[{"data":1,"prerenderedAt":24886},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-08-19-yan-jiu-guan-xi-de-san-zhong-shi-jiao":3,"i-mingcute:time-line":24880,"i-ci:tag":24884},{"post":4,"nextPost":5269,"prevPost":24611},{"id":5,"title":6,"body":7,"description":13,"draft":5256,"enableComment":5257,"extension":5258,"image":5247,"important":5256,"location":5259,"meta":5260,"navigation":5257,"ogImage":5259,"path":5261,"seo":5262,"stem":5263,"summary":5264,"tags":5265,"time":5267,"__hash__":5268},"blog\u002Fblog\u002F2026\u002F2026-08-19-yan-jiu-guan-xi-de-san-zhong-shi-jiao.md","研究关系的三种视角",{"type":8,"value":9,"toc":5246},"minimark",[10,14,17,20,25,28,296,975,1170,1174,1177,1579,1583,1586,2063,2342,3155,3158,3161,3221,3569,3664,3939,3942,3945,3948,4059,4447,4450,4453,4526,4529,5240,5243],[11,12,13],"p",{},"哲学里有一种命题：「存在即关系」，在数理研究中同样也有很深的体现。听起来很抽象，但它其实是对世界本质是什么的一种深刻回答。理解事物不是先存在、再去建立关系，而是事物在建立关系的过程中，才得以成为事物。它最著名的提出者是 20 世纪法国哲学家怀特海（过程哲学），但其思想渊源可以追溯到德国哲学家莱布尼茨（单子论）和古希腊的赫拉克利特（万物流动）。",[11,15,16],{},"所以，研究「存在」也是研究「关系」。而关系通常有三种视角：几何、拓扑和因果。",[11,18,19],{},"拓扑提供稳健的骨架，几何赋予精细的尺度，因果指明干预的方向。三者共同构成智能系统理解世界的先验地基。更准确的说法或许是：几何和拓扑是关于「关系的静态结构」（分别是度量化的和非度量化的），而因果是关于「关系的动态\u002F生成机制」，它回答的不是「这两者如何关联」，而是「如果我改变一个，另一个会怎样」。",[21,22,24],"h2",{"id":23},"几何度量结构","几何（度量结构）",[11,26,27],{},"几何关注的是「测量」。基本问题是多接近、多相似，或者一个具体的度量数值。——它需要一个度量结构（距离、内积、相似度函数），通常要求满足对称性、三角不等式等公理。典型例子：嵌入空间里两个向量的余弦相似度、社会心理学里的「亲密度量表」。研究的是关系的强度。几何关系是最重的——信息量最大，但也最脆弱，度量方式一变，结论可能就变。",[11,29,30,31,114,115,295],{},"一个度量空间是二元组 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\\|x-y\\|",[32,1098,1100,1136,1157],{"className":1099,"ariaHidden":62},[78],[32,1101,1103,1106,1109,1112,1115,1118,1121,1124,1127,1130,1133],{"className":1102},[82],[32,1104],{"className":1105,"style":87},[86],[32,1107,66],{"className":1108},[95,96],[32,1110,55],{"className":1111},[91],[32,1113,358],{"className":1114},[95,96],[32,1116,63],{"className":1117},[101],[32,1119],{"className":1120,"style":106},[105],[32,1122,363],{"className":1123,"style":740},[95,96],[32,1125,69],{"className":1126},[113],[32,1128],{"className":1129,"style":177},[105],[32,1131,368],{"className":1132},[181],[32,1134],{"className":1135,"style":177},[105],[32,1137,1139,1142,1145,1148,1151,1154],{"className":1138},[82],[32,1140],{"className":1141,"style":87},[86],[32,1143,1084],{"className":1144},[95],[32,1146,358],{"className":1147},[95,96],[32,1149],{"className":1150,"style":198},[105],[32,1152,1089],{"className":1153},[202],[32,1155],{"className":1156,"style":198},[105],[32,1158,1160,1163,1166],{"className":1159},[82],[32,1161],{"className":1162,"style":87},[86],[32,1164,363],{"className":1165,"style":740},[95,96],[32,1167,1084],{"className":1168},[95],"，这才有了余弦相似度、点积注意力这些操作。度量结构的信息量最大，因为它不仅告诉你「是否有关系」，还告诉你「关系有多强」——但代价是它对表示方式极度敏感：换一个嵌入模型、换一个归一化方式，同一批数据算出来的相似度可以完全不同。",[21,1171,1173],{"id":1172},"拓扑连通结构","拓扑（连通结构）",[11,1175,1176],{},"拓扑关注的是「连通」。把度量忘掉，只问是否有路径可达、结构是否连通、是否存在环\u002F洞。在同胚变换（连续拉伸、不撕裂）下不变的东西才算拓扑性质。典型例子：社交网络里的连通分量、图的最短路径存在性（而非具体长度）。拓扑关系比几何轻，研究的是关系的骨架，但换来的是更强的稳健性——不在乎具体尺度，只在乎结构骨架。",[11,1178,1179,1180,1234,1235,1265,1266,1294,1295,1362,1363,1367,1368,1444,1445,1473,1474,1578],{},"一个拓扑空间是二元组 ",[32,1181,1183,1206],{"className":1182,"translate":36},[35],[32,1184,1186],{"className":1185},[40],[42,1187,1188],{"xmlns":44},[46,1189,1190,1203],{},[49,1191,1192,1194,1196,1198,1201],{},[52,1193,55],{"stretchy":54},[57,1195,59],{},[52,1197,63],{"separator":62},[57,1199,1200],{},"τ",[52,1202,69],{"stretchy":54},[71,1204,1205],{"encoding":73},"(X, \\tau)",[32,1207,1209],{"className":1208,"ariaHidden":62},[78],[32,1210,1212,1215,1218,1221,1224,1227,1231],{"className":1211},[82],[32,1213],{"className":1214,"style":87},[86],[32,1216,55],{"className":1217},[91],[32,1219,59],{"className":1220,"style":97},[95,96],[32,1222,63],{"className":1223},[101],[32,1225],{"className":1226,"style":106},[105],[32,1228,1200],{"className":1229,"style":1230},[95,96],"margin-right:0.1132em;",[32,1232,69],{"className":1233},[113],"，",[32,1236,1238,1252],{"className":1237,"translate":36},[35],[32,1239,1241],{"className":1240},[40],[42,1242,1243],{"xmlns":44},[46,1244,1245,1249],{},[49,1246,1247],{},[57,1248,1200],{},[71,1250,1251],{"encoding":73},"\\tau",[32,1253,1255],{"className":1254,"ariaHidden":62},[78],[32,1256,1258,1262],{"className":1257},[82],[32,1259],{"className":1260,"style":1261},[86],"height:0.4306em;",[32,1263,1200],{"className":1264,"style":1230},[95,96]," 是 ",[32,1267,1269,1282],{"className":1268,"translate":36},[35],[32,1270,1272],{"className":1271},[40],[42,1273,1274],{"xmlns":44},[46,1275,1276,1280],{},[49,1277,1278],{},[57,1279,59],{},[71,1281,59],{"encoding":73},[32,1283,1285],{"className":1284,"ariaHidden":62},[78],[32,1286,1288,1291],{"className":1287},[82],[32,1289],{"className":1290,"style":212},[86],[32,1292,59],{"className":1293,"style":97},[95,96]," 的一族「开集」，满足对任意并、有限交封闭，且 ",[32,1296,1298,1322],{"className":1297,"translate":36},[35],[32,1299,1301],{"className":1300},[40],[42,1302,1303],{"xmlns":44},[46,1304,1305,1319],{},[49,1306,1307,1310,1312,1314,1317],{},[57,1308,1309],{"mathvariant":1083},"∅",[52,1311,63],{"separator":62},[57,1313,59],{},[52,1315,1316],{},"∈",[57,1318,1200],{},[71,1320,1321],{"encoding":73},"\\emptyset, X \\in \\tau",[32,1323,1325,1353],{"className":1324,"ariaHidden":62},[78],[32,1326,1328,1332,1335,1338,1341,1344,1347,1350],{"className":1327},[82],[32,1329],{"className":1330,"style":1331},[86],"height:0.9444em;vertical-align:-0.1944em;",[32,1333,1309],{"className":1334},[95],[32,1336,63],{"className":1337},[101],[32,1339],{"className":1340,"style":106},[105],[32,1342,59],{"className":1343,"style":97},[95,96],[32,1345],{"className":1346,"style":177},[105],[32,1348,1316],{"className":1349},[181],[32,1351],{"className":1352,"style":177},[105],[32,1354,1356,1359],{"className":1355},[82],[32,1357],{"className":1358,"style":1261},[86],[32,1360,1200],{"className":1361,"style":1230},[95,96],"。拓扑结构不需要度量：任何度量空间都能诱导出一个拓扑（用开球生成开集），但反过来不成立——拓扑空间不一定可度量化。关键的不变量是在",[1364,1365,1366],"strong",{},"同胚","（连续双射且逆映射也连续）下保持不变的性质，比如连通性、紧致性、亏格（洞的个数）。用代数拓扑的语言，贝蒂数 ",[32,1369,1371,1391],{"className":1370,"translate":36},[35],[32,1372,1374],{"className":1373},[40],[42,1375,1376],{"xmlns":44},[46,1377,1378,1388],{},[49,1379,1380],{},[143,1381,1382,1385],{},[57,1383,1384],{},"b",[57,1386,1387],{},"k",[71,1389,1390],{"encoding":73},"b_k",[32,1392,1394],{"className":1393,"ariaHidden":62},[78],[32,1395,1397,1401],{"className":1396},[82],[32,1398],{"className":1399,"style":1400},[86],"height:0.8444em;vertical-align:-0.15em;",[32,1402,1404,1407],{"className":1403},[95],[32,1405,1384],{"className":1406},[95,96],[32,1408,1410],{"className":1409},[242],[32,1411,1413,1435],{"className":1412},[246,247],[32,1414,1416,1432],{"className":1415},[251],[32,1417,1420],{"className":1418,"style":1419},[255],"height:0.3361em;",[32,1421,1422,1425],{"style":259},[32,1423],{"className":1424,"style":264},[263],[32,1426,1428],{"className":1427},[268,269,270,271],[32,1429,1387],{"className":1430,"style":1431},[95,96,271],"margin-right:0.0315em;",[32,1433,285],{"className":1434},[284],[32,1436,1438],{"className":1437},[251],[32,1439,1442],{"className":1440,"style":1441},[255],"height:0.15em;",[32,1443],{}," 数的是 ",[32,1446,1448,1461],{"className":1447,"translate":36},[35],[32,1449,1451],{"className":1450},[40],[42,1452,1453],{"xmlns":44},[46,1454,1455,1459],{},[49,1456,1457],{},[57,1458,1387],{},[71,1460,1387],{"encoding":73},[32,1462,1464],{"className":1463,"ariaHidden":62},[78],[32,1465,1467,1470],{"className":1466},[82],[32,1468],{"className":1469,"style":170},[86],[32,1471,1387],{"className":1472,"style":1431},[95,96]," 维「洞」的个数，这是持久同调（persistent homology）在数据分析里常用的工具——它告诉你数据云的「形状骨架」，而完全不关心具体的尺度参数。这解释了为什么拓扑比几何「轻」：它是几何结构经过遗忘函子 ",[32,1475,1477,1518],{"className":1476,"translate":36},[35],[32,1478,1480],{"className":1479},[40],[42,1481,1482],{"xmlns":44},[46,1483,1484,1515],{},[49,1485,1486,1489,1491,1503,1505],{},[57,1487,1488],{},"U",[52,1490,131],{},[49,1492,1493,1497,1500],{},[57,1494,1496],{"mathvariant":1495},"bold","M",[57,1498,1499],{"mathvariant":1495},"e",[57,1501,1502],{"mathvariant":1495},"t",[52,1504,141],{},[49,1506,1507,1510,1513],{},[57,1508,1509],{"mathvariant":1495},"T",[57,1511,1512],{"mathvariant":1495},"o",[57,1514,11],{"mathvariant":1495},[71,1516,1517],{"encoding":73},"U: \\mathbf{Met} \\to \\mathbf{Top}",[32,1519,1521,1540,1564],{"className":1520,"ariaHidden":62},[78],[32,1522,1524,1527,1531,1534,1537],{"className":1523},[82],[32,1525],{"className":1526,"style":212},[86],[32,1528,1488],{"className":1529,"style":1530},[95,96],"margin-right:0.109em;",[32,1532],{"className":1533,"style":177},[105],[32,1535,131],{"className":1536},[181],[32,1538],{"className":1539,"style":177},[105],[32,1541,1543,1547,1555,1558,1561],{"className":1542},[82],[32,1544],{"className":1545,"style":1546},[86],"height:0.6861em;",[32,1548,1550],{"className":1549},[95],[32,1551,1554],{"className":1552},[95,1553],"mathbf","Met",[32,1556],{"className":1557,"style":177},[105],[32,1559,141],{"className":1560},[181],[32,1562],{"className":1563,"style":177},[105],[32,1565,1567,1571],{"className":1566},[82],[32,1568],{"className":1569,"style":1570},[86],"height:0.8805em;vertical-align:-0.1944em;",[32,1572,1574],{"className":1573},[95],[32,1575,1577],{"className":1576},[95,1553],"Top"," 投影之后剩下的东西。",[21,1580,1582],{"id":1581},"因果约束结构","因果（约束结构）",[11,1584,1585],{},"因果关注的是「约束」。这一支和前两支有本质区别：几何和拓扑通常是对称的（A 到 B 的距离=B 到 A 的距离；A 连着 B 等价于 B 连着 A），而因果关系本质上是非对称的——「X 决定\u002F限制 Y 的取值范围」不等于反过来成立。因果结构需要额外的信息才能识别（干预、反事实、时间先后），单纯的相关性数据（哪怕是几何或拓扑意义上完整的）不足以确定因果方向。这也是 Pearl 那套 do-calculus 要解决的问题。在 ML 中，几何\u002F拓扑模型在独立同分布（i.i.d.）下表现很好，但只要测试分布变了（协变量偏移），它们就会崩。而因果结构（比如有向无环图 DAG）告诉你的是「机制」——只要干预机制不变，哪怕输入分布变了，模型依然能泛化。这也是为什么说因果是稳健性的终极来源。",[11,1587,1588,1589,114,1679,1707,1708,1886,1887,2062],{},"结构因果模型（SCM）是一个三元组 ",[32,1590,1592,1624],{"className":1591,"translate":36},[35],[32,1593,1595],{"className":1594},[40],[42,1596,1597],{"xmlns":44},[46,1598,1599,1621],{},[49,1600,1601,1603,1605,1607,1609,1611,1614,1616,1619],{},[57,1602,1496],{},[52,1604,368],{},[52,1606,998],{"stretchy":54},[57,1608,1488],{},[52,1610,63],{"separator":62},[57,1612,1613],{},"V",[52,1615,63],{"separator":62},[57,1617,1618],{},"F",[52,1620,1008],{"stretchy":54},[71,1622,1623],{"encoding":73},"M = \\langle U, V, F \\rangle",[32,1625,1627,1645],{"className":1626,"ariaHidden":62},[78],[32,1628,1630,1633,1636,1639,1642],{"className":1629},[82],[32,1631],{"className":1632,"style":212},[86],[32,1634,1496],{"className":1635,"style":1530},[95,96],[32,1637],{"className":1638,"style":177},[105],[32,1640,368],{"className":1641},[181],[32,1643],{"className":1644,"style":177},[105],[32,1646,1648,1651,1654,1657,1660,1663,1666,1669,1672,1676],{"className":1647},[82],[32,1649],{"className":1650,"style":87},[86],[32,1652,998],{"className":1653},[91],[32,1655,1488],{"className":1656,"style":1530},[95,96],[32,1658,63],{"className":1659},[101],[32,1661],{"className":1662,"style":106},[105],[32,1664,1613],{"className":1665,"style":198},[95,96],[32,1667,63],{"className":1668},[101],[32,1670],{"className":1671,"style":106},[105],[32,1673,1618],{"className":1674,"style":1675},[95,96],"margin-right:0.1389em;",[32,1677,1008],{"className":1678},[113],[32,1680,1682,1695],{"className":1681,"translate":36},[35],[32,1683,1685],{"className":1684},[40],[42,1686,1687],{"xmlns":44},[46,1688,1689,1693],{},[49,1690,1691],{},[57,1692,1488],{},[71,1694,1488],{"encoding":73},[32,1696,1698],{"className":1697,"ariaHidden":62},[78],[32,1699,1701,1704],{"className":1700},[82],[32,1702],{"className":1703,"style":212},[86],[32,1705,1488],{"className":1706,"style":1530},[95,96]," 是外生变量，",[32,1709,1711,1753],{"className":1710,"translate":36},[35],[32,1712,1714],{"className":1713},[40],[42,1715,1716],{"xmlns":44},[46,1717,1718,1750],{},[49,1719,1720,1722,1724,1726,1733,1735,1738,1740,1747],{},[57,1721,1613],{},[52,1723,368],{},[52,1725,315],{"stretchy":54},[143,1727,1728,1730],{},[57,1729,1613],{},[155,1731,1732],{},"1",[52,1734,63],{"separator":62},[52,1736,1737],{},"…",[52,1739,63],{"separator":62},[143,1741,1742,1744],{},[57,1743,1613],{},[57,1745,1746],{},"n",[52,1748,1749],{"stretchy":54},"}",[71,1751,1752],{"encoding":73},"V = \\{V_1, \\dots, V_n\\}",[32,1754,1756,1774],{"className":1755,"ariaHidden":62},[78],[32,1757,1759,1762,1765,1768,1771],{"className":1758},[82],[32,1760],{"className":1761,"style":212},[86],[32,1763,1613],{"className":1764,"style":198},[95,96],[32,1766],{"className":1767,"style":177},[105],[32,1769,368],{"className":1770},[181],[32,1772],{"className":1773,"style":177},[105],[32,1775,1777,1780,1783,1824,1827,1830,1833,1836,1839,1842,1883],{"className":1776},[82],[32,1778],{"className":1779,"style":87},[86],[32,1781,315],{"className":1782},[91],[32,1784,1786,1789],{"className":1785},[95],[32,1787,1613],{"className":1788,"style":198},[95,96],[32,1790,1792],{"className":1791},[242],[32,1793,1795,1816],{"className":1794},[246,247],[32,1796,1798,1813],{"className":1797},[251],[32,1799,1801],{"className":1800,"style":256},[255],[32,1802,1804,1807],{"style":1803},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[32,1805],{"className":1806,"style":264},[263],[32,1808,1810],{"className":1809},[268,269,270,271],[32,1811,1732],{"className":1812},[95,271],[32,1814,285],{"className":1815},[284],[32,1817,1819],{"className":1818},[251],[32,1820,1822],{"className":1821,"style":1441},[255],[32,1823],{},[32,1825,63],{"className":1826},[101],[32,1828],{"className":1829,"style":106},[105],[32,1831,1737],{"className":1832},[507],[32,1834],{"className":1835,"style":106},[105],[32,1837,63],{"className":1838},[101],[32,1840],{"className":1841,"style":106},[105],[32,1843,1845,1848],{"className":1844},[95],[32,1846,1613],{"className":1847,"style":198},[95,96],[32,1849,1851],{"className":1850},[242],[32,1852,1854,1875],{"className":1853},[246,247],[32,1855,1857,1872],{"className":1856},[251],[32,1858,1861],{"className":1859,"style":1860},[255],"height:0.1514em;",[32,1862,1863,1866],{"style":1803},[32,1864],{"className":1865,"style":264},[263],[32,1867,1869],{"className":1868},[268,269,270,271],[32,1870,1746],{"className":1871},[95,96,271],[32,1873,285],{"className":1874},[284],[32,1876,1878],{"className":1877},[251],[32,1879,1881],{"className":1880,"style":1441},[255],[32,1882],{},[32,1884,1749],{"className":1885},[113]," 是内生变量，",[32,1888,1890,1929],{"className":1889,"translate":36},[35],[32,1891,1893],{"className":1892},[40],[42,1894,1895],{"xmlns":44},[46,1896,1897,1926],{},[49,1898,1899,1901,1903,1905,1912,1914,1916,1918,1924],{},[57,1900,1618],{},[52,1902,368],{},[52,1904,315],{"stretchy":54},[143,1906,1907,1910],{},[57,1908,1909],{},"f",[155,1911,1732],{},[52,1913,63],{"separator":62},[52,1915,1737],{},[52,1917,63],{"separator":62},[143,1919,1920,1922],{},[57,1921,1909],{},[57,1923,1746],{},[52,1925,1749],{"stretchy":54},[71,1927,1928],{"encoding":73},"F = \\{f_1, \\dots, f_n\\}",[32,1930,1932,1950],{"className":1931,"ariaHidden":62},[78],[32,1933,1935,1938,1941,1944,1947],{"className":1934},[82],[32,1936],{"className":1937,"style":212},[86],[32,1939,1618],{"className":1940,"style":1675},[95,96],[32,1942],{"className":1943,"style":177},[105],[32,1945,368],{"className":1946},[181],[32,1948],{"className":1949,"style":177},[105],[32,1951,1953,1956,1959,2001,2004,2007,2010,2013,2016,2019,2059],{"className":1952},[82],[32,1954],{"className":1955,"style":87},[86],[32,1957,315],{"className":1958},[91],[32,1960,1962,1966],{"className":1961},[95],[32,1963,1909],{"className":1964,"style":1965},[95,96],"margin-right:0.1076em;",[32,1967,1969],{"className":1968},[242],[32,1970,1972,1993],{"className":1971},[246,247],[32,1973,1975,1990],{"className":1974},[251],[32,1976,1978],{"className":1977,"style":256},[255],[32,1979,1981,1984],{"style":1980},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[32,1982],{"className":1983,"style":264},[263],[32,1985,1987],{"className":1986},[268,269,270,271],[32,1988,1732],{"className":1989},[95,271],[32,1991,285],{"className":1992},[284],[32,1994,1996],{"className":1995},[251],[32,1997,1999],{"className":1998,"style":1441},[255],[32,2000],{},[32,2002,63],{"className":2003},[101],[32,2005],{"className":2006,"style":106},[105],[32,2008,1737],{"className":2009},[507],[32,2011],{"className":2012,"style":106},[105],[32,2014,63],{"className":2015},[101],[32,2017],{"className":2018,"style":106},[105],[32,2020,2022,2025],{"className":2021},[95],[32,2023,1909],{"className":2024,"style":1965},[95,96],[32,2026,2028],{"className":2027},[242],[32,2029,2031,2051],{"className":2030},[246,247],[32,2032,2034,2048],{"className":2033},[251],[32,2035,2037],{"className":2036,"style":1860},[255],[32,2038,2039,2042],{"style":1980},[32,2040],{"className":2041,"style":264},[263],[32,2043,2045],{"className":2044},[268,269,270,271],[32,2046,1746],{"className":2047},[95,96,271],[32,2049,285],{"className":2050},[284],[32,2052,2054],{"className":2053},[251],[32,2055,2057],{"className":2056,"style":1441},[255],[32,2058],{},[32,2060,1749],{"className":2061},[113]," 是一组函数，满足",[32,2064,2066],{"className":2065,"translate":36},[299],[32,2067,2069,2128],{"className":2068,"translate":36},[35],[32,2070,2072],{"className":2071},[40],[42,2073,2074],{"xmlns":44,"display":308},[46,2075,2076,2125],{},[49,2077,2078,2085,2087,2089,2095,2097,2105,2107,2113,2115,2117,2123],{},[143,2079,2080,2082],{},[57,2081,1613],{},[57,2083,2084],{},"i",[52,2086,131],{},[52,2088,368],{},[143,2090,2091,2093],{},[57,2092,1909],{},[57,2094,2084],{},[52,2096,55],{"stretchy":54},[49,2098,2099,2102],{},[57,2100,2101],{"mathvariant":1083},"P",[57,2103,2104],{"mathvariant":1083},"a",[52,2106,55],{"stretchy":54},[143,2108,2109,2111],{},[57,2110,1613],{},[57,2112,2084],{},[52,2114,69],{"stretchy":54},[52,2116,63],{"separator":62},[143,2118,2119,2121],{},[57,2120,1488],{},[57,2122,2084],{},[52,2124,69],{"stretchy":54},[71,2126,2127],{"encoding":73},"V_i := f_i(\\mathrm{Pa}(V_i), U_i)",[32,2129,2131,2189],{"className":2130,"ariaHidden":62},[78],[32,2132,2134,2138,2179,2182,2186],{"className":2133},[82],[32,2135],{"className":2136,"style":2137},[86],"height:0.8333em;vertical-align:-0.15em;",[32,2139,2141,2144],{"className":2140},[95],[32,2142,1613],{"className":2143,"style":198},[95,96],[32,2145,2147],{"className":2146},[242],[32,2148,2150,2171],{"className":2149},[246,247],[32,2151,2153,2168],{"className":2152},[251],[32,2154,2157],{"className":2155,"style":2156},[255],"height:0.3117em;",[32,2158,2159,2162],{"style":1803},[32,2160],{"className":2161,"style":264},[263],[32,2163,2165],{"className":2164},[268,269,270,271],[32,2166,2084],{"className":2167},[95,96,271],[32,2169,285],{"className":2170},[284],[32,2172,2174],{"className":2173},[251],[32,2175,2177],{"className":2176,"style":1441},[255],[32,2178],{},[32,2180],{"className":2181,"style":177},[105],[32,2183,2185],{"className":2184},[181],":=",[32,2187],{"className":2188,"style":177},[105],[32,2190,2192,2195,2235,2238,2246,2249,2289,2292,2295,2298,2339],{"className":2191},[82],[32,2193],{"className":2194,"style":87},[86],[32,2196,2198,2201],{"className":2197},[95],[32,2199,1909],{"className":2200,"style":1965},[95,96],[32,2202,2204],{"className":2203},[242],[32,2205,2207,2227],{"className":2206},[246,247],[32,2208,2210,2224],{"className":2209},[251],[32,2211,2213],{"className":2212,"style":2156},[255],[32,2214,2215,2218],{"style":1980},[32,2216],{"className":2217,"style":264},[263],[32,2219,2221],{"className":2220},[268,269,270,271],[32,2222,2084],{"className":2223},[95,96,271],[32,2225,285],{"className":2226},[284],[32,2228,2230],{"className":2229},[251],[32,2231,2233],{"className":2232,"style":1441},[255],[32,2234],{},[32,2236,55],{"className":2237},[91],[32,2239,2241],{"className":2240},[95],[32,2242,2245],{"className":2243},[95,2244],"mathrm","Pa",[32,2247,55],{"className":2248},[91],[32,2250,2252,2255],{"className":2251},[95],[32,2253,1613],{"className":2254,"style":198},[95,96],[32,2256,2258],{"className":2257},[242],[32,2259,2261,2281],{"className":2260},[246,247],[32,2262,2264,2278],{"className":2263},[251],[32,2265,2267],{"className":2266,"style":2156},[255],[32,2268,2269,2272],{"style":1803},[32,2270],{"className":2271,"style":264},[263],[32,2273,2275],{"className":2274},[268,269,270,271],[32,2276,2084],{"className":2277},[95,96,271],[32,2279,285],{"className":2280},[284],[32,2282,2284],{"className":2283},[251],[32,2285,2287],{"className":2286,"style":1441},[255],[32,2288],{},[32,2290,69],{"className":2291},[113],[32,2293,63],{"className":2294},[101],[32,2296],{"className":2297,"style":106},[105],[32,2299,2301,2304],{"className":2300},[95],[32,2302,1488],{"className":2303,"style":1530},[95,96],[32,2305,2307],{"className":2306},[242],[32,2308,2310,2331],{"className":2309},[246,247],[32,2311,2313,2328],{"className":2312},[251],[32,2314,2316],{"className":2315,"style":2156},[255],[32,2317,2319,2322],{"style":2318},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[32,2320],{"className":2321,"style":264},[263],[32,2323,2325],{"className":2324},[268,269,270,271],[32,2326,2084],{"className":2327},[95,96,271],[32,2329,285],{"className":2330},[284],[32,2332,2334],{"className":2333},[251],[32,2335,2337],{"className":2336,"style":1441},[255],[32,2338],{},[32,2340,69],{"className":2341},[113],[11,2343,2344,2345,2375,2376,2405,2406,2409,2410,2439,2440,2577,2578,2733,2734,2786,2787,2897,2898,2926,2927,2955,2956,2984,2985,3073,3074,3125,3126,3154],{},"赋值符号 ",[32,2346,2348,2363],{"className":2347,"translate":36},[35],[32,2349,2351],{"className":2350},[40],[42,2352,2353],{"xmlns":44},[46,2354,2355,2361],{},[49,2356,2357,2359],{},[52,2358,131],{},[52,2360,368],{},[71,2362,2185],{"encoding":73},[32,2364,2366],{"className":2365,"ariaHidden":62},[78],[32,2367,2369,2372],{"className":2368},[82],[32,2370],{"className":2371,"style":1261},[86],[32,2373,2185],{"className":2374},[181],"（而非 ",[32,2377,2379,2392],{"className":2378,"translate":36},[35],[32,2380,2382],{"className":2381},[40],[42,2383,2384],{"xmlns":44},[46,2385,2386,2390],{},[49,2387,2388],{},[52,2389,368],{},[71,2391,368],{"encoding":73},[32,2393,2395],{"className":2394,"ariaHidden":62},[78],[32,2396,2398,2402],{"className":2397},[82],[32,2399],{"className":2400,"style":2401},[86],"height:0.3669em;",[32,2403,368],{"className":2404},[181],"）是关键——它表示这是一个",[1364,2407,2408],{},"机制","，不是一个可逆的代数等式。