,57280],{"className":57279,"style":1862},[150],[15,57281],{},[15,57283],{"className":57284},[195,1803],[15,57286,57288],{"className":57287},[188],[15,57289,53],{"className":57290},[228,24450],[15,57292,318],{"className":57293},[129],[15,57295],{"className":57296,"style":3277},[93],[15,57298,315],{"className":57299},[2436],[15,57301],{"className":57302,"style":3277},[93],[15,57304,57306,57309,57346,57403,57406,57409],{"className":57305},[120],[15,57307],{"className":57308,"style":213},[124],[15,57310,57312],{"className":57311},[188],[15,57313,57315],{"className":57314},[228,4433],[15,57316,57318,57338],{"className":57317},[141,142],[15,57319,57321,57335],{"className":57320},[146],[15,57322,57324],{"className":57323,"style":4443},[150],[15,57325,57326,57329],{"style":4446},[15,57327],{"className":57328,"style":4450},[158],[15,57330,57331],{"style":53804},[4455,57332,57333],{"xmlns":4457,"width":53807,"height":73,"viewBox":53808},[4461,57334],{"d":53811},[15,57336,175],{"className":57337},[174],[15,57339,57341],{"className":57340},[146],[15,57342,57344],{"className":57343,"style":4473},[150],[15,57345],{},[15,57347,57349,57352],{"className":57348},[129],[15,57350,1388],{"className":57351,"style":12443},[129,133],[15,57353,57355],{"className":57354},[137],[15,57356,57358,57395],{"className":57357},[141,142],[15,57359,57361,57392],{"className":57360},[146],[15,57362,57364,57375],{"className":57363,"style":49554},[150],[15,57365,57366,57369],{"style":53843},[15,57367],{"className":57368,"style":159},[158],[15,57370,57372],{"className":57371},[163,164,165,166],[15,57373,541],{"className":57374},[129,166],[15,57376,57377,57380],{"style":1272},[15,57378],{"className":57379,"style":159},[158],[15,57381,57383],{"className":57382},[163,164,165,166],[15,57384,57386,57389],{"className":57385},[129,166],[15,57387,49355],{"className":57388},[129,166],[15,57390,39314],{"className":57391},[129,133,166],[15,57393,175],{"className":57394},[174],[15,57396,57398],{"className":57397},[146],[15,57399,57401],{"className":57400,"style":53879},[150],[15,57402],{},[15,57404],{"className":57405,"style":3277},[93],[15,57407,315],{"className":57408},[2436],[15,57410],{"className":57411,"style":3277},[93],[15,57413,57415,57418,57475,57555,57561],{"className":57414},[120],[15,57416],{"className":57417,"style":45821},[124],[15,57419,57421,57424],{"className":57420},[129],[15,57422,1388],{"className":57423,"style":12443},[129,133],[15,57425,57427],{"className":57426},[137],[15,57428,57430,57467],{"className":57429},[141,142],[15,57431,57433,57464],{"className":57432},[146],[15,57434,57436,57447],{"className":57435,"style":49554},[150],[15,57437,57438,57441],{"style":53843},[15,57439],{"className":57440,"style":159},[158],[15,57442,57444],{"className":57443},[163,164,165,166],[15,57445,318],{"className":57446},[129,166],[15,57448,57449,57452],{"style":1272},[15,57450],{"className":57451,"style":159},[158],[15,57453,57455],{"className":57454},[163,164,165,166],[15,57456,57458,57461],{"className":57457},[129,166],[15,57459,49355],{"className":57460},[129,166],[15,57462,39314],{"className":57463},[129,133,166],[15,57465,175],{"className":57466},[174],[15,57468,57470],{"className":57469},[146],[15,57471,57473],{"className":57472,"style":53879},[150],[15,57474],{},[15,57476,57478,57515],{"className":57477},[195],[15,57479,57481],{"className":57480},[195],[15,57482,57484],{"className":57483},[228,4433],[15,57485,57487,57507],{"className":57486},[141,142],[15,57488,57490,57504],{"className":57489},[146],[15,57491,57493],{"className":57492,"style":4443},[150],[15,57494,57495,57498],{"style":4446},[15,57496],{"className":57497,"style":4450},[158],[15,57499,57500],{"style":53804},[4455,57501,57502],{"xmlns":4457,"width":53807,"height":73,"viewBox":53808},[4461,57503],{"d":53811},[15,57505,175],{"className":57506},[174],[15,57508,57510],{"className":57509},[146],[15,57511,57513],{"className":57512,"style":4473},[150],[15,57514],{},[15,57516,57518],{"className":57517},[137],[15,57519,57521,57547],{"className":57520},[141,142],[15,57522,57524,57544],{"className":57523},[146],[15,57525,57527],{"className":57526,"style":54006},[150],[15,57528,57529,57532],{"style":54009},[15,57530],{"className":57531,"style":159},[158],[15,57533,57535],{"className":57534},[163,164,165,166],[15,57536,57538],{"className":57537},[129,166],[15,57539,57541],{"className":57540},[129,166],[15,57542,54025],{"className":57543,"style":2246},[129,221,166],[15,57545,175],{"className":57546},[174],[15,57548,57550],{"className":57549},[146],[15,57551,57553],{"className":57552,"style":54035},[150],[15,57554],{},[15,57556,57558],{"className":57557},[195],[15,57559,59],{"className":57560},[228,24450],[15,57562,107],{"className":57563},[129],[11,57565,57566,57567,57602,57603,57736,57737,57769],{},"这个式子的意思是：",[698,57568,57569,57570,1313],{},"两个分布的总变差越小，任何检验的错误率就越接近 ",[15,57571,57573,57590],{"className":57572,"translate":19},[23],[15,57574,57576],{"className":57575},[27],[29,57577,57578],{"xmlns":31},[34,57579,57580,57588],{},[37,57581,57582,57584,57586],{},[308,57583,318],{},[43,57585,544],{"mathvariant":65},[308,57587,310],{},[109,57589,56720],{"encoding":111},[15,57591,57593],{"className":57592,"ariaHidden":72},[116],[15,57594,57596,57599],{"className":57595},[120],[15,57597],{"className":57598,"style":125},[124],[15,57600,56720],{"className":57601},[129]," 当 ",[15,57604,57606,57632],{"className":57605,"translate":19},[23],[15,57607,57609],{"className":57608},[27],[29,57610,57611],{"xmlns":31},[34,57612,57613,57629],{},[37,57614,57615,57621,57623],{},[40,57616,57617,57619],{},[43,57618,1388],{},[308,57620,541],{},[50,57622,62],{},[40,57624,57625,57627],{},[43,57626,1388],{},[308,57628,318],{},[109,57630,57631],{"encoding":111},"P_0=P_1",[15,57633,57635,57690],{"className":57634,"ariaHidden":72},[116],[15,57636,57638,57641,57681,57684,57687],{"className":57637},[120],[15,57639],{"className":57640,"style":20220},[124],[15,57642,57644,57647],{"className":57643},[129],[15,57645,1388],{"className":57646,"style":12443},[129,133],[15,57648,57650],{"className":57649},[137],[15,57651,57653,57673],{"className":57652},[141,142],[15,57654,57656,57670],{"className":57655},[146],[15,57657,57659],{"className":57658,"style":1051},[150],[15,57660,57661,57664],{"style":34226},[15,57662],{"className":57663,"style":159},[158],[15,57665,57667],{"className":57666},[163,164,165,166],[15,57668,541],{"className":57669},[129,166],[15,57671,175],{"className":57672},[174],[15,57674,57676],{"className":57675},[146],[15,57677,57679],{"className":57678,"style":182},[150],[15,57680],{},[15,57682],{"className":57683,"style":199},[93],[15,57685,62],{"className":57686},[203],[15,57688],{"className":57689,"style":199},[93],[15,57691,57693,57696],{"className":57692},[120],[15,57694],{"className":57695,"style":20220},[124],[15,57697,57699,57702],{"className":57698},[129],[15,57700,1388],{"className":57701,"style":12443},[129,133],[15,57703,57705],{"className":57704},[137],[15,57706,57708,57728],{"className":57707},[141,142],[15,57709,57711,57725],{"className":57710},[146],[15,57712,57714],{"className":57713,"style":1051},[150],[15,57715,57716,57719],{"style":34226},[15,57717],{"className":57718,"style":159},[158],[15,57720,57722],{"className":57721},[163,164,165,166],[15,57723,318],{"className":57724},[129,166],[15,57726,175],{"className":57727},[174],[15,57729,57731],{"className":57730},[146],[15,57732,57734],{"className":57733,"style":182},[150],[15,57735],{}," 时，错误率至少是 ",[15,57738,57740,57757],{"className":57739,"translate":19},[23],[15,57741,57743],{"className":57742},[27],[29,57744,57745],{"xmlns":31},[34,57746,57747,57755],{},[37,57748,57749,57751,57753],{},[308,57750,318],{},[43,57752,544],{"mathvariant":65},[308,57754,310],{},[109,57756,56720],{"encoding":111},[15,57758,57760],{"className":57759,"ariaHidden":72},[116],[15,57761,57763,57766],{"className":57762},[120],[15,57764],{"className":57765,"style":125},[124],[15,57767,56720],{"className":57768},[129],"，这当然——数据里根本没有信息。",[11,57771,57772,57773,57801,57802,22158],{},"定理 3.1 要求 ",[15,57774,57776,57789],{"className":57775,"translate":19},[23],[15,57777,57779],{"className":57778},[27],[29,57780,57781],{"xmlns":31},[34,57782,57783,57787],{},[37,57784,57785],{},[43,57786,27518],{"mathvariant":65},[109,57788,53193],{"encoding":111},[15,57790,57792],{"className":57791,"ariaHidden":72},[116],[15,57793,57795,57798],{"className":57794},[120],[15,57796],{"className":57797,"style":522},[124],[15,57799,27518],{"className":57800},[129]," 满足三角不等式，平方损失并不满足。对平方损失，有一个常数稍有不同的版本。设 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",[15,59393,59395,59449],{"className":59394,"translate":19},[23],[15,59396,59398],{"className":59397},[27],[29,59399,59400],{"xmlns":31},[34,59401,59402,59446],{},[37,59403,59404,59406,59408,59410,59412,59414,59416,59422,59424,59426,59428,59430,59432,59434,59440,59442,59444],{},[43,59405,48342],{"mathvariant":65},[6852,59407,15781],{},[43,59409,26563],{"mathvariant":103},[50,59411,1407],{"stretchy":52},[43,59413,15756],{},[50,59415,53],{"stretchy":52},[305,59417,59418,59420],{},[43,59419,597],{},[43,59421,39314],{},[50,59423,59],{"stretchy":52},[50,59425,53],{"stretchy":52},[308,59427,318],{},[50,59429,315],{},[43,59431,15756],{},[50,59433,53],{"stretchy":52},[305,59435,59436,59438],{},[43,59437,597],{},[43,59439,39314],{},[50,59441,59],{"stretchy":52},[50,59443,59],{"stretchy":52},[50,59445,1423],{"stretchy":52},[109,59447,59448],{"encoding":111},"\\Delta\\,\\mathbb E[\\pi(X^n)(1-\\pi(X^n))]",[15,59450,59452,59523],{"className":59451,"ariaHidden":72},[116],[15,59453,59455,59458,59461,59464,59467,59470,59473,59476,59505,59508,59511,59514,59517,59520],{"className":59454},[120],[15,59456],{"className":59457,"style":125},[124],[15,59459,48342],{"className":59460},[129],[15,59462],{"className":59463,"style":262},[93],[15,59465,26563],{"className":59466},[129,285],[15,59468,1407],{"className":59469},[188],[15,59471,15756],{"className":59472,"style":887},[129,133],[15,59474,53],{"className":59475},[188],[15,59477,59479,59482],{"className":59478},[129],[15,59480,597],{"className":59481,"style":47958},[129,133],[15,59483,59485],{"className":59484},[137],[15,59486,59488],{"className":59487},[141],[15,59489,59491],{"className":59490},[146],[15,59492,59494],{"className":59493,"style":6377},[150],[15,59495,59496,59499],{"style":383},[15,59497],{"className":59498,"style":159},[158],[15,59500,59502],{"className":59501},[163,164,165,166],[15,59503,39314],{"className":59504},[129,133,166],[15,59506,59],{"className":59507},[195],[15,59509,53],{"className":59510},[188],[15,59512,318],{"className":59513},[129],[15,59515],{"className":59516,"style":3277},[93],[15,59518,315],{"className":59519},[2436],[15,59521],{"className":59522,"style":3277},[93],[15,59524,59526,59529,59532,59535,59564],{"className":59525},[120],[15,59527],{"className":59528,"style":125},[124],[15,59530,15756],{"className":59531,"style":887},[129,133],[15,59533,53],{"className":59534},[188],[15,59536,59538,59541],{"className":59537},[129],[15,59539,597],{"className":59540,"style":47958},[129,133],[15,59542,59544],{"className":59543},[137],[15,59545,59547],{"className":59546},[141],[15,59548,59550],{"className":59549},[146],[15,59551,59553],{"className":59552,"style":6377},[150],[15,59554,59555,59558],{"style":383},[15,59556],{"className":59557,"style":159},[158],[15,59559,59561],{"className":59560},[163,164,165,166],[15,59562,39314],{"className":59563},[129,133,166],[15,59565,59567],{"className":59566},[195],"))]","。由 ",[15,59570,59572,59622],{"className":59571,"translate":19},[23],[15,59573,59575],{"className":59574},[27],[29,59576,59577],{"xmlns":31},[34,59578,59579,59619],{},[37,59580,59581,59583,59585,59587,59589,59591,59593,59595,59601,59603,59605,59607,59609,59611,59613,59615,59617],{},[43,59582,15756],{},[50,59584,53],{"stretchy":52},[308,59586,318],{},[50,59588,315],{},[43,59590,15756],{},[50,59592,59],{"stretchy":52},[50,59594,7469],{},[1443,59596,59597,59599],{},[308,59598,318],{},[308,59600,310],{},[43,59602,2146],{},[50,59604,69],{},[50,59606,53],{"stretchy":52},[43,59608,15756],{},[50,59610,91],{"separator":72},[308,59612,318],{},[50,59614,315],{},[43,59616,15756],{},[50,59618,59],{"stretchy":52},[109,59620,59621],{"encoding":111},"\\pi(1-\\pi)\\ge\\frac12\\min(\\pi,1-\\pi)",[15,59623,59625,59649,59670,59778],{"className":59624,"ariaHidden":72},[116],[15,59626,59628,59631,59634,59637,59640,59643,59646],{"className":59627},[120],[15,59629],{"className":59630,"style":125},[124],[15,59632,15756],{"className":59633,"style":887},[129,133],[15,59635,53],{"className":59636},[188],[15,59638,318],{"className":59639},[129],[15,59641],{"className":59642,"style":3277},[93],[15,59644,315],{"className":59645},[2436],[15,59647],{"className":59648,"style":3277},[93],[15,59650,59652,59655,59658,59661,59664,59667],{"className":59651},[120],[15,59653],{"className":59654,"style":125},[124],[15,59656,15756],{"className":59657,"style":887},[129,133],[15,59659,59],{"className":59660},[195],[15,59662],{"className":59663,"style":199},[93],[15,59665,7469],{"className":59666},[203],[15,59668],{"className":59669,"style":199},[93],[15,59671,59673,59677,59748,59751,59754,59757,59760,59763,59766,59769,59772,59775],{"className":59672},[120],[15,59674],{"className":59675,"style":59676},[124],"height:1.1901em;vertical-align:-0.345em;",[15,59678,59680,59683,59745],{"className":59679},[129],[15,59681],{"className":59682},[188,1803],[15,59684,59686],{"className":59685},[1443],[15,59687,59689,59737],{"className":59688},[141,142],[15,59690,59692,59734],{"className":59691},[146],[15,59693,59696,59711,59719],{"className":59694,"style":59695},[150],"height:0.8451em;",[15,59697,59699,59702],{"style":59698},"top:-2.655em;",[15,59700],{"className":59701,"style":1823},[158],[15,59703,59705],{"className":59704},[163,164,165,166],[15,59706,59708],{"className":59707},[129,166],[15,59709,310],{"className":59710},[129,166],[15,59712,59713,59716],{"style":1832},[15,59714],{"className":59715,"style":1823},[158],[15,59717],{"className":59718,"style":1840},[1839],[15,59720,59722,59725],{"style":59721},"top:-3.394em;",[15,59723],{"className":59724,"style":1823},[158],[15,59726,59728],{"className":59727},[163,164,165,166],[15,59729,59731],{"className":59730},[129,166],[15,59732,318],{"className":59733},[129,166],[15,59735,175],{"className":59736},[174],[15,59738,59740],{"className":59739},[146],[15,59741,59743],{"className":59742,"style":23479},[150],[15,59744],{},[15,59746],{"className":59747},[195,1803],[15,59749],{"className":59750,"style":262},[93],[15,59752,2146],{"className":59753},[217],[15,59755,53],{"className":59756},[188],[15,59758,15756],{"className":59759,"style":887},[129,133],[15,59761,91],{"className":59762},[254],[15,59764],{"className":59765,"style":262},[93],[15,59767,318],{"className":59768},[129],[15,59770],{"className":59771,"style":3277},[93],[15,59773,315],{"className":59774},[2436],[15,59776],{"className":59777,"style":3277},[93],[15,59779,59781,59784,59787],{"className":59780},[120],[15,59782],{"className":59783,"style":125},[124],[15,59785,15756],{"className":59786,"style":887},[129,133],[15,59788,59],{"className":59789},[195]," 与均匀先验下贝叶斯检验误差的下界 ",[15,59792,59794,59852],{"className":59793,"translate":19},[23],[15,59795,59797],{"className":59796},[27],[29,59798,59799],{"xmlns":31},[34,59800,59801,59849],{},[37,59802,59803,59805,59807,59809,59811,59813,59815,59817,59819,59821,59823,59825,59827,59829,59835,59837,59839,59841,59847],{},[43,59804,26563],{"mathvariant":103},[50,59806,1407],{"stretchy":52},[43,59808,2146],{},[50,59810,69],{},[50,59812,53],{"stretchy":52},[43,59814,15756],{},[50,59816,91],{"separator":72},[308,59818,318],{},[50,59820,315],{},[43,59822,15756],{},[50,59824,59],{"stretchy":52},[50,59826,1423],{"stretchy":52},[50,59828,7469],{},[1443,59830,59831,59833],{},[308,59832,318],{},[308,59834,310],{},[50,59836,53],{"stretchy":52},[308,59838,318],{},[50,59840,315],{},[37,59842,59843,59845],{},[43,59844,53394],{"mathvariant":65},[43,59846,53397],{"mathvariant":65},[50,59848,59],{"stretchy":52},[109,59850,59851],{"encoding":111},"\\mathbb E[\\min(\\pi,1-\\pi)]\\ge\\frac12(1-\\mathrm{TV})",[15,59853,59855,59894,59916,60005],{"className":59854,"ariaHidden":72},[116],[15,59856,59858,59861,59864,59867,59870,59873,59876,59879,59882,59885,59888,59891],{"className":59857},[120],[15,59859],{"className":59860,"style":125},[124],[15,59862,26563],{"className":59863},[129,285],[15,59865,1407],{"className":59866},[188],[15,59868,2146],{"className":59869},[217],[15,59871,53],{"className":59872},[188],[15,59874,15756],{"className":59875,"style":887},[129,133],[15,59877,91],{"className":59878},[254],[15,59880],{"className":59881,"style":262},[93],[15,59883,318],{"className":59884},[129],[15,59886],{"className":59887,"style":3277},[93],[15,59889,315],{"className":59890},[2436],[15,59892],{"className":59893,"style":3277},[93],[15,59895,59897,59900,59903,59907,59910,59913],{"className":59896},[120],[15,59898],{"className":59899,"style":125},[124],[15,59901,15756],{"className":59902,"style":887},[129,133],[15,59904,59906],{"className":59905},[195],")]",[15,59908],{"className":59909,"style":199},[93],[15,59911,7469],{"className":59912},[203],[15,59914],{"className":59915,"style":199},[93],[15,59917,59919,59922,59990,59993,59996,59999,60002],{"className":59918},[120],[15,59920],{"className":59921,"style":59676},[124],[15,59923,59925,59928,59987],{"className":59924},[129],[15,59926],{"className":59927},[188,1803],[15,59929,59931],{"className":59930},[1443],[15,59932,59934,59979],{"className":59933},[141,142],[15,59935,59937,59976],{"className":59936},[146],[15,59938,59940,59954,59962],{"className":59939,"style":59695},[150],[15,59941,59942,59945],{"style":59698},[15,59943],{"className":59944,"style":1823},[158],[15,59946,59948],{"className":59947},[163,164,165,166],[15,59949,59951],{"className":59950},[129,166],[15,59952,310],{"className":59953},[129,166],[15,59955,59956,59959],{"style":1832},[15,59957],{"className":59958,"style":1823},[158],[15,59960],{"className":59961,"style":1840},[1839],[15,59963,59964,59967],{"style":59721},[15,59965],{"className":59966,"style":1823},[158],[15,59968,59970],{"className":59969},[163,164,165,166],[15,59971,59973],{"className":59972},[129,166],[15,59974,318],{"className":59975},[129,166],[15,59977,175],{"className":59978},[174],[15,59980,59982],{"className":59981},[146],[15,59983,59985],{"className":59984,"style":23479},[150],[15,59986],{},[15,59988],{"className":59989},[195,1803],[15,59991,53],{"className":59992},[188],[15,59994,318],{"className":59995},[129],[15,59997],{"className":59998,"style":3277},[93],[15,60000,315],{"className":60001},[2436],[15,60003],{"className":60004,"style":3277},[93],[15,60006,60008,60011,60017],{"className":60007},[120],[15,60009],{"className":60010,"style":125},[124],[15,60012,60014],{"className":60013},[129],[15,60015,54025],{"className":60016,"style":2246},[129,221],[15,60018,59],{"className":60019},[195],"，风险至少是 ",[15,60022,60024,60056],{"className":60023,"translate":19},[23],[15,60025,60027],{"className":60026},[27],[29,60028,60029],{"xmlns":31},[34,60030,60031,60053],{},[37,60032,60033,60039,60041,60043,60045,60051],{},[1443,60034,60035,60037],{},[43,60036,48342],{"mathvariant":65},[308,60038,14502],{},[50,60040,53],{"stretchy":52},[308,60042,318],{},[50,60044,315],{},[37,60046,60047,60049],{},[43,60048,53394],{"mathvariant":65},[43,60050,53397],{"mathvariant":65},[50,60052,59],{"stretchy":52},[109,60054,60055],{"encoding":111},"\\frac{\\Delta}{4}(1-\\mathrm{TV})",[15,60057,60059,60150],{"className":60058,"ariaHidden":72},[116],[15,60060,60062,60066,60135,60138,60141,60144,60147],{"className":60061},[120],[15,60063],{"className":60064,"style":60065},[124],"height:1.2173em;vertical-align:-0.345em;",[15,60067,60069,60072,60132],{"className":60068},[129],[15,60070],{"className":60071},[188,1803],[15,60073,60075],{"className":60074},[1443],[15,60076,60078,60124],{"className":60077},[141,142],[15,60079,60081,60121],{"className":60080},[146],[15,60082,60085,60099,60107],{"className":60083,"style":60084},[150],"height:0.8723em;",[15,60086,60087,60090],{"style":59698},[15,60088],{"className":60089,"style":1823},[158],[15,60091,60093],{"className":60092},[163,164,165,166],[15,60094,60096],{"className":60095},[129,166],[15,60097,14502],{"className":60098},[129,166],[15,60100,60101,60104],{"style":1832},[15,60102],{"className":60103,"style":1823},[158],[15,60105],{"className":60106,"style":1840},[1839],[15,60108,60109,60112],{"style":59721},[15,60110],{"className":60111,"style":1823},[158],[15,60113,60115],{"className":60114},[163,164,165,166],[15,60116,60118],{"className":60117},[129,166],[15,60119,48342],{"className":60120},[129,166],[15,60122,175],{"className":60123},[174],[15,60125,60127],{"className":60126},[146],[15,60128,60130],{"className":60129,"style":23479},[150],[15,60131],{},[15,60133],{"className":60134},[195,1803],[15,60136,53],{"className":60137},[188],[15,60139,318],{"className":60140},[129],[15,60142],{"className":60143,"style":3277},[93],[15,60145,315],{"className":60146},[2436],[15,60148],{"className":60149,"style":3277},[93],[15,60151,60153,60156,60162],{"className":60152},[120],[15,60154],{"className":60155,"style":125},[124],[15,60157,60159],{"className":60158},[129],[15,60160,54025],{"className":60161,"style":2246},[129,221],[15,60163,59],{"className":60164},[195],"。极小极大风险不小于贝叶斯风险，得证。",[575,60167,60169],{"id":60168},"_32-从-tv-到-kl","3.2 从 TV 到 KL",[11,60171,60172],{},"总变差通常不好直接算，尤其对乘积分布。Pinsker 不等式把它换成 KL 散度：",[15,60174,60176],{"className":60175,"translate":19},[18],[15,60177,60179,60243],{"className":60178,"translate":19},[23],[15,60180,60182],{"className":60181},[27],[29,60183,60184],{"xmlns":31,"display":32},[34,60185,60186,60240],{},[37,60187,60188,60190,60192,60194,60196,60206,60208,60238],{},[50,60189,33810],{"stretchy":52},[43,60191,1388],{},[50,60193,315],{},[43,60195,34341],{},[40,60197,60198,60200],{},[50,60199,33810],{"stretchy":52},[37,60201,60202,60204],{},[43,60203,53394],{"mathvariant":65},[43,60205,53397],{"mathvariant":65},[50,60207,2717],{},[29004,60209,60210],{},[37,60211,60212,60218,60228,60230,60232,60234,60236],{},[1443,60213,60214,60216],{},[308,60215,318],{},[308,60217,310],{},[40,60219,60220,60222],{},[43,60221,906],{},[37,60223,60224,60226],{},[43,60225,34511],{"mathvariant":65},[43,60227,1373],{"mathvariant":65},[50,60229,53],{"stretchy":52},[43,60231,1388],{},[43,60233,33810],{"mathvariant":65},[43,60235,34341],{},[50,60237,59],{"stretchy":52},[43,60239,107],{"mathvariant":65},[109,60241,60242],{"encoding":111},"\\lVert P-Q\\rVert_{\\mathrm{TV}}\\le\\sqrt{\\frac12 D_{\\mathrm{KL}}(P\\Vert Q)}.",[15,60244,60246,60267,60331],{"className":60245,"ariaHidden":72},[116],[15,60247,60249,60252,60255,60258,60261,60264],{"className":60248},[120],[15,60250],{"className":60251,"style":125},[124],[15,60253,33810],{"className":60254},[188],[15,60256,1388],{"className":60257,"style":12443},[129,133],[15,60259],{"className":60260,"style":3277},[93],[15,60262,315],{"className":60263},[2436],[15,60265],{"className":60266,"style":3277},[93],[15,60268,60270,60273,60276,60322,60325,60328],{"className":60269},[120],[15,60271],{"className":60272,"style":125},[124],[15,60274,34341],{"className":60275},[129,133],[15,60277,60279,60282],{"className":60278},[195],[15,60280,33810],{"className":60281},[195],[15,60283,60285],{"className":60284},[137],[15,60286,60288,60314],{"className":60287},[141,142],[15,60289,60291,60311],{"className":60290},[146],[15,60292,60294],{"className":60293,"style":1525},[150],[15,60295,60296,60299],{"style":154},[15,60297],{"className":60298,"style":159},[158],[15,60300,60302],{"className":60301},[163,164,165,166],[15,60303,60305],{"className":60304},[129,166],[15,60306,60308],{"className":60307},[129,166],[15,60309,54025],{"className":60310,"style":2246},[129,221,166],[15,60312,175],{"className":60313},[174],[15,60315,60317],{"className":60316},[146],[15,60318,60320],{"className":60319,"style":182},[150],[15,60321],{},[15,60323],{"className":60324,"style":199},[93],[15,60326,2717],{"className":60327},[203],[15,60329],{"className":60330,"style":199},[93],[15,60332,60334,60338,60508],{"className":60333},[120],[15,60335],{"className":60336,"style":60337},[124],"height:2.44em;vertical-align:-0.7884em;",[15,60339,60341],{"className":60340},[129,29112],[15,60342,60344,60499],{"className":60343},[141,142],[15,60345,60347,60496],{"className":60346},[146],[15,60348,60351,60483],{"className":60349,"style":60350},[150],"height:1.6516em;",[15,60352,60354,60357],{"className":60353,"style":29127},[29126],[15,60355],{"className":60356,"style":29131},[158],[15,60358,60360,60422,60468,60471,60474,60477,60480],{"className":60359,"style":29135},[129],[15,60361,60363,60366,60419],{"className":60362},[129],[15,60364],{"className":60365},[188,1803],[15,60367,60369],{"className":60368},[1443],[15,60370,60372,60411],{"className":60371},[141,142],[15,60373,60375,60408],{"className":60374},[146],[15,60376,60378,60389,60397],{"className":60377,"style":1816},[150],[15,60379,60380,60383],{"style":1819},[15,60381],{"className":60382,"style":1823},[158],[15,60384,60386],{"className":60385},[129],[15,60387,310],{"className":60388},[129],[15,60390,60391,60394],{"style":1832},[15,60392],{"className":60393,"style":1823},[158],[15,60395],{"className":60396,"style":1840},[1839],[15,60398,60399,60402],{"style":1843},[15,60400],{"className":60401,"style":1823},[158],[15,60403,60405],{"className":60404},[129],[15,60406,318],{"className":60407},[129],[15,60409,175],{"className":60410},[174],[15,60412,60414],{"className":60413},[146],[15,60415,60417],{"className":60416,"style":1862},[150],[15,60418],{},[15,60420],{"className":60421},[195,1803],[15,60423,60425,60428],{"className":60424},[129],[15,60426,906],{"className":60427,"style":170},[129,133],[15,60429,60431],{"className":60430},[137],[15,60432,60434,60460],{"className":60433},[141,142],[15,60435,60437,60457],{"className":60436},[146],[15,60438,60440],{"className":60439,"style":1525},[150],[15,60441,60442,60445],{"style":48281},[15,60443],{"className":60444,"style":159},[158],[15,60446,60448],{"className":60447},[163,164,165,166],[15,60449,60451],{"className":60450},[129,166],[15,60452,60454],{"className":60453},[129,166],[15,60455,34915],{"className":60456},[129,221,166],[15,60458,175],{"className":60459},[174],[15,60461,60463],{"className":60462},[146],[15,60464,60466],{"className":60465,"style":182},[150],[15,60467],{},[15,60469,53],{"className":60470},[188],[15,60472,1388],{"className":60473,"style":12443},[129,133],[15,60475,33810],{"className":60476},[129],[15,60478,34341],{"className":60479},[129,133],[15,60481,59],{"className":60482},[195],[15,60484,60486,60489],{"style":60485},"top:-3.6116em;",[15,60487],{"className":60488,"style":29131},[158],[15,60490,60492],{"className":60491,"style":29272},[29271],[4455,60493,60494],{"xmlns":4457,"width":29275,"height":29276,"viewBox":29277,"preserveAspectRatio":29278},[4461,60495],{"d":29281},[15,60497,175],{"className":60498},[174],[15,60500,60502],{"className":60501},[146],[15,60503,60506],{"className":60504,"style":60505},[150],"height:0.7884em;",[15,60507],{},[15,60509,107],{"className":60510},[129],[11,60512,60513],{},"而 KL 对乘积分布是可加的：",[15,60515,60517],{"className":60516,"translate":19},[18],[15,60518,60520,60608],{"className":60519,"translate":19},[23],[15,60521,60523],{"className":60522},[27],[29,60524,60525],{"xmlns":31,"display":32},[34,60526,60527,60605],{},[37,60528,60529,60539,60541,60553,60555,60567,60569,60571,60573,60575,60585,60587,60593,60595,60601,60603],{},[40,60530,60531,60533],{},[43,60532,906],{},[37,60534,60535,60537],{},[43,60536,34511],{"mathvariant":65},[43,60538,1373],{"mathvariant":65},[50,60540,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[23149,60542,60543,60545,60547],{},[43,60544,1388],{},[308,60546,541],{},[37,60548,60549,60551],{},[50,60550,49355],{},[43,60552,39314],{},[43,60554,33810],{"mathvariant":65},[23149,60556,60557,60559,60561],{},[43,60558,1388],{},[308,60560,318],{},[37,60562,60563,60565],{},[50,60564,49355],{},[43,60566,39314],{},[50,60568,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,60570,62],{},[43,60572,39314],{},[6852,60574,15781],{},[40,60576,60577,60579],{},[43,60578,906],{},[37,60580,60581,60583],{},[43,60582,34511],{"mathvariant":65},[43,60584,1373],{"mathvariant":65},[50,60586,53],{"stretchy":52},[40,60588,60589,60591],{},[43,60590,1388],{},[308,60592,541],{},[43,60594,33810],{"mathvariant":65},[40,60596,60597,60599],{},[43,60598,1388],{},[308,60600,318],{},[50,60602,59],{"stretchy":52},[43,60604,107],{"mathvariant":65},[109,60606,60607],{"encoding":111},"D_{\\mathrm{KL}}\\bigl(P_0^{\\otimes n}\\Vert P_1^{\\otimes n}\\bigr)=n\\,D_{\\mathrm{KL}}(P_0\\Vert 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,61681,61684],{"className":61682,"style":61683},[150],"height:0.8953em;",[15,61685],{},[15,61687,61689],{"className":61688,"style":5052},[195,5051],[15,61690,59],{"className":61691},[228,165],[15,61693],{"className":61694,"style":262},[93],[15,61696,107],{"className":61697},[129],[11,61699,61700,61701,804,61704,61733,61734,804,61737,61822,61823,62001],{},"这个式子把下界拆成了两个因子：",[698,61702,61703],{},"损失间隔",[15,61705,61707,61721],{"className":61706,"translate":19},[23],[15,61708,61710],{"className":61709},[27],[29,61711,61712],{"xmlns":31},[34,61713,61714,61718],{},[37,61715,61716],{},[43,61717,29564],{},[109,61719,61720],{"encoding":111},"\\delta",[15,61722,61724],{"className":61723,"ariaHidden":72},[116],[15,61725,61727,61730],{"className":61726},[120],[15,61728],{"className":61729,"style":522},[124],[15,61731,29564],{"className":61732,"style":29845},[129,133],"，和",[698,61735,61736],{},"信息量",[15,61738,61740,61764],{"className":61739,"translate":19},[23],[15,61741,61743],{"className":61742},[27],[29,61744,61745],{"xmlns":31},[34,61746,61747,61761],{},[37,61748,61749,61751],{},[43,61750,39314],{},[40,61752,61753,61755],{},[43,61754,906],{},[37,61756,61757,61759],{},[43,61758,34511],{"mathvariant":65},[43,61760,1373],{"mathvariant":65},[109,61762,61763],{"encoding":111},"nD_{\\mathrm{KL}}",[15,61765,61767],{"className":61766,"ariaHidden":72},[116],[15,61768,61770,61773,61776],{"className":61769},[120],[15,61771],{"className":61772,"style":20220},[124],[15,61774,39314],{"className":61775},[129,133],[15,61777,61779,61782],{"className":61778},[129],[15,61780,906],{"className":61781,"style":170},[129,133],[15,61783,61785],{"className":61784},[137],[15,61786,61788,61814],{"className":61787},[141,142],[15,61789,61791,61811],{"className":61790},[146],[15,61792,61794],{"className":61793,"style":1525},[150],[15,61795,61796,61799],{"style":48281},[15,61797],{"className":61798,"style":159},[158],[15,61800,61802],{"className":61801},[163,164,165,166],[15,61803,61805],{"className":61804},[129,166],[15,61806,61808],{"className":61807},[129,166],[15,61809,34915],{"className":61810},[129,221,166],[15,61812,175],{"className":61813},[174],[15,61815,61817],{"className":61816},[146],[15,61818,61820],{"className":61819,"style":182},[150],[15,61821],{},"。要让下界为正，必须 ",[15,61824,61826,61858],{"className":61825,"translate":19},[23],[15,61827,61829],{"className":61828},[27],[29,61830,61831],{"xmlns":31},[34,61832,61833,61855],{},[37,61834,61835,61841,61851,61853],{},[1443,61836,61837,61839],{},[43,61838,39314],{},[308,61840,310],{},[40,61842,61843,61845],{},[43,61844,906],{},[37,61846,61847,61849],{},[43,61848,34511],{"mathvariant":65},[43,61850,1373],{"mathvariant":65},[50,61852,7906],{},[308,61854,318],{},[109,61856,61857],{"encoding":111},"\\frac n2D_{\\mathrm{KL}}\u003C1",[15,61859,61861,61992],{"className":61860,"ariaHidden":72},[116],[15,61862,61864,61868,61937,61983,61986,61989],{"className":61863},[120],[15,61865],{"className":61866,"style":61867},[124],"height:1.0404em;vertical-align:-0.345em;",[15,61869,61871,61874,61934],{"className":61870},[129],[15,61872],{"className":61873},[188,1803],[15,61875,61877],{"className":61876},[1443],[15,61878,61880,61926],{"className":61879},[141,142],[15,61881,61883,61923],{"className":61882},[146],[15,61884,61887,61901,61909],{"className":61885,"style":61886},[150],"height:0.6954em;",[15,61888,61889,61892],{"style":59698},[15,61890],{"className":61891,"style":1823},[158],[15,61893,61895],{"className":61894},[163,164,165,166],[15,61896,61898],{"className":61897},[129,166],[15,61899,310],{"className":61900},[129,166],[15,61902,61903,61906],{"style":1832},[15,61904],{"className":61905,"style":1823},[158],[15,61907],{"className":61908,"style":1840},[1839],[15,61910,61911,61914],{"style":59721},[15,61912],{"className":61913,"style":1823},[158],[15,61915,61917],{"className":61916},[163,164,165,166],[15,61918,61920],{"className":61919},[129,166],[15,61921,39314],{"className":61922},[129,133,166],[15,61924,175],{"className":61925},[174],[15,61927,61929],{"className":61928},[146],[15,61930,61932],{"className":61931,"style":23479},[150],[15,61933],{},[15,61935],{"className":61936},[195,1803],[15,61938,61940,61943],{"className":61939},[129],[15,61941,906],{"className":61942,"style":170},[129,133],[15,61944,61946],{"className":61945},[137],[15,61947,61949,61975],{"className":61948},[141,142],[15,61950,61952,61972],{"className":61951},[146],[15,61953,61955],{"className":61954,"style":1525},[150],[15,61956,61957,61960],{"style":48281},[15,61958],{"className":61959,"style":159},[158],[15,61961,61963],{"className":61962},[163,164,165,166],[15,61964,61966],{"className":61965},[129,166],[15,61967,61969],{"className":61968},[129,166],[15,61970,34915],{"className":61971},[129,221,166],[15,61973,175],{"className":61974},[174],[15,61976,61978],{"className":61977},[146],[15,61979,61981],{"className":61980,"style":182},[150],[15,61982],{},[15,61984],{"className":61985,"style":199},[93],[15,61987,7906],{"className":61988},[203],[15,61990],{"className":61991,"style":199},[93],[15,61993,61995,61998],{"className":61994},[120],[15,61996],{"className":61997,"style":4886},[124],[15,61999,318],{"className":62000},[129],"；换句话说，如果两个分布在这个样本量下已经能被可靠地区分，两点方法就给不出非平凡的下界了。",[11,62003,62004,62005,62039],{},"这也解释了为什么下界里的样本量总以「",[15,62006,62008,62024],{"className":62007,"translate":19},[23],[15,62009,62011],{"className":62010},[27],[29,62012,62013],{"xmlns":31},[34,62014,62015,62021],{},[37,62016,62017,62019],{},[43,62018,39314],{},[50,62020,11418],{},[109,62022,62023],{"encoding":111},"n\\times",[15,62025,62027],{"className":62026,"ariaHidden":72},[116],[15,62028,62030,62033,62036],{"className":62029},[120],[15,62031],{"className":62032,"style":8514},[124],[15,62034,39314],{"className":62035},[129,133],[15,62037,11418],{"className":62038},[129]," 单样本信息」的形式出现：独立样本的信息是线性累加的。",[575,62041,62043],{"id":62042},"_33-一个例子bernoulli-均值","3.3 一个例子：Bernoulli 均值",[11,62045,21927,62046,7606,62164,62268],{},[15,62047,62049,62084],{"className":62048,"translate":19},[23],[15,62050,62052],{"className":62051},[27],[29,62053,62054],{"xmlns":31},[34,62055,62056,62081],{},[37,62057,62058,62064,62066,62075,62077,62079],{},[40,62059,62060,62062],{},[43,62061,597],{},[43,62063,1459],{},[50,62065,991],{},[37,62067,62068,62070,62072],{},[43,62069,33882],{"mathvariant":65},[43,62071,10695],{"mathvariant":65},[43,62073,62074],{"mathvariant":65},"r",[50,62076,53],{"stretchy":52},[43,62078,11],{},[50,62080,59],{"stretchy":52},[109,62082,62083],{"encoding":111},"X_i\\sim\\mathrm{Ber}(p)",[15,62085,62087,62142],{"className":62086,"ariaHidden":72},[116],[15,62088,62090,62093,62133,62136,62139],{"className":62089},[120],[15,62091],{"className":62092,"style":20220},[124],[15,62094,62096,62099],{"className":62095},[129],[15,62097,597],{"className":62098,"style":47958},[129,133],[15,62100,62102],{"className":62101},[137],[15,62103,62105,62125],{"className":62104},[141,142],[15,62106,62108,62122],{"className":62107},[146],[15,62109,62111],{"className":62110,"style":1981},[150],[15,62112,62113,62116],{"style":47973},[15,62114],{"className":62115,"style":159},[158],[15,62117,62119],{"className":62118},[163,164,165,166],[15,62120,1459],{"className":62121},[129,133,166],[15,62123,175],{"className":62124},[174],[15,62126,62128],{"className":62127},[146],[15,62129,62131],{"className":62130,"style":182},[150],[15,62132],{},[15,62134],{"className":62135,"style":199},[93],[15,62137,991],{"className":62138},[203],[15,62140],{"className":62141,"style":199},[93],[15,62143,62145,62148,62155,62158,62161],{"className":62144},[120],[15,62146],{"className":62147,"style":125},[124],[15,62149,62151],{"className":62150},[129],[15,62152,62154],{"className":62153},[129,221],"Ber",[15,62156,53],{"className":62157},[188],[15,62159,11],{"className":62160},[129,133],[15,62162,59],{"className":62163},[195],[15,62165,62167,62205],{"className":62166,"translate":19},[23],[15,62168,62170],{"className":62169},[27],[29,62171,62172],{"xmlns":31},[34,62173,62174,62202],{},[37,62175,62176,62178,62180,62182,62184,62186,62188,62190,62192,62194,62196,62198,62200],{},[43,62177,11],{},[50,62179,100],{},[50,62181,636],{"stretchy":52},[308,62183,318],{},[43,62185,544],{"mathvariant":65},[308,62187,310],{},[50,62189,91],{"separator":72},[308,62191,318],{},[43,62193,544],{"mathvariant":65},[308,62195,310],{},[50,62197,2682],{},[43,62199,48342],{"mathvariant":65},[50,62201,645],{"stretchy":52},[109,62203,62204],{"encoding":111},"p\\in\\{1\u002F2,1\u002F2+\\Delta\\}",[15,62206,62208,62226,62256],{"className":62207,"ariaHidden":72},[116],[15,62209,62211,62214,62217,62220,62223],{"className":62210},[120],[15,62212],{"className":62213,"style":10955},[124],[15,62215,11],{"className":62216},[129,133],[15,62218],{"className":62219,"style":199},[93],[15,62221,100],{"className":62222},[203],[15,62224],{"className":62225,"style":199},[93],[15,62227,62229,62232,62235,62238,62241,62244,62247,62250,62253],{"className":62228},[120],[15,62230],{"className":62231,"style":125},[124],[15,62233,636],{"className":62234},[188],[15,62236,56720],{"className":62237},[129],[15,62239,91],{"className":62240},[254],[15,62242],{"className":62243,"style":262},[93],[15,62245,56720],{"className":62246},[129],[15,62248],{"className":62249,"style":3277},[93],[15,62251,2682],{"className":62252},[2436],[15,62254],{"className":62255,"style":3277},[93],[15,62257,62259,62262,62265],{"className":62258},[120],[15,62260],{"className":62261,"style":125},[124],[15,62263,48342],{"className":62264},[129],[15,62266,645],{"className":62267},[195],"，损失为平方损失。两个分布的单样本 KL 是",[15,62270,62272],{"className":62271,"translate":19},[18],[15,62273,62275,62403],{"className":62274,"translate":19},[23],[15,62276,62278],{"className":62277},[27],[29,62279,62280],{"xmlns":31,"display":32},[34,62281,62282,62400],{},[37,62283,62284,62294,62296,62304,62306,62314,62316,62318,62326,62328,62336,62338,62340,62342,62344,62346,62348,62350,62356,62358,62360,62362,62364,62366,62368,62374,62376,62378,62380,62386,62388,62390,62392,62394,62396,62398],{},[40,62285,62286,62288],{},[43,62287,906],{},[37,62289,62290,62292],{},[43,62291,34511],{"mathvariant":65},[43,62293,1373],{"mathvariant":65},[50,62295,53],{"fence":72,"stretchy":72,"minsize":45693,"maxsize":45693},[37,62297,62298,62300,62302],{},[43,62299,33882],{"mathvariant":65},[43,62301,10695],{"mathvariant":65},[43,62303,62074],{"mathvariant":65},[50,62305,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[2612,62307,62308],{"scriptlevel":541,"displaystyle":52},[1443,62309,62310,62312],{},[308,62311,318],{},[308,62313,310],{},[50,62315,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,62317,33810],{"fence":52,"stretchy":72,"minsize":45693,"maxsize":45693},[37,62319,62320,62322,62324],{},[43,62321,33882],{"mathvariant":65},[43,62323,10695],{"mathvariant":65},[43,62325,62074],{"mathvariant":65},[50,62327,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[2612,62329,62330],{"scriptlevel":541,"displaystyle":52},[1443,62331,62332,62334],{},[308,62333,318],{},[308,62335,310],{},[50,62337,2682],{},[43,62339,48342],{"mathvariant":65},[50,62341,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,62343,59],{"fence":72,"stretchy":72,"minsize":45693,"maxsize":45693},[93,62345],{"linebreak":24710},[50,62347,62],{},[50,62349,315],{},[1443,62351,62352,62354],{},[308,62353,318],{},[308,62355,310],{},[43,62357,14721],{},[50,62359,69],{},[50,62361,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[308,62363,318],{},[50,62365,315],{},[308,62367,14502],{},[305,62369,62370,62372],{},[43,62371,48342],{"mathvariant":65},[308,62373,310],{},[50,62375,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,62377,32392],{},[308,62379,310],{},[305,62381,62382,62384],{},[43,62383,48342],{"mathvariant":65},[308,62385,310],{},[93,62387],{"width":95},[50,62389,53],{"stretchy":52},[43,62391,48342],{"mathvariant":65},[50,62393,723],{},[308,62395,541],{},[50,62397,59],{"stretchy":52},[43,62399,107],{"mathvariant":65},[109,62401,62402],{"encoding":111},"D_{\\mathrm{KL}}\\Bigl(\\mathrm{Ber}\\bigl(\\tfrac12\\bigr)\\Big\\Vert\\mathrm{Ber}\\bigl(\\tfrac12+\\Delta\\bigr)\\Bigr) \\\\\n=-\\frac12\\log\\bigl(1-4\\Delta^2\\bigr)\n\\approx 2\\Delta^2\n\\qquad(\\Delta\\to0).",[15,62404,62406,62683,62704,62707,62719,62816,62870,62926],{"className":62405,"ariaHidden":72},[116],[15,62407,62409,62412,62458,62464,62470,62476,62544,62550,62594,62600,62606,62674,62677,62680],{"className":62408},[120],[15,62410],{"className":62411,"style":45821},[124],[15,62413,62415,62418],{"className":62414},[129],[15,62416,906],{"className":62417,"style":170},[129,133],[15,62419,62421],{"className":62420},[137],[15,62422,62424,62450],{"className":62423},[141,142],[15,62425,62427,62447],{"className":62426},[146],[15,62428,62430],{"className":62429,"style":1525},[150],[15,62431,62432,62435],{"style":48281},[15,62433],{"className":62434,"style":159},[158],[15,62436,62438],{"className":62437},[163,164,165,166],[15,62439,62441],{"className":62440},[129,166],[15,62442,62444],{"className":62443},[129,166],[15,62445,34915],{"className":62446},[129,221,166],[15,62448,175],{"className":62449},[174],[15,62451,62453],{"className":62452},[146],[15,62454,62456],{"className":62455,"style":182},[150],[15,62457],{},[15,62459,62461],{"className":62460},[188],[15,62462,53],{"className":62463},[228,24450],[15,62465,62467],{"className":62466},[129],[15,62468,62154],{"className":62469},[129,221],[15,62471,62473],{"className":62472},[188],[15,62474,53],{"className":62475},[228,229],[15,62477,62479,62482,62541],{"className":62478},[129],[15,62480],{"className":62481},[188,1803],[15,62483,62485],{"className":62484},[1443],[15,62486,62488,62533],{"className":62487},[141,142],[15,62489,62491,62530],{"className":62490},[146],[15,62492,62494,62508,62516],{"className":62493,"style":59695},[150],[15,62495,62496,62499],{"style":59698},[15,62497],{"className":62498,"style":1823},[158],[15,62500,62502],{"className":62501},[163,164,165,166],[15,62503,62505],{"className":62504},[129,166],[15,62506,310],{"className":62507},[129,166],[15,62509,62510,62513],{"style":1832},[15,62511],{"className":62512,"style":1823},[158],[15,62514],{"className":62515,"style":1840},[1839],[15,62517,62518,62521],{"style":59721},[15,62519],{"className":62520,"style":1823},[158],[15,62522,62524],{"className":62523},[163,164,165,166],[15,62525,62527],{"className":62526},[129,166],[15,62528,318],{"className":62529},[129,166],[15,62531,175],{"className":62532},[174],[15,62534,62536],{"className":62535},[146],[15,62537,62539],{"className":62538,"style":23479},[150],[15,62540],{},[15,62542],{"className":62543},[195,1803],[15,62545,62547],{"className":62546},[195],[15,62548,59],{"className":62549},[228,229],[15,62551,62553],{"className":62552},[129],[15,62554,62556],{"className":62555},[228,4433],[15,62557,62559,62585],{"className":62558},[141,142],[15,62560,62562,62582],{"className":62561},[146],[15,62563,62566],{"className":62564,"style":62565},[150],"height:1.15em;",[15,62567,62569,62573],{"style":62568},"top:-3.15em;",[15,62570],{"className":62571,"style":62572},[158],"height:3.8em;",[15,62574,62576],{"style":62575},"width:0.556em;height:1.8em;",[4455,62577,62579],{"xmlns":4457,"width":53807,"height":45693,"viewBox":62578},"0 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E\\bigl[(p_i-\\hat p)^2\\bigr]\n\\ge\n\\frac{\\Delta^2}{4}\\left(1-\\frac12\\right)\n=\\frac{1}{32n}.",[15,63372,63374,63490,63583,63780],{"className":63373,"ariaHidden":72},[116],[15,63375,63377,63381,63426,63429,63432,63438,63441,63481,63484,63487],{"className":63376},[120],[15,63378],{"className":63379,"style":63380},[124],"height:1.5777em;vertical-align:-0.7277em;",[15,63382,63384],{"className":63383},[217,1874],[15,63385,63387,63417],{"className":63386},[141,142],[15,63388,63390,63414],{"className":63389},[146],[15,63391,63393,63404],{"className":63392,"style":4383},[150],[15,63394,63395,63398],{"style":53429},[15,63396],{"className":63397,"style":1823},[158],[15,63399,63401],{"className":63400},[163,164,165,166],[15,63402,1459],{"className":63403},[129,133,166],[15,63405,63406,63409],{"style":2287},[15,63407],{"className":63408,"style":1823},[158],[15,63410,63411],{},[15,63412,19696],{"className":63413},[217],[15,63415,175],{"className":63416},[174],[15,63418,63420],{"className":63419},[146],[15,63421,63424],{"className":63422,"style":63423},[150],"height:0.7277em;",[15,63425],{},[15,63427],{"className":63428,"style":262},[93],[15,63430,26563],{"className":63431},[129,285],[15,63433,63435],{"className":63434},[188],[15,63436,1407],{"className":63437},[228,229],[15,63439,53],{"className":63440},[188],[15,63442,63444,63447],{"className":63443},[129],[15,63445,11],{"className":63446},[129,133],[15,63448,63450],{"className":63449},[137],[15,63451,63453,63473],{"className":63452},[141,142],[15,63454,63456,63470],{"className":63455},[146],[15,63457,63459],{"className":63458,"style":1981},[150],[15,63460,63461,63464],{"style":154},[15,63462],{"className":63463,"style":159},[158],[15,63465,63467],{"className":63466},[163,164,165,166],[15,63468,1459],{"className":63469},[129,133,166],[15,63471,175],{"className":63472},[174],[15,63474,63476],{"className":63475},[146],[15,63477,63479],{"className":63478,"style":182},[150],[15,63480],{},[15,63482],{"className":63483,"style":3277},[93],[15,63485,315],{"className":63486},[2436],[15,63488],{"className":63489,"style":3277},[93],[15,63491,63493,63496,63539,63568,63574,63577,63580],{"className":63492},[120],[15,63494],{"className":63495,"style":62822},[124],[15,63497,63499],{"className":63498},[129,48488],[15,63500,63502,63530],{"className":63501},[141,142],[15,63503,63505,63527],{"className":63504},[146],[15,63506,63508,63516],{"className":63507,"style":522},[150],[15,63509,63510,63513],{"style":2287},[15,63511],{"className":63512,"style":1823},[158],[15,63514,11],{"className":63515},[129,133],[15,63517,63518,63521],{"style":2287},[15,63519],{"className":63520,"style":1823},[158],[15,63522,63524],{"className":63523,"style":48516},[48515],[15,63525,48471],{"className":63526},[129],[15,63528,175],{"className":63529},[174],[15,63531,63533],{"className":63532},[146],[15,63534,63537],{"className":63535,"style":63536},[150],"height:0.1944em;",[15,63538],{},[15,63540,63542,63545],{"className":63541},[195],[15,63543,59],{"className":63544},[195],[15,63546,63548],{"className":63547},[137],[15,63549,63551],{"className":63550},[141],[15,63552,63554],{"className":63553},[146],[15,63555,63557],{"className":63556,"style":40979},[150],[15,63558,63559,63562],{"style":1272},[15,63560],{"className":63561,"style":159},[158],[15,63563,63565],{"className":63564},[163,164,165,166],[15,63566,310],{"className":63567},[129,166],[15,63569,63571],{"className":63570},[195],[15,63572,1423],{"className":63573},[228,229],[15,63575],{"className":63576,"style":199},[93],[15,63578,7469],{"className":63579},[203],[15,63581],{"className":63582,"style":199},[93],[15,63584,63586,63590,63679,63682,63771,63774,63777],{"className":63585},[120],[15,63587],{"className":63588,"style":63589},[124],"height:2.4411em;vertical-align:-0.95em;",[15,63591,63593,63596,63676],{"className":63592},[129],[15,63594],{"className":63595},[188,1803],[15,63597,63599],{"className":63598},[1443],[15,63600,63602,63668],{"className":63601},[141,142],[15,63603,63605,63665],{"className":63604},[146],[15,63606,63609,63620,63628],{"className":63607,"style":63608},[150],"height:1.4911em;",[15,63610,63611,63614],{"style":1819},[15,63612],{"className":63613,"style":1823},[158],[15,63615,63617],{"className":63616},[129],[15,63618,14502],{"className":63619},[129],[15,63621,63622,63625],{"style":1832},[15,63623],{"className":63624,"style":1823},[158],[15,63626],{"className":63627,"style":1840},[1839],[15,63629,63630,63633],{"style":1843},[15,63631],{"className":63632,"style":1823},[158],[15,63634,63636],{"className":63635},[129],[15,63637,63639,63642],{"className":63638},[129],[15,63640,48342],{"className":63641},[129],[15,63643,63645],{"className":63644},[137],[15,63646,63648],{"className":63647},[141],[15,63649,63651],{"className":63650},[146],[15,63652,63654],{"className":63653,"style":380},[150],[15,63655,63656,63659],{"style":383},[15,63657],{"className":63658,"style":159},[158],[15,63660,63662],{"className":63661},[163,164,165,166],[15,63663,310],{"className":63664},[129,166],[15,63666,175],{"className":63667},[174],[15,63669,63671],{"className":63670},[146],[15,63672,63674],{"className":63673,"style":1862},[150],[15,63675],{},[15,63677],{"className":63678},[195,1803],[15,63680],{"className":63681,"style":262},[93],[15,63683,63685,63691,63694,63697,63700,63703,63765],{"className":63684},[449],[15,63686,63688],{"className":63687,"style":5052},[188,5051],[15,63689,53],{"className":63690},[228,165],[15,63692,318],{"className":63693},[129],[15,63695],{"className":63696,"style":3277},[93],[15,63698,315],{"className":63699},[2436],[15,63701],{"className":63702,"style":3277},[93],[15,63704,63706,63709,63762],{"className":63705},[129],[15,63707],{"className":63708},[188,1803],[15,63710,63712],{"className":63711},[1443],[15,63713,63715,63754],{"className":63714},[141,142],[15,63716,63718,63751],{"className":63717},[146],[15,63719,63721,63732,63740],{"className":63720,"style":1816},[150],[15,63722,63723,63726],{"style":1819},[15,63724],{"className":63725,"style":1823},[158],[15,63727,63729],{"className":63728},[129],[15,63730,310],{"className":63731},[129],[15,63733,63734,63737],{"style":1832},[15,63735],{"className":63736,"style":1823},[158],[15,63738],{"className":63739,"style":1840},[1839],[15,63741,63742,63745],{"style":1843},[15,63743],{"className":63744,"style":1823},[158],[15,63746,63748],{"className":63747},[129],[15,63749,318],{"className":63750},[129],[15,63752,175],{"className":63753},[174],[15,63755,63757],{"className":63756},[146],[15,63758,63760],{"className":63759,"style":1862},[150],[15,63761],{},[15,63763],{"className":63764},[195,1803],[15,63766,63768],{"className":63767,"style":5052},[195,5051],[15,63769,59],{"className":63770},[228,165],[15,63772],{"className":63773,"style":199},[93],[15,63775,62],{"className":63776},[203],[15,63778],{"className":63779,"style":199},[93],[15,63781,63783,63786,63851],{"className":63782},[120],[15,63784],{"className":63785,"style":57222},[124],[15,63787,63789,63792,63848],{"className":63788},[129],[15,63790],{"className":63791},[188,1803],[15,63793,63795],{"className":63794},[1443],[15,63796,63798,63840],{"className":63797},[141,142],[15,63799,63801,63837],{"className":63800},[146],[15,63802,63804,63818,63826],{"className":63803,"style":1816},[150],[15,63805,63806,63809],{"style":1819},[15,63807],{"className":63808,"style":1823},[158],[15,63810,63812,63815],{"className":63811},[129],[15,63813,63363],{"className":63814},[129],[15,63816,39314],{"className":63817},[129,133],[15,63819,63820,63823],{"style":1832},[15,63821],{"className":63822,"style":1823},[158],[15,63824],{"className":63825,"style":1840},[1839],[15,63827,63828,63831],{"style":1843},[15,63829],{"className":63830,"style":1823},[158],[15,63832,63834],{"className":63833},[129],[15,63835,318],{"className":63836},[129],[15,63838,175],{"className":63839},[174],[15,63841,63843],{"className":63842},[146],[15,63844,63846],{"className":63845,"style":1862},[150],[15,63847],{},[15,63849],{"className":63850},[195,1803],[15,63852,107],{"className":63853},[129],[11,63855,63856,63857,63892,63893],{},"所以 Bernoulli 均值的极小极大速率是 ",[15,63858,63860,63877],{"className":63859,"translate":19},[23],[15,63861,63863],{"className":63862},[27],[29,63864,63865],{"xmlns":31},[34,63866,63867,63875],{},[37,63868,63869,63871,63873],{},[308,63870,318],{},[43,63872,544],{"mathvariant":65},[43,63874,39314],{},[109,63876,48138],{"encoding":111},[15,63878,63880],{"className":63879,"ariaHidden":72},[116],[15,63881,63883,63886,63889],{"className":63882},[120],[15,63884],{"className":63885,"style":125},[124],[15,63887,29841],{"className":63888},[129],[15,63890,39314],{"className":63891},[129,133],"。这个结论不稀奇，但推导方式值得记住：",[698,63894,63895,63896,63931],{},"我们只用了两个点，就把「",[15,63897,63899,63916],{"className":63898,"translate":19},[23],[15,63900,63902],{"className":63901},[27],[29,63903,63904],{"xmlns":31},[34,63905,63906,63914],{},[37,63907,63908,63910,63912],{},[308,63909,318],{},[43,63911,544],{"mathvariant":65},[43,63913,39314],{},[109,63915,48138],{"encoding":111},[15,63917,63919],{"className":63918,"ariaHidden":72},[116],[15,63920,63922,63925,63928],{"className":63921},[120],[15,63923],{"className":63924,"style":125},[124],[15,63926,29841],{"className":63927},[129],[15,63929,39314],{"className":63930},[129,133]," 不可能被突破」这件事证明完了。",[575,63933,63935],{"id":63934},"_34-两点方法的局限","3.4 两点方法的局限",[11,63937,63938,63939,63967],{},"两点方法简单、常数清楚，但它只能看见一个方向。考虑 ",[15,63940,63942,63955],{"className":63941,"translate":19},[23],[15,63943,63945],{"className":63944},[27],[29,63946,63947],{"xmlns":31},[34,63948,63949,63953],{},[37,63950,63951],{},[43,63952,10372],{},[109,63954,10372],{"encoding":111},[15,63956,63958],{"className":63957,"ariaHidden":72},[116],[15,63959,63961,63964],{"className":63960},[120],[15,63962],{"className":63963,"style":522},[124],[15,63965,10372],{"className":63966},[129,133]," 维高斯均值：",[15,63969,63971],{"className":63970,"translate":19},[18],[15,63972,63974,64045],{"className":63973,"translate":19},[23],[15,63975,63977],{"className":63976},[27],[29,63978,63979],{"xmlns":31,"display":32},[34,63980,63981,64042],{},[37,63982,63983,63989,63991,63993,63995,64001,64003,64005,64007,64009,64011,64017,64024,64026,64028,64030,64032,64034,64040],{},[40,63984,63985,63987],{},[43,63986,597],{},[308,63988,318],{},[50,63990,91],{"separator":72},[50,63992,335],{},[50,63994,91],{"separator":72},[40,63996,63997,63999],{},[43,63998,597],{},[43,64000,39314],{},[50,64002,991],{},[43,64004,12555],{},[50,64006,53],{"stretchy":52},[43,64008,48],{},[50,64010,91],{"separator":72},[305,64012,64013,64015],{},[43,64014,27283],{},[308,64016,310],{},[40,64018,64019,64022],{},[43,64020,64021],{},"I",[43,64023,10372],{},[50,64025,59],{"stretchy":52},[50,64027,91],{"separator":72},[93,64029],{"width":95},[43,64031,48],{},[50,64033,100],{},[305,64035,64036,64038],{},[43,64037,104],{"mathvariant":103},[43,64039,10372],{},[43,64041,107],{"mathvariant":65},[109,64043,64044],{"encoding":111},"X_1,\\dots,X_n\\sim N(\\theta,\\sigma^2 I_d),\\qquad \\theta\\in\\mathbb R^d.",[15,64046,64048,64161,64275],{"className":64047,"ariaHidden":72},[116],[15,64049,64051,64054,64094,64097,64100,64103,64106,64109,64112,64152,64155,64158],{"className":64050},[120],[15,64052],{"className":64053,"style":34353},[124],[15,64055,64057,64060],{"className":64056},[129],[15,64058,597],{"className":64059,"style":47958},[129,133],[15,64061,64063],{"className":64062},[137],[15,64064,64066,64086],{"className":64065},[141,142],[15,64067,64069,64083],{"className":64068},[146],[15,64070,64072],{"className":64071,"style":1051},[150],[15,64073,64074,64077],{"style":47973},[15,64075],{"className":64076,"style":159},[158],[15,64078,64080],{"className":64079},[163,164,165,166],[15,64081,318],{"className":64082},[129,166],[15,64084,175],{"className":64085},[174],[15,64087,64089],{"className":64088},[146],[15,64090,64092],{"className":64091,"style":182},[150],[15,64093],{},[15,64095,91],{"className":64096},[254],[15,64098],{"className":64099,"style":262},[93],[15,64101,335],{"className":64102},[449],[15,64104],{"className":64105,"style":262},[93],[15,64107,91],{"className":64108},[254],[15,64110],{"className":64111,"style":262},[93],[15,64113,64115,64118],{"className":64114},[129],[15,64116,597],{"className":64117,"style":47958},[129,133],[15,64119,64121],{"className":64120},[137],[15,64122,64124,64144],{"className":64123},[141,142],[15,64125,64127,64141],{"className":64126},[146],[15,64128,64130],{"className":64129,"style":1163},[150],[15,64131,64132,64135],{"style":47973},[15,64133],{"className":64134,"style":159},[158],[15,64136,64138],{"className":64137},[163,164,165,166],[15,64139,39314],{"className":64140},[129,133,166],[15,64142,175],{"className":64143},[174],[15,64145,64147],{"className":64146},[146],[15,64148,64150],{"className":64149,"style":182},[150],[15,64151],{},[15,64153],{"className":64154,"style":199},[93],[15,64156,991],{"className":64157},[203],[15,64159],{"className":64160,"style":199},[93],[15,64162,64164,64167,64170,64173,64176,64179,64182,64211,64251,64254,64257,64260,64263,64266,64269,64272],{"className":64163},[120],[15,64165],{"className":64166,"style":40908},[124],[15,64168,12555],{"className":64169,"style":12591},[129,133],[15,64171,53],{"className":64172},[188],[15,64174,48],{"className":64175,"style":170},[129,133],[15,64177,91],{"className":64178},[254],[15,64180],{"className":64181,"style":262},[93],[15,64183,64185,64188],{"className":64184},[129],[15,64186,27283],{"className":64187,"style":887},[129,133],[15,64189,64191],{"className":64190},[137],[15,64192,64194],{"className":64193},[141],[15,64195,64197],{"className":64196},[146],[15,64198,64200],{"className":64199,"style":40979},[150],[15,64201,64202,64205],{"style":1272},[15,64203],{"className":64204,"style":159},[158],[15,64206,64208],{"className":64207},[163,164,165,166],[15,64209,310],{"className":64210},[129,166],[15,64212,64214,64217],{"className":64213},[129],[15,64215,64021],{"className":64216,"style":47958},[129,133],[15,64218,64220],{"className":64219},[137],[15,64221,64223,64243],{"className":64222},[141,142],[15,64224,64226,64240],{"className":64225},[146],[15,64227,64229],{"className":64228,"style":151},[150],[15,64230,64231,64234],{"style":47973},[15,64232],{"className":64233,"style":159},[158],[15,64235,64237],{"className":64236},[163,164,165,166],[15,64238,10372],{"className":64239},[129,133,166],[15,64241,175],{"className":64242},[174],[15,64244,64246],{"className":64245},[146],[15,64247,64249],{"className":64248,"style":182},[150],[15,64250],{},[15,64252,59],{"className":64253},[195],[15,64255,91],{"className":64256},[254],[15,64258],{"className":64259,"style":258},[93],[15,64261],{"className":64262,"style":262},[93],[15,64264,48],{"className":64265,"style":170},[129,133],[15,64267],{"className":64268,"style":199},[93],[15,64270,100],{"className":64271},[203],[15,64273],{"className":64274,"style":199},[93],[15,64276,64278,64282,64311],{"className":64277},[120],[15,64279],{"className":64280,"style":64281},[124],"height:0.8991em;",[15,64283,64285,64288],{"className":64284},[129],[15,64286,104],{"className":64287},[129,285],[15,64289,64291],{"className":64290},[137],[15,64292,64294],{"className":64293},[141],[15,64295,64297],{"className":64296},[146],[15,64298,64300],{"className":64299,"style":64281},[150],[15,64301,64302,64305],{"style":1272},[15,64303],{"className":64304,"style":159},[158],[15,64306,64308],{"className":64307},[163,164,165,166],[15,64309,10372],{"className":64310},[129,133,166],[15,64312,107],{"className":64313},[129],[11,64315,64316,64317,33859,64408,64546,64547,64666,64667,64734,64735,64841,64842,64877,64878,64952,64953,1313],{},"如果只取 ",[15,64318,64320,64341],{"className":64319,"translate":19},[23],[15,64321,64323],{"className":64322},[27],[29,64324,64325],{"xmlns":31},[34,64326,64327,64339],{},[37,64328,64329,64335,64337],{},[40,64330,64331,64333],{},[43,64332,48],{},[308,64334,541],{},[50,64336,62],{},[308,64338,541],{},[109,64340,48251],{"encoding":111},[15,64342,64344,64399],{"className":64343,"ariaHidden":72},[116],[15,64345,64347,64350,64390,64393,64396],{"className":64346},[120],[15,64348],{"className":64349,"style":2183},[124],[15,64351,64353,64356],{"className":64352},[129],[15,64354,48],{"className":64355,"style":170},[129,133],[15,64357,64359],{"className":64358},[137],[15,64360,64362,64382],{"className":64361},[141,142],[15,64363,64365,64379],{"className":64364},[146],[15,64366,64368],{"className":64367,"style":1051},[150],[15,64369,64370,64373],{"style":48281},[15,64371],{"className":64372,"style":159},[158],[15,64374,64376],{"className":64375},[163,164,165,166],[15,64377,541],{"className":64378},[129,166],[15,64380,175],{"className":64381},[174],[15,64383,64385],{"className":64384},[146],[15,64386,64388],{"className":64387,"style":182},[150],[15,64389],{},[15,64391],{"className":64392,"style":199},[93],[15,64394,62],{"className":64395},[203],[15,64397],{"className":64398,"style":199},[93],[15,64400,64402,64405],{"className":64401},[120],[15,64403],{"className":64404,"style":4886},[124],[15,64406,541],{"className":64407},[129],[15,64409,64411,64439],{"className":64410,"translate":19},[23],[15,64412,64414],{"className":64413},[27],[29,64415,64416],{"xmlns":31},[34,64417,64418,64436],{},[37,64419,64420,64426,64428,64430],{},[40,64421,64422,64424],{},[43,64423,48],{},[308,64425,318],{},[50,64427,62],{},[43,64429,8686],{},[40,64431,64432,64434],{},[43,64433,10695],{},[308,64435,318],{},[109,64437,64438],{"encoding":111},"\\theta_1=a e_1",[15,64440,64442,64497],{"className":64441,"ariaHidden":72},[116],[15,64443,64445,64448,64488,64491,64494],{"className":64444},[120],[15,64446],{"className":64447,"style":2183},[124],[15,64449,64451,64454],{"className":64450},[129],[15,64452,48],{"className":64453,"style":170},[129,133],[15,64455,64457],{"className":64456},[137],[15,64458,64460,64480],{"className":64459},[141,142],[15,64461,64463,64477],{"className":64462},[146],[15,64464,64466],{"className":64465,"style":1051},[150],[15,64467,64468,64471],{"style":48281},[15,64469],{"className":64470,"style":159},[158],[15,64472,64474],{"className":64473},[163,164,165,166],[15,64475,318],{"className":64476},[129,166],[15,64478,175],{"className":64479},[174],[15,64481,64483],{"className":64482},[146],[15,64484,64486],{"className":64485,"style":182},[150],[15,64487],{},[15,64489],{"className":64490,"style":199},[93],[15,64492,62],{"className":64493},[203],[15,64495],{"className":64496,"style":199},[93],[15,64498,64500,64503,64506],{"className":64499},[120],[15,64501],{"className":64502,"style":7981},[124],[15,64504,8686],{"className":64505},[129,133],[15,64507,64509,64512],{"className":64508},[129],[15,64510,10695],{"className":64511},[129,133],[15,64513,64515],{"className":64514},[137],[15,64516,64518,64538],{"className":64517},[141,142],[15,64519,64521,64535],{"className":64520},[146],[15,64522,64524],{"className":64523,"style":1051},[150],[15,64525,64526,64529],{"style":154},[15,64527],{"className":64528,"style":159},[158],[15,64530,64532],{"className":64531},[163,164,165,166],[15,64533,318],{"className":64534},[129,166],[15,64536,175],{"className":64537},[174],[15,64539,64541],{"className":64540},[146],[15,64542,64544],{"className":64543,"style":182},[150],[15,64545],{},"，那么 KL 是 ",[15,64548,64550,64584],{"className":64549,"translate":19},[23],[15,64551,64553],{"className":64552},[27],[29,64554,64555],{"xmlns":31},[34,64556,64557,64581],{},[37,64558,64559,64561,64567,64569,64571,64573,64579],{},[43,64560,39314],{},[305,64562,64563,64565],{},[43,64564,8686],{},[308,64566,310],{},[43,64568,544],{"mathvariant":65},[50,64570,53],{"stretchy":52},[308,64572,310],{},[305,64574,64575,64577],{},[43,64576,27283],{},[308,64578,310],{},[50,64580,59],{"stretchy":52},[109,64582,64583],{"encoding":111},"n a^2\u002F(2\\sigma^2)",[15,64585,64587],{"className":64586,"ariaHidden":72},[116],[15,64588,64590,64593,64596,64625,64628,64631,64634,64663],{"className":64589},[120],[15,64591],{"className":64592,"style":33996},[124],[15,64594,39314],{"className":64595},[129,133],[15,64597,64599,64602],{"className":64598},[129],[15,64600,8686],{"className":64601},[129,133],[15,64603,64605],{"className":64604},[137],[15,64606,64608],{"className":64607},[141],[15,64609,64611],{"className":64610},[146],[15,64612,64614],{"className":64613,"style":380},[150],[15,64615,64616,64619],{"style":383},[15,64617],{"className":64618,"style":159},[158],[15,64620,64622],{"className":64621},[163,164,165,166],[15,64623,310],{"className":64624},[129,166],[15,64626,544],{"className":64627},[129],[15,64629,53],{"className":64630},[188],[15,64632,310],{"className":64633},[129],[15,64635,64637,64640],{"className":64636},[129],[15,64638,27283],{"className":64639,"style":887},[129,133],[15,64641,64643],{"className":64642},[137],[15,64644,64646],{"className":64645},[141],[15,64647,64649],{"className":64648},[146],[15,64650,64652],{"className":64651,"style":380},[150],[15,64653,64654,64657],{"style":383},[15,64655],{"className":64656,"style":159},[158],[15,64658,64660],{"className":64659},[163,164,165,166],[15,64661,310],{"className":64662},[129,166],[15,64664,59],{"className":64665},[195],"，Le Cam 给出的下界是 ",[15,64668,64670,64692],{"className":64669,"translate":19},[23],[15,64671,64673],{"className":64672},[27],[29,64674,64675],{"xmlns":31},[34,64676,64677,64689],{},[37,64678,64679,64685,64687],{},[305,64680,64681,64683],{},[43,64682,8686],{},[308,64684,310],{},[43,64686,544],{"mathvariant":65},[308,64688,14502],{},[109,64690,64691],{"encoding":111},"a^2\u002F4",[15,64693,64695],{"className":64694,"ariaHidden":72},[116],[15,64696,64698,64701,64730],{"className":64697},[120],[15,64699],{"className":64700,"style":33996},[124],[15,64702,64704,64707],{"className":64703},[129],[15,64705,8686],{"className":64706},[129,133],[15,64708,64710],{"className":64709},[137],[15,64711,64713],{"className":64712},[141],[15,64714,64716],{"className":64715},[146],[15,64717,64719],{"className":64718,"style":380},[150],[15,64720,64721,64724],{"style":383},[15,64722],{"className":64723,"style":159},[158],[15,64725,64727],{"className":64726},[163,164,165,166],[15,64728,310],{"className":64729},[129,166],[15,64731,64733],{"className":64732},[129],"\u002F4"," 的量级，取 ",[15,64736,64738,64763],{"className":64737,"translate":19},[23],[15,64739,64741],{"className":64740},[27],[29,64742,64743],{"xmlns":31},[34,64744,64745,64760],{},[37,64746,64747,64749,64752,64754,64756],{},[43,64748,8686],{},[50,64750,64751],{},"≍",[43,64753,27283],{},[43,64755,544],{"mathvariant":65},[29004,64757,64758],{},[43,64759,39314],{},[109,64761,64762],{"encoding":111},"a\\asymp\\sigma\u002F\\sqrt n",[15,64764,64766,64785],{"className":64765,"ariaHidden":72},[116],[15,64767,64769,64773,64776,64779,64782],{"className":64768},[120],[15,64770],{"className":64771,"style":64772},[124],"height:0.4637em;",[15,64774,8686],{"className":64775},[129,133],[15,64777],{"className":64778,"style":199},[93],[15,64780,64751],{"className":64781},[203],[15,64783],{"className":64784,"style":199},[93],[15,64786,64788,64791,64794,64797],{"className":64787},[120],[15,64789],{"className":64790,"style":30436},[124],[15,64792,27283],{"className":64793,"style":887},[129,133],[15,64795,544],{"className":64796},[129],[15,64798,64800],{"className":64799},[129,29112],[15,64801,64803,64833],{"className":64802},[141,142],[15,64804,64806,64830],{"className":64805},[146],[15,64807,64809,64818],{"className":64808,"style":30452},[150],[15,64810,64812,64815],{"className":64811,"style":2287},[29126],[15,64813],{"className":64814,"style":1823},[158],[15,64816,39314],{"className":64817,"style":30462},[129,133],[15,64819,64820,64823],{"style":30465},[15,64821],{"className":64822,"style":1823},[158],[15,64824,64826],{"className":64825,"style":30472},[29271],[4455,64827,64828],{"xmlns":4457,"width":29275,"height":30475,"viewBox":30476,"preserveAspectRatio":29278},[4461,64829],{"d":30479},[15,64831,175],{"className":64832},[174],[15,64834,64836],{"className":64835},[146],[15,64837,64839],{"className":64838,"style":30489},[150],[15,64840],{}," 后只有 ",[15,64843,64845,64862],{"className":64844,"translate":19},[23],[15,64846,64848],{"className":64847},[27],[29,64849,64850],{"xmlns":31},[34,64851,64852,64860],{},[37,64853,64854,64856,64858],{},[308,64855,318],{},[43,64857,544],{"mathvariant":65},[43,64859,39314],{},[109,64861,48138],{"encoding":111},[15,64863,64865],{"className":64864,"ariaHidden":72},[116],[15,64866,64868,64871,64874],{"className":64867},[120],[15,64869],{"className":64870,"style":125},[124],[15,64872,29841],{"className":64873},[129],[15,64875,39314],{"className":64876},[129,133],"。但真实的极小极大风险是 ",[15,64879,64881,64905],{"className":64880,"translate":19},[23],[15,64882,64884],{"className":64883},[27],[29,64885,64886],{"xmlns":31},[34,64887,64888,64902],{},[37,64889,64890,64896,64898,64900],{},[305,64891,64892,64894],{},[43,64893,27283],{},[308,64895,310],{},[43,64897,10372],{},[43,64899,544],{"mathvariant":65},[43,64901,39314],{},[109,64903,64904],{"encoding":111},"\\sigma^2d\u002Fn",[15,64906,64908],{"className":64907,"ariaHidden":72},[116],[15,64909,64911,64914,64943,64946,64949],{"className":64910},[120],[15,64912],{"className":64913,"style":33996},[124],[15,64915,64917,64920],{"className":64916},[129],[15,64918,27283],{"className":64919,"style":887},[129,133],[15,64921,64923],{"className":64922},[137],[15,64924,64926],{"className":64925},[141],[15,64927,64929],{"className":64928},[146],[15,64930,64932],{"className":64931,"style":380},[150],[15,64933,64934,64937],{"style":383},[15,64935],{"className":64936,"style":159},[158],[15,64938,64940],{"className":64939},[163,164,165,166],[15,64941,310],{"className":64942},[129,166],[15,64944,10372],{"className":64945},[129,133],[15,64947,544],{"className":64948},[129],[15,64950,39314],{"className":64951},[129,133],"——差了整整一个 ",[15,64954,64956,64969],{"className":64955,"translate":19},[23],[15,64957,64959],{"className":64958},[27],[29,64960,64961],{"xmlns":31},[34,64962,64963,64967],{},[37,64964,64965],{},[43,64966,10372],{},[109,64968,10372],{"encoding":111},[15,64970,64972],{"className":64971,"ariaHidden":72},[116],[15,64973,64975,64978],{"className":64974},[120],[15,64976],{"className":64977,"style":522},[124],[15,64979,10372],{"className":64980},[129,133],[11,64982,64983,64984,65012,65013,65041,65042,65070],{},"原因很直接：两点方法只探测了一个坐标方向，而 ",[15,64985,64987,65000],{"className":64986,"translate":19},[23],[15,64988,64990],{"className":64989},[27],[29,64991,64992],{"xmlns":31},[34,64993,64994,64998],{},[37,64995,64996],{},[43,64997,10372],{},[109,64999,10372],{"encoding":111},[15,65001,65003],{"className":65002,"ariaHidden":72},[116],[15,65004,65006,65009],{"className":65005},[120],[15,65007],{"className":65008,"style":522},[124],[15,65010,10372],{"className":65011},[129,133]," 维问题的困难来自 ",[15,65014,65016,65029],{"className":65015,"translate":19},[23],[15,65017,65019],{"className":65018},[27],[29,65020,65021],{"xmlns":31},[34,65022,65023,65027],{},[37,65024,65025],{},[43,65026,10372],{},[109,65028,10372],{"encoding":111},[15,65030,65032],{"className":65031,"ariaHidden":72},[116],[15,65033,65035,65038],{"className":65034},[120],[15,65036],{"className":65037,"style":522},[124],[15,65039,10372],{"className":65040},[129,133]," 个方向同时存在的不确定性。要捕捉这一点，需要让假设集合随着 ",[15,65043,65045,65058],{"className":65044,"translate":19},[23],[15,65046,65048],{"className":65047},[27],[29,65049,65050],{"xmlns":31},[34,65051,65052,65056],{},[37,65053,65054],{},[43,65055,10372],{},[109,65057,10372],{"encoding":111},[15,65059,65061],{"className":65060,"ariaHidden":72},[116],[15,65062,65064,65067],{"className":65063},[120],[15,65065],{"className":65066,"style":522},[124],[15,65068,10372],{"className":65069},[129,133]," 指数增长。这就是 Fano 不等式和 Assouad 引理的用武之地。",[570,65072,65074],{"id":65073},"四fano-不等式","四、Fano 不等式",[575,65076,65078],{"id":65077},"_41-信息论形式","4.1 信息论形式",[6559,65080,65082,65569,65829],{"title":65081},"定理 4.1（Fano 不等式）",[11,65083,21927,65084,5831,65112,65180,65181,65209,65210,7606,65238,65372,65373,22158],{},[15,65085,65087,65100],{"className":65086,"translate":19},[23],[15,65088,65090],{"className":65089},[27],[29,65091,65092],{"xmlns":31},[34,65093,65094,65098],{},[37,65095,65096],{},[43,65097,34511],{},[109,65099,34511],{"encoding":111},[15,65101,65103],{"className":65102,"ariaHidden":72},[116],[15,65104,65106,65109],{"className":65105},[120],[15,65107],{"className":65108,"style":610},[124],[15,65110,34511],{"className":65111,"style":5665},[129,133],[15,65113,65115,65141],{"className":65114,"translate":19},[23],[15,65116,65118],{"className":65117},[27],[29,65119,65120],{"xmlns":31},[34,65121,65122,65138],{},[37,65123,65124,65126,65128,65130,65132,65134,65136],{},[50,65125,636],{"stretchy":52},[308,65127,318],{},[50,65129,91],{"separator":72},[50,65131,335],{},[50,65133,91],{"separator":72},[43,65135,43234],{},[50,65137,645],{"stretchy":52},[109,65139,65140],{"encoding":111},"\\{1,\\dots,M\\}",[15,65142,65144],{"className":65143,"ariaHidden":72},[116],[15,65145,65147,65150,65153,65156,65159,65162,65165,65168,65171,65174,65177],{"className":65146},[120],[15,65148],{"className":65149,"style":125},[124],[15,65151,636],{"className":65152},[188],[15,65154,318],{"className":65155},[129],[15,65157,91],{"className":65158},[254],[15,65160],{"className":65161,"style":262},[93],[15,65163,335],{"className":65164},[449],[15,65166],{"className":65167,"style":262},[93],[15,65169,91],{"className":65170},[254],[15,65172],{"className":65173,"style":262},[93],[15,65175,43234],{"className":65176,"style":12591},[129,133],[15,65178,645],{"className":65179},[195]," 上均匀分布，",[15,65182,65184,65197],{"className":65183,"translate":19},[23],[15,65185,65187],{"className":65186},[27],[29,65188,65189],{"xmlns":31},[34,65190,65191,65195],{},[37,65192,65193],{},[43,65194,597],{},[109,65196,597],{"encoding":111},[15,65198,65200],{"className":65199,"ariaHidden":72},[116],[15,65201,65203,65206],{"className":65202},[120],[15,65204],{"className":65205,"style":610},[124],[15,65207,597],{"className":65208,"style":47958},[129,133]," 的分布依赖于 ",[15,65211,65213,65226],{"className":65212,"translate":19},[23],[15,65214,65216],{"className":65215},[27],[29,65217,65218],{"xmlns":31},[34,65219,65220,65224],{},[37,65221,65222],{},[43,65223,34511],{},[109,65225,34511],{"encoding":111},[15,65227,65229],{"className":65228,"ariaHidden":72},[116],[15,65230,65232,65235],{"className":65231},[120],[15,65233],{"className":65234,"style":610},[124],[15,65236,34511],{"className":65237,"style":5665},[129,133],[15,65239,65241,65273],{"className":65240,"translate":19},[23],[15,65242,65244],{"className":65243},[27],[29,65245,65246],{"xmlns":31},[34,65247,65248,65270],{},[37,65249,65250,65256,65258,65264,65266,65268],{},[46835,65251,65252,65254],{"accent":72},[43,65253,34511],{},[50,65255,48471],{},[50,65257,62],{},[46835,65259,65260,65262],{"accent":72},[43,65261,34511],{},[50,65263,48471],{},[50,65265,53],{"stretchy":52},[43,65267,597],{},[50,65269,59],{"stretchy":52},[109,65271,65272],{"encoding":111},"\\hat K=\\hat K(X)",[15,65274,65276,65325],{"className":65275,"ariaHidden":72},[116],[15,65277,65279,65283,65316,65319,65322],{"className":65278},[120],[15,65280],{"className":65281,"style":65282},[124],"height:0.9468em;",[15,65284,65286],{"className":65285},[129,48488],[15,65287,65289],{"className":65288},[141],[15,65290,65292],{"className":65291},[146],[15,65293,65295,65303],{"className":65294,"style":65282},[150],[15,65296,65297,65300],{"style":2287},[15,65298],{"className":65299,"style":1823},[158],[15,65301,34511],{"className":65302,"style":5665},[129,133],[15,65304,65306,65309],{"style":65305},"top:-3.2523em;",[15,65307],{"className":65308,"style":1823},[158],[15,65310,65313],{"className":65311,"style":65312},[48515],"left:-0.1944em;",[15,65314,48471],{"className":65315},[129],[15,65317],{"className":65318,"style":199},[93],[15,65320,62],{"className":65321},[203],[15,65323],{"className":65324,"style":199},[93],[15,65326,65328,65332,65363,65366,65369],{"className":65327},[120],[15,65329],{"className":65330,"style":65331},[124],"height:1.1968em;vertical-align:-0.25em;",[15,65333,65335],{"className":65334},[129,48488],[15,65336,65338],{"className":65337},[141],[15,65339,65341],{"className":65340},[146],[15,65342,65344,65352],{"className":65343,"style":65282},[150],[15,65345,65346,65349],{"style":2287},[15,65347],{"className":65348,"style":1823},[158],[15,65350,34511],{"className":65351,"style":5665},[129,133],[15,65353,65354,65357],{"style":65305},[15,65355],{"className":65356,"style":1823},[158],[15,65358,65360],{"className":65359,"style":65312},[48515],[15,65361,48471],{"className":65362},[129],[15,65364,53],{"className":65365},[188],[15,65367,597],{"className":65368,"style":47958},[129,133],[15,65370,59],{"className":65371},[195]," 是任意估计。记 ",[15,65374,65376,65414],{"className":65375,"translate":19},[23],[15,65377,65379],{"className":65378},[27],[29,65380,65381],{"xmlns":31},[34,65382,65383,65411],{},[37,65384,65385,65391,65393,65395,65397,65399,65405,65407,65409],{},[40,65386,65387,65389],{},[43,65388,1388],{},[43,65390,10695],{},[50,65392,62],{},[43,65394,4961],{},[50,65396,69],{},[50,65398,53],{"stretchy":52},[46835,65400,65401,65403],{"accent":72},[43,65402,34511],{},[50,65404,48471],{},[50,65406,1418],{"mathvariant":65},[43,65408,34511],{},[50,65410,59],{"stretchy":52},[109,65412,65413],{"encoding":111},"P_e=\\Pr(\\hat K\\neq 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1-\\frac{I(K;X)+\\log 2}{\\log 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是互信息。",[11,65890,65891],{},"证明分两步。第一步是条件熵的界：",[15,65893,65895],{"className":65894,"translate":19},[18],[15,65896,65898,65986],{"className":65897,"translate":19},[23],[15,65899,65901],{"className":65900},[27],[29,65902,65903],{"xmlns":31,"display":32},[34,65904,65905,65983],{},[37,65906,65907,65909,65911,65913,65915,65921,65923,65925,65927,65929,65935,65937,65939,65945,65947,65949,65951,65953,65955,65957,65959,65961,65963,65965,65967,65969,65975,65977,65979,65981],{},[43,65908,819],{},[50,65910,53],{"stretchy":52},[43,65912,34511],{},[50,65914,2945],{},[46835,65916,65917,65919],{"accent":72},[43,65918,34511],{},[50,65920,48471],{},[50,65922,59],{"stretchy":52},[50,65924,2717],{},[43,65926,45],{},[50,65928,53],{"stretchy":52},[40,65930,65931,65933],{},[43,65932,1388],{},[43,65934,10695],{},[50,65936,59],{"stretchy":52},[50,65938,2682],{},[40,65940,65941,65943],{},[43,65942,1388],{},[43,65944,10695],{},[43,65946,14721],{},[50,65948,69],{},[50,65950,53],{"stretchy":52},[43,65952,43234],{},[50,65954,315],{},[308,65956,318],{},[50,65958,59],{"stretchy":52},[50,65960,2717],{},[43,65962,14721],{},[50,65964,69],{},[308,65966,310],{},[50,65968,2682],{},[40,65970,65971,65973],{},[43,65972,1388],{},[43,65974,10695],{},[43,65976,14721],{},[50,65978,69],{},[43,65980,43234],{},[50,65982,91],{"separator":72},[109,65984,65985],{"encoding":111},"H(K\\mid 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2",[15,66344,66346,66410],{"className":66345,"ariaHidden":72},[116],[15,66347,66349,66352,66355,66358,66398,66401,66404,66407],{"className":66348},[120],[15,66350],{"className":66351,"style":125},[124],[15,66353,45],{"className":66354},[129,133],[15,66356,53],{"className":66357},[188],[15,66359,66361,66364],{"className":66360},[129],[15,66362,1388],{"className":66363,"style":12443},[129,133],[15,66365,66367],{"className":66366},[137],[15,66368,66370,66390],{"className":66369},[141,142],[15,66371,66373,66387],{"className":66372},[146],[15,66374,66376],{"className":66375,"style":1163},[150],[15,66377,66378,66381],{"style":34226},[15,66379],{"className":66380,"style":159},[158],[15,66382,66384],{"className":66383},[163,164,165,166],[15,66385,10695],{"className":66386},[129,133,166],[15,66388,175],{"className":66389},[174],[15,66391,66393],{"className":66392},[146],[15,66394,66396],{"className":66395,"style":182},[150],[15,66397],{},[15,66399,59],{"className":66400},[195],[15,66402],{"className":66403,"style":199},[93],[15,66405,2717],{"className":66406},[203],[15,66408],{"className":66409,"style":199},[93],[15,66411,66413,66416,66421,66424],{"className":66412},[120],[15,66414],{"className":66415,"style":1682},[124],[15,66417,14771,66419],{"className":66418},[217],[15,66420,2247],{"style":2246},[15,66422],{"className":66423,"style":262},[93],[15,66425,310],{"className":66426},[129]," 是二元熵的最大值。