这组机制天然诱导一张有向无环图（DAG）",[32,2411,2413,2427],{"className":2412,"translate":36},[35],[32,2414,2416],{"className":2415},[40],[42,2417,2418],{"xmlns":44},[46,2419,2420,2425],{},[49,2421,2422],{},[57,2423,2424],{},"G",[71,2426,2424],{"encoding":73},[32,2428,2430],{"className":2429,"ariaHidden":62},[78],[32,2431,2433,2436],{"className":2432},[82],[32,2434],{"className":2435,"style":212},[86],[32,2437,2424],{"className":2438},[95,96],"，边 ",[32,2441,2443,2470],{"className":2442,"translate":36},[35],[32,2444,2446],{"className":2445},[40],[42,2447,2448],{"xmlns":44},[46,2449,2450,2467],{},[49,2451,2452,2459,2461],{},[143,2453,2454,2456],{},[57,2455,1613],{},[57,2457,2458],{},"j",[52,2460,141],{},[143,2462,2463,2465],{},[57,2464,1613],{},[57,2466,2084],{},[71,2468,2469],{"encoding":73},"V_j \\to V_i",[32,2471,2473,2531],{"className":2472,"ariaHidden":62},[78],[32,2474,2476,2480,2522,2525,2528],{"className":2475},[82],[32,2477],{"className":2478,"style":2479},[86],"height:0.9694em;vertical-align:-0.2861em;",[32,2481,2483,2486],{"className":2482},[95],[32,2484,1613],{"className":2485,"style":198},[95,96],[32,2487,2489],{"className":2488},[242],[32,2490,2492,2513],{"className":2491},[246,247],[32,2493,2495,2510],{"className":2494},[251],[32,2496,2498],{"className":2497,"style":2156},[255],[32,2499,2500,2503],{"style":1803},[32,2501],{"className":2502,"style":264},[263],[32,2504,2506],{"className":2505},[268,269,270,271],[32,2507,2458],{"className":2508,"style":2509},[95,96,271],"margin-right:0.0572em;",[32,2511,285],{"className":2512},[284],[32,2514,2516],{"className":2515},[251],[32,2517,2520],{"className":2518,"style":2519},[255],"height:0.2861em;",[32,2521],{},[32,2523],{"className":2524,"style":177},[105],[32,2526,141],{"className":2527},[181],[32,2529],{"className":2530,"style":177},[105],[32,2532,2534,2537],{"className":2533},[82],[32,2535],{"className":2536,"style":2137},[86],[32,2538,2540,2543],{"className":2539},[95],[32,2541,1613],{"className":2542,"style":198},[95,96],[32,2544,2546],{"className":2545},[242],[32,2547,2549,2569],{"className":2548},[246,247],[32,2550,2552,2566],{"className":2551},[251],[32,2553,2555],{"className":2554,"style":2156},[255],[32,2556,2557,2560],{"style":1803},[32,2558],{"className":2559,"style":264},[263],[32,2561,2563],{"className":2562},[268,269,270,271],[32,2564,2084],{"className":2565},[95,96,271],[32,2567,285],{"className":2568},[284],[32,2570,2572],{"className":2571},[251],[32,2573,2575],{"className":2574,"style":1441},[255],[32,2576],{}," 存在当且仅当 ",[32,2579,2581,2617],{"className":2580,"translate":36},[35],[32,2582,2584],{"className":2583},[40],[42,2585,2586],{"xmlns":44},[46,2587,2588,2614],{},[49,2589,2590,2596,2598,2604,2606,2612],{},[143,2591,2592,2594],{},[57,2593,1613],{},[57,2595,2458],{},[52,2597,1316],{},[49,2599,2600,2602],{},[57,2601,2101],{"mathvariant":1083},[57,2603,2104],{"mathvariant":1083},[52,2605,55],{"stretchy":54},[143,2607,2608,2610],{},[57,2609,1613],{},[57,2611,2084],{},[52,2613,69],{"stretchy":54},[71,2615,2616],{"encoding":73},"V_j \\in \\mathrm{Pa}(V_i)",[32,2618,2620,2675],{"className":2619,"ariaHidden":62},[78],[32,2621,2623,2626,2666,2669,2672],{"className":2622},[82],[32,2624],{"className":2625,"style":2479},[86],[32,2627,2629,2632],{"className":2628},[95],[32,2630,1613],{"className":2631,"style":198},[95,96],[32,2633,2635],{"className":2634},[242],[32,2636,2638,2658],{"className":2637},[246,247],[32,2639,2641,2655],{"className":2640},[251],[32,2642,2644],{"className":2643,"style":2156},[255],[32,2645,2646,2649],{"style":1803},[32,2647],{"className":2648,"style":264},[263],[32,2650,2652],{"className":2651},[268,269,270,271],[32,2653,2458],{"className":2654,"style":2509},[95,96,271],[32,2656,285],{"className":2657},[284],[32,2659,2661],{"className":2660},[251],[32,2662,2664],{"className":2663,"style":2519},[255],[32,2665],{},[32,2667],{"className":2668,"style":177},[105],[32,2670,1316],{"className":2671},[181],[32,2673],{"className":2674,"style":177},[105],[32,2676,2678,2681,2687,2690,2730],{"className":2677},[82],[32,2679],{"className":2680,"style":87},[86],[32,2682,2684],{"className":2683},[95],[32,2685,2245],{"className":2686},[95,2244],[32,2688,55],{"className":2689},[91],[32,2691,2693,2696],{"className":2692},[95],[32,2694,1613],{"className":2695,"style":198},[95,96],[32,2697,2699],{"className":2698},[242],[32,2700,2702,2722],{"className":2701},[246,247],[32,2703,2705,2719],{"className":2704},[251],[32,2706,2708],{"className":2707,"style":2156},[255],[32,2709,2710,2713],{"style":1803},[32,2711],{"className":2712,"style":264},[263],[32,2714,2716],{"className":2715},[268,269,270,271],[32,2717,2084],{"className":2718},[95,96,271],[32,2720,285],{"className":2721},[284],[32,2723,2725],{"className":2724},[251],[32,2726,2728],{"className":2727,"style":1441},[255],[32,2729],{},[32,2731,69],{"className":2732},[113],"。因果推断的核心操作是 ",[32,2735,2737,2761],{"className":2736,"translate":36},[35],[32,2738,2740],{"className":2739},[40],[42,2741,2742],{"xmlns":44},[46,2743,2744,2758],{},[49,2745,2746,2752,2754,2756],{},[49,2747,2748,2750],{},[57,2749,66],{"mathvariant":1083},[57,2751,1512],{"mathvariant":1083},[52,2753,55],{"stretchy":54},[52,2755,1001],{},[52,2757,69],{"stretchy":54},[71,2759,2760],{"encoding":73},"\\mathrm{do}(\\cdot)",[32,2762,2764],{"className":2763,"ariaHidden":62},[78],[32,2765,2767,2770,2777,2780,2783],{"className":2766},[82],[32,2768],{"className":2769,"style":87},[86],[32,2771,2773],{"className":2772},[95],[32,2774,2776],{"className":2775},[95,2244],"do",[32,2778,55],{"className":2779},[91],[32,2781,1001],{"className":2782},[95],[32,2784,69],{"className":2785},[113]," 算子：",[32,2788,2790,2830],{"className":2789,"translate":36},[35],[32,2791,2793],{"className":2792},[40],[42,2794,2795],{"xmlns":44},[46,2796,2797,2827],{},[49,2798,2799,2801,2803,2806,2809,2815,2817,2819,2821,2823,2825],{},[57,2800,2101],{},[52,2802,55],{"stretchy":54},[57,2804,2805],{},"Y",[52,2807,2808],{},"∣",[49,2810,2811,2813],{},[57,2812,66],{"mathvariant":1083},[57,2814,1512],{"mathvariant":1083},[52,2816,55],{"stretchy":54},[57,2818,59],{},[52,2820,368],{},[57,2822,358],{},[52,2824,69],{"stretchy":54},[52,2826,69],{"stretchy":54},[71,2828,2829],{"encoding":73},"P(Y \\mid \\mathrm{do}(X=x))",[32,2831,2833,2857,2884],{"className":2832,"ariaHidden":62},[78],[32,2834,2836,2839,2842,2845,2848,2851,2854],{"className":2835},[82],[32,2837],{"className":2838,"style":87},[86],[32,2840,2101],{"className":2841,"style":1675},[95,96],[32,2843,55],{"className":2844},[91],[32,2846,2805],{"className":2847,"style":198},[95,96],[32,2849],{"className":2850,"style":177},[105],[32,2852,2808],{"className":2853},[181],[32,2855],{"className":2856,"style":177},[105],[32,2858,2860,2863,2869,2872,2875,2878,2881],{"className":2859},[82],[32,2861],{"className":2862,"style":87},[86],[32,2864,2866],{"className":2865},[95],[32,2867,2776],{"className":2868},[95,2244],[32,2870,55],{"className":2871},[91],[32,2873,59],{"className":2874,"style":97},[95,96],[32,2876],{"className":2877,"style":177},[105],[32,2879,368],{"className":2880},[181],[32,2882],{"className":2883,"style":177},[105],[32,2885,2887,2890,2893],{"className":2886},[82],[32,2888],{"className":2889,"style":87},[86],[32,2891,358],{"className":2892},[95,96],[32,2894,2896],{"className":2895},[113],"))"," 表示「把 ",[32,2899,2901,2914],{"className":2900,"translate":36},[35],[32,2902,2904],{"className":2903},[40],[42,2905,2906],{"xmlns":44},[46,2907,2908,2912],{},[49,2909,2910],{},[57,2911,59],{},[71,2913,59],{"encoding":73},[32,2915,2917],{"className":2916,"ariaHidden":62},[78],[32,2918,2920,2923],{"className":2919},[82],[32,2921],{"className":2922,"style":212},[86],[32,2924,59],{"className":2925,"style":97},[95,96]," 的赋值机制替换为常数 ",[32,2928,2930,2943],{"className":2929,"translate":36},[35],[32,2931,2933],{"className":2932},[40],[42,2934,2935],{"xmlns":44},[46,2936,2937,2941],{},[49,2938,2939],{},[57,2940,358],{},[71,2942,358],{"encoding":73},[32,2944,2946],{"className":2945,"ariaHidden":62},[78],[32,2947,2949,2952],{"className":2948},[82],[32,2950],{"className":2951,"style":1261},[86],[32,2953,358],{"className":2954},[95,96],"，切断它与所有父节点的连接」之后 ",[32,2957,2959,2972],{"className":2958,"translate":36},[35],[32,2960,2962],{"className":2961},[40],[42,2963,2964],{"xmlns":44},[46,2965,2966,2970],{},[49,2967,2968],{},[57,2969,2805],{},[71,2971,2805],{"encoding":73},[32,2973,2975],{"className":2974,"ariaHidden":62},[78],[32,2976,2978,2981],{"className":2977},[82],[32,2979],{"className":2980,"style":212},[86],[32,2982,2805],{"className":2983,"style":198},[95,96]," 的分布，这与条件概率 ",[32,2986,2988,3016],{"className":2987,"translate":36},[35],[32,2989,2991],{"className":2990},[40],[42,2992,2993],{"xmlns":44},[46,2994,2995,3013],{},[49,2996,2997,2999,3001,3003,3005,3007,3009,3011],{},[57,2998,2101],{},[52,3000,55],{"stretchy":54},[57,3002,2805],{},[52,3004,2808],{},[57,3006,59],{},[52,3008,368],{},[57,3010,358],{},[52,3012,69],{"stretchy":54},[71,3014,3015],{"encoding":73},"P(Y \\mid X=x)",[32,3017,3019,3043,3061],{"className":3018,"ariaHidden":62},[78],[32,3020,3022,3025,3028,3031,3034,3037,3040],{"className":3021},[82],[32,3023],{"className":3024,"style":87},[86],[32,3026,2101],{"className":3027,"style":1675},[95,96],[32,3029,55],{"className":3030},[91],[32,3032,2805],{"className":3033,"style":198},[95,96],[32,3035],{"className":3036,"style":177},[105],[32,3038,2808],{"className":3039},[181],[32,3041],{"className":3042,"style":177},[105],[32,3044,3046,3049,3052,3055,3058],{"className":3045},[82],[32,3047],{"className":3048,"style":212},[86],[32,3050,59],{"className":3051,"style":97},[95,96],[32,3053],{"className":3054,"style":177},[105],[32,3056,368],{"className":3057},[181],[32,3059],{"className":3060,"style":177},[105],[32,3062,3064,3067,3070],{"className":3063},[82],[32,3065],{"className":3066,"style":87},[86],[32,3068,358],{"className":3069},[95,96],[32,3071,69],{"className":3072},[113],"——「观察到 ",[32,3075,3077,3095],{"className":3076,"translate":36},[35],[32,3078,3080],{"className":3079},[40],[42,3081,3082],{"xmlns":44},[46,3083,3084,3092],{},[49,3085,3086,3088,3090],{},[57,3087,59],{},[52,3089,368],{},[57,3091,358],{},[71,3093,3094],{"encoding":73},"X=x",[32,3096,3098,3116],{"className":3097,"ariaHidden":62},[78],[32,3099,3101,3104,3107,3110,3113],{"className":3100},[82],[32,3102],{"className":3103,"style":212},[86],[32,3105,59],{"className":3106,"style":97},[95,96],[32,3108],{"className":3109,"style":177},[105],[32,3111,368],{"className":3112},[181],[32,3114],{"className":3115,"style":177},[105],[32,3117,3119,3122],{"className":3118},[82],[32,3120],{"className":3121,"style":1261},[86],[32,3123,358],{"className":3124},[95,96]," 之后 ",[32,3127,3129,3142],{"className":3128,"translate":36},[35],[32,3130,3132],{"className":3131},[40],[42,3133,3134],{"xmlns":44},[46,3135,3136,3140],{},[49,3137,3138],{},[57,3139,2805],{},[71,3141,2805],{"encoding":73},[32,3143,3145],{"className":3144,"ariaHidden":62},[78],[32,3146,3148,3151],{"className":3147},[82],[32,3149],{"className":3150,"style":212},[86],[32,3152,2805],{"className":3153,"style":198},[95,96]," 的分布」——一般是不相等的。两者的差异，正是相关不等于因果的形式化版本。",[21,3156,3157],{"id":3157},"因果与拓扑",[11,3159,3160],{},"一个自然的问题是：几何遗忘度量得到拓扑，那因果是不是遗忘更多东西之后剩下的更轻的骨架？答案是否定的。",[11,3162,3163,3164,3167,3168,3171,3172,3175,3176,3220],{},"几何到拓扑的「变轻」是",[1364,3165,3166],{},"同一个信息维度上的压缩","——都是对称关系，只是分辨率从「连续的数值」降到了「是否连通」的二值\u002F离散判断。但因果结构引入了一个几何和拓扑都没有的新维度：",[1364,3169,3170],{},"方向性\u002F非对称性","，并且这个方向性原则上",[1364,3173,3174],{},"不能","从纯观测的联合分布 ",[32,3177,3179,3199],{"className":3178,"translate":36},[35],[32,3180,3182],{"className":3181},[40],[42,3183,3184],{"xmlns":44},[46,3185,3186,3196],{},[49,3187,3188,3190,3192,3194],{},[57,3189,2101],{},[52,3191,55],{"stretchy":54},[57,3193,1613],{},[52,3195,69],{"stretchy":54},[71,3197,3198],{"encoding":73},"P(V)",[32,3200,3202],{"className":3201,"ariaHidden":62},[78],[32,3203,3205,3208,3211,3214,3217],{"className":3204},[82],[32,3206],{"className":3207,"style":87},[86],[32,3209,2101],{"className":3210,"style":1675},[95,96],[32,3212,55],{"className":3213},[91],[32,3215,1613],{"className":3216,"style":198},[95,96],[32,3218,69],{"className":3219},[113]," 中单独用几何或拓扑手段识别出来。这就是可识别性问题（identifiability）：",[3222,3223,3224],"blockquote",{},[11,3225,3226,3229,3230,3273,3274,3331,3332,3383,3384,3383,3435,3511,3512,3568],{},[1364,3227,3228],{},"命题（因果不可从观测唯一识别）。"," 对于两个变量 ",[32,3231,3233,3251],{"className":3232,"translate":36},[35],[32,3234,3236],{"className":3235},[40],[42,3237,3238],{"xmlns":44},[46,3239,3240,3248],{},[49,3241,3242,3244,3246],{},[57,3243,59],{},[52,3245,63],{"separator":62},[57,3247,2805],{},[71,3249,3250],{"encoding":73},"X, Y",[32,3252,3254],{"className":3253,"ariaHidden":62},[78],[32,3255,3257,3261,3264,3267,3270],{"className":3256},[82],[32,3258],{"className":3259,"style":3260},[86],"height:0.8778em;vertical-align:-0.1944em;",[32,3262,59],{"className":3263,"style":97},[95,96],[32,3265,63],{"className":3266},[101],[32,3268],{"className":3269,"style":106},[105],[32,3271,2805],{"className":3272,"style":198},[95,96],"，仅凭观测联合分布 ",[32,3275,3277,3301],{"className":3276,"translate":36},[35],[32,3278,3280],{"className":3279},[40],[42,3281,3282],{"xmlns":44},[46,3283,3284,3298],{},[49,3285,3286,3288,3290,3292,3294,3296],{},[57,3287,2101],{},[52,3289,55],{"stretchy":54},[57,3291,59],{},[52,3293,63],{"separator":62},[57,3295,2805],{},[52,3297,69],{"stretchy":54},[71,3299,3300],{"encoding":73},"P(X,Y)",[32,3302,3304],{"className":3303,"ariaHidden":62},[78],[32,3305,3307,3310,3313,3316,3319,3322,3325,3328],{"className":3306},[82],[32,3308],{"className":3309,"style":87},[86],[32,3311,2101],{"className":3312,"style":1675},[95,96],[32,3314,55],{"className":3315},[91],[32,3317,59],{"className":3318,"style":97},[95,96],[32,3320,63],{"className":3321},[101],[32,3323],{"className":3324,"style":106},[105],[32,3326,2805],{"className":3327,"style":198},[95,96],[32,3329,69],{"className":3330},[113],"，一般无法在 ",[32,3333,3335,3353],{"className":3334,"translate":36},[35],[32,3336,3338],{"className":3337},[40],[42,3339,3340],{"xmlns":44},[46,3341,3342,3350],{},[49,3343,3344,3346,3348],{},[57,3345,59],{},[52,3347,141],{},[57,3349,2805],{},[71,3351,3352],{"encoding":73},"X \\to Y",[32,3354,3356,3374],{"className":3355,"ariaHidden":62},[78],[32,3357,3359,3362,3365,3368,3371],{"className":3358},[82],[32,3360],{"className":3361,"style":212},[86],[32,3363,59],{"className":3364,"style":97},[95,96],[32,3366],{"className":3367,"style":177},[105],[32,3369,141],{"className":3370},[181],[32,3372],{"className":3373,"style":177},[105],[32,3375,3377,3380],{"className":3376},[82],[32,3378],{"className":3379,"style":212},[86],[32,3381,2805],{"className":3382,"style":198},[95,96],"、",[32,3385,3387,3405],{"className":3386,"translate":36},[35],[32,3388,3390],{"className":3389},[40],[42,3391,3392],{"xmlns":44},[46,3393,3394,3402],{},[49,3395,3396,3398,3400],{},[57,3397,2805],{},[52,3399,141],{},[57,3401,59],{},[71,3403,3404],{"encoding":73},"Y \\to X",[32,3406,3408,3426],{"className":3407,"ariaHidden":62},[78],[32,3409,3411,3414,3417,3420,3423],{"className":3410},[82],[32,3412],{"className":3413,"style":212},[86],[32,3415,2805],{"className":3416,"style":198},[95,96],[32,3418],{"className":3419,"style":177},[105],[32,3421,141],{"className":3422},[181],[32,3424],{"className":3425,"style":177},[105],[32,3427,3429,3432],{"className":3428},[82],[32,3430],{"className":3431,"style":212},[86],[32,3433,59],{"className":3434,"style":97},[95,96],[32,3436,3438,3462],{"className":3437,"translate":36},[35],[32,3439,3441],{"className":3440},[40],[42,3442,3443],{"xmlns":44},[46,3444,3445,3459],{},[49,3446,3447,3449,3452,3455,3457],{},[57,3448,59],{},[52,3450,3451],{},"←",[57,3453,3454],{},"Z",[52,3456,141],{},[57,3458,2805],{},[71,3460,3461],{"encoding":73},"X \\leftarrow Z \\to Y",[32,3463,3465,3483,3502],{"className":3464,"ariaHidden":62},[78],[32,3466,3468,3471,3474,3477,3480],{"className":3467},[82],[32,3469],{"className":3470,"style":212},[86],[32,3472,59],{"className":3473,"style":97},[95,96],[32,3475],{"className":3476,"style":177},[105],[32,3478,3451],{"className":3479},[181],[32,3481],{"className":3482,"style":177},[105],[32,3484,3486,3489,3493,3496,3499],{"className":3485},[82],[32,3487],{"className":3488,"style":212},[86],[32,3490,3454],{"className":3491,"style":3492},[95,96],"margin-right:0.0715em;",[32,3494],{"className":3495,"style":177},[105],[32,3497,141],{"className":3498},[181],[32,3500],{"className":3501,"style":177},[105],[32,3503,3505,3508],{"className":3504},[82],[32,3506],{"className":3507,"style":212},[86],[32,3509,2805],{"className":3510,"style":198},[95,96],"（混杂）等因果结构之间做出唯一判定——它们可以诱导出完全相同的 ",[32,3513,3515,3538],{"className":3514,"translate":36},[35],[32,3516,3518],{"className":3517},[40],[42,3519,3520],{"xmlns":44},[46,3521,3522,3536],{},[49,3523,3524,3526,3528,3530,3532,3534],{},[57,3525,2101],{},[52,3527,55],{"stretchy":54},[57,3529,59],{},[52,3531,63],{"separator":62},[57,3533,2805],{},[52,3535,69],{"stretchy":54},[71,3537,3300],{"encoding":73},[32,3539,3541],{"className":3540,"ariaHidden":62},[78],[32,3542,3544,3547,3550,3553,3556,3559,3562,3565],{"className":3543},[82],[32,3545],{"className":3546,"style":87},[86],[32,3548,2101],{"className":3549,"style":1675},[95,96],[32,3551,55],{"className":3552},[91],[32,3554,59],{"className":3555,"style":97},[95,96],[32,3557,63],{"className":3558},[101],[32,3560],{"className":3561,"style":106},[105],[32,3563,2805],{"className":3564,"style":198},[95,96],[32,3566,69],{"className":3567},[113],"。",[11,3570,3571,3572,3606,3607,1234,3635,3663],{},"要打破这种不可识别性，需要额外的信息源：随机对照实验（真正的 ",[32,3573,3575,3591],{"className":3574,"translate":36},[35],[32,3576,3578],{"className":3577},[40],[42,3579,3580],{"xmlns":44},[46,3581,3582,3588],{},[49,3583,3584,3586],{},[57,3585,66],{"mathvariant":1083},[57,3587,1512],{"mathvariant":1083},[71,3589,3590],{"encoding":73},"\\mathrm{do}",[32,3592,3594],{"className":3593,"ariaHidden":62},[78],[32,3595,3597,3600],{"className":3596},[82],[32,3598],{"className":3599,"style":170},[86],[32,3601,3603],{"className":3602},[95],[32,3604,2776],{"className":3605},[95,2244],"）、时间先后顺序、结构假设（如可加噪声模型、非高斯性，如 LiNGAM 方法利用非高斯噪声打破对称性）。这说明因果结构不是几何\u002F拓扑那条压缩链上的下一站，而是一个正交的轴——即便你拥有关于系统的完整几何信息（所有点对的精确距离）和完整拓扑信息（所有连通关系），你依然可能无法回答「如果我强行改变 ",[32,3608,3610,3623],{"className":3609,"translate":36},[35],[32,3611,3613],{"className":3612},[40],[42,3614,3615],{"xmlns":44},[46,3616,3617,3621],{},[49,3618,3619],{},[57,3620,59],{},[71,3622,59],{"encoding":73},[32,3624,3626],{"className":3625,"ariaHidden":62},[78],[32,3627,3629,3632],{"className":3628},[82],[32,3630],{"className":3631,"style":212},[86],[32,3633,59],{"className":3634,"style":97},[95,96],[32,3636,3638,3651],{"className":3637,"translate":36},[35],[32,3639,3641],{"className":3640},[40],[42,3642,3643],{"xmlns":44},[46,3644,3645,3649],{},[49,3646,3647],{},[57,3648,2805],{},[71,3650,2805],{"encoding":73},[32,3652,3654],{"className":3653,"ariaHidden":62},[78],[32,3655,3657,3660],{"className":3656},[82],[32,3658],{"className":3659,"style":212},[86],[32,3661,2805],{"className":3662,"style":198},[95,96]," 会怎样」这个问题。这正是 Pearl 提出因果之梯（Ladder of Causation）的意义所在：",[3665,3666,3667,3686],"table",{},[3668,3669,3670],"thead",{},[3671,3672,3673,3677,3680,3683],"tr",{},[3674,3675,3676],"th",{},"层级",[3674,3678,3679],{},"问题类型",[3674,3681,3682],{},"例子",[3674,3684,3685],{},"所需信息",[3687,3688,3689,3769,3867],"tbody",{},[3671,3690,3691,3695,3698,3766],{},[3692,3693,3694],"td",{},"1. 关联",[3692,3696,3697],{},"看见（seeing）",[3692,3699,3700],{},[32,3701,3703,3727],{"className":3702,"translate":36},[35],[32,3704,3706],{"className":3705},[40],[42,3707,3708],{"xmlns":44},[46,3709,3710,3724],{},[49,3711,3712,3714,3716,3718,3720,3722],{},[57,3713,2101],{},[52,3715,55],{"stretchy":54},[57,3717,2805],{},[52,3719,2808],{},[57,3721,59],{},[52,3723,69],{"stretchy":54},[71,3725,3726],{"encoding":73},"P(Y\\mid X)",[32,3728,3730,3754],{"className":3729,"ariaHidden":62},[78],[32,3731,3733,3736,3739,3742,3745,3748,3751],{"className":3732},[82],[32,3734],{"className":3735,"style":87},[86],[32,3737,2101],{"className":3738,"style":1675},[95,96],[32,3740,55],{"className":3741},[91],[32,3743,2805],{"className":3744,"style":198},[95,96],[32,3746],{"className":3747,"style":177},[105],[32,3749,2808],{"className":3750},[181],[32,3752],{"className":3753,"style":177},[105],[32,3755,3757,3760,3763],{"className":3756},[82],[32,3758],{"className":3759,"style":87},[86],[32,3761,59],{"className":3762,"style":97},[95,96],[32,3764,69],{"className":3765},[113],[3692,3767,3768],{},"观测数据即可，几何\u002F拓扑视角覆盖此层",[3671,3770,3771,3774,3777,3864],{},[3692,3772,3773],{},"2. 干预",[3692,3775,3776],{},"行动（doing）",[3692,3778,3779],{},[32,3780,3782,3816],{"className":3781,"translate":36},[35],[32,3783,3785],{"className":3784},[40],[42,3786,3787],{"xmlns":44},[46,3788,3789,3813],{},[49,3790,3791,3793,3795,3797,3799,3805,3807,3809,3811],{},[57,3792,2101],{},[52,3794,55],{"stretchy":54},[57,3796,2805],{},[52,3798,2808],{},[49,3800,3801,3803],{},[57,3802,66],{"mathvariant":1083},[57,3804,1512],{"mathvariant":1083},[52,3806,55],{"stretchy":54},[57,3808,59],{},[52,3810,69],{"stretchy":54},[52,3812,69],{"stretchy":54},[71,3814,3815],{"encoding":73},"P(Y\\mid \\mathrm{do}(X))",[32,3817,3819,3843],{"className":3818,"ariaHidden":62},[78],[32,3820,3822,3825,3828,3831,3834,3837,3840],{"className":3821},[82],[32,3823],{"className":3824,"style":87},[86],[32,3826,2101],{"className":3827,"style":1675},[95,96],[32,3829,55],{"className":3830},[91],[32,3832,2805],{"className":3833,"style":198},[95,96],[32,3835],{"className":3836,"style":177},[105],[32,3838,2808],{"className":3839},[181],[32,3841],{"className":3842,"style":177},[105],[32,3844,3846,3849,3855,3858,3861],{"className":3845},[82],[32,3847],{"className":3848,"style":87},[86],[32,3850,3852],{"className":3851},[95],[32,3853,2776],{"className":3854},[95,2244],[32,3856,55],{"className":3857},[91],[32,3859,59],{"className":3860,"style":97},[95,96],[32,3862,2896],{"className":3863},[113],[3692,3865,3866],{},"需要实验或因果图假设",[3671,3868,3869,3872,3875,3936],{},[3692,3870,3871],{},"3. 反事实",[3692,3873,3874],{},"想象（imagining）",[3692,3876,3877,3878,3906,3907,3935],{},"若 ",[32,3879,3881,3894],{"className":3880,"translate":36},[35],[32,3882,3884],{"className":3883},[40],[42,3885,3886],{"xmlns":44},[46,3887,3888,3892],{},[49,3889,3890],{},[57,3891,59],{},[71,3893,59],{"encoding":73},[32,3895,3897],{"className":3896,"ariaHidden":62},[78],[32,3898,3900,3903],{"className":3899},[82],[32,3901],{"className":3902,"style":212},[86],[32,3904,59],{"className":3905,"style":97},[95,96]," 不同，",[32,3908,3910,3923],{"className":3909,"translate":36},[35],[32,3911,3913],{"className":3912},[40],[42,3914,3915],{"xmlns":44},[46,3916,3917,3921],{},[49,3918,3919],{},[57,3920,2805],{},[71,3922,2805],{"encoding":73},[32,3924,3926],{"className":3925,"ariaHidden":62},[78],[32,3927,3929,3932],{"className":3928},[82],[32,3930],{"className":3931,"style":212},[86],[32,3933,2805],{"className":3934,"style":198},[95,96]," 会怎样",[3692,3937,3938],{},"需要完整 SCM，包括外生噪声的联合分布",[11,3940,3941],{},"几何和拓扑视角，本质上被限制在第一层；只有引入因果结构，才能爬上第二、第三层。这也是为什么「因果关系需要额外信息才能识别」这句话不是技术细节，而是三种视角在存在论意义上的分界线。",[21,3943,3944],{"id":3944},"机器学习视角",[11,3946,3947],{},"回到机器学习的语境，三种视角之间的差异直接决定了模型在分布偏移下的表现。可以按「随分布变化保持不变的程度」给三者排一个序：",[32,3949,3951],{"className":3950,"translate":36},[299],[32,3952,3954,3988],{"className":3953,"translate":36},[35],[32,3955,3957],{"className":3956},[40],[42,3958,3959],{"xmlns":44,"display":308},[46,3960,3961,3985],{},[49,3962,3963,3966,3968,3971,3973,3976,3978,3980,3982],{},[372,3964,3965],{},"Geometry",[372,3967,374],{},[52,3969,3970],{},"≺",[372,3972,374],{},[372,3974,3975],{},"Topology Invariance",[372,3977,374],{},[52,3979,3970],{},[372,3981,374],{},[372,3983,3984],{},"Causality",[71,3986,3987],{"encoding":73},"\\text{Geometry} \\;\\prec\\; \\text{Topology Invariance} \\;\\prec\\; \\text{Causality}",[32,3989,3991,4019,4047],{"className":3990,"ariaHidden":62},[78],[32,3992,3994,3997,4004,4007,4010,4013,4016],{"className":3993},[82],[32,3995],{"className":3996,"style":3260},[86],[32,3998,4001],{"className":3999},[95,4000],"text",[32,4002,3965],{"className":4003},[95],[32,4005],{"className":4006,"style":177},[105],[32,4008],{"className":4009,"style":177},[105],[32,4011,3970],{"className":4012},[181],[32,4014],{"className":4015,"style":177},[105],[32,4017],{"className":4018,"style":177},[105],[32,4020,4022,4026,4032,4035,4038,4041,4044],{"className":4021},[82],[32,4023],{"className":4024,"style":4025},[86],"height:0.8889em;vertical-align:-0.1944em;",[32,4027,4029],{"className":4028},[95,4000],[32,4030,3975],{"className":4031},[95],[32,4033],{"className":4034,"style":177},[105],[32,4036],{"className":4037,"style":177},[105],[32,4039,3970],{"className":4040},[181],[32,4042],{"className":4043,"style":177},[105],[32,4045],{"className":4046,"style":177},[105],[32,4048,4050,4053],{"className":4049},[82],[32,4051],{"className":4052,"style":4025},[86],[32,4054,4056],{"className":4055},[95,4000],[32,4057,3984],{"className":4058},[95],[4060,4061,4062,4069,4119],"ul",{},[4063,4064,4065,4068],"li",{},[1364,4066,4067],{},"几何层面的失败","：训练集和测试集哪怕只是特征做了不同的归一化，相似度排序就可能整体偏移——这是最脆弱的一层，任何非本质的表示变化都会渗透进度量数值本身。",[4063,4070,4071,4074,4075,4118],{},[1364,4072,4073],{},"拓扑层面的稳健性","：持久同调等方法对噪声、尺度变化天然不敏感，因为它们本来就是在滤除度量细节之后提取「形状」。