第二步是数据处理不等式：",[15,66429,66431],{"className":66430,"translate":19},[18],[15,66432,66434,66524],{"className":66433,"translate":19},[23],[15,66435,66437],{"className":66436},[27],[29,66438,66439],{"xmlns":31,"display":32},[34,66440,66441,66521],{},[37,66442,66443,66445,66447,66449,66451,66457,66459,66461,66463,66465,66467,66469,66471,66473,66475,66477,66479,66481,66483,66485,66487,66489,66491,66493,66495,66497,66499,66501,66503,66505,66507,66509,66511,66513,66515,66517,66519],{},[43,66444,819],{},[50,66446,53],{"stretchy":52},[43,66448,34511],{},[50,66450,2945],{},[46835,66452,66453,66455],{"accent":72},[43,66454,34511],{},[50,66456,48471],{},[50,66458,59],{"stretchy":52},[50,66460,7469],{},[43,66462,819],{},[50,66464,53],{"stretchy":52},[43,66466,34511],{},[50,66468,2945],{},[43,66470,597],{},[50,66472,59],{"stretchy":52},[50,66474,62],{},[43,66476,819],{},[50,66478,53],{"stretchy":52},[43,66480,34511],{},[50,66482,59],{"stretchy":52},[50,66484,315],{},[43,66486,64021],{},[50,66488,53],{"stretchy":52},[43,66490,34511],{},[50,66492,65608],{"separator":72},[43,66494,597],{},[50,66496,59],{"stretchy":52},[50,66498,62],{},[43,66500,14721],{},[50,66502,69],{},[43,66504,43234],{},[50,66506,315],{},[43,66508,64021],{},[50,66510,53],{"stretchy":52},[43,66512,34511],{},[50,66514,65608],{"separator":72},[43,66516,597],{},[50,66518,59],{"stretchy":52},[43,66520,107],{"mathvariant":65},[109,66522,66523],{"encoding":111},"H(K\\mid \\hat K)\\ge H(K\\mid X)=H(K)-I(K;X)=\\log M-I(K;X).",[15,66525,66527,66551,66600,66624,66645,66672,66708,66734],{"className":66526,"ariaHidden":72},[116],[15,66528,66530,66533,66536,66539,66542,66545,66548],{"className":66529},[120],[15,66531],{"className":66532,"style":125},[124],[15,66534,819],{"className":66535,"style":65998},[129,133],[15,66537,53],{"className":66538},[188],[15,66540,34511],{"className":66541,"style":5665},[129,133],[15,66543],{"className":66544,"style":199},[93],[15,66546,2945],{"className":66547},[203],[15,66549],{"className":66550,"style":199},[93],[15,66552,66554,66557,66588,66591,66594,66597],{"className":66553},[120],[15,66555],{"className":66556,"style":65331},[124],[15,66558,66560],{"className":66559},[129,48488],[15,66561,66563],{"className":66562},[141],[15,66564,66566],{"className":66565},[146],[15,66567,66569,66577],{"className":66568,"style":65282},[150],[15,66570,66571,66574],{"style":2287},[15,66572],{"className":66573,"style":1823},[158],[15,66575,34511],{"className":66576,"style":5665},[129,133],[15,66578,66579,66582],{"style":65305},[15,66580],{"className":66581,"style":1823},[158],[15,66583,66585],{"className":66584,"style":65312},[48515],[15,66586,48471],{"className":66587},[129],[15,66589,59],{"className":66590},[195],[15,66592],{"className":66593,"style":199},[93],[15,66595,7469],{"className":66596},[203],[15,66598],{"className":66599,"style":199},[93],[15,66601,66603,66606,66609,66612,66615,66618,66621],{"className":66602},[120],[15,66604],{"className":66605,"style":125},[124],[15,66607,819],{"className":66608,"style":65998},[129,133],[15,66610,53],{"className":66611},[188],[15,66613,34511],{"className":66614,"style":5665},[129,133],[15,66616],{"className":66617,"style":199},[93],[15,66619,2945],{"className":66620},[203],[15,66622],{"className":66623,"style":199},[93],[15,66625,66627,66630,66633,66636,66639,66642],{"className":66626},[120],[15,66628],{"className":66629,"style":125},[124],[15,66631,597],{"className":66632,"style":47958},[129,133],[15,66634,59],{"className":66635},[195],[15,66637],{"className":66638,"style":199},[93],[15,66640,62],{"className":66641},[203],[15,66643],{"className":66644,"style":199},[93],[15,66646,66648,66651,66654,66657,66660,66663,66666,66669],{"className":66647},[120],[15,66649],{"className":66650,"style":125},[124],[15,66652,819],{"className":66653,"style":65998},[129,133],[15,66655,53],{"className":66656},[188],[15,66658,34511],{"className":66659,"style":5665},[129,133],[15,66661,59],{"className":66662},[195],[15,66664],{"className":66665,"style":3277},[93],[15,66667,315],{"className":66668},[2436],[15,66670],{"className":66671,"style":3277},[93],[15,66673,66675,66678,66681,66684,66687,66690,66693,66696,66699,66702,66705],{"className":66674},[120],[15,66676],{"className":66677,"style":125},[124],[15,66679,64021],{"className":66680,"style":47958},[129,133],[15,66682,53],{"className":66683},[188],[15,66685,34511],{"className":66686,"style":5665},[129,133],[15,66688,65608],{"className":66689},[254],[15,66691],{"className":66692,"style":262},[93],[15,66694,597],{"className":66695,"style":47958},[129,133],[15,66697,59],{"className":66698},[195],[15,66700],{"className":66701,"style":199},[93],[15,66703,62],{"className":66704},[203],[15,66706],{"className":66707,"style":199},[93],[15,66709,66711,66714,66719,66722,66725,66728,66731],{"className":66710},[120],[15,66712],{"className":66713,"style":1682},[124],[15,66715,14771,66717],{"className":66716},[217],[15,66718,2247],{"style":2246},[15,66720],{"className":66721,"style":262},[93],[15,66723,43234],{"className":66724,"style":12591},[129,133],[15,66726],{"className":66727,"style":3277},[93],[15,66729,315],{"className":66730},[2436],[15,66732],{"className":66733,"style":3277},[93],[15,66735,66737,66740,66743,66746,66749,66752,66755,66758,66761],{"className":66736},[120],[15,66738],{"className":66739,"style":125},[124],[15,66741,64021],{"className":66742,"style":47958},[129,133],[15,66744,53],{"className":66745},[188],[15,66747,34511],{"className":66748,"style":5665},[129,133],[15,66750,65608],{"className":66751},[254],[15,66753],{"className":66754,"style":262},[93],[15,66756,597],{"className":66757,"style":47958},[129,133],[15,66759,59],{"className":66760},[195],[15,66762,107],{"className":66763},[129],[11,66765,66766],{},"两式合并，整理即得。",[11,66768,66769,66770,67027,67028,67069,67070,67110,67111,67139],{},"这个不等式有一个很直观的读法：",[698,66771,66772,66773,66801,66802,66830,66831,66882,66883,4890,66969,66997,66998,67026],{},"如果 ",[15,66774,66776,66789],{"className":66775,"translate":19},[23],[15,66777,66779],{"className":66778},[27],[29,66780,66781],{"xmlns":31},[34,66782,66783,66787],{},[37,66784,66785],{},[43,66786,597],{},[109,66788,597],{"encoding":111},[15,66790,66792],{"className":66791,"ariaHidden":72},[116],[15,66793,66795,66798],{"className":66794},[120],[15,66796],{"className":66797,"style":610},[124],[15,66799,597],{"className":66800,"style":47958},[129,133]," 几乎不包含关于 ",[15,66803,66805,66818],{"className":66804,"translate":19},[23],[15,66806,66808],{"className":66807},[27],[29,66809,66810],{"xmlns":31},[34,66811,66812,66816],{},[37,66813,66814],{},[43,66815,34511],{},[109,66817,34511],{"encoding":111},[15,66819,66821],{"className":66820,"ariaHidden":72},[116],[15,66822,66824,66827],{"className":66823},[120],[15,66825],{"className":66826,"style":610},[124],[15,66828,34511],{"className":66829,"style":5665},[129,133]," 的信息（",[15,66832,66834,66852],{"className":66833,"translate":19},[23],[15,66835,66837],{"className":66836},[27],[29,66838,66839],{"xmlns":31},[34,66840,66841,66849],{},[37,66842,66843,66845,66847],{},[43,66844,64021],{},[50,66846,32392],{},[308,66848,541],{},[109,66850,66851],{"encoding":111},"I\\approx0",[15,66853,66855,66873],{"className":66854,"ariaHidden":72},[116],[15,66856,66858,66861,66864,66867,66870],{"className":66857},[120],[15,66859],{"className":66860,"style":610},[124],[15,66862,64021],{"className":66863,"style":47958},[129,133],[15,66865],{"className":66866,"style":199},[93],[15,66868,32392],{"className":66869},[203],[15,66871],{"className":66872,"style":199},[93],[15,66874,66876,66879],{"className":66875},[120],[15,66877],{"className":66878,"style":4886},[124],[15,66880,541],{"className":66881},[129],"），那么任何猜测的错误率至少是 ",[15,66884,66886,66916],{"className":66885,"translate":19},[23],[15,66887,66889],{"className":66888},[27],[29,66890,66891],{"xmlns":31},[34,66892,66893,66913],{},[37,66894,66895,66897,66899,66901,66903,66905,66907,66909,66911],{},[308,66896,318],{},[50,66898,315],{},[43,66900,14721],{},[50,66902,69],{},[308,66904,310],{},[43,66906,544],{"mathvariant":65},[43,66908,14721],{},[50,66910,69],{},[43,66912,43234],{},[109,66914,66915],{"encoding":111},"1-\\log 2\u002F\\log M",[15,66917,66919,66937],{"className":66918,"ariaHidden":72},[116],[15,66920,66922,66925,66928,66931,66934],{"className":66921},[120],[15,66923],{"className":66924,"style":16477},[124],[15,66926,318],{"className":66927},[129],[15,66929],{"className":66930,"style":3277},[93],[15,66932,315],{"className":66933},[2436],[15,66935],{"className":66936,"style":3277},[93],[15,66938,66940,66943,66948,66951,66955,66958,66963,66966],{"className":66939},[120],[15,66941],{"className":66942,"style":125},[124],[15,66944,14771,66946],{"className":66945},[217],[15,66947,2247],{"style":2246},[15,66949],{"className":66950,"style":262},[93],[15,66952,66954],{"className":66953},[129],"2\u002F",[15,66956],{"className":66957,"style":262},[93],[15,66959,14771,66961],{"className":66960},[217],[15,66962,2247],{"style":2246},[15,66964],{"className":66965,"style":262},[93],[15,66967,43234],{"className":66968,"style":12591},[129,133],[15,66970,66972,66985],{"className":66971,"translate":19},[23],[15,66973,66975],{"className":66974},[27],[29,66976,66977],{"xmlns":31},[34,66978,66979,66983],{},[37,66980,66981],{},[43,66982,43234],{},[109,66984,43234],{"encoding":111},[15,66986,66988],{"className":66987,"ariaHidden":72},[116],[15,66989,66991,66994],{"className":66990},[120],[15,66992],{"className":66993,"style":610},[124],[15,66995,43234],{"className":66996,"style":12591},[129,133]," 很大时接近 ",[15,66999,67001,67014],{"className":67000,"translate":19},[23],[15,67002,67004],{"className":67003},[27],[29,67005,67006],{"xmlns":31},[34,67007,67008,67012],{},[37,67009,67010],{},[308,67011,318],{},[109,67013,318],{"encoding":111},[15,67015,67017],{"className":67016,"ariaHidden":72},[116],[15,67018,67020,67023],{"className":67019},[120],[15,67021],{"className":67022,"style":4886},[124],[15,67024,318],{"className":67025},[129],"，也就是接近「瞎猜」。"," 反过来，要让错误率小，互信息必须达到 ",[15,67029,67031,67049],{"className":67030,"translate":19},[23],[15,67032,67034],{"className":67033},[27],[29,67035,67036],{"xmlns":31},[34,67037,67038,67046],{},[37,67039,67040,67042,67044],{},[43,67041,14721],{},[50,67043,69],{},[43,67045,43234],{},[109,67047,67048],{"encoding":111},"\\log M",[15,67050,67052],{"className":67051,"ariaHidden":72},[116],[15,67053,67055,67058,67063,67066],{"className":67054},[120],[15,67056],{"className":67057,"style":1682},[124],[15,67059,14771,67061],{"className":67060},[217],[15,67062,2247],{"style":2246},[15,67064],{"className":67065,"style":262},[93],[15,67067,43234],{"className":67068,"style":12591},[129,133]," 的量级——你需要至少 ",[15,67071,67073,67090],{"className":67072,"translate":19},[23],[15,67074,67076],{"className":67075},[27],[29,67077,67078],{"xmlns":31},[34,67079,67080,67088],{},[37,67081,67082,67084,67086],{},[43,67083,14721],{},[50,67085,69],{},[43,67087,43234],{},[109,67089,67048],{"encoding":111},[15,67091,67093],{"className":67092,"ariaHidden":72},[116],[15,67094,67096,67099,67104,67107],{"className":67095},[120],[15,67097],{"className":67098,"style":1682},[124],[15,67100,14771,67102],{"className":67101},[217],[15,67103,2247],{"style":2246},[15,67105],{"className":67106,"style":262},[93],[15,67108,43234],{"className":67109,"style":12591},[129,133]," 奈特的信息才能区分 ",[15,67112,67114,67127],{"className":67113,"translate":19},[23],[15,67115,67117],{"className":67116},[27],[29,67118,67119],{"xmlns":31},[34,67120,67121,67125],{},[37,67122,67123],{},[43,67124,43234],{},[109,67126,43234],{"encoding":111},[15,67128,67130],{"className":67129,"ariaHidden":72},[116],[15,67131,67133,67136],{"className":67132},[120],[15,67134],{"className":67135,"style":610},[124],[15,67137,43234],{"className":67138,"style":12591},[129,133]," 个等可能的选择。",[575,67141,67143],{"id":67142},"_42-从检验到估计","4.2 从检验到估计",[11,67145,67146],{},"把 Fano 用在极小极大下界上，需要一个把「损失」翻译成「猜错」的分离条件。",[6559,67148,67150,67372,67622,67933,68536,68618],{"title":67149},"定理 4.2（Fano 方法，度量形式）",[11,67151,21927,67152,13757,67180,67208,67209,67371],{},[15,67153,67155,67168],{"className":67154,"translate":19},[23],[15,67156,67158],{"className":67157},[27],[29,67159,67160],{"xmlns":31},[34,67161,67162,67166],{},[37,67163,67164],{},[43,67165,10372],{},[109,67167,10372],{"encoding":111},[15,67169,67171],{"className":67170,"ariaHidden":72},[116],[15,67172,67174,67177],{"className":67173},[120],[15,67175],{"className":67176,"style":522},[124],[15,67178,10372],{"className":67179},[129,133],[15,67181,67183,67196],{"className":67182,"translate":19},[23],[15,67184,67186],{"className":67185},[27],[29,67187,67188],{"xmlns":31},[34,67189,67190,67194],{},[37,67191,67192],{},[43,67193,30862],{"mathvariant":65},[109,67195,31291],{"encoding":111},[15,67197,67199],{"className":67198,"ariaHidden":72},[116],[15,67200,67202,67205],{"className":67201},[120],[15,67203],{"className":67204,"style":610},[124],[15,67206,30862],{"className":67207},[129]," 上的度量，",[15,67210,67212,67246],{"className":67211,"translate":19},[23],[15,67213,67215],{"className":67214},[27],[29,67216,67217],{"xmlns":31},[34,67218,67219,67243],{},[37,67220,67221,67227,67229,67231,67233,67239,67241],{},[40,67222,67223,67225],{},[43,67224,48],{},[308,67226,318],{},[50,67228,91],{"separator":72},[50,67230,335],{},[50,67232,91],{"separator":72},[40,67234,67235,67237],{},[43,67236,48],{},[43,67238,43234],{},[50,67240,100],{},[43,67242,30862],{"mathvariant":65},[109,67244,67245],{"encoding":111},"\\theta_1,\\dots,\\theta_M\\in\\Theta",[15,67247,67249,67362],{"className":67248,"ariaHidden":72},[116],[15,67250,67252,67255,67295,67298,67301,67304,67307,67310,67313,67353,67356,67359],{"className":67251},[120],[15,67253],{"className":67254,"style":1682},[124],[15,67256,67258,67261],{"className":67257},[129],[15,67259,48],{"className":67260,"style":170},[129,133],[15,67262,67264],{"className":67263},[137],[15,67265,67267,67287],{"className":67266},[141,142],[15,67268,67270,67284],{"className":67269},[146],[15,67271,67273],{"className":67272,"style":1051},[150],[15,67274,67275,67278],{"style":48281},[15,67276],{"className":67277,"style":159},[158],[15,67279,67281],{"className":67280},[163,164,165,166],[15,67282,318],{"className":67283},[129,166],[15,67285,175],{"className":67286},[174],[15,67288,67290],{"className":67289},[146],[15,67291,67293],{"className":67292,"style":182},[150],[15,67294],{},[15,67296,91],{"className":67297},[254],[15,67299],{"className":67300,"style":262},[93],[15,67302,335],{"className":67303},[449],[15,67305],{"className":67306,"style":262},[93],[15,67308,91],{"className":67309},[254],[15,67311],{"className":67312,"style":262},[93],[15,67314,67316,67319],{"className":67315},[129],[15,67317,48],{"className":67318,"style":170},[129,133],[15,67320,67322],{"className":67321},[137],[15,67323,67325,67345],{"className":67324},[141,142],[15,67326,67328,67342],{"className":67327},[146],[15,67329,67331],{"className":67330,"style":1525},[150],[15,67332,67333,67336],{"style":48281},[15,67334],{"className":67335,"style":159},[158],[15,67337,67339],{"className":67338},[163,164,165,166],[15,67340,43234],{"className":67341,"style":12591},[129,133,166],[15,67343,175],{"className":67344},[174],[15,67346,67348],{"className":67347},[146],[15,67349,67351],{"className":67350,"style":182},[150],[15,67352],{},[15,67354],{"className":67355,"style":199},[93],[15,67357,100],{"className":67358},[203],[15,67360],{"className":67361,"style":199},[93],[15,67363,67365,67368],{"className":67364},[120],[15,67366],{"className":67367,"style":610},[124],[15,67369,30862],{"className":67370},[129]," 满足",[15,67373,67375],{"className":67374,"translate":19},[18],[15,67376,67378,67430],{"className":67377,"translate":19},[23],[15,67379,67381],{"className":67380},[27],[29,67382,67383],{"xmlns":31,"display":32},[34,67384,67385,67427],{},[37,67386,67387,67389,67391,67397,67399,67405,67407,67409,67411,67413,67415,67417,67419,67421,67423,67425],{},[43,67388,10372],{},[50,67390,53],{"stretchy":52},[40,67392,67393,67395],{},[43,67394,48],{},[43,67396,1459],{},[50,67398,91],{"separator":72},[40,67400,67401,67403],{},[43,67402,48],{},[43,67404,13949],{},[50,67406,59],{"stretchy":52},[50,67408,7469],{},[308,67410,310],{},[43,67412,29564],{},[93,67414],{"width":95},[50,67416,53],{"stretchy":52},[43,67418,1459],{},[50,67420,1418],{"mathvariant":65},[43,67422,13949],{},[50,67424,59],{"stretchy":52},[43,67426,107],{"mathvariant":65},[109,67428,67429],{"encoding":111},"d(\\theta_i,\\theta_j)\\ge 2\\delta\\qquad(i\\neq 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",[15,70194,70196,70209],{"className":70195,"translate":19},[23],[15,70197,70199],{"className":70198},[27],[29,70200,70201],{"xmlns":31},[34,70202,70203,70207],{},[37,70204,70205],{},[43,70206,34511],{},[109,70208,34511],{"encoding":111},[15,70210,70212],{"className":70211,"ariaHidden":72},[116],[15,70213,70215,70218],{"className":70214},[120],[15,70216],{"className":70217,"style":610},[124],[15,70219,34511],{"className":70220,"style":5665},[129,133]," 是唯一的最小值点，",[15,70223,70225,70247],{"className":70224,"translate":19},[23],[15,70226,70228],{"className":70227},[27],[29,70229,70230],{"xmlns":31},[34,70231,70232,70244],{},[37,70233,70234,70240,70242],{},[46835,70235,70236,70238],{"accent":72},[43,70237,34511],{},[50,70239,48471],{},[50,70241,62],{},[43,70243,34511],{},[109,70245,70246],{"encoding":111},"\\hat K=K",[15,70248,70250,70296],{"className":70249,"ariaHidden":72},[116],[15,70251,70253,70256,70287,70290,70293],{"className":70252},[120],[15,70254],{"className":70255,"style":65282},[124],[15,70257,70259],{"className":70258},[129,48488],[15,70260,70262],{"className":70261},[141],[15,70263,70265],{"className":70264},[146],[15,70266,70268,70276],{"className":70267,"style":65282},[150],[15,70269,70270,70273],{"style":2287},[15,70271],{"className":70272,"style":1823},[158],[15,70274,34511],{"className":70275,"style":5665},[129,133],[15,70277,70278,70281],{"style":65305},[15,70279],{"className":70280,"style":1823},[158],[15,70282,70284],{"className":70283,"style":65312},[48515],[15,70285,48471],{"className":70286},[129],[15,70288],{"className":70289,"style":199},[93],[15,70291,62],{"className":70292},[203],[15,70294],{"className":70295,"style":199},[93],[15,70297,70299,70302],{"className":70298},[120],[15,70300],{"className":70301,"style":610},[124],[15,70303,34511],{"className":70304,"style":5665},[129,133],"。因此",[15,70307,70309],{"className":70308,"translate":19},[18],[15,70310,70312,70370],{"className":70311,"translate":19},[23],[15,70313,70315],{"className":70314},[27],[29,70316,70317],{"xmlns":31,"display":32},[34,70318,70319,70367],{},[37,70320,70321,70323,70325,70327,70333,70335,70341,70343,70345,70347,70349,70351,70353,70359,70361,70363,70365],{},[50,70322,636],{"stretchy":52},[43,70324,10372],{},[50,70326,53],{"stretchy":52},[40,70328,70329,70331],{},[43,70330,48],{},[43,70332,34511],{},[50,70334,91],{"separator":72},[46835,70336,70337,70339],{"accent":72},[43,70338,48],{},[50,70340,48471],{},[50,70342,59],{"stretchy":52},[50,70344,7906],{},[43,70346,29564],{},[50,70348,645],{"stretchy":52},[50,70350,5647],{},[50,70352,636],{"stretchy":52},[46835,70354,70355,70357],{"accent":72},[43,70356,34511],{},[50,70358,48471],{},[50,70360,62],{},[43,70362,34511],{},[50,70364,645],{"stretchy":52},[50,70366,91],{"separator":72},[109,70368,70369],{"encoding":111},"\\{d(\\theta_K,\\hat\\theta)\u003C\\delta\\}\\subseteq\\{\\hat 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",[15,71291,71293,71306],{"className":71292,"translate":19},[23],[15,71294,71296],{"className":71295},[27],[29,71297,71298],{"xmlns":31},[34,71299,71300,71304],{},[37,71301,71302],{},[43,71303,43234],{},[109,71305,43234],{"encoding":111},[15,71307,71309],{"className":71308,"ariaHidden":72},[116],[15,71310,71312,71315],{"className":71311},[120],[15,71313],{"className":71314,"style":610},[124],[15,71316,43234],{"className":71317,"style":12591},[129,133]," 个风险的平均，平均不超过最大，定理得证。",[11,71320,71321,71322,71403,71404,30201,71432,71511,71512,71541,71542,71753,71754,71859],{},"对平方损失 ",[15,71323,71325,71347],{"className":71324,"translate":19},[23],[15,71326,71328],{"className":71327},[27],[29,71329,71330],{"xmlns":31},[34,71331,71332,71344],{},[37,71333,71334,71336,71338],{},[43,71335,27518],{"mathvariant":65},[50,71337,62],{},[305,71339,71340,71342],{},[43,71341,10372],{},[308,71343,310],{},[109,71345,71346],{"encoding":111},"\\ell=d^2",[15,71348,71350,71368],{"className":71349,"ariaHidden":72},[116],[15,71351,71353,71356,71359,71362,71365],{"className":71352},[120],[15,71354],{"className":71355,"style":522},[124],[15,71357,27518],{"className":71358},[129],[15,71360],{"className":71361,"style":199},[93],[15,71363,62],{"className":71364},[203],[15,71366],{"className":71367,"style":199},[93],[15,71369,71371,71374],{"className":71370},[120],[15,71372],{"className":71373,"style":380},[124],[15,71375,71377,71380],{"className":71376},[129],[15,71378,10372],{"className":71379},[129,133],[15,71381,71383],{"className":71382},[137],[15,71384,71386],{"className":71385},[141],[15,71387,71389],{"className":71388},[146],[15,71390,71392],{"className":71391,"style":380},[150],[15,71393,71394,71397],{"style":383},[15,71395],{"className":71396,"style":159},[158],[15,71398,71400],{"className":71399},[163,164,165,166],[15,71401,310],{"className":71402},[129,166],"，同样的解码论证把 ",[15,71405,71407,71420],{"className":71406,"translate":19},[23],[15,71408,71410],{"className":71409},[27],[29,71411,71412],{"xmlns":31},[34,71413,71414,71418],{},[37,71415,71416],{},[43,71417,10372],{},[109,71419,10372],{"encoding":111},[15,71421,71423],{"className":71422,"ariaHidden":72},[116],[15,71424,71426,71429],{"className":71425},[120],[15,71427],{"className":71428,"style":522},[124],[15,71430,10372],{"className":71431},[129,133],[15,71433,71435,71451],{"className":71434,"translate":19},[23],[15,71436,71438],{"className":71437},[27],[29,71439,71440],{"xmlns":31},[34,71441,71442,71448],{},[37,71443,71444],{},[29004,71445,71446],{},[43,71447,27518],{"mathvariant":65},[109,71449,71450],{"encoding":111},"\\sqrt{\\ell}",[15,71452,71454],{"className":71453,"ariaHidden":72},[116],[15,71455,71457,71461],{"className":71456},[120],[15,71458],{"className":71459,"style":71460},[124],"height:1.04em;vertical-align:-0.1078em;",[15,71462,71464],{"className":71463},[129,29112],[15,71465,71467,71502],{"className":71466},[141,142],[15,71468,71470,71499],{"className":71469},[146],[15,71471,71474,71486],{"className":71472,"style":71473},[150],"height:0.9322em;",[15,71475,71477,71480],{"className":71476,"style":2287},[29126],[15,71478],{"className":71479,"style":1823},[158],[15,71481,71483],{"className":71482,"style":30462},[129],[15,71484,27518],{"className":71485},[129],[15,71487,71489,71492],{"style":71488},"top:-2.8922em;",[15,71490],{"className":71491,"style":1823},[158],[15,71493,71495],{"className":71494,"style":30472},[29271],[4455,71496,71497],{"xmlns":4457,"width":29275,"height":30475,"viewBox":30476,"preserveAspectRatio":29278},[4461,71498],{"d":30479},[15,71500,175],{"className":71501},[174],[15,71503,71505],{"className":71504},[146],[15,71506,71509],{"className":71507,"style":71508},[150],"height:0.1078em;",[15,71510],{},"。若平方间隔为 ",[15,71513,71515,71529],{"className":71514,"translate":19},[23],[15,71516,71518],{"className":71517},[27],[29,71519,71520],{"xmlns":31},[34,71521,71522,71526],{},[37,71523,71524],{},[43,71525,48342],{"mathvariant":65},[109,71527,71528],{"encoding":111},"\\Delta",[15,71530,71532],{"className":71531,"ariaHidden":72},[116],[15,71533,71535,71538],{"className":71534},[120],[15,71536],{"className":71537,"style":610},[124],[15,71539,48342],{"className":71540},[129],"，即 ",[15,71543,71545,71583],{"className":71544,"translate":19},[23],[15,71546,71548],{"className":71547},[27],[29,71549,71550],{"xmlns":31},[34,71551,71552,71580],{},[37,71553,71554,71556,71558,71564,71566,71572,71574,71576],{},[43,71555,10372],{},[50,71557,53],{"stretchy":52},[40,71559,71560,71562],{},[43,71561,48],{},[43,71563,1459],{},[50,71565,91],{"separator":72},[40,71567,71568,71570],{},[43,71569,48],{},[43,71571,13949],{},[50,71573,59],{"stretchy":52},[50,71575,7469],{},[29004,71577,71578],{},[43,71579,48342],{"mathvariant":65},[109,71581,71582],{"encoding":111},"d(\\theta_i,\\theta_j)\\ge\\sqrt{\\Delta}",[15,71584,71586,71696],{"className":71585,"ariaHidden":72},[116],[15,71587,71589,71592,71595,71598,71638,71641,71644,71684,71687,71690,71693],{"className":71588},[120],[15,71590],{"className":71591,"style":67439},[124],[15,71593,10372],{"className":71594},[129,133],[15,71596,53],{"className":71597},[188],[15,71599,71601,71604],{"className":71600},[129],[15,71602,48],{"className":71603,"style":170},[129,133],[15,71605,71607],{"className":71606},[137],[15,71608,71610,71630],{"className":71609},[141,142],[15,71611,71613,71627],{"className":71612},[146],[15,71614,71616],{"className":71615,"style":1981},[150],[15,71617,71618,71621],{"style":48281},[15,71619],{"className":71620,"style":159},[158],[15,71622,71624],{"className":71623},[163,164,165,166],[15,71625,1459],{"className":71626},[129,133,166],[15,71628,175],{"className":71629},[174],[15,71631,71633],{"className":71632},[146],[15,71634,71636],{"className":71635,"style":182},[150],[15,71637],{},[15,71639,91],{"className":71640},[254],[15,71642],{"className":71643,"style":262},[93],[15,71645,71647,71650],{"className":71646},[129],[15,71648,48],{"className":71649,"style":170},[129,133],[15,71651,71653],{"className":71652},[137],[15,71654,71656,71676],{"className":71655},[141,142],[15,71657,71659,71673],{"className":71658},[146],[15,71660,71662],{"className":71661,"style":1981},[150],[15,71663,71664,71667],{"style":48281},[15,71665],{"className":71666,"style":159},[158],[15,71668,71670],{"className":71669},[163,164,165,166],[15,71671,13949],{"className":71672,"style":13964},[129,133,166],[15,71674,175],{"className":71675},[174],[15,71677,71679],{"className":71678},[146],[15,71680,71682],{"className":71681,"style":14035},[150],[15,71683],{},[15,71685,59],{"className":71686},[195],[15,71688],{"className":71689,"style":199},[93],[15,71691,7469],{"className":71692},[203],[15,71694],{"className":71695,"style":199},[93],[15,71697,71699,71703],{"className":71698},[120],[15,71700],{"className":71701,"style":71702},[124],"height:1.04em;vertical-align:-0.1133em;",[15,71704,71706],{"className":71705},[129,29112],[15,71707,71709,71744],{"className":71708},[141,142],[15,71710,71712,71741],{"className":71711},[146],[15,71713,71716,71728],{"className":71714,"style":71715},[150],"height:0.9267em;",[15,71717,71719,71722],{"className":71718,"style":2287},[29126],[15,71720],{"className":71721,"style":1823},[158],[15,71723,71725],{"className":71724,"style":30462},[129],[15,71726,48342],{"className":71727},[129],[15,71729,71731,71734],{"style":71730},"top:-2.8867em;",[15,71732],{"className":71733,"style":1823},[158],[15,71735,71737],{"className":71736,"style":30472},[29271],[4455,71738,71739],{"xmlns":4457,"width":29275,"height":30475,"viewBox":30476,"preserveAspectRatio":29278},[4461,71740],{"d":30479},[15,71742,175],{"className":71743},[174],[15,71745,71747],{"className":71746},[146],[15,71748,71751],{"className":71749,"style":71750},[150],"height:0.1133em;",[15,71752],{},"，则取 ",[15,71755,71757,71781],{"className":71756,"translate":19},[23],[15,71758,71760],{"className":71759},[27],[29,71761,71762],{"xmlns":31},[34,71763,71764,71778],{},[37,71765,71766,71768,71770,71774,71776],{},[43,71767,29564],{},[50,71769,62],{},[29004,71771,71772],{},[43,71773,48342],{"mathvariant":65},[43,71775,544],{"mathvariant":65},[308,71777,310],{},[109,71779,71780],{"encoding":111},"\\delta=\\sqrt{\\Delta}\u002F2",[15,71782,71784,71802],{"className":71783,"ariaHidden":72},[116],[15,71785,71787,71790,71793,71796,71799],{"className":71786},[120],[15,71788],{"className":71789,"style":522},[124],[15,71791,29564],{"className":71792,"style":29845},[129,133],[15,71794],{"className":71795,"style":199},[93],[15,71797,62],{"className":71798},[203],[15,71800],{"className":71801,"style":199},[93],[15,71803,71805,71809,71856],{"className":71804},[120],[15,71806],{"className":71807,"style":71808},[124],"height:1.1767em;vertical-align:-0.25em;",[15,71810,71812],{"className":71811},[129,29112],[15,71813,71815,71848],{"className":71814},[141,142],[15,71816,71818,71845],{"className":71817},[146],[15,71819,71821,71833],{"className":71820,"style":71715},[150],[15,71822,71824,71827],{"className":71823,"style":2287},[29126],[15,71825],{"className":71826,"style":1823},[158],[15,71828,71830],{"className":71829,"style":30462},[129],[15,71831,48342],{"className":71832},[129],[15,71834,71835,71838],{"style":71730},[15,71836],{"className":71837,"style":1823},[158],[15,71839,71841],{"className":71840,"style":30472},[29271],[4455,71842,71843],{"xmlns":4457,"width":29275,"height":30475,"viewBox":30476,"preserveAspectRatio":29278},[4461,71844],{"d":30479},[15,71846,175],{"className":71847},[174],[15,71849,71851],{"className":71850},[146],[15,71852,71854],{"className":71853,"style":71750},[150],[15,71855],{},[15,71857,56603],{"className":71858},[129],"，得到",[15,71861,71863],{"className":71862,"translate":19},[18],[15,71864,71866,71968],{"className":71865,"translate":19},[23],[15,71867,71869],{"className":71868},[27],[29,71870,71871],{"xmlns":31,"display":32},[34,71872,71873,71965],{},[37,71874,71875,71881,71887,71889,71891,71893,71895,71901,71903,71909,71911,71913,71915,71921,71963],{},[1443,71876,71877,71879],{},[308,71878,318],{},[43,71880,43234],{},[2140,71882,71883,71885],{},[50,71884,1454],{},[43,71886,1459],{},[43,71888,26563],{"mathvariant":103},[50,71890,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,71892,27518],{"mathvariant":65},[50,71894,53],{"stretchy":52},[40,71896,71897,71899],{},[43,71898,48],{},[43,71900,1459],{},[50,71902,91],{"separator":72},[46835,71904,71905,71907],{"accent":72},[43,71906,48],{},[50,71908,48471],{},[50,71910,59],{"stretchy":52},[50,71912,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,71914,7469],{},[1443,71916,71917,71919],{},[43,71918,48342],{"mathvariant":65},[308,71920,14502],{},[37,71922,71923,71925,71927,71929,71961],{},[50,71924,53],{"fence":72},[308,71926,318],{},[50,71928,315],{},[1443,71930,71931,71953],{},[37,71932,71933,71935,71937,71939,71941,71943,71945,71947,71949,71951],{},[43,71934,64021],{},[50,71936,53],{"stretchy":52},[43,71938,34511],{},[50,71940,65608],{"separator":72},[43,71942,597],{},[50,71944,59],{"stretchy":52},[50,71946,2682],{},[43,71948,14721],{},[50,71950,69],{},[308,71952,310],{},[37,71954,71955,71957,71959],{},[43,71956,14721],{},[50,71958,69],{},[43,71960,43234],{},[50,71962,59],{"fence":72},[43,71964,107],{"mathvariant":65},[109,71966,71967],{"encoding":111},"\\frac1M\\sum_i\\mathbb E\\bigl[\\ell(\\theta_i,\\hat\\theta)\\bigr]\n\\ge\n\\frac{\\Delta}{4}\\left(1-\\frac{I(K;X)+\\log 2}{\\log M}\\right).",[15,71969,71971,72200],{"className":71970,"ariaHidden":72},[116],[15,71972,71974,71978,72040,72043,72087,72090,72093,72099,72102,72105,72145,72148,72151,72182,72185,72191,72194,72197],{"className":71973},[120],[15,71975],{"className":71976,"style":71977},[124],"height:2.5991em;vertical-align:-1.2777em;",[15,71979,71981,71984,72037],{"className":71980},[129],[15,71982],{"className":71983},[188,1803],[15,71985,71987],{"className":71986},[1443],[15,71988,71990,72029],{"className":71989},[141,142],[15,71991,71993,72026],{"className":71992},[146],[15,71994,71996,72007,72015],{"className":71995,"style":1816},[150],[15,71997,71998,72001],{"style":1819},[15,71999],{"className":72000,"style":1823},[158],[15,72002,72004],{"className":72003},[129],[15,72005,43234],{"className":72006,"style":12591},[129,133],[15,72008,72009,72012],{"style":1832},[15,72010],{"className":72011,"style":1823},[158],[15,72013],{"className":72014,"style":1840},[1839],[15,72016,72017,72020],{"style":1843},[15,72018],{"className":72019,"style":1823},[158],[15,72021,72023],{"className":72022},[129],[15,72024,318],{"className":72025},[129],[15,72027,175],{"className":72028},[174],[15,72030,72032],{"className":72031},[146],[15,72033,72035],{"className":72034,"style":1862},[150],[15,72036],{},[15,72038],{"className":72039},[195,1803],[15,72041],{"className":72042,"style":262},[93],[15,72044,72046],{"className":72045},[217,1874],[15,72047,72049,72079],{"className":72048},[141,142],[15,72050,72052,72076],{"className":72051},[146],[15,72053,72055,72066],{"className":72054,"style":12143},[150],[15,72056,72057,72060],{"style":1887},[15,72058],{"className":72059,"style":1891},[158],[15,72061,72063],{"className":72062},[163,164,165,166],[15,72064,1459],{"className":72065},[129,133,166],[15,72067,72068,72071],{"style":1909},[15,72069],{"className":72070,"style":1891},[158],[15,72072,72073],{},[15,72074,1454],{"className":72075},[217,1918,1919],[15,72077,175],{"className":72078},[174],[15,72080,72082],{"className":72081},[146],[15,72083,72085],{"className":72084,"style":1941},[150],[15,72086],{},[15,72088],{"className":72089,"style":262},[93],[15,72091,26563],{"className":72092},[129,285],[15,72094,72096],{"className":72095},[188],[15,72097,1407],{"className":72098},[228,229],[15,72100,27518],{"className":72101},[129],[15,72103,53],{"className":72104},[188],[15,72106,72108,72111],{"className":72107},[129],[15,72109,48],{"className":72110,"style":170},[129,133],[15,72112,72114],{"className":72113},[137],[15,72115,72117,72137],{"className":72116},[141,142],[15,72118,72120,72134],{"className":72119},[146],[15,72121,72123],{"className":72122,"style":1981},[150],[15,72124,72125,72128],{"style":48281},[15,72126],{"className":72127,"style":159},[158],[15,72129,72131],{"className":72130},[163,164,165,166],[15,72132,1459],{"className":72133},[129,133,166],[15,72135,175],{"className":72136},[174],[15,72138,72140],{"className":72139},[146],[15,72141,72143],{"className":72142,"style":182},[150],[15,72144],{},[15,72146,91],{"className":72147},[254],[15,72149],{"className":72150,"style":262},[93],[15,72152,72154],{"className":72153},[129,48488],[15,72155,72157],{"className":72156},[141],[15,72158,72160],{"className":72159},[146],[15,72161,72163,72171],{"className":72162,"style":48484},[150],[15,72164,72165,72168],{"style":2287},[15,72166],{"className":72167,"style":1823},[158],[15,72169,48],{"className":72170,"style":170},[129,133],[15,72172,72173,72176],{"style":48508},[15,72174],{"className":72175,"style":1823},[158],[15,72177,72179],{"className":72178,"style":48516},[48515],[15,72180,48471],{"className":72181},[129],[15,72183,59],{"className":72184},[195],[15,72186,72188],{"className":72187},[195],[15,72189,1423],{"className":72190},[228,229],[15,72192],{"className":72193,"style":199},[93],[15,72195,7469],{"className":72196},[203],[15,72198],{"className":72199,"style":199},[93],[15,72201,72203,72206,72268,72271,72406,72409],{"className":72202},[120],[15,72204],{"className":72205,"style":17007},[124],[15,72207,72209,72212,72265],{"className":72208},[129],[15,72210],{"className":72211},[188,1803],[15,72213,72215],{"className":72214},[1443],[15,72216,72218,72257],{"className":72217},[141,142],[15,72219,72221,72254],{"className":72220},[146],[15,72222,72224,72235,72243],{"className":72223,"style":58499},[150],[15,72225,72226,72229],{"style":1819},[15,72227],{"className":72228,"style":1823},[158],[15,72230,72232],{"className":72231},[129],[15,72233,14502],{"className":72234},[129],[15,72236,72237,72240],{"style":1832},[15,72238],{"className":72239,"style":1823},[158],[15,72241],{"className":72242,"style":1840},[1839],[15,72244,72245,72248],{"style":1843},[15,72246],{"className":72247,"style":1823},[158],[15,72249,72251],{"className":72250},[129],[15,72252,48342],{"className":72253},[129],[15,72255,175],{"className":72256},[174],[15,72258,72260],{"className":72259},[146],[15,72261,72263],{"className":72262,"style":1862},[150],[15,72264],{},[15,72266],{"className":72267},[195,1803],[15,72269],{"className":72270,"style":262},[93],[15,72272,72274,72280,72283,72286,72289,72292,72400],{"className":72273},[449],[15,72275,72277],{"className":72276,"style":5052},[188,5051],[15,72278,53],{"className":72279},[228,165],[15,72281,318],{"className":72282},[129],[15,72284],{"className":72285,"style":3277},[93],[15,72287,315],{"className":72288},[2436],[15,72290],{"className":72291,"style":3277},[93],[15,72293,72295,72298,72397],{"className":72294},[129],[15,72296],{"className":72297},[188,1803],[15,72299,72301],{"className":72300},[1443],[15,72302,72304,72389],{"className":72303},[141,142],[15,72305,72307,72386],{"className":72306},[146],[15,72308,72310,72329,72337],{"className":72309,"style":29154},[150],[15,72311,72312,72315],{"style":1819},[15,72313],{"className":72314,"style":1823},[158],[15,72316,72318,72323,72326],{"className":72317},[129],[15,72319,14771,72321],{"className":72320},[217],[15,72322,2247],{"style":2246},[15,72324],{"className":72325,"style":262},[93],[15,72327,43234],{"className":72328,"style":12591},[129,133],[15,72330,72331,72334],{"style":1832},[15,72332],{"className":72333,"style":1823},[158],[15,72335],{"className":72336,"style":1840},[1839],[15,72338,72339,72342],{"style":1843},[15,72340],{"className":72341,"style":1823},[158],[15,72343,72345,72348,72351,72354,72357,72360,72363,72366,72369,72372,72375,72380,72383],{"className":72344},[129],[15,72346,64021],{"className":72347,"style":47958},[129,133],[15,72349,53],{"className":72350},[188],[15,72352,34511],{"className":72353,"style":5665},[129,133],[15,72355,65608],{"className":72356},[254],[15,72358],{"className":72359,"style":262},[93],[15,72361,597],{"className":72362,"style":47958},[129,133],[15,72364,59],{"className":72365},[195],[15,72367],{"className":72368,"style":3277},[93],[15,72370,2682],{"className":72371},[2436],[15,72373],{"className":72374,"style":3277},[93],[15,72376,14771,72378],{"className":72377},[217],[15,72379,2247],{"style":2246},[15,72381],{"className":72382,"style":262},[93],[15,72384,310],{"className":72385},[129],[15,72387,175],{"className":72388},[174],[15,72390,72392],{"className":72391},[146],[15,72393,72395],{"className":72394,"style":65820},[150],[15,72396],{},[15,72398],{"className":72399},[195,1803],[15,72401,72403],{"className":72402,"style":5052},[195,5051],[15,72404,59],{"className":72405},[228,165],[15,72407],{"className":72408,"style":262},[93],[15,72410,107],{"className":72411},[129],[11,72413,72414,72415,72476,72477,1313,72749,72786],{},"这里用了 ",[15,72416,72418,72440],{"className":72417,"translate":19},[23],[15,72419,72421],{"className":72420},[27],[29,72422,72423],{"xmlns":31},[34,72424,72425,72437],{},[37,72426,72427,72429,72431,72433,72435],{},[50,72428,636],{"stretchy":52},[43,72430,10372],{},[50,72432,7906],{},[43,72434,29564],{},[50,72436,645],{"stretchy":52},[109,72438,72439],{"encoding":111},"\\{d\u003C\\delta\\}",[15,72441,72443,72464],{"className":72442,"ariaHidden":72},[116],[15,72444,72446,72449,72452,72455,72458,72461],{"className":72445},[120],[15,72447],{"className":72448,"style":125},[124],[15,72450,636],{"className":72451},[188],[15,72453,10372],{"className":72454},[129,133],[15,72456],{"className":72457,"style":199},[93],[15,72459,7906],{"className":72460},[203],[15,72462],{"className":72463,"style":199},[93],[15,72465,72467,72470,72473],{"className":72466},[120],[15,72468],{"className":72469,"style":125},[124],[15,72471,29564],{"className":72472,"style":29845},[129,133],[15,72474,645],{"className":72475},[195]," 蕴含解码正确，因此 ",[15,72478,72480,72540],{"className":72479,"translate":19},[23],[15,72481,72483],{"className":72482},[27],[29,72484,72485],{"xmlns":31},[34,72486,72487,72537],{},[37,72488,72489,72491,72493,72499,72501,72503,72509,72511,72513,72515,72517,72519,72521,72523,72525,72531],{},[43,72490,26563],{"mathvariant":103},[50,72492,1407],{"stretchy":52},[305,72494,72495,72497],{},[43,72496,10372],{},[308,72498,310],{},[50,72500,1423],{"stretchy":52},[50,72502,7469],{},[305,72504,72505,72507],{},[43,72506,29564],{},[308,72508,310],{},[43,72510,4961],{},[50,72512,69],{},[50,72514,53],{"stretchy":52},[43,72516,10372],{},[50,72518,7469],{},[43,72520,29564],{},[50,72522,59],{"stretchy":52},[50,72524,7469],{},[305,72526,72527,72529],{},[43,72528,29564],{},[308,72530,310],{},[40,72532,72533,72535],{},[43,72534,1388],{},[43,72536,10695],{},[109,72538,72539],{"encoding":111},"\\mathbb E[d^2]\\ge\\delta^2\\Pr(d\\ge\\delta)\\ge\\delta^2P_e",[15,72541,72543,72596,72652,72673],{"className":72542,"ariaHidden":72},[116],[15,72544,72546,72549,72552,72555,72584,72587,72590,72593],{"className":72545},[120],[15,72547],{"className":72548,"style":33996},[124],[15,72550,26563],{"className":72551},[129,285],[15,72553,1407],{"className":72554},[188],[15,72556,72558,72561],{"className":72557},[129],[15,72559,10372],{"className":72560},[129,133],[15,72562,72564],{"className":72563},[137],[15,72565,72567],{"className":72566},[141],[15,72568,72570],{"className":72569},[146],[15,72571,72573],{"className":72572,"style":380},[150],[15,72574,72575,72578],{"style":383},[15,72576],{"className":72577,"style":159},[158],[15,72579,72581],{"className":72580},[163,164,165,166],[15,72582,310],{"className":72583},[129,166],[15,72585,1423],{"className":72586},[195],[15,72588],{"className":72589,"style":199},[93],[15,72591,7469],{"className":72592},[203],[15,72594],{"className":72595,"style":199},[93],[15,72597,72599,72602,72631,72634,72637,72640,72643,72646,72649],{"className":72598},[120],[15,72600],{"className":72601,"style":33996},[124],[15,72603,72605,72608],{"className":72604},[129],[15,72606,29564],{"className":72607,"style":29845},[129,133],[15,72609,72611],{"className":72610},[137],[15,72612,72614],{"className":72613},[141],[15,72615,72617],{"className":72616},[146],[15,72618,72620],{"className":72619,"style":380},[150],[15,72621,72622,72625],{"style":383},[15,72623],{"className":72624,"style":159},[158],[15,72626,72628],{"className":72627},[163,164,165,166],[15,72629,310],{"className":72630},[129,166],[15,72632],{"className":72633,"style":262},[93],[15,72635,4961],{"className":72636},[217],[15,72638,53],{"className":72639},[188],[15,72641,10372],{"className":72642},[129,133],[15,72644],{"className":72645,"style":199},[93],[15,72647,7469],{"className":72648},[203],[15,72650],{"className":72651,"style":199},[93],[15,72653,72655,72658,72661,72664,72667,72670],{"className":72654},[120],[15,72656],{"className":72657,"style":125},[124],[15,72659,29564],{"className":72660,"style":29845},[129,133],[15,72662,59],{"className":72663},[195],[15,72665],{"className":72666,"style":199},[93],[15,72668,7469],{"className":72669},[203],[15,72671],{"className":72672,"style":199},[93],[15,72674,72676,72680,72709],{"className":72675},[120],[15,72677],{"className":72678,"style":72679},[124],"height:0.9641em;vertical-align:-0.15em;",[15,72681,72683,72686],{"className":72682},[129],[15,72684,29564],{"className":72685,"style":29845},[129,133],[15,72687,72689],{"className":72688},[137],[15,72690,72692],{"className":72691},[141],[15,72693,72695],{"className":72694},[146],[15,72696,72698],{"className":72697,"style":380},[150],[15,72699,72700,72703],{"style":383},[15,72701],{"className":72702,"style":159},[158],[15,72704,72706],{"className":72705},[163,164,165,166],[15,72707,310],{"className":72708},[129,166],[15,72710,72712,72715],{"className":72711},[129],[15,72713,1388],{"className":72714,"style":12443},[129,133],[15,72716,72718],{"className":72717},[137],[15,72719,72721,72741],{"className":72720},[141,142],[15,72722,72724,72738],{"className":72723},[146],[15,72725,72727],{"className":72726,"style":1163},[150],[15,72728,72729,72732],{"style":34226},[15,72730],{"className":72731,"style":159},[158],[15,72733,72735],{"className":72734},[163,164,165,166],[15,72736,10695],{"className":72737},[129,133,166],[15,72739,175],{"className":72740},[174],[15,72742,72744],{"className":72743},[146],[15,72745,72747],{"className":72746,"style":182},[150],[15,72748],{},[698,72750,72751,72752],{},"平方损失的常数是 ",[15,72753,72755,72773],{"className":72754,"translate":19},[23],[15,72756,72758],{"className":72757},[27],[29,72759,72760],{"xmlns":31},[34,72761,72762,72770],{},[37,72763,72764,72766,72768],{},[43,72765,48342],{"mathvariant":65},[43,72767,544],{"mathvariant":65},[308,72769,14502],{},[109,72771,72772],{"encoding":111},"\\Delta\u002F4",[15,72774,72776],{"className":72775,"ariaHidden":72},[116],[15,72777,72779,72782],{"className":72778},[120],[15,72780],{"className":72781,"style":125},[124],[15,72783,72785],{"className":72784},[129],"Δ\u002F4","，与 Le Cam 的平方损失版本一致。本文后面的例子只关心速率，常数不影响结论。",[11,72788,72789],{},"剩下的任务是把互信息控制住。由定义，",[15,72791,72793],{"className":72792,"translate":19},[18],[15,72794,72796,72911],{"className":72795,"translate":19},[23],[15,72797,72799],{"className":72798},[27],[29,72800,72801],{"xmlns":31,"display":32},[34,72802,72803,72908],{},[37,72804,72805,72807,72809,72811,72813,72815,72817,72819,72825,72839,72849,72851,72857,72859,72866,72868,72870,72872,72878,72880,72886,72900,72906],{},[43,72806,64021],{},[50,72808,53],{"stretchy":52},[43,72810,34511],{},[50,72812,65608],{"separator":72},[43,72814,597],{},[50,72816,59],{"stretchy":52},[50,72818,62],{},[1443,72820,72821,72823],{},[308,72822,318],{},[43,72824,43234],{},[1450,72826,72827,72829,72837],{},[50,72828,1454],{},[37,72830,72831,72833,72835],{},[43,72832,1459],{},[50,72834,62],{},[308,72836,318],{},[43,72838,43234],{},[40,72840,72841,72843],{},[43,72842,906],{},[37,72844,72845,72847],{},[43,72846,34511],{"mathvariant":65},[43,72848,1373],{"mathvariant":65},[50,72850,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,72852,72853,72855],{},[43,72854,1388],{},[43,72856,1459],{},[43,72858,33810],{"mathvariant":65},[46835,72860,72861,72863],{"accent":72},[43,72862,1388],{},[50,72864,72865],{},"ˉ",[50,72867,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,72869,91],{"separator":72},[93,72871],{"width":95},[46835,72873,72874,72876],{"accent":72},[43,72875,1388],{},[50,72877,72865],{},[50,72879,62],{},[1443,72881,72882,72884],{},[308,72883,318],{},[43,72885,43234],{},[1450,72887,72888,72890,72898],{},[50,72889,1454],{},[37,72891,72892,72894,72896],{},[43,72893,1459],{},[50,72895,62],{},[308,72897,318],{},[43,72899,43234],{},[40,72901,72902,72904],{},[43,72903,1388],{},[43,72905,1459],{},[43,72907,107],{"mathvariant":65},[109,72909,72910],{"encoding":111},"I(K;X)=\\frac1M\\sum_{i=1}^M D_{\\mathrm{KL}}\\bigl(P_i\\Vert \\bar P\\bigr),\n\\qquad \\bar P=\\frac1M\\sum_{i=1}^M P_i.",[15,72912,72914,72950,73270],{"className":72913,"ariaHidden":72},[116],[15,72915,72917,72920,72923,72926,72929,72932,72935,72938,72941,72944,72947],{"className":72916},[120],[15,72918],{"className":72919,"style":125},[124],[15,72921,64021],{"className":72922,"style":47958},[129,133],[15,72924,53],{"className":72925},[188],[15,72927,34511],{"className":72928,"style":5665},[129,133],[15,72930,65608],{"className":72931},[254],[15,72933],{"className":72934,"style":262},[93],[15,72936,597],{"className":72937,"style":47958},[129,133],[15,72939,59],{"className":72940},[195],[15,72942],{"className":72943,"style":199},[93],[15,72945,62],{"className":72946},[203],[15,72948],{"className":72949,"style":199},[93],[15,72951,72953,72956,73018,73021,73085,73088,73134,73140,73180,73183,73215,73221,73224,73227,73230,73261,73264,73267],{"className":72952},[120],[15,72954],{"className":72955,"style":68062},[124],[15,72957,72959,72962,73015],{"className":72958},[129],[15,72960],{"className":72961},[188,1803],[15,72963,72965],{"className":72964},[1443],[15,72966,72968,73007],{"className":72967},[141,142],[15,72969,72971,73004],{"className":72970},[146],[15,72972,72974,72985,72993],{"className":72973,"style":1816},[150],[15,72975,72976,72979],{"style":1819},[15,72977],{"className":72978,"style":1823},[158],[15,72980,72982],{"className":72981},[129],[15,72983,43234],{"className":72984,"style":12591},[129,133],[15,72986,72987,72990],{"style":1832},[15,72988],{"className":72989,"style":1823},[158],[15,72991],{"className":72992,"style":1840},[1839],[15,72994,72995,72998],{"style":1843},[15,72996],{"className":72997,"style":1823},[158],[15,72999,73001],{"className":73000},[129],[15,73002,318],{"className":73003},[129],[15,73005,175],{"className":73006},[174],[15,73008,73010],{"className":73009},[146],[15,73011,73013],{"className":73012,"style":1862},[150],[15,73014],{},[15,73016],{"className":73017},[195,1803],[15,73019],{"className":73020,"style":262},[93],[15,73022,73024],{"className":73023},[217,1874],[15,73025,73027,73077],{"className":73026},[141,142],[15,73028,73030,73074],{"className":73029},[146],[15,73031,73033,73053,73063],{"className":73032,"style":68140},[150],[15,73034,73035,73038],{"style":1887},[15,73036],{"className":73037,"style":1891},[158],[15,73039,73041],{"className":73040},[163,164,165,166],[15,73042,73044,73047,73050],{"className":73043},[129,166],[15,73045,1459],{"className":73046},[129,133,166],[15,73048,62],{"className":73049},[203,166],[15,73051,318],{"className":73052},[129,166],[15,73054,73055,73058],{"style":1909},[15,73056],{"className":73057,"style":1891},[158],[15,73059,73060],{},[15,73061,1454],{"className":73062},[217,1918,1919],[15,73064,73065,73068],{"style":1922},[15,73066],{"className":73067,"style":1891},[158],[15,73069,73071],{"className":73070},[163,164,165,166],[15,73072,43234],{"className":73073,"style":12591},[129,133,166],[15,73075,175],{"className":73076},[174],[15,73078,73080],{"className":73079},[146],[15,73081,73083],{"className":73082,"style":1941},[150],[15,73084],{},[15,73086],{"className":73087,"style":262},[93],[15,73089,73091,73094],{"className":73090},[129],[15,73092,906],{"className":73093,"style":170},[129,133],[15,73095,73097],{"className":73096},[137],[15,73098,73100,73126],{"className":73099},[141,142],[15,73101,73103,73123],{"className":73102},[146],[15,73104,73106],{"className":73105,"style":1525},[150],[15,73107,73108,73111],{"style":48281},[15,73109],{"className":73110,"style":159},[158],[15,73112,73114],{"className":73113},[163,164,165,166],[15,73115,73117],{"className":73116},[129,166],[15,73118,73120],{"className":73119},[129,166],[15,73121,34915],{"className":73122},[129,221,166],[15,73124,175],{"className":73125},[174],[15,73127,73129],{"className":73128},[146],[15,73130,73132],{"className":73131,"style":182},[150],[15,73133],{},[15,73135,73137],{"className":73136},[188],[15,73138,53],{"className":73139},[228,229],[15,73141,73143,73146],{"className":73142},[129],[15,73144,1388],{"className":73145,"style":12443},[129,133],[15,73147,73149],{"className":73148},[137],[15,73150,73152,73172],{"className":73151},[141,142],[15,73153,73155,73169],{"className":73154},[146],[15,73156,73158],{"className":73157,"style":1981},[150],[15,73159,73160,73163],{"style":34226},[15,73161],{"className":73162,"style":159},[158],[15,73164,73166],{"className":73165},[163,164,165,166],[15,73167,1459],{"className":73168},[129,133,166],[15,73170,175],{"className":73171},[174],[15,73173,73175],{"className":73174},[146],[15,73176,73178],{"className":73177,"style":182},[150],[15,73179],{},[15,73181,33810],{"className":73182},[129],[15,73184,73186],{"className":73185},[129,48488],[15,73187,73189],{"className":73188},[141],[15,73190,73192],{"className":73191},[146],[15,73193,73196,73204],{"className":73194,"style":73195},[150],"height:0.8201em;",[15,73197,73198,73201],{"style":2287},[15,73199],{"className":73200,"style":1823},[158],[15,73202,1388],{"className":73203,"style":12443},[129,133],[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KL 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\\frac1{M^2}\\sum_{i,j=1}^M D_{\\mathrm{KL}}(P_i\\Vert P_j)\n\\le \\max_{i,j}D_{\\mathrm{KL}}(P_i\\Vert P_j).",[15,73591,73593,73629,73944],{"className":73592,"ariaHidden":72},[116],[15,73594,73596,73599,73602,73605,73608,73611,73614,73617,73620,73623,73626],{"className":73595},[120],[15,73597],{"className":73598,"style":125},[124],[15,73600,64021],{"className":73601,"style":47958},[129,133],[15,73603,53],{"className":73604},[188],[15,73606,34511],{"className":73607,"style":5665},[129,133],[15,73609,65608],{"className":73610},[254],[15,73612],{"className":73613,"style":262},[93],[15,73615,597],{"className":73616,"style":47958},[129,133],[15,73618,59],{"className":73619},[195],[15,73621],{"className":73622,"style":199},[93],[15,73624,2717],{"className":73625},[203],[15,73627],{"className":73628,"style":199},[93],[15,73630,73632,73636,73724,73727,73797,73800,73846,73849,73889,73892,73932,73935,73938,73941],{"className":73631},[120],[15,73633],{"className":73634,"style":73635},[124],"height:3.2421em;vertical-align:-1.4138em;",[15,73637,73639,73642,73721],{"className":73638},[129],[15,73640],{"className":73641},[188,1803],[15,73643,73645],{"className":73644},[1443],[15,73646,73648,73713],{"className":73647},[141,142],[15,73649,73651,73710],{"className":73650},[146],[15,73652,73654,73691,73699],{"className":73653,"style":1816},[150],[15,73655,73656,73659],{"style":1819},[15,73657],{"className":73658,"style":1823},[158],[15,73660,73662],{"className":73661},[129],[15,73663,73665,73668],{"className":73664},[129],[15,73666,43234],{"className":73667,"style":12591},[129,133],[15,73669,73671],{"className":73670},[137],[15,73672,73674],{"className":73673},[141],[15,73675,73677],{"className":73676},[146],[15,73678,73680],{"className":73679,"style":30988},[150],[15,73681,73682,73685],{"style":17194},[15,73683],{"className":73684,"style":159},[158],[15,73686,73688],{"className":73687},[163,164,165,166],[15,73689,310],{"className":73690},[129,166],[15,73692,73693,73696],{"style":1832},[15,73694],{"className":73695,"style":1823},[158],[15,73697],{"className":73698,"style":1840},[1839],[15,73700,73701,73704],{"style":1843},[15,73702],{"className":73703,"style":1823},[158],[15,73705,73707],{"className":73706},[129],[15,73708,318],{"className":73709},[129],[15,73711,175],{"className":73712},[174],[15,73714,73716],{"className":73715},[146],[15,73717,73719],{"className":73718,"style":1862},[150],[15,73720],{},[15,73722],{"className":73723},[195,1803],[15,73725],{"className":73726,"style":262},[93],[15,73728,73730],{"className":73729},[217,1874],[15,73731,73733,73789],{"className":73732},[141,142],[15,73734,73736,73786],{"className":73735},[146],[15,73737,73739,73765,73775],{"className":73738,"style":68140},[150],[15,73740,73741,73744],{"style":1887},[15,73742],{"className":73743,"style":1891},[158],[15,73745,73747],{"className":73746},[163,164,165,166],[15,73748,73750,73753,73756,73759,73762],{"className":73749},[129,166],[15,73751,1459],{"className":73752},[129,133,166],[15,73754,91],{"className":73755},[254,166],[15,73757,13949],{"className":73758,"style":13964},[129,133,166],[15,73760,62],{"className":73761},[203,166],[15,73763,318],{"className":73764},[129,166],[15,73766,73767,73770],{"style":1909},[15,73768],{"className":73769,"style":1891},[158],[15,73771,73772],{},[15,73773,1454],{"className":73774},[217,1918,1919],[15,73776,73777,73780],{"style":1922},[15,73778],{"className":73779,"style":1891},[158],[15,73781,73783],{"className":73782},[163,164,165,166],[15,73784,43234],{"className":73785,"style":12591},[129,133,166],[15,73787,175],{"className":73788},[174],[15,73790,73792],{"className":73791},[146],[15,73793,73795],{"className":73794,"style":16028},[150],[15,73796],{},[15,73798],{"className":73799,"style":262},[93],[15,73801,73803,73806],{"className":73802},[129],[15,73804,906],{"className":73805,"style":170},[129,133],[15,73807,73809],{"className":73808},[137],[15,73810,73812,73838],{"className":73811},[141,142],[15,73813,73815,73835],{"className":73814},[146],[15,73816,73818],{"className":73817,"style":1525},[150],[15,73819,73820,73823],{"style":48281},[15,73821],{"className":73822,"style":159},[158],[15,73824,73826],{"className":73825},[163,164,165,166],[15,73827,73829],{"className":73828},[129,166],[15,73830,73832],{"className":73831},[129,166],[15,73833,34915],{"className":73834},[129,221,166],[15,73836,175],{"className":73837},[174],[15,73839,73841],{"className":73840},[146],[15,73842,73844],{"className":73843,"style":182},[150],[15,73845],{},[15,73847,53],{"className":73848},[188],[15,73850,73852,73855],{"className":73851},[129],[15,73853,1388],{"className":73854,"style":12443},[129,133],[15,73856,73858],{"className":73857},[137],[15,73859,73861,73881],{"className":73860},[141,142],[15,73862,73864,73878],{"className":73863},[146],[15,73865,73867],{"className":73866,"style":1981},[150],[15,73868,73869,73872],{"style":34226},[15,73870],{"className":73871,"style":159},[158],[15,73873,73875],{"className":73874},[163,164,165,166],[15,73876,1459],{"className":73877},[129,133,166],[15,73879,175],{"className":73880},[174],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",[15,74148,74150,74168],{"className":74149,"translate":19},[23],[15,74151,74153],{"className":74152},[27],[29,74154,74155],{"xmlns":31},[34,74156,74157,74165],{},[37,74158,74159],{},[40,74160,74161,74163],{},[43,74162,1388],{},[43,74164,1459],{},[109,74166,74167],{"encoding":111},"P_i",[15,74169,74171],{"className":74170,"ariaHidden":72},[116],[15,74172,74174,74177],{"className":74173},[120],[15,74175],{"className":74176,"style":20220},[124],[15,74178,74180,74183],{"className":74179},[129],[15,74181,1388],{"className":74182,"style":12443},[129,133],[15,74184,74186],{"className":74185},[137],[15,74187,74189,74209],{"className":74188},[141,142],[15,74190,74192,74206],{"className":74191},[146],[15,74193,74195],{"className":74194,"style":1981},[150],[15,74196,74197,74200],{"style":34226},[15,74198],{"className":74199,"style":159},[158],[15,74201,74203],{"className":74202},[163,164,165,166],[15,74204,1459],{"className":74205},[129,133,166],[15,74207,175],{"className":74208},[174],[15,74210,74212],{"className":74211},[146],[15,74213,74215],{"className":74214,"style":182},[150],[15,74216],{}," 都是 ",[15,74219,74221,74234],{"className":74220,"translate":19},[23],[15,74222,74224],{"className":74223},[27],[29,74225,74226],{"xmlns":31},[34,74227,74228,74232],{},[37,74229,74230],{},[43,74231,39314],{},[109,74233,39314],{"encoding":111},[15,74235,74237],{"className":74236,"ariaHidden":72},[116],[15,74238,74240,74243],{"className":74239},[120],[15,74241],{"className":74242,"style":4383},[124],[15,74244,39314],{"className":74245},[129,133]," 个独立样本的分布，那么成对 KL 是单样本 KL 的 ",[15,74248,74250,74263],{"className":74249,"translate":19},[23],[15,74251,74253],{"className":74252},[27],[29,74254,74255],{"xmlns":31},[34,74256,74257,74261],{},[37,74258,74259],{},[43,74260,39314],{},[109,74262,39314],{"encoding":111},[15,74264,74266],{"className":74265,"ariaHidden":72},[116],[15,74267,74269,74272],{"className":74268},[120],[15,74270],{"className":74271,"style":4383},[124],[15,74273,39314],{"className":74274},[129,133]," 倍。这样，Fano 下界就变成了「",[15,74277,74279,74296],{"className":74278,"translate":19},[23],[15,74280,74282],{"className":74281},[27],[29,74283,74284],{"xmlns":31},[34,74285,74286,74294],{},[37,74287,74288,74290,74292],{},[43,74289,14721],{},[50,74291,69],{},[43,74293,43234],{},[109,74295,67048],{"encoding":111},[15,74297,74299],{"className":74298,"ariaHidden":72},[116],[15,74300,74302,74305,74310,74313],{"className":74301},[120],[15,74303],{"className":74304,"style":1682},[124],[15,74306,14771,74308],{"className":74307},[217],[15,74309,2247],{"style":2246},[15,74311],{"className":74312,"style":262},[93],[15,74314,43234],{"className":74315,"style":12591},[129,133],[15,74317,74319,74334],{"className":74318,"translate":19},[23],[15,74320,74322],{"className":74321},[27],[29,74323,74324],{"xmlns":31},[34,74325,74326,74332],{},[37,74327,74328,74330],{},[43,74329,39314],{},[50,74331,11418],{},[109,74333,62023],{"encoding":111},[15,74335,74337],{"className":74336,"ariaHidden":72},[116],[15,74338,74340,74343,74346],{"className":74339},[120],[15,74341],{"className":74342,"style":8514},[124],[15,74344,39314],{"className":74345},[129,133],[15,74347,11418],{"className":74348},[129],"KL 的竞争」。",[575,74351,74353],{"id":74352},"_43-广义-fano-与局部版本","4.3 广义 Fano 与局部版本",[11,74355,74356,74357,74360],{},"定理 4.2 常被称为",[698,74358,74359],{},"广义 Fano 不等式","。它的使用可以总结成一句口诀：",[15,74362,74364],{"className":74363,"translate":19},[18],[15,74365,74367,74475],{"className":74366,"translate":19},[23],[15,74368,74370],{"className":74369},[27],[29,74371,74372],{"xmlns":31,"display":32},[34,74373,74374,74472],{},[37,74375,74376],{},[74377,74378,74380],"menclose",{"notation":74379},"box",[2612,74381,74382],{"scriptlevel":541,"displaystyle":52},[2612,74383,74384],{"scriptlevel":541,"displaystyle":52},[2612,74385,74386],{"scriptlevel":541,"displaystyle":72},[37,74387,74388,74391,74393,74395,74397,74399,74402,74404,74406,74408,74424,74434,74436,74442,74444,74450,74452,74454,74457,74463,74466,74468,74470],{},[6852,74389,74390],{}," 只要 ",[43,74392,14721],{},[50,74394,69],{},[43,74396,43234],{},[6852,74398,6854],{},[50,74400,74401],{},"≫",[6852,74403,6854],{},[43,74405,39314],{},[50,74407,10549],{},[2140,74409,74410,74416],{},[37,74411,74412,74414],{},[43,74413,19696],{},[50,74415,69],{},[37,74417,74418,74420,74422],{},[43,74419,1459],{},[50,74421,91],{"separator":72},[43,74423,13949],{},[40,74425,74426,74428],{},[43,74427,906],{},[37,74429,74430,74432],{},[43,74431,34511],{"mathvariant":65},[43,74433,1373],{"mathvariant":65},[50,74435,53],{"stretchy":52},[40,74437,74438,74440],{},[43,74439,1388],{},[43,74441,1459],{},[43,74443,33810],{"mathvariant":65},[40,74445,74446,74448],{},[43,74447,1388],{},[43,74449,13949],{},[50,74451,59],{"stretchy":52},[50,74453,91],{"separator":72},[6852,74455,74456],{}," 就有 ",[40,74458,74459,74461],{},[43,74460,43234],{"mathvariant":28899},[43,74462,39314],{},[50,74464,74465],{},"≳",[43,74467,29564],{},[43,74469,107],{"mathvariant":65},[6852,74471,6854],{},[109,74473,74474],{"encoding":111},"\\boxed{\\ \\text{只要}\\ \\log M\\ \\gg\\ n\\cdot \\max_{i,j}D_{\\mathrm{KL}}(P_i\\Vert P_j),\\ \\text{就有}\\ \\mathfrak M_n\\gtrsim\\delta.\\ }",[15,74476,74478],{"className":74477,"ariaHidden":72},[116],[15,74479,74481,74485],{"className":74480},[120],[15,74482],{"className":74483,"style":74484},[124],"height:2.2938em;vertical-align:-1.2038em;",[15,74486,74488],{"className":74487},[129],[15,74489,74491,74852],{"className":74490},[141,142],[15,74492,74494,74849],{"className":74493},[146],[15,74495,74498,74837],{"className":74496,"style":74497},[150],"height:1.09em;",[15,74499,74501,74505],{"style":74500},"top:-4.2938em;",[15,74502],{"className":74503,"style":74504},[158],"height:4.2938em;",[15,74506,74509],{"className":74507},[74508],"boxpad",[15,74510,74512],{"className":74511},[129],[15,74513,74515,74518,74525,74528,74531,74536,74539,74542,74545,74548,74551,74554,74557,74560,74563,74566,74569,74622,74625,74671,74674,74714,74717,74757,74760,74763,74766,74769,74776,74779,74819,74822,74825,74828,74831,74834],{"className":74514},[129],[15,74516,6854],{"className":74517},[93],[15,74519,74521],{"className":74520},[129,37670],[15,74522,74524],{"className":74523},[129,37678],"只要",[15,74526,6854],{"className":74527},[93],[15,74529],{"className":74530,"style":262},[93],[15,74532,14771,74534],{"className":74533},[217],[15,74535,2247],{"style":2246},[15,74537],{"className":74538,"style":262},[93],[15,74540,43234],{"className":74541,"style":12591},[129,133],[15,74543,6854],{"className":74544},[93],[15,74546],{"className":74547,"style":199},[93],[15,74549,74401],{"className":74550},[203],[15,74552,6854],{"className":74553},[93],[15,74555],{"className":74556,"style":199},[93],[15,74558,39314],{"className":74559},[129,133],[15,74561],{"className":74562,"style":3277},[93],[15,74564,10549],{"className":74565},[2436],[15,74567],{"className":74568,"style":3277},[93],[15,74570,74572],{"className":74571},[217,1874],[15,74573,74575,74614],{"className":74574},[141,142],[15,74576,74578,74611],{"className":74577},[146],[15,74579,74581,74601],{"className":74580,"style":4383},[150],[15,74582,74583,74586],{"style":53429},[15,74584],{"className":74585,"style":1823},[158],[15,74587,74589],{"className":74588},[163,164,165,166],[15,74590,74592,74595,74598],{"className":74591},[129,166],[15,74593,1459],{"className":74594},[129,133,166],[15,74596,91],{"className":74597},[254,166],[15,74599,13949],{"className":74600,"style":13964},[129,133,166],[15,74602,74603,74606],{"style":2287},[15,74604],{"className":74605,"style":1823},[158],[15,74607,74608],{},[15,74609,19696],{"className":74610},[217],[15,74612,175],{"className":74613},[174],[15,74615,74617],{"className":74616},[146],[15,74618,74620],{"className":74619,"style":53473},[150],[15,74621],{},[15,74623],{"className":74624,"style":262},[93],[15,74626,74628,74631],{"className":74627},[129],[15,74629,906],{"className":74630,"style":170},[129,133],[15,74632,74634],{"className":74633},[137],[15,74635,74637,74663],{"className":74636},[141,142],[15,74638,74640,74660],{"className":74639},[146],[15,74641,74643],{"className":74642,"style":1525},[150],[15,74644,74645,74648],{"style":48281},[15,74646],{"className":74647,"style":159},[158],[15,74649,74651],{"className":74650},[163,164,165,166],[15,74652,74654],{"className":74653},[129,166],[15,74655,74657],{"className":74656},[129,166],[15,74658,34915],{"className":74659},[129,221,166],[15,74661,175],{"className":74662},[174],[15,74664,74666],{"className":74665},[146],[15,74667,74669],{"className":74668,"style":182},[150],[15,74670],{},[15,74672,53],{"className":74673},[188],[15,74675,74677,74680],{"className":74676},[129],[15,74678,1388],{"className":74679,"style":12443},[129,133],[15,74681,74683],{"className":74682},[137],[15,74684,74686,74706],{"className":74685},[141,142],[15,74687,74689,74703],{"className":74688},[146],[15,74690,74692],{"className":74691,"style":1981},[150],[15,74693,74694,74697],{"style":34226},[15,74695],{"className":74696,"style":159},[158],[15,74698,74700],{"className":74699},[163,164,165,166],[15,74701,1459],{"className":74702},[129,133,166],[15,74704,175],{"className":74705},[174],[15,74707,74709],{"className":74708},[146],[15,74710,74712],{"className":74711,"style":182},[150],[15,74713],{},[15,74715,33810],{"className":74716},[129],[15,74718,74720,74723],{"className":74719},[129],[15,74721,1388],{"className":74722,"style":12443},[129,133],[15,74724,74726],{"className":74725},[137],[15,74727,74729,74749],{"className":74728},[141,142],[15,74730,74732,74746],{"className":74731},[146],[15,74733,74735],{"className":74734,"style":1981},[150],[15,74736,74737,74740],{"style":34226},[15,74738],{"className":74739,"style":159},[158],[15,74741,74743],{"className":74742},[163,164,165,166],[15,74744,13949],{"className":74745,"style":13964},[129,133,166],[15,74747,175],{"className":74748},[174],[15,74750,74752],{"className":74751},[146],[15,74753,74755],{"className":74754,"style":14035},[150],[15,74756],{},[15,74758,59],{"className":74759},[195],[15,74761,91],{"className":74762},[254],[15,74764,6854],{"className":74765},[93],[15,74767],{"className":74768,"style":262},[93],[15,74770,74772],{"className":74771},[129,37670],[15,74773,74775],{"className":74774},[129,37678],"就有",[15,74777,6854],{"className":74778},[93],[15,74780,74782,74785],{"className":74781},[129],[15,74783,43234],{"className":74784},[129,28926],[15,74786,74788],{"className":74787},[137],[15,74789,74791,74811],{"className":74790},[141,142],[15,74792,74794,74808],{"className":74793},[146],[15,74795,74797],{"className":74796,"style":1163},[150],[15,74798,74799,74802],{"style":154},[15,74800],{"className":74801,"style":159},[158],[15,74803,74805],{"className":74804},[163,164,165,166],[15,74806,39314],{"className":74807},[129,133,166],[15,74809,175],{"className":74810},[174],[15,74812,74814],{"className":74813},[146],[15,74815,74817],{"className":74816,"style":182},[150],[15,74818],{},[15,74820],{"className":74821,"style":199},[93],[15,74823,74465],{"className":74824},[203,13310],[15,74826],{"className":74827,"style":199},[93],[15,74829,29564],{"className":74830,"style":29845},[129,133],[15,74832,107],{"className":74833},[129],[15,74835,6854],{"className":74836},[93],[15,74838,74840,74843],{"style":74839},"top:-3.09em;",[15,74841],{"className":74842,"style":74504},[158],[15,74844],{"className":74845,"style":74848},[74846,74847],"stretchy","fbox","height:2.2938em;border-style:solid;border-width:0.04em;",[15,74850,175],{"className":74851},[174],[15,74853,74855],{"className":74854},[146],[15,74856,74859],{"className":74857,"style":74858},[150],"height:1.2038em;",[15,74860],{},[11,74862,74863,74864,74892,74893,74921,74922,74925,74926,74954],{},"这里 ",[15,74865,74867,74880],{"className":74866,"translate":19},[23],[15,74868,74870],{"className":74869},[27],[29,74871,74872],{"xmlns":31},[34,74873,74874,74878],{},[37,74875,74876],{},[43,74877,43234],{},[109,74879,43234],{"encoding":111},[15,74881,74883],{"className":74882,"ariaHidden":72},[116],[15,74884,74886,74889],{"className":74885},[120],[15,74887],{"className":74888,"style":610},[124],[15,74890,43234],{"className":74891,"style":12591},[129,133]," 是「用 ",[15,74894,74896,74909],{"className":74895,"translate":19},[23],[15,74897,74899],{"className":74898},[27],[29,74900,74901],{"xmlns":31},[34,74902,74903,74907],{},[37,74904,74905],{},[43,74906,29564],{},[109,74908,61720],{"encoding":111},[15,74910,74912],{"className":74911,"ariaHidden":72},[116],[15,74913,74915,74918],{"className":74914},[120],[15,74916],{"className":74917,"style":522},[124],[15,74919,29564],{"className":74920,"style":29845},[129,133]," 分开的假设个数」，也就是损失空间里的",[698,74923,74924],{},"打包数","（packing number）。于是下界问题变成了一个几何问题：在参数空间里能放多少个半径为 ",[15,74927,74929,74942],{"className":74928,"translate":19},[23],[15,74930,74932],{"className":74931},[27],[29,74933,74934],{"xmlns":31},[34,74935,74936,74940],{},[37,74937,74938],{},[43,74939,29564],{},[109,74941,61720],{"encoding":111},[15,74943,74945],{"className":74944,"ariaHidden":72},[116],[15,74946,74948,74951],{"className":74947},[120],[15,74949],{"className":74950,"style":522},[124],[15,74952,29564],{"className":74953,"style":29845},[129,133]," 的小球，同时又让它们在 KL 意义下足够接近？",[11,74956,74957,74958,75313,75314,75346],{},"Yang 与 Barron 的版本直接用 ",[15,74959,74961,75027],{"className":74960,"translate":19},[23],[15,74962,74964],{"className":74963},[27],[29,74965,74966],{"xmlns":31},[34,74967,74968,75024],{},[37,74969,74970,74972,74974,74976,74978,74980,74982,74984,74990,74996,75006,75008,75014,75016,75022],{},[43,74971,64021],{},[50,74973,53],{"stretchy":52},[43,74975,34511],{},[50,74977,65608],{"separator":72},[43,74979,597],{},[50,74981,59],{"stretchy":52},[50,74983,62],{},[1443,74985,74986,74988],{},[308,74987,318],{},[43,74989,43234],{},[40,74991,74992,74994],{},[50,74993,1454],{},[43,74995,1459],{},[40,74997,74998,75000],{},[43,74999,906],{},[37,75001,75002,75004],{},[43,75003,34511],{"mathvariant":65},[43,75005,1373],{"mathvariant":65},[50,75007,53],{"stretchy":52},[40,75009,75010,75012],{},[43,75011,1388],{},[43,75013,1459],{},[43,75015,33810],{"mathvariant":65},[46835,75017,75018,75020],{"accent":72},[43,75019,1388],{},[50,75021,72865],{},[50,75023,59],{"stretchy":52},[109,75025,75026],{"encoding":111},"I(K;X)=\\frac1M\\sum_i D_{\\mathrm{KL}}(P_i\\Vert\\bar P)",[15,75028,75030,75066],{"className":75029,"ariaHidden":72},[116],[15,75031,75033,75036,75039,75042,75045,75048,75051,75054,75057,75060,75063],{"className":75032},[120],[15,75034],{"className":75035,"style":125},[124],[15,75037,64021],{"className":75038,"style":47958},[129,133],[15,75040,53],{"className":75041},[188],[15,75043,34511],{"className":75044,"style":5665},[129,133],[15,75046,65608],{"className":75047},[254],[15,75049],{"className":75050,"style":262},[93],[15,75052,597],{"className":75053,"style":47958},[129,133],[15,75055,59],{"className":75056},[195],[15,75058],{"className":75059,"style":199},[93],[15,75061,62],{"className":75062},[203],[15,75064],{"className":75065,"style":199},[93],[15,75067,75069,75072,75140,75143,75184,75187,75233,75236,75276,75279,75310],{"className":75068},[120],[15,75070],{"className":75071,"style":59676},[124],[15,75073,75075,75078,75137],{"className":75074},[129],[15,75076],{"className":75077},[188,1803],[15,75079,75081],{"className":75080},[1443],[15,75082,75084,75129],{"className":75083},[141,142],[15,75085,75087,75126],{"className":75086},[146],[15,75088,75090,75104,75112],{"className":75089,"style":59695},[150],[15,75091,75092,75095],{"style":59698},[15,75093],{"className":75094,"style":1823},[158],[15,75096,75098],{"className":75097},[163,164,165,166],[15,75099,75101],{"className":75100},[129,166],[15,75102,43234],{"className":75103,"style":12591},[129,133,166],[15,75105,75106,75109],{"style":1832},[15,75107],{"className":75108,"style":1823},[158],[15,75110],{"className":75111,"style":1840},[1839],[15,75113,75114,75117],{"style":59721},[15,75115],{"className":75116,"style":1823},[158],[15,75118,75120],{"className":75119},[163,164,165,166],[15,75121,75123],{"className":75122},[129,166],[15,75124,318],{"className":75125},[129,166],[15,75127,175],{"className":75128},[174],[15,75130,75132],{"className":75131},[146],[15,75133,75135],{"className":75134,"style":23479},[150],[15,75136],{},[15,75138],{"className":75139},[195,1803],[15,75141],{"className":75142,"style":262},[93],[15,75144,75146,75149],{"className":75145},[217],[15,75147,1454],{"className":75148,"style":23341},[217,1918,23340],[15,75150,75152],{"className":75151},[137],[15,75153,75155,75176],{"className":75154},[141,142],[15,75156,75158,75173],{"className":75157},[146],[15,75159,75162],{"className":75160,"style":75161},[150],"height:0.162em;",[15,75163,75164,75167],{"style":23357},[15,75165],{"className":75166,"style":159},[158],[15,75168,75170],{"className":75169},[163,164,165,166],[15,75171,1459],{"className":75172},[129,133,166],[15,75174,175],{"className":75175},[174],[15,75177,75179],{"className":75178},[146],[15,75180,75182],{"className":75181,"style":23406},[150],[15,75183],{},[15,75185],{"className":75186,"style":262},[93],[15,75188,75190,75193],{"className":75189},[129],[15,75191,906],{"className":75192,"style":170},[129,133],[15,75194,75196],{"className":75195},[137],[15,75197,75199,75225],{"className":75198},[141,142],[15,75200,75202,75222],{"className":75201},[146],[15,75203,75205],{"className":75204,"style":1525},[150],[15,75206,75207,75210],{"style":48281},[15,75208],{"className":75209,"style":159},[158],[15,75211,75213],{"className":75212},[163,164,165,166],[15,75214,75216],{"className":75215},[129,166],[15,75217,75219],{"className":75218},[129,166],[15,75220,34915],{"className":75221},[129,221,166],[15,75223,175],{"className":75224},[174],[15,75226,75228],{"className":75227},[146],[15,75229,75231],{"className":75230,"style":182},[150],[15,75232],{},[15,75234,53],{"className":75235},[188],[15,75237,75239,75242],{"className":75238},[129],[15,75240,1388],{"className":75241,"style":12443},[129,133],[15,75243,75245],{"className":75244},[137],[15,75246,75248,75268],{"className":75247},[141,142],[15,75249,75251,75265],{"className":75250},[146],[15,75252,75254],{"className":75253,"style":1981},[150],[15,75255,75256,75259],{"style":34226},[15,75257],{"className":75258,"style":159},[158],[15,75260,75262],{"className":75261},[163,164,165,166],[15,75263,1459],{"className":75264},[129,133,166],[15,75266,175],{"className":75267},[174],[15,75269,75271],{"className":75270},[146],[15,75272,75274],{"className":75273,"style":182},[150],[15,75275],{},[15,75277,33810],{"className":75278},[129],[15,75280,75282],{"className":75281},[129,48488],[15,75283,75285],{"className":75284},[141],[15,75286,75288],{"className":75287},[146],[15,75289,75291,75299],{"className":75290,"style":73195},[150],[15,75292,75293,75296],{"style":2287},[15,75294],{"className":75295,"style":1823},[158],[15,75297,1388],{"className":75298,"style":12443},[129,133],[15,75300,75301,75304],{"style":65305},[15,75302],{"className":75303,"style":1823},[158],[15,75305,75307],{"className":75306,"style":48516},[48515],[15,75308,72865],{"className":75309},[129],[15,75311,59],{"className":75312},[195],"，而不是把它放缩成最大成对 KL。这在局部问题里更精确，常常能去掉多余的 ",[15,75315,75317,75332],{"className":75316,"translate":19},[23],[15,75318,75320],{"className":75319},[27],[29,75321,75322],{"xmlns":31},[34,75323,75324,75330],{},[37,75325,75326,75328],{},[43,75327,14721],{},[50,75329,69],{},[109,75331,30119],{"encoding":111},[15,75333,75335],{"className":75334,"ariaHidden":72},[116],[15,75336,75338,75341],{"className":75337},[120],[15,75339],{"className":75340,"style":1682},[124],[15,75342,14771,75344],{"className":75343},[217],[15,75345,2247],{"style":2246}," 因子。其思想是：不要要求所有假设两两都接近，只要它们整体上离它们的「重心」接近就够了。",[11,75348,75349,75350,75353,75354,75424,75425,75453,75454,75563,75564,75592,75593,75621,75622,1313],{},"Fano 还有一个",[698,75351,75352],{},"局部版本","：不要求 ",[15,75355,75357,75375],{"className":75356,"translate":19},[23],[15,75358,75360],{"className":75359},[27],[29,75361,75362],{"xmlns":31},[34,75363,75364,75372],{},[37,75365,75366],{},[40,75367,75368,75370],{},[43,75369,48],{},[43,75371,1459],{},[109,75373,75374],{"encoding":111},"\\theta_i",[15,75376,75378],{"className":75377,"ariaHidden":72},[116],[15,75379,75381,75384],{"className":75380},[120],[15,75382],{"className":75383,"style":2183},[124],[15,75385,75387,75390],{"className":75386},[129],[15,75388,48],{"className":75389,"style":170},[129,133],[15,75391,75393],{"className":75392},[137],[15,75394,75396,75416],{"className":75395},[141,142],[15,75397,75399,75413],{"className":75398},[146],[15,75400,75402],{"className":75401,"style":1981},[150],[15,75403,75404,75407],{"style":48281},[15,75405],{"className":75406,"style":159},[158],[15,75408,75410],{"className":75409},[163,164,165,166],[15,75411,1459],{"className":75412},[129,133,166],[15,75414,175],{"className":75415},[174],[15,75417,75419],{"className":75418},[146],[15,75420,75422],{"className":75421,"style":182},[150],[15,75423],{}," 覆盖整个参数空间，只要求它们落在一个半径 ",[15,75426,75428,75441],{"className":75427,"translate":19},[23],[15,75429,75431],{"className":75430},[27],[29,75432,75433],{"xmlns":31},[34,75434,75435,75439],{},[37,75436,75437],{},[43,75438,62074],{},[109,75440,62074],{"encoding":111},[15,75442,75444],{"className":75443,"ariaHidden":72},[116],[15,75445,75447,75450],{"className":75446},[120],[15,75448],{"className":75449,"style":4383},[124],[15,75451,62074],{"className":75452,"style":170},[129,133]," 的小球里。非参数速率 ",[15,75455,75457,75495],{"className":75456,"translate":19},[23],[15,75458,75460],{"className":75459},[27],[29,75461,75462],{"xmlns":31},[34,75463,75464,75492],{},[37,75465,75466],{},[305,75467,75468,75470],{},[43,75469,39314],{},[37,75471,75472,75474,75476,75478,75480,75482,75484,75486,75488,75490],{},[50,75473,315],{},[308,75475,310],{},[43,75477,36264],{},[43,75479,544],{"mathvariant":65},[50,75481,53],{"stretchy":52},[308,75483,310],{},[43,75485,36264],{},[50,75487,2682],{},[43,75489,10372],{},[50,75491,59],{"stretchy":52},[109,75493,75494],{"encoding":111},"n^{-2\\beta\u002F(2\\beta+d)}",[15,75496,75498],{"className":75497,"ariaHidden":72},[116],[15,75499,75501,75504],{"className":75500},[120],[15,75502],{"className":75503,"style":14941},[124],[15,75505,75507,75510],{"className":75506},[129],[15,75508,39314],{"className":75509},[129,133],[15,75511,75513],{"className":75512},[137],[15,75514,75516],{"className":75515},[141],[15,75517,75519],{"className":75518},[146],[15,75520,75522],{"className":75521,"style":14941},[150],[15,75523,75524,75527],{"style":383},[15,75525],{"className":75526,"style":159},[158],[15,75528,75530],{"className":75529},[163,164,165,166],[15,75531,75533,75536,75539,75542,75545,75548,75551,75554,75557,75560],{"className":75532},[129,166],[15,75534,315],{"className":75535},[129,166],[15,75537,310],{"className":75538},[129,166],[15,75540,36264],{"className":75541,"style":36280},[129,133,166],[15,75543,544],{"className":75544},[129,166],[15,75546,53],{"className":75547},[188,166],[15,75549,310],{"className":75550},[129,166],[15,75552,36264],{"className":75553,"style":36280},[129,133,166],[15,75555,2682],{"className":75556},[2436,166],[15,75558,10372],{"className":75559},[129,133,166],[15,75561,59],{"className":75562},[195,166]," 正是这样来的——局部的打包数随 ",[15,75565,75567,75580],{"className":75566,"translate":19},[23],[15,75568,75570],{"className":75569},[27],[29,75571,75572],{"xmlns":31},[34,75573,75574,75578],{},[37,75575,75576],{},[43,75577,62074],{},[109,75579,62074],{"encoding":111},[15,75581,75583],{"className":75582,"ariaHidden":72},[116],[15,75584,75586,75589],{"className":75585},[120],[15,75587],{"className":75588,"style":4383},[124],[15,75590,62074],{"className":75591,"style":170},[129,133]," 增长，而 KL 随 ",[15,75594,75596,75609],{"className":75595,"translate":19},[23],[15,75597,75599],{"className":75598},[27],[29,75600,75601],{"xmlns":31},[34,75602,75603,75607],{},[37,75604,75605],{},[43,75606,62074],{},[109,75608,62074],{"encoding":111},[15,75610,75612],{"className":75611,"ariaHidden":72},[116],[15,75613,75615,75618],{"className":75614},[120],[15,75616],{"className":75617,"style":4383},[124],[15,75619,62074],{"className":75620,"style":170},[129,133]," 增长，两者平衡给出最优的 ",[15,75623,75625,75638],{"className":75624,"translate":19},[23],[15,75626,75628],{"className":75627},[27],[29,75629,75630],{"xmlns":31},[34,75631,75632,75636],{},[37,75633,75634],{},[43,75635,62074],{},[109,75637,62074],{"encoding":111},[15,75639,75641],{"className":75640,"ariaHidden":72},[116],[15,75642,75644,75647],{"className":75643},[120],[15,75645],{"className":75646,"style":4383},[124],[15,75648,62074],{"className":75649,"style":170},[129,133],[570,75651,75653],{"id":75652},"五assouad-引理","五、Assouad 引理",[575,75655,75657],{"id":75656},"_51-超立方体","5.1 超立方体",[11,75659,75660,75661,75689],{},"Fano 处理的是「",[15,75662,75664,75677],{"className":75663,"translate":19},[23],[15,75665,75667],{"className":75666},[27],[29,75668,75669],{"xmlns":31},[34,75670,75671,75675],{},[37,75672,75673],{},[43,75674,43234],{},[109,75676,43234],{"encoding":111},[15,75678,75680],{"className":75679,"ariaHidden":72},[116],[15,75681,75683,75686],{"className":75682},[120],[15,75684],{"className":75685,"style":610},[124],[15,75687,43234],{"className":75688,"style":12591},[129,133]," 个任意假设」，Assouad 处理的是一种特殊但极有用的结构：超立方体。",[6559,75691,75693,75898,76667,76670],{"title":75692},"定理 5.1（Assouad 引理，平方损失）",[11,75694,21927,75695,7606,75799,75869,75870,15734],{},[15,75696,75698,75728],{"className":75697,"translate":19},[23],[15,75699,75701],{"className":75700},[27],[29,75702,75703],{"xmlns":31},[34,75704,75705,75725],{},[37,75706,75707,75709,75711,75713,75715,75717,75719],{},[43,75708,30862],{"mathvariant":65},[50,75710,62],{},[50,75712,636],{"stretchy":52},[308,75714,541],{},[50,75716,91],{"separator":72},[308,75718,318],{},[305,75720,75721,75723],{},[50,75722,645],{"stretchy":52},[43,75724,348],{},[109,75726,75727],{"encoding":111},"\\Theta=\\{0,1\\}^m",[15,75729,75731,75749],{"className":75730,"ariaHidden":72},[116],[15,75732,75734,75737,75740,75743,75746],{"className":75733},[120],[15,75735],{"className":75736,"style":610},[124],[15,75738,30862],{"className":75739},[129],[15,75741],{"className":75742,"style":199},[93],[15,75744,62],{"className":75745},[203],[15,75747],{"className":75748,"style":199},[93],[15,75750,75752,75755,75758,75761,75764,75767,75770],{"className":75751},[120],[15,75753],{"className":75754,"style":125},[124],[15,75756,636],{"className":75757},[188],[15,75759,541],{"className":75760},[129],[15,75762,91],{"className":75763},[254],[15,75765],{"className":75766,"style":262},[93],[15,75768,318],{"className":75769},[129],[15,75771,75773,75776],{"className":75772},[195],[15,75774,645],{"className":75775},[195],[15,75777,75779],{"className":75778},[137],[15,75780,75782],{"className":75781},[141],[15,75783,75785],{"className":75784},[146],[15,75786,75788],{"className":75787,"style":6377},[150],[15,75789,75790,75793],{"style":383},[15,75791],{"className":75792,"style":159},[158],[15,75794,75796],{"className":75795},[163,164,165,166],[15,75797,348],{"className":75798},[129,133,166],[15,75800,75802,75820],{"className":75801,"translate":19},[23],[15,75803,75805],{"className":75804},[27],[29,75806,75807],{"xmlns":31},[34,75808,75809,75817],{},[37,75810,75811],{},[40,75812,75813,75815],{},[43,75814,1388],{},[43,75816,48],{},[109,75818,75819],{"encoding":111},"P_\\theta",[15,75821,75823],{"className":75822,"ariaHidden":72},[116],[15,75824,75826,75829],{"className":75825},[120],[15,75827],{"className":75828,"style":20220},[124],[15,75830,75832,75835],{"className":75831},[129],[15,75833,1388],{"className":75834,"style":12443},[129,133],[15,75836,75838],{"className":75837},[137],[15,75839,75841,75861],{"className":75840},[141,142],[15,75842,75844,75858],{"className":75843},[146],[15,75845,75847],{"className":75846,"style":151},[150],[15,75848,75849,75852],{"style":34226},[15,75850],{"className":75851,"style":159},[158],[15,75853,75855],{"className":75854},[163,164,165,166],[15,75856,48],{"className":75857,"style":170},[129,133,166],[15,75859,175],{"className":75860},[174],[15,75862,75864],{"className":75863},[146],[15,75865,75867],{"className":75866,"style":182},[150],[15,75868],{}," 是观测分布，损失为平方欧氏距离。对每个坐标 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X]",[15,77551,77553,77645,77669],{"className":77552,"ariaHidden":72},[116],[15,77554,77556,77559,77627,77630,77633,77636,77639,77642],{"className":77555},[120],[15,77557],{"className":77558,"style":50020},[124],[15,77560,77562,77593],{"className":77561},[129],[15,77563,77565],{"className":77564},[129,48488],[15,77566,77568],{"className":77567},[141],[15,77569,77571],{"className":77570},[146],[15,77572,77574,77582],{"className":77573,"style":48484},[150],[15,77575,77576,77579],{"style":2287},[15,77577],{"className":77578,"style":1823},[158],[15,77580,48],{"className":77581,"style":170},[129,133],[15,77583,77584,77587],{"style":48508},[15,77585],{"className":77586,"style":1823},[158],[15,77588,77590],{"className":77589,"style":48516},[48515],[15,77591,48471],{"className":77592},[129],[15,77594,77596],{"className":77595},[137],[15,77597,77599,77619],{"className":77598},[141,142],[15,77600,77602,77616],{"className":77601},[146],[15,77603,77605],{"className":77604,"style":1525},[150],[15,77606,77607,77610],{"style":48281},[15,77608],{"className":77609,"style":159},[158],[15,77611,77613],{"className":77612},[163,164,165,166],[15,77614,33882],{"className":77615,"style":33922},[129,133,166],[15,77617,175],{"className":77618},[174],[15,77620,77622],{"className":77621},[146],[15,77623,77625],{"className":77624,"style":182},[150],[15,77626],{},[15,77628,53],{"className":77629},[188],[15,77631,597],{"className":77632,"style":47958},[129,133],[15,77634,59],{"className":77635},[195],[15,77637],{"className":77638,"style":199},[93],[15,77640,62],{"className":77641},[203],[15,77643],{"className":77644,"style":199},[93],[15,77646,77648,77651,77654,77657,77660,77663,77666],{"className":77647},[120],[15,77649],{"className":77650,"style":125},[124],[15,77652,26563],{"className":77653},[129,285],[15,77655,1407],{"className":77656},[188],[15,77658,48],{"className":77659,"style":170},[129,133],[15,77661],{"className":77662,"style":199},[93],[15,77664,2945],{"className":77665},[203],[15,77667],{"className":77668,"style":199},[93],[15,77670,77672,77675,77678],{"className":77671},[120],[15,77673],{"className":77674,"style":125},[124],[15,77676,597],{"className":77677,"style":47958},[129,133],[15,77679,1423],{"className":77680},[195],"，其贝叶斯风险为",[15,77683,77685],{"className":77684,"translate":19},[18],[15,77686,77688,77794],{"className":77687,"translate":19},[23],[15,77689,77691],{"className":77690},[27],[29,77692,77693],{"xmlns":31,"display":32},[34,77694,77695,77791],{},[37,77696,77697,77711,77713,77715,77721,77723,77725,77727,77729,77731,77733,77739,77741,77743,77745,77747,77749,77751,77753,77759,77761,77763,77765,77767,77769,77771,77773,77779,77781,77783,77785,77787,77789],{},[1450,77698,77699,77701,77709],{},[50,77700,1454],{},[37,77702,77703,77705,77707],{},[43,77704,13949],{},[50,77706,62],{},[308,77708,318],{},[43,77710,348],{},[43,77712,26563],{"mathvariant":103},[50,77714,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,77716,77717,77719],{},[43,77718,15756],{},[43,77720,13949],{},[50,77722,53],{"stretchy":52},[43,77724,597],{},[50,77726,59],{"stretchy":52},[50,77728,53],{"stretchy":52},[308,77730,318],{},[50,77732,315],{},[40,77734,77735,77737],{},[43,77736,15756],{},[43,77738,13949],{},[50,77740,53],{"stretchy":52},[43,77742,597],{},[50,77744,59],{"stretchy":52},[50,77746,59],{"stretchy":52},[50,77748,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,77750,91],{"separator":72},[93,77752],{"width":95},[40,77754,77755,77757],{},[43,77756,15756],{},[43,77758,13949],{},[50,77760,53],{"stretchy":52},[43,77762,597],{},[50,77764,59],{"stretchy":52},[50,77766,62],{},[43,77768,4961],{},[50,77770,69],{},[50,77772,53],{"stretchy":52},[40,77774,77775,77777],{},[43,77776,48],{},[43,77778,13949],{},[50,77780,62],{},[308,77782,318],{},[50,77784,2945],{},[43,77786,597],{},[50,77788,59],{"stretchy":52},[43,77790,107],{"mathvariant":65},[109,77792,77793],{"encoding":111},"\\sum_{j=1}^m\\mathbb E\\bigl[\\pi_j(X)(1-\\pi_j(X))\\bigr],\n\\qquad\n\\pi_j(X)=\\Pr(\\theta_j=1\\mid X).",[15,77795,77797,77944,78072,78133,78151],{"className":77796,"ariaHidden":72},[116],[15,77798,77800,77804,77868,77871,77874,77880,77920,77923,77926,77929,77932,77935,77938,77941],{"className":77799},[120],[15,77801],{"className":77802,"style":77803},[124],"height:3.0652em;vertical-align:-1.4138em;",[15,77805,77807],{"className":77806},[217,1874],[15,77808,77810,77860],{"className":77809},[141,142],[15,77811,77813,77857],{"className":77812},[146],[15,77814,77816,77836,77846],{"className":77815,"style":1884},[150],[15,77817,77818,77821],{"style":1887},[15,77819],{"className":77820,"style":1891},[158],[15,77822,77824],{"className":77823},[163,164,165,166],[15,77825,77827,77830,77833],{"className":77826},[129,166],[15,77828,13949],{"className":77829,"style":13964},[129,133,166],[15,77831,62],{"className":77832},[203,166],[15,77834,318],{"className":77835},[129,166],[15,77837,77838,77841],{"style":1909},[15,77839],{"className":77840,"style":1891},[158],[15,77842,77843],{},[15,77844,1454],{"className":77845},[217,1918,1919],[15,77847,77848,77851],{"style":1922},[15,77849],{"className":77850,"style":1891},[158],[15,77852,77854],{"className":77853},[163,164,165,166],[15,77855,348],{"className":77856},[129,133,166],[15,77858,175],{"className":77859},[174],[15,77861,77863],{"className":77862},[146],[15,77864,77866],{"className":77865,"style":16028},[150],[15,77867],{},[15,77869],{"className":77870,"style":262},[93],[15,77872,26563],{"className":77873},[129,285],[15,77875,77877],{"className":77876},[188],[15,77878,1407],{"className":77879},[228,229],[15,77881,77883,77886],{"className":77882},[129],[15,77884,15756],{"className":77885,"style":887},[129,133],[15,77887,77889],{"className":77888},[137],[15,77890,77892,77912],{"className":77891},[141,142],[15,77893,77895,77909],{"className":77894},[146],[15,77896,77898],{"className":77897,"style":1981},[150],[15,77899,77900,77903],{"style":1100},[15,77901],{"className":77902,"style":159},[158],[15,77904,77906],{"className":77905},[163,164,165,166],[15,77907,13949],{"className":77908,"style":13964},[129,133,166],[15,77910,175],{"className":77911},[174],[15,77913,77915],{"className":77914},[146],[15,77916,77918],{"className":77917,"style":14035},[150],[15,77919],{},[15,77921,53],{"className":77922},[188],[15,77924,597],{"className":77925,"style":47958},[129,133],[15,77927,59],{"className":77928},[195],[15,77930,53],{"className":77931},[188],[15,77933,318],{"className":77934},[129],[15,77936],{"className":77937,"style":3277},[93],[15,77939,315],{"className":77940},[2436],[15,77942],{"className":77943,"style":3277},[93],[15,77945,77947,77950,77990,77993,77996,77999,78005,78008,78011,78014,78054,78057,78060,78063,78066,78069],{"className":77946},[120],[15,77948],{"className":77949,"style":213},[124],[15,77951,77953,77956],{"className":77952},[129],[15,77954,15756],{"className":77955,"style":887},[129,133],[15,77957,77959],{"className":77958},[137],[15,77960,77962,77982],{"className":77961},[141,142],[15,77963,77965,77979],{"className":77964},[146],[15,77966,77968],{"className":77967,"style":1981},[150],[15,77969,77970,77973],{"style":1100},[15,77971],{"className":77972,"style":159},[158],[15,77974,77976],{"className":77975},[163,164,165,166],[15,77977,13949],{"className":77978,"style":13964},[129,133,166],[15,77980,175],{"className":77981},[174],[15,77983,77985],{"className":77984},[146],[15,77986,77988],{"className":77987,"style":14035},[150],[15,77989],{},[15,77991,53],{"className":77992},[188],[15,77994,597],{"className":77995,"style":47958},[129,133],[15,77997,6149],{"className":77998},[195],[15,78000,78002],{"className":78001},[195],[15,78003,1423],{"className":78004},[228,229],[15,78006,91],{"className":78007},[254],[15,78009],{"className":78010,"style":258},[93],[15,78012],{"className":78013,"style":262},[93],[15,78015,78017,78020],{"className":78016},[129],[15,78018,15756],{"className":78019,"style":887},[129,133],[15,78021,78023],{"className":78022},[137],[15,78024,78026,78046],{"className":78025},[141,142],[15,78027,78029,78043],{"className":78028},[146],[15,78030,78032],{"className":78031,"style":1981},[150],[15,78033,78034,78037],{"style":1100},[15,78035],{"className":78036,"style":159},[158],[15,78038,78040],{"className":78039},[163,164,165,166],[15,78041,13949],{"className":78042,"style":13964},[129,133,166],[15,78044,175],{"className":78045},[174],[15,78047,78049],{"className":78048},[146],[15,78050,78052],{"className":78051,"style":14035},[150],[15,78053],{},[15,78055,53],{"className":78056},[188],[15,78058,597],{"className":78059,"style":47958},[129,133],[15,78061,59],{"className":78062},[195],[15,78064],{"className":78065,"style":199},[93],[15,78067,62],{"className":78068},[203],[15,78070],{"className":78071,"style":199},[93],[15,78073,78075,78078,78081,78084,78124,78127,78130],{"className":78074},[120],[15,78076],{"className":78077,"style":67439},[124],[15,78079,4961],{"className":78080},[217],[15,78082,53],{"className":78083},[188],[15,78085,78087,78090],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",[15,78170,78172,78185],{"className":78171,"translate":19},[23],[15,78173,78175],{"className":78174},[27],[29,78176,78177],{"xmlns":31},[34,78178,78179,78183],{},[37,78180,78181],{},[43,78182,13949],{},[109,78184,13949],{"encoding":111},[15,78186,78188],{"className":78187,"ariaHidden":72},[116],[15,78189,78191,78194],{"className":78190},[120],[15,78192],{"className":78193,"style":11187},[124],[15,78195,13949],{"className":78196,"style":13964},[129,133],"。