但拓扑结构依然是观测分布 ",[32,4076,4078,4097],{"className":4077,"translate":36},[35],[32,4079,4081],{"className":4080},[40],[42,4082,4083],{"xmlns":44},[46,4084,4085,4095],{},[49,4086,4087,4089,4091,4093],{},[57,4088,2101],{},[52,4090,55],{"stretchy":54},[57,4092,1613],{},[52,4094,69],{"stretchy":54},[71,4096,3198],{"encoding":73},[32,4098,4100],{"className":4099,"ariaHidden":62},[78],[32,4101,4103,4106,4109,4112,4115],{"className":4102},[82],[32,4104],{"className":4105,"style":87},[86],[32,4107,2101],{"className":4108,"style":1675},[95,96],[32,4110,55],{"className":4111},[91],[32,4113,1613],{"className":4114,"style":198},[95,96],[32,4116,69],{"className":4117},[113]," 的函数——一旦分布本身发生结构性改变（不只是尺度变化，而是变量间依赖关系变了），拓扑不变量也会随之改变，因为它仍然停留在因果之梯的第一层。",[4063,4120,4121,4124,4125,4175,4176,4316,4317,4361,4362,4446],{},[1364,4122,4123],{},"因果层面的稳健性","：如果 ",[32,4126,4128,4145],{"className":4127,"translate":36},[35],[32,4129,4131],{"className":4130},[40],[42,4132,4133],{"xmlns":44},[46,4134,4135,4143],{},[49,4136,4137,4139,4141],{},[57,4138,59],{},[52,4140,141],{},[57,4142,2805],{},[71,4144,3352],{"encoding":73},[32,4146,4148,4166],{"className":4147,"ariaHidden":62},[78],[32,4149,4151,4154,4157,4160,4163],{"className":4150},[82],[32,4152],{"className":4153,"style":212},[86],[32,4155,59],{"className":4156,"style":97},[95,96],[32,4158],{"className":4159,"style":177},[105],[32,4161,141],{"className":4162},[181],[32,4164],{"className":4165,"style":177},[105],[32,4167,4169,4172],{"className":4168},[82],[32,4170],{"className":4171,"style":212},[86],[32,4173,2805],{"className":4174,"style":198},[95,96]," 的机制 ",[32,4177,4179,4211],{"className":4178,"translate":36},[35],[32,4180,4182],{"className":4181},[40],[42,4183,4184],{"xmlns":44},[46,4185,4186,4208],{},[49,4187,4188,4194,4196,4198,4200,4206],{},[143,4189,4190,4192],{},[57,4191,1909],{},[57,4193,2805],{},[52,4195,55],{"stretchy":54},[57,4197,59],{},[52,4199,63],{"separator":62},[143,4201,4202,4204],{},[57,4203,1488],{},[57,4205,2805],{},[52,4207,69],{"stretchy":54},[71,4209,4210],{"encoding":73},"f_Y(X, U_Y)",[32,4212,4214],{"className":4213,"ariaHidden":62},[78],[32,4215,4217,4220,4261,4264,4267,4270,4273,4313],{"className":4216},[82],[32,4218],{"className":4219,"style":87},[86],[32,4221,4223,4226],{"className":4222},[95],[32,4224,1909],{"className":4225,"style":1965},[95,96],[32,4227,4229],{"className":4228},[242],[32,4230,4232,4253],{"className":4231},[246,247],[32,4233,4235,4250],{"className":4234},[251],[32,4236,4239],{"className":4237,"style":4238},[255],"height:0.3283em;",[32,4240,4241,4244],{"style":1980},[32,4242],{"className":4243,"style":264},[263],[32,4245,4247],{"className":4246},[268,269,270,271],[32,4248,2805],{"className":4249,"style":198},[95,96,271],[32,4251,285],{"className":4252},[284],[32,4254,4256],{"className":4255},[251],[32,4257,4259],{"className":4258,"style":1441},[255],[32,4260],{},[32,4262,55],{"className":4263},[91],[32,4265,59],{"className":4266,"style":97},[95,96],[32,4268,63],{"className":4269},[101],[32,4271],{"className":4272,"style":106},[105],[32,4274,4276,4279],{"className":4275},[95],[32,4277,1488],{"className":4278,"style":1530},[95,96],[32,4280,4282],{"className":4281},[242],[32,4283,4285,4305],{"className":4284},[246,247],[32,4286,4288,4302],{"className":4287},[251],[32,4289,4291],{"className":4290,"style":4238},[255],[32,4292,4293,4296],{"style":2318},[32,4294],{"className":4295,"style":264},[263],[32,4297,4299],{"className":4298},[268,269,270,271],[32,4300,2805],{"className":4301,"style":198},[95,96,271],[32,4303,285],{"className":4304},[284],[32,4306,4308],{"className":4307},[251],[32,4309,4311],{"className":4310,"style":1441},[255],[32,4312],{},[32,4314,69],{"className":4315},[113]," 本身不随环境改变（这是很多领域里的物理假设——比如「重力加速度与颜色无关」），那么哪怕 ",[32,4318,4320,4340],{"className":4319,"translate":36},[35],[32,4321,4323],{"className":4322},[40],[42,4324,4325],{"xmlns":44},[46,4326,4327,4337],{},[49,4328,4329,4331,4333,4335],{},[57,4330,2101],{},[52,4332,55],{"stretchy":54},[57,4334,59],{},[52,4336,69],{"stretchy":54},[71,4338,4339],{"encoding":73},"P(X)",[32,4341,4343],{"className":4342,"ariaHidden":62},[78],[32,4344,4346,4349,4352,4355,4358],{"className":4345},[82],[32,4347],{"className":4348,"style":87},[86],[32,4350,2101],{"className":4351,"style":1675},[95,96],[32,4353,55],{"className":4354},[91],[32,4356,59],{"className":4357,"style":97},[95,96],[32,4359,69],{"className":4360},[113]," 的边际分布因为采样环境不同而剧烈变化，",[32,4363,4365,4398],{"className":4364,"translate":36},[35],[32,4366,4368],{"className":4367},[40],[42,4369,4370],{"xmlns":44},[46,4371,4372,4396],{},[49,4373,4374,4376,4378,4380,4382,4388,4390,4392,4394],{},[57,4375,2101],{},[52,4377,55],{"stretchy":54},[57,4379,2805],{},[52,4381,2808],{},[49,4383,4384,4386],{},[57,4385,66],{"mathvariant":1083},[57,4387,1512],{"mathvariant":1083},[52,4389,55],{"stretchy":54},[57,4391,59],{},[52,4393,69],{"stretchy":54},[52,4395,69],{"stretchy":54},[71,4397,3815],{"encoding":73},[32,4399,4401,4425],{"className":4400,"ariaHidden":62},[78],[32,4402,4404,4407,4410,4413,4416,4419,4422],{"className":4403},[82],[32,4405],{"className":4406,"style":87},[86],[32,4408,2101],{"className":4409,"style":1675},[95,96],[32,4411,55],{"className":4412},[91],[32,4414,2805],{"className":4415,"style":198},[95,96],[32,4417],{"className":4418,"style":177},[105],[32,4420,2808],{"className":4421},[181],[32,4423],{"className":4424,"style":177},[105],[32,4426,4428,4431,4437,4440,4443],{"className":4427},[82],[32,4429],{"className":4430,"style":87},[86],[32,4432,4434],{"className":4433},[95],[32,4435,2776],{"className":4436},[95,2244],[32,4438,55],{"className":4439},[91],[32,4441,59],{"className":4442,"style":97},[95,96],[32,4444,2896],{"className":4445},[113]," 这个机制仍然成立，模型依然可以泛化。这正是 Arjovsky 等人提出不变风险最小化方法的出发点：不去拟合在各个环境里都表现最好的相关性，而是寻找在所有环境下都保持不变的预测机制——本质上是把学习目标从「几何\u002F统计上的最优拟合」换成「因果机制上的不变性」。",[11,4448,4449],{},"用一句话概括：几何模型学的是「数据长什么样」，拓扑模型学的是「数据的骨架怎么连」，而因果模型学的是「数据为什么会长成这样、什么力量在背后生成它」。前两者是对现象的描述，后者试图逼近生成现象的机制——这也是为什么协变量偏移能摧毁前两者，却不一定能摧毁后者。",[21,4451,4452],{"id":4452},"递进的先验",[11,4454,4455,4456,4525],{},"如果用范畴论的语言简单勾勒一下三者的关系：存在遗忘函子 ",[32,4457,4459,4489],{"className":4458,"translate":36},[35],[32,4460,4462],{"className":4461},[40],[42,4463,4464],{"xmlns":44},[46,4465,4466,4486],{},[49,4467,4468,4476,4478],{},[49,4469,4470,4472,4474],{},[57,4471,1496],{"mathvariant":1495},[57,4473,1499],{"mathvariant":1495},[57,4475,1502],{"mathvariant":1495},[52,4477,141],{},[49,4479,4480,4482,4484],{},[57,4481,1509],{"mathvariant":1495},[57,4483,1512],{"mathvariant":1495},[57,4485,11],{"mathvariant":1495},[71,4487,4488],{"encoding":73},"\\mathbf{Met} \\to \\mathbf{Top}",[32,4490,4492,4513],{"className":4491,"ariaHidden":62},[78],[32,4493,4495,4498,4504,4507,4510],{"className":4494},[82],[32,4496],{"className":4497,"style":1546},[86],[32,4499,4501],{"className":4500},[95],[32,4502,1554],{"className":4503},[95,1553],[32,4505],{"className":4506,"style":177},[105],[32,4508,141],{"className":4509},[181],[32,4511],{"className":4512,"style":177},[105],[32,4514,4516,4519],{"className":4515},[82],[32,4517],{"className":4518,"style":1570},[86],[32,4520,4522],{"className":4521},[95],[32,4523,1577],{"className":4524},[95,1553],"，把度量空间的态射（等距同构）松弛成拓扑空间的态射（同胚），信息在这一步单调减少，但依然停留在「对称关系」的范畴里。因果结构则不在这条链上——它是在对象上额外加了一层有向、非对称的生成机制，这层机制无法通过对已有对称结构做进一步遗忘得到，只能通过引入新的认识论工具（干预、时间、结构假设）来「添加」。",[11,4527,4528],{},"所以更准确的图景不是一条线性的「由重到轻」的链条，而是：",[32,4530,4532],{"className":4531,"translate":36},[299],[32,4533,4535,4637],{"className":4534,"translate":36},[35],[32,4536,4538],{"className":4537},[40],[42,4539,4540],{"xmlns":44,"display":308},[46,4541,4542,4634],{},[317,4543,4546,4566,4583,4601,4616],{"rowspacing":4544,"columnalign":4545,"columnspacing":321},"0.16em","center",[323,4547,4548],{},[326,4549,4550],{},[329,4551,4552],{"scriptlevel":157,"displaystyle":54},[4553,4554,4555,4563],"munder",{},[4553,4556,4557,4560],{},[372,4558,4559],{},"几何",[52,4561,4562],{"stretchy":62},"⏟",[372,4564,4565],{},"对称，度量化，最重",[323,4567,4568],{},[326,4569,4570],{},[329,4571,4572],{"scriptlevel":157,"displaystyle":54},[49,4573,4574,4578],{},[52,4575,4577],{"fence":54,"stretchy":62,"minsize":4576,"maxsize":4576},"1.8em","↓",[329,4579,4580],{"scriptlevel":1732,"displaystyle":54},[372,4581,4582],{},"遗忘度量",[323,4584,4585],{},[326,4586,4587],{},[329,4588,4589],{"scriptlevel":157,"displaystyle":54},[4553,4590,4591,4598],{},[4553,4592,4593,4596],{},[372,4594,4595],{},"拓扑",[52,4597,4562],{"stretchy":62},[372,4599,4600],{},"对称，非度量化，中等",[323,4602,4603],{},[326,4604,4605],{},[329,4606,4607],{"scriptlevel":157,"displaystyle":54},[49,4608,4609,4611],{},[52,4610,4577],{"fence":54,"stretchy":62,"minsize":4576,"maxsize":4576},[329,4612,4613],{"scriptlevel":1732,"displaystyle":54},[372,4614,4615],{},"额外的干预\u002F时序信息",[323,4617,4618],{},[326,4619,4620],{},[329,4621,4622],{"scriptlevel":157,"displaystyle":54},[4553,4623,4624,4631],{},[4553,4625,4626,4629],{},[372,4627,4628],{},"因果",[52,4630,4562],{"stretchy":62},[372,4632,4633],{},"非对称，机制化，最稳健",[71,4635,4636],{"encoding":73},"\\begin{array}{c}\n\\underbrace{\\text{几何}}_{\\text{对称，度量化，最重}} \\\\[8pt]\n\\Big\\downarrow \\scriptstyle \\text{遗忘度量} \\\\[8pt]\n\\underbrace{\\text{拓扑}}_{\\text{对称，非度量化，中等}} \\\\[8pt]\n\\Big\\downarrow \\scriptstyle \\text{额外的干预\u002F时序信息} 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representation learning）想做的事情，正是把几何\u002F拓扑意义上学到的表示，进一步「提纯」出背后满足机制不变性的潜变量和因果图——这也是几何、拓扑、因果这三种视角在方法论上真正会师的地方。",{"title":5247,"searchDepth":5248,"depth":5248,"links":5249},"",2,[5250,5251,5252,5253,5254,5255],{"id":23,"depth":5248,"text":24},{"id":1172,"depth":5248,"text":1173},{"id":1581,"depth":5248,"text":1582},{"id":3157,"depth":5248,"text":3157},{"id":3944,"depth":5248,"text":3944},{"id":4452,"depth":5248,"text":4452},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F2026-08-19-yan-jiu-guan-xi-de-san-zhong-shi-jiao",{"title":6,"description":13},"blog\u002F2026\u002F2026-08-19-yan-jiu-guan-xi-de-san-zhong-shi-jiao","研究「关系」有三种互补视角：几何，拓扑和因果。分别关注测量、连通和约束。它们不是并列的学科，而是看待「存在」的三个层层递进的维度。三者共同构成智能系统理解世界的先验地基。",[5266],"machine-learning","2026-08-19T00:00:00+08:00","A1vSrI3BiIRmUZD1R2c8bUvp1iXhI9JF4wAlPI7utpA",{"id":5270,"title":5271,"body":5272,"description":5276,"draft":5256,"enableComment":5257,"extension":5258,"image":5259,"important":5256,"location":24602,"meta":24603,"navigation":5257,"ogImage":5259,"path":24604,"seo":24605,"stem":24606,"summary":24607,"tags":24608,"time":24609,"__hash__":24610},"blog\u002Fblog\u002F2026\u002F2026-08-22-are-local-minima-rare-in-high-dimensions.md","高维非凸优化的「鞍点主导」",{"type":8,"value":5273,"toc":24583},[5274,5277,5280,5296,5299,5302,5305,5370,5373,5377,5678,6568,6854,8210,8331,8989,9063,9560,9975,10215,10218,10634,10670,11983,12044,12325,12328,12331,12334,12908,13684,13750,13755,13758,13761,13764,13904,14066,14154,14438,14506,14804,15169,15172,15519,15521,16019,16022,16239,16300,16311,16314,16317,16320,16324,16361,16671,16703,16707,16864,16987,16990,17117,17120,18003,18140,18273,18276,18348,18769,18956,19152,19213,19695,19701,19705,19710,19902,20234,20295,20948,21063,21117,21430,21529,21604,21636,21639,21644,21648,21651,21785,21973,22134,22140,22427,22647,22650,22654,22661,23458,23461,23687,23690,23916,24029,24032,24035,24404,24411,24414,24417,24420,24431,24434,24437,24497,24558,24561,24566,24569,24573],[11,5275,5276],{},"深度学习中，Roon 说过一句话，被很多人当作信条：最大的 whitepill 是：局部极小值在高维空间中很罕见。也就是说，你的模型很难卡在局部最优里。但它到底有没有道理？如果有道理，道理又在哪里？",[11,5278,5279],{},"这句观点并非对每个光滑函数都字面成立。比如凸函数可能只有一个全局极小值，压根没有鞍点。它的精确版本需要一个「景观的概率模型」，以及退化与非退化临界点之间的区分。这篇文章会把这句直觉，分三步严格化：",[5281,5282,5283,5286,5293],"ol",{},[4063,5284,5285],{},"对 Morse 函数而言，临界点是孤立的，因此在所处区域中占据零体积。",[4063,5287,5288,5289,5292],{},"在非退化临界点的条件下，局部极小值要求 Hessian ",[1364,5290,5291],{},"正定","，即所有 N 个独立曲率方向都为正。",[4063,5294,5295],{},"在 Hessian 表现得像大型中心化随机对称矩阵的随机景观模型中，正定是指数级罕见的事件，鞍点则压倒性地常见。",[11,5297,5298],{},"Kac–Rice 公式为这一现象在高斯随机场中提供了严格的计数框架，Morse 理论则解释了为什么「指数」是临界点的自然拓扑不变量。下面我们一步步把这条链路走完。",[21,5300,5301],{"id":5301},"高维的地形景观",[11,5303,5304],{},"我们习惯把优化问题想象成一幅二维或三维的地形图：有山峰（极大值）、山谷（极小值）、还有山口（鞍点，像马鞍一样一边上一边下）。这种直觉在低维时很管用。",[11,5306,5307,5308,5337,5338,5366,5367],{},"但当维度 ",[32,5309,5311,5325],{"className":5310,"translate":36},[35],[32,5312,5314],{"className":5313},[40],[42,5315,5316],{"xmlns":44},[46,5317,5318,5323],{},[49,5319,5320],{},[57,5321,5322],{},"N",[71,5324,5322],{"encoding":73},[32,5326,5328],{"className":5327,"ariaHidden":62},[78],[32,5329,5331,5334],{"className":5330},[82],[32,5332],{"className":5333,"style":212},[86],[32,5335,5322],{"className":5336,"style":1530},[95,96]," 变得很大——几千、几百万，甚至更多——这幅图景就开始「失真」。想象你站在高维空间的一个点上：此时脚下不再只有「前后左右」两个方向，而是有 ",[32,5339,5341,5354],{"className":5340,"translate":36},[35],[32,5342,5344],{"className":5343},[40],[42,5345,5346],{"xmlns":44},[46,5347,5348,5352],{},[49,5349,5350],{},[57,5351,5322],{},[71,5353,5322],{"encoding":73},[32,5355,5357],{"className":5356,"ariaHidden":62},[78],[32,5358,5360,5363],{"className":5359},[82],[32,5361],{"className":5362,"style":212},[86],[32,5364,5322],{"className":5365,"style":1530},[95,96]," 个相互独立的方向可以移动。要说这个点是一个真正的谷底（局部极小值），你必须保证：",[1364,5368,5369],{},"无论往哪个方向迈一小步，海拔都会升高。",[11,5371,5372],{},"这个直觉在数学上如何精确化？我们从最基础的定义开始。",[21,5374,5376],{"id":5375},"临界点梯度与-hessian","临界点、梯度与 Hessian",[11,5378,5379,5380,5487,5488,3568,5618,5677],{},"设 ",[32,5381,5383,5410],{"className":5382,"translate":36},[35],[32,5384,5386],{"className":5385},[40],[42,5387,5388],{"xmlns":44},[46,5389,5390,5407],{},[49,5391,5392,5394,5396,5403,5405],{},[57,5393,1909],{},[52,5395,131],{},[5397,5398,5399,5401],"msup",{},[57,5400,148],{"mathvariant":147},[57,5402,5322],{},[52,5404,141],{},[57,5406,148],{"mathvariant":147},[71,5408,5409],{"encoding":73},"f: \\mathbb{R}^N \\to \\mathbb{R}",[32,5411,5413,5431,5477],{"className":5412,"ariaHidden":62},[78],[32,5414,5416,5419,5422,5425,5428],{"className":5415},[82],[32,5417],{"className":5418,"style":4025},[86],[32,5420,1909],{"className":5421,"style":1965},[95,96],[32,5423],{"className":5424,"style":177},[105],[32,5426,131],{"className":5427},[181],[32,5429],{"className":5430,"style":177},[105],[32,5432,5434,5438,5468,5471,5474],{"className":5433},[82],[32,5435],{"className":5436,"style":5437},[86],"height:0.8413em;",[32,5439,5441,5444],{"className":5440},[95],[32,5442,148],{"className":5443},[95,238],[32,5445,5447],{"className":5446},[242],[32,5448,5450],{"className":5449},[246],[32,5451,5453],{"className":5452},[251],[32,5454,5456],{"className":5455,"style":5437},[255],[32,5457,5459,5462],{"style":5458},"top:-3.063em;margin-right:0.05em;",[32,5460],{"className":5461,"style":264},[263],[32,5463,5465],{"className":5464},[268,269,270,271],[32,5466,5322],{"className":5467,"style":1530},[95,96,271],[32,5469],{"className":5470,"style":177},[105],[32,5472,141],{"className":5473},[181],[32,5475],{"className":5476,"style":177},[105],[32,5478,5480,5484],{"className":5479},[82],[32,5481],{"className":5482,"style":5483},[86],"height:0.6889em;",[32,5485,148],{"className":5486},[95,238]," 是二阶连续可微函数，记作 ",[32,5489,5491,5525],{"className":5490,"translate":36},[35],[32,5492,5494],{"className":5493},[40],[42,5495,5496],{"xmlns":44},[46,5497,5498,5522],{},[49,5499,5500,5502,5504,5512,5514,5520],{},[57,5501,1909],{},[52,5503,1316],{},[5397,5505,5506,5509],{},[57,5507,5508],{},"C",[155,5510,5511],{},"2",[52,5513,55],{"stretchy":54},[5397,5515,5516,5518],{},[57,5517,148],{"mathvariant":147},[57,5519,5322],{},[52,5521,69],{"stretchy":54},[71,5523,5524],{"encoding":73},"f \\in C^2(\\mathbb{R}^N)",[32,5526,5528,5546],{"className":5527,"ariaHidden":62},[78],[32,5529,5531,5534,5537,5540,5543],{"className":5530},[82],[32,5532],{"className":5533,"style":4025},[86],[32,5535,1909],{"className":5536,"style":1965},[95,96],[32,5538],{"className":5539,"style":177},[105],[32,5541,1316],{"className":5542},[181],[32,5544],{"className":5545,"style":177},[105],[32,5547,5549,5553,5583,5586,5615],{"className":5548},[82],[32,5550],{"className":5551,"style":5552},[86],"height:1.0913em;vertical-align:-0.25em;",[32,5554,5556,5559],{"className":5555},[95],[32,5557,5508],{"className":5558,"style":3492},[95,96],[32,5560,5562],{"className":5561},[242],[32,5563,5565],{"className":5564},[246],[32,5566,5568],{"className":5567},[251],[32,5569,5572],{"className":5570,"style":5571},[255],"height:0.8141em;",[32,5573,5574,5577],{"style":5458},[32,5575],{"className":5576,"style":264},[263],[32,5578,5580],{"className":5579},[268,269,270,271],[32,5581,5511],{"className":5582},[95,271],[32,5584,55],{"className":5585},[91],[32,5587,5589,5592],{"className":5588},[95],[32,5590,148],{"className":5591},[95,238],[32,5593,5595],{"className":5594},[242],[32,5596,5598],{"className":5597},[246],[32,5599,5601],{"className":5600},[251],[32,5602,5604],{"className":5603,"style":5437},[255],[32,5605,5606,5609],{"style":5458},[32,5607],{"className":5608,"style":264},[263],[32,5610,5612],{"className":5611},[268,269,270,271],[32,5613,5322],{"className":5614,"style":1530},[95,96,271],[32,5616,69],{"className":5617},[113],[32,5619,5621,5639],{"className":5620,"translate":36},[35],[32,5622,5624],{"className":5623},[40],[42,5625,5626],{"xmlns":44},[46,5627,5628,5636],{},[49,5629,5630],{},[5397,5631,5632,5634],{},[57,5633,5508],{},[155,5635,5511],{},[71,5637,5638],{"encoding":73},"C^2",[32,5640,5642],{"className":5641,"ariaHidden":62},[78],[32,5643,5645,5648],{"className":5644},[82],[32,5646],{"className":5647,"style":5571},[86],[32,5649,5651,5654],{"className":5650},[95],[32,5652,5508],{"className":5653,"style":3492},[95,96],[32,5655,5657],{"className":5656},[242],[32,5658,5660],{"className":5659},[246],[32,5661,5663],{"className":5662},[251],[32,5664,5666],{"className":5665,"style":5571},[255],[32,5667,5668,5671],{"style":5458},[32,5669],{"className":5670,"style":264},[263],[32,5672,5674],{"className":5673},[268,269,270,271],[32,5675,5511],{"className":5676},[95,271]," 假设保证了 Hessian 良定义且对称。",[3222,5679,5680,5940],{},[11,5681,5682,5685,5686,5714,5715,5939],{},[1364,5683,5684],{},"定义 2.1（梯度）","。函数 ",[32,5687,5689,5702],{"className":5688,"translate":36},[35],[32,5690,5692],{"className":5691},[40],[42,5693,5694],{"xmlns":44},[46,5695,5696,5700],{},[49,5697,5698],{},[57,5699,1909],{},[71,5701,1909],{"encoding":73},[32,5703,5705],{"className":5704,"ariaHidden":62},[78],[32,5706,5708,5711],{"className":5707},[82],[32,5709],{"className":5710,"style":4025},[86],[32,5712,1909],{"className":5713,"style":1965},[95,96]," 在 ",[32,5716,5718,5764],{"className":5717,"translate":36},[35],[32,5719,5721],{"className":5720},[40],[42,5722,5723],{"xmlns":44},[46,5724,5725,5761],{},[49,5726,5727,5729,5731,5733,5739,5741,5743,5745,5751,5753,5755],{},[57,5728,358],{},[52,5730,368],{},[52,5732,55],{"stretchy":54},[143,5734,5735,5737],{},[57,5736,358],{},[155,5738,1732],{},[52,5740,63],{"separator":62},[52,5742,1737],{},[52,5744,63],{"separator":62},[143,5746,5747,5749],{},[57,5748,358],{},[57,5750,5322],{},[52,5752,69],{"stretchy":54},[52,5754,1316],{},[5397,5756,5757,5759],{},[57,5758,148],{"mathvariant":147},[57,5760,5322],{},[71,5762,5763],{"encoding":73},"x=(x_1,\\dots,x_N)\\in\\mathbb{R}^N",[32,5765,5767,5785,5904],{"className":5766,"ariaHidden":62},[78],[32,5768,5770,5773,5776,5779,5782],{"className":5769},[82],[32,5771],{"className":5772,"style":1261},[86],[32,5774,358],{"className":5775},[95,96],[32,5777],{"className":5778,"style":177},[105],[32,5780,368],{"className":5781},[181],[32,5783],{"className":5784,"style":177},[105],[32,5786,5788,5791,5794,5834,5837,5840,5843,5846,5849,5852,5892,5895,5898,5901],{"className":5787},[82],[32,5789],{"className":5790,"style":87},[86],[32,5792,55],{"className":5793},[91],[32,5795,5797,5800],{"className":5796},[95],[32,5798,358],{"className":5799},[95,96],[32,5801,5803],{"className":5802},[242],[32,5804,5806,5826],{"className":5805},[246,247],[32,5807,5809,5823],{"className":5808},[251],[32,5810,5812],{"className":5811,"style":256},[255],[32,5813,5814,5817],{"style":259},[32,5815],{"className":5816,"style":264},[263],[32,5818,5820],{"className":5819},[268,269,270,271],[32,5821,1732],{"className":5822},[95,271],[32,5824,285],{"className":5825},[284],[32,5827,5829],{"className":5828},[251],[32,5830,5832],{"className":5831,"style":1441},[255],[32,5833],{},[32,5835,63],{"className":5836},[101],[32,5838],{"className":5839,"style":106},[105],[32,5841,1737],{"className":5842},[507],[32,5844],{"className":5845,"style":106},[105],[32,5847,63],{"className":5848},[101],[32,5850],{"className":5851,"style":106},[105],[32,5853,5855,5858],{"className":5854},[95],[32,5856,358],{"className":5857},[95,96],[32,5859,5861],{"className":5860},[242],[32,5862,5864,5884],{"className":5863},[246,247],[32,5865,5867,5881],{"className":5866},[251],[32,5868,5870],{"className":5869,"style":4238},[255],[32,5871,5872,5875],{"style":259},[32,5873],{"className":5874,"style":264},[263],[32,5876,5878],{"className":5877},[268,269,270,271],[32,5879,5322],{"className":5880,"style":1530},[95,96,271],[32,5882,285],{"className":5883},[284],[32,5885,5887],{"className":5886},[251],[32,5888,5890],{"className":5889,"style":1441},[255],[32,5891],{},[32,5893,69],{"className":5894},[113],[32,5896],{"className":5897,"style":177},[105],[32,5899,1316],{"className":5900},[181],[32,5902],{"className":5903,"style":177},[105],[32,5905,5907,5910],{"className":5906},[82],[32,5908],{"className":5909,"style":5437},[86],[32,5911,5913,5916],{"className":5912},[95],[32,5914,148],{"className":5915},[95,238],[32,5917,5919],{"className":5918},[242],[32,5920,5922],{"className":5921},[246],[32,5923,5925],{"className":5924},[251],[32,5926,5928],{"className":5927,"style":5437},[255],[32,5929,5930,5933],{"style":5458},[32,5931],{"className":5932,"style":264},[263],[32,5934,5936],{"className":5935},[268,269,270,271],[32,5937,5322],{"className":5938,"style":1530},[95,96,271]," 处的梯度是向量",[32,5941,5943],{"className":5942,"translate":36},[299],[32,5944,5946,6080],{"className":5945,"translate":36},[35],[32,5947,5949],{"className":5948},[40],[42,5950,5951],{"xmlns":44,"display":308},[46,5952,5953,6077],{},[49,5954,5955,5958,5960,5962,5964,5966,5968,5970,6066,6068,6074],{},[57,5956,5957],{"mathvariant":1083},"∇",[57,5959,1909],{},[52,5961,55],{"stretchy":54},[57,5963,358],{},[52,5965,69],{"stretchy":54},[52,5967,131],{},[52,5969,368],{},[49,5971,5972,5974,6064],{},[52,5973,55],{"fence":62},[317,5975,5976,6012,6030],{"rowspacing":4544,"columnalign":4545,"columnspacing":321},[323,5977,5978],{},[326,5979,5980],{},[329,5981,5982],{"scriptlevel":157,"displaystyle":54},[49,5983,5984,6006,6008,6010],{},[329,5985,5986],{"scriptlevel":157,"displaystyle":62},[5987,5988,5989,5996],"mfrac",{},[49,5990,5991,5994],{},[57,5992,5993],{"mathvariant":1083},"∂",[57,5995,1909],{},[49,5997,5998,6000],{},[57,5999,5993],{"mathvariant":1083},[143,6001,6002,6004],{},[57,6003,358],{},[155,6005,1732],{},[52,6007,55],{"stretchy":54},[57,6009,358],{},[52,6011,69],{"stretchy":54},[323,6013,6014],{},[326,6015,6016],{},[329,6017,6018],{"scriptlevel":157,"displaystyle":54},[49,6019,6020,6023],{},[57,6021,6022],{"mathvariant":1083},"⋮",[6024,6025,6026],"mpadded",{"height":335,"voffset":335},[105,6027],{"mathbackground":6028,"width":335,"height":6029},"black","1.5em",[323,6031,6032],{},[326,6033,6034],{},[329,6035,6036],{"scriptlevel":157,"displaystyle":54},[49,6037,6038,6058,6060,6062],{},[329,6039,6040],{"scriptlevel":157,"displaystyle":62},[5987,6041,6042,6048],{},[49,6043,6044,6046],{},[57,6045,5993],{"mathvariant":1083},[57,6047,1909],{},[49,6049,6050,6052],{},[57,6051,5993],{"mathvariant":1083},[143,6053,6054,6056],{},[57,6055,358],{},[57,6057,5322],{},[52,6059,55],{"stretchy":54},[57,6061,358],{},[52,6063,69],{"stretchy":54},[52,6065,69],{"fence":62},[52,6067,1316],{},[5397,6069,6070,6072],{},[57,6071,148],{"mathvariant":147},[57,6073,5322],{},[57,6075,6076],{"mathvariant":1083},".",[71,6078,6079],{"encoding":73},"\\nabla f(x) := \\left(\\begin{array}{c}\\dfrac{\\partial f}{\\partial x_1}(x)\\\\ \\vdots \\\\ \\dfrac{\\partial f}{\\partial 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2.3（Hessian）",[32,6862,6864,6877],{"className":6863,"translate":36},[35],[32,6865,6867],{"className":6866},[40],[42,6868,6869],{"xmlns":44},[46,6870,6871,6875],{},[49,6872,6873],{},[57,6874,1909],{},[71,6876,1909],{"encoding":73},[32,6878,6880],{"className":6879,"ariaHidden":62},[78],[32,6881,6883,6886],{"className":6882},[82],[32,6884],{"className":6885,"style":4025},[86],[32,6887,1909],{"className":6888,"style":1965},[95,96],[32,6890,6892,6914],{"className":6891,"translate":36},[35],[32,6893,6895],{"className":6894},[40],[42,6896,6897],{"xmlns":44},[46,6898,6899,6911],{},[49,6900,6901,6903,6905],{},[57,6902,358],{},[52,6904,1316],{},[5397,6906,6907,6909],{},[57,6908,148],{"mathvariant":147},[57,6910,5322],{},[71,6912,6913],{"encoding":73},"x\\in\\mathbb{R}^N",[32,6915,6917,6936],{"className":6916,"ariaHidden":62},[78],[32,6918,6920,6924,6927,6930,6933],{"className":6919},[82],[32,6921],{"className":6922,"style":6923},[86],"height:0.5782em;vertical-align:-0.0391em;",[32,6925,358],{"className":6926},[95,96],[32,6928],{"className":6929,"style":177},[105],[32,6931,1316],{"className":6932},[181],[32,6934],{"className":6935,"style":177},[105],[32,6937,6939,6942],{"className":6938},[82],[32,6940],{"className":6941,"style":5437},[86],[32,6943,6945,6948],{"className":6944},[95],[32,6946,148],{"className":6947},[95,238],[32,6949,6951],{"className":6950},[242],[32,6952,6954],{"className":6953},[246],[32,6955,6957],{"className":6956},[251],[32,6958,6960],{"className":6959,"style":5437},[255],[32,6961,6962,6965],{"style":5458},[32,6963],{"className":6964,"style":264},[263],[32,6966,6968],{"className":6967},[268,269,270,271],[32,6969,5322],{"className":6970,"style":1530},[95,96,271]," 处的 Hessian 是 ",[32,6973,6975,6993],{"className":6974,"translate":36},[35],[32,6976,6978],{"className":6977},[40],[42,6979,6980],{"xmlns":44},[46,6981,6982,6990],{},[49,6983,6984,6986,6988],{},[57,6985,5322],{},[52,6987,136],{},[57,6989,5322],{},[71,6991,6992],{"encoding":73},"N\\times N",[32,6994,6996,7014],{"className":6995,"ariaHidden":62},[78],[32,6997,6999,7002,7005,7008,7011],{"className":6998},[82],[32,7000],{"className":7001,"style":191},[86],[32,7003,5322],{"className":7004,"style":1530},[95,96],[32,7006],{"className":7007,"style":198},[105],[32,7009,136],{"className":7010},[202],[32,7012],{"className":7013,"style":198},[105],[32,7015,7017,7020],{"className":7016},[82],[32,7018],{"className":7019,"style":212},[86],[32,7021,5322],{"className":7022,"style":1530},[95,96]," 矩阵",[32,7025,7027],{"className":7026,"translate":36},[299],[32,7028,7030,7138],{"className":7029,"translate":36},[35],[32,7031,7033],{"className":7032},[40],[42,7034,7035],{"xmlns":44,"display":308},[46,7036,7037,7135],{},[49,7038,7039,7046,7048,7050,7052,7054,7060,7062,7064,7066,7068,7070,7072,7133],{},[143,7040,7041,7044],{},[57,7042,7043],{},"H",[57,7045,1909],{},[52,7047,55],{"stretchy":54},[57,7049,358],{},[52,7051,69],{"stretchy":54},[52,7053,368],{},[5397,7055,7056,7058],{},[57,7057,5957],{"mathvariant":1083},[155,7059,5511],{},[57,7061,1909],{},[52,7063,55],{"stretchy":54},[57,7065,358],{},[52,7067,69],{"stretchy":54},[52,7069,131],{},[52,7071,368],{},[7073,7074,7075,7119,7131],"msubsup",{},[49,7076,7077,7080,7110,7112,7114,7116],{},[52,7078,7079],{"fence":62},"[",[5987,7081,7082,7092],{},[49,7083,7084,7090],{},[5397,7085,7086,7088],{},[57,7087,5993],{"mathvariant":1083},[155,7089,5511],{},[57,7091,1909],{},[49,7093,7094,7096,7102,7104],{},[57,7095,5993],{"mathvariant":1083},[143,7097,7098,7100],{},[57,7099,358],{},[57,7101,2084],{},[57,7103,5993],{"mathvariant":1083},[143,7105,7106,7108],{},[57,7107,358],{},[57,7109,2458],{},[52,7111,55],{"stretchy":54},[57,7113,358],{},[52,7115,69],{"stretchy":54},[52,7117,7118],{"fence":62},"]",[49,7120,7121,7123,7125,7127,7129],{},[57,7122,2084],{},[52,7124,63],{"separator":62},[57,7126,2458],{},[52,7128,368],{},[155,7130,1732],{},[57,7132,5322],{},[57,7134,6076],{"mathvariant":1083},[71,7136,7137],{"encoding":73},"H_f(x) = \\nabla^2 f(x) := \\left[\\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(x)\\right]_{i,j=1}^N.",[32,7139,7141,7208,7266],{"className":7140,"ariaHidden":62},[78],[32,7142,7144,7148,7190,7193,7196,7199,7202,7205],{"className":7143},[82],[32,7145],{"className":7146,"style":7147},[86],"height:1.0361em;vertical-align:-0.2861em;",[32,7149,7151,7155],{"className":7150},[95],[32,7152,7043],{"className":7153,"style":7154},[95,96],"margin-right:0.0813em;",[32,7156,7158],{"className":7157},[242],[32,7159,7161,7182],{"className":7160},[246,247],[32,7162,7164,7179],{"className":7163},[251],[32,7165,7167],{"className":7166,"style":1419},[255],[32,7168,7170,7173],{"style":7169},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[32,7171],{"className":7172,"style":264},[263],[32,7174,7176],{"className":7175},[268,269,270,271],[32,7177,1909],{"className":7178,"style":1965},[95,96,271],[32,7180,285],{"className":7181},[284],[32,7183,7185],{"className":7184},[251],[32,7186,7188],{"className":7187,"style":2519},[255],[32,7189],{},[32,7191,55],{"className":7192},[91],[32,7194,358],{"className":7195},[95,96],[32,7197,69],{"className":7198},[113],[32,7200],{"className":7201,"style":177},[105],[32,7203,368],{"className":7204},[181],[32,7206],{"className":7207,"style":177},[105],[32,7209,7211,7215,7245,7248,7251,7254,7257,7260,7263],{"className":7210},[82],[32,7212],{"className":7213,"style":7214},[86],"height:1.1141em;vertical-align:-0.25em;",[32,7216,7218,7221],{"className":7217},[95],[32,7219,5957],{"className":7220},[95],[32,7222,7224],{"className":7223},[242],[32,7225,7227],{"className":7226},[246],[32,7228,7230],{"className":7229},[251],[32,7231,7234],{"className":7232,"style":7233},[255],"height:0.8641em;",[32,7235,7236,7239],{"style":6555},[32,7237],{"className":7238,"style":264},[263],[32,7240,7242],{"className":7241},[268,269,270,271],[32,7243,5511],{"className":7244},[95,271],[32,7246,1909],{"className":7247,"style":1965},[95,96],[32,7249,55],{"className":7250},[91],[32,7252,358],{"className":7253},[95,96],[32,7255,69],{"className":7256},[113],[32,7258],{"className":7259,"style":177},[105],[32,7261,2185],{"className":7262},[181],[32,7264],{"className":7265,"style":177},[105],[32,7267,7269,7273,7542,7545],{"className":7268},[82],[32,7270],{"className":7271,"style":7272},[86],"height:2.8803em;vertical-align:-1.1579em;",[32,7274,7276,7478],{"className":7275},[507],[32,7277,7279,7287,7463,7466,7469,7472],{"className":7278},[507],[32,7280,7284],{"className":7281,"style":7283},[91,7282],"delimcenter","top:0em;",[32,7285,7079],{"className":7286},[514,270],[32,7288,7290,7293,7460],{"className":7289},[95],[32,7291],{"className":7292},[91,974],[32,7294,7296],{"className":7295},[5987],[32,7297,7299,7451],{"className":7298},[246,247],[32,7300,7302,7448],{"className":7301},[251],[32,7303,7306,7400,7408],{"className":7304,"style":7305},[255],"height:1.4911em;",[32,7307,7308,7311],{"style":6222},[32,7309],{"className":7310,"style":715},[263],[32,7312,7314,7317,7357,7360],{"className":7313},[95],[32,7315,5993],{"className":7316,"style":6232},[95],[32,7318,7320,7323],{"className":7319},[95],[32,7321,358],{"className":7322},[95,96],[32,7324,7326],{"className":7325},[242],[32,7327,7329,7349],{"className":7328},[246,247],[32,7330,7332,7346],{"className":7331},[251],[32,7333,7335],{"className":7334,"style":2156},[255],[32,7336,7337,7340],{"style":259},[32,7338],{"className":7339,"style":264},[263],[32,7341,7343],{"className":7342},[268,269,270,271],[32,7344,2084],{"className":7345},[95,96,271],[32,7347,285],{"className":7348},[284],[32,7350,7352],{"className":7351},[251],[32,7353,7355],{"className":7354,"style":1441},[255],[32,7356],{},[32,7358,5993],{"className":7359,"style":6232},[95],[32,7361,7363,7366],{"className":7362},[95],[32,7364,358],{"className":7365},[95,96],[32,7367,7369],{"className":7368},[242],[32,7370,7372,7392],{"className":7371},[246,247],[32,7373,7375,7389],{"className":7374},[251],[32,7376,7378],{"className":7377,"style":2156},[255],[32,7379,7380,7383],{"style":259},[32,7381],{"className":7382,"style":264},[263],[32,7384,7386],{"className":7385},[268,269,270,271],[32,7387,2458],{"className":7388,"style":2509},[95,96,271],[32,7390,285],{"className":7391},[284],[32,7393,7395],{"className":7394},[251],[32,7396,7398],{"className":7397,"style":2519},[255],[32,7399],{},[32,7401,7402,7405],{"style":6275},[32,7403],{"className":7404,"style":715},[263],[32,7406],{"className":7407,"style":6283},[6282],[32,7409,7410,7413],{"style":6286},[32,7411],{"className":7412,"style":715},[263],[32,7414,7416,7445],{"className":7415},[95],[32,7417,7419,7422],{"className":7418},[95],[32,7420,5993],{"className":7421,"style":6232},[95],[32,7423,7425],{"className":7424},[242],[32,7426,7428],{"className":7427},[246],[32,7429,7431],{"className":7430},[251],[32,7432,7434],{"className":7433,"style":5571},[255],[32,7435,7436,7439],{"style":5458},[32,7437],{"className":7438,"style":264},[263],[32,7440,7442],{"className":7441},[268,269,270,271],[32,7443,5511],{"className":7444},[95,271],[32,7446,1909],{"className":7447,"style":1965},[95,96],[32,7449,285],{"className":7450},[284],[32,7452,7454],{"className":7453},[251],[32,7455,7458],{"className":7456,"style":7457},[255],"height:0.9721em;",[32,7459],{},[32,7461],{"className":7462},[113,974],[32,7464,55],{"className":7465},[91],[32,7467,358],{"className":7468},[95,96],[32,7470,69],{"className":7471},[113],[32,7473,7475],{"className":7474,"style":7283},[113,7282],[32,7476,7118],{"className":7477},[514,270],[32,7479,7481],{"className":7480},[242],[32,7482,7484,7533],{"className":7483},[246,247],[32,7485,7487,7530],{"className":7486},[251],[32,7488,7491,7518],{"className":7489,"style":7490},[255],"height:1.7223em;",[32,7492,7494,7497],{"style":7493},"top:-1.6782em;margin-right:0.05em;",[32,7495],{"className":7496,"style":264},[263],[32,7498,7500],{"className":7499},[268,269,270,271],[32,7501,7503,7506,7509,7512,7515],{"className":7502},[95,271],[32,7504,2084],{"className":7505},[95,96,271],[32,7507,63],{"className":7508},[101,271],[32,7510,2458],{"className":7511,"style":2509},[95,96,271],[32,7513,368],{"className":7514},[181,271],[32,7516,1732],{"className":7517},[95,271],[32,7519,7521,7524],{"style":7520},"top:-3.944em;margin-right:0.05em;",[32,7522],{"className":7523,"style":264},[263],[32,7525,7527],{"className":7526},[268,269,270,271],[32,7528,5322],{"className":7529,"style":1530},[95,96,271],[32,7531,285],{"className":7532},[284],[32,7534,7536],{"className":7535},[251],[32,7537,7540],{"className":7538,"style":7539},[255],"height:1.1579em;",[32,7541],{},[32,7543],{"className":7544,"style":106},[105],[32,7546,6076],{"className":7547},[95],[11,7549,7550,7551,7632],{},"由于 ",[32,7552,7554,7576],{"className":7553,"translate":36},[35],[32,7555,7557],{"className":7556},[40],[42,7558,7559],{"xmlns":44},[46,7560,7561,7573],{},[49,7562,7563,7565,7567],{},[57,7564,1909],{},[52,7566,1316],{},[5397,7568,7569,7571],{},[57,7570,5508],{},[155,7572,5511],{},[71,7574,7575],{"encoding":73},"f\\in C^2",[32,7577,7579,7597],{"className":7578,"ariaHidden":62},[78],[32,7580,7582,7585,7588,7591,7594],{"className":7581},[82],[32,7583],{"className":7584,"style":4025},[86],[32,7586,1909],{"className":7587,"style":1965},[95,96],[32,7589],{"className":7590,"style":177},[105],[32,7592,1316],{"className":7593},[181],[32,7595],{"className":7596,"style":177},[105],[32,7598,7600,7603],{"className":7599},[82],[32,7601],{"className":7602,"style":5571},[86],[32,7604,7606,7609],{"className":7605},[95],[32,7607,5508],{"className":7608,"style":3492},[95,96],[32,7610,7612],{"className":7611},[242],[32,7613,7615],{"className":7614},[246],[32,7616,7618],{"className":7617},[251],[32,7619,7621],{"className":7620,"style":5571},[255],[32,7622,7623,7626],{"style":5458},[32,7624],{"className":7625,"style":264},[263],[32,7627,7629],{"className":7628},[268,269,270,271],[32,7630,5511],{"className":7631},[95,271],"，由 Clairaut 定理可得",[32,7634,7636],{"className":7635,"translate":36},[299],[32,7637,7639,7727],{"className":7638,"translate":36},[35],[32,7640,7642],{"className":7641},[40],[42,7643,7644],{"xmlns":44,"display":308},[46,7645,7646,7724],{},[49,7647,7648,7678,7680,7682,7684,7686,7716,7718,7720,7722],{},[5987,7649,7650,7660],{},[49,7651,7652,7658],{},[5397,7653,7654,7656],{},[57,7655,5993],{"mathvariant":1083},[155,7657,5511],{},[57,7659,1909],{},[49,7661,7662,7664,7670,7672],{},[57,7663,5993],{"mathvariant":1083},[143,7665,7666,7668],{},[57,7667,358],{},[57,7669,2084],{},[57,7671,5993],{"mathvariant":1083},[143,7673,7674,7676],{},[57,7675,358],{},[57,7677,2458],{},[52,7679,55],{"stretchy":54},[57,7681,358],{},[52,7683,69],{"stretchy":54},[52,7685,368],{},[5987,7687,7688,7698],{},[49,7689,7690,7696],{},[5397,7691,7692,7694],{},[57,7693,5993],{"mathvariant":1083},[155,7695,5511],{},[57,7697,1909],{},[49,7699,7700,7702,7708,7710],{},[57,7701,5993],{"mathvariant":1083},[143,7703,7704,7706],{},[57,7705,358],{},[57,7707,2458],{},[57,7709,5993],{"mathvariant":1083},[143,7711,7712,7714],{},[57,7713,358],{},[57,7715,2084],{},[52,7717,55],{"stretchy":54},[57,7719,358],{},[52,7721,69],{"stretchy":54},[52,7723,63],{"separator":62},[71,7725,7726],{"encoding":73},"\\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(x) = \\frac{\\partial^2 f}{\\partial x_j \\partial 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= f(x_0) + \\nabla f(x_0)^T h + \\frac{1}{2}h^T H_f(x_0) h + o(\\|h\\|^2). 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处，线性项消失，因此局部几何由下式决定到二阶：",[32,9064,9066],{"className":9065,"translate":36},[299],[32,9067,9069,9163],{"className":9068,"translate":36},[35],[32,9070,9072],{"className":9071},[40],[42,9073,9074],{"xmlns":44,"display":308},[46,9075,9076,9160],{},[49,9077,9078,9080,9082,9088,9090,9092,9094,9096,9098,9100,9106,9108,9110,9116,9122,9128,9130,9136,9138,9140,9142,9144,9146,9148,9150,9156,9158],{},[57,9079,1909],{},[52,9081,55],{"stretchy":54},[143,9083,9084,9086],{},[57,9085,358],{},[57,9087,6594],{},[52,9089,478],{},[57,9091,8227],{},[52,9093,69],{"stretchy":54},[52,9095,1089],{},[57,9097,1909],{},[52,9099,55],{"stretchy":54},[143,9101,9102,9104],{},[57,9103,358],{},[57,9105,6594],{},[52,9107,69],{"stretchy":54},[52,9109,368],{},[5987,9111,9112,9114],{},[155,9113,1732],{},[155,9115,5511],{},[5397,9117,9118,9120],{},[57,9119,8227],{},[57,9121,1509],{},[143,9123,9124,9126],{},[57,9125,7043],{},[57,9127,1909],{},[52,9129,55],{"stretchy":54},[143,9131,9132,9134],{},[57,9133,358],{},[57,9135,6594],{},[52,9137,69],{"stretchy":54},[57,9139,8227],{},[52,9141,478],{},[57,9143,1512],{},[52,9145,55],{"stretchy":54},[57,9147,1084],{"mathvariant":1083},[57,9149,8227],{},[5397,9151,9152,9154],{},[57,9153,1084],{"mathvariant":1083},[155,9155,5511],{},[52,9157,69],{"stretchy":54},[57,9159,6076],{"mathvariant":1083},[71,9161,9162],{"encoding":73},"f(x_c+h) - f(x_c) = \\frac{1}{2}h^T H_f(x_c)h + o(\\|h\\|^2).",[32,9164,9166,9227,9248,9312,9507],{"className":9165,"ariaHidden":62},[78],[32,9167,9169,9172,9175,9178,9218,9221,9224],{"className":9168},[82],[32,9170],{"className":9171,"style":87},[86],[32,9173,1909],{"className":9174,"style":1965},[95,96],[32,9176,55],{"className":9177},[91],[32,9179,9181,9184],{"className":9180},[95],[32,9182,358],{"className":9183},[95,96],[32,9185,9187],{"className":9186},[242],[32,9188,9190,9210],{"className":9189},[246,247],[32,9191,9193,9207],{"className":9192},[251],[32,9194,9196],{"className":9195,"style":1860},[255],[32,9197,9198,9201],{"style":259},[32,9199],{"className":9200,"style":264},[263],[32,9202,9204],{"className":9203},[268,269,270,271],[32,9205,6594],{"className":9206},[95,96,271],[32,9208,285],{"className":9209},[284],[32,9211,9213],{"className":9212},[251],[32,9214,9216],{"className":9215,"style":1441},[255],[32,9217],{},[32,9219],{"className":9220,"style":198},[105],[32,9222,478],{"className":9223},[202],[32,9225],{"className":9226,"style":198},[105],[32,9228,9230,9233,9236,9239,9242,9245],{"className":9229},[82],[32,9231],{"className":9232,"style":87},[86],[32,9234,8227],{"className":9235},[95,96],[32,9237,69],{"className":9238},[113],[32,9240],{"className":9241,"style":198},[105],[32,9243,1089],{"className":9244},[202],[32,9246],{"className":9247,"style":198},[105],[32,9249,9251,9254,9257,9260,9300,9303,9306,9309],{"className":9250},[82],[32,9252],{"className":9253,"style":87},[86],[32,9255,1909],{"className":9256,"style":1965},[95,96],[32,9258,55],{"className":9259},[91],[32,9261,9263,9266],{"className":9262},[95],[32,9264,358],{"className":9265},[95,96],[32,9267,9269],{"className":9268},[242],[32,9270,9272,9292],{"className":9271},[246,247],[32,9273,9275,9289],{"className":9274},[251],[32,9276,9278],{"className":9277,"style":1860},[255],[32,9279,9280,9283],{"style":259},[32,9281],{"className":9282,"style":264},[263],[32,9284,9286],{"className":9285},[268,269,270,271],[32,9287,6594],{"className":9288},[95,96,271],[32,9290,285],{"className":9291},[284],[32,9293,9295],{"className":9294},[251],[32,9296,9298],{"className":9297,"style":1441},[255],[32,9299],{},[32,9301,69],{"className":9302},[113],[32,9304],{"className":9305,"style":177},[105],[32,9307,368],{"className":9308},[181],[32,9310],{"className":9311,"style":177},[105],[32,9313,9315,9318,9380,9409,9449,9452,9492,9495,9498,9501,9504],{"className":9314},[82],[32,9316],{"className":9317,"style":8722},[86],[32,9319,9321,9324,9377],{"className":9320},[95],[32,9322],{"className":9323},[91,974],[32,9325,9327],{"className":9326},[5987],[32,9328,9330,9369],{"className":9329},[246,247],[32,9331,9333,9366],{"className":9332},[251],[32,9334,9336,9347,9355],{"className":9335,"style":8741},[255],[32,9337,9338,9341],{"style":6222},[32,9339],{"className":9340,"style":715},[263],[32,9342,9344],{"className":9343},[95],[32,9345,5511],{"className":9346},[95],[32,9348,9349,9352],{"style":6275},[32,9350],{"className":9351,"style":715},[263],[32,9353],{"className":9354,"style":6283},[6282],[32,9356,9357,9360],{"style":6286},[32,9358],{"className":9359,"style":715},[263],[32,9361,9363],{"className":9362},[95],[32,9364,1732],{"className":9365},[95],[32,9367,285],{"className":9368},[284],[32,9370,9372],{"className":9371},[251],[32,9373,9375],{"className":9374,"style":8781},[255],[32,9376],{},[32,9378],{"className":9379},[113,974],[32,9381,9383,9386],{"className":9382},[95],[32,9384,8227],{"className":9385},[95,96],[32,9387,9389],{"className":9388},[242],[32,9390,9392],{"className":9391},[246],[32,9393,9395],{"className":9394},[251],[32,9396,9398],{"className":9397,"style":6534},[255],[32,9399,9400,9403],{"style":6555},[32,9401],{"className":9402,"style":264},[263],[32,9404,9406],{"className":9405},[268,269,270,271],[32,9407,1509],{"className":9408,"style":1675},[95,96,271],[32,9410,9412,9415],{"className":9411},[95],[32,9413,7043],{"className":9414,"style":7154},[95,96],[32,9416,9418],{"className":9417},[242],[32,9419,9421,9441],{"className":9420},[246,247],[32,9422,9424,9438],{"className":9423},[251],[32,9425,9427],{"className":9426,"style":1419},[255],[32,9428,9429,9432],{"style":7169},[32,9430],{"className":9431,"style":264},[263],[32,9433,9435],{"className":9434},[268,269,270,271],[32,9436,1909],{"className":9437,"style":1965},[95,96,271],[32,9439,285],{"className":9440},[284],[32,9442,9444],{"className":9443},[251],[32,9445,9447],{"className":9446,"style":2519},[255],[32,9448],{},[32,9450,55],{"className":9451},[91],[32,9453,9455,9458],{"className":9454},[95],[32,9456,358],{"className":9457},[95,96],[32,9459,9461],{"className":9460},[242],[32,9462,9464,9484],{"className":9463},[246,247],[32,9465,9467,9481],{"className":9466},[251],[32,9468,9470],{"className":9469,"style":1860},[255],[32,9471,9472,9475],{"style":259},[32,9473],{"className":9474,"style":264},[263],[32,9476,9478],{"className":9477},[268,269,270,271],[32,9479,6594],{"className":9480},[95,96,271],[32,9482,285],{"className":9483},[284],[32,9485,9487],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是实对称矩阵，谱定理给出由特征向量 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组成的标准正交基，对应的实特征值为 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= \\sum_{i=1}^N a_i v_i,",[32,10028,10030,10048],{"className":10029,"ariaHidden":62},[78],[32,10031,10033,10036,10039,10042,10045],{"className":10032},[82],[32,10034],{"className":10035,"style":170},[86],[32,10037,8227],{"className":10038},[95,96],[32,10040],{"className":10041,"style":177},[105],[32,10043,368],{"className":10044},[181],[32,10046],{"className":10047,"style":177},[105],[32,10049,10051,10055,10129,10132,10172,10212],{"className":10050},[82],[32,10052],{"className":10053,"style":10054},[86],"height:3.106em;vertical-align:-1.2777em;",[32,10056,10060],{"className":10057},[10058,10059],"mop","op-limits",[32,10061,10063,10120],{"className":10062},[246,247],[32,10064,10066,10117],{"className":10065},[251],[32,10067,10070,10092,10105],{"className":10068,"style":10069},[255],"height:1.8283em;",[32,10071,10073,10077],{"style":10072},"top:-1.8723em;margin-left:0em;",[32,10074],{"className":10075,"style":10076},[263],"height:3.05em;",[32,10078,10080],{"className":10079},[268,269,270,271],[32,10081,10083,10086,10089],{"className":10082},[95,271],[32,10084,2084],{"className":10085},[95,96,271],[32,10087,368],{"className":10088},[181,271],[32,10090,1732],{"className":10091},[95,271],[32,10093,10095,10098],{"style":10094},"top:-3.05em;",[32,10096],{"className":10097,"style":10076},[263],[32,10099,10100],{},[32,10101,9999],{"className":10102},[10058,10103,10104],"op-symbol","large-op",[32,10106,10108,10111],{"style":10107},"top:-4.3em;margin-left:0em;",[32,10109],{"className":10110,"style":10076},[263],[32,10112,10114],{"className":10113},[268,269,270,271],[32,10115,5322],{"className":10116,"style":1530},[95,96,271],[32,10118,285],{"className":10119},[284],[32,10121,10123],{"className":10122},[251],[32,10124,10127],{"className":10125,"style":10126},[255],"height:1.2777em;",[32,10128],{},[32,10130],{"className":10131,"style":106},[105],[32,10133,10135,10138],{"className":10134},[95],[32,10136,2104],{"className":10137},[95,96],[32,10139,10141],{"className":10140},[242],[32,10142,10144,10164],{"className":10143},[246,247],[32,10145,10147,10161],{"className":10146},[251],[32,10148,10150],{"className":10149,"style":2156},[255],[32,10151,10152,10155],{"style":259},[32,10153],{"className":10154,"style":264},[263],[32,10156,10158],{"className":10157},[268,269,270,271],[32,10159,2084],{"className":10160},[95,96,271],[32,10162,285],{"className":10163},[284],[32,10165,10167],{"className":10166},[251],[32,10168,10170],{"className":10169,"style":1441},[255],[32,10171],{},[32,10173,10175,10178],{"className":10174},[95],[32,10176,9705],{"className":10177,"style":740},[95,96],[32,10179,10181],{"className":10180},[242],[32,10182,10184,10204],{"className":10183},[246,247],[32,10185,10187,10201],{"className":10186},[251],[32,10188,10190],{"className":10189,"style":2156},[255],[32,10191,10192,10195],{"style":9753},[32,10193],{"className":10194,"style":264},[263],[32,10196,10198],{"className":10197},[268,269,270,271],[32,10199,2084],{"className":10200},[95,96,271],[32,10202,285],{"className":10203},[284],[32,10205,10207],{"className":10206},[251],[32,10208,10210],{"className":10209,"style":1441},[255],[32,10211],{},[32,10213,63],{"className":10214},[101],[11,10216,10217],{},"则",[32,10219,10221],{"className":10220,"translate":36},[299],[32,10222,10224,10307],{"className":10223,"translate":36},[35],[32,10225,10227],{"className":10226},[40],[42,10228,10229],{"xmlns":44,"display":308},[46,10230,10231,10304],{},[317,10232,10233],{"width":8346},[323,10234,10235,10237,10297,10299],{},[326,10236],{"width":8351},[326,10238,10239],{},[49,10240,10241,10247,10253,10255,10261,10263,10265,10267,10281,10287,10295],{},[5397,10242,10243,10245],{},[57,10244,8227],{},[57,10246,1509],{},[143,10248,10249,10251],{},[57,10250,7043],{},[57,10252,1909],{},[52,10254,55],{"stretchy":54},[143,10256,10257,10259],{},[57,10258,358],{},[57,10260,6594],{},[52,10262,69],{"stretchy":54},[57,10264,8227],{},[52,10266,368],{},[9995,10268,10269,10271,10279],{},[52,10270,9999],{},[49,10272,10273,10275,10277],{},[57,10274,2084],{},[52,10276,368],{},[155,10278,1732],{},[57,10280,5322],{},[143,10282,10283,10285],{},[57,10284,9849],{},[57,10286,2084],{},[7073,10288,10289,10291,10293],{},[57,10290,2104],{},[57,10292,2084],{},[155,10294,5511],{},[57,10296,6076],{"mathvariant":1083},[326,10298],{"width":8351},[326,10300,10301],{},[372,10302,10303],{},"(2)",[71,10305,10306],{"encoding":73},"h^T H_f(x_c) h = \\sum_{i=1}^N \\lambda_i