边际后验 ",[15,78199,78201,78219],{"className":78200,"translate":19},[23],[15,78202,78204],{"className":78203},[27],[29,78205,78206],{"xmlns":31},[34,78207,78208,78216],{},[37,78209,78210],{},[40,78211,78212,78214],{},[43,78213,15756],{},[43,78215,13949],{},[109,78217,78218],{"encoding":111},"\\pi_j",[15,78220,78222],{"className":78221,"ariaHidden":72},[116],[15,78223,78225,78228],{"className":78224},[120],[15,78226],{"className":78227,"style":13996},[124],[15,78229,78231,78234],{"className":78230},[129],[15,78232,15756],{"className":78233,"style":887},[129,133],[15,78235,78237],{"className":78236},[137],[15,78238,78240,78260],{"className":78239},[141,142],[15,78241,78243,78257],{"className":78242},[146],[15,78244,78246],{"className":78245,"style":1981},[150],[15,78247,78248,78251],{"style":1100},[15,78249],{"className":78250,"style":159},[158],[15,78252,78254],{"className":78253},[163,164,165,166],[15,78255,13949],{"className":78256,"style":13964},[129,133,166],[15,78258,175],{"className":78259},[174],[15,78261,78263],{"className":78262},[146],[15,78264,78266],{"className":78265,"style":14035},[150],[15,78267],{}," 正是 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在均匀先验下的后验概率。对任意 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3.1 的检验形式。于是每个坐标贡献至少 ",[15,79248,79250,79286],{"className":79249,"translate":19},[23],[15,79251,79253],{"className":79252},[27],[29,79254,79255],{"xmlns":31},[34,79256,79257,79283],{},[37,79258,79259,79265,79267,79269,79271,79281],{},[1443,79260,79261,79263],{},[308,79262,318],{},[308,79264,14502],{},[50,79266,53],{"stretchy":52},[308,79268,318],{},[50,79270,315],{},[40,79272,79273,79279],{},[37,79274,79275,79277],{},[43,79276,53394],{"mathvariant":65},[43,79278,53397],{"mathvariant":65},[43,79280,13949],{},[50,79282,59],{"stretchy":52},[109,79284,79285],{"encoding":111},"\\frac14(1-\\mathrm{TV}_j)",[15,79287,79289,79378],{"className":79288,"ariaHidden":72},[116],[15,79290,79292,79295,79363,79366,79369,79372,79375],{"className":79291},[120],[15,79293],{"className":79294,"style":59676},[124],[15,79296,79298,79301,79360],{"className":79297},[129],[15,79299],{"className":79300},[188,1803],[15,79302,79304],{"className":79303},[1443],[15,79305,79307,79352],{"className":79306},[141,142],[15,79308,79310,79349],{"className":79309},[146],[15,79311,79313,79327,79335],{"className":79312,"style":59695},[150],[15,79314,79315,79318],{"style":59698},[15,79316],{"className":79317,"style":1823},[158],[15,79319,79321],{"className":79320},[163,164,165,166],[15,79322,79324],{"className":79323},[129,166],[15,79325,14502],{"className":79326},[129,166],[15,79328,79329,79332],{"style":1832},[15,79330],{"className":79331,"style":1823},[158],[15,79333],{"className":79334,"style":1840},[1839],[15,79336,79337,79340],{"style":59721},[15,79338],{"className":79339,"style":1823},[158],[15,79341,79343],{"className":79342},[163,164,165,166],[15,79344,79346],{"className":79345},[129,166],[15,79347,318],{"className":79348},[129,166],[15,79350,175],{"className":79351},[174],[15,79353,79355],{"className":79354},[146],[15,79356,79358],{"className":79357,"style":23479},[150],[15,79359],{},[15,79361],{"className":79362},[195,1803],[15,79364,53],{"className":79365},[188],[15,79367,318],{"className":79368},[129],[15,79370],{"className":79371,"style":3277},[93],[15,79373,315],{"className":79374},[2436],[15,79376],{"className":79377,"style":3277},[93],[15,79379,79381,79384,79428],{"className":79380},[120],[15,79382],{"className":79383,"style":67439},[124],[15,79385,79387,79393],{"className":79386},[129],[15,79388,79390],{"className":79389},[129],[15,79391,54025],{"className":79392,"style":2246},[129,221],[15,79394,79396],{"className":79395},[137],[15,79397,79399,79420],{"className":79398},[141,142],[15,79400,79402,79417],{"className":79401},[146],[15,79403,79405],{"className":79404,"style":1981},[150],[15,79406,79408,79411],{"style":79407},"top:-2.55em;margin-right:0.05em;",[15,79409],{"className":79410,"style":159},[158],[15,79412,79414],{"className":79413},[163,164,165,166],[15,79415,13949],{"className":79416,"style":13964},[129,133,166],[15,79418,175],{"className":79419},[174],[15,79421,79423],{"className":79422},[146],[15,79424,79426],{"className":79425,"style":14035},[150],[15,79427],{},[15,79429,59],{"className":79430},[195],"，求和得",[15,79433,79435],{"className":79434,"translate":19},[18],[15,79436,79438,79524],{"className":79437,"translate":19},[23],[15,79439,79441],{"className":79440},[27],[29,79442,79443],{"xmlns":31,"display":32},[34,79444,79445,79521],{},[37,79446,79447,79453,79455,79461,79519],{},[40,79448,79449,79451],{},[43,79450,33882],{"mathvariant":28899},[43,79452,39314],{},[50,79454,7469],{},[1443,79456,79457,79459],{},[43,79458,348],{},[308,79460,14502],{},[37,79462,79463,79465,79467,79469,79479,79481,79493,79495,79507,79517],{},[50,79464,53],{"fence":72},[308,79466,318],{},[50,79468,315],{},[2140,79470,79471,79477],{},[37,79472,79473,79475],{},[43,79474,19696],{},[50,79476,69],{},[43,79478,13949],{},[50,79480,33810],{"stretchy":52},[40,79482,79483,79485],{},[43,79484,34341],{},[37,79486,79487,79489,79491],{},[43,79488,13949],{},[50,79490,91],{"separator":72},[308,79492,541],{},[50,79494,315],{},[40,79496,79497,79499],{},[43,79498,34341],{},[37,79500,79501,79503,79505],{},[43,79502,13949],{},[50,79504,91],{"separator":72},[308,79506,318],{},[40,79508,79509,79511],{},[50,79510,33810],{"stretchy":52},[37,79512,79513,79515],{},[43,79514,53394],{"mathvariant":65},[43,79516,53397],{"mathvariant":65},[50,79518,59],{"fence":72},[43,79520,107],{"mathvariant":65},[109,79522,79523],{"encoding":111},"\\mathfrak B_n\\ge\\frac m4\\left(1-\\max_j\\lVert Q_{j,0}-Q_{j,1}\\rVert_{\\mathrm{TV}}\\right).",[15,79525,79527,79583],{"className":79526,"ariaHidden":72},[116],[15,79528,79530,79534,79574,79577,79580],{"className":79529},[120],[15,79531],{"className":79532,"style":79533},[124],"height:0.8414em;vertical-align:-0.15em;",[15,79535,79537,79540],{"className":79536},[129],[15,79538,33882],{"className":79539},[129,28926],[15,79541,79543],{"className":79542},[137],[15,79544,79546,79566],{"className":79545},[141,142],[15,79547,79549,79563],{"className":79548},[146],[15,79550,79552],{"className":79551,"style":1163},[150],[15,79553,79554,79557],{"style":154},[15,79555],{"className":79556,"style":159},[158],[15,79558,79560],{"className":79559},[163,164,165,166],[15,79561,39314],{"className":79562},[129,133,166],[15,79564,175],{"className":79565},[174],[15,79567,79569],{"className":79568},[146],[15,79570,79572],{"className":79571,"style":182},[150],[15,79573],{},[15,79575],{"className":79576,"style":199},[93],[15,79578,7469],{"className":79579},[203],[15,79581],{"className":79582,"style":199},[93],[15,79584,79586,79589,79651,79654,79881,79884],{"className":79585},[120],[15,79587],{"className":79588,"style":17007},[124],[15,79590,79592,79595,79648],{"className":79591},[129],[15,79593],{"className":79594},[188,1803],[15,79596,79598],{"className":79597},[1443],[15,79599,79601,79640],{"className":79600},[141,142],[15,79602,79604,79637],{"className":79603},[146],[15,79605,79607,79618,79626],{"className":79606,"style":21210},[150],[15,79608,79609,79612],{"style":1819},[15,79610],{"className":79611,"style":1823},[158],[15,79613,79615],{"className":79614},[129],[15,79616,14502],{"className":79617},[129],[15,79619,79620,79623],{"style":1832},[15,79621],{"className":79622,"style":1823},[158],[15,79624],{"className":79625,"style":1840},[1839],[15,79627,79628,79631],{"style":1843},[15,79629],{"className":79630,"style":1823},[158],[15,79632,79634],{"className":79633},[129],[15,79635,348],{"className":79636},[129,133],[15,79638,175],{"className":79639},[174],[15,79641,79643],{"className":79642},[146],[15,79644,79646],{"className":79645,"style":1862},[150],[15,79647],{},[15,79649],{"className":79650},[195,1803],[15,79652],{"className":79653,"style":262},[93],[15,79655,79657,79663,79666,79669,79672,79675,79719,79722,79771,79774,79777,79780,79829,79875],{"className":79656},[449],[15,79658,79660],{"className":79659,"style":5052},[188,5051],[15,79661,53],{"className":79662},[228,165],[15,79664,318],{"className":79665},[129],[15,79667],{"className":79668,"style":3277},[93],[15,79670,315],{"className":79671},[2436],[15,79673],{"className":79674,"style":3277},[93],[15,79676,79678],{"className":79677},[217,1874],[15,79679,79681,79711],{"className":79680},[141,142],[15,79682,79684,79708],{"className":79683},[146],[15,79685,79687,79698],{"className":79686,"style":4383},[150],[15,79688,79689,79692],{"style":53429},[15,79690],{"className":79691,"style":1823},[158],[15,79693,79695],{"className":79694},[163,164,165,166],[15,79696,13949],{"className":79697,"style":13964},[129,133,166],[15,79699,79700,79703],{"style":2287},[15,79701],{"className":79702,"style":1823},[158],[15,79704,79705],{},[15,79706,19696],{"className":79707},[217],[15,79709,175],{"className":79710},[174],[15,79712,79714],{"className":79713},[146],[15,79715,79717],{"className":79716,"style":53473},[150],[15,79718],{},[15,79720,33810],{"className":79721},[188],[15,79723,79725,79728],{"className":79724},[129],[15,79726,34341],{"className":79727},[129,133],[15,79729,79731],{"className":79730},[137],[15,79732,79734,79763],{"className":79733},[141,142],[15,79735,79737,79760],{"className":79736},[146],[15,79738,79740],{"className":79739,"style":1981},[150],[15,79741,79742,79745],{"style":154},[15,79743],{"className":79744,"style":159},[158],[15,79746,79748],{"className":79747},[163,164,165,166],[15,79749,79751,79754,79757],{"className":79750},[129,166],[15,79752,13949],{"className":79753,"style":13964},[129,133,166],[15,79755,91],{"className":79756},[254,166],[15,79758,541],{"className":79759},[129,166],[15,79761,175],{"className":79762},[174],[15,79764,79766],{"className":79765},[146],[15,79767,79769],{"className":79768,"style":14035},[150],[15,79770],{},[15,79772],{"className":79773,"style":3277},[93],[15,79775,315],{"className":79776},[2436],[15,79778],{"className":79779,"style":3277},[93],[15,79781,79783,79786],{"className":79782},[129],[15,79784,34341],{"className":79785},[129,133],[15,79787,79789],{"className":79788},[137],[15,79790,79792,79821],{"className":79791},[141,142],[15,79793,79795,79818],{"className":79794},[146],[15,79796,79798],{"className":79797,"style":1981},[150],[15,79799,79800,79803],{"style":154},[15,79801],{"className":79802,"style":159},[158],[15,79804,79806],{"className":79805},[163,164,165,166],[15,79807,79809,79812,79815],{"className":79808},[129,166],[15,79810,13949],{"className":79811,"style":13964},[129,133,166],[15,79813,91],{"className":79814},[254,166],[15,79816,318],{"className":79817},[129,166],[15,79819,175],{"className":79820},[174],[15,79822,79824],{"className":79823},[146],[15,79825,79827],{"className":79826,"style":14035},[150],[15,79828],{},[15,79830,79832,79835],{"className":79831},[195],[15,79833,33810],{"className":79834},[195],[15,79836,79838],{"className":79837},[137],[15,79839,79841,79867],{"className":79840},[141,142],[15,79842,79844,79864],{"className":79843},[146],[15,79845,79847],{"className":79846,"style":1525},[150],[15,79848,79849,79852],{"style":154},[15,79850],{"className":79851,"style":159},[158],[15,79853,79855],{"className":79854},[163,164,165,166],[15,79856,79858],{"className":79857},[129,166],[15,79859,79861],{"className":79860},[129,166],[15,79862,54025],{"className":79863,"style":2246},[129,221,166],[15,79865,175],{"className":79866},[174],[15,79868,79870],{"className":79869},[146],[15,79871,79873],{"className":79872,"style":182},[150],[15,79874],{},[15,79876,79878],{"className":79877,"style":5052},[195,5051],[15,79879,59],{"className":79880},[228,165],[15,79882],{"className":79883,"style":262},[93],[15,79885,107],{"className":79886},[129],[11,79888,79889],{},"极小极大风险不小于贝叶斯风险，定理得证。",[575,79891,79893],{"id":79892},"_52-逐坐标下界","5.2 逐坐标下界",[11,79895,79896,79897,79925,79926,79954,79955,79983,79984,80012,80013,80041],{},"Assouad 的威力在于它把 ",[15,79898,79900,79913],{"className":79899,"translate":19},[23],[15,79901,79903],{"className":79902},[27],[29,79904,79905],{"xmlns":31},[34,79906,79907,79911],{},[37,79908,79909],{},[43,79910,348],{},[109,79912,348],{"encoding":111},[15,79914,79916],{"className":79915,"ariaHidden":72},[116],[15,79917,79919,79922],{"className":79918},[120],[15,79920],{"className":79921,"style":4383},[124],[15,79923,348],{"className":79924},[129,133]," 维问题拆成了 ",[15,79927,79929,79942],{"className":79928,"translate":19},[23],[15,79930,79932],{"className":79931},[27],[29,79933,79934],{"xmlns":31},[34,79935,79936,79940],{},[37,79937,79938],{},[43,79939,348],{},[109,79941,348],{"encoding":111},[15,79943,79945],{"className":79944,"ariaHidden":72},[116],[15,79946,79948,79951],{"className":79947},[120],[15,79949],{"className":79950,"style":4383},[124],[15,79952,348],{"className":79953},[129,133]," 个一维问题。只要每个坐标上的两个混合分布足够接近，",[15,79956,79958,79971],{"className":79957,"translate":19},[23],[15,79959,79961],{"className":79960},[27],[29,79962,79963],{"xmlns":31},[34,79964,79965,79969],{},[37,79966,79967],{},[43,79968,348],{},[109,79970,348],{"encoding":111},[15,79972,79974],{"className":79973,"ariaHidden":72},[116],[15,79975,79977,79980],{"className":79976},[120],[15,79978],{"className":79979,"style":4383},[124],[15,79981,348],{"className":79982},[129,133]," 个坐标就会贡献 ",[15,79985,79987,80000],{"className":79986,"translate":19},[23],[15,79988,79990],{"className":79989},[27],[29,79991,79992],{"xmlns":31},[34,79993,79994,79998],{},[37,79995,79996],{},[43,79997,348],{},[109,79999,348],{"encoding":111},[15,80001,80003],{"className":80002,"ariaHidden":72},[116],[15,80004,80006,80009],{"className":80005},[120],[15,80007],{"className":80008,"style":4383},[124],[15,80010,348],{"className":80011},[129,133]," 倍的下界——这正是高维速率里 ",[15,80014,80016,80029],{"className":80015,"translate":19},[23],[15,80017,80019],{"className":80018},[27],[29,80020,80021],{"xmlns":31},[34,80022,80023,80027],{},[37,80024,80025],{},[43,80026,10372],{},[109,80028,10372],{"encoding":111},[15,80030,80032],{"className":80031,"ariaHidden":72},[116],[15,80033,80035,80038],{"className":80034},[120],[15,80036],{"className":80037,"style":522},[124],[15,80039,10372],{"className":80040},[129,133]," 的来源。",[11,80043,80044,80045,80149,80150,16656,80239,80377],{},"对 ",[15,80046,80048,80078],{"className":80047,"translate":19},[23],[15,80049,80051],{"className":80050},[27],[29,80052,80053],{"xmlns":31},[34,80054,80055,80075],{},[37,80056,80057,80059,80061,80063,80065,80067,80069],{},[43,80058,30862],{"mathvariant":65},[50,80060,62],{},[50,80062,636],{"stretchy":52},[308,80064,541],{},[50,80066,91],{"separator":72},[43,80068,4849],{},[305,80070,80071,80073],{},[50,80072,645],{"stretchy":52},[43,80074,348],{},[109,80076,80077],{"encoding":111},"\\Theta=\\{0,\\varepsilon\\}^m",[15,80079,80081,80099],{"className":80080,"ariaHidden":72},[116],[15,80082,80084,80087,80090,80093,80096],{"className":80083},[120],[15,80085],{"className":80086,"style":610},[124],[15,80088,30862],{"className":80089},[129],[15,80091],{"className":80092,"style":199},[93],[15,80094,62],{"className":80095},[203],[15,80097],{"className":80098,"style":199},[93],[15,80100,80102,80105,80108,80111,80114,80117,80120],{"className":80101},[120],[15,80103],{"className":80104,"style":125},[124],[15,80106,636],{"className":80107},[188],[15,80109,541],{"className":80110},[129],[15,80112,91],{"className":80113},[254],[15,80115],{"className":80116,"style":262},[93],[15,80118,4849],{"className":80119},[129,133],[15,80121,80123,80126],{"className":80122},[195],[15,80124,645],{"className":80125},[195],[15,80127,80129],{"className":80128},[137],[15,80130,80132],{"className":80131},[141],[15,80133,80135],{"className":80134},[146],[15,80136,80138],{"className":80137,"style":6377},[150],[15,80139,80140,80143],{"style":383},[15,80141],{"className":80142,"style":159},[158],[15,80144,80146],{"className":80145},[163,164,165,166],[15,80147,348],{"className":80148},[129,133,166],"，令 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_2^2\n=\\varepsilon^2\\bigl\\lVert\\hat\\theta\u002F\\varepsilon-\\theta'\\bigr\\rVert_2^2,",[15,80455,80457,80506,80575,80693],{"className":80456,"ariaHidden":72},[116],[15,80458,80460,80463,80466,80497,80500,80503],{"className":80459},[120],[15,80461],{"className":80462,"style":50020},[124],[15,80464,33810],{"className":80465},[188],[15,80467,80469],{"className":80468},[129,48488],[15,80470,80472],{"className":80471},[141],[15,80473,80475],{"className":80474},[146],[15,80476,80478,80486],{"className":80477,"style":48484},[150],[15,80479,80480,80483],{"style":2287},[15,80481],{"className":80482,"style":1823},[158],[15,80484,48],{"className":80485,"style":170},[129,133],[15,80487,80488,80491],{"style":48508},[15,80489],{"className":80490,"style":1823},[158],[15,80492,80494],{"className":80493,"style":48516},[48515],[15,80495,48471],{"className":80496},[129],[15,80498],{"className":80499,"style":3277},[93],[15,80501,315],{"className":80502},[2436],[15,80504],{"className":80505,"style":3277},[93],[15,80507,80509,80512,80515,80566,80569,80572],{"className":80508},[120],[15,80510],{"className":80511,"style":40908},[124],[15,80513,48],{"className":80514,"style":170},[129,133],[15,80516,80518,80521],{"className":80517},[195],[15,80519,33810],{"className":80520},[195],[15,80522,80524],{"className":80523},[137],[15,80525,80527,80558],{"className":80526},[141,142],[15,80528,80530,80555],{"className":80529},[146],[15,80531,80533,80544],{"className":80532,"style":40979},[150],[15,80534,80535,80538],{"style":58433},[15,80536],{"className":80537,"style":159},[158],[15,80539,80541],{"className":80540},[163,164,165,166],[15,80542,310],{"className":80543},[129,166],[15,80545,80546,80549],{"style":1272},[15,80547],{"className":80548,"style":159},[158],[15,80550,80552],{"className":80551},[163,164,165,166],[15,80553,310],{"className":80554},[129,166],[15,80556,175],{"className":80557},[174],[15,80559,80561],{"className":80560},[146],[15,80562,80564],{"className":80563,"style":28162},[150],[15,80565],{},[15,80567],{"className":80568,"style":199},[93],[15,80570,62],{"className":80571},[203],[15,80573],{"className":80574,"style":199},[93],[15,80576,80578,80581,80610,80647,80678,80681,80684,80687,80690],{"className":80577},[120],[15,80579],{"className":80580,"style":70920},[124],[15,80582,80584,80587],{"className":80583},[129],[15,80585,4849],{"className":80586},[129,133],[15,80588,80590],{"className":80589},[137],[15,80591,80593],{"className":80592},[141],[15,80594,80596],{"className":80595},[146],[15,80597,80599],{"className":80598,"style":40979},[150],[15,80600,80601,80604],{"style":1272},[15,80602],{"className":80603,"style":159},[158],[15,80605,80607],{"className":80606},[163,164,165,166],[15,80608,310],{"className":80609},[129,166],[15,80611,80613],{"className":80612},[188],[15,80614,80616],{"className":80615},[228,4433],[15,80617,80619,80639],{"className":80618},[141,142],[15,80620,80622,80636],{"className":80621},[146],[15,80623,80625],{"className":80624,"style":4443},[150],[15,80626,80627,80630],{"style":4446},[15,80628],{"className":80629,"style":4450},[158],[15,80631,80632],{"style":53804},[4455,80633,80634],{"xmlns":4457,"width":53807,"height":73,"viewBox":53808},[4461,80635],{"d":53811},[15,80637,175],{"className":80638},[174],[15,80640,80642],{"className":80641},[146],[15,80643,80645],{"className":80644,"style":4473},[150],[15,80646],{},[15,80648,80650],{"className":80649},[129,48488],[15,80651,80653],{"className":80652},[141],[15,80654,80656],{"className":80655},[146],[15,80657,80659,80667],{"className":80658,"style":48484},[150],[15,80660,80661,80664],{"style":2287},[15,80662],{"className":80663,"style":1823},[158],[15,80665,48],{"className":80666,"style":170},[129,133],[15,80668,80669,80672],{"style":48508},[15,80670],{"className":80671,"style":1823},[158],[15,80673,80675],{"className":80674,"style":48516},[48515],[15,80676,48471],{"className":80677},[129],[15,80679,544],{"className":80680},[129],[15,80682,4849],{"className":80683},[129,133],[15,80685],{"className":80686,"style":3277},[93],[15,80688,315],{"className":80689},[2436],[15,80691],{"className":80692,"style":3277},[93],[15,80694,80696,80700,80732,80819],{"className":80695},[120],[15,80697],{"className":80698,"style":80699},[124],"height:1.4537em;vertical-align:-0.3997em;",[15,80701,80703,80706],{"className":80702},[129],[15,80704,48],{"className":80705,"style":170},[129,133],[15,80707,80709],{"className":80708},[137],[15,80710,80712],{"className":80711},[141],[15,80713,80715],{"className":80714},[146],[15,80716,80718],{"className":80717,"style":28126},[150],[15,80719,80720,80723],{"style":1272},[15,80721],{"className":80722,"style":159},[158],[15,80724,80726],{"className":80725},[163,164,165,166],[15,80727,80729],{"className":80728},[129,166],[15,80730,21499],{"className":80731},[129,166],[15,80733,80735,80772],{"className":80734},[195],[15,80736,80738],{"className":80737},[195],[15,80739,80741],{"className":80740},[228,4433],[15,80742,80744,80764],{"c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5.1 直接给出",[15,80826,80828],{"className":80827,"translate":19},[18],[15,80829,80831,80983],{"className":80830,"translate":19},[23],[15,80832,80834],{"className":80833},[27],[29,80835,80836],{"xmlns":31,"display":32},[34,80837,80838,80980],{},[37,80839,80840,80854,80882,80884,80886,80892,80894,80896,80904,80906,80920,80978],{},[2140,80841,80842,80848],{},[37,80843,80844,80846],{},[43,80845,48696],{},[50,80847,69],{},[46835,80849,80850,80852],{"accent":72},[43,80851,48],{},[50,80853,48471],{},[2140,80855,80856,80862],{},[37,80857,80858,80860],{},[43,80859,19696],{},[50,80861,69],{},[37,80863,80864,80866,80868,80870,80872,80874,80876],{},[43,80865,48],{},[50,80867,100],{},[50,80869,636],{"stretchy":52},[308,80871,541],{},[50,80873,91],{"separator":72},[43,80875,4849],{},[305,80877,80878,80880],{},[50,80879,645],{"stretchy":52},[43,80881,348],{},[43,80883,26563],{"mathvariant":103},[50,80885,33810],{"stretchy":52},[46835,80887,80888,80890],{"accent":72},[43,80889,48],{},[50,80891,48471],{},[50,80893,315],{},[43,80895,48],{},[23149,80897,80898,80900,80902],{},[50,80899,33810],{"stretchy":52},[308,80901,310],{},[308,80903,310],{},[50,80905,7469],{},[1443,80907,80908,80918],{},[37,80909,80910,80916],{},[305,80911,80912,80914],{},[43,80913,4849],{},[308,80915,310],{},[43,80917,348],{},[308,80919,14502],{},[37,80921,80922,80924,80926,80928,80938,80940,80952,80954,80966,80976],{},[50,80923,53],{"fence":72},[308,80925,318],{},[50,80927,315],{},[2140,80929,80930,80936],{},[37,80931,80932,80934],{},[43,80933,19696],{},[50,80935,69],{},[43,80937,13949],{},[50,80939,33810],{"stretchy":52},[40,80941,80942,80944],{},[43,80943,34341],{},[37,80945,80946,80948,80950],{},[43,80947,13949],{},[50,80949,91],{"separator":72},[308,80951,541],{},[50,80953,315],{},[40,80955,80956,80958],{},[43,80957,34341],{},[37,80959,80960,80962,80964],{},[43,80961,13949],{},[50,80963,91],{"separator":72},[308,80965,318],{},[40,80967,80968,80970],{},[50,80969,33810],{"stretchy":52},[37,80971,80972,80974],{},[43,80973,53394],{"mathvariant":65},[43,80975,53397],{"mathvariant":65},[50,80977,59],{"fence":72},[43,80979,107],{"mathvariant":65},[109,80981,80982],{"encoding":111},"\\inf_{\\hat\\theta}\\max_{\\theta\\in\\{0,\\varepsilon\\}^m}\\mathbb 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Hamming 距离，同样的逐坐标论证给出常数 ",[15,81620,81622,81639],{"className":81621,"translate":19},[23],[15,81623,81625],{"className":81624},[27],[29,81626,81627],{"xmlns":31},[34,81628,81629,81637],{},[37,81630,81631,81633,81635],{},[308,81632,318],{},[43,81634,544],{"mathvariant":65},[308,81636,310],{},[109,81638,56720],{"encoding":111},[15,81640,81642],{"className":81641,"ariaHidden":72},[116],[15,81643,81645,81648],{"className":81644},[120],[15,81646],{"className":81647,"style":125},[124],[15,81649,56720],{"className":81650},[129],"：把后验均值换成阈值检验即可。",[570,81653,81655],{"id":81654},"六两个完整的例子","六、两个完整的例子",[575,81657,81659,81660],{"id":81658},"_61-高斯均值σ2dnsigma2dnσ2dn","6.1 高斯均值：",[15,81661,81663,81686],{"className":81662,"translate":19},[23],[15,81664,81666],{"className":81665},[27],[29,81667,81668],{"xmlns":31},[34,81669,81670,81684],{},[37,81671,81672,81678,81680,81682],{},[305,81673,81674,81676],{},[43,81675,27283],{},[308,81677,310],{},[43,81679,10372],{},[43,81681,544],{"mathvariant":65},[43,81683,39314],{},[109,81685,64904],{"encoding":111},[15,81687,81689],{"className":81688,"ariaHidden":72},[116],[15,81690,81692,81695,81724,81727,81730],{"className":81691},[120],[15,81693],{"className":81694,"style":33996},[124],[15,81696,81698,81701],{"className":81697},[129],[15,81699,27283],{"className":81700,"style":887},[129,133],[15,81702,81704],{"className":81703},[137],[15,81705,81707],{"className":81706},[141],[15,81708,81710],{"className":81709},[146],[15,81711,81713],{"className":81712,"style":380},[150],[15,81714,81715,81718],{"style":383},[15,81716],{"className":81717,"style":159},[158],[15,81719,81721],{"className":81720},[163,164,165,166],[15,81722,310],{"className":81723},[129,166],[15,81725,10372],{"className":81726},[129,133],[15,81728,544],{"className":81729},[129],[15,81731,39314],{"className":81732},[129,133],[11,81734,81735],{},"模型：",[15,81737,81739],{"className":81738,"translate":19},[18],[15,81740,81742,81812],{"className":81741,"translate":19},[23],[15,81743,81745],{"className":81744},[27],[29,81746,81747],{"xmlns":31,"display":32},[34,81748,81749,81809],{},[37,81750,81751,81757,81759,81761,81763,81769,81771,81773,81775,81777,81779,81785,81791,81793,81795,81797,81799,81801,81807],{},[40,81752,81753,81755],{},[43,81754,597],{},[308,81756,318],{},[50,81758,91],{"separator":72},[50,81760,335],{},[50,81762,91],{"separator":72},[40,81764,81765,81767],{},[43,81766,597],{},[43,81768,39314],{},[50,81770,991],{},[43,81772,12555],{},[50,81774,53],{"stretchy":52},[43,81776,48],{},[50,81778,91],{"separator":72},[305,81780,81781,81783],{},[43,81782,27283],{},[308,81784,310],{},[40,81786,81787,81789],{},[43,81788,64021],{},[43,81790,10372],{},[50,81792,59],{"stretchy":52},[50,81794,91],{"separator":72},[93,81796],{"width":95},[43,81798,48],{},[50,81800,100],{},[305,81802,81803,81805],{},[43,81804,104],{"mathvariant":103},[43,81806,10372],{},[50,81808,91],{"separator":72},[109,81810,81811],{"encoding":111},"X_1,\\dots,X_n\\sim N(\\theta,\\sigma^2 I_d),\\qquad \\theta\\in\\mathbb R^d,",[15,81813,81815,81928,82042],{"className":81814,"ariaHidden":72},[116],[15,81816,81818,81821,81861,81864,81867,81870,81873,81876,81879,81919,81922,81925],{"className":81817},[120],[15,81819],{"className":81820,"style":34353},[124],[15,81822,81824,81827],{"className":81823},[129],[15,81825,597],{"className":81826,"style":47958},[129,133],[15,81828,81830],{"className":81829},[137],[15,81831,81833,81853],{"className":81832},[141,142],[15,81834,81836,81850],{"className":81835},[146],[15,81837,81839],{"className":81838,"style":1051},[150],[15,81840,81841,81844],{"style":47973},[15,81842],{"className":81843,"style":159},[158],[15,81845,81847],{"className":81846},[163,164,165,166],[15,81848,318],{"className":81849},[129,166],[15,81851,175],{"className":81852},[174],[15,81854,81856],{"className":81855},[146],[15,81857,81859],{"className":81858,"style":182},[150],[15,81860],{},[15,81862,91],{"className":81863},[254],[15,81865],{"className":81866,"style":262},[93],[15,81868,335],{"className":81869},[449],[15,81871],{"className":81872,"style":262},[93],[15,81874,91],{"className":81875},[254],[15,81877],{"className":81878,"style":262},[93],[15,81880,81882,81885],{"className":81881},[129],[15,81883,597],{"className":81884,"style":47958},[129,133],[15,81886,81888],{"className":81887},[137],[15,81889,81891,81911],{"className":81890},[141,142],[15,81892,81894,81908],{"className":81893},[146],[15,81895,81897],{"className":81896,"style":1163},[150],[15,81898,81899,81902],{"style":47973},[15,81900],{"className":81901,"style":159},[158],[15,81903,81905],{"className":81904},[163,164,165,166],[15,81906,39314],{"className":81907},[129,133,166],[15,81909,175],{"className":81910},[174],[15,81912,81914],{"className":81913},[146],[15,81915,81917],{"className":81916,"style":182},[150],[15,81918],{},[15,81920],{"className":81921,"style":199},[93],[15,81923,991],{"className":81924},[203],[15,81926],{"className":81927,"style":199},[93],[15,81929,81931,81934,81937,81940,81943,81946,81949,81978,82018,82021,82024,82027,82030,82033,82036,82039],{"className":81930},[120],[15,81932],{"className":81933,"style":40908},[124],[15,81935,12555],{"className":81936,"style":12591},[129,133],[15,81938,53],{"className":81939},[188],[15,81941,48],{"className":81942,"style":170},[129,133],[15,81944,91],{"className":81945},[254],[15,81947],{"className":81948,"style":262},[93],[15,81950,81952,81955],{"className":81951},[129],[15,81953,27283],{"className":81954,"style":887},[129,133],[15,81956,81958],{"className":81957},[137],[15,81959,81961],{"className":81960},[141],[15,81962,81964],{"className":81963},[146],[15,81965,81967],{"className":81966,"style":40979},[150],[15,81968,81969,81972],{"style":1272},[15,81970],{"className":81971,"style":159},[158],[15,81973,81975],{"className":81974},[163,164,165,166],[15,81976,310],{"className":81977},[129,166],[15,81979,81981,81984],{"className":81980},[129],[15,81982,64021],{"className":81983,"style":47958},[129,133],[15,81985,81987],{"className":81986},[137],[15,81988,81990,82010],{"className":81989},[141,142],[15,81991,81993,82007],{"className":81992},[146],[15,81994,81996],{"className":81995,"style":151},[150],[15,81997,81998,82001],{"style":47973},[15,81999],{"className":82000,"style":159},[158],[15,82002,82004],{"className":82003},[163,164,165,166],[15,82005,10372],{"className":82006},[129,133,166],[15,82008,175],{"className":82009},[174],[15,82011,82013],{"className":82012},[146],[15,82014,82016],{"className":82015,"style":182},[150],[15,82017],{},[15,82019,59],{"className":82020},[195],[15,82022,91],{"className":82023},[254],[15,82025],{"className":82026,"style":258},[93],[15,82028],{"className":82029,"style":262},[93],[15,82031,48],{"className":82032,"style":170},[129,133],[15,82034],{"className":82035,"style":199},[93],[15,82037,100],{"className":82038},[203],[15,82040],{"className":82041,"style":199},[93],[15,82043,82045,82049,82078],{"className":82044},[120],[15,82046],{"className":82047,"style":82048},[124],"height:1.0935em;vertical-align:-0.1944em;",[15,82050,82052,82055],{"className":82051},[129],[15,82053,104],{"className":82054},[129,285],[15,82056,82058],{"className":82057},[137],[15,82059,82061],{"className":82060},[141],[15,82062,82064],{"className":82063},[146],[15,82065,82067],{"className":82066,"style":64281},[150],[15,82068,82069,82072],{"style":1272},[15,82070],{"className":82071,"style":159},[158],[15,82073,82075],{"className":82074},[163,164,165,166],[15,82076,10372],{"className":82077},[129,133,166],[15,82079,91],{"className":82080},[254],[11,82082,82083,82084,1313],{},"损失为 ",[15,82085,82087,82119],{"className":82086,"translate":19},[23],[15,82088,82090],{"className":82089},[27],[29,82091,82092],{"xmlns":31},[34,82093,82094,82116],{},[37,82095,82096,82098,82104,82106,82108],{},[50,82097,33810],{"stretchy":52},[46835,82099,82100,82102],{"accent":72},[43,82101,48],{},[50,82103,48471],{},[50,82105,315],{},[43,82107,48],{},[23149,82109,82110,82112,82114],{},[50,82111,33810],{"stretchy":52},[308,82113,310],{},[308,82115,310],{},[109,82117,82118],{"encoding":111},"\\lVert\\hat\\theta-\\theta\\rVert_2^2",[15,82120,82122,82171],{"className":82121,"ariaHidden":72},[116],[15,82123,82125,82128,82131,82162,82165,82168],{"className":82124},[120],[15,82126],{"className":82127,"style":50020},[124],[15,82129,33810],{"className":82130},[188],[15,82132,82134],{"className":82133},[129,48488],[15,82135,82137],{"className":82136},[141],[15,82138,82140],{"className":82139},[146],[15,82141,82143,82151],{"className":82142,"style":48484},[150],[15,82144,82145,82148],{"style":2287},[15,82146],{"className":82147,"style":1823},[158],[15,82149,48],{"className":82150,"style":170},[129,133],[15,82152,82153,82156],{"style":48508},[15,82154],{"className":82155,"style":1823},[158],[15,82157,82159],{"className":82158,"style":48516},[48515],[15,82160,48471],{"className":82161},[129],[15,82163],{"className":82164,"style":3277},[93],[15,82166,315],{"className":82167},[2436],[15,82169],{"className":82170,"style":3277},[93],[15,82172,82174,82177,82180],{"className":82173},[120],[15,82175],{"className":82176,"style":33996},[124],[15,82178,48],{"className":82179,"style":170},[129,133],[15,82181,82183,82186],{"className":82182},[195],[15,82184,33810],{"className":82185},[195],[15,82187,82189],{"className":82188},[137],[15,82190,82192,82223],{"className":82191},[141,142],[15,82193,82195,82220],{"className":82194},[146],[15,82196,82198,82209],{"className":82197,"style":380},[150],[15,82199,82200,82203],{"style":50157},[15,82201],{"className":82202,"style":159},[158],[15,82204,82206],{"className":82205},[163,164,165,166],[15,82207,310],{"className":82208},[129,166],[15,82210,82211,82214],{"style":383},[15,82212],{"className":82213,"style":159},[158],[15,82215,82217],{"className":82216},[163,164,165,166],[15,82218,310],{"className":82219},[129,166],[15,82221,175],{"className":82222},[174],[15,82224,82226],{"className":82225},[146],[15,82227,82229],{"className":82228,"style":38936},[150],[15,82230],{},[11,82232,82233,82236,82237,82298],{},[698,82234,82235],{},"上界。"," 样本均值 ",[15,82238,82240,82258],{"className":82239,"translate":19},[23],[15,82241,82243],{"className":82242},[27],[29,82244,82245],{"xmlns":31},[34,82246,82247,82255],{},[37,82248,82249],{},[46835,82250,82251,82253],{"accent":72},[43,82252,597],{},[50,82254,72865],{},[109,82256,82257],{"encoding":111},"\\bar X",[15,82259,82261],{"className":82260,"ariaHidden":72},[116],[15,82262,82264,82267],{"className":82263},[120],[15,82265],{"className":82266,"style":73195},[124],[15,82268,82270],{"className":82269},[129,48488],[15,82271,82273],{"className":82272},[141],[15,82274,82276],{"className":82275},[146],[15,82277,82279,82287],{"className":82278,"style":73195},[150],[15,82280,82281,82284],{"style":2287},[15,82282],{"className":82283,"style":1823},[158],[15,82285,597],{"className":82286,"style":47958},[129,133],[15,82288,82289,82292],{"style":65305},[15,82290],{"className":82291,"style":1823},[158],[15,82293,82295],{"className":82294,"style":48516},[48515],[15,82296,72865],{"className":82297},[129]," 的风险是",[15,82300,82302],{"className":82301,"translate":19},[18],[15,82303,82305,82357],{"className":82304,"translate":19},[23],[15,82306,82308],{"className":82307},[27],[29,82309,82310],{"xmlns":31,"display":32},[34,82311,82312,82354],{},[37,82313,82314,82316,82318,82324,82326,82328,82336,82338,82352],{},[43,82315,26563],{"mathvariant":103},[50,82317,33810],{"stretchy":52},[46835,82319,82320,82322],{"accent":72},[43,82321,597],{},[50,82323,72865],{},[50,82325,315],{},[43,82327,48],{},[23149,82329,82330,82332,82334],{},[50,82331,33810],{"stretchy":52},[308,82333,310],{},[308,82335,310],{},[50,82337,62],{},[1443,82339,82340,82350],{},[37,82341,82342,82348],{},[305,82343,82344,82346],{},[43,82345,27283],{},[308,82347,310],{},[43,82349,10372],{},[43,82351,39314],{},[43,82353,107],{"mathvariant":65},[109,82355,82356],{"encoding":111},"\\mathbb E\\lVert\\bar X-\\theta\\rVert_2^2=\\frac{\\sigma^2d}{n}.",[15,82358,82360,82413,82482],{"className":82359,"ariaHidden":72},[116],[15,82361,82363,82367,82370,82373,82404,82407,82410],{"className":82362},[120],[15,82364],{"className":82365,"style":82366},[124],"height:1.0701em;vertical-align:-0.25em;",[15,82368,26563],{"className":82369},[129,285],[15,82371,33810],{"className":82372},[188],[15,82374,82376],{"className":82375},[129,48488],[15,82377,82379],{"className":82378},[141],[15,82380,82382],{"className":82381},[146],[15,82383,82385,82393],{"className":82384,"style":73195},[150],[15,82386,82387,82390],{"style":2287},[15,82388],{"className":82389,"style":1823},[158],[15,82391,597],{"className":82392,"style":47958},[129,133],[15,82394,82395,82398],{"style":65305},[15,82396],{"className":82397,"style":1823},[158],[15,82399,82401],{"className":82400,"style":48516},[48515],[15,82402,72865],{"className":82403},[129],[15,82405],{"className":82406,"style":3277},[93],[15,82408,315],{"className":82409},[2436],[15,82411],{"className":82412,"style":3277},[93],[15,82414,82416,82419,82422,82473,82476,82479],{"className":82415},[120],[15,82417],{"className":82418,"style":40908},[124],[15,82420,48],{"className":82421,"style":170},[129,133],[15,82423,82425,82428],{"className":82424},[195],[15,82426,33810],{"className":82427},[195],[15,82429,82431],{"className":82430},[137],[15,82432,82434,82465],{"className":82433},[141,142],[15,82435,82437,82462],{"className":82436},[146],[15,82438,82440,82451],{"className":82439,"style":40979},[150],[15,82441,82442,82445],{"style":58433},[15,82443],{"className":82444,"style":159},[158],[15,82446,82448],{"className":82447},[163,164,165,166],[15,82449,310],{"className":82450},[129,166],[15,82452,82453,82456],{"style":1272},[15,82454],{"className":82455,"style":159},[158],[15,82457,82459],{"className":82458},[163,164,165,166],[15,82460,310],{"className":82461},[129,166],[15,82463,175],{"className":82464},[174],[15,82466,82468],{"className":82467},[146],[15,82469,82471],{"className":82470,"style":28162},[150],[15,82472],{},[15,82474],{"className":82475,"style":199},[93],[15,82477,62],{"className":82478},[203],[15,82480],{"className":82481,"style":199},[93],[15,82483,82485,82489,82580],{"className":82484},[120],[15,82486],{"className":82487,"style":82488},[124],"height:2.1771em;vertical-align:-0.686em;",[15,82490,82492,82495,82577],{"className":82491},[129],[15,82493],{"className":82494},[188,1803],[15,82496,82498],{"className":82497},[1443],[15,82499,82501,82569],{"className":82500},[141,142],[15,82502,82504,82566],{"className":82503},[146],[15,82505,82507,82518,82526],{"className":82506,"style":63608},[150],[15,82508,82509,82512],{"style":1819},[15,82510],{"className":82511,"style":1823},[158],[15,82513,82515],{"className":82514},[129],[15,82516,39314],{"className":82517},[129,133],[15,82519,82520,82523],{"style":1832},[15,82521],{"className":82522,"style":1823},[158],[15,82524],{"className":82525,"style":1840},[1839],[15,82527,82528,82531],{"style":1843},[15,82529],{"className":82530,"style":1823},[158],[15,82532,82534,82563],{"className":82533},[129],[15,82535,82537,82540],{"className":82536},[129],[15,82538,27283],{"className":82539,"style":887},[129,133],[15,82541,82543],{"className":82542},[137],[15,82544,82546],{"className":82545},[141],[15,82547,82549],{"className":82548},[146],[15,82550,82552],{"className":82551,"style":380},[150],[15,82553,82554,82557],{"style":383},[15,82555],{"className":82556,"style":159},[158],[15,82558,82560],{"className":82559},[163,164,165,166],[15,82561,310],{"className":82562},[129,166],[15,82564,10372],{"className":82565},[129,133],[15,82567,175],{"className":82568},[174],[15,82570,82572],{"className":82571},[146],[15,82573,82575],{"className":82574,"style":1862},[150],[15,82576],{},[15,82578],{"className":82579},[195,1803],[15,82581,107],{"className":82582},[129],[11,82584,82585,82588,82589,82734,82735,82763,82764,10175,82848,82932,82933,82961],{},[698,82586,82587],{},"下界。"