a_i^2. \\tag{2}",[32,10308,10310,10444,10613],{"className":10309,"ariaHidden":62},[78],[32,10311,10313,10317,10346,10386,10389,10429,10432,10435,10438,10441],{"className":10312},[82],[32,10314],{"className":10315,"style":10316},[86],"height:1.1774em;vertical-align:-0.2861em;",[32,10318,10320,10323],{"className":10319},[95],[32,10321,8227],{"className":10322},[95,96],[32,10324,10326],{"className":10325},[242],[32,10327,10329],{"className":10328},[246],[32,10330,10332],{"className":10331},[251],[32,10333,10335],{"className":10334,"style":6534},[255],[32,10336,10337,10340],{"style":6555},[32,10338],{"className":10339,"style":264},[263],[32,10341,10343],{"className":10342},[268,269,270,271],[32,10344,1509],{"className":10345,"style":1675},[95,96,271],[32,10347,10349,10352],{"className":10348},[95],[32,10350,7043],{"className":10351,"style":7154},[95,96],[32,10353,10355],{"className":10354},[242],[32,10356,10358,10378],{"className":10357},[246,247],[32,10359,10361,10375],{"className":10360},[251],[32,10362,10364],{"className":10363,"style":1419},[255],[32,10365,10366,10369],{"style":7169},[32,10367],{"className":10368,"style":264},[263],[32,10370,10372],{"className":10371},[268,269,270,271],[32,10373,1909],{"className":10374,"style":1965},[95,96,271],[32,10376,285],{"className":10377},[284],[32,10379,10381],{"className":10380},[251],[32,10382,10384],{"className":10383,"style":2519},[255],[32,10385],{},[32,10387,55],{"className":10388},[91],[32,10390,10392,10395],{"className":10391},[95],[32,10393,358],{"className":10394},[95,96],[32,10396,10398],{"className":10397},[242],[32,10399,10401,10421],{"className":10400},[246,247],[32,10402,10404,10418],{"className":10403},[251],[32,10405,10407],{"className":10406,"style":1860},[255],[32,10408,10409,10412],{"style":259},[32,10410],{"className":10411,"style":264},[263],[32,10413,10415],{"className":10414},[268,269,270,271],[32,10416,6594],{"className":10417},[95,96,271],[32,10419,285],{"className":10420},[284],[32,10422,10424],{"className":10423},[251],[32,10425,10427],{"className":10426,"style":1441},[255],[32,10428],{},[32,10430,69],{"className":10431},[113],[32,10433,8227],{"className":10434},[95,96],[32,10436],{"className":10437,"style":177},[105],[32,10439,368],{"className":10440},[181],[32,10442],{"className":10443,"style":177},[105],[32,10445,10447,10450,10514,10517,10557,10610],{"className":10446},[82],[32,10448],{"className":10449,"style":10054},[86],[32,10451,10453],{"className":10452},[10058,10059],[32,10454,10456,10506],{"className":10455},[246,247],[32,10457,10459,10503],{"className":10458},[251],[32,10460,10462,10482,10492],{"className":10461,"style":10069},[255],[32,10463,10464,10467],{"style":10072},[32,10465],{"className":10466,"style":10076},[263],[32,10468,10470],{"className":10469},[268,269,270,271],[32,10471,10473,10476,10479],{"className":10472},[95,271],[32,10474,2084],{"className":10475},[95,96,271],[32,10477,368],{"className":10478},[181,271],[32,10480,1732],{"className":10481},[95,271],[32,10483,10484,10487],{"style":10094},[32,10485],{"className":10486,"style":10076},[263],[32,10488,10489],{},[32,10490,9999],{"className":10491},[10058,10103,10104],[32,10493,10494,10497],{"style":10107},[32,10495],{"className":10496,"style":10076},[263],[32,10498,10500],{"className":10499},[268,269,270,271],[32,10501,5322],{"className":10502,"style":1530},[95,96,271],[32,10504,285],{"className":10505},[284],[32,10507,10509],{"className":10508},[251],[32,10510,10512],{"className":10511,"style":10126},[255],[32,10513],{},[32,10515],{"className":10516,"style":106},[105],[32,10518,10520,10523],{"className":10519},[95],[32,10521,9849],{"className":10522},[95,96],[32,10524,10526],{"className":10525},[242],[32,10527,10529,10549],{"className":10528},[246,247],[32,10530,10532,10546],{"className":10531},[251],[32,10533,10535],{"className":10534,"style":2156},[255],[32,10536,10537,10540],{"style":259},[32,10538],{"className":10539,"style":264},[263],[32,10541,10543],{"className":10542},[268,269,270,271],[32,10544,2084],{"className":10545},[95,96,271],[32,10547,285],{"className":10548},[284],[32,10550,10552],{"className":10551},[251],[32,10553,10555],{"className":10554,"style":1441},[255],[32,10556],{},[32,10558,10560,10563],{"className":10559},[95],[32,10561,2104],{"className":10562},[95,96],[32,10564,10566],{"className":10565},[242],[32,10567,10569,10601],{"className":10568},[246,247],[32,10570,10572,10598],{"className":10571},[251],[32,10573,10575,10587],{"className":10574,"style":7233},[255],[32,10576,10578,10581],{"style":10577},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[32,10579],{"className":10580,"style":264},[263],[32,10582,10584],{"className":10583},[268,269,270,271],[32,10585,2084],{"className":10586},[95,96,271],[32,10588,10589,10592],{"style":6555},[32,10590],{"className":10591,"style":264},[263],[32,10593,10595],{"className":10594},[268,269,270,271],[32,10596,5511],{"className":10597},[95,271],[32,10599,285],{"className":10600},[284],[32,10602,10604],{"className":10603},[251],[32,10605,10608],{"className":10606,"style":10607},[255],"height:0.247em;",[32,10609],{},[32,10611,6076],{"className":10612},[95],[32,10614,10616,10619],{"className":10615},[8970],[32,10617],{"className":10618,"style":10054},[86],[32,10620,10622,10625,10631],{"className":10621},[95,4000],[32,10623,55],{"className":10624},[95],[32,10626,10628],{"className":10627},[95],[32,10629,5511],{"className":10630},[95],[32,10632,69],{"className":10633},[95],[11,10635,10636,10637,10669],{},"因此，",[1364,10638,10639,10640,10668],{},"Hessian 的特征值就是 ",[32,10641,10643,10656],{"className":10642,"translate":36},[35],[32,10644,10646],{"className":10645},[40],[42,10647,10648],{"xmlns":44},[46,10649,10650,10654],{},[49,10651,10652],{},[57,10653,1909],{},[71,10655,1909],{"encoding":73},[32,10657,10659],{"className":10658,"ariaHidden":62},[78],[32,10660,10662,10665],{"className":10661},[82],[32,10663],{"className":10664,"style":4025},[86],[32,10666,1909],{"className":10667,"style":1965},[95,96]," 在临界点处的主二阶曲率","——每个特征值的符号，直接告诉我们函数沿对应方向「向上弯」还是「向下弯」。",[3222,10671,10672,10874,11470,11672,11898],{},[11,10673,10674,10677,10678,10747,10748,10873],{},[1364,10675,10676],{},"定义 2.4（非退化极小值、极大值与鞍点）","。设 ",[32,10679,10681,10698],{"className":10680,"translate":36},[35],[32,10682,10684],{"className":10683},[40],[42,10685,10686],{"xmlns":44},[46,10687,10688,10696],{},[49,10689,10690],{},[143,10691,10692,10694],{},[57,10693,358],{},[57,10695,6594],{},[71,10697,9012],{"encoding":73},[32,10699,10701],{"className":10700,"ariaHidden":62},[78],[32,10702,10704,10707],{"className":10703},[82],[32,10705],{"className":10706,"style":8289},[86],[32,10708,10710,10713],{"className":10709},[95],[32,10711,358],{"className":10712},[95,96],[32,10714,10716],{"className":10715},[242],[32,10717,10719,10739],{"className":10718},[246,247],[32,10720,10722,10736],{"className":10721},[251],[32,10723,10725],{"className":10724,"style":1860},[255],[32,10726,10727,10730],{"style":259},[32,10728],{"className":10729,"style":264},[263],[32,10731,10733],{"className":10732},[268,269,270,271],[32,10734,6594],{"className":10735},[95,96,271],[32,10737,285],{"className":10738},[284],[32,10740,10742],{"className":10741},[251],[32,10743,10745],{"className":10744,"style":1441},[255],[32,10746],{}," 是非退化临界点，即 ",[32,10749,10751,10778],{"className":10750,"translate":36},[35],[32,10752,10754],{"className":10753},[40],[42,10755,10756],{"xmlns":44},[46,10757,10758,10776],{},[49,10759,10760,10766,10768,10774],{},[143,10761,10762,10764],{},[57,10763,7043],{},[57,10765,1909],{},[52,10767,55],{"stretchy":54},[143,10769,10770,10772],{},[57,10771,358],{},[57,10773,6594],{},[52,10775,69],{"stretchy":54},[71,10777,9592],{"encoding":73},[32,10779,10781],{"className":10780,"ariaHidden":62},[78],[32,10782,10784,10787,10827,10830,10870],{"className":10783},[82],[32,10785],{"className":10786,"style":7147},[86],[32,10788,10790,10793],{"className":10789},[95],[32,10791,7043],{"className":10792,"style":7154},[95,96],[32,10794,10796],{"className":10795},[242],[32,10797,10799,10819],{"className":10798},[246,247],[32,10800,10802,10816],{"className":10801},[251],[32,10803,10805],{"className":10804,"style":1419},[255],[32,10806,10807,10810],{"style":7169},[32,10808],{"className":10809,"style":264},[263],[32,10811,10813],{"className":10812},[268,269,270,271],[32,10814,1909],{"className":10815,"style":1965},[95,96,271],[32,10817,285],{"className":10818},[284],[32,10820,10822],{"className":10821},[251],[32,10823,10825],{"className":10824,"style":2519},[255],[32,10826],{},[32,10828,55],{"className":10829},[91],[32,10831,10833,10836],{"className":10832},[95],[32,10834,358],{"className":10835},[95,96],[32,10837,10839],{"className":10838},[242],[32,10840,10842,10862],{"className":10841},[246,247],[32,10843,10845,10859],{"className":10844},[251],[32,10846,10848],{"className":10847,"style":1860},[255],[32,10849,10850,10853],{"style":259},[32,10851],{"className":10852,"style":264},[263],[32,10854,10856],{"className":10855},[268,269,270,271],[32,10857,6594],{"className":10858},[95,96,271],[32,10860,285],{"className":10861},[284],[32,10863,10865],{"className":10864},[251],[32,10866,10868],{"className":10867,"style":1441},[255],[32,10869],{},[32,10871,69],{"className":10872},[113]," 没有零特征值。",[4060,10875,10876,11074,11272],{},[4063,10877,3877,10878,11003,11004,11073],{},[32,10879,10881,10908],{"className":10880,"translate":36},[35],[32,10882,10884],{"className":10883},[40],[42,10885,10886],{"xmlns":44},[46,10887,10888,10906],{},[49,10889,10890,10896,10898,10904],{},[143,10891,10892,10894],{},[57,10893,7043],{},[57,10895,1909],{},[52,10897,55],{"stretchy":54},[143,10899,10900,10902],{},[57,10901,358],{},[57,10903,6594],{},[52,10905,69],{"stretchy":54},[71,10907,9592],{"encoding":73},[32,10909,10911],{"className":10910,"ariaHidden":62},[78],[32,10912,10914,10917,10957,10960,11000],{"className":10913},[82],[32,10915],{"className":10916,"style":7147},[86],[32,10918,10920,10923],{"className":10919},[95],[32,10921,7043],{"className":10922,"style":7154},[95,96],[32,10924,10926],{"className":10925},[242],[32,10927,10929,10949],{"className":10928},[246,247],[32,10930,10932,10946],{"className":10931},[251],[32,10933,10935],{"className":10934,"style":1419},[255],[32,10936,10937,10940],{"style":7169},[32,10938],{"className":10939,"style":264},[263],[32,10941,10943],{"className":10942},[268,269,270,271],[32,10944,1909],{"className":10945,"style":1965},[95,96,271],[32,10947,285],{"className":10948},[284],[32,10950,10952],{"className":10951},[251],[32,10953,10955],{"className":10954,"style":2519},[255],[32,10956],{},[32,10958,55],{"className":10959},[91],[32,10961,10963,10966],{"className":10962},[95],[32,10964,358],{"className":10965},[95,96],[32,10967,10969],{"className":10968},[242],[32,10970,10972,10992],{"className":10971},[246,247],[32,10973,10975,10989],{"className":10974},[251],[32,10976,10978],{"className":10977,"style":1860},[255],[32,10979,10980,10983],{"style":259},[32,10981],{"className":10982,"style":264},[263],[32,10984,10986],{"className":10985},[268,269,270,271],[32,10987,6594],{"className":10988},[95,96,271],[32,10990,285],{"className":10991},[284],[32,10993,10995],{"className":10994},[251],[32,10996,10998],{"className":10997,"style":1441},[255],[32,10999],{},[32,11001,69],{"className":11002},[113]," 的所有特征值都为正，则 ",[32,11005,11007,11024],{"className":11006,"translate":36},[35],[32,11008,11010],{"className":11009},[40],[42,11011,11012],{"xmlns":44},[46,11013,11014,11022],{},[49,11015,11016],{},[143,11017,11018,11020],{},[57,11019,358],{},[57,11021,6594],{},[71,11023,9012],{"encoding":73},[32,11025,11027],{"className":11026,"ariaHidden":62},[78],[32,11028,11030,11033],{"className":11029},[82],[32,11031],{"className":11032,"style":8289},[86],[32,11034,11036,11039],{"className":11035},[95],[32,11037,358],{"className":11038},[95,96],[32,11040,11042],{"className":11041},[242],[32,11043,11045,11065],{"className":11044},[246,247],[32,11046,11048,11062],{"className":11047},[251],[32,11049,11051],{"className":11050,"style":1860},[255],[32,11052,11053,11056],{"style":259},[32,11054],{"className":11055,"style":264},[263],[32,11057,11059],{"className":11058},[268,269,270,271],[32,11060,6594],{"className":11061},[95,96,271],[32,11063,285],{"className":11064},[284],[32,11066,11068],{"className":11067},[251],[32,11069,11071],{"className":11070,"style":1441},[255],[32,11072],{}," 是非退化局部极小值。",[4063,11075,3877,11076,11201,11202,11271],{},[32,11077,11079,11106],{"className":11078,"translate":36},[35],[32,11080,11082],{"className":11081},[40],[42,11083,11084],{"xmlns":44},[46,11085,11086,11104],{},[49,11087,11088,11094,11096,11102],{},[143,11089,11090,11092],{},[57,11091,7043],{},[57,11093,1909],{},[52,11095,55],{"stretchy":54},[143,11097,11098,11100],{},[57,11099,358],{},[57,11101,6594],{},[52,11103,69],{"stretchy":54},[71,11105,9592],{"encoding":73},[32,11107,11109],{"className":11108,"ariaHidden":62},[78],[32,11110,11112,11115,11155,11158,11198],{"className":11111},[82],[32,11113],{"className":11114,"style":7147},[86],[32,11116,11118,11121],{"className":11117},[95],[32,11119,7043],{"className":11120,"style":7154},[95,96],[32,11122,11124],{"className":11123},[242],[32,11125,11127,11147],{"className":11126},[246,247],[32,11128,11130,11144],{"className":11129},[251],[32,11131,11133],{"className":11132,"style":1419},[255],[32,11134,11135,11138],{"style":7169},[32,11136],{"className":11137,"style":264},[263],[32,11139,11141],{"className":11140},[268,269,270,271],[32,11142,1909],{"className":11143,"style":1965},[95,96,271],[32,11145,285],{"className":11146},[284],[32,11148,11150],{"className":11149},[251],[32,11151,11153],{"className":11152,"style":2519},[255],[32,11154],{},[32,11156,55],{"className":11157},[91],[32,11159,11161,11164],{"className":11160},[95],[32,11162,358],{"className":11163},[95,96],[32,11165,11167],{"className":11166},[242],[32,11168,11170,11190],{"className":11169},[246,247],[32,11171,11173,11187],{"className":11172},[251],[32,11174,11176],{"className":11175,"style":1860},[255],[32,11177,11178,11181],{"style":259},[32,11179],{"className":11180,"style":264},[263],[32,11182,11184],{"className":11183},[268,269,270,271],[32,11185,6594],{"className":11186},[95,96,271],[32,11188,285],{"className":11189},[284],[32,11191,11193],{"className":11192},[251],[32,11194,11196],{"className":11195,"style":1441},[255],[32,11197],{},[32,11199,69],{"className":11200},[113]," 的所有特征值都为负，则 ",[32,11203,11205,11222],{"className":11204,"translate":36},[35],[32,11206,11208],{"className":11207},[40],[42,11209,11210],{"xmlns":44},[46,11211,11212,11220],{},[49,11213,11214],{},[143,11215,11216,11218],{},[57,11217,358],{},[57,11219,6594],{},[71,11221,9012],{"encoding":73},[32,11223,11225],{"className":11224,"ariaHidden":62},[78],[32,11226,11228,11231],{"className":11227},[82],[32,11229],{"className":11230,"style":8289},[86],[32,11232,11234,11237],{"className":11233},[95],[32,11235,358],{"className":11236},[95,96],[32,11238,11240],{"className":11239},[242],[32,11241,11243,11263],{"className":11242},[246,247],[32,11244,11246,11260],{"className":11245},[251],[32,11247,11249],{"className":11248,"style":1860},[255],[32,11250,11251,11254],{"style":259},[32,11252],{"className":11253,"style":264},[263],[32,11255,11257],{"className":11256},[268,269,270,271],[32,11258,6594],{"className":11259},[95,96,271],[32,11261,285],{"className":11262},[284],[32,11264,11266],{"className":11265},[251],[32,11267,11269],{"className":11268,"style":1441},[255],[32,11270],{}," 是非退化局部极大值。",[4063,11273,3877,11274,11399,11400,11469],{},[32,11275,11277,11304],{"className":11276,"translate":36},[35],[32,11278,11280],{"className":11279},[40],[42,11281,11282],{"xmlns":44},[46,11283,11284,11302],{},[49,11285,11286,11292,11294,11300],{},[143,11287,11288,11290],{},[57,11289,7043],{},[57,11291,1909],{},[52,11293,55],{"stretchy":54},[143,11295,11296,11298],{},[57,11297,358],{},[57,11299,6594],{},[52,11301,69],{"stretchy":54},[71,11303,9592],{"encoding":73},[32,11305,11307],{"className":11306,"ariaHidden":62},[78],[32,11308,11310,11313,11353,11356,11396],{"className":11309},[82],[32,11311],{"className":11312,"style":7147},[86],[32,11314,11316,11319],{"className":11315},[95],[32,11317,7043],{"className":11318,"style":7154},[95,96],[32,11320,11322],{"className":11321},[242],[32,11323,11325,11345],{"className":11324},[246,247],[32,11326,11328,11342],{"className":11327},[251],[32,11329,11331],{"className":11330,"style":1419},[255],[32,11332,11333,11336],{"style":7169},[32,11334],{"className":11335,"style":264},[263],[32,11337,11339],{"className":11338},[268,269,270,271],[32,11340,1909],{"className":11341,"style":1965},[95,96,271],[32,11343,285],{"className":11344},[284],[32,11346,11348],{"className":11347},[251],[32,11349,11351],{"className":11350,"style":2519},[255],[32,11352],{},[32,11354,55],{"className":11355},[91],[32,11357,11359,11362],{"className":11358},[95],[32,11360,358],{"className":11361},[95,96],[32,11363,11365],{"className":11364},[242],[32,11366,11368,11388],{"className":11367},[246,247],[32,11369,11371,11385],{"className":11370},[251],[32,11372,11374],{"className":11373,"style":1860},[255],[32,11375,11376,11379],{"style":259},[32,11377],{"className":11378,"style":264},[263],[32,11380,11382],{"className":11381},[268,269,270,271],[32,11383,6594],{"className":11384},[95,96,271],[32,11386,285],{"className":11387},[284],[32,11389,11391],{"className":11390},[251],[32,11392,11394],{"className":11393,"style":1441},[255],[32,11395],{},[32,11397,69],{"className":11398},[113]," 至少有一个正特征值和至少一个负特征值，则 ",[32,11401,11403,11420],{"className":11402,"translate":36},[35],[32,11404,11406],{"className":11405},[40],[42,11407,11408],{"xmlns":44},[46,11409,11410,11418],{},[49,11411,11412],{},[143,11413,11414,11416],{},[57,11415,358],{},[57,11417,6594],{},[71,11419,9012],{"encoding":73},[32,11421,11423],{"className":11422,"ariaHidden":62},[78],[32,11424,11426,11429],{"className":11425},[82],[32,11427],{"className":11428,"style":8289},[86],[32,11430,11432,11435],{"className":11431},[95],[32,11433,358],{"className":11434},[95,96],[32,11436,11438],{"className":11437},[242],[32,11439,11441,11461],{"className":11440},[246,247],[32,11442,11444,11458],{"className":11443},[251],[32,11445,11447],{"className":11446,"style":1860},[255],[32,11448,11449,11452],{"style":259},[32,11450],{"className":11451,"style":264},[263],[32,11453,11455],{"className":11454},[268,269,270,271],[32,11456,6594],{"className":11457},[95,96,271],[32,11459,285],{"className":11460},[284],[32,11462,11464],{"className":11463},[251],[32,11465,11467],{"className":11466,"style":1441},[255],[32,11468],{}," 是鞍点。",[11,11471,11472,11475,11476,11545,11546,11671],{},[1364,11473,11474],{},"定义 2.5（Morse 指数）","。非退化临界点 ",[32,11477,11479,11496],{"className":11478,"translate":36},[35],[32,11480,11482],{"className":11481},[40],[42,11483,11484],{"xmlns":44},[46,11485,11486,11494],{},[49,11487,11488],{},[143,11489,11490,11492],{},[57,11491,358],{},[57,11493,6594],{},[71,11495,9012],{"encoding":73},[32,11497,11499],{"className":11498,"ariaHidden":62},[78],[32,11500,11502,11505],{"className":11501},[82],[32,11503],{"className":11504,"style":8289},[86],[32,11506,11508,11511],{"className":11507},[95],[32,11509,358],{"className":11510},[95,96],[32,11512,11514],{"className":11513},[242],[32,11515,11517,11537],{"className":11516},[246,247],[32,11518,11520,11534],{"className":11519},[251],[32,11521,11523],{"className":11522,"style":1860},[255],[32,11524,11525,11528],{"style":259},[32,11526],{"className":11527,"style":264},[263],[32,11529,11531],{"className":11530},[268,269,270,271],[32,11532,6594],{"className":11533},[95,96,271],[32,11535,285],{"className":11536},[284],[32,11538,11540],{"className":11539},[251],[32,11541,11543],{"className":11542,"style":1441},[255],[32,11544],{}," 的 Morse 指数是 ",[32,11547,11549,11576],{"className":11548,"translate":36},[35],[32,11550,11552],{"className":11551},[40],[42,11553,11554],{"xmlns":44},[46,11555,11556,11574],{},[49,11557,11558,11564,11566,11572],{},[143,11559,11560,11562],{},[57,11561,7043],{},[57,11563,1909],{},[52,11565,55],{"stretchy":54},[143,11567,11568,11570],{},[57,11569,358],{},[57,11571,6594],{},[52,11573,69],{"stretchy":54},[71,11575,9592],{"encoding":73},[32,11577,11579],{"className":11578,"ariaHidden":62},[78],[32,11580,11582,11585,11625,11628,11668],{"className":11581},[82],[32,11583],{"className":11584,"style":7147},[86],[32,11586,11588,11591],{"className":11587},[95],[32,11589,7043],{"className":11590,"style":7154},[95,96],[32,11592,11594],{"className":11593},[242],[32,11595,11597,11617],{"className":11596},[246,247],[32,11598,11600,11614],{"className":11599},[251],[32,11601,11603],{"className":11602,"style":1419},[255],[32,11604,11605,11608],{"style":7169},[32,11606],{"className":11607,"style":264},[263],[32,11609,11611],{"className":11610},[268,269,270,271],[32,11612,1909],{"className":11613,"style":1965},[95,96,271],[32,11615,285],{"className":11616},[284],[32,11618,11620],{"className":11619},[251],[32,11621,11623],{"className":11622,"style":2519},[255],[32,11624],{},[32,11626,55],{"className":11627},[91],[32,11629,11631,11634],{"className":11630},[95],[32,11632,358],{"className":11633},[95,96],[32,11635,11637],{"className":11636},[242],[32,11638,11640,11660],{"className":11639},[246,247],[32,11641,11643,11657],{"className":11642},[251],[32,11644,11646],{"className":11645,"style":1860},[255],[32,11647,11648,11651],{"style":259},[32,11649],{"className":11650,"style":264},[263],[32,11652,11654],{"className":11653},[268,269,270,271],[32,11655,6594],{"className":11656},[95,96,271],[32,11658,285],{"className":11659},[284],[32,11661,11663],{"className":11662},[251],[32,11664,11666],{"className":11665,"style":1441},[255],[32,11667],{},[32,11669,69],{"className":11670},[113]," 的负特征值个数",[32,11673,11675],{"className":11674,"translate":36},[299],[32,11676,11678,11734],{"className":11677,"translate":36},[35],[32,11679,11681],{"className":11680},[40],[42,11682,11683],{"xmlns":44,"display":308},[46,11684,11685,11731],{},[49,11686,11687,11690,11693,11695,11701,11703,11705,11707,11710,11712,11714,11716,11722,11725,11727,11729],{},[57,11688,11689],{"mathvariant":1083},"index",[52,11691,11692],{},"⁡",[52,11694,55],{"stretchy":54},[143,11696,11697,11699],{},[57,11698,358],{},[57,11700,6594],{},[52,11702,69],{"stretchy":54},[52,11704,131],{},[52,11706,368],{},[57,11708,11709],{"mathvariant":1083},"#",[52,11711,315],{"stretchy":54},[57,11713,2084],{},[52,11715,131],{},[143,11717,11718,11720],{},[57,11719,9849],{},[57,11721,2084],{},[52,11723,11724],{},"\u003C",[155,11726,157],{},[52,11728,1749],{"stretchy":54},[57,11730,6076],{"mathvariant":1083},[71,11732,11733],{"encoding":73},"\\operatorname{index}(x_c) := \\#\\{i : \\lambda_i \u003C 0\\}.",[32,11735,11737,11804,11828,11883],{"className":11736,"ariaHidden":62},[78],[32,11738,11740,11743,11749,11752,11792,11795,11798,11801],{"className":11739},[82],[32,11741],{"className":11742,"style":87},[86],[32,11744,11746],{"className":11745},[10058],[32,11747,11689],{"className":11748},[95,2244],[32,11750,55],{"className":11751},[91],[32,11753,11755,11758],{"className":11754},[95],[32,11756,358],{"className":11757},[95,96],[32,11759,11761],{"className":11760},[242],[32,11762,11764,11784],{"className":11763},[246,247],[32,11765,11767,11781],{"className":11766},[251],[32,11768,11770],{"className":11769,"style":1860},[255],[32,11771,11772,11775],{"style":259},[32,11773],{"className":11774,"style":264},[263],[32,11776,11778],{"className":11777},[268,269,270,271],[32,11779,6594],{"className":11780},[95,96,271],[32,11782,285],{"className":11783},[284],[32,11785,11787],{"className":11786},[251],[32,11788,11790],{"className":11789,"style":1441},[255],[32,11791],{},[32,11793,69],{"className":11794},[113],[32,11796],{"className":11797,"style":177},[105],[32,11799,2185],{"className":11800},[181],[32,11802],{"className":11803,"style":177},[105],[32,11805,11807,11810,11813,11816,11819,11822,11825],{"className":11806},[82],[32,11808],{"className":11809,"style":87},[86],[32,11811,11709],{"className":11812},[95],[32,11814,315],{"className":11815},[91],[32,11817,2084],{"className":11818},[95,96],[32,11820],{"className":11821,"style":177},[105],[32,11823,131],{"className":11824},[181],[32,11826],{"className":11827,"style":177},[105],[32,11829,11831,11834,11874,11877,11880],{"className":11830},[82],[32,11832],{"className":11833,"style":1400},[86],[32,11835,11837,11840],{"className":11836},[95],[32,11838,9849],{"className":11839},[95,96],[32,11841,11843],{"className":11842},[242],[32,11844,11846,11866],{"className":11845},[246,247],[32,11847,11849,11863],{"className":11848},[251],[32,11850,11852],{"className":11851,"style":2156},[255],[32,11853,11854,11857],{"style":259},[32,11855],{"className":11856,"style":264},[263],[32,11858,11860],{"className":11859},[268,269,270,271],[32,11861,2084],{"className":11862},[95,96,271],[32,11864,285],{"className":11865},[284],[32,11867,11869],{"className":11868},[251],[32,11870,11872],{"className":11871,"style":1441},[255],[32,11873],{},[32,11875],{"className":11876,"style":177},[105],[32,11878,11724],{"className":11879},[181],[32,11881],{"className":11882,"style":177},[105],[32,11884,11886,11889,11892,11895],{"className":11885},[82],[32,11887],{"className":11888,"style":87},[86],[32,11890,157],{"className":11891},[95],[32,11893,1749],{"className":11894},[113],[32,11896,6076],{"className":11897},[95],[11,11899,11900,11901,11929,11930,11982],{},"非退化局部极小值的指数为 0，非退化局部极大值的指数为 ",[32,11902,11904,11917],{"className":11903,"translate":36},[35],[32,11905,11907],{"className":11906},[40],[42,11908,11909],{"xmlns":44},[46,11910,11911,11915],{},[49,11912,11913],{},[57,11914,5322],{},[71,11916,5322],{"encoding":73},[32,11918,11920],{"className":11919,"ariaHidden":62},[78],[32,11921,11923,11926],{"className":11922},[82],[32,11924],{"className":11925,"style":212},[86],[32,11927,5322],{"className":11928,"style":1530},[95,96],"，鞍点的指数在 1 到 ",[32,11931,11933,11951],{"className":11932,"translate":36},[35],[32,11934,11936],{"className":11935},[40],[42,11937,11938],{"xmlns":44},[46,11939,11940,11948],{},[49,11941,11942,11944,11946],{},[57,11943,5322],{},[52,11945,1089],{},[155,11947,1732],{},[71,11949,11950],{"encoding":73},"N-1",[32,11952,11954,11972],{"className":11953,"ariaHidden":62},[78],[32,11955,11957,11960,11963,11966,11969],{"className":11956},[82],[32,11958],{"className":11959,"style":191},[86],[32,11961,5322],{"className":11962,"style":1530},[95,96],[32,11964],{"className":11965,"style":198},[105],[32,11967,1089],{"className":11968},[202],[32,11970],{"className":11971,"style":198},[105],[32,11973,11975,11979],{"className":11974},[82],[32,11976],{"className":11977,"style":11978},[86],"height:0.6444em;",[32,11980,1732],{"className":11981},[95]," 之间。",[11,11984,11985,11986,12014,12015,12043],{},"高维的困难此刻清晰可见。