," 把参数限制到超立方体 ",[15,82590,82592,82626],{"className":82591,"translate":19},[23],[15,82593,82595],{"className":82594},[27],[29,82596,82597],{"xmlns":31},[34,82598,82599,82623],{},[37,82600,82601,82607,82609,82611,82613,82615,82617],{},[40,82602,82603,82605],{},[43,82604,30862],{"mathvariant":65},[308,82606,541],{},[50,82608,62],{},[50,82610,636],{"stretchy":52},[308,82612,541],{},[50,82614,91],{"separator":72},[43,82616,4849],{},[305,82618,82619,82621],{},[50,82620,645],{"stretchy":52},[43,82622,10372],{},[109,82624,82625],{"encoding":111},"\\Theta_0=\\{0,\\varepsilon\\}^d",[15,82627,82629,82684],{"className":82628,"ariaHidden":72},[116],[15,82630,82632,82635,82675,82678,82681],{"className":82631},[120],[15,82633],{"className":82634,"style":20220},[124],[15,82636,82638,82641],{"className":82637},[129],[15,82639,30862],{"className":82640},[129],[15,82642,82644],{"className":82643},[137],[15,82645,82647,82667],{"className":82646},[141,142],[15,82648,82650,82664],{"className":82649},[146],[15,82651,82653],{"className":82652,"style":1051},[150],[15,82654,82655,82658],{"style":154},[15,82656],{"className":82657,"style":159},[158],[15,82659,82661],{"className":82660},[163,164,165,166],[15,82662,541],{"className":82663},[129,166],[15,82665,175],{"className":82666},[174],[15,82668,82670],{"className":82669},[146],[15,82671,82673],{"className":82672,"style":182},[150],[15,82674],{},[15,82676],{"className":82677,"style":199},[93],[15,82679,62],{"className":82680},[203],[15,82682],{"className":82683,"style":199},[93],[15,82685,82687,82690,82693,82696,82699,82702,82705],{"className":82686},[120],[15,82688],{"className":82689,"style":10974},[124],[15,82691,636],{"className":82692},[188],[15,82694,541],{"className":82695},[129],[15,82697,91],{"className":82698},[254],[15,82700],{"className":82701,"style":262},[93],[15,82703,4849],{"className":82704},[129,133],[15,82706,82708,82711],{"className":82707},[195],[15,82709,645],{"className":82710},[195],[15,82712,82714],{"className":82713},[137],[15,82715,82717],{"className":82716},[141],[15,82718,82720],{"className":82719},[146],[15,82721,82723],{"className":82722,"style":10486},[150],[15,82724,82725,82728],{"style":383},[15,82726],{"className":82727,"style":159},[158],[15,82729,82731],{"className":82730},[163,164,165,166],[15,82732,10372],{"className":82733},[129,133,166],"。对坐标 ",[15,82736,82738,82751],{"className":82737,"translate":19},[23],[15,82739,82741],{"className":82740},[27],[29,82742,82743],{"xmlns":31},[34,82744,82745,82749],{},[37,82746,82747],{},[43,82748,13949],{},[109,82750,13949],{"encoding":111},[15,82752,82754],{"className":82753,"ariaHidden":72},[116],[15,82755,82757,82760],{"className":82756},[120],[15,82758],{"className":82759,"style":11187},[124],[15,82761,13949],{"className":82762,"style":13964},[129,133],"，混合分布 ",[15,82765,82767,82790],{"className":82766,"translate":19},[23],[15,82768,82770],{"className":82769},[27],[29,82771,82772],{"xmlns":31},[34,82773,82774,82788],{},[37,82775,82776],{},[40,82777,82778,82780],{},[43,82779,34341],{},[37,82781,82782,82784,82786],{},[43,82783,13949],{},[50,82785,91],{"separator":72},[308,82787,541],{},[109,82789,78295],{"encoding":111},[15,82791,82793],{"className":82792,"ariaHidden":72},[116],[15,82794,82796,82799],{"className":82795},[120],[15,82797],{"className":82798,"style":76043},[124],[15,82800,82802,82805],{"className":82801},[129],[15,82803,34341],{"className":82804},[129,133],[15,82806,82808],{"className":82807},[137],[15,82809,82811,82840],{"className":82810},[141,142],[15,82812,82814,82837],{"className":82813},[146],[15,82815,82817],{"className":82816,"style":1981},[150],[15,82818,82819,82822],{"style":154},[15,82820],{"className":82821,"style":159},[158],[15,82823,82825],{"className":82824},[163,164,165,166],[15,82826,82828,82831,82834],{"className":82827},[129,166],[15,82829,13949],{"className":82830,"style":13964},[129,133,166],[15,82832,91],{"className":82833},[254,166],[15,82835,541],{"className":82836},[129,166],[15,82838,175],{"className":82839},[174],[15,82841,82843],{"className":82842},[146],[15,82844,82846],{"className":82845,"style":14035},[150],[15,82847],{},[15,82849,82851,82874],{"className":82850,"translate":19},[23],[15,82852,82854],{"className":82853},[27],[29,82855,82856],{"xmlns":31},[34,82857,82858,82872],{},[37,82859,82860],{},[40,82861,82862,82864],{},[43,82863,34341],{},[37,82865,82866,82868,82870],{},[43,82867,13949],{},[50,82869,91],{"separator":72},[308,82871,318],{},[109,82873,78380],{"encoding":111},[15,82875,82877],{"className":82876,"ariaHidden":72},[116],[15,82878,82880,82883],{"className":82879},[120],[15,82881],{"className":82882,"style":76043},[124],[15,82884,82886,82889],{"className":82885},[129],[15,82887,34341],{"className":82888},[129,133],[15,82890,82892],{"className":82891},[137],[15,82893,82895,82924],{"className":82894},[141,142],[15,82896,82898,82921],{"className":82897},[146],[15,82899,82901],{"className":82900,"style":1981},[150],[15,82902,82903,82906],{"style":154},[15,82904],{"className":82905,"style":159},[158],[15,82907,82909],{"className":82908},[163,164,165,166],[15,82910,82912,82915,82918],{"className":82911},[129,166],[15,82913,13949],{"className":82914,"style":13964},[129,133,166],[15,82916,91],{"className":82917},[254,166],[15,82919,318],{"className":82920},[129,166],[15,82922,175],{"className":82923},[174],[15,82925,82927],{"className":82926},[146],[15,82928,82930],{"className":82929,"style":14035},[150],[15,82931],{}," 只在这一维的均值上相差 ",[15,82934,82936,82949],{"className":82935,"translate":19},[23],[15,82937,82939],{"className":82938},[27],[29,82940,82941],{"xmlns":31},[34,82942,82943,82947],{},[37,82944,82945],{},[43,82946,4849],{},[109,82948,34307],{"encoding":111},[15,82950,82952],{"className":82951,"ariaHidden":72},[116],[15,82953,82955,82958],{"className":82954},[120],[15,82956],{"className":82957,"style":4383},[124],[15,82959,4849],{"className":82960},[129,133],"。由 KL 的联合凸性，",[15,82963,82965],{"className":82964,"translate":19},[18],[15,82966,82968,83046],{"className":82967,"translate":19},[23],[15,82969,82971],{"className":82970},[27],[29,82972,82973],{"xmlns":31,"display":32},[34,82974,82975,83043],{},[37,82976,82977,82987,82989,83001,83003,83015,83017,83019,83041],{},[40,82978,82979,82981],{},[43,82980,906],{},[37,82982,82983,82985],{},[43,82984,34511],{"mathvariant":65},[43,82986,1373],{"mathvariant":65},[50,82988,53],{"stretchy":52},[40,82990,82991,82993],{},[43,82992,34341],{},[37,82994,82995,82997,82999],{},[43,82996,13949],{},[50,82998,91],{"separator":72},[308,83000,541],{},[43,83002,33810],{"mathvariant":65},[40,83004,83005,83007],{},[43,83006,34341],{},[37,83008,83009,83011,83013],{},[43,83010,13949],{},[50,83012,91],{"separator":72},[308,83014,318],{},[50,83016,59],{"stretchy":52},[50,83018,2717],{},[1443,83020,83021,83031],{},[37,83022,83023,83025],{},[43,83024,39314],{},[305,83026,83027,83029],{},[43,83028,4849],{},[308,83030,310],{},[37,83032,83033,83035],{},[308,83034,310],{},[305,83036,83037,83039],{},[43,83038,27283],{},[308,83040,310],{},[43,83042,107],{"mathvariant":65},[109,83044,83045],{"encoding":111},"D_{\\mathrm{KL}}(Q_{j,0}\\Vert Q_{j,1})\n\\le \\frac{n\\varepsilon^2}{2\\sigma^2}.",[15,83047,83049,83217],{"className":83048,"ariaHidden":72},[116],[15,83050,83052,83055,83101,83104,83153,83156,83205,83208,83211,83214],{"className":83051},[120],[15,83053],{"className":83054,"style":67439},[124],[15,83056,83058,83061],{"className":83057},[129],[15,83059,906],{"className":83060,"style":170},[129,133],[15,83062,83064],{"className":83063},[137],[15,83065,83067,83093],{"className":83066},[141,142],[15,83068,83070,83090],{"className":83069},[146],[15,83071,83073],{"className":83072,"style":1525},[150],[15,83074,83075,83078],{"style":48281},[15,83076],{"className":83077,"style":159},[158],[15,83079,83081],{"className":83080},[163,164,165,166],[15,83082,83084],{"className":83083},[129,166],[15,83085,83087],{"className":83086},[129,166],[15,83088,34915],{"className":83089},[129,221,166],[15,83091,175],{"className":83092},[174],[15,83094,83096],{"className":83095},[146],[15,83097,83099],{"className":83098,"style":182},[150],[15,83100],{},[15,83102,53],{"className":83103},[188],[15,83105,83107,83110],{"className":83106},[129],[15,83108,34341],{"className":83109},[129,133],[15,83111,83113],{"className":83112},[137],[15,83114,83116,83145],{"className":83115},[141,142],[15,83117,83119,83142],{"className":83118},[146],[15,83120,83122],{"className":83121,"style":1981},[150],[15,83123,83124,83127],{"style":154},[15,83125],{"className":83126,"style":159},[158],[15,83128,83130],{"className":83129},[163,164,165,166],[15,83131,83133,83136,83139],{"className":83132},[129,166],[15,83134,13949],{"className":83135,"style":13964},[129,133,166],[15,83137,91],{"className":83138},[254,166],[15,83140,541],{"className":83141},[129,166],[15,83143,175],{"className":83144},[174],[15,83146,83148],{"className":83147},[146],[15,83149,83151],{"className":83150,"style":14035},[150],[15,83152],{},[15,83154,33810],{"className":83155},[129],[15,83157,83159,83162],{"className":83158},[129],[15,83160,34341],{"className":83161},[129,133],[15,83163,83165],{"className":83164},[137],[15,83166,83168,83197],{"className":83167},[141,142],[15,83169,83171,83194],{"className":83170},[146],[15,83172,83174],{"className":83173,"style":1981},[150],[15,83175,83176,83179],{"style":154},[15,83177],{"className":83178,"style":159},[158],[15,83180,83182],{"className":83181},[163,164,165,166],[15,83183,83185,83188,83191],{"className":83184},[129,166],[15,83186,13949],{"className":83187,"style":13964},[129,133,166],[15,83189,91],{"className":83190},[254,166],[15,83192,318],{"className":83193},[129,166],[15,83195,175],{"className":83196},[174],[15,83198,83200],{"className":83199},[146],[15,83201,83203],{"className":83202,"style":14035},[150],[15,83204],{},[15,83206,59],{"className":83207},[195],[15,83209],{"className":83210,"style":199},[93],[15,83212,2717],{"className":83213},[203],[15,83215],{"className":83216,"style":199},[93],[15,83218,83220,83223,83343],{"className":83219},[120],[15,83221],{"className":83222,"style":82488},[124],[15,83224,83226,83229,83340],{"className":83225},[129],[15,83227],{"className":83228},[188,1803],[15,83230,83232],{"className":83231},[1443],[15,83233,83235,83332],{"className":83234},[141,142],[15,83236,83238,83329],{"className":83237},[146],[15,83239,83241,83281,83289],{"className":83240,"style":63608},[150],[15,83242,83243,83246],{"style":1819},[15,83244],{"className":83245,"style":1823},[158],[15,83247,83249,83252],{"className":83248},[129],[15,83250,310],{"className":83251},[129],[15,83253,83255,83258],{"className":83254},[129],[15,83256,27283],{"className":83257,"style":887},[129,133],[15,83259,83261],{"className":83260},[137],[15,83262,83264],{"className":83263},[141],[15,83265,83267],{"className":83266},[146],[15,83268,83270],{"className":83269,"style":30988},[150],[15,83271,83272,83275],{"style":17194},[15,83273],{"className":83274,"style":159},[158],[15,83276,83278],{"className":83277},[163,164,165,166],[15,83279,310],{"className":83280},[129,166],[15,83282,83283,83286],{"style":1832},[15,83284],{"className":83285,"style":1823},[158],[15,83287],{"className":83288,"style":1840},[1839],[15,83290,83291,83294],{"style":1843},[15,83292],{"className":83293,"style":1823},[158],[15,83295,83297,83300],{"className":83296},[129],[15,83298,39314],{"className":83299},[129,133],[15,83301,83303,83306],{"className":83302},[129],[15,83304,4849],{"className":83305},[129,133],[15,83307,83309],{"className":83308},[137],[15,83310,83312],{"className":83311},[141],[15,83313,83315],{"className":83314},[146],[15,83316,83318],{"className":83317,"style":380},[150],[15,83319,83320,83323],{"style":383},[15,83321],{"className":83322,"style":159},[158],[15,83324,83326],{"className":83325},[163,164,165,166],[15,83327,310],{"className":83328},[129,166],[15,83330,175],{"className":83331},[174],[15,83333,83335],{"className":83334},[146],[15,83336,83338],{"className":83337,"style":1862},[150],[15,83339],{},[15,83341],{"className":83342},[195,1803],[15,83344,107],{"className":83345},[129],[11,83347,83348],{},"Pinsker 给出",[15,83350,83352],{"className":83351,"translate":19},[18],[15,83353,83355,83425],{"className":83354,"translate":19},[23],[15,83356,83358],{"className":83357},[27],[29,83359,83360],{"xmlns":31,"display":32},[34,83361,83362,83422],{},[37,83363,83364,83366,83378,83380,83392,83402,83404,83420],{},[50,83365,33810],{"stretchy":52},[40,83367,83368,83370],{},[43,83369,34341],{},[37,83371,83372,83374,83376],{},[43,83373,13949],{},[50,83375,91],{"separator":72},[308,83377,541],{},[50,83379,315],{},[40,83381,83382,83384],{},[43,83383,34341],{},[37,83385,83386,83388,83390],{},[43,83387,13949],{},[50,83389,91],{"separator":72},[308,83391,318],{},[40,83393,83394,83396],{},[50,83395,33810],{"stretchy":52},[37,83397,83398,83400],{},[43,83399,53394],{"mathvariant":65},[43,83401,53397],{"mathvariant":65},[50,83403,2717],{},[1443,83405,83406,83414],{},[37,83407,83408,83410],{},[43,83409,4849],{},[29004,83411,83412],{},[43,83413,39314],{},[37,83415,83416,83418],{},[308,83417,310],{},[43,83419,27283],{},[43,83421,107],{"mathvariant":65},[109,83423,83424],{"encoding":111},"\\lVert Q_{j,0}-Q_{j,1}\\rVert_{\\mathrm{TV}}\n\\le \\frac{\\varepsilon\\sqrt n}{2\\sigma}.",[15,83426,83428,83495,83605],{"className":83427,"ariaHidden":72},[116],[15,83429,83431,83434,83437,83486,83489,83492],{"className":83430},[120],[15,83432],{"className":83433,"style":67439},[124],[15,83435,33810],{"className":83436},[188],[15,83438,83440,83443],{"className":83439},[129],[15,83441,34341],{"className":83442},[129,133],[15,83444,83446],{"className":83445},[137],[15,83447,83449,83478],{"className":83448},[141,142],[15,83450,83452,83475],{"className":83451},[146],[15,83453,83455],{"className":83454,"style":1981},[150],[15,83456,83457,83460],{"style":154},[15,83458],{"className":83459,"style":159},[158],[15,83461,83463],{"className":83462},[163,164,165,166],[15,83464,83466,83469,83472],{"className":83465},[129,166],[15,83467,13949],{"className":83468,"style":13964},[129,133,166],[15,83470,91],{"className":83471},[254,166],[15,83473,541],{"className":83474},[129,166],[15,83476,175],{"className":83477},[174],[15,83479,83481],{"className":83480},[146],[15,83482,83484],{"className":83483,"style":14035},[150],[15,83485],{},[15,83487],{"className":83488,"style":3277},[93],[15,83490,315],{"className":83491},[2436],[15,83493],{"className":83494,"style":3277},[93],[15,83496,83498,83501,83550,83596,83599,83602],{"className":83497},[120],[15,83499],{"className":83500,"style":67439},[124],[15,83502,83504,83507],{"className":83503},[129],[15,83505,34341],{"className":83506},[129,133],[15,83508,83510],{"className":83509},[137],[15,83511,83513,83542],{"className":83512},[141,142],[15,83514,83516,83539],{"className":83515},[146],[15,83517,83519],{"className":83518,"style":1981},[150],[15,83520,83521,83524],{"style":154},[15,83522],{"className":83523,"style":159},[158],[15,83525,83527],{"className":83526},[163,164,165,166],[15,83528,83530,83533,83536],{"className":83529},[129,166],[15,83531,13949],{"className":83532,"style":13964},[129,133,166],[15,83534,91],{"className":83535},[254,166],[15,83537,318],{"className":83538},[129,166],[15,83540,175],{"className":83541},[174],[15,83543,83545],{"className":83544},[146],[15,83546,83548],{"className":83547,"style":14035},[150],[15,83549],{},[15,83551,83553,83556],{"className":83552},[195],[15,83554,33810],{"className":83555},[195],[15,83557,83559],{"className":83558},[137],[15,83560,83562,83588],{"className":83561},[141,142],[15,83563,83565,83585],{"className":83564},[146],[15,83566,83568],{"className":83567,"style":1525},[150],[15,83569,83570,83573],{"style":154},[15,83571],{"className":83572,"style":159},[158],[15,83574,83576],{"className":83575},[163,164,165,166],[15,83577,83579],{"className":83578},[129,166],[15,83580,83582],{"className":83581},[129,166],[15,83583,54025],{"className":83584,"style":2246},[129,221,166],[15,83586,175],{"className":83587},[174],[15,83589,83591],{"className":83590},[146],[15,83592,83594],{"className":83593,"style":182},[150],[15,83595],{},[15,83597],{"className":83598,"style":199},[93],[15,83600,2717],{"className":83601},[203],[15,83603],{"className":83604,"style":199},[93],[15,83606,83608,83612,83722],{"className":83607},[120],[15,83609],{"className":83610,"style":83611},[124],"height:2.1633em;vertical-align:-0.686em;",[15,83613,83615,83618,83719],{"className":83614},[129],[15,83616],{"className":83617},[188,1803],[15,83619,83621],{"className":83620},[1443],[15,83622,83624,83711],{"className":83623},[141,142],[15,83625,83627,83708],{"className":83626},[146],[15,83628,83631,83645,83653],{"className":83629,"style":83630},[150],"height:1.4773em;",[15,83632,83633,83636],{"style":1819},[15,83634],{"className":83635,"style":1823},[158],[15,83637,83639,83642],{"className":83638},[129],[15,83640,310],{"className":83641},[129],[15,83643,27283],{"className":83644,"style":887},[129,133],[15,83646,83647,83650],{"style":1832},[15,83648],{"className":83649,"style":1823},[158],[15,83651],{"className":83652,"style":1840},[1839],[15,83654,83655,83658],{"style":1843},[15,83656],{"className":83657,"style":1823},[158],[15,83659,83661,83664],{"className":83660},[129],[15,83662,4849],{"className":83663},[129,133],[15,83665,83667],{"className":83666},[129,29112],[15,83668,83670,83700],{"className":83669},[141,142],[15,83671,83673,83697],{"className":83672},[146],[15,83674,83676,83685],{"className":83675,"style":30452},[150],[15,83677,83679,83682],{"className":83678,"style":2287},[29126],[15,83680],{"className":83681,"style":1823},[158],[15,83683,39314],{"className":83684,"style":30462},[129,133],[15,83686,83687,83690],{"style":30465},[15,83688],{"className":83689,"style":1823},[158],[15,83691,83693],{"className":83692,"style":30472},[29271],[4455,83694,83695],{"xmlns":4457,"width":29275,"height":30475,"viewBox":30476,"preserveAspectRatio":29278},[4461,83696],{"d":30479},[15,83698,175],{"className":83699},[174],[15,83701,83703],{"className":83702},[146],[15,83704,83706],{"className":83705,"style":30489},[150],[15,83707],{},[15,83709,175],{"className":83710},[174],[15,83712,83714],{"className":83713},[146],[15,83715,83717],{"className":83716,"style":1862},[150],[15,83718],{},[15,83720],{"className":83721},[195,1803],[15,83723,107],{"className":83724},[129],[11,83726,7437,83727,83831,83832,83864],{},[15,83728,83730,83754],{"className":83729,"translate":19},[23],[15,83731,83733],{"className":83732},[27],[29,83734,83735],{"xmlns":31},[34,83736,83737,83751],{},[37,83738,83739,83741,83743,83745,83747],{},[43,83740,4849],{},[50,83742,62],{},[43,83744,27283],{},[43,83746,544],{"mathvariant":65},[29004,83748,83749],{},[43,83750,39314],{},[109,83752,83753],{"encoding":111},"\\varepsilon=\\sigma\u002F\\sqrt n",[15,83755,83757,83775],{"className":83756,"ariaHidden":72},[116],[15,83758,83760,83763,83766,83769,83772],{"className":83759},[120],[15,83761],{"className":83762,"style":4383},[124],[15,83764,4849],{"className":83765},[129,133],[15,83767],{"className":83768,"style":199},[93],[15,83770,62],{"className":83771},[203],[15,83773],{"className":83774,"style":199},[93],[15,83776,83778,83781,83784,83787],{"className":83777},[120],[15,83779],{"className":83780,"style":30436},[124],[15,83782,27283],{"className":83783,"style":887},[129,133],[15,83785,544],{"className":83786},[129],[15,83788,83790],{"className":83789},[129,29112],[15,83791,83793,83823],{"className":83792},[141,142],[15,83794,83796,83820],{"className":83795},[146],[15,83797,83799,83808],{"className":83798,"style":30452},[150],[15,83800,83802,83805],{"className":83801,"style":2287},[29126],[15,83803],{"className":83804,"style":1823},[158],[15,83806,39314],{"className":83807,"style":30462},[129,133],[15,83809,83810,83813],{"style":30465},[15,83811],{"className":83812,"style":1823},[158],[15,83814,83816],{"className":83815,"style":30472},[29271],[4455,83817,83818],{"xmlns":4457,"width":29275,"height":30475,"viewBox":30476,"preserveAspectRatio":29278},[4461,83819],{"d":30479},[15,83821,175],{"className":83822},[174],[15,83824,83826],{"className":83825},[146],[15,83827,83829],{"className":83828,"style":30489},[150],[15,83830],{},"，则 TV 至多 ",[15,83833,83835,83852],{"className":83834,"translate":19},[23],[15,83836,83838],{"className":83837},[27],[29,83839,83840],{"xmlns":31},[34,83841,83842,83850],{},[37,83843,83844,83846,83848],{},[308,83845,318],{},[43,83847,544],{"mathvariant":65},[308,83849,310],{},[109,83851,56720],{"encoding":111},[15,83853,83855],{"className":83854,"ariaHidden":72},[116],[15,83856,83858,83861],{"className":83857},[120],[15,83859],{"className":83860,"style":125},[124],[15,83862,56720],{"className":83863},[129],"。Assouad：",[15,83866,83868],{"className":83867,"translate":19},[18],[15,83869,83871,83979],{"className":83870,"translate":19},[23],[15,83872,83874],{"className":83873},[27],[29,83875,83876],{"xmlns":31,"display":32},[34,83877,83878,83976],{},[37,83879,83880,83900,83902,83904,83910,83912,83914,83922,83924,83938,83954,83956,83974],{},[2140,83881,83882,83888],{},[37,83883,83884,83886],{},[43,83885,19696],{},[50,83887,69],{},[37,83889,83890,83892,83894],{},[43,83891,48],{},[50,83893,100],{},[40,83895,83896,83898],{},[43,83897,30862],{"mathvariant":65},[308,83899,541],{},[43,83901,26563],{"mathvariant":103},[50,83903,33810],{"stretchy":52},[46835,83905,83906,83908],{"accent":72},[43,83907,48],{},[50,83909,48471],{},[50,83911,315],{},[43,83913,48],{},[23149,83915,83916,83918,83920],{},[50,83917,33810],{"stretchy":52},[308,83919,310],{},[308,83921,310],{},[50,83923,7469],{},[1443,83925,83926,83936],{},[37,83927,83928,83934],{},[305,83929,83930,83932],{},[43,83931,4849],{},[308,83933,310],{},[43,83935,10372],{},[308,83937,14502],{},[37,83939,83940,83942,83944,83946,83952],{},[50,83941,53],{"fence":72},[308,83943,318],{},[50,83945,315],{},[1443,83947,83948,83950],{},[308,83949,318],{},[308,83951,310],{},[50,83953,59],{"fence":72},[50,83955,62],{},[1443,83957,83958,83968],{},[37,83959,83960,83966],{},[305,83961,83962,83964],{},[43,83963,27283],{},[308,83965,310],{},[43,83967,10372],{},[37,83969,83970,83972],{},[308,83971,32902],{},[43,83973,39314],{},[43,83975,107],{"mathvariant":65},[109,83977,83978],{"encoding":111},"\\max_{\\theta\\in\\Theta_0}\\mathbb E\\lVert\\hat\\theta-\\theta\\rVert_2^2\n\\ge\n\\frac{\\varepsilon^2d}{4}\\left(1-\\frac12\\right)\n=\\frac{\\sigma^2d}{8n}.",[15,83980,83982,84130,84199,84397],{"className":83981,"ariaHidden":72},[116],[15,83983,83985,83989,84081,84084,84087,84090,84121,84124,84127],{"className":83984},[120],[15,83986],{"className":83987,"style":83988},[124],"height:1.8101em;vertical-align:-0.8522em;",[15,83990,83992],{"className":83991},[217,1874],[15,83993,83995,84072],{"className":83994},[141,142],[15,83996,83998,84069],{"className":83997},[146],[15,83999,84001,84059],{"className":84000,"style":4383},[150],[15,84002,84003,84006],{"style":2266},[15,84004],{"className":84005,"style":1823},[158],[15,84007,84009],{"className":84008},[163,164,165,166],[15,84010,84012,84015,84018],{"className":84011},[129,166],[15,84013,48],{"className":84014,"style":170},[129,133,166],[15,84016,100],{"className":84017},[203,166],[15,84019,84021,84024],{"className":84020},[129,166],[15,84022,30862],{"className":84023},[129,166],[15,84025,84027],{"className":84026},[137],[15,84028,84030,84051],{"className":84029},[141,142],[15,84031,84033,84048],{"className":84032},[146],[15,84034,84036],{"className":84035,"style":46983},[150],[15,84037,84039,84042],{"style":84038},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[15,84040],{"className":84041,"style":15980},[158],[15,84043,84045],{"className":84044},[163,15984,229,166],[15,84046,541],{"className":84047},[129,166],[15,84049,175],{"className":84050},[174],[15,84052,84054],{"className":84053},[146],[15,84055,84057],{"className":84056,"style":15997},[150],[15,84058],{},[15,84060,84061,84064],{"style":2287},[15,84062],{"className":84063,"style":1823},[158],[15,84065,84066],{},[15,84067,19696],{"className":84068},[217],[15,84070,175],{"className":84071},[174],[15,84073,84075],{"className":84074},[146],[15,84076,84079],{"className":84077,"style":84078},[150],"height:0.8522em;",[15,84080],{},[15,84082],{"className":84083,"style":262},[93],[15,84085,26563],{"className":84086},[129,285],[15,84088,33810],{"className":84089},[188],[15,84091,84093],{"className":84092},[129,48488],[15,84094,84096],{"className":84095},[141],[15,84097,84099],{"className":84098},[146],[15,84100,84102,84110],{"className":84101,"style":48484},[150],[15,84103,84104,84107],{"style":2287},[15,84105],{"className":84106,"style":1823},[158],[15,84108,48],{"className":84109,"style":170},[129,133],[15,84111,84112,84115],{"style":48508},[15,84113],{"className":84114,"style":1823},[158],[15,84116,84118],{"className":84117,"style":48516},[48515],[15,84119,48471],{"className":84120},[129],[15,84122],{"className":84123,"style":3277},[93],[15,84125,315],{"className":84126},[2436],[15,84128],{"className":84129,"style":3277},[93],[15,84131,84133,84136,84139,84190,84193,84196],{"className":84132},[120],[15,84134],{"className":84135,"style":40908},[124],[15,84137,48],{"className":84138,"style":170},[129,133],[15,84140,84142,84145],{"className":84141},[195],[15,84143,33810],{"className":84144},[195],[15,84146,84148],{"className":84147},[137],[15,84149,84151,84182],{"className":84150},[141,142],[15,84152,84154,84179],{"className":84153},[146],[15,84155,84157,84168],{"className":84156,"style":40979},[150],[15,84158,84159,84162],{"style":58433},[15,84160],{"className":84161,"style":159},[158],[15,84163,84165],{"className":84164},[163,164,165,166],[15,84166,310],{"className":84167},[129,166],[15,84169,84170,84173],{"style":1272},[15,84171],{"className":84172,"style":159},[158],[15,84174,84176],{"className":84175},[163,164,165,166],[15,84177,310],{"className":84178},[129,166],[15,84180,175],{"className":84181},[174],[15,84183,84185],{"className":84184},[146],[15,84186,84188],{"className":84187,"style":28162},[150],[15,84189],{},[15,84191],{"className":84192,"style":199},[93],[15,84194,7469],{"className":84195},[203],[15,84197],{"className":84198,"style":199},[93],[15,84200,84202,84205,84296,84299,84388,84391,84394],{"className":84201},[120],[15,84203],{"className":84204,"style":63589},[124],[15,84206,84208,84211,84293],{"className":84207},[129],[15,84209],{"className":84210},[188,1803],[15,84212,84214],{"className":84213},[1443],[15,84215,84217,84285],{"className":84216},[141,142],[15,84218,84220,84282],{"className":84219},[146],[15,84221,84223,84234,84242],{"className":84222,"style":63608},[150],[15,84224,84225,84228],{"style":1819},[15,84226],{"className":84227,"style":1823},[158],[15,84229,84231],{"className":84230},[129],[15,84232,14502],{"className":84233},[129],[15,84235,84236,84239],{"style":1832},[15,84237],{"className":84238,"style":1823},[158],[15,84240],{"className":84241,"style":1840},[1839],[15,84243,84244,84247],{"style":1843},[15,84245],{"className":84246,"style":1823},[158],[15,84248,84250,84279],{"className":84249},[129],[15,84251,84253,84256],{"className":84252},[129],[15,84254,4849],{"className":84255},[129,133],[15,84257,84259],{"className":84258},[137],[15,84260,84262],{"className":84261},[141],[15,84263,84265],{"className":84264},[146],[15,84266,84268],{"className":84267,"style":380},[150],[15,84269,84270,84273],{"style":383},[15,84271],{"className":84272,"style":159},[158],[15,84274,84276],{"className":84275},[163,164,165,166],[15,84277,310],{"className":84278},[129,166],[15,84280,10372],{"className":84281},[129,133],[15,84283,175],{"className":84284},[174],[15,84286,84288],{"className":84287},[146],[15,84289,84291],{"className":84290,"style":1862},[150],[15,84292],{},[15,84294],{"className":84295},[195,1803],[15,84297],{"className":84298,"style":262},[93],[15,84300,84302,84308,84311,84314,84317,84320,84382],{"className":84301},[449],[15,84303,84305],{"className":84304,"style":5052},[188,5051],[15,84306,53],{"className":84307},[228,165],[15,84309,318],{"className":84310},[129],[15,84312],{"className":84313,"style":3277},[93],[15,84315,315],{"className":84316},[2436],[15,84318],{"className":84319,"style":3277},[93],[15,84321,84323,84326,84379],{"className":84322},[129],[15,84324],{"className":84325},[188,1803],[15,84327,84329],{"className":84328},[1443],[15,84330,84332,84371],{"className":84331},[141,142],[15,84333,84335,84368],{"className":84334},[146],[15,84336,84338,84349,84357],{"className":84337,"style":1816},[150],[15,84339,84340,84343],{"style":1819},[15,84341],{"className":84342,"style":1823},[158],[15,84344,84346],{"className":84345},[129],[15,84347,310],{"className":84348},[129],[15,84350,84351,84354],{"style":1832},[15,84352],{"className":84353,"style":1823},[158],[15,84355],{"className":84356,"style":1840},[1839],[15,84358,84359,84362],{"style":1843},[15,84360],{"className":84361,"style":1823},[158],[15,84363,84365],{"className":84364},[129],[15,84366,318],{"className":84367},[129],[15,84369,175],{"className":84370},[174],[15,84372,84374],{"className":84373},[146],[15,84375,84377],{"className":84376,"style":1862},[150],[15,84378],{},[15,84380],{"className":84381},[195,1803],[15,84383,84385],{"className":84384,"style":5052},[195,5051],[15,84386,59],{"className":84387},[228,165],[15,84389],{"className":84390,"style":199},[93],[15,84392,62],{"className":84393},[203],[15,84395],{"className":84396,"style":199},[93],[15,84398,84400,84403,84497],{"className":84399},[120],[15,84401],{"className":84402,"style":82488},[124],[15,84404,84406,84409,84494],{"className":84405},[129],[15,84407],{"className":84408},[188,1803],[15,84410,84412],{"className":84411},[1443],[15,84413,84415,84486],{"className":84414},[141,142],[15,84416,84418,84483],{"className":84417},[146],[15,84419,84421,84435,84443],{"className":84420,"style":63608},[150],[15,84422,84423,84426],{"style":1819},[15,84424],{"className":84425,"style":1823},[158],[15,84427,84429,84432],{"className":84428},[129],[15,84430,32902],{"className":84431},[129],[15,84433,39314],{"className":84434},[129,133],[15,84436,84437,84440],{"style":1832},[15,84438],{"className":84439,"style":1823},[158],[15,84441],{"className":84442,"style":1840},[1839],[15,84444,84445,84448],{"style":1843},[15,84446],{"className":84447,"style":1823},[158],[15,84449,84451,84480],{"className":84450},[129],[15,84452,84454,84457],{"className":84453},[129],[15,84455,27283],{"className":84456,"style":887},[129,133],[15,84458,84460],{"className":84459},[137],[15,84461,84463],{"className":84462},[141],[15,84464,84466],{"className":84465},[146],[15,84467,84469],{"className":84468,"style":380},[150],[15,84470,84471,84474],{"style":383},[15,84472],{"className":84473,"style":159},[158],[15,84475,84477],{"className":84476},[163,164,165,166],[15,84478,310],{"className":84479},[129,166],[15,84481,10372],{"className":84482},[129,133],[15,84484,175],{"className":84485},[174],[15,84487,84489],{"className":84488},[146],[15,84490,84492],{"className":84491,"style":1862},[150],[15,84493],{},[15,84495],{"className":84496},[195,1803],[15,84498,107],{"className":84499},[129],[11,84501,84502,84503,84531,84532,84605],{},"上下界只差常数 ",[15,84504,84506,84519],{"className":84505,"translate":19},[23],[15,84507,84509],{"className":84508},[27],[29,84510,84511],{"xmlns":31},[34,84512,84513,84517],{},[37,84514,84515],{},[308,84516,32902],{},[109,84518,32902],{"encoding":111},[15,84520,84522],{"className":84521,"ariaHidden":72},[116],[15,84523,84525,84528],{"className":84524},[120],[15,84526],{"className":84527,"style":4886},[124],[15,84529,32902],{"className":84530},[129],"，速率 ",[15,84533,84535,84558],{"className":84534,"translate":19},[23],[15,84536,84538],{"className":84537},[27],[29,84539,84540],{"xmlns":31},[34,84541,84542,84556],{},[37,84543,84544,84550,84552,84554],{},[305,84545,84546,84548],{},[43,84547,27283],{},[308,84549,310],{},[43,84551,10372],{},[43,84553,544],{"mathvariant":65},[43,84555,39314],{},[109,84557,64904],{"encoding":111},[15,84559,84561],{"className":84560,"ariaHidden":72},[116],[15,84562,84564,84567,84596,84599,84602],{"className":84563},[120],[15,84565],{"className":84566,"style":33996},[124],[15,84568,84570,84573],{"className":84569},[129],[15,84571,27283],{"className":84572,"style":887},[129,133],[15,84574,84576],{"className":84575},[137],[15,84577,84579],{"className":84578},[141],[15,84580,84582],{"className":84581},[146],[15,84583,84585],{"className":84584,"style":380},[150],[15,84586,84587,84590],{"style":383},[15,84588],{"className":84589,"style":159},[158],[15,84591,84593],{"className":84592},[163,164,165,166],[15,84594,310],{"className":84595},[129,166],[15,84597,10372],{"className":84598},[129,133],[15,84600,544],{"className":84601},[129],[15,84603,39314],{"className":84604},[129,133]," 被夹住了。",[11,84607,84608,84609,84637],{},"这个例子也说明了 Fano 与 Assouad 的分工：Fano 需要构造一个 ",[15,84610,84612,84625],{"className":84611,"translate":19},[23],[15,84613,84615],{"className":84614},[27],[29,84616,84617],{"xmlns":31},[34,84618,84619,84623],{},[37,84620,84621],{},[43,84622,29564],{},[109,84624,61720],{"encoding":111},[15,84626,84628],{"className":84627,"ariaHidden":72},[116],[15,84629,84631,84634],{"className":84630},[120],[15,84632],{"className":84633,"style":522},[124],[15,84635,29564],{"className":84636,"style":29845},[129,133],"-分离的打包，Assouad 则直接利用乘积结构，把打包问题化成了逐坐标的 TV 计算。",[575,84639,84641,84642],{"id":84640},"_62-非参数回归n2β2βdn-2beta2betadn2β2βd","6.2 非参数回归：",[15,84643,84645,84682],{"className":84644,"translate":19},[23],[15,84646,84648],{"className":84647},[27],[29,84649,84650],{"xmlns":31},[34,84651,84652,84680],{},[37,84653,84654],{},[305,84655,84656,84658],{},[43,84657,39314],{},[37,84659,84660,84662,84664,84666,84668,84670,84672,84674,84676,84678],{},[50,84661,315],{},[308,84663,310],{},[43,84665,36264],{},[43,84667,544],{"mathvariant":65},[50,84669,53],{"stretchy":52},[308,84671,310],{},[43,84673,36264],{},[50,84675,2682],{},[43,84677,10372],{},[50,84679,59],{"stretchy":52},[109,84681,75494],{"encoding":111},[15,84683,84685],{"className":84684,"ariaHidden":72},[116],[15,84686,84688,84691],{"className":84687},[120],[15,84689],{"className":84690,"style":14941},[124],[15,84692,84694,84697],{"className":84693},[129],[15,84695,39314],{"className":84696},[129,133],[15,84698,84700],{"className":84699},[137],[15,84701,84703],{"className":84702},[141],[15,84704,84706],{"className":84705},[146],[15,84707,84709],{"className":84708,"style":14941},[150],[15,84710,84711,84714],{"style":383},[15,84712],{"className":84713,"style":159},[158],[15,84715,84717],{"className":84716},[163,164,165,166],[15,84718,84720,84723,84726,84729,84732,84735,84738,84741,84744,84747],{"className":84719},[129,166],[15,84721,315],{"className":84722},[129,166],[15,84724,310],{"className":84725},[129,166],[15,84727,36264],{"className":84728,"style":36280},[129,133,166],[15,84730,544],{"className":84731},[129,166],[15,84733,53],{"className":84734},[188,166],[15,84736,310],{"className":84737},[129,166],[15,84739,36264],{"className":84740,"style":36280},[129,133,166],[15,84742,2682],{"className":84743},[2436,166],[15,84745,10372],{"className":84746},[129,133,166],[15,84748,59],{"className":84749},[195,166],[11,84751,81735],{},[15,84753,84755],{"className":84754,"translate":19},[18],[15,84756,84758,84869],{"className":84757,"translate":19},[23],[15,84759,84761],{"className":84760},[27],[29,84762,84763],{"xmlns":31,"display":32},[34,84764,84765,84866],{},[37,84766,84767,84773,84775,84777,84779,84785,84787,84789,84796,84798,84800,84806,84808,84818,84820,84822,84824,84826,84828,84834,84836,84838,84840,84846,84848,84850,84852,84854,84856,84862,84864],{},[40,84768,84769,84771],{},[43,84770,631],{},[43,84772,1459],{},[50,84774,62],{},[43,84776,36556],{},[50,84778,53],{"stretchy":52},[40,84780,84781,84783],{},[43,84782,597],{},[43,84784,1459],{},[50,84786,59],{"stretchy":52},[50,84788,2682],{},[40,84790,84791,84794],{},[43,84792,84793],{},"ξ",[43,84795,1459],{},[50,84797,91],{"separator":72},[93,84799],{"width":95},[40,84801,84802,84804],{},[43,84803,597],{},[43,84805,1459],{},[50,84807,991],{},[37,84809,84810,84812,84814,84816],{},[43,84811,67644],{"mathvariant":65},[43,84813,39314],{"mathvariant":65},[43,84815,1459],{"mathvariant":65},[43,84817,36556],{"mathvariant":65},[50,84819,53],{"stretchy":52},[50,84821,1407],{"stretchy":52},[308,84823,541],{},[50,84825,91],{"separator":72},[308,84827,318],{},[305,84829,84830,84832],{},[50,84831,1423],{"stretchy":52},[43,84833,10372],{},[50,84835,59],{"stretchy":52},[50,84837,91],{"separator":72},[93,84839],{"width":95},[40,84841,84842,84844],{},[43,84843,84793],{},[43,84845,1459],{},[50,84847,991],{},[43,84849,12555],{},[50,84851,53],{"stretchy":52},[308,84853,541],{},[50,84855,91],{"separator":72},[305,84857,84858,84860],{},[43,84859,27283],{},[308,84861,310],{},[50,84863,59],{"stretchy":52},[50,84865,91],{"separator":72},[109,84867,84868],{"encoding":111},"Y_i=f(X_i)+\\xi_i,\\qquad X_i\\sim\\mathrm{Unif}([0,1]^d),\\qquad \\xi_i\\sim N(0,\\sigma^2),",[15,84870,84872,84928,84992,85098,85217],{"className":84871,"ariaHidden":72},[116],[15,84873,84875,84878,84919,84922,84925],{"className":84874},[120],[15,84876],{"className":84877,"style":20220},[124],[15,84879,84881,84884],{"className":84880},[129],[15,84882,631],{"className":84883,"style":3277},[129,133],[15,84885,84887],{"className":84886},[137],[15,84888,84890,84911],{"className":84889},[141,142],[15,84891,84893,84908],{"className":84892},[146],[15,84894,84896],{"className":84895,"style":1981},[150],[15,84897,84899,84902],{"style":84898},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[15,84900],{"className":84901,"style":159},[158],[15,84903,84905],{"className":84904},[163,164,165,166],[15,84906,1459],{"className":84907},[129,133,166],[15,84909,175],{"className":84910},[174],[15,84912,84914],{"className":84913},[146],[15,84915,84917],{"className":84916,"style":182},[150],[15,84918],{},[15,84920],{"className":84921,"style":199},[93],[15,84923,62],{"className":84924},[203],[15,84926],{"className":84927,"style":199},[93],[15,84929,84931,84934,84937,84940,84980,84983,84986,84989],{"className":84930},[120],[15,84932],{"className":84933,"style":125},[124],[15,84935,36556],{"className":84936,"style":36571},[129,133],[15,84938,53],{"className":84939},[188],[15,84941,84943,84946],{"className":84942},[129],[15,84944,597],{"className":84945,"style":47958},[129,133],[15,84947,84949],{"className":84948},[137],[15,84950,84952,84972],{"className":84951},[141,142],[15,84953,84955,84969],{"className":84954},[146],[15,84956,84958],{"className":84957,"style":1981},[150],[15,84959,84960,84963],{"style":47973},[15,84961],{"className":84962,"style":159},[158],[15,84964,84966],{"className":84965},[163,164,165,166],[15,84967,1459],{"className":84968},[129,133,166],[15,84970,175],{"className":84971},[174],[15,84973,84975],{"className":84974},[146],[15,84976,84978],{"className":84977,"style":182},[150],[15,84979],{},[15,84981,59],{"className":84982},[195],[15,84984],{"className":84985,"style":3277},[93],[15,84987,2682],{"className":84988},[2436],[15,84990],{"className":84991,"style":3277},[93],[15,84993,84995,84998,85040,85043,85046,85049,85089,85092,85095],{"className":84994},[120],[15,84996],{"className":84997,"style":1682},[124],[15,84999,85001,85005],{"className":85000},[129],[15,85002,84793],{"className":85003,"style":85004},[129,133],"margin-right:0.046em;",[15,85006,85008],{"className":85007},[137],[15,85009,85011,85032],{"className":85010},[141,142],[15,85012,85014,85029],{"className":85013},[146],[15,85015,85017],{"className":85016,"style":1981},[150],[15,85018,85020,85023],{"style":85019},"top:-2.55em;margin-left:-0.046em;margin-right:0.05em;",[15,85021],{"className":85022,"style":159},[158],[15,85024,85026],{"className":85025},[163,164,165,166],[15,85027,1459],{"className":85028},[129,133,166],[15,85030,175],{"className":85031},[174],[15,85033,85035],{"className":85034},[146],[15,85036,85038],{"className":85037,"style":182},[150],[15,85039],{},[15,85041,91],{"className":85042},[254],[15,85044],{"className":85045,"style":258},[93],[15,85047],{"className":85048,"style":262},[93],[15,85050,85052,85055],{"className":85051},[129],[15,85053,597],{"className":85054,"style":47958},[129,133],[15,85056,85058],{"className":85057},[137],[15,85059,85061,85081],{"className":85060},[141,142],[15,85062,85064,85078],{"className":85063},[146],[15,85065,85067],{"className":85066,"style":1981},[150],[15,85068,85069,85072],{"style":47973},[15,85070],{"className":85071,"style":159},[158],[15,85073,85075],{"className":85074},[163,164,165,166],[15,85076,1459],{"className":85077},[129,133,166],[15,85079,175],{"className":85080},[174],[15,85082,85084],{"className":85083},[146],[15,85085,85087],{"className":85086,"style":182},[150],[15,85088],{},[15,85090],{"className":85091,"style":199},[93],[15,85093,991],{"className":85094},[203],[15,85096],{"className":85097,"style":199},[93],[15,85099,85101,85105,85111,85115,85118,85121,85124,85127,85156,85159,85162,85165,85168,85208,85211,85214],{"className":85100},[120],[15,85102],{"className":85103,"style":85104},[124],"height:1.1491em;vertical-align:-0.25em;",[15,85106,85108],{"className":85107},[129],[15,85109,67701],{"className":85110,"style":48838},[129,221],[15,85112,85114],{"className":85113},[188],"([",[15,85116,541],{"className":85117},[129],[15,85119,91],{"className":85120},[254],[15,85122],{"className":85123,"style":262},[93],[15,85125,318],{"className":85126},[129],[15,85128,85130,85133],{"className":85129},[195],[15,85131,1423],{"className":85132},[195],[15,85134,85136],{"className":85135},[137],[15,85137,85139],{"className":85138},[141],[15,85140,85142],{"className":85141},[146],[15,85143,85145],{"className":85144,"style":64281},[150],[15,85146,85147,85150],{"style":1272},[15,85148],{"className":85149,"style":159},[158],[15,85151,85153],{"className":85152},[163,164,165,166],[15,85154,10372],{"className":85155},[129,133,166],[15,85157,59],{"className":85158},[195],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属于 ",[15,85305,85307,85320],{"className":85306,"translate":19},[23],[15,85308,85310],{"className":85309},[27],[29,85311,85312],{"xmlns":31},[34,85313,85314,85318],{},[37,85315,85316],{},[43,85317,36264],{},[109,85319,36267],{"encoding":111},[15,85321,85323],{"className":85322,"ariaHidden":72},[116],[15,85324,85326,85329],{"className":85325},[120],[15,85327],{"className":85328,"style":1682},[124],[15,85330,36264],{"className":85331,"style":36280},[129,133],"-Hölder 球 ",[15,85334,85336,85361],{"className":85335,"translate":19},[23],[15,85337,85339],{"className":85338},[27],[29,85340,85341],{"xmlns":31},[34,85342,85343,85358],{},[37,85344,85345,85348,85350,85352,85354,85356],{},[43,85346,85347],{"mathvariant":65},"Σ",[50,85349,53],{"stretchy":52},[43,85351,36264],{},[50,85353,91],{"separator":72},[43,85355,1373],{},[50,85357,59],{"stretchy":52},[109,85359,85360],{"encoding":111},"\\Sigma(\\beta,L)",[15,85362,85364],{"className":85363,"ariaHidden":72},[116],[15,85365,85367,85370,85373,85376,85379,85382,85385,85388],{"className":85366},[120],[15,85368],{"className":85369,"style":125},[124],[15,85371,85347],{"className":85372},[129],[15,85374,53],{"className":85375},[188],[15,85377,36264],{"className":85378,"style":36280},[129,133],[15,85380,91],{"className":85381},[254],[15,85383],{"className":85384,"style":262},[93],[15,85386,1373],{"className":85387},[129,133],[15,85389,59],{"className":85390},[195],"，损失为 ",[15,85393,85395,85431],{"className":85394,"translate":19},[23],[15,85396,85398],{"className":85397},[27],[29,85399,85400],{"xmlns":31},[34,85401,85402,85428],{},[37,85403,85404,85406,85412,85414,85416],{},[50,85405,33810],{"stretchy":52},[46835,85407,85408,85410],{"accent":72},[43,85409,36556],{},[50,85411,48471],{},[50,85413,315],{},[43,85415,36556],{},[23149,85417,85418,85420,85426],{},[50,85419,33810],{"stretchy":52},[305,85421,85422,85424],{},[43,85423,1373],{},[308,85425,310],{},[308,85427,310],{},[109,85429,85430],{"encoding":111},"\\lVert\\hat 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核估计或局部多项式估计可以达到",[15,85594,85596],{"className":85595,"translate":19},[18],[15,85597,85599,85637],{"className":85598,"translate":19},[23],[15,85600,85602],{"className":85601},[27],[29,85603,85604],{"xmlns":31,"display":32},[34,85605,85606,85634],{},[37,85607,85608],{},[305,85609,85610,85612],{},[43,85611,39314],{},[37,85613,85614,85616],{},[50,85615,315],{},[1443,85617,85618,85624],{},[37,85619,85620,85622],{},[308,85621,310],{},[43,85623,36264],{},[37,85625,85626,85628,85630,85632],{},[308,85627,310],{},[43,85629,36264],{},[50,85631,2682],{},[43,85633,10372],{},[109,85635,85636],{"encoding":111},"n^{-\\frac{2\\beta}{2\\beta+d}}",[15,85638,85640],{"className":85639,"ariaHidden":72},[116],[15,85641,85643,85647],{"className":85642},[120],[15,85644],{"className":85645,"style":85646},[124],"height:1.1167em;",[15,85648,85650,85653],{"className":85649},[129],[15,85651,39314],{"className":85652},[129,133],[15,85654,85656],{"className":85655},[137],[15,85657,85659],{"className":85658},[141],[15,85660,85662],{"className":85661},[146],[15,85663,85665],{"className":85664,"style":85646},[150],[15,85666,85668,85671],{"style":85667},"top:-3.4458em;margin-right:0.05em;",[15,85669],{"className":85670,"style":1823},[158],[15,85672,85674],{"className":85673},[163,164,165,166],[15,85675,85677,85680],{"className":85676},[129,166],[15,85678,315],{"className":85679},[129,166],[15,85681,85683,85687,85764],{"className":85682},[129,166],[15,85684],{"className":85685},[188,1803,163,15984,85686],"size6",[15,85688,85690],{"className":85689},[1443],[15,85691,85693,85755],{"className":85692},[141,142],[15,85694,85696,85752],{"className":85695},[146],[15,85697,85700,85724,85734],{"className":85698,"style":85699},[150],"height:0.9584em;",[15,85701,85703,85706],{"style":85702},"top:-2.656em;",[15,85704],{"className":85705,"style":1823},[158],[15,85707,85709],{"className":85708},[163,15984,229,166],[15,85710,85712,85715,85718,85721],{"className":85711},[129,166],[15,85713,310],{"className":85714},[129,166],[15,85716,36264],{"className":85717,"style":36280},[129,133,166],[15,85719,2682],{"className":85720},[2436,166],[15,85722,10372],{"className":85723},[129,133,166],[15,85725,85727,85730],{"style":85726},"top:-3.2255em;",[15,85728],{"className":85729,"style":1823},[158],[15,85731],{"className":85732,"style":85733},[1839,166],"border-bottom-width:0.049em;",[15,85735,85737,85740],{"style":85736},"top:-3.4624em;",[15,85738],{"className":85739,"style":1823},[158],[15,85741,85743],{"className":85742},[163,15984,229,166],[15,85744,85746,85749],{"className":85745},[129,166],[15,85747,310],{"className":85748},[129,166],[15,85750,36264],{"className":85751,"style":36280},[129,133,166],[15,85753,175],{"className":85754},[174],[15,85756,85758],{"className":85757},[146],[15,85759,85762],{"className":85760,"style":85761},[150],"height:0.4829em;",[15,85763],{},[15,85765],{"className":85766},[195,1803,163,15984,85686],[11,85768,85769,85770,85772],{},"（见 Tsybakov 第 1 章）。",[698,85771,10667],{},"用 Fano。",[11,85774,7437,85775,2455,85849,85931,85932,86023,86024,86052,86053,86083],{},[15,85776,85778,85804],{"className":85777,"translate":19},[23],[15,85779,85781],{"className":85780},[27],[29,85782,85783],{"xmlns":31},[34,85784,85785,85801],{},[37,85786,85787,85789,85791,85793,85795,85797,85799],{},[43,85788,45],{},[50,85790,100],{},[50,85792,53],{"stretchy":52},[308,85794,541],{},[50,85796,91],{"separator":72},[308,85798,318],{},[50,85800,59],{"stretchy":52},[109,85802,85803],{"encoding":111},"h\\in(0,1)",[15,85805,85807,85825],{"className":85806,"ariaHidden":72},[116],[15,85808,85810,85813,85816,85819,85822],{"className":85809},[120],[15,85811],{"className":85812,"style":2405},[124],[15,85814,45],{"className":85815},[129,133],[15,85817],{"className":85818,"style":199},[93],[15,85820,100],{"className":85821},[203],[15,85823],{"className":85824,"style":199},[93],[15,85826,85828,85831,85834,85837,85840,85843,85846],{"className":85827},[120],[15,85829],{"className":85830,"style":125},[124],[15,85832,53],{"className":85833},[188],[15,85835,541],{"className":85836},[129],[15,85838,91],{"className":85839},[254],[15,85841],{"className":85842,"style":262},[93],[15,85844,318],{"className":85845},[129],[15,85847,59],{"className":85848},[195],[15,85850,85852,85878],{"className":85851,"translate":19},[23],[15,85853,85855],{"className":85854},[27],[29,85856,85857],{"xmlns":31},[34,85858,85859,85875],{},[37,85860,85861,85863,85865,85867,85869],{},[50,85862,1407],{"stretchy":52},[308,85864,541],{},[50,85866,91],{"separator":72},[308,85868,318],{},[305,85870,85871,85873],{},[50,85872,1423],{"stretchy":52},[43,85874,10372],{},[109,85876,85877],{"encoding":111},"[0,1]^d",[15,85879,85881],{"className":85880,"ariaHidden":72},[116],[15,85882,85884,85887,85890,85893,85896,85899,85902],{"className":85883},[120],[15,85885],{"className":85886,"style":10974},[124],[15,85888,1407],{"className":85889},[188],[15,85891,541],{"className":85892},[129],[15,85894,91],{"className":85895},[254],[15,85897],{"className":85898,"style":262},[93],[15,85900,318],{"className":85901},[129],[15,85903,85905,85908],{"className":85904},[195],[15,85906,1423],{"className":85907},[195],[15,85909,85911],{"className":85910},[137],[15,85912,85914],{"className":85913},[141],[15,85915,85917],{"className":85916},[146],[15,85918,85920],{"className":85919,"style":10486},[150],[15,85921,85922,85925],{"style":383},[15,85923],{"className":85924,"style":159},[158],[15,85926,85928],{"className":85927},[163,164,165,166],[15,85929,10372],{"className":85930},[129,133,166]," 分成 ",[15,85933,85935,85961],{"className":85934,"translate":19},[23],[15,85936,85938],{"className":85937},[27],[29,85939,85940],{"xmlns":31},[34,85941,85942,85958],{},[37,85943,85944,85946,85948],{},[43,85945,348],{},[50,85947,62],{},[305,85949,85950,85952],{},[43,85951,45],{},[37,85953,85954,85956],{},[50,85955,315],{},[43,85957,10372],{},[109,85959,85960],{"encoding":111},"m=h^{-d}",[15,85962,85964,85982],{"className":85963,"ariaHidden":72},[116],[15,85965,85967,85970,85973,85976,85979],{"className":85966},[120],[15,85968],{"className":85969,"style":4383},[124],[15,85971,348],{"className":85972},[129,133],[15,85974],{"className":85975,"style":199},[93],[15,85977,62],{"className":85978},[203],[15,85980],{"className":85981,"style":199},[93],[15,85983,85985,85988],{"className":85984},[120],[15,85986],{"className":85987,"style":10486},[124],[15,85989,85991,85994],{"className":85990},[129],[15,85992,45],{"className":85993},[129,133],[15,85995,85997],{"className":85996},[137],[15,85998,86000],{"className":85999},[141],[15,86001,86003],{"className":86002},[146],[15,86004,86006],{"className":86005,"style":10486},[150],[15,86007,86008,86011],{"style":383},[15,86009],{"className":86010,"style":159},[158],[15,86012,86014],{"className":86013},[163,164,165,166],[15,86015,86017,86020],{"className":86016},[129,166],[15,86018,315],{"className":86019},[129,166],[15,86021,10372],{"className":86022},[129,133,166]," 个边长为 ",[15,86025,86027,86040],{"className":86026,"translate":19},[23],[15,86028,86030],{"className":86029},[27],[29,86031,86032],{"xmlns":31},[34,86033,86034,86038],{},[37,86035,86036],{},[43,86037,45],{},[109,86039,45],{"encoding":111},[15,86041,86043],{"className":86042,"ariaHidden":72},[116],[15,86044,86046,86049],{"className":86045},[120],[15,86047],{"className":86048,"style":522},[124],[15,86050,45],{"className":86051},[129,133]," 的小方块。取一个光滑的鼓包函数 ",[15,86054,86056,86071],{"className":86055,"translate":19},[23],[15,86057,86059],{"className":86058},[27],[29,86060,86061],{"xmlns":31},[34,86062,86063,86068],{},[37,86064,86065],{},[43,86066,86067],{},"φ",[109,86069,86070],{"encoding":111},"\\varphi",[15,86072,86074],{"className":86073,"ariaHidden":72},[116],[15,86075,86077,86080],{"className":86076},[120],[15,86078],{"className":86079,"style":5399},[124],[15,86081,86067],{"className":86082},[129,133],"，支撑在单位方块内，令",[15,86085,86087],{"className":86086,"translate":19},[18],[15,86088,86090,86140],{"className":86089,"translate":19},[23],[15,86091,86093],{"className":86092},[27],[29,86094,86095],{"xmlns":31,"display":32},[34,86096,86097,86137],{},[37,86098,86099,86105,86107,86109,86111,86113,86115,86117,86133,86135],{},[40,86100,86101,86103],{},[43,86102,86067],{},[43,86104,13949],{},[50,86106,53],{"stretchy":52},[43,86108,56],{},[50,86110,59],{"stretchy":52},[50,86112,62],{},[43,86114,86067],{},[50,86116,53],{"fence":72,"stretchy":72,"minsize":45693,"maxsize":45693},[1443,86118,86119,86131],{},[37,86120,86121,86123,86125],{},[43,86122,56],{},[50,86124,315],{},[40,86126,86127,86129],{},[43,86128,56],{},[43,86130,13949],{},[43,86132,45],{},[50,86134,59],{"fence":72,"stretchy":72,"minsize":45693,"maxsize":45693},[50,86136,91],{"separator":72},[109,86138,86139],{"encoding":111},"\\varphi_j(x)=\\varphi\\Bigl(\\frac{x-x_j}{h}\\Bigr),",[15,86141,86143,86207],{"className":86142,"ariaHidden":72},[116],[15,86144,86146,86149,86189,86192,86195,86198,86201,86204],{"className":86145},[120],[15,86147],{"className":86148,"style":67439},[124],[15,86150,86152,86155],{"className":86151},[129],[15,86153,86067],{"className":86154},[129,133],[15,86156,86158],{"className":86157},[137],[15,86159,86161,86181],{"className":86160},[141,142],[15,86162,86164,86178],{"className":86163},[146],[15,86165,86167],{"className":86166,"style":1981},[150],[15,86168,86169,86172],{"style":154},[15,86170],{"className":86171,"style":159},[158],[15,86173,86175],{"className":86174},[163,164,165,166],[15,86176,13949],{"className":86177,"style":13964},[129,133,166],[15,86179,175],{"className":86180},[174],[15,86182,86184],{"className":86183},[146],[15,86185,86187],{"className":86186,"style":14035},[150],[15,86188],{},[15,86190,53],{"className":86191},[188],[15,86193,56],{"className":86194},[129,133],[15,86196,59],{"className":86197},[195],[15,86199],{"className":86200,"style":199},[93],[15,86202,62],{"className":86203},[203],[15,86205],{"className":86206,"style":199},[93],[15,86208,86210,86214,86217,86223,86335,86341],{"className":86209},[120],[15,86211],{"className":86212,"style":86213},[124],"height:1.9463em;vertical-align:-0.686em;",[15,86215,86067],{"className":86216},[129,133],[15,86218,86220],{"className":86219},[188],[15,86221,53],{"className":86222},[228,24450],[15,86224,86226,86229,86332],{"className":86225},[129],[15,86227],{"className":86228},[188,1803],[15,86230,86232],{"className":86231},[1443],[15,86233,86235,86324],{"className":86234},[141,142],[15,86236,86238,86321],{"className":86237},[146],[15,86239,86242,86253,86261],{"className":86240,"style":86241},[150],"height:1.2603em;",[15,86243,86244,86247],{"style":1819},[15,86245],{"className":86246,"style":1823},[158],[15,86248,86250],{"className":86249},[129],[15,86251,45],{"className":86252},[129,133],[15,86254,86255,86258],{"style":1832},[15,86256],{"className":86257,"style":1823},[158],[15,86259],{"className":86260,"style":1840},[1839],[15,86262,86263,86266],{"style":1843},[15,86264],{"className":86265,"style":1823},[158],[15,86267,86269,86272,86275,86278,86281],{"className":86268},[129],[15,86270,56],{"className":86271},[129,133],[15,86273],{"className":86274,"style":3277},[93],[15,86276,315],{"className":86277},[2436],[15,86279],{"className":86280,"style":3277},[93],[15,86282,86284,86287],{"className":86283},[129],[15,86285,56],{"className":86286},[129,133],[15,86288,86290],{"className":86289},[137],[15,86291,86293,86313],{"className":86292},[141,142],[15,86294,86296,86310],{"className":86295},[146],[15,86297,86299],{"className":86298,"style":1981},[150],[15,86300,86301,86304],{"style":154},[15,86302],{"className":86303,"style":159},[158],[15,86305,86307],{"className":86306},[163,164,165,166],[15,86308,13949],{"className":86309,"style":13964},[129,133,166],[15,86311,175],{"className":86312},[174],[15,86314,86316],{"className":86315},[146],[15,86317,86319],{"className":86318,"style":14035},[150],[15,86320],{},[15,86322,175],{"className":86323},[174],[15,86325,86327],{"className":86326},[146],[15,86328,86330],{"className":86329,"style":1862},[150],[15,86331],{},[15,86333],{"className":86334},[195,1803],[15,86336,86338],{"className":86337},[195],[15,86339,59],{"className":86340},[228,24450],[15,86342,91],{"className":86343},[254],[11,86345,28686,86346,86416,86417,86445,86446,86537],{},[15,86347,86349,86367],{"className":86348,"translate":19},[23],[15,86350,86352],{"className":86351},[27],[29,86353,86354],{"xmlns":31},[34,86355,86356,86364],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是第 ",[15,86418,86420,86433],{"className":86419,"translate":19},[23],[15,86421,86423],{"className":86422},[27],[29,86424,86425],{"xmlns":31},[34,86426,86427,86431],{},[37,86428,86429],{},[43,86430,13949],{},[109,86432,13949],{"encoding":111},[15,86434,86436],{"className":86435,"ariaHidden":72},[116],[15,86437,86439,86442],{"className":86438},[120],[15,86440],{"className":86441,"style":11187},[124],[15,86443,13949],{"className":86444,"style":13964},[129,133]," 个方块的中心。对任意子集 ",[15,86447,86449,86479],{"className":86448,"translate":19},[23],[15,86450,86452],{"className":86451},[27],[29,86453,86454],{"xmlns":31},[34,86455,86456,86476],{},[37,86457,86458,86460,86462,86464,86466,86468,86470,86472,86474],{},[43,86459,940],{},[50,86461,5647],{},[50,86463,636],{"stretchy":52},[308,86465,318],{},[50,86467,91],{"separator":72},[50,86469,335],{},[50,86471,91],{"separator":72},[43,86473,348],{},[50,86475,645],{"stretchy":52},[109,86477,86478],{"encoding":111},"S\\subseteq\\{1,\\dots,m\\}",[15,86480,86482,86501],{"className":86481,"ariaHidden":72},[116],[15,86483,86485,86489,86492,86495,86498],{"className":86484},[120],[15,86486],{"className":86487,"style":86488},[124],"height:0.8193em;vertical-align:-0.136em;",[15,86490,940],{"className":86491,"style":1013},[129,133],[15,86493],{"className":86494,"style":199},[93],[15,86496,5647],{"className":86497},[203],[15,86499],{"className":86500,"style":199},[93],[15,86502,86504,86507,86510,86513,86516,86519,86522,86525,86528,86531,86534],{"className":86503},[120],[15,86505],{"className":86506,"style":125},[124],[15,86508,636],{"className":86509},[188],[15,86511,318],{"className":86512},[129],[15,86514,91],{"className":86515},[254],[15,86517],{"className":86518,"style":262},[93],[15,86520,335],{"className":86521},[449],[15,86523],{"className":86524,"style":262},[93],[15,86526,91],{"className":86527},[254],[15,86529],{"className":86530,"style":262},[93],[15,86532,348],{"className":86533},[129,133],[15,86535,645],{"className":86536},[195],"，定义",[15,86539,86541],{"className":86540,"translate":19},[18],[15,86542,86544,86604],{"className":86543,"translate":19},[23],[15,86545,86547],{"className":86546},[27],[29,86548,86549],{"xmlns":31,"display":32},[34,86550,86551,86601],{},[37,86552,86553,86559,86561,86563,86575,86581,86583,86585,86587,86589,86591,86593,86599],{},[40,86554,86555,86557],{},[43,86556,36556],{},[43,86558,940],{},[50,86560,62],{},[43,86562,8686],{},[2140,86564,86565,86567],{},[50,86566,1454],{},[37,86568,86569,86571,86573],{},[43,86570,13949],{},[50,86572,100],{},[43,86574,940],{},[40,86576,86577,86579],{},[43,86578,86067],{},[43,86580,13949],{},[50,86582,91],{"separator":72},[93,86584],{"width":95},[43,86586,8686],{},[50,86588,62],{},[43,86590,5628],{},[6852,86592,15781],{},[305,86594,86595,86597],{},[43,86596,45],{},[43,86598,36264],{},[43,86600,107],{"mathvariant":65},[109,86602,86603],{"encoding":111},"f_S=a\\sum_{j\\in S}\\varphi_j,\\qquad a=c\\,h^\\beta.",[15,86605,86607,86662,86793],{"className":86606,"ariaHidden":72},[116],[15,86608,86610,86613,86653,86656,86659],{"className":86609},[120],[15,86611],{"className":86612,"style":1682},[124],[15,86614,86616,86619],{"className":86615},[129],[15,86617,36556],{"className":86618,"style":36571},[129,133],[15,86620,86622],{"className":86621},[137],[15,86623,86625,86645],{"className":86624},[141,142],[15,86626,86628,86642],{"className":86627},[146],[15,86629,86631],{"className":86630,"style":1525},[150],[15,86632,86633,86636],{"style":39810},[15,86634],{"className":86635,"style":159},[158],[15,86637,86639],{"className":86638},[163,164,165,166],[15,86640,940],{"className":86641,"style":1013},[129,133,166],[15,86643,175],{"className":86644},[174],[15,86646,86648],{"className":86647},[146],[15,86649,86651],{"className":86650,"style":182},[150],[15,86652],{},[15,86654],{"className":86655,"style":199},[93],[15,86657,62],{"className":86658},[203],[15,86660],{"className":86661,"style":199},[93],[15,86663,86665,86669,86672,86675,86729,86732,86772,86775,86778,86781,86784,86787,86790],{"className":86664},[120],[15,86666],{"className":86667,"style":86668},[124],"height:2.4804em;vertical-align:-1.4304em;",[15,86670,8686],{"className":86671},[129,133],[15,86673],{"className":86674,"style":262},[93],[15,86676,86678],{"className":86677},[217,1874],[15,86679,86681,86720],{"className":86680},[141,142],[15,86682,86684,86717],{"className":86683},[146],[15,86685,86687,86707],{"className":86686,"style":12143},[150],[15,86688,86689,86692],{"style":12790},[15,86690],{"className":86691,"style":1891},[158],[15,86693,86695],{"className":86694},[163,164,165,166],[15,86696,86698,86701,86704],{"className":86697},[129,166],[15,86699,13949],{"className":86700,"style":13964},[129,133,166],[15,86702,100],{"className":86703},[203,166],[15,86705,940],{"className":86706,"style":1013},[129,133,166],[15,86708,86709,86712],{"style":1909},[15,86710],{"className":86711,"style":1891},[158],[15,86713,86714],{},[15,86715,1454],{"className":86716},[217,1918,1919],[15,86718,175],{"className":86719},[174],[15,86721,86723],{"className":86722},[146],[15,86724,86727],{"className":86725,"style":86726},[150],"height:1.4304em;",[15,86728],{},[15,86730],{"className":86731,"style":262},[93],[15,86733,86735,86738],{"className":86734},[129],[15,86736,86067],{"className":86737},[129,133],[15,86739,86741],{"className":86740},[137],[15,86742,86744,86764],{"className":86743},[141,142],[15,86745,86747,86761],{"className":86746},[146],[15,86748,86750],{"className":86749,"style":1981},[150],[15,86751,86752,86755],{"style":154},[15,86753],{"className":86754,"style":159},[158],[15,86756,86758],{"className":86757},[163,164,165,166],[15,86759,13949],{"className":86760,"style":13964},[129,133,166],[15,86762,175],{"className":86763},[174],[15,86765,86767],{"className":86766},[146],[15,86768,86770],{"className":86769,"style":14035},[150],[15,86771],{},[15,86773,91],{"className":86774},[254],[15,86776],{"className":86777,"style":258},[93],[15,86779],{"className":86780,"style":262},[93],[15,86782,8686],{"className":86783},[129,133],[15,86785],{"className":86786,"style":199},[93],[15,86788,62],{"className":86789},[203],[15,86791],{"className":86792,"style":199},[93],[15,86794,86796,86799,86802,86805,86834],{"className":86795},[120],[15,86797],{"className":86798,"style":64281},[124],[15,86800,5628],{"className":86801},[129,133],[15,86803],{"className":86804,"style":262},[93],[15,86806,86808,86811],{"className":86807},[129],[15,86809,45],{"className":86810},[129,133],[15,86812,86814],{"className":86813},[137],[15,86815,86817],{"className":86816},[141],[15,86818,86820],{"className":86819},[146],[15,86821,86823],{"className":86822,"style":64281},[150],[15,86824,86825,86828],{"style":1272},[15,86826],{"className":86827,"style":159},[158],[15,86829,86831],{"className":86830},[163,164,165,166],[15,86832,36264],{"className":86833,"style":36280},[129,133,166],[15,86835,107],{"className":86836},[129],[11,86838,17328,86839,86867,86868,86988,86989,87048,87049,15068,87077,87105,87106,1313],{},[15,86840,86842,86855],{"className":86841,"translate":19},[23],[15,86843,86845],{"className":86844},[27],[29,86846,86847],{"xmlns":31},[34,86848,86849,86853],{},[37,86850,86851],{},[43,86852,5628],{},[109,86854,5628],{"encoding":111},[15,86856,86858],{"className":86857,"ariaHidden":72},[116],[15,86859,86861,86864],{"className":86860},[120],[15,86862],{"className":86863,"style":4383},[124],[15,86865,5628],{"className":86866},[129,133]," 足够小时，",[15,86869,86871,86903],{"className":86870,"translate":19},[23],[15,86872,86874],{"className":86873},[27],[29,86875,86876],{"xmlns":31},[34,86877,86878,86900],{},[37,86879,86880,86886,86888,86890,86892,86894,86896,86898],{},[40,86881,86882,86884],{},[43,86883,36556],{},[43,86885,940],{},[50,86887,100],{},[43,86889,85347],{"mathvariant":65},[50,86891,53],{"stretchy":52},[43,86893,36264],{},[50,86895,91],{"separator":72},[43,86897,1373],{},[50,86899,59],{"stretchy":52},[109,86901,86902],{"encoding":111},"f_S\\in\\Sigma(\\beta,L)",[15,86904,86906,86961],{"className":86905,"ariaHidden":72},[116],[15,86907,86909,86912,86952,86955,86958],{"className":86908},[120],[15,86910],{"className":86911,"style":1682},[124],[15,86913,86915,86918],{"className":86914},[129],[15,86916,36556],{"className":86917,"style":36571},[129,133],[15,86919,86921],{"className":86920},[137],[15,86922,86924,86944],{"className":86923},[141,142],[15,86925,86927,86941],{"className":86926},[146],[15,86928,86930],{"className":86929,"style":1525},[150],[15,86931,86932,86935],{"style":39810},[15,86933],{"className":86934,"style":159},[158],[15,86936,86938],{"className":86937},[163,164,165,166],[15,86939,940],{"className":86940,"style":1013},[129,133,166],[15,86942,175],{"className":86943},[174],[15,86945,86947],{"className":86946},[146],[15,86948,86950],{"className":86949,"style":182},[150],[15,86951],{},[15,86953],{"className":86954,"style":199},[93],[15,86956,100],{"className":86957},[203],[15,86959],{"className":86960,"style":199},[93],[15,86962,86964,86967,86970,86973,86976,86979,86982,86985],{"className":86963},[120],[15,86965],{"className":86966,"style":125},[124],[15,86968,85347],{"className":86969},[129],[15,86971,53],{"className":86972},[188],[15,86974,36264],{"className":86975,"style":36280},[129,133],[15,86977,91],{"className":86978},[254],[15,86980],{"className":86981,"style":262},[93],[15,86983,1373],{"className":86984},[129,133],[15,86986,59],{"className":86987},[195],"：鼓包的高度是 ",[15,86990,86992,87010],{"className":86991,"translate":19},[23],[15,86993,86995],{"className":86994},[27],[29,86996,86997],{"xmlns":31},[34,86998,86999,87007],{},[37,87000,87001],{},[305,87002,87003,87005],{},[43,87004,45],{},[43,87006,36264],{},[109,87008,87009],{"encoding":111},"h^\\beta",[15,87011,87013],{"className":87012,"ariaHidden":72},[116],[15,87014,87016,87019],{"className":87015},[120],[15,87017],{"className":87018,"style":10486},[124],[15,87020,87022,87025],{"className":87021},[129],[15,87023,45],{"className":87024},[129,133],[15,87026,87028],{"className":87027},[137],[15,87029,87031],{"className":87030},[141],[15,87032,87034],{"className":87033},[146],[15,87035,87037],{"className":87036,"style":10486},[150],[15,87038,87039,87042],{"style":383},[15,87040],{"className":87041,"style":159},[158],[15,87043,87045],{"className":87044},[163,164,165,166],[15,87046,36264],{"className":87047,"style":36280},[129,133,166],"、宽度是 ",[15,87050,87052,87065],{"className":87051,"translate":19},[23],[15,87053,87055],{"className":87054},[27],[29,87056,87057],{"xmlns":31},[34,87058,87059,87063],{},[37,87060,87061],{},[43,87062,45],{},[109,87064,45],{"encoding":111},[15,87066,87068],{"className":87067,"ariaHidden":72},[116],[15,87069,87071,87074],{"className":87070},[120],[15,87072],{"className":87073,"style":522},[124],[15,87075,45],{"className":87076},[129,133],[15,87078,87080,87093],{"className":87079,"translate":19},[23],[15,87081,87083],{"className":87082},[27],[29,87084,87085],{"xmlns":31},[34,87086,87087,87091],{},[37,87088,87089],{},[43,87090,36264],{},[109,87092,36267],{"encoding":111},[15,87094,87096],{"className":87095,"ariaHidden":72},[116],[15,87097,87099,87102],{"className":87098},[120],[15,87100],{"className":87101,"style":1682},[124],[15,87103,36264],{"className":87104,"style":36280},[129,133]," 阶导数的量级是 ",[15,87107,87109,87139],{"className":87108,"translate":19},[23],[15,87110,87112],{"className":87111},[27],[29,87113,87114],{"xmlns":31},[34,87115,87116,87136],{},[37,87117,87118,87124,87126,87132,87134],{},[305,87119,87120,87122],{},[43,87121,45],{},[43,87123,36264],{},[43,87125,544],{"mathvariant":65},[305,87127,87128,87130],{},[43,87129,45],{},[43,87131,36264],{},[50,87133,62],{},[308,87135,318],{},[109,87137,87138],{"encoding":111},"h^\\beta\u002Fh^\\beta=1",[15,87140,87142,87218],{"className":87141,"ariaHidden":72},[116],[15,87143,87145,87148,87177,87180,87209,87212,87215],{"className":87144},[120],[15,87146],{"className":87147,"style":10974},[124],[15,87149,87151,87154],{"className":87150},[129],[15,87152,45],{"className":87153},[129,133],[15,87155,87157],{"className":87156},[137],[15,87158,87160],{"className":87159},[141],[15,87161,87163],{"className":87162},[146],[15,87164,87166],{"className":87165,"style":10486},[150],[15,87167,87168,87171],{"style":383},[15,87169],{"className":87170,"style":159},[158],[15,87172,87174],{"className":87173},[163,164,165,166],[15,87175,36264],{"className":87176,"style":36280},[129,133,166],[15,87178,544],{"className":87179},[129],[15,87181,87183,87186],{"className":87182},[129],[15,87184,45],{"className":87185},[129,133],[15,87187,87189],{"className":87188},[137],[15,87190,87192],{"className":87191},[141],[15,87193,87195],{"className":87194},[146],[15,87196,87198],{"className":87197,"style":10486},[150],[15,87199,87200,87203],{"style":383},[15,87201],{"className":87202,"style":159},[158],[15,87204,87206],{"className":87205},[163,164,165,166],[15,87207,36264],{"className":87208,"style":36280},[129,133,166],[15,87210],{"className":87211,"style":199},[93],[15,87213,62],{"className":87214},[203],[15,87216],{"className":87217,"style":199},[93],[15,87219,87221,87224],{"className":87220},[120],[15,87222],{"className":87223,"style":4886},[124],[15,87225,318],{"className":87226},[129],[11,87228,87229,87230,6797,87390,87449,87450,22158],{},"两个函数 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距离只取决于它们不同的那些鼓包。若 ",[15,87451,87453,87484],{"className":87452,"translate":19},[23],[15,87454,87456],{"className":87455},[27],[29,87457,87458],{"xmlns":31},[34,87459,87460,87481],{},[37,87461,87462,87464,87466,87469,87475,87477,87479],{},[43,87463,2945],{"mathvariant":65},[43,87465,940],{},[43,87467,87468],{"mathvariant":65},"△",[305,87470,87471,87473],{},[43,87472,940],{},[50,87474,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[43,87476,2945],{"mathvariant":65},[50,87478,62],{},[43,87480,16555],{},[109,87482,87483],{"encoding":111},"|S\\triangle 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",[15,88328,88330,88360],{"className":88329,"translate":19},[23],[15,88331,88333],{"className":88332},[27],[29,88334,88335],{"xmlns":31},[34,88336,88337,88357],{},[37,88338,88339,88341,88343],{},[43,88340,43234],{},[50,88342,62],{},[305,88344,88345,88347],{},[308,88346,310],{},[37,88348,88349,88355],{},[40,88350,88351,88353],{},[43,88352,5628],{},[308,88354,541],{},[43,88356,348],{},[109,88358,88359],{"encoding":111},"M=2^{c_0m}",[15,88361,88363,88381],{"className":88362,"ariaHidden":72},[116],[15,88364,88366,88369,88372,88375,88378],{"className":88365},[120],[15,88367],{"className":88368,"style":610},[124],[15,88370,43234],{"className":88371,"style":12591},[129,133],[15,88373],{"className":88374,"style":199},[93],[15,88376,62],{"className":88377},[203],[15,88379],{"className":88380,"style":199},[93],[15,88382,88384,88387],{"className":88383},[120],[15,88385],{"className":88386,"style":6377},[124],[15,88388,88390,88393],{"className":88389},[129],[15,88391,310],{"className":88392},[129],[15,88394,88396],{"className":88395},[137],[15,88397,88399],{"className":88398},[141],[15,88400,88402],{"className":88401},[146],[15,88403,88405],{"className":88404,"style":6377},[150],[15,88406,88407,88410],{"style":383},[15,88408],{"className":88409,"style":159},[158],[15,88411,88413],{"className":88412},[163,164,165,166],[15,88414,88416,88456],{"className":88415},[129,166],[15,88417,88419,88422],{"className":88418},[129,166],[15,88420,5628],{"className":88421},[129,133,166],[15,88423,88425],{"className":88424},[137],[15,88426,88428,88448],{"className":88427},[141,142],[15,88429,88431,88445],{"className":88430},[146],[15,88432,88434],{"className":88433,"style":46983},[150],[15,88435,88436,88439],{"style":84038},[15,88437],{"className":88438,"style":15980},[158],[15,88440,88442],{"className":88441},[163,15984,229,166],[15,88443,541],{"className":88444},[129,166],[15,88446,175],{"className":88447},[174],[15,88449,88451],{"className":88450},[146],[15,88452,88454],{"className":88453,"style":15997},[150],[15,88455],{},[15,88457,348],{"className":88458},[129,133,166]," 个码字（",[15,88461,88463,88485],{"className":88462,"translate":19},[23],[15,88464,88466],{"className":88465},[27],[29,88467,88468],{"xmlns":31},[34,88469,88470,88482],{},[37,88471,88472,88478,88480],{},[40,88473,88474,88476],{},[43,88475,5628],{},[308,88477,541],{},[50,88479,4852],{},[308,88481,541],{},[109,88483,88484],{"encoding":111},"c_0>0",[15,88486,88488,88543],{"className":88487,"ariaHidden":72},[116],[15,88489,88491,88494,88534,88537,88540],{"className":88490},[120],[15,88492],{"className":88493,"style":7925},[124],[15,88495,88497,88500],{"className":88496},[129],[15,88498,5628],{"className":88499},[129,133],[15,88501,88503],{"className":88502},[137],[15,88504,88506,88526],{"className":88505},[141,142],[15,88507,88509,88523],{"className":88508},[146],[15,88510,88512],{"className":88511,"style":1051},[150],[15,88513,88514,88517],{"style":154},[15,88515],{"className":88516,"style":159},[158],[15,88518,88520],{"className":88519},[163,164,165,166],[15,88521,541],{"className":88522},[129,166],[15,88524,175],{"className":88525},[174],[15,88527,88529],{"className":88528},[146],[15,88530,88532],{"className":88531,"style":182},[150],[15,88533],{},[15,88535],{"className":88536,"style":199},[93],[15,88538,4852],{"className":88539},[203],[15,88541],{"className":88542,"style":199},[93],[15,88544,88546,88549],{"className":88545},[120],[15,88547],{"className":88548,"style":4886},[124],[15,88550,541],{"className":88551},[129]," 是常数），任意两个码字至少在 ",[15,88554,88556,88574],{"className":88555,"translate":19},[23],[15,88557,88559],{"className":88558},[27],[29,88560,88561],{"xmlns":31},[34,88562,88563,88571],{},[37,88564,88565,88567,88569],{},[43,88566,348],{},[43,88568,544],{"mathvariant":65},[308,88570,14502],{},[109,88572,88573],{"encoding":111},"m\u002F4",[15,88575,88577],{"className":88576,"ariaHidden":72},[116],[15,88578,88580,88583,88586],{"className":88579},[120],[15,88581],{"className":88582,"style":125},[124],[15,88584,348],{"className":88585},[129,133],[15,88587,64733],{"className":88588},[129]," 个位置不同。于是",[15,88591,88593],{"className":88592,"translate":19},[18],[15,88594,88596,88660],{"className":88595,"translate":19},[23],[15,88597,88599],{"className":88598},[27],[29,88600,88601],{"xmlns":31,"display":32},[34,88602,88603,88657],{},[37,88604,88605,88607,88613,88615,88625,88637,88639,88645,88655],{},[50,88606,33810],{"stretchy":52},[40,88608,88609,88611],{},[43,88610,36556],{},[43,88612,940],{},[50,88614,315],{},[40,88616,88617,88619],{},[43,88618,36556],{},[305,88620,88621,88623],{},[43,88622,940],{},[50,88624,21499],{"mathvariant":65,"lspace":2604,"rspace":2604},[23149,88626,88627,88629,88635],{},[50,88628,33810],{"stretchy":52},[305,88630,88631,88633],{},[43,88632,1373],{},[308,88634,310],{},[308,88636,310],{},[50,88638,7469],{},[40,88640,88641,88643],{},[43,88642,5628],{},[308,88644,318],{},[305,88646,88647,88649],{},[43,88648,45],{},[37,88650,88651,88653],{},[308,88652,310],{},[43,88654,36264],{},[50,88656,91],{"separator":72},[109,88658,88659],{"encoding":111},"\\lVert f_S-f_{S'}\\rVert_{L^2}^2\n\\ge c_1h^{2\\beta},",[15,88661,88663,88721,88888],{"className":88662,"ariaHidden":72},[116],[15,88664,88666,88669,88672,88712,88715,88718],{"className":88665},[120],[15,88667],{"className":88668,"style":125},[124],[15,88670,33810],{"className":88671},[188],[15,88673,88675,88678],{"className":88674},[129],[15,88676,36556],{"className":88677,"style":36571},[129,133],[15,88679,88681],{"className":88680},[137],[15,88682,88684,88704],{"className":88683},[141,142],[15,88685,88687,88701],{"className":88686},[146],[15,88688,88690],{"className":88689,"style":1525},[150],[15,88691,88692,88695],{"style":39810},[15,88693],{"className":88694,"style":159},[158],[15,88696,88698],{"className":88697},[163,164,165,166],[15,88699,940],{"className":88700,"style":1013},[129,133,166],[15,88702,175],{"className":88703},[174],[15,88705,88707],{"className":88706},[146],[15,88708,88710],{"className":88709,"style":182},[150],[15,88711],{},[15,88713],{"className":88714,"style":3277},[93],[15,88716,315],{"className":88717},[2436],[15,88719],{"className":88720,"style":3277},[93],[15,88722,88724,88727,88799,88879,88882,88885],{"className":88723},[120],[15,88725],{"className":88726,"style":87754},[124],[15,88728,88730,88733],{"className":88729},[129],[15,88731,36556],{"className":88732,"style":36571},[129,133],[15,88734,88736],{"className":88735},[137],[15,88737,88739,88791],{"className":88738},[141,142],[15,88740,88742,88788],{"className":88741},[146],[15,88743,88745],{"className":88744,"style":1525},[150],[15,88746,88747,88750],{"style":39810},[15,88748],{"className":88749,"style":159},[158],[15,88751,88753],{"className":88752},[163,164,165,166],[15,88754,88756],{"className":88755},[129,166],[15,88757,88759,88762],{"className":88758},[129,166],[15,88760,940],{"className":88761,"style":1013},[129,133,166],[15,88763,88765],{"className":88764},[137],[15,88766,88768],{"className":88767},[141],[15,88769,88771],{"className":88770},[146],[15,88772,88774],{"className":88773,"style":23264},[150],[15,88775,88776,88779],{"style":23267},[15,88777],{"className":88778,"style":15980},[158],[15,88780,88782],{"className":88781},[163,15984,229,166],[15,88783,88785],{"className":88784},[129,166],[15,88786,21499],{"className":88787},[129,166],[15,88789,175],{"className":88790},[174],[15,88792,88794],{"className":88793},[146],[15,88795,88797],{"className":88796,"style":182},[150],[15,88798],{},[15,88800,88802,88805],{"className":88801},[195],[15,88803,33810],{"className":88804},[195],[15,88806,88808],{"className":88807},[137],[15,88809,88811,88871],{"className":88810},[141,142],[15,88812,88814,88868],{"className":88813},[146],[15,88815,88817,88857],{"className":88816,"style":40979},[150],[15,88818,88819,88822],{"style":87847},[15,88820],{"className":88821,"style":159},[158],[15,88823,88825],{"className":88824},[163,164,165,166],[15,88826,88828],{"className":88827},[129,166],[15,88829,88831,88834],{"className":88830},[129,166],[15,88832,1373],{"className":88833},[129,133,166],[15,88835,88837],{"className":88836},[137],[15,88838,88840],{"className":88839},[141],[15,88841,88843],{"className":88842},[146],[15,88844,88846],{"className":88845,"style":85553},[150],[15,88847,88848,88851],{"style":23267},[15,88849],{"className":88850,"style":15980},[158],[15,88852,88854],{"className":88853},[163,15984,229,166],[15,88855,310],{"className":88856},[129,166],[15,88858,88859,88862],{"style":1272},[15,88860],{"className":88861,"style":159},[158],[15,88863,88865],{"className":88864},[163,164,165,166],[15,88866,310],{"className":88867},[129,166],[15,88869,175],{"className":88870},[174],[15,88872,88874],{"className":88873},[146],[15,88875,88877],{"className":88876,"style":87906},[150],[15,88878],{},[15,88880],{"className":88881,"style":199},[93],[15,88883,7469],{"className":88884},[203],[15,88886],{"className":88887,"style":199},[93],[15,88889,88891,88894,88934,88969],{"className":88890},[120],[15,88892],{"className":88893,"style":82048},[124],[15,88895,88897,88900],{"className":88896},[129],[15,88898,5628],{"className":88899},[129,133],[15,88901,88903],{"className":88902},[137],[15,88904,88906,88926],{"className":88905},[141,142],[15,88907,88909,88923],{"className":88908},[146],[15,88910,88912],{"className":88911,"style":1051},[150],[15,88913,88914,88917],{"style":154},[15,88915],{"className":88916,"style":159},[158],[15,88918,88920],{"className":88919},[163,164,165,166],[15,88921,318],{"className":88922},[129,166],[15,88924,175],{"className":88925},[174],[15,88927,88929],{"className":88928},[146],[15,88930,88932],{"className":88931,"style":182},[150],[15,88933],{},[15,88935,88937,88940],{"className":88936},[129],[15,88938,45],{"className":88939},[129,133],[15,88941,88943],{"className":88942},[137],[15,88944,88946],{"className":88945},[141],[15,88947,88949],{"className":88948},[146],[15,88950,88952],{"className":88951,"style":64281},[150],[15,88953,88954,88957],{"style":1272},[15,88955],{"className":88956,"style":159},[158],[15,88958,88960],{"className":88959},[163,164,165,166]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要求 ",[15,90073,90075,90107],{"className":90074,"translate":19},[23],[15,90076,90078],{"className":90077},[27],[29,90079,90080],{"xmlns":31},[34,90081,90082,90104],{},[37,90083,90084,90086,90088,90090,90092,90094],{},[43,90085,14721],{},[50,90087,69],{},[43,90089,43234],{},[50,90091,74401],{},[43,90093,39314],{},[305,90095,90096,90098],{},[43,90097,45],{},[37,90099,90100,90102],{},[308,90101,310],{},[43,90103,36264],{},[109,90105,90106],{"encoding":111},"\\log M\\gg n h^{2\\beta}",[15,90108,90110,90136],{"className":90109,"ariaHidden":72},[116],[15,90111,90113,90116,90121,90124,90127,90130,90133],{"className":90112},[120],[15,90114],{"className":90115,"style":1682},[124],[15,90117,14771,90119],{"className":90118},[217],[15,90120,2247],{"style":2246},[15,90122],{"className":90123,"style":262},[93],[15,90125,43234],{"className":90126,"style":12591},[129,133],[15,90128],{"className":90129,"style":199},[93],[15,90131,74401],{"className":90132},[203],[15,90134],{"className":90135,"style":199},[93],[15,90137,90139,90142,90145],{"className":90138},[120],[15,90140],{"className":90141,"style":10486},[124],[15,90143,39314],{"className":90144},[129,133],[15,90146,90148,90151],{"className":90147},[129],[15,90149,45],{"className":90150},[129,133],[15,90152,90154],{"className":90153},[137],[15,90155,90157],{"className":90156},[141],[15,90158,90160],{"className":90159},[146],[15,90161,90163],{"className":90162,"style":10486},[150],[15,90164,90165,90168],{"style":383},[15,90166],{"className":90167,"style":159},[158],[15,90169,90171],{"className":90170},[163,164,165,166],[15,90172,90174,90177],{"className":90173},[129,166],[15,90175,310],{"className":90176},[129,166],[15,90178,36264],{"className":90179,"style":36280},[129,133,166],"，即",[15,90182,90184],{"className":90183,"translate":19},[18],[15,90185,90187,90258],{"className":90186,"translate":19},[23],[15,90188,90190],{"className":90189},[27],[29,90191,90192],{"xmlns":31,"display":32},[34,90193,90194,90255],{},[37,90195,90196,90206,90208,90210,90220,90222,90225,90227,90229,90231,90253],{},[305,90197,90198,90200],{},[43,90199,45],{},[37,90201,90202,90204],{},[50,90203,315],{},[43,90205,10372],{},[50,90207,74465],{},[43,90209,39314],{},[305,90211,90212,90214],{},[43,90213,45],{},[37,90215,90216,90218],{},[308,90217,310],{},[43,90219,36264],{},[93,90221],{"width":16914},[50,90223,90224],{},"⟺",[93,90226],{"width":16914},[43,90228,45],{},[50,90230,26556],{},[305,90232,90233,90235],{},[43,90234,39314],{},[37,90236,90237,90239],{},[50,90238,315],{},[1443,90240,90241,90243],{},[308,90242,318],{},[37,90244,90245,90247,90249,90251],{},[308,90246,310],{},[43,90248,36264],{},[50,90250,2682],{},[43,90252,10372],{},[43,90254,107],{"mathvariant":65},[109,90256,90257],{"encoding":111},"h^{-d}\\gtrsim n h^{2\\beta}\n\\quad\\Longleftrightarrow\\quad\nh\\lesssim n^{-\\frac{1}{2\\beta+d}}.",[15,90259,90261,90312,90372,90391],{"className":90260,"ariaHidden":72},[116],[15,90262,90264,90268,90303,90306,90309],{"className":90263},[120],[15,90265],{"className":90266,"style":90267},[124],"height:1.1287em;vertical-align:-0.2296em;",[15,90269,90271,90274],{"className":90270},[129],[15,90272,45],{"className":90273},[129,133],[15,90275,90277],{"className":90276},[137],[15,90278,90280],{"className":90279},[141],[15,90281,90283],{"className":90282},[146],[15,90284,90286],{"className":90285,"style":64281},[150],[15,90287,90288,90291],{"style":1272},[15,90289],{"className":90290,"style":159},[158],[15,90292,90294],{"className":90293},[163,164,165,166],[15,90295,90297,90300],{"className":90296},[129,166],[15,90298,315],{"className":90299},[129,166],[15,90301,10372],{"className":90302},[129,133,166],[15,90304],{"className":90305,"style":199},[93],[15,90307,74465],{"className":90308},[203,13310],[15,90310],{"className":90311,"style":199},[93],[15,90313,90315,90319,90322,90357,90360,90363,90366,90369],{"className":90314},[120],[15,90316],{"className":90317,"style":90318},[124],"height:0.9231em;vertical-align:-0.024em;",[15,90320,39314],{"className":90321},[129,133],[15,90323,90325,90328],{"className":90324},[129],[15,90326,45],{"className":90327},[129,133],[15,90329,90331],{"className":90330},[137],[15,90332,90334],{"className":90333},[141],[15,90335,90337],{"className":90336},[146],[15,90338,90340],{"className":90339,"style":64281},[150],[15,90341,90342,90345],{"style":1272},[15,90343],{"className":90344,"style":159},[158],[15,90346,90348],{"className":90347},[163,164,165,166],[15,90349,90351,90354],{"className":90350},[129,166],[15,90352,310],{"className":90353},[129,166],[15,90355,36264],{"className":90356,"style":36280},[129,133,166],[15,90358],{"className":90359,"style":17287},[93],[15,90361],{"className":90362,"style":199},[93],[15,90364,90224],{"className":90365},[203],[15,90367],{"className":90368,"style":17287},[93],[15,90370],{"className":90371,"style":199},[93],[15,90373,90375,90379,90382,90385,90388],{"className":90374},[120],[15,90376],{"className":90377,"style":90378},[124],"height:0.9592em;vertical-align:-0.2296em;",[15,90380,45],{"className":90381},[129,133],[15,90383],{"className":90384,"style":199},[93],[15,90386,26556],{"className":90387},[203,13310],[15,90389],{"className":90390,"style":199},[93],[15,90392,90394,90398,90509],{"className":90393},[120],[15,90395],{"className":90396,"style":90397},[124],"height:1.0368em;",[15,90399,90401,90404],{"className":90400},[129],[15,90402,39314],{"className":90403},[129,133],[15,90405,90407],{"className":90406},[137],[15,90408,90410],{"className":90409},[141],[15,90411,90413],{"className":90412},[146],[15,90414,90416],{"className":90415,"style":90397},[150],[15,90417,90418,90421],{"style":85667},[15,90419],{"className":90420,"style":1823},[158],[15,90422,90424],{"className":90423},[163,164,165,166],[15,90425,90427,90430],{"className":90426},[129,166],[15,90428,315],{"className":90429},[129,166],[15,90431,90433,90436,90506],{"className":90432},[129,166],[15,90434],{"className":90435},[188,1803,163,15984,85686],[15,90437,90439],{"className":90438},[1443],[15,90440,90442,90498],{"className":90441},[141,142],[15,90443,90445,90495],{"className":90444},[146],[15,90446,90449,90472,90480],{"className":90447,"style":90448},[150],"height:0.8443em;",[15,90450,90451,90454],{"style":85702},[15,90452],{"className":90453,"style":1823},[158],[15,90455,90457],{"className":90456},[163,15984,229,166],[15,90458,90460,90463,90466,90469],{"className":90459},[129,166],[15,90461,310],{"className":90462},[129,166],[15,90464,36264],{"className":90465,"style":36280},[129,133,166],[15,90467,2682],{"className":90468},[2436,166],[15,90470,10372],{"className":90471},[129,133,166],[15,90473,90474,90477],{"style":85726},[15,90475],{"className":90476,"style":1823},[158],[15,90478],{"className":90479,"style":85733},[1839,166],[15,90481,90483,90486],{"style":90482},"top:-3.384em;",[15,90484],{"className":90485,"style":1823},[158],[15,90487,90489],{"className":90488},[163,15984,229,166],[15,90490,90492],{"className":90491},[129,166],[15,90493,318],{"className":90494},[129,166],[15,90496,175],{"className":90497},[174],[15,90499,90501],{"className":90500},[146],[15,90502,90504],{"className":90503,"style":85761},[150],[15,90505],{},[15,90507],{"className":90508},[195,1803,163,15984,85686],[15,90510,107],{"className":90511},[129],[11,90513,7437,90514,90637],{},[15,90515,90517,90557],{"className":90516,"translate":19},[23],[15,90518,90520],{"className":90519},[27],[29,90521,90522],{"xmlns":31},[34,90523,90524,90554],{},[37,90525,90526,90528,90530],{},[43,90527,45],{},[50,90529,64751],{},[305,90531,90532,90534],{},[43,90533,39314],{},[37,90535,90536,90538,90540,90542,90544,90546,90548,90550,90552],{},[50,90537,315],{},[308,90539,318],{},[43,90541,544],{"mathvariant":65},[50,90543,53],{"stretchy":52},[308,90545,310],{},[43,90547,36264],{},[50,90549,2682],{},[43,90551,1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n^{-1\u002F(2\\beta+d)}",[15,90558,90560,90578],{"className":90559,"ariaHidden":72},[116],[15,90561,90563,90566,90569,90572,90575],{"className":90562},[120],[15,90564],{"className":90565,"style":522},[124],[15,90567,45],{"className":90568},[129,133],[15,90570],{"className":90571,"style":199},[93],[15,90573,64751],{"className":90574},[203],[15,90576],{"className":90577,"style":199},[93],[15,90579,90581,90584],{"className":90580},[120],[15,90582],{"className":90583,"style":14941},[124],[15,90585,90587,90590],{"className":90586},[129],[15,90588,39314],{"className":90589},[129,133],[15,90591,90593],{"className":90592},[137],[15,90594,90596],{"className":90595},[141],[15,90597,90599],{"className":90598},[146],[15,90600,90602],{"className":90601,"style":14941},[150],[15,90603,90604,90607],{"style":383},[15,90605],{"className":90606,"style":159},[158],[15,90608,90610],{"className":90609},[163,164,165,166],[15,90611,90613,90616,90619,90622,90625,90628,90631,90634],{"className":90612},[129,166],[15,90614,315],{"className":90615},[129,166],[15,90617,29841],{"className":90618},[129,166],[15,90620,53],{"className":90621},[188,166],[15,90623,310],{"className":90624},[129,166],[15,90626,36264],{"className":90627,"style":36280},[129,133,166],[15,90629,2682],{"className":90630},[2436,166],[15,90632,10372],{"className":90633},[129,133,166],[15,90635,59],{"className":90636},[195,166],"，下界就是",[15,90639,90641],{"className":90640,"translate":19},[18],[15,90642,90644,90704],{"className":90643,"translate":19},[23],[15,90645,90647],{"className":90646},[27],[29,90648,90649],{"xmlns":31,"display":32},[34,90650,90651,90701],{},[37,90652,90653,90659,90661,90671,90673,90699],{},[40,90654,90655,90657],{},[43,90656,43234],{"mathvariant":28899},[43,90658,39314],{},[50,90660,74465],{},[305,90662,90663,90665],{},[43,90664,45],{},[37,90666,90667,90669],{},[308,90668,310],{},[43,90670,36264],{},[50,90672,64751],{},[305,90674,90675,90677],{},[43,90676,39314],{},[37,90678,90679,90681],{},[50,90680,315],{},[1443,90682,90683,90689],{},[37,90684,90685,90687],{},[308,90686,310],{},[43,90688,36264],{},[37,90690,90691,90693,90695,90697],{},[308,90692,310],{},[43,90694,36264],{},[50,90696,2682],{},[43,90698,10372],{},[50,90700,91],{"separator":72},[109,90702,90703],{"encoding":111},"\\mathfrak M_n\\gtrsim h^{2\\beta}\n\\asymp n^{-\\frac{2\\beta}{2\\beta+d}},",[15,90705,90707,90762,90812],{"className":90706,"ariaHidden":72},[116],[15,90708,90710,90713,90753,90756,90759],{"className":90709},[120],[15,90711],{"className":90712,"style":90378},[124],[15,90714,90716,90719],{"className":90715},[129],[15,90717,43234],{"className":90718},[129,28926],[15,90720,90722],{"className":90721},[137],[15,90723,90725,90745],{"className":90724},[141,142],[15,90726,90728,90742],{"className":90727},[146],[15,90729,90731],{"className":90730,"style":1163},[150],[15,90732,90733,90736],{"style":154},[15,90734],{"className":90735,"style":159},[158],[15,90737,90739],{"className":90738},[163,164,165,166],[15,90740,39314],{"className":90741},[129,133,166],[15,90743,175],{"className":90744},[174],[15,90746,90748],{"className":90747},[146],[15,90749,90751],{"className":90750,"style":182},[150],[15,90752],{},[15,90754],{"className":90755,"style":199},[93],[15,90757,74465],{"className":90758},[203,13310],[15,90760],{"className":90761,"style":199},[93],[15,90763,90765,90768,90803,90806,90809],{"className":90764},[120],[15,90766],{"className":90767,"style":64281},[124],[15,90769,90771,90774],{"className":90770},[129],[15,90772,45],{"className":90773},[129,133],[15,90775,90777],{"className":90776},[137],[15,90778,90780],{"className":90779},[141],[15,90781,90783],{"className":90782},[146],[15,90784,90786],{"className":90785,"style":64281},[150],[15,90787,90788,90791],{"style":1272},[15,90789],{"className":90790,"style":159},[158],[15,90792,90794],{"className":90793},[163,164,165,166],[15,90795,90797,90800],{"className":90796},[129,166],[15,90798,310],{"className":90799},[129,166],[15,90801,36264],{"className":90802,"style":36280},[129,133,166],[15,90804],{"className":90805,"style":199},[93],[15,90807,64751],{"className":90808},[203],[15,90810],{"className":90811,"style":199},[93],[15,90813,90815,90819,90931],{"className":90814},[120],[15,90816],{"className":90817,"style":90818},[124],"height:1.3111em;vertical-align:-0.1944em;",[15,90820,90822,90825],{"className":90821},[129],[15,90823,39314],{"className":90824},[129,133],[15,90826,90828],{"className":90827},[137],[15,90829,90831],{"className":90830},[141],[15,90832,90834],{"className":90833},[146],[15,90835,90837],{"className":90836,"style":85646},[150],[15,90838,90839,90842],{"style":85667},[15,90840],{"className":90841,"style":1823},[158],[15,90843,90845],{"className":90844},[163,164,165,166],[15,90846,90848,90851],{"className":90847},[129,166],[15,90849,315],{"className":90850},[129,166],[15,90852,90854,90857,90928],{"className":90853},[129,166],[15,90855],{"className":90856},[188,1803,163,15984,85686],[15,90858,90860],{"className":90859},[1443],[15,90861,90863,90920],{"className":90862},[141,142],[15,90864,90866,90917],{"className":90865},[146],[15,90867,90869,90892,90900],{"className":90868,"style":85699},[150],[15,90870,90871,90874],{"style":85702},[15,90872],{"className":90873,"style":1823},[158],[15,90875,90877],{"className":90876},[163,15984,229,166],[15,90878,90880,90883,90886,90889],{"className":90879},[129,166],[15,90881,310],{"className":90882},[129,166],[15,90884,36264],{"className":90885,"style":36280},[129,133,166],[15,90887,2682],{"className":90888},[2436,166],[15,90890,10372],{"className":90891},[129,133,166],[15,90893,90894,90897],{"style":85726},[15,90895],{"className":90896,"style":1823},[158],[15,90898],{"className":90899,"style":85733},[1839,166],[15,90901,90902,90905],{"style":85736},[15,90903],{"className":90904,"style":1823},[158],[15,90906,90908],{"className":90907},[163,15984,229,166],[15,90909,90911,90914],{"className":90910},[129,166],[15,90912,310],{"className":90913},[129,166],[15,90915,36264],{"className":90916,"style":36280},[129,133,166],[15,90918,175],{"className":90919},[174],[15,90921,90923],{"className":90922},[146],[15,90924,90926],{"className":90925,"style":85761},[150],[15,90927],{},[15,90929],{"className":90930},[195,1803,163,15984,85686],[15,90932,91],{"className":90933},[254],[11,90935,90936],{},"与上界同阶。证毕。",[11,90938,90939,90940,91187,91188,91216],{},"这个构造值得反复看：",[698,90941,90942,90943,90971,90972,91041,91042,91116,91117,91186],{},"鼓包的宽度 ",[15,90944,90946,90959],{"className":90945,"translate":19},[23],[15,90947,90949],{"className":90948},[27],[29,90950,90951],{"xmlns":31},[34,90952,90953,90957],{},[37,90954,90955],{},[43,90956,45],{},[109,90958,45],{"encoding":111},[15,90960,90962],{"className":90961,"ariaHidden":72},[116],[15,90963,90965,90968],{"className":90964},[120],[15,90966],{"className":90967,"style":522},[124],[15,90969,45],{"className":90970},[129,133]," 控制两件事——损失间隔（",[15,90973,90975,90997],{"className":90974,"translate":19},[23],[15,90976,90978],{"className":90977},[27],[29,90979,90980],{"xmlns":31},[34,90981,90982,90994],{},[37,90983,90984],{},[305,90985,90986,90988],{},[43,90987,45],{},[37,90989,90990,90992],{},[308,90991,310],{},[43,90993,36264],{},[109,90995,90996],{"encoding":111},"h^{2\\beta}",[15,90998,91000],{"className":90999,"ariaHidden":72},[116],[15,91001,91003,91006],{"className":91002},[120],[15,91004],{"className":91005,"style":10486},[124],[15,91007,91009,91012],{"className":91008},[129],[15,91010,45],{"className":91011},[129,133],[15,91013,91015],{"className":91014},[137],[15,91016,91018],{"className":91017},[141],[15,91019,91021],{"className":91020},[146],[15,91022,91024],{"className":91023,"style":10486},[150],[15,91025,91026,91029],{"style":383},[15,91027],{"className":91028,"style":159},[158],[15,91030,91032],{"className":91031},[163,164,165,166],[15,91033,91035,91038],{"className":91034},[129,166],[15,91036,310],{"className":91037},[129,166],[15,91039,36264],{"className":91040,"style":36280},[129,133,166],"）和 KL（",[15,91043,91045,91069],{"className":91044,"translate":19},[23],[15,91046,91048],{"className":91047},[27],[29,91049,91050],{"xmlns":31},[34,91051,91052,91066],{},[37,91053,91054,91056],{},[43,91055,39314],{},[305,91057,91058,91060],{},[43,91059,45],{},[37,91061,91062,91064],{},[308,91063,310],{},[43,91065,36264],{},[109,91067,91068],{"encoding":111},"n h^{2\\beta}",[15,91070,91072],{"className":91071,"ariaHidden":72},[116],[15,91073,91075,91078,91081],{"className":91074},[120],[15,91076],{"className":91077,"style":10486},[124],[15,91079,39314],{"className":91080},[129,133],[15,91082,91084,91087],{"className":91083},[129],[15,91085,45],{"className":91086},[129,133],[15,91088,91090],{"className":91089},[137],[15,91091,91093],{"className":91092},[141],[15,91094,91096],{"className":91095},[146],[15,91097,91099],{"className":91098,"style":10486},[150],[15,91100,91101,91104],{"style":383},[15,91102],{"className":91103,"style":159},[158],[15,91105,91107],{"className":91106},[163,164,165,166],[15,91108,91110,91113],{"className":91109},[129,166],[15,91111,310],{"className":91112},[129,166],[15,91114,36264],{"className":91115,"style":36280},[129,133,166],"），而鼓包的个数 ",[15,91118,91120,91142],{"className":91119,"translate":19},[23],[15,91121,91123],{"className":91122},[27],[29,91124,91125],{"xmlns":31},[34,91126,91127,91139],{},[37,91128,91129],{},[305,91130,91131,91133],{},[43,91132,45],{},[37,91134,91135,91137],{},[50,91136,315],{},[43,91138,10372],{},[109,91140,91141],{"encoding":111},"h^{-d}",[15,91143,91145],{"className":91144,"ariaHidden":72},[116],[15,91146,91148,91151],{"className":91147},[120],[15,91149],{"className":91150,"style":10486},[124],[15,91152,91154,91157],{"className":91153},[129],[15,91155,45],{"className":91156},[129,133],[15,91158,91160],{"className":91159},[137],[15,91161,91163],{"className":91162},[141],[15,91164,91166],{"className":91165},[146],[15,91167,91169],{"className":91168,"style":10486},[150],[15,91170,91171,91174],{"style":383},[15,91172],{"className":91173,"style":159},[158],[15,91175,91177],{"className":91176},[163,164,165,166],[15,91178,91180,91183],{"className":91179},[129,166],[15,91181,315],{"className":91182},[129,166],[15,91184,10372],{"className":91185},[129,133,166]," 控制假设的个数。"," 三者平衡，就得到非参数速率。维度 ",[15,91189,91191,91204],{"className":91190,"translate":19},[23],[15,91192,91194],{"className":91193},[27],[29,91195,91196],{"xmlns":31},[34,91197,91198,91202],{},[37,91199,91200],{},[43,91201,10372],{},[109,91203,10372],{"encoding":111},[15,91205,91207],{"className":91206,"ariaHidden":72},[116],[15,91208,91210,91213],{"className":91209},[120],[15,91211],{"className":91212,"style":522},[124],[15,91214,10372],{"className":91215},[129,133]," 出现在指数上，正是因为它决定了「同样宽度下能放多少个鼓包」。",[575,91218,91220],{"id":91219},"_63-一些经典速率","6.3 一些经典速率",[18033,91222,91223,91236],{},[18036,91224,91225],{},[18039,91226,91227,91230,91233],{},[18042,91228,91229],{},"问题",[18042,91231,91232],{},"参数类",[18042,91234,91235],{},"极小极大速率",[18050,91237,91238,91450,91624,91797,92175],{},[18039,91239,91240,91271,91375],{},[18055,91241,91242,91270],{},[15,91243,91245,91258],{"className":91244,"translate":19},[23],[15,91246,91248],{"className":91247},[27],[29,91249,91250],{"xmlns":31},[34,91251,91252,91256],{},[37,91253,91254],{},[43,91255,10372],{},[109,91257,10372],{"encoding":111},[15,91259,91261],{"className":91260,"ariaHidden":72},[116],[15,91262,91264,91267],{"className":91263},[120],[15,91265],{"className":91266,"style":522},[124],[15,91268,10372],{"className":91269},[129,133]," 维高斯均值",[18055,91272,91273],{},[15,91274,91276,91302],{"className":91275,"translate":19},[23],[15,91277,91279],{"className":91278},[27],[29,91280,91281],{"xmlns":31},[34,91282,91283,91299],{},[37,91284,91285,91287,91289,91295,91297],{},[50,91286,33810],{"stretchy":52},[43,91288,48],{},[40,91290,91291,91293],{},[50,91292,33810],{"stretchy":52},[308,91294,310],{},[50,91296,2717],{},[43,91298,104],{},[109,91300,91301],{"encoding":111},"\\lVert\\theta\\rVert_2\\le R",[15,91303,91305,91366],{"className":91304,"ariaHidden":72},[116],[15,91306,91308,91311,91314,91317,91357,91360,91363],{"className":91307},[120],[15,91309],{"className":91310,"style":125},[124],[15,91312,33810],{"className":91313},[188],[15,91315,48],{"className":91316,"style":170},[129,133],[15,91318,91320,91323],{"className":91319},[195],[15,91321,33810],{"className":91322},[195],[15,91324,91326],{"className":91325},[137],[15,91327,91329,91349],{"className":91328},[141,142],[15,91330,91332,91346],{"className":91331},[146],[15,91333,91335],{"className":91334,"style":1051},[150],[15,91336,91337,91340],{"style":154},[15,91338],{"className":91339,"style":159},[158],[15,91341,91343],{"className":91342},[163,164,165,166],[15,91344,310],{"className":91345},[129,166],[15,91347,175],{"className":91348},[174],[15,91350,91352],{"className":91351},[146],[15,91353,91355],{"className":91354,"style":182},[150],[15,91356],{},[15,91358],{"className":91359,"style":199},[93],[15,91361,2717],{"className":91362},[203],[15,91364],{"className":91365,"style":199},[93],[15,91367,91369,91372],{"className":91368},[120],[15,91370],{"className":91371,"style":610},[124],[15,91373,104],{"className":91374,"style":33858},[129,133],[18055,91376,91377],{},[15,91378,91380,91403],{"className":91379,"translate":19},[23],[15,91381,91383],{"className":91382},[27],[29,91384,91385],{"xmlns":31},[34,91386,91387,91401],{},[37,91388,91389,91395,91397,91399],{},[305,91390,91391,91393],{},[43,91392,27283],{},[308,91394,310],{},[43,91396,10372],{},[43,91398,544],{"mathvariant":65},[43,91400,39314],{},[109,91402,64904],{"encoding":111},[15,91404,91406],{"className":91405,"ariaHidden":72},[116],[15,91407,91409,91412,91441,91444,91447],{"className":91408},[120],[15,91410],{"className":91411,"style":33996},[124],[15,91413,91415,91418],{"className":91414},[129],[15,91416,27283],{"className":91417,"style":887},[129,133],[15,91419,91421],{"className":91420},[137],[15,91422,91424],{"className":91423},[141],[15,91425,91427],{"className":91426},[146],[15,91428,91430],{"className":91429,"style":380},[150],[15,91431,91432,91435],{"style":383},[15,91433],{"className":91434,"style":159},[158],[15,91436,91438],{"className":91437},[163,164,165,166],[15,91439,310],{"className":91440},[129,166],[15,91442,10372],{"className":91443},[129,133],[15,91445,544],{"className":91446},[129],[15,91448,39314],{"className":91449},[129,133],[18039,91451,91452,91455,91514],{},[18055,91453,91454],{},"非参数回归",[18055,91456,91457,91485,91486],{},[15,91458,91460,91473],{"className":91459,"translate":19},[23],[15,91461,91463],{"className":91462},[27],[29,91464,91465],{"xmlns":31},[34,91466,91467,91471],{},[37,91468,91469],{},[43,91470,36264],{},[109,91472,36267],{"encoding":111},[15,91474,91476],{"className":91475,"ariaHidden":72},[116],[15,91477,91479,91482],{"className":91478},[120],[15,91480],{"className":91481,"style":1682},[124],[15,91483,36264],{"className":91484,"style":36280},[129,133],"-Hölder，",[15,91487,91489,91502],{"className":91488,"translate":19},[23],[15,91490,91492],{"className":91491},[27],[29,91493,91494],{"xmlns":31},[34,91495,91496,91500],{},[37,91497,91498],{},[43,91499,1373],{},[109,91501,1373],{"encoding":111},[15,91503,91505],{"className":91504,"ariaHidden":72},[116],[15,91506,91508,91511],{"className":91507},[120],[15,91509],{"className":91510,"style":610},[124],[15,91512,1373],{"className":91513},[129,133],[18055,91515,91516],{},[15,91517,91519,91556],{"className":91518,"translate":19},[23],[15,91520,91522],{"className":91521},[27],[29,91523,91524],{"xmlns":31},[34,91525,91526,91554],{},[37,91527,91528],{},[305,91529,91530,91532],{},[43,91531,39314],{},[37,91533,91534,91536,91538,91540,91542,91544,91546,91548,91550,91552],{},[50,91535,315],{},[308,91537,310],{},[43,91539,36264],{},[43,91541,544],{"mathvariant":65},[50,91543,53],{"stretchy":52},[308,91545,310],{},[43,91547,36264],{},[50,91549,2682],{},[43,91551,10372],{},[50,91553,59],{"stretchy":52},[109,91555,75494],{"encoding":111},[15,91557,91559],{"className":91558,"ariaHidden":72},[116],[15,91560,91562,91565],{"className":91561},[120],[15,91563],{"className":91564,"style":14941},[124],[15,91566,91568,91571],{"className":91567},[129],[15,91569,39314],{"className":91570},[129,133],[15,91572,91574],{"className":91573},[137],[15,91575,91577],{"className":91576},[141],[15,91578,91580],{"className":91579},[146],[15,91581,91583],{"className":91582,"style":14941},[150],[15,91584,91585,91588],{"style":383},[15,91586],{"className":91587,"style":159},[158],[15,91589,91591],{"className":91590},[163,164,165,166],[15,91592,91594,91597,91600,91603,91606,91609,91612,91615,91618,91621],{"className":91593},[129,166],[15,91595,315],{"className":91596},[129,166],[15,91598,310],{"className":91599},[129,166],[15,91601,36264],{"className":91602,"style":36280},[129,133,166],[15,91604,544],{"className":91605},[129,166],[15,91607,53],{"className":91608},[188,166],[15,91610,310],{"className":91611},[129,166],[15,91613,36264],{"classNa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s\\ll d",[15,91867,91869,91930,91948],{"className":91868,"ariaHidden":72},[116],[15,91870,91872,91875,91878,91881,91921,91924,91927],{"className":91871},[120],[15,91873],{"className":91874,"style":125},[124],[15,91876,33810],{"className":91877},[188],[15,91879,48],{"className":91880,"style":170},[129,133],[15,91882,91884,91887],{"className":91883},[195],[15,91885,33810],{"className":91886},[195],[15,91888,91890],{"className":91889},[137],[15,91891,91893,91913],{"className":91892},[141,142],[15,91894,91896,91910],{"className":91895},[146],[15,91897,91899],{"className":91898,"style":1051},[150],[15,91900,91901,91904],{"style":154},[15,91902],{"className":91903,"style":159},[158],[15,91905,91907],{"className":91906},[163,164,165,166],[15,91908,541],{"className":91909},[129,166],[15,91911,175],{"className":91912},[174],[15,91914,91916],{"className":91915},[146],[15,91917,91919],{"className":91918,"style":182},[150],[15,91920],{},[15,91922],{"className":91923,"style":199},[93],[15,91925,2717],{"className":91926},[203],[15,91928],{"className":91929,"style":199},[93],[15,91931,91933,91936,91939,91942,91945],{"className":91932},[120],[15,91934],{"className":91935,"style":4867},[124],[15,91937,22143],{"className":91938},[129,133],[15,91940],{"className":91941,"style":199},[93],[15,91943,91860],{"className":91944},[203],[15,91946],{"className":91947,"style":199},[93],[15,91949,91951,91954],{"className":91950},[120],[15,91952],{"className":91953,"style":522},[124],[15,91955,10372],{"className":91956},[129,133],[18055,91958,91959],{},[15,91960,91962,92002],{"className":91961,"translate":19},[23],[15,91963,91965],{"className":91964},[27],[29,91966,91967],{"xmlns":31},[34,91968,91969,91999],{},[37,91970,91971,91987,91989,91991],{},[2612,91972,91973],{"scriptlevel":541,"displaystyle":72},[1443,91974,91975,91985],{},[37,91976,91977,91983],{},[305,91978,91979,91981],{},[43,91980,27283],{},[308,91982,310],{},[43,91984,22143],{},[43,91986,39314],{},[43,91988,14721],{},[50,91990,69],{},[2612,91992,91993],{"scriptlevel":541,"displaystyle":72},[1443,91994,91995,91997],{},[43,91996,10372],{},[43,91998,22143],{},[109,92000,92001],{"encoding":111},"\\dfrac{\\sigma^2s}{n}\\log\\dfrac ds",[15,92003,92005],{"className":92004,"ariaHidden":72},[116],[15,92006,92008,92011,92102,92105,92110,92113],{"className":92007},[120],[15,92009],{"className":92010,"style":82488},[124],[15,92012,92014,92017,92099],{"className":92013},[129],[15,92015],{"className":92016},[188,1803],[15,92018,92020],{"className":92019},[1443],[15,92021,92023,92091],{"className":92022},[141,142],[15,92024,92026,92088],{"className":92025},[146],[15,92027,92029,92040,92048],{"className":92028,"style":63608},[150],[15,92030,92031,92034],{"style":1819},[15,92032],{"className":92033,"style":1823},[158],[15,92035,92037],{"className":92036},[129],[15,92038,39314],{"className":92039},[129,133],[15,92041,92042,92045],{"style":1832},[15,92043],{"className":92044,"style":1823},[158],[15,92046],{"className":92047,"style":1840},[1839],[15,92049,92050,92053],{"style":1843},[15,92051],{"className":92052,"style":1823},[158],[15,92054,92056,92085],{"className":92055},[129],[15,92057,92059,92062],{"className":92058},[129],[15,92060,27283],{"className":92061,"style":887},[129,133],[15,92063,92065],{"className":92064},[137],[15,92066,92068],{"className":92067},[141],[15,92069,92071],{"className":92070},[146],[15,92072,92074],{"className":92073,"style":380},[150],[15,92075,92076,92079],{"style":383},[15,92077],{"className":92078,"style":159},[158],[15,92080,92082],{"className":92081},[163,164,165,166],[15,92083,310],{"className":92084},[129,166],[15,92086,22143],{"className":92087},[129,133],[15,92089,175],{"className":92090},[174],[15,92092,92094],{"className":92093},[146],[15,92095,92097],{"className":92096,"style":1862},[150],[15,92098],{},[15,92100],{"className":92101},[195,1803],[15,92103],{"className":92104,"style":262},[93],[15,92106,14771,92108],{"className":92107},[217],[15,92109,2247],{"style":2246},[15,92111],{"className":92112,"style":262},[93],[15,92114,92116,92119,92172],{"className":92115},[129],[15,92117],{"className":92118},[188,1803],[15,92120,92122],{"className":92121},[1443],[15,92123,92125,92164],{"className":92124},[141,142],[15,92126,92128,92161],{"className":92127},[146],[15,92129,92131,92142,92150],{"className":92130,"style":17164},[150],[15,92132,92133,92136],{"style":1819},[15,92134],{"className":92135,"style":1823},[158],[15,92137,92139],{"className":92138},[129],[15,92140,22143],{"className":92141},[129,133],[15,92143,92144,92147],{"style":1832},[15,92145],{"className":92146,"style":1823},[158],[15,92148],{"className":92149,"style":1840},[1839],[15,92151,92152,92155],{"style":1843},[15,92153],{"className":92154,"style":1823},[158],[15,92156,92158],{"className":92157},[129],[15,92159,10372],{"className":92160},[129,133],[15,92162,175],{"className":92163},[174],[15,92165,92167],{"className":92166},[146],[15,92168,92170],{"className":92169,"style":1862},[150],[15,92171],{},[15,92173],{"className":92174},[195,1803],[18039,92176,92177,92180,92183],{},[18055,92178,92179],{},"一维单调回归",[18055,92181,92182],{},"单调、有界",[18055,92184,92185],{},[15,92186,92188,92214],{"className":92187,"translate":19},[23],[15,92189,92191],{"className":92190},[27],[29,92192,92193],{"xmlns":31},[34,92194,92195,92211],{},[37,92196,92197],{},[305,92198,92199,92201],{},[43,92200,39314],{},[37,92202,92203,92205,92207,92209],{},[50,92204,315],{},[308,92206,310],{},[43,92208,544],{"mathvariant":65},[308,92210,9875],{},[109,92212,92213],{"encoding":111},"n^{-2\u002F3}",[15,92215,92217],{"className":92216,"ariaHidden":72},[116],[15,92218,92220,92223],{"className":92219},[120],[15,92221],{"className":92222,"style":14941},[124],[15,92224,92226,92229],{"className":92225},[129],[15,92227,39314],{"className":92228},[129,133],[15,92230,92232],{"className":92231},[137],[15,92233,92235],{"className":92234},[141],[15,92236,92238],{"className":92237},[146],[15,92239,92241],{"className":92240,"style":14941},[150],[15,92242,92243,92246],{"style":383},[15,92244],{"className":92245,"style":159},[158],[15,92247,92249],{"className":92248},[163,164,165,166],[15,92250,92252,92255],{"className":92251},[129,166],[15,92253,315],{"className":92254},[129,166],[15,92256,92258],{"className":92257},[129,166],"2\u002F3",[11,92260,92261],{},"这张表里的每一个速率，都可以用本文的三种方法之一推出来。稀疏均值的下界通常用 Fano 加一个随机支撑的打包；单调回归则用 Le Cam 加一个局部的「台阶」构造。",[570,92263,92265],{"id":92264},"七三种方法的比较","七、三种方法的比较",[18033,92267,92268,92287],{},[18036,92269,92270],{},[18039,92271,92272,92275,92278,92281,92284],{},[18042,92273,92274],{},"方法",[18042,92276,92277],{},"假设个数",[18042,92279,92280],{},"距离",[18042,92282,92283],{},"优势",[18042,92285,92286],{},"局限",[18050,92288,92289,92333,92377],{},[18039,92290,92291,92294,92324,92327,92330],{},[18055,92292,92293],{},"Le Cam",[18055,92295,92296],{},[15,92297,92299,92312],{"className":92298,"translate":19},[23],[15,92300,92302],{"className":92301},[27],[29,92303,92304],{"xmlns":31},[34,92305,92306,92310],{},[37,92307,92308],{},[308,92309,310],{},[109,92311,310],{"encoding":111},[15,92313,92315],{"className":92314,"ariaHidden":72},[116],[15,92316,92318,92321],{"className":92317},[120],[15,92319],{"className":92320,"style":4886},[124],[15,92322,310],{"className":92323},[129],[18055,92325,92326],{},"TV \u002F KL",[18055,92328,92329],{},"简单，常数清楚",[18055,92331,92332],{},"只探测一个方向，高维问题会松",[18039,92334,92335,92338,92368,92371,92374],{},[18055,92336,92337],{},"Fano",[18055,92339,92340],{},[15,92341,92343,92356],{"className":92342,"translate":19},[23],[15,92344,92346],{"className":92345},[27],[29,92347,92348],{"xmlns":31},[34,92349,92350,92354],{},[37,92351,92352],{},[43,92353,43234],{},[109,92355,43234],{"encoding":111},[15,92357,92359],{"className":92358,"ariaHidden":72},[116],[15,92360,92362,92365],{"className":92361},[120],[15,92363],{"className":92364,"style":610},[124],[15,92366,43234],{"className":92367,"style":12591},[129,133],[18055,92369,92370],{},"互信息 \u002F KL",[18055,92372,92373],{},"通用，适合高维与非参数",[18055,92375,92376],{},"需要构造打包，常数较松",[18039,92378,92379,92382,92442,92445,92448],{},[18055,92380,92381],{},"Assouad",[18055,92383,92384],{},[15,92385,92387,92404],{"className":92386,"translate":19},[23],[15,92388,92390],{"className":92389},[27],[29,92391,92392],{"xmlns":31},[34,92393,92394,92402],{},[37,92395,92396],{},[305,92397,92398,92400],{},[308,92399,310],{},[43,92401,348],{},[109,92403,6367],{"encoding":111},[15,92405,92407],{"className":92406,"ariaHidden":72},[116],[15,92408,92410,92413],{"className":92409},[120],[15,92411],{"className":92412,"style":6377},[124],[15,92414,92416,92419],{"className":92415},[129],[15,92417,310],{"className":92418},[129],[15,92420,92422],{"className":92421},[137],[15,92423,92425],{"className":92424},[141],[15,92426,92428],{"className":92427},[146],[15,92429,92431],{"className":92430,"style":6377},[150],[15,92432,92433,92436],{"style":383},[15,92434],{"className":92435,"style":159},[158],[15,92437,92439],{"className":92438},[163,164,165,166],[15,92440,348],{"className":92441},[129,133,166],[18055,92443,92444],{},"逐坐标 TV",[18055,92446,92447],{},"乘积结构下最锋利",[18055,92449,92450],{},"只适用于超立方体结构",[11,92452,92453],{},"三者的关系可以这样理解：",[21719,92455,92456,92462,92497],{},[21722,92457,92458,92461],{},[698,92459,92460],{},"Le Cam 是 Fano 的两点版本。"," 当只有两个假设时，Fano 退化为两点检验；Le Cam 用 TV 给出了更干净的常数。",[21722,92463,92464,92467,92468,92496],{},[698,92465,92466],{},"Fano 是 Le Cam 的多假设推广。"," 它把「两个点分不清」推广为「",[15,92469,92471,92484],{"className":92470,"translate":19},[23],[15,92472,92474],{"className":92473},[27],[29,92475,92476],{"xmlns":31},[34,92477,92478,92482],{},[37,92479,92480],{},[43,92481,43234],{},[109,92483,43234],{"encoding":111},[15,92485,92487],{"className":92486,"ariaHidden":72},[116],[15,92488,92490,92493],{"className":92489},[120],[15,92491],{"className":92492,"style":610},[124],[15,92494,43234],{"className":92495,"style":12591},[129,133]," 个点平均分不清」。",[21722,92498,92499,92502,92503,92561,92562,92590],{},[698,92500,92501],{},"Assouad 是 Fano 在乘积结构上的精细化。"," 它不直接对 ",[15,92504,92506,92523],{"className":92505,"translate":19},[23],[15,92507,92509],{"className":92508},[27],[29,92510,92511],{"xmlns":31},[34,92512,92513,92521],{},[37,92514,92515],{},[305,92516,92517,92519],{},[308,92518,310],{},[43,92520,348],{},[109,92522,6367],{"encoding":111},[15,92524,92526],{"className":92525,"ariaHidden":72},[116],[15,92527,92529,92532],{"className":92528},[120],[15,92530],{"className":92531,"style":6377},[124],[15,92533,92535,92538],{"className":92534},[129],[15,92536,310],{"className":92537},[129],[15,92539,92541],{"className":92540},[137],[15,92542,92544],{"className":92543},[141],[15,92545,92547],{"className":92546},[146],[15,92548,92550],{"className":92549,"style":6377},[150],[15,92551,92552,92555],{"style":383},[15,92553],{"className":92554,"style":159},[158],[15,92556,92558],{"className":92557},[163,164,165,166],[15,92559,348],{"className":92560},[129,133,166]," 个假设做 Fano，而是逐坐标做 Le Cam，再把 ",[15,92563,92565,92578],{"className":92564,"translate":19},[23],[15,92566,92568],{"className":92567},[27],[29,92569,92570],{"xmlns":31},[34,92571,92572,92576],{},[37,92573,92574],{},[43,92575,348],{},[109,92577,348],{"encoding":111},[15,92579,92581],{"className":92580,"ariaHidden":72},[116],[15,92582,92584,92587],{"className":92583},[120],[15,92585],{"className":92586,"style":4383},[124],[15,92588,348],{"className":92589},[129,133]," 个下界加起来。",[11,92592,92593],{},"实际使用时，选择哪种方法取决于参数空间的结构：",[21719,92595,92596,92599,92643],{},[21722,92597,92598],{},"一维、局部、两样本问题：Le Cam 通常足够，而且常数最干净。",[21722,92600,92601,92602,92642],{},"高维、非参数、需要 ",[15,92603,92605,92622],{"className":92604,"translate":19},[23],[15,92606,92608],{"className":92607},[27],[29,92609,92610],{"xmlns":31},[34,92611,92612,92620],{},[37,92613,92614,92616,92618],{},[43,92615,14721],{},[50,92617,69],{},[43,92619,43234],{},[109,92621,67048],{"encoding":111},[15,92623,92625],{"className":92624,"ariaHidden":72},[116],[15,92626,92628,92631,92636,92639],{"className":92627},[120],[15,92629],{"className":92630,"style":1682},[124],[15,92632,14771,92634],{"className":92633},[217],[15,92635,2247],{"style":2246},[15,92637],{"className":92638,"style":262},[93],[15,92640,43234],{"className":92641,"style":12591},[129,133]," 的地方：Fano 是标准工具。",[21722,92644,92645,92646,39034,92716,92786],{},"乘积结构、",[15,92647,92649,92667],{"className":92648,"translate":19},[23],[15,92650,92652],{"className":92651},[27],[29,92653,92654],{"xmlns":31},[34,92655,92656,92664],{},[37,92657,92658],{},[40,92659,92660,92662],{},[43,92661,27518],{"mathvariant":65},[308,92663,310],{},[109,92665,92666],{"encoding":111},"\\ell_2",[15,92668,92670],{"className":92669,"ariaHidden":72},[116],[15,92671,92673,92676],{"className":92672},[120],[15,92674],{"className":92675,"style":2183},[124],[15,92677,92679,92682],{"className":92678},[129],[15,92680,27518],{"className":92681},[129],[15,92683,92685],{"className":92684},[137],[15,92686,92688,92708],{"className":92687},[141,142],[15,92689,92691,92705],{"className":92690},[146],[15,92692,92694],{"className":92693,"style":1051},[150],[15,92695,92696,92699],{"style":154},[15,92697],{"className":92698,"style":159},[158],[15,92700,92702],{"className":92701},[163,164,165,166],[15,92703,310],{"className":92704},[129,166],[15,92706,175],{"className":92707},[174],[15,92709,92711],{"className":92710},[146],[15,92712,92714],{"className":92713,"style":182},[150],[15,92715],{},[15,92717,92719,92737],{"className":92718,"translate":19},[23],[15,92720,92722],{"className":92721},[27],[29,92723,92724],{"xmlns":31},[34,92725,92726,92734],{},[37,92727,92728],{},[40,92729,92730,92732],{},[43,92731,27518],{"mathvariant":65},[43,92733,4909],{"mathvariant":65},[109,92735,92736],{"encoding":111},"\\ell_\\infty",[15,92738,92740],{"className":92739,"ariaHidden":72},[116],[15,92741,92743,92746],{"className":92742},[120],[15,92744],{"className":92745,"style":2183},[124],[15,92747,92749,92752],{"className":92748},[129],[15,92750,27518],{"className":92751},[129],[15,92753,92755],{"className":92754},[137],[15,92756,92758,92778],{"className":92757},[141,142],[15,92759,92761,92775],{"className":92760},[146],[15,92762,92764],{"className":92763,"style":1163},[150],[15,92765,92766,92769],{"style":154},[15,92767],{"className":92768,"style":159},[158],[15,92770,92772],{"className":92771},[163,164,165,166],[15,92773,4909],{"className":92774},[129,166],[15,92776,175],{"className":92777},[174],[15,92779,92781],{"className":92780},[146],[15,92782,92784],{"className":92783,"style":182},[150],[15,92785],{}," 速率：Assouad 最方便。",[570,92788,36411],{"id":36411},[11,92790,92791,92792,92827,92828,92856],{},"回到开头的问题：为什么 ",[15,92793,92795,92812],{"className":92794,"translate":19},[23],[15,92796,92798],{"className":92797},[27],[29,92799,92800],{"xmlns":31},[34,92801,92802,92810],{},[37,92803,92804,92806,92808],{},[308,92805,318],{},[43,92807,544],{"mathvariant":65},[43,92809,39314],{},[109,92811,48138],{"encoding":111},[15,92813,92815],{"className":92814,"ariaHidden":72},[116],[15,92816,92818,92821,92824],{"className":92817},[120],[15,92819],{"className":92820,"style":125},[124],[15,92822,29841],{"className":92823},[129],[15,92825,39314],{"className":92826},[129,133]," 不能再小？因为两个足够接近的分布，在 ",[15,92829,92831,92844],{"className":92830,"translate":19},[23],[15,92832,92834],{"className":92833},[27],[29,92835,92836],{"xmlns":31},[34,92837,92838,92842],{},[37,92839,92840],{},[43,92841,39314],{},[109,92843,39314],{"encoding":111},[15,92845,92847],{"className":92846,"ariaHidden":72},[116],[15,92848,92850,92853],{"className":92849},[120],[15,92851],{"className":92852,"style":4383},[124],[15,92854,39314],{"className":92855},[129,133]," 个样本下仍然难以区分；而任何估计器都必须先区分它们。这个论证只需要两个点，所以叫两点方法。",[11,92858,92859,92860,92888,92889,92917,92918,92946],{},"但当困难来自许多方向时，两个点就不够了。Fano 把「两个点」换成「",[15,92861,92863,92876],{"className":92862,"translate":19},[23],[15,92864,92866],{"className":92865},[27],[29,92867,92868],{"xmlns":31},[34,92869,92870,92874],{},[37,92871,92872],{},[43,92873,43234],{},[109,92875,43234],{"encoding":111},[15,92877,92879],{"className":92878,"ariaHidden":72},[116],[15,92880,92882,92885],{"className":92881},[120],[15,92883],{"className":92884,"style":610},[124],[15,92886,43234],{"className":92887,"style":12591},[129,133]," 个点」，用互信息度量整体的不可区分性；Assouad 则把 ",[15,92890,92892,92905],{"className":92891,"translate":19},[23],[15,92893,92895],{"className":92894},[27],[29,92896,92897],{"xmlns":31},[34,92898,92899,92903],{},[37,92900,92901],{},[43,92902,348],{},[109,92904,348],{"encoding":111},[15,92906,92908],{"className":92907,"ariaHidden":72},[116],[15,92909,92911,92914],{"className":92910},[120],[15,92912],{"className":92913,"style":4383},[124],[15,92915,348],{"className":92916},[129,133]," 维问题拆成 ",[15,92919,92921,92934],{"className":92920,"translate":19},[23],[15,92922,92924],{"className":92923},[27],[29,92925,92926],{"xmlns":31},[34,92927,92928,92932],{},[37,92929,92930],{},[43,92931,348],{},[109,92933,348],{"encoding":111},[15,92935,92937],{"className":92936,"ariaHidden":72},[116],[15,92938,92940,92943],{"className":92939},[120],[15,92941],{"className":92942,"style":4383},[124],[15,92944,348],{"className":92945},[129,133]," 个一维问题，逐坐标地累加下界。",[11,92948,92949],{},"如果只记三句话：",[30659,92951,92952,92955,93046],{},[21722,92953,92954],{},"下界证明的第一步永远是归约：把估计问题换成一个有限检验问题。",[21722,92956,92957,92958,92986,92987,93045],{},"Le Cam 用两个点，Fano 用 ",[15,92959,92961,92974],{"className":92960,"translate":19},[23],[15,92962,92964],{"className":92963},[27],[29,92965,92966],{"xmlns":31},[34,92967,92968,92972],{},[37,92969,92970],{},[43,92971,43234],{},[109,92973,43234],{"encoding":111},[15,92975,92977],{"className":92976,"ariaHidden":72},[116],[15,92978,92980,92983],{"className":92979},[120],[15,92981],{"className":92982,"style":610},[124],[15,92984,43234],{"className":92985,"style":12591},[129,133]," 个点，Assouad 用 ",[15,92988,92990,93007],{"className":92989,"translate":19},[23],[15,92991,92993],{"className":92992},[27],[29,92994,92995],{"xmlns":31},[34,92996,92997,93005],{},[37,92998,92999],{},[305,93000,93001,93003],{},[308,93002,310],{},[43,93004,348],{},[109,93006,6367],{"encoding":111},[15,93008,93010],{"className":93009,"ariaHidden":72},[116],[15,93011,93013,93016],{"className":93012},[120],[15,93014],{"className":93015,"style":6377},[124],[15,93017,93019,93022],{"className":93018},[129],[15,93020,310],{"className":93021},[129],[15,93023,93025],{"className":93024},[137],[15,93026,93028],{"className":93027},[141],[15,93029,93031],{"className":93030},[146],[15,93032,93034],{"className":93033,"style":6377},[150],[15,93035,93036,93039],{"style":383},[15,93037],{"className":93038,"style":159},[158],[15,93040,93042],{"className":93041},[163,164,165,166],[15,93043,348],{"className":93044},[129,133,166]," 个点；工具在变，思想都是「构造一个在损失上离得远、在信息上离得近的假设集合」。",[21722,93047,93048],{},"信息论下界告诉你统计上不可能；它不告诉你计算上不可能。深度学习时代真正难的，往往是后一个问题。",{"title":36419,"searchDepth":36420,"depth":36420,"links":93050},[93051,93052,93057,93063,93068,93072,93079,93080],{"id":48448,"depth":36420,"text":48449},{"id":49078,"depth":36420,"text":49079,"children":93053},[93054,93055,93056],{"id":49082,"depth":36425,"text":49083},{"id":51610,"depth":36425,"text":51611},{"id":52207,"depth":36425,"text":52208},{"id":52851,"depth":36420,"text":52852,"children":93058},[93059,93060,93061,93062],{"id":52855,"depth":36425,"text":52856},{"id":60168,"depth":36425,"text":60169},{"id":62042,"depth":36425,"text":62043},{"id":63934,"depth":36425,"text":63935},{"id":65073,"depth":36420,"text":65074,"children":93064},[93065,93066,93067],{"id":65077,"depth":36425,"text":65078},{"id":67142,"depth":36425,"text":67143},{"id":74352,"depth":36425,"text":74353},{"id":75652,"depth":36420,"text":75653,"children":93069},[93070,93071],{"id":75656,"depth":36425,"text":75657},{"id":79892,"depth":36425,"text":79893},{"id":81654,"depth":36420,"text":81655,"children":93073},[93074,93076,93078],{"id":81658,"depth":36425,"text":93075},"6.1 高斯均值：σ2d\u002Fn\\sigma^2d\u002Fnσ2d\u002Fn",{"id":84640,"depth":36425,"text":93077},"6.2 非参数回归：n−2β\u002F(2β+d)n^{-2\\beta\u002F(2\\beta+d)}n−2β\u002F(2β+d)",{"id":91219,"depth":36425,"text":91220},{"id":92264,"depth":36420,"text":92265},{"id":36411,"depth":36420,"text":36411},{},"\u002Fblog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa",{"title":47888,"description":47893},"blog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa","极小极大下界的三种基本工具：Le Cam 两点方法、Fano 不等式与 Assouad 引理。本文从决策论框架出发，给出完整证明与常数，并用高斯均值、非参数回归等例子展示如何把信息论不等式变成速率下界。",[36463,36464],"2025-09-27T16:28:39+08:00","UddcaRIuRTyzw3VagGFZOj0rS3lFxJ-QDtX0NVJjl5Q",1794483713164]