在 1 维中，非退化极小值只要求一个正的二阶导数。在 ",[32,11987,11989,12002],{"className":11988,"translate":36},[35],[32,11990,11992],{"className":11991},[40],[42,11993,11994],{"xmlns":44},[46,11995,11996,12000],{},[49,11997,11998],{},[57,11999,5322],{},[71,12001,5322],{"encoding":73},[32,12003,12005],{"className":12004,"ariaHidden":62},[78],[32,12006,12008,12011],{"className":12007},[82],[32,12009],{"className":12010,"style":212},[86],[32,12012,5322],{"className":12013,"style":1530},[95,96]," 维中，非退化局部极小值要求 ",[32,12016,12018,12031],{"className":12017,"translate":36},[35],[32,12019,12021],{"className":12020},[40],[42,12022,12023],{"xmlns":44},[46,12024,12025,12029],{},[49,12026,12027],{},[57,12028,5322],{},[71,12030,5322],{"encoding":73},[32,12032,12034],{"className":12033,"ariaHidden":62},[78],[32,12035,12037,12040],{"className":12036},[82],[32,12038],{"className":12039,"style":212},[86],[32,12041,5322],{"className":12042,"style":1530},[95,96]," 个同时成立的正曲率条件：",[32,12045,12047],{"className":12046,"translate":36},[299],[32,12048,12050,12109],{"className":12049,"translate":36},[35],[32,12051,12053],{"className":12052},[40],[42,12054,12055],{"xmlns":44,"display":308},[46,12056,12057,12106],{},[49,12058,12059,12065,12068,12070,12072,12075,12081,12083,12085,12087,12089,12091,12093,12095,12101,12103],{},[143,12060,12061,12063],{},[57,12062,9849],{},[155,12064,1732],{},[52,12066,12067],{},">",[155,12069,157],{},[52,12071,63],{"separator":62},[372,12073,12074],{}," ",[143,12076,12077,12079],{},[57,12078,9849],{},[155,12080,5511],{},[52,12082,12067],{},[155,12084,157],{},[52,12086,63],{"separator":62},[372,12088,12074],{},[52,12090,1737],{},[52,12092,63],{"separator":62},[372,12094,12074],{},[143,12096,12097,12099],{},[57,12098,9849],{},[57,12100,5322],{},[52,12102,12067],{},[155,12104,12105],{},"0.",[71,12107,12108],{"encoding":73},"\\lambda_1 > 0,\\ \\lambda_2 > 0,\\ \\dots,\\ \\lambda_N > 0.",[32,12110,12112,12167,12234,12316],{"className":12111,"ariaHidden":62},[78],[32,12113,12115,12118,12158,12161,12164],{"className":12114},[82],[32,12116],{"className":12117,"style":1400},[86],[32,12119,12121,12124],{"className":12120},[95],[32,12122,9849],{"className":12123},[95,96],[32,12125,12127],{"className":12126},[242],[32,12128,12130,12150],{"className":12129},[246,247],[32,12131,12133,12147],{"className":12132},[251],[32,12134,12136],{"className":12135,"style":256},[255],[32,12137,12138,12141],{"style":259},[32,12139],{"className":12140,"style":264},[263],[32,12142,12144],{"className":12143},[268,269,270,271],[32,12145,1732],{"className":12146},[95,271],[32,12148,285],{"className":12149},[284],[32,12151,12153],{"className":12152},[251],[32,12154,12156],{"className":12155,"style":1441},[255],[32,12157],{},[32,12159],{"className":12160,"style":177},[105],[32,12162,12067],{"className":12163},[181],[32,12165],{"className":12166,"style":177},[105],[32,12168,12170,12173,12176,12179,12182,12185,12225,12228,12231],{"className":12169},[82],[32,12171],{"className":12172,"style":4025},[86],[32,12174,157],{"className":12175},[95],[32,12177,63],{"className":12178},[101],[32,12180,12074],{"className":12181},[105],[32,12183],{"className":12184,"style":106},[105],[32,12186,12188,12191],{"className":12187},[95],[32,12189,9849],{"className":12190},[95,96],[32,12192,12194],{"className":12193},[242],[32,12195,12197,12217],{"className":12196},[246,247],[32,12198,12200,12214],{"className":12199},[251],[32,12201,12203],{"className":12202,"style":256},[255],[32,12204,12205,12208],{"style":259},[32,12206],{"className":12207,"style":264},[263],[32,12209,12211],{"className":12210},[268,269,270,271],[32,12212,5511],{"className":12213},[95,271],[32,12215,285],{"className":12216},[284],[32,12218,12220],{"className":12219},[251],[32,12221,12223],{"className":12222,"style":1441},[255],[32,12224],{},[32,12226],{"className":12227,"style":177},[105],[32,12229,12067],{"className":12230},[181],[32,12232],{"className":12233,"style":177},[105],[32,12235,12237,12240,12243,12246,12249,12252,12255,12258,12261,12264,12267,12307,12310,12313],{"className":12236},[82],[32,12238],{"className":12239,"style":4025},[86],[32,12241,157],{"className":12242},[95],[32,12244,63],{"className":12245},[101],[32,12247,12074],{"className":12248},[105],[32,12250],{"className":12251,"style":106},[105],[32,12253,1737],{"className":12254},[507],[32,12256],{"className":12257,"style":106},[105],[32,12259,63],{"className":12260},[101],[32,12262,12074],{"className":12263},[105],[32,12265],{"className":12266,"style":106},[105],[32,12268,12270,12273],{"className":12269},[95],[32,12271,9849],{"className":12272},[95,96],[32,12274,12276],{"className":12275},[242],[32,12277,12279,12299],{"className":12278},[246,247],[32,12280,12282,12296],{"className":12281},[251],[32,12283,12285],{"className":12284,"style":4238},[255],[32,12286,12287,12290],{"style":259},[32,12288],{"className":12289,"style":264},[263],[32,12291,12293],{"className":12292},[268,269,270,271],[32,12294,5322],{"className":12295,"style":1530},[95,96,271],[32,12297,285],{"className":12298},[284],[32,12300,12302],{"className":12301},[251],[32,12303,12305],{"className":12304,"style":1441},[255],[32,12306],{},[32,12308],{"className":12309,"style":177},[105],[32,12311,12067],{"className":12312},[181],[32,12314],{"className":12315,"style":177},[105],[32,12317,12319,12322],{"className":12318},[82],[32,12320],{"className":12321,"style":11978},[86],[32,12323,12105],{"className":12324},[95],[11,12326,12327],{},"相比之下，鞍点只要求至少一个正特征值和至少一个负特征值。当 Hessian 是随机的、且对正曲率没有强偏好时，后者显然容易满足得多。",[21,12329,12330],{"id":12330},"临界点本身就稀少",[11,12332,12333],{},"在讨论「临界点是不是极小值」之前，先确认一个更基础、但常被忽视的事实：临界点本身在空间中就是「测度为零」的稀疏集合。这个结论不难证明，但它还不是本文要讲的高维现象——它对一维函数同样成立。",[3222,12335,12336,12574,12782,12814],{},[11,12337,12338,6575,12341,12399,12400,12503,12504,12573],{},[1364,12339,12340],{},"定义 3.1（Morse 函数）",[32,12342,12344,12361],{"className":12343,"translate":36},[35],[32,12345,12347],{"className":12346},[40],[42,12348,12349],{"xmlns":44},[46,12350,12351,12359],{},[49,12352,12353],{},[5397,12354,12355,12357],{},[57,12356,5508],{},[155,12358,5511],{},[71,12360,5638],{"encoding":73},[32,12362,12364],{"className":12363,"ariaHidden":62},[78],[32,12365,12367,12370],{"className":12366},[82],[32,12368],{"className":12369,"style":5571},[86],[32,12371,12373,12376],{"className":12372},[95],[32,12374,5508],{"className":12375,"style":3492},[95,96],[32,12377,12379],{"className":12378},[242],[32,12380,12382],{"className":12381},[246],[32,12383,12385],{"className":12384},[251],[32,12386,12388],{"className":12387,"style":5571},[255],[32,12389,12390,12393],{"style":5458},[32,12391],{"className":12392,"style":264},[263],[32,12394,12396],{"className":12395},[268,269,270,271],[32,12397,5511],{"className":12398},[95,271]," 函数 ",[32,12401,12403,12429],{"className":12402,"translate":36},[35],[32,12404,12406],{"className":12405},[40],[42,12407,12408],{"xmlns":44},[46,12409,12410,12426],{},[49,12411,12412,12414,12416,12422,12424],{},[57,12413,1909],{},[52,12415,131],{},[5397,12417,12418,12420],{},[57,12419,148],{"mathvariant":147},[57,12421,5322],{},[52,12423,141],{},[57,12425,148],{"mathvariant":147},[71,12427,12428],{"encoding":73},"f:\\mathbb{R}^N\\to\\mathbb{R}",[32,12430,12432,12450,12494],{"className":12431,"ariaHidden":62},[78],[32,12433,12435,12438,12441,12444,12447],{"className":12434},[82],[32,12436],{"className":12437,"style":4025},[86],[32,12439,1909],{"className":12440,"style":1965},[95,96],[32,12442],{"className":12443,"style":177},[105],[32,12445,131],{"className":12446},[181],[32,12448],{"className":12449,"style":177},[105],[32,12451,12453,12456,12485,12488,12491],{"className":12452},[82],[32,12454],{"className":12455,"style":5437},[86],[32,12457,12459,12462],{"className":12458},[95],[32,12460,148],{"className":12461},[95,238],[32,12463,12465],{"className":12464},[242],[32,12466,12468],{"className":12467},[246],[32,12469,12471],{"className":12470},[251],[32,12472,12474],{"className":12473,"style":5437},[255],[32,12475,12476,12479],{"style":5458},[32,12477],{"className":12478,"style":264},[263],[32,12480,12482],{"className":12481},[268,269,270,271],[32,12483,5322],{"className":12484,"style":1530},[95,96,271],[32,12486],{"className":12487,"style":177},[105],[32,12489,141],{"className":12490},[181],[32,12492],{"className":12493,"style":177},[105],[32,12495,12497,12500],{"className":12496},[82],[32,12498],{"className":12499,"style":5483},[86],[32,12501,148],{"className":12502},[95,238]," 的每个临界点都是非退化的，即对每个临界点 ",[32,12505,12507,12524],{"className":12506,"translate":36},[35],[32,12508,12510],{"className":12509},[40],[42,12511,12512],{"xmlns":44},[46,12513,12514,12522],{},[49,12515,12516],{},[143,12517,12518,12520],{},[57,12519,358],{},[57,12521,6594],{},[71,12523,9012],{"encoding":73},[32,12525,12527],{"className":12526,"ariaHidden":62},[78],[32,12528,12530,12533],{"className":12529},[82],[32,12531],{"className":12532,"style":8289},[86],[32,12534,12536,12539],{"className":12535},[95],[32,12537,358],{"className":12538},[95,96],[32,12540,12542],{"className":12541},[242],[32,12543,12545,12565],{"className":12544},[246,247],[32,12546,12548,12562],{"className":12547},[251],[32,12549,12551],{"className":12550,"style":1860},[255],[32,12552,12553,12556],{"style":259},[32,12554],{"className":12555,"style":264},[263],[32,12557,12559],{"className":12558},[268,269,270,271],[32,12560,6594],{"className":12561},[95,96,271],[32,12563,285],{"className":12564},[284],[32,12566,12568],{"className":12567},[251],[32,12569,12571],{"className":12570,"style":1441},[255],[32,12572],{}," 有",[32,12575,12577],{"className":12576,"translate":36},[299],[32,12578,12580,12620],{"className":12579,"translate":36},[35],[32,12581,12583],{"className":12582},[40],[42,12584,12585],{"xmlns":44,"display":308},[46,12586,12587,12617],{},[49,12588,12589,12592,12594,12600,12602,12608,12610,12613,12615],{},[57,12590,12591],{},"det",[52,12593,11692],{},[143,12595,12596,12598],{},[57,12597,7043],{},[57,12599,1909],{},[52,12601,55],{"stretchy":54},[143,12603,12604,12606],{},[57,12605,358],{},[57,12607,6594],{},[52,12609,69],{"stretchy":54},[52,12611,12612],{"mathvariant":1083},"≠",[155,12614,157],{},[52,12616,63],{"separator":62},[71,12618,12619],{"encoding":73},"\\det H_f(x_c) \\neq 0,",[32,12621,12623,12770],{"className":12622,"ariaHidden":62},[78],[32,12624,12626,12629,12632,12635,12675,12678,12718,12721,12724,12767],{"className":12625},[82],[32,12627],{"className":12628,"style":7147},[86],[32,12630,12591],{"className":12631},[10058],[32,12633],{"className":12634,"style":106},[105],[32,12636,12638,12641],{"className":12637},[95],[32,12639,7043],{"className":12640,"style":7154},[95,96],[32,12642,12644],{"className":12643},[242],[32,12645,12647,12667],{"className":12646},[246,247],[32,12648,12650,12664],{"className":12649},[251],[32,12651,12653],{"className":12652,"style":1419},[255],[32,12654,12655,12658],{"style":7169},[32,12656],{"className":12657,"style":264},[263],[32,12659,12661],{"className":12660},[268,269,270,271],[32,12662,1909],{"className":12663,"style":1965},[95,96,271],[32,12665,285],{"className":12666},[284],[32,12668,12670],{"className":12669},[251],[32,12671,12673],{"className":12672,"style":2519},[255],[32,12674],{},[32,12676,55],{"className":12677},[91],[32,12679,12681,12684],{"className":12680},[95],[32,12682,358],{"className":12683},[95,96],[32,12685,12687],{"className":12686},[242],[32,12688,12690,12710],{"className":12689},[246,247],[32,12691,12693,12707],{"className":12692},[251],[32,12694,12696],{"className":12695,"style":1860},[255],[32,12697,12698,12701],{"style":259},[32,12699],{"className":12700,"style":264},[263],[32,12702,12704],{"className":12703},[268,269,270,271],[32,12705,6594],{"className":12706},[95,96,271],[32,12708,285],{"className":12709},[284],[32,12711,12713],{"className":12712},[251],[32,12714,12716],{"className":12715,"style":1441},[255],[32,12717],{},[32,12719,69],{"className":12720},[113],[32,12722],{"className":12723,"style":177},[105],[32,12725,12727,12760,12764],{"className":12726},[181],[32,12728,12730],{"className":12729},[181],[32,12731,12734],{"className":12732},[95,12733],"vbox",[32,12735,12738],{"className":12736},[12737],"thinbox",[32,12739,12742,12745,12756],{"className":12740},[12741],"rlap",[32,12743],{"className":12744,"style":4025},[86],[32,12746,12749],{"className":12747},[12748],"inner",[32,12750,12752],{"className":12751},[95],[32,12753,12755],{"className":12754},[181],"",[32,12757],{"className":12758},[12759],"fix",[32,12761],{"className":12762},[105,12763],"nobreak",[32,12765,368],{"className":12766},[181],[32,12768],{"className":12769,"style":177},[105],[32,12771,12773,12776,12779],{"className":12772},[82],[32,12774],{"className":12775,"style":6815},[86],[32,12777,157],{"className":12778},[95],[32,12780,63],{"className":12781},[101],[11,12783,12784,12785,12813],{},"则称 ",[32,12786,12788,12801],{"className":12787,"translate":36},[35],[32,12789,12791],{"className":12790},[40],[42,12792,12793],{"xmlns":44},[46,12794,12795,12799],{},[49,12796,12797],{},[57,12798,1909],{},[71,12800,1909],{"encoding":73},[32,12802,12804],{"className":12803,"ariaHidden":62},[78],[32,12805,12807,12810],{"className":12806},[82],[32,12808],{"className":12809,"style":4025},[86],[32,12811,1909],{"className":12812,"style":1965},[95,96]," 为 Morse 函数。",[11,12815,12816,6575,12819,12847,12848,12907],{},[1364,12817,12818],{},"命题 3.2",[32,12820,12822,12835],{"className":12821,"translate":36},[35],[32,12823,12825],{"className":12824},[40],[42,12826,12827],{"xmlns":44},[46,12828,12829,12833],{},[49,12830,12831],{},[57,12832,1909],{},[71,12834,1909],{"encoding":73},[32,12836,12838],{"className":12837,"ariaHidden":62},[78],[32,12839,12841,12844],{"className":12840},[82],[32,12842],{"className":12843,"style":4025},[86],[32,12845,1909],{"className":12846,"style":1965},[95,96]," 是 Morse 函数，则其临界点是孤立的。特别地，在 ",[32,12849,12851,12869],{"className":12850,"translate":36},[35],[32,12852,12854],{"className":12853},[40],[42,12855,12856],{"xmlns":44},[46,12857,12858,12866],{},[49,12859,12860],{},[5397,12861,12862,12864],{},[57,12863,148],{"mathvariant":147},[57,12865,5322],{},[71,12867,12868],{"encoding":73},"\\mathbb{R}^N",[32,12870,12872],{"className":12871,"ariaHidden":62},[78],[32,12873,12875,12878],{"className":12874},[82],[32,12876],{"className":12877,"style":5437},[86],[32,12879,12881,12884],{"className":12880},[95],[32,12882,148],{"className":12883},[95,238],[32,12885,12887],{"className":12886},[242],[32,12888,12890],{"className":12889},[246],[32,12891,12893],{"className":12892},[251],[32,12894,12896],{"className":12895,"style":5437},[255],[32,12897,12898,12901],{"style":5458},[32,12899],{"className":12900,"style":264},[263],[32,12902,12904],{"className":12903},[268,269,270,271],[32,12905,5322],{"className":12906,"style":1530},[95,96,271]," 的每个紧子集中，只有有限多个临界点，因此临界点集具有 Lebesgue 测度零。",[11,12909,12910,12914,12915,13053,13054,13123,13124,6575,13249,13318,13319,13444,13445,5714,13479,13548,13549,1265,13618,13651,13652],{},[12911,12912,12913],"em",{},"证明","：临界点是梯度映射 ",[32,12916,12918,12950],{"className":12917,"translate":36},[35],[32,12919,12921],{"className":12920},[40],[42,12922,12923],{"xmlns":44},[46,12924,12925,12947],{},[49,12926,12927,12929,12931,12933,12939,12941],{},[57,12928,5957],{"mathvariant":1083},[57,12930,1909],{},[52,12932,131],{},[5397,12934,12935,12937],{},[57,12936,148],{"mathvariant":147},[57,12938,5322],{},[52,12940,141],{},[5397,12942,12943,12945],{},[57,12944,148],{"mathvariant":147},[57,12946,5322],{},[71,12948,12949],{"encoding":73},"\\nabla f: \\mathbb{R}^N \\to \\mathbb{R}^N",[32,12951,12953,12974,13018],{"className":12952,"ariaHidden":62},[78],[32,12954,12956,12959,12962,12965,12968,12971],{"className":12955},[82],[32,12957],{"className":12958,"style":4025},[86],[32,12960,5957],{"className":12961},[95],[32,12963,1909],{"className":12964,"style":1965},[95,96],[32,12966],{"className":12967,"style":177},[105],[32,12969,131],{"className":12970},[181],[32,12972],{"className":12973,"style":177},[105],[32,12975,12977,12980,13009,13012,13015],{"className":12976},[82],[32,12978],{"className":12979,"style":5437},[86],[32,12981,12983,12986],{"className":12982},[95],[32,12984,148],{"className":12985},[95,238],[32,12987,12989],{"className":12988},[242],[32,12990,12992],{"className":12991},[246],[32,12993,12995],{"className":12994},[251],[32,12996,12998],{"className":12997,"style":5437},[255],[32,12999,13000,13003],{"style":5458},[32,13001],{"className":13002,"style":264},[263],[32,13004,13006],{"className":13005},[268,269,270,271],[32,13007,5322],{"className":13008,"style":1530},[95,96,271],[32,13010],{"className":13011,"style":177},[105],[32,13013,141],{"className":13014},[181],[32,13016],{"className":13017,"style":177},[105],[32,13019,13021,13024],{"className":13020},[82],[32,13022],{"className":13023,"style":5437},[86],[32,13025,13027,13030],{"className":13026},[95],[32,13028,148],{"className":13029},[95,238],[32,13031,13033],{"className":13032},[242],[32,13034,13036],{"className":13035},[246],[32,13037,13039],{"className":13038},[251],[32,13040,13042],{"className":13041,"style":5437},[255],[32,13043,13044,13047],{"style":5458},[32,13045],{"className":13046,"style":264},[263],[32,13048,13050],{"className":13049},[268,269,270,271],[32,13051,5322],{"className":13052,"style":1530},[95,96,271]," 的零点。该映射在 ",[32,13055,13057,13074],{"className":13056,"translate":36},[35],[32,13058,13060],{"className":13059},[40],[42,13061,13062],{"xmlns":44},[46,13063,13064,13072],{},[49,13065,13066],{},[143,13067,13068,13070],{},[57,13069,358],{},[57,13071,6594],{},[71,13073,9012],{"encoding":73},[32,13075,13077],{"className":13076,"ariaHidden":62},[78],[32,13078,13080,13083],{"className":13079},[82],[32,13081],{"className":13082,"style":8289},[86],[32,13084,13086,13089],{"className":13085},[95],[32,13087,358],{"className":13088},[95,96],[32,13090,13092],{"className":13091},[242],[32,13093,13095,13115],{"className":13094},[246,247],[32,13096,13098,13112],{"className":13097},[251],[32,13099,13101],{"className":13100,"style":1860},[255],[32,13102,13103,13106],{"style":259},[32,13104],{"className":13105,"style":264},[263],[32,13107,13109],{"className":13108},[268,269,270,271],[32,13110,6594],{"className":13111},[95,96,271],[32,13113,285],{"className":13114},[284],[32,13116,13118],{"className":13117},[251],[32,13119,13121],{"className":13120,"style":1441},[255],[32,13122],{}," 处的导数就是 Hessian ",[32,13125,13127,13154],{"className":13126,"translate":36},[35],[32,13128,13130],{"className":13129},[40],[42,13131,13132],{"xmlns":44},[46,13133,13134,13152],{},[49,13135,13136,13142,13144,13150],{},[143,13137,13138,13140],{},[57,13139,7043],{},[57,13141,1909],{},[52,13143,55],{"stretchy":54},[143,13145,13146,13148],{},[57,13147,358],{},[57,13149,6594],{},[52,13151,69],{"stretchy":54},[71,13153,9592],{"encoding":73},[32,13155,13157],{"className":13156,"ariaHidden":62},[78],[32,13158,13160,13163,13203,13206,13246],{"className":13159},[82],[32,13161],{"className":13162,"style":7147},[86],[32,13164,13166,13169],{"className":13165},[95],[32,13167,7043],{"className":13168,"style":7154},[95,96],[32,13170,13172],{"className":13171},[242],[32,13173,13175,13195],{"className":13174},[246,247],[32,13176,13178,13192],{"className":13177},[251],[32,13179,13181],{"className":13180,"style":1419},[255],[32,13182,13183,13186],{"style":7169},[32,13184],{"className":13185,"style":264},[263],[32,13187,13189],{"className":13188},[268,269,270,271],[32,13190,1909],{"className":13191,"style":1965},[95,96,271],[32,13193,285],{"className":13194},[284],[32,13196,13198],{"className":13197},[251],[32,13199,13201],{"className":13200,"style":2519},[255],[32,13202],{},[32,13204,55],{"className":13205},[91],[32,13207,13209,13212],{"className":13208},[95],[32,13210,358],{"className":13211},[95,96],[32,13213,13215],{"className":13214},[242],[32,13216,13218,13238],{"className":13217},[246,247],[32,13219,13221,13235],{"className":13220},[251],[32,13222,13224],{"className":13223,"style":1860},[255],[32,13225,13226,13229],{"style":259},[32,13227],{"className":13228,"style":264},[263],[32,13230,13232],{"className":13231},[268,269,270,271],[32,13233,6594],{"className":13234},[95,96,271],[32,13236,285],{"className":13237},[284],[32,13239,13241],{"className":13240},[251],[32,13242,13244],{"className":13243,"style":1441},[255],[32,13245],{},[32,13247,69],{"className":13248},[113],[32,13250,13252,13269],{"className":13251,"translate":36},[35],[32,13253,13255],{"className":13254},[40],[42,13256,13257],{"xmlns":44},[46,13258,13259,13267],{},[49,13260,13261],{},[143,13262,13263,13265],{},[57,13264,358],{},[57,13266,6594],{},[71,13268,9012],{"encoding":73},[32,13270,13272],{"className":13271,"ariaHidden":62},[78],[32,13273,13275,13278],{"className":13274},[82],[32,13276],{"className":13277,"style":8289},[86],[32,13279,13281,13284],{"className":13280},[95],[32,13282,358],{"className":13283},[95,96],[32,13285,13287],{"className":13286},[242],[32,13288,13290,13310],{"className":13289},[246,247],[32,13291,13293,13307],{"className":13292},[251],[32,13294,13296],{"className":13295,"style":1860},[255],[32,13297,13298,13301],{"style":259},[32,13299],{"className":13300,"style":264},[263],[32,13302,13304],{"className":13303},[268,269,270,271],[32,13305,6594],{"className":13306},[95,96,271],[32,13308,285],{"className":13309},[284],[32,13311,13313],{"className":13312},[251],[32,13314,13316],{"className":13315,"style":1441},[255],[32,13317],{}," 非退化，则 ",[32,13320,13322,13349],{"className":13321,"translate":36},[35],[32,13323,13325],{"className":13324},[40],[42,13326,13327],{"xmlns":44},[46,13328,13329,13347],{},[49,13330,13331,13337,13339,13345],{},[143,13332,13333,13335],{},[57,13334,7043],{},[57,13336,1909],{},[52,13338,55],{"stretchy":54},[143,13340,13341,13343],{},[57,13342,358],{},[57,13344,6594],{},[52,13346,69],{"stretchy":54},[71,13348,9592],{"encoding":73},[32,13350,13352],{"className":13351,"ariaHidden":62},[78],[32,13353,13355,13358,13398,13401,13441],{"className":13354},[82],[32,13356],{"className":13357,"style":7147},[86],[32,13359,13361,13364],{"className":13360},[95],[32,13362,7043],{"className":13363,"style":7154},[95,96],[32,13365,13367],{"className":13366},[242],[32,13368,13370,13390],{"className":13369},[246,247],[32,13371,13373,13387],{"className":13372},[251],[32,13374,13376],{"className":13375,"style":1419},[255],[32,13377,13378,13381],{"style":7169},[32,13379],{"className":13380,"style":264},[263],[32,13382,13384],{"className":13383},[268,269,270,271],[32,13385,1909],{"className":13386,"style":1965},[95,96,271],[32,13388,285],{"className":13389},[284],[32,13391,13393],{"className":13392},[251],[32,13394,13396],{"className":13395,"style":2519},[255],[32,13397],{},[32,13399,55],{"className":13400},[91],[32,13402,13404,13407],{"className":13403},[95],[32,13405,358],{"className":13406},[95,96],[32,13408,13410],{"className":13409},[242],[32,13411,13413,13433],{"className":13412},[246,247],[32,13414,13416,13430],{"className":13415},[251],[32,13417,13419],{"className":13418,"style":1860},[255],[32,13420,13421,13424],{"style":259},[32,13422],{"className":13423,"style":264},[263],[32,13425,13427],{"className":13426},[268,269,270,271],[32,13428,6594],{"className":13429},[95,96,271],[32,13431,285],{"className":13432},[284],[32,13434,13436],{"className":13435},[251],[32,13437,13439],{"className":13438,"style":1441},[255],[32,13440],{},[32,13442,69],{"className":13443},[113]," 可逆。由反函数定理，",[32,13446,13448,13464],{"className":13447,"translate":36},[35],[32,13449,13451],{"className":13450},[40],[42,13452,13453],{"xmlns":44},[46,13454,13455,13461],{},[49,13456,13457,13459],{},[57,13458,5957],{"mathvariant":1083},[57,13460,1909],{},[71,13462,13463],{"encoding":73},"\\nabla f",[32,13465,13467],{"className":13466,"ariaHidden":62},[78],[32,13468,13470,13473,13476],{"className":13469},[82],[32,13471],{"className":13472,"style":4025},[86],[32,13474,5957],{"className":13475},[95],[32,13477,1909],{"className":13478,"style":1965},[95,96],[32,13480,13482,13499],{"className":13481,"translate":36},[35],[32,13483,13485],{"className":13484},[40],[42,13486,13487],{"xmlns":44},[46,13488,13489,13497],{},[49,13490,13491],{},[143,13492,13493,13495],{},[57,13494,358],{},[57,13496,6594],{},[71,13498,9012],{"encoding":73},[32,13500,13502],{"className":13501,"ariaHidden":62},[78],[32,13503,13505,13508],{"className":13504},[82],[32,13506],{"className":13507,"style":8289},[86],[32,13509,13511,13514],{"className":13510},[95],[32,13512,358],{"className":13513},[95,96],[32,13515,13517],{"className":13516},[242],[32,13518,13520,13540],{"className":13519},[246,247],[32,13521,13523,13537],{"className":13522},[251],[32,13524,13526],{"className":13525,"style":1860},[255],[32,13527,13528,13531],{"style":259},[32,13529],{"className":13530,"style":264},[263],[32,13532,13534],{"className":13533},[268,269,270,271],[32,13535,6594],{"className":13536},[95,96,271],[32,13538,285],{"className":13539},[284],[32,13541,13543],{"className":13542},[251],[32,13544,13546],{"className":13545,"style":1441},[255],[32,13547],{}," 附近局部可逆，因此 ",[32,13550,13552,13569],{"className":13551,"translate":36},[35],[32,13553,13555],{"className":13554},[40],[42,13556,13557],{"xmlns":44},[46,13558,13559,13567],{},[49,13560,13561],{},[143,13562,13563,13565],{},[57,13564,358],{},[57,13566,6594],{},[71,13568,9012],{"encoding":73},[32,13570,13572],{"className":13571,"ariaHidden":62},[78],[32,13573,13575,13578],{"className":13574},[82],[32,13576],{"className":13577,"style":8289},[86],[32,13579,13581,13584],{"className":13580},[95],[32,13582,358],{"className":13583},[95,96],[32,13585,13587],{"className":13586},[242],[32,13588,13590,13610],{"className":13589},[246,247],[32,13591,13593,13607],{"className":13592},[251],[32,13594,13596],{"className":13595,"style":1860},[255],[32,13597,13598,13601],{"style":259},[32,13599],{"className":13600,"style":264},[263],[32,13602,13604],{"className":13603},[268,269,270,271],[32,13605,6594],{"className":13606},[95,96,271],[32,13608,285],{"className":13609},[284],[32,13611,13613],{"className":13612},[251],[32,13614,13616],{"className":13615,"style":1441},[255],[32,13617],{},[32,13619,13621,13636],{"className":13620,"translate":36},[35],[32,13622,13624],{"className":13623},[40],[42,13625,13626],{"xmlns":44},[46,13627,13628,13634],{},[49,13629,13630,13632],{},[57,13631,5957],{"mathvariant":1083},[57,13633,1909],{},[71,13635,13463],{"encoding":73},[32,13637,13639],{"className":13638,"ariaHidden":62},[78],[32,13640,13642,13645,13648],{"className":13641},[82],[32,13643],{"className":13644,"style":4025},[86],[32,13646,5957],{"className":13647},[95],[32,13649,1909],{"className":13650,"style":1965},[95,96]," 在某个邻域内的唯一零点。于是每个临界点都是孤立的。由于临界点集是闭集，紧区域内若有无限多个临界点，就会有一个同样为临界的聚点，这与孤立性矛盾。因此任何紧区域中只有有限多个临界点。有限集具有 Lebesgue 测度零。",[32,13653,13655,13670],{"className":13654,"translate":36},[35],[32,13656,13658],{"className":13657},[40],[42,13659,13660],{"xmlns":44},[46,13661,13662,13667],{},[49,13663,13664],{},[57,13665,13666],{"mathvariant":1083},"■",[71,13668,13669],{"encoding":73},"\\blacksquare",[32,13671,13673],{"className":13672,"ariaHidden":62},[78],[32,13674,13676,13680],{"className":13675},[82],[32,13677],{"className":13678,"style":13679},[86],"height:0.675em;",[32,13681,13666],{"className":13682},[95,13683],"amsrm",[11,13685,13686,13687,13745,13746,13749],{},"这说明：如果从 ",[32,13688,13690,13707],{"className":13689,"translate":36},[35],[32,13691,13693],{"className":13692},[40],[42,13694,13695],{"xmlns":44},[46,13696,13697,13705],{},[49,13698,13699],{},[5397,13700,13701,13703],{},[57,13702,148],{"mathvariant":147},[57,13704,5322],{},[71,13706,12868],{"encoding":73},[32,13708,13710],{"className":13709,"ariaHidden":62},[78],[32,13711,13713,13716],{"className":13712},[82],[32,13714],{"className":13715,"style":5437},[86],[32,13717,13719,13722],{"className":13718},[95],[32,13720,148],{"className":13721},[95,238],[32,13723,13725],{"className":13724},[242],[32,13726,13728],{"className":13727},[246],[32,13729,13731],{"className":13730},[251],[32,13732,13734],{"className":13733,"style":5437},[255],[32,13735,13736,13739],{"style":5458},[32,13737],{"className":13738,"style":264},[263],[32,13740,13742],{"className":13741},[268,269,270,271],[32,13743,5322],{"className":13744,"style":1530},[95,96,271]," 中的一个盒子里均匀采样一点，恰好落在临界点上的概率为零。但真正有趣的问题是",[1364,13747,13748],{},"条件性的","：",[3222,13751,13752],{},[11,13753,13754],{},"既然我们已经身处某个高维随机函数的临界点，那么这个临界点是非退化局部极小值的概率是多少？",[11,13756,13757],{},"这个问题的答案，由 Hessian 特征值的符号来回答。",[21,13759,13760],{"id":13760},"马鞍面",[11,13762,13763],{},"在进入随机矩阵论证之前，先看一个最简单的鞍点例子，建立手感。",[11,13765,13766,13767,5714,13851,13903],{},"函数 ",[32,13768,13770,13800],{"className":13769,"translate":36},[35],[32,13771,13773],{"className":13772},[40],[42,13774,13775],{"xmlns":44},[46,13776,13777,13797],{},[49,13778,13779,13781,13783,13785,13787,13789,13791,13793,13795],{},[57,13780,1909],{},[52,13782,55],{"stretchy":54},[57,13784,358],{},[52,13786,63],{"separator":62},[57,13788,363],{},[52,13790,69],{"stretchy":54},[52,13792,368],{},[57,13794,358],{},[57,13796,363],{},[71,13798,13799],{"encoding":73},"f(x,y) = xy",[32,13801,13803,13839],{"className":13802,"ariaHidden":62},[78],[32,13804,13806,13809,13812,13815,13818,13821,13824,13827,13830,13833,13836],{"className":13805},[82],[32,13807],{"className":13808,"style":87},[86],[32,13810,1909],{"className":13811,"style":1965},[95,96],[32,13813,55],{"className":13814},[91],[32,13816,358],{"className":13817},[95,96],[32,13819,63],{"className":13820},[101],[32,13822],{"className":13823,"style":106},[105],[32,13825,363],{"className":13826,"style":740},[95,96],[32,13828,69],{"className":13829},[113],[32,13831],{"className":13832,"style":177},[105],[32,13834,368],{"className":13835},[181],[32,13837],{"className":13838,"style":177},[105],[32,13840,13842,13845,13848],{"className":13841},[82],[32,13843],{"className":13844,"style":9732},[86],[32,13846,358],{"className":13847},[95,96],[32,13849,363],{"className":13850,"style":740},[95,96],[32,13852,13854,13876],{"className":13853,"translate":36},[35],[32,13855,13857],{"className":13856},[40],[42,13858,13859],{"xmlns":44},[46,13860,13861,13873],{},[49,13862,13863,13865,13867,13869,13871],{},[52,13864,55],{"stretchy":54},[155,13866,157],{},[52,13868,63],{"separator":62},[155,13870,157],{},[52,13872,69],{"stretchy":54},[71,13874,13875],{"encoding":73},"(0,0)",[32,13877,13879],{"className":13878,"ariaHidden":62},[78],[32,13880,13882,13885,13888,13891,13894,13897,13900],{"className":13881},[82],[32,13883],{"className":13884,"style":87},[86],[32,13886,55],{"className":13887},[91],[32,13889,157],{"className":13890},[95],[32,13892,63],{"className":13893},[101],[32,13895],{"className":13896,"style":106},[105],[32,13898,157],{"className":13899},[95],[32,13901,69],{"className":13902},[113]," 处有一个临界点，因为",[32,13905,13907],{"className":13906,"translate":36},[299],[32,13908,13910,13953],{"className":13909,"translate":36},[35],[32,13911,13913],{"className":13912},[40],[42,13914,13915],{"xmlns":44,"display":308},[46,13916,13917,13950],{},[49,13918,13919,13921,13923,13925,13927,13929,13931,13933,13935,13948],{},[57,13920,5957],{"mathvariant":1083},[57,13922,1909],{},[52,13924,55],{"stretchy":54},[57,13926,358],{},[52,13928,63],{"separator":62},[57,13930,363],{},[52,13932,69],{"stretchy":54},[52,13934,368],{},[49,13936,13937,13939,13946],{},[52,13938,55],{"fence":62},[5987,13940,13942,13944],{"linethickness":13941},"0px",[57,13943,363],{},[57,13945,358],{},[52,13947,69],{"fence":62},[52,13949,63],{"separator":62},[71,13951,13952],{"encoding":73},"\\nabla f(x,y) = \\binom{y}{x},",[32,13954,13956,13995],{"className":13955,"ariaHidden":62},[78],[32,13957,13959,13962,13965,13968,13971,13974,13977,13980,13983,13986,13989,13992],{"className":13958},[82],[32,13960],{"className":13961,"style":87},[86],[32,13963,5957],{"className":13964},[95],[32,13966,1909],{"className":13967,"style":1965},[95,96],[32,13969,55],{"className":13970},[91],[32,13972,358],{"className":13973},[95,96],[32,13975,63],{"className":13976},[101],[32,13978],{"className":13979,"style":106},[105],[32,13981,363],{"className":13982,"style":740},[95,96],[32,13984,69],{"className":13985},[113],[32,13987],{"className":13988,"style":177},[105],[32,13990,368],{"className":13991},[181],[32,13993],{"className":13994,"style":177},[105],[32,13996,13998,14002,14063],{"className":13997},[82],[32,13999],{"className":14000,"style":14001},[86],"height:2.4em;vertical-align:-0.95em;",[32,14003,14005,14011,14057],{"className":14004},[95],[32,14006,14008],{"className":14007,"style":7283},[91,7282],[32,14009,55],{"className":14010},[514,270],[32,14012,14014],{"className":14013},[5987],[32,14015,14017,14049],{"className":14016},[246,247],[32,14018,14020,14046],{"className":14019},[251],[32,14021,14024,14035],{"className":14022,"style":14023},[255],"height:1.1076em;",[32,14025,14026,14029],{"style":6222},[32,14027],{"className":14028,"style":715},[263],[32,14030,14032],{"className":14031},[95],[32,14033,358],{"className":14034},[95,96],[32,14036,14037,14040],{"style":6286},[32,14038],{"className":14039,"style":715},[263],[32,14041,14043],{"className":14042},[95],[32,14044,363],{"className":14045,"style":740},[95,96],[32,14047,285],{"className":14048},[284],[32,14050,14052],{"className":14051},[251],[32,14053,14055],{"className":14054,"style":8781},[255],[32,14056],{},[32,14058,14060],{"className":14059,"style":7283},[113,7282],[32,14061,69],{"className":14062},[514,270],[32,14064,63],{"className":14065},[101],[11,14067,14068,14069,14153],{},"所以 ",[32,14070,14072,14102],{"className":14071,"translate":36},[35],[32,14073,14075],{"className":14074},[40],[42,14076,14077],{"xmlns":44},[46,14078,14079,14099],{},[49,14080,14081,14083,14085,14087,14089,14091,14093,14095,14097],{},[57,14082,5957],{"mathvariant":1083},[57,14084,1909],{},[52,14086,55],{"stretchy":54},[155,14088,157],{},[52,14090,63],{"separator":62},[155,14092,157],{},[52,14094,69],{"stretchy":54},[52,14096,368],{},[155,14098,157],{},[71,14100,14101],{"encoding":73},"\\nabla f(0,0) = 0",[32,14103,14105,14144],{"className":14104,"ariaHidden":62},[78],[32,14106,14108,14111,14114,14117,14120,14123,14126,14129,14132,14135,14138,14141],{"className":14107},[82],[32,14109],{"className":14110,"style":87},[86],[32,14112,5957],{"className":14113},[95],[32,14115,1909],{"className":14116,"style":1965},[95,96],[32,14118,55],{"className":14119},[91],[32,14121,157],{"className":14122},[95],[32,14124,63],{"className":14125},[101],[32,14127],{"className":14128,"style":106},[105],[32,14130,157],{"className":14131},[95],[32,14133,69],{"className":14134},[113],[32,14136],{"className":14137,"style":177},[105],[32,14139,368],{"className":14140},[181],[32,14142],{"className":14143,"style":177},[105],[32,14145,14147,14150],{"className":14146},[82],[32,14148],{"className":14149,"style":11978},[86],[32,14151,157],{"className":14152},[95],"。它的 Hessian 是",[32,14155,14157],{"className":14156,"translate":36},[299],[32,14158,14160,14229],{"className":14159,"translate":36},[35],[32,14161,14163],{"className":14162},[40],[42,14164,14165],{"xmlns":44,"display":308},[46,14166,14167,14226],{},[49,14168,14169,14175,14177,14179,14181,14183,14185,14187,14224],{},[143,14170,14171,14173],{},[57,14172,7043],{},[57,14174,1909],{},[52,14176,55],{"stretchy":54},[155,14178,157],{},[52,14180,63],{"separator":62},[155,14182,157],{},[52,14184,69],{"stretchy":54},[52,14186,368],{},[49,14188,14189,14191,14222],{},[52,14190,55],{"fence":62},[317,14192,14194,14208],{"rowspacing":4544,"columnalign":14193,"columnspacing":321},"center center",[323,14195,14196,14202],{},[326,14197,14198],{},[329,14199,14200],{"scriptlevel":157,"displaystyle":54},[155,14201,157],{},[326,14203,14204],{},[329,14205,14206],{"scriptlevel":157,"displaystyle":54},[155,14207,1732],{},[323,14209,14210,14216],{},[326,14211,14212],{},[329,14213,14214],{"scriptlevel":157,"displaystyle":54},[155,14215,1732],{},[326,14217,14218],{},[329,14219,14220],{"scriptlevel":157,"displaystyle":54},[155,14221,157],{},[52,14223,69],{"fence":62},[57,14225,6076],{"mathvariant":1083},[71,14227,14228],{"encoding":73},"H_f(0,0) = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}.",[32,14230,14232,14305],{"className":14231,"ariaHidden":62},[78],[32,14233,14235,14238,14278,14281,14284,14287,14290,14293,14296,14299,14302],{"className":14234},[82],[32,14236],{"className":14237,"style":7147},[86],[32,14239,14241,14244],{"className":14240},[95],[32,14242,7043],{"className":14243,"style":7154},[95,96],[32,14245,14247],{"className":14246},[242],[32,14248,14250,14270],{"className":14249},[246,247],[32,14251,14253,14267],{"className":14252},[251],[32,14254,14256],{"className":14255,"style":1419},[255],[32,14257,14258,14261],{"style":7169},[32,14259],{"className":14260,"style":264},[263],[32,14262,14264],{"className":14263},[268,269,270,271],[32,14265,1909],{"className":14266,"style":1965},[95,96,271],[32,14268,285],{"className":14269},[284],[32,14271,14273],{"className":14272},[251],[32,14274,14276],{"className":14275,"style":2519},[255],[32,14277],{},[32,14279,55],{"className":14280},[91],[32,14282,157],{"className":14283},[95],[32,14285,63],{"className":14286},[101],[32,14288],{"className":14289,"style":106},[105],[32,14291,157],{"className":14292},[95],[32,14294,69],{"className":14295},[113],[32,14297],{"className":14298,"style":177},[105],[32,14300,368],{"className":14301},[181],[32,14303],{"className":14304,"style":177},[105],[32,14306,14308,14311,14432,14435],{"className":14307},[82],[32,14309],{"className":14310,"style":14001},[86],[32,14312,14314,14320,14426],{"className":14313},[507],[32,14315,14317],{"className":14316,"style":7283},[91,7282],[32,14318,55],{"className":14319},[514,270],[32,14321,14323],{"className":14322},[95],[32,14324,14326,14375,14378,14381],{"className":14325},[317],[32,14327,14329],{"className":14328},[4661],[32,14330,14332,14366],{"className":14331},[246,247],[32,14333,14335,14363],{"className":14334},[251],[32,14336,14339,14351],{"className":14337,"style":14338},[255],"height:1.45em;",[32,14340,14342,14345],{"style":14341},"top:-3.61em;",[32,14343],{"className":14344,"style":715},[263],[32,14346,14348],{"className":14347},[95],[32,14349,157],{"className":14350},[95],[32,14352,14354,14357],{"style":14353},"top:-2.41em;",[32,14355],{"className":14356,"style":715},[263],[32,14358,14360],{"className":14359},[95],[32,14361,1732],{"className":14362},[95],[32,14364,285],{"className":14365},[284],[32,14367,14369],{"className":14368},[251],[32,14370,14373],{"className":14371,"style":14372},[255],"height:0.95em;",[32,14374],{},[32,14376],{"className":14377,"style":4657},[4656],[32,14379],{"className":14380,"style":4657},[4656],[32,14382,14384],{"className":14383},[4661],[32,14385,14387,14418],{"className":14386},[246,247],[32,14388,14390,14415],{"className":14389},[251],[32,14391,14393,14404],{"className":14392,"style":14338},[255],[32,14394,14395,14398],{"style":14341},[32,14396],{"className":14397,"style":715},[263],[32,14399,14401],{"className":14400},[95],[32,14402,1732],{"className":14403},[95],[32,14405,14406,14409],{"style":14353},[32,14407],{"className":14408,"style":715},[263],[32,14410,14412],{"className":14411},[95],[32,14413,157],{"className":14414},[95],[32,14416,285],{"className":14417},[284],[32,14419,14421],{"className":14420},[251],[32,14422,14424],{"className":14423,"style":14372},[255],[32,14425],{},[32,14427,14429],{"className":14428,"style":7283},[113,7282],[32,14430,69],{"className":14431},[514,270],[32,14433],{"className":14434,"style":106},[105],[32,14436,6076],{"className":14437},[95],[11,14439,14440,14441,14469,14470,14505],{},"特征值为 ",[32,14442,14444,14457],{"className":14443,"translate":36},[35],[32,14445,14447],{"className":14446},[40],[42,14448,14449],{"xmlns":44},[46,14450,14451,14455],{},[49,14452,14453],{},[155,14454,1732],{},[71,14456,1732],{"encoding":73},[32,14458,14460],{"className":14459,"ariaHidden":62},[78],[32,14461,14463,14466],{"className":14462},[82],[32,14464],{"className":14465,"style":11978},[86],[32,14467,1732],{"className":14468},[95]," 和 ",[32,14471,14473,14489],{"className":14472,"translate":36},[35],[32,14474,14476],{"className":14475},[40],[42,14477,14478],{"xmlns":44},[46,14479,14480,14486],{},[49,14481,14482,14484],{},[52,14483,1089],{},[155,14485,1732],{},[71,14487,14488],{"encoding":73},"-1",[32,14490,14492],{"className":14491,"ariaHidden":62},[78],[32,14493,14495,14499,14502],{"className":14494},[82],[32,14496],{"className":14497,"style":14498},[86],"height:0.7278em;vertical-align:-0.0833em;",[32,14500,1089],{"className":14501},[95],[32,14503,1732],{"className":14504},[95],"，因此原点是一个鞍点。",[11,14507,14508,14509,14561,14562,14640,14641,14722,14723,14803],{},"这个例子也说明了为什么混合偏导数项会遮蔽几何。在 ",[32,14510,14512,14534],{"className":14511,"translate":36},[35],[32,14513,14515],{"className":14514},[40],[42,14516,14517],{"xmlns":44},[46,14518,14519,14531],{},[49,14520,14521,14523,14525,14527,14529],{},[52,14522,55],{"stretchy":54},[57,14524,358],{},[52,14526,63],{"separator":62},[57,14528,363],{},[52,14530,69],{"stretchy":54},[71,14532,14533],{"encoding":73},"(x,y)",[32,14535,14537],{"className":14536,"ariaHidden":62},[78],[32,14538,14540,14543,14546,14549,14552,14555,14558],{"className":14539},[82],[32,14541],{"className":14542,"style":87},[86],[32,14544,55],{"className":14545},[91],[32,14547,358],{"className":14548},[95,96],[32,14550,63],{"className":14551},[101],[32,14553],{"className":14554,"style":106},[105],[32,14556,363],{"className":14557,"style":740},[95,96],[32,14559,69],{"className":14560},[113]," 坐标中，二阶导数 ",[32,14563,14565,14587],{"className":14564,"translate":36},[35],[32,14566,14568],{"className":14567},[40],[42,14569,14570],{"xmlns":44},[46,14571,14572,14584],{},[49,14573,14574],{},[143,14575,14576,14578],{},[57,14577,1909],{},[49,14579,14580,14582],{},[57,14581,358],{},[57,14583,358],{},[71,14585,14586],{"encoding":73},"f_{xx}",[32,14588,14590],{"className":14589,"ariaHidden":62},[78],[32,14591,14593,14596],{"className":14592},[82],[32,14594],{"className":14595,"style":4025},[86],[32,14597,14599,14602],{"className":14598},[95],[32,14600,1909],{"className":14601,"style":1965},[95,96],[32,14603,14605],{"className":14604},[242],[32,14606,14608,14632],{"className":14607},[246,247],[32,14609,14611,14629],{"className":14610},[251],[32,14612,14614],{"className":14613,"style":1860},[255],[32,14615,14616,14619],{"style":1980},[32,14617],{"className":14618,"style":264},[263],[32,14620,14622],{"className":14621},[268,269,270,271],[32,14623,14625],{"className":14624},[95,271],[32,14626,14628],{"className":14627},[95,96,271],"xx",[32,14630,285],{"className":14631},[284],[32,14633,14635],{"className":14634},[251],[32,14636,14638],{"className":14637,"style":1441},[255],[32,14639],{}," 与 ",[32,14642,14644,14666],{"className":14643,"translate":36},[35],[32,14645,14647],{"className":14646},[40],[42,14648,14649],{"xmlns":44},[46,14650,14651,14663],{},[49,14652,14653],{},[143,14654,14655,14657],{},[57,14656,1909],{},[49,14658,14659,14661],{},[57,14660,363],{},[57,14662,363],{},[71,14664,14665],{"encoding":73},"f_{yy}",[32,14667,14669],{"className":14668,"ariaHidden":62},[78],[32,14670,14672,14676],{"className":14671},[82],[32,14673],{"className":14674,"style":14675},[86],"height:0.9805em;vertical-align:-0.2861em;",[32,14677,14679,14682],{"className":14678},[95],[32,14680,1909],{"className":14681,"style":1965},[95,96],[32,14683,14685],{"className":14684},[242],[32,14686,14688,14714],{"className":14687},[246,247],[32,14689,14691,14711],{"className":14690},[251],[32,14692,14694],{"className":14693,"style":1860},[255],[32,14695,14696,14699],{"style":1980},[32,14697],{"className":14698,"style":264},[263],[32,14700,14702],{"className":14701},[268,269,270,271],[32,14703,14705,14708],{"className":14704},[95,271],[32,14706,363],{"className":14707,"style":740},[95,96,271],[32,14709,363],{"className":14710,"style":740},[95,96,271],[32,14712,285],{"className":14713},[284],[32,14715,14717],{"className":14716},[251],[32,14718,14720],{"className":14719,"style":2519},[255],[32,14721],{}," 都为零，而混合导数 ",[32,14724,14726,14748],{"className":14725,"translate":36},[35],[32,14727,14729],{"className":14728},[40],[42,14730,14731],{"xmlns":44},[46,14732,14733,14745],{},[49,14734,14735],{},[143,14736,14737,14739],{},[57,14738,1909],{},[49,14740,14741,14743],{},[57,14742,358],{},[57,14744,363],{},[71,14746,14747],{"encoding":73},"f_{xy}",[32,14749,14751],{"className":14750,"ariaHidden":62},[78],[32,14752,14754,14757],{"className":14753},[82],[32,14755],{"className":14756,"style":14675},[86],[32,14758,14760,14763],{"className":14759},[95],[32,14761,1909],{"className":14762,"style":1965},[95,96],[32,14764,14766],{"className":14765},[242],[32,14767,14769,14795],{"className":14768},[246,247],[32,14770,14772,14792],{"className":14771},[251],[32,14773,14775],{"className":14774,"style":1860},[255],[32,14776,14777,14780],{"style":1980},[32,14778],{"className":14779,"style":264},[263],[32,14781,14783],{"className":14782},[268,269,270,271],[32,14784,14786,14789],{"className":14785},[95,271],[32,14787,358],{"className":14788},[95,96,271],[32,14790,363],{"className":14791,"style":740},[95,96,271],[32,14793,285],{"className":14794},[284],[32,14796,14798],{"className":14797},[251],[32,14799,14801],{"className":14800,"style":2519},[255],[32,14802],{}," 非零——主曲率方向并不是坐标轴。定义旋转后的坐标",[32,14805,14807],{"className":14806,"translate":36},[299],[32,14808,14810,14867],{"className":14809,"translate":36},[35],[32,14811,14813],{"className":14812},[40],[42,14814,14815],{"xmlns":44,"display":308},[46,14816,14817,14864],{},[49,14818,14819,14822,14824,14839,14841,14844,14846,14848,14862],{},[57,14820,14821],{},"u",[52,14823,368],{},[5987,14825,14826,14834],{},[49,14827,14828,14830,14832],{},[57,14829,358],{},[52,14831,478],{},[57,14833,363],{},[14835,14836,14837],"msqrt",{},[155,14838,5511],{},[52,14840,63],{"separator":62},[105,14842],{"width":14843},"2em",[57,14845,9705],{},[52,14847,368],{},[5987,14849,14850,14858],{},[49,14851,14852,14854,14856],{},[57,14853,358],{},[52,14855,1089],{},[57,14857,363],{},[14835,14859,14860],{},[155,14861,5511],{},[52,14863,63],{"separator":62},[71,14865,14866],{"encoding":73},"u = \\frac{x+y}{\\sqrt2}, \\qquad v = \\frac{x-y}{\\sqrt2},",[32,14868,14870,14888,15045],{"className":14869,"ariaHidden":62},[78],[32,14871,14873,14876,14879,14882,14885],{"className":14872},[82],[32,14874],{"className":14875,"style":1261},[86],[32,14877,14821],{"className":14878},[95,96],[32,14880],{"className":14881,"style":177},[105],[32,14883,368],{"className":14884},[181],[32,14886],{"className":14887,"style":177},[105],[32,14889,14891,14895,15023,15026,15030,15033,15036,15039,15042],{"className":14890},[82],[32,14892],{"className":14893,"style":14894},[86],"height:2.1903em;vertical-align:-0.93em;",[32,14896,14898,14901,15020],{"className":14897},[95],[32,14899],{"className":14900},[91,974],[32,14902,14904],{"className":14903},[5987],[32,14905,14907,15011],{"className":14906},[246,247],[32,14908,14910,15008],{"className":14909},[251],[32,14911,14914,14977,14985],{"className":14912,"style":14913},[255],"height:1.2603em;",[32,14915,14917,14920],{"style":14916},"top:-2.2028em;",[32,14918],{"className":14919,"style":715},[263],[32,14921,14923],{"className":14922},[95],[32,14924,14927],{"className":14925},[95,14926],"sqrt",[32,14928,14930,14968],{"className":14929},[246,247],[32,14931,14933,14965],{"className":14932},[251],[32,14934,14937,14947],{"className":14935,"style":14936},[255],"height:0.9072em;",[32,14938,14940,14943],{"className":14939,"style":4715},[4734],[32,14941],{"className":14942,"style":715},[263],[32,14944,5511],{"className":14945,"style":14946},[95],"padding-left:0.833em;",[32,14948,14950,14953],{"style":14949},"top:-2.8672em;",[32,14951],{"className":14952,"style":715},[263],[32,14954,14958],{"className":14955,"style":14957},[14956],"hide-tail","min-width:0.853em;height:1.08em;",[551,14959,14962],{"xmlns":553,"width":4751,"height":14960,"viewBox":14961,"preserveAspectRatio":4754},"1.08em","0 0 400000 1080",[560,14963],{"d":14964},"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",[32,14966,285],{"className":14967},[284],[32,14969,14971],{"className":14970},[251],[32,14972,14975],{"className":14973,"style":14974},[255],"height:0.1328em;",[32,14976],{},[32,14978,14979,14982],{"style":6275},[32,14980],{"className":14981,"style":715},[263],[32,14983],{"className":14984,"style":6283},[6282],[32,14986,14987,14990],{"style":6286},[32,14988],{"className":14989,"style":715},[263],[32,14991,14993,14996,14999,15002,15005],{"className":14992},[95],[32,14994,358],{"className":14995},[95,96],[32,14997],{"className":14998,"style":198},[105],[32,15000,478],{"className":15001},[202],[32,15003],{"className":15004,"style":198},[105],[32,15006,363],{"className":15007,"style":740},[95,96],[32,15009,285],{"className":15010},[284],[32,15012,15014],{"className":15013},[251],[32,15015,15018],{"className":15016,"style":15017},[255],"height:0.93em;",[32,15019],{},[32,15021],{"className":15022},[113,974],[32,15024,63],{"className":15025},[101],[32,15027],{"className":15028,"style":15029},[105],"margin-right:2em;",[32,15031],{"className":15032,"style":106},[105],[32,15034,9705],{"className":15035,"style":740},[95,96],[32,15037],{"className":15038,"style":177},[105],[32,15040,368],{"className":15041},[181],[32,15043],{"className":15044,"style":177},[105],[32,15046,15048,15051,15166],{"className":15047},[82],[32,15049],{"className":15050,"style":14894},[86],[32,15052,15054,15057,15163],{"className":15053},[95],[32,15055],{"className":15056},[91,974],[32,15058,15060],{"className":15059},[5987],[32,15061,15063,15155],{"className":15062},[246,247],[32,15064,15066,15152],{"className":15065},[251],[32,15067,15069,15121,15129],{"className":15068,"style":14913},[255],[32,15070,15071,15074],{"style":14916},[32,15072],{"className":15073,"style":715},[263],[32,15075,15077],{"className":15076},[95],[32,15078,15080],{"className":15079},[95,14926],[32,15081,15083,15113],{"className":15082},[246,247],[32,15084,15086,15110],{"className":15085},[251],[32,15087,15089,15098],{"className":15088,"style":14936},[255],[32,15090,15092,15095],{"className":15091,"style":4715},[4734],[32,15093],{"className":15094,"style":715},[263],[32,15096,5511],{"className":15097,"style":14946},[95],[32,15099,15100,15103],{"style":14949},[32,15101],{"className":15102,"style":715},[263],[32,15104,15106],{"className":15105,"style":14957},[14956],[551,15107,15108],{"xmlns":553,"width":4751,"height":14960,"viewBox":14961,"preserveAspectRatio":4754},[560,15109],{"d":14964},[32,15111,285],{"className":15112},[284],[32,15114,15116],{"className":15115},[251],[32,15117,15119],{"className":15118,"style":14974},[255],[32,15120],{},[32,15122,15123,15126],{"style":6275},[32,15124],{"className":15125,"style":715},[263],[32,15127],{"className":15128,"style":6283},[6282],[32,15130,15131,15134],{"style":6286},[32,15132],{"className":15133,"style":715},[263],[32,15135,15137,15140,15143,15146,15149],{"className":15136},[95],[32,15138,358],{"className":15139},[95,96],[32,15141],{"className":15142,"style":198},[105],[32,15144,1089],{"className":15145},[202],[32,15147],{"className":15148,"style":198},[105],[32,15150,363],{"className":15151,"style":740},[95,96],[32,15153,285],{"className":15154},[284],[32,15156,15158],{"className":15157},[251],[32,15159,15161],{"className":15160,"style":15017},[255],[32,15162],{},[32,15164],{"className":15165},[113,974],[32,15167,63],{"className":15168},[101],[11,15170,15171],{},"等价地",[32,15173,15175],{"className":15174,"translate":36},[299],[32,15176,15178,15232],{"className":15177,"translate":36},[35],[32,15179,15181],{"className":15180},[40],[42,15182,15183],{"xmlns":44,"display":308},[46,15184,15185,15229],{},[49,15186,15187,15189,15191,15205,15207,15209,15211,15213,15227],{},[57,15188,358],{},[52,15190,368],{},[5987,15192,15193,15201],{},[49,15194,15195,15197,15199],{},[57,15196,14821],{},[52,15198,478],{},[57,15200,9705],{},[14835,15202,15203],{},[155,15204,5511],{},[52,15206,63],{"separator":62},[105,15208],{"width":14843},[57,15210,363],{},[52,15212,368],{},[5987,15214,15215,15223],{},[49,15216,15217,15219,15221],{},[57,15218,14821],{},[52,15220,1089],{},[57,15222,9705],{},[14835,15224,15225],{},[155,15226,5511],{},[57,15228,6076],{"mathvariant":1083},[71,15230,15231],{"encoding":73},"x = \\frac{u+v}{\\sqrt2}, \\qquad y = \\frac{u-v}{\\sqrt2}.",[32,15233,15235,15253,15395],{"className":15234,"ariaHidden":62},[78],[32,15236,15238,15241,15244,15247,15250],{"className":15237},[82],[32,15239],{"className":15240,"style":1261},[86],[32,15242,358],{"className":15243},[95,96],[32,15245],{"className":15246,"style":177},[105],[32,15248,368],{"className":15249},[181],[32,15251],{"className":15252,"style":177},[105],[32,15254,15256,15259,15374,15377,15380,15383,15386,15389,15392],{"className":15255},[82],[32,15257],{"className":15258,"style":14894},[86],[32,15260,15262,15265,15371],{"className":15261},[95],[32,15263],{"className":15264},[91,974],[32,15266,15268],{"className":15267},[5987],[32,15269,15271,15363],{"className":15270},[246,247],[32,15272,15274,15360],{"className":15273},[251],[32,15275,15277,15329,15337],{"className":15276,"style":14913},[255],[32,15278,15279,15282],{"style":14916},[32,15280],{"className":15281,"style":715},[263],[32,15283,15285],{"className":15284},[95],[32,15286,15288],{"className":15287},[95,14926],[32,15289,15291,15321],{"className":15290},[246,247],[32,15292,15294,15318],{"className":15293},[251],[32,15295,15297,15306],{"className":15296,"style":14936},[255],[32,15298,15300,15303],{"className":15299,"style":4715},[4734],[32,15301],{"className":15302,"style":715},[263],[32,15304,5511],{"className":15305,"style":14946},[95],[32,15307,15308,15311],{"style":14949},[32,15309],{"className":15310,"style":715},[263],[32,15312,15314],{"className":15313,"style":14957},[14956],[551,15315,15316],{"xmlns":553,"width":4751,"height":14960,"viewBox":14961,"preserveAspectRatio":4754},[560,15317],{"d":14964},[32,15319,285],{"className":15320},[284],[32,15322,15324],{"className":15323},[251],[32,15325,15327],{"className":15326,"style":14974},[255],[32,15328],{},[32,15330,15331,15334],{"style":6275},[32,15332],{"className":15333,"style":715},[263],[32,15335],{"className":15336,"style":6283},[6282],[32,15338,15339,15342],{"style":6286},[32,15340],{"className":15341,"style":715},[263],[32,15343,15345,15348,15351,15354,15357],{"className":15344},[95],[32,15346,14821],{"className":15347},[95,96],[32,15349],{"className":15350,"style":198},[105],[32,15352,478],{"className":15353},[202],[32,15355],{"className":15356,"style":198},[105],[32,15358,9705],{"className":15359,"style":740},[95,96],[32,15361,285],{"className":15362},[284],[32,15364,15366],{"className":15365},[251],[32,15367,15369],{"className":15368,"style":15017},[255],[32,15370],{},[32,15372],{"className":15373},[113,974],[32,15375,63],{"className":15376},[101],[32,15378],{"className":15379,"style":15029},[105],[32,15381],{"className":15382,"style":106},[105],[32,15384,363],{"className":15385,"style":740},[95,96],[32,15387],{"className":15388,"style":177},[105],[32,15390,368],{"className":15391},[181],[32,15393],{"className":15394,"style":177},[105],[32,15396,15398,15401,15516],{"className":15397},[82],[32,15399],{"className":15400,"style":14894},[86],[32,15402,15404,15407,15513],{"className":15403},[95],[32,15405],{"className":15406},[91,974],[32,15408,15410],{"className":15409},[5987],[32,15411,15413,15505],{"className":15412},[246,247],[32,15414,15416,15502],{"className":15415},[251],[32,15417,15419,15471,15479],{"className":15418,"style":14913},[255],[32,15420,15421,15424],{"style":14916},[32,15422],{"className":15423,"style":715},[263],[32,15425,15427],{"className":15426},[95],[32,15428,15430],{"className":15429},[95,14926],[32,15431,15433,15463],{"className":15432},[246,247],[32,15434,15436,15460],{"className":15435},[251],[32,15437,15439,15448],{"className":15438,"style":14936},[255],[32,15440,15442,15445],{"className":15441,"style":4715},[4734],[32,15443],{"className":15444,"style":715},[263],[32,15446,5511],{"className":15447,"style":14946},[95],[32,15449,15450,15453],{"style":14949},[32,15451],{"className":15452,"style":715},[263],[32,15454,15456],{"className":15455,"style":14957},[14956],[551,15457,15458],{"xmlns":553,"width":4751,"height":14960,"viewBox":14961,"preserveAspectRatio":4754},[560,15459],{"d":14964},[32,15461,285],{"className":15462},[284],[32,15464,15466],{"className":15465},[251],[32,15467,15469],{"className":15468,"style":14974},[255],[32,15470],{},[32,15472,15473,15476],{"style":6275},[32,15474],{"className":15475,"style":715},[263],[32,15477],{"className":15478,"style":6283},[6282],[32,15480,15481,15484],{"style":6286},[32,15482],{"className":15483,"style":715},[263],[32,15485,15487,15490,15493,15496,15499],{"className":15486},[95],[32,15488,14821],{"className":15489},[95,96],[32,15491],{"className":15492,"style":198},[105],[32,15494,1089],{"className":15495},[202],[32,15497],{"className":15498,"style":198},[105],[32,15500,9705],{"className":15501,"style":740},[95,96],[32,15503,285],{"className":15504},[284],[32,15506,15508],{"className":15507},[251],[32,15509,15511],{"className":15510,"style":15017},[255],[32,15512],{},[32,15514],{"className":15515},[113,974],[32,15517,6076],{"className":15518},[95],[11,15520,10217],{},[32,15522,15524],{"c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= \\frac{u+v}{\\sqrt2}\\cdot\\frac{u-v}{\\sqrt2} = \\frac{u^2-v^2}{2}.",[32,15600,15602,15623,15753,15883],{"className":15601,"ariaHidden":62},[78],[32,15603,15605,15608,15611,15614,15617,15620],{"className":15604},[82],[32,15606],{"className":15607,"style":9732},[86],[32,15609,358],{"className":15610},[95,96],[32,15612,363],{"className":15613,"style":740},[95,96],[32,15615],{"className":15616,"style":177},[105],[32,15618,368],{"className":15619},[181],[32,15621],{"className":15622,"style":177},[105],[32,15624,15626,15629,15744,15747,15750],{"className":15625},[82],[32,15627],{"className":15628,"style":14894},[86],[32,15630,15632,15635,15741],{"className":15631},[95],[32,15633],{"className":15634},[91,974],[32,15636,15638],{"className":15637},[5987],[32,15639,15641,15733],{"className":15640},[246,247],[32,15642,15644,15730],{"className":15643},[251],[32,15645,15647,15699,15707],{"className":15646,"style":14913},[255],[32,15648,15649,15652],{"style":14916},[32,15650],{"className":15651,"style":715},[263],[32,15653,15655],{"className":15654},[95],[32,15656,15658],{"className":15657},[95,14926],[32,15659,15661,15691],{"className":15660},[246,247],[32,15662,15664,15688],{"className":15663},[251],[32,15665,15667,15676],{"className":15666,"style":14936},[255],[32,15668,15670,15673],{"className":15669,"style":4715},[4734],[32,15671],{"className":15672,"style":715},[263],[32,15674,5511],{"className":15675,"style":14946},[95],[32,15677,15678,15681],{"style":14949},[32,15679],{"className":15680,"style":715},[263],[32,15682,15684],{"className":15683,"style":14957},[14956],[551,15685,15686],{"xmlns":553,"width":4751,"height":14960,"viewBox":14961,"preserveAspectRatio":4754},[560,15687],{"d":14964},[32,15689,285],{"className":15690},[284],[32,15692,15694],{"className":15693},[251],[32,15695,15697],{"className":15696,"style":14974},[255],[32,15698],{},[32,15700,15701,15704],{"style":6275},[32,15702],{"className":15703,"style":715},[263],[32,15705],{"className":15706,"style":6283},[6282],[32,15708,15709,15712],{"style":6286},[32,15710],{"className":15711,"style":715},[263],[32,15713,15715,15718,15721,15724,15727],{"className":15714},[95],[32,15716,14821],{"className":15717},[95,96],[32,15719],{"className":15720,"style":198},[105],[32,15722,478],{"className":15723},[202],[32,15725],{"className":15726,"style":198},[105],[32,15728,9705],{"className":15729,"style":740},[95,96],[32,15731,285],{"className":15732},[284],[32,15734,15736],{"className":15735},[251],[32,15737,15739],{"className":15738,"style":15017},[255],[32,15740],{},[32,15742],{"className":15743},[113,974],[32,15745],{"className":15746,"style":198},[105],[32,15748,1001],{"className":15749},[202],[32,15751],{"className":15752,"style":198},[105],[32,15754,15756,15759,15874,15877,15880],{"className":15755},[82],[32,15757],{"className":15758,"style":14894},[86],[32,15760,15762,15765,15871],{"className":15761},[95],[32,15763],{"className":15764},[91,974],[32,15766,15768],{"className":15767},[5987],[32,15769,15771,15863],{"className":15770},[246,247],[32,15772,15774,15860],{"className":15773},[251],[32,15775,15777,15829,15837],{"className":15776,"style":14913},[255],[32,15778,15779,15782],{"style":14916},[32,15780],{"className":15781,"style":715},[263],[32,15783,15785],{"className":15784},[95],[32,15786,15788],{"className":15787},[95,14926],[32,15789,15791,15821],{"className":15790},[246,247],[32,15792,15794,15818],{"className":15793},[251],[32,15795,15797,15806],{"className":15796,"style":14936},[255],[32,15798,15800,15803],{"className":15799,"style":4715},[4734],[32,15801],{"className":15802,"style":715},[263],[32,15804,5511],{"className":15805,"style":14946},[95],[32,15807,15808,15811],{"style":14949},[32,15809],{"className":15810,"style":715},[263],[32,15812,15814],{"className":15813,"style":14957},[14956],[551,15815,15816],{"xmlns":553,"width":4751,"height":14960,"viewBox":14961,"preserveAspectRatio":4754},[560,15817],{"d":14964},[32,15819,285],{"className":15820},[284],[32,15822,15824],{"className":15823},[251],[32,15825,15827],{"className":15826,"style":14974},[255],[32,15828],{},[32,15830,15831,15834],{"style":6275},[32,15832],{"className":15833,"style":715},[263],[32,15835],{"className":15836,"style":6283},[6282],[32,15838,15839,15842],{"style":6286},[32,15840],{"className":15841,"style":715},[263],[32,15843,15845,15848,15851,15854,15857],{"className":15844},[95],[32,15846,14821],{"className":15847},[95,96],[32,15849],{"className":15850,"style":198},[105],[32,15852,1089],{"className":15853},[202],[32,15855],{"className":15856,"style":198},[105],[32,15858,9705],{"className":15859,"style":740},[95,96],[32,15861,285],{"className":15862},[284],[32,15864,15866],{"className":15865},[251],[32,15867,15869],{"className":15868,"style":15017},[255],[32,15870],{},[32,15872],{"className":15873},[113,974],[32,15875],{"className":15876,"style":177},[105],[32,15878,368],{"className":15879},[181],[32,15881],{"className":15882,"style":177},[105],[32,15884,15886,15890,16016],{"className":15885},[82],[32,15887],{"className":15888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是对角的：",[32,16023,16025],{"className":16024,"translate":36},[299],[32,16026,16028,16086],{"className":16027,"translate":36},[35],[32,16029,16031],{"className":16030},[40],[42,16032,16033],{"xmlns":44,"display":308},[46,16034,16035,16083],{},[49,16036,16037,16039,16041,16081],{},[57,16038,7043],{},[52,16040,368],{},[49,16042,16043,16045,16079],{},[52,16044,55],{"fence":62},[317,16046,16047,16061],{"rowspacing":4544,"columnalign":14193,"columnspacing":321},[323,16048,16049,16055],{},[326,16050,16051],{},[329,16052,16053],{"scriptlevel":157,"displaystyle":54},[155,16054,1732],{},[326,16056,16057],{},[329,16058,16059],{"scriptlevel":157,"displaystyle":54},[155,16060,157],{},[323,16062,16063,16069],{},[326,16064,16065],{},[329,16066,16067],{"scriptlevel":157,"displaystyle":54},[155,16068,157],{},[326,16070,16071],{},[329,16072,16073],{"scriptlevel":157,"displaystyle":54},[49,16074,16075,16077],{},[52,16076,1089],{},[155,16078,1732],{},[52,16080,69],{"fence":62},[57,16082,6076],{"mathvariant":1083},[71,16084,16085],{"encoding":73},"H = \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}.",[32,16087,16089,16107],{"className":16088,"ariaHidden":62},[78],[32,16090,16092,16095,16098,16101,16104],{"className":16091},[82],[32,16093],{"className":16094,"style":212},[86],[32,16096,7043],{"className":16097,"style":7154},[95,96],[32,16099],{"className":16100,"style":177},[105],[32,16102,368],{"className":16103},[181],[32,16105],{"className":16106,"style":177},[105],[32,16108,16110,16113,16233,16236],{"className":16109},[82],[32,16111],{"className":16112,"style":14001},[86],[32,16114,16116,16122,16227],{"className":16115},[507],[32,16117,16119],{"className":16118,"style":7283},[91,7282],[32,16120,55],{"className":16121},[514,270],[32,16123,16125],{"className":16124},[95],[32,16126,16128,16173,16176,16179],{"className":16127},[317],[32,16129,16131],{"className":16130},[4661],[32,16132,16134,16165],{"className":16133},[246,247],[32,16135,16137,16162],{"className":16136},[251],[32,16138,16140,16151],{"className":16139,"style":14338},[255],[32,16141,16142,16145],{"style":14341},[32,16143],{"className":16144,"style":715},[263],[32,16146,16148],{"className":16147},[95],[32,16149,1732],{"className":16150},[95],[32,16152,16153,16156],{"style":14353},[32,16154],{"className":16155,"style":715},[263],[32,16157,16159],{"className":16158},[95],[32,16160,157],{"className":16161},[95],[32,16163,285],{"className":16164},[284],[32,16166,16168],{"className":16167},[251],[32,16169,16171],{"className":16170,"style":14372},[255],[32,16172],{},[32,16174],{"className":16175,"style":4657},[4656],[32,16177],{"className":16178,"style":4657},[4656],[32,16180,16182],{"className":16181},[4661],[32,16183,16185,16219],{"className":16184},[246,247],[32,16186,16188,16216],{"className":16187},[251],[32,16189,16191,16202],{"className":16190,"style":14338},[255],[32,16192,16193,16196],{"style":14341},[32,16194],{"className":16195,"style":715},[263],[32,16197,16199],{"className":16198},[95],[32,16200,157],{"className":16201},[95],[32,16203,16204,16207],{"style":14353},[32,16205],{"className":16206,"style":715},[263],[32,16208,16210,16213],{"className":16209},[95],[32,16211,1089],{"className":16212},[95],[32,16214,1732],{"className":16215},[95],[32,16217,285],{"className":16218},[284],[32,16220,16222],{"className":16221},[251],[32,16223,16225],{"className":16224,"style":14372},[255],[32,16226],{},[32,16228,16230],{"className":16229,"style":7283},[113,7282],[32,16231,69],{"className":16232},[514,270],[32,16234],{"className":16235,"style":106},[105],[32,16237,6076],{"className":16238},[95],[11,16240,16241,16242,16270,16271,16299],{},"此刻鞍点结构一目了然：函数在 ",[32,16243,16245,16258],{"className":16244,"translate":36},[35],[32,16246,16248],{"className":16247},[40],[42,16249,16250],{"xmlns":44},[46,16251,16252,16256],{},[49,16253,16254],{},[57,16255,14821],{},[71,16257,14821],{"encoding":73},[32,16259,16261],{"className":16260,"ariaHidden":62},[78],[32,16262,16264,16267],{"className":16263},[82],[32,16265],{"className":16266,"style":1261},[86],[32,16268,14821],{"className":16269},[95,96]," 方向向上弯曲，在 ",[32,16272,16274,16287],{"className":16273,"translate":36},[35],[32,16275,16277],{"className":16276},[40],[42,16278,16279],{"xmlns":44},[46,16280,16281,16285],{},[49,16282,16283],{},[57,16284,9705],{},[71,16286,9705],{"encoding":73},[32,16288,16290],{"className":16289,"ariaHidden":62},[78],[32,16291,16293,16296],{"className":16292},[82],[32,16294],{"className":16295,"style":1261},[86],[32,16297,9705],{"className":16298,"style":740},[95,96]," 方向向下弯曲。",[11,16301,16302,16303,16306,16307,16310],{},"不变的事实并不是 Hessian 在某个选定坐标系中的具体矩阵元素，而是它的",[1364,16304,16305],{},"符号分布","：一个正特征值、一个负特征值。这是",[1364,16308,16309],{},"西尔维斯特惯性定律（Sylvester's law of inertia）"," 的一个特例——该定律说，实二次型的正、负、零方向的数目，在可逆线性坐标变换下保持不变。",[21,16312,16313],{"id":16313},"随机矩阵的视角",[11,16315,16316],{},"现在回到主线问题：假设我们随机站在高维景观的某个临界点上，它是局部极小值的概率有多大？",[11,16318,16319],{},"第一个启发式是把典型临界点处的 Hessian 建模为一个随机实对称矩阵。这个模型对每个随机函数并非字面上都正确——Hessian 可能与函数值、梯度或问题中其它结构相关——但它抓住了核心机制：如果 Hessian 没有强的正偏向，它的特征值谱就应当同时含有正负号。",[16321,16322,16323],"h3",{"id":16323},"简化启发",[11,16325,16326,16327,16360],{},"仅仅出于直觉，假设每个 Hessian 特征值取正的概率为 ",[32,16328,16330,16348],{"className":16329,"translate":36},[35],[32,16331,16333],{"className":16332},[40],[42,16334,16335],{"xmlns":44},[46,16336,16337,16345],{},[49,16338,16339,16341,16343],{},[155,16340,1732],{},[57,16342,5099],{"mathvariant":1083},[155,16344,5511],{},[71,16346,16347],{"encoding":73},"1\u002F2",[32,16349,16351],{"className":16350,"ariaHidden":62},[78],[32,16352,16354,16357],{"className":16353},[82],[32,16355],{"className":16356,"style":87},[86],[32,16358,16347],{"className":16359},[95],"，且这些符号相互独立。那么",[32,16362,16364],{"className":16363,"translate":36},[299],[32,16365,16367,16436],{"className":16366,"translate":36},[35],[32,16368,16370],{"className":16369},[40],[42,16371,16372],{"xmlns":44,"display":308},[46,16373,16374,16433],{},[49,16375,16376,16378,16380,16383,16385,16387,16389,16391,16397,16399,16401,16403,16405,16407,16413,16415,16417,16419,16421,16431],{},[57,16377,2101],{"mathvariant":147},[52,16379,55],{"stretchy":54},[372,16381,16382],{},"非退化局部极小值",[52,16384,69],{"stretchy":54},[52,16386,368],{},[57,16388,2101],{"mathvariant":147},[52,16390,55],{"stretchy":54},[143,16392,16393,16395],{},[57,16394,9849],{},[155,16396,1732],{},[52,16398,12067],{},[155,16400,157],{},[52,16402,63],{"separator":62},[52,16404,1737],{},[52,16406,63],{"separator":62},[143,16408,16409,16411],{},[57,16410,9849],{},[57,16412,5322],{},[52,16414,12067],{},[155,16416,157],{},[52,16418,69],{"stretchy":54},[52,16420,368],{},[5397,16422,16423,16425],{},[155,16424,5511],{},[49,16426,16427,16429],{},[52,16428,1089],{},[57,16430,5322],{},[57,16432,6076],{"mathvariant":1083},[71,16434,16435],{"encoding":73},"\\mathbb{P}(\\text{非退化局部极小值}) = \\mathbb{P}(\\lambda_1>0,\\dots,\\lambda_N>0) = 2^{-N}.",[32,16437,16439,16469,16530,16606,16627],{"className":16438,"ariaHidden":62},[78],[32,16440,16442,16445,16448,16451,16457,16460,16463,16466],{"className":16441},[82],[32,16443],{"className":16444,"style":87},[86],[32,16446,2101],{"className":16447},[95,238],[32,16449,55],{"className":16450},[91],[32,16452,16454],{"className":16453},[95,4000],[32,16455,16382],{"className":16456},[95,4712],[32,16458,69],{"className":16459},[113],[32,16461],{"className":16462,"style":177},[105],[32,16464,368],{"className":16465},[181],[32,16467],{"className":16468,"style":177},[105],[32,16470,16472,16475,16478,16481,16521,16524,16527],{"className":16471},[82],[32,16473],{"className":16474,"style":87},[86],[32,16476,2101],{"className":16477},[95,238],[32,16479,55],{"className":16480},[91],[32,16482,16484,16487],{"className":16483},[95],[32,16485,9849],{"className":16486},[95,96],[32,16488,16490],{"className":16489},[242],[32,16491,16493,16513],{"className":16492},[246,247],[32,16494,16496,16510],{"className":16495},[251],[32,16497,16499],{"className":16498,"style":256},[255],[32,16500,16501,16504],{"style":259},[32,16502],{"className":16503,"style":264},[263],[32,16505,16507],{"className":16506},[268,269,270,271],[32,16508,1732],{"className":16509},[95,271],[32,16511,285],{"className":16512},[284],[32,16514,16516],{"className":16515},[251],[32,16517,16519],{"className":16518,"style":1441},[255],[32,16520],{},[32,16522],{"className":16523,"style":177},[105],[32,16525,12067],{"className":16526},[181],[32,16528],{"className":16529,"style":177},[105],[32,16531,16533,16536,16539,16542,16545,16548,16551,16554,16557,16597,16600,16603],{"className":16532},[82],[32,16534],{"className":16535,"style":4025},[86],[32,16537,157],{"className":16538},[95],[32,16540,63],{"className":16541},[101],[32,16543],{"className":16544,"style":106},[105],[32,16546,1737],{"className":16547},[507],[32,16549],{"className":16550,"style":106},[105],[32,16552,63],{"className":16553},[101],[32,16555],{"className":16556,"style":106},[105],[32,16558,16560,16563],{"className":16559},[95],[32,16561,9849],{"className":16562},[95,96],[32,16564,16566],{"className":16565},[242],[32,16567,16569,16589],{"className":16568},[246,247],[32,16570,16572,16586],{"className":16571},[251],[32,16573,16575],{"className":16574,"style":4238},[255],[32,16576,16577,16580],{"style":259},[32,16578],{"className":16579,"style":264},[263],[32,16581,16583],{"className":16582},[268,269,270,271],[32,16584,5322],{"className":16585,"style":1530},[95,96,271],[32,16587,285],{"className":16588},[284],[32,16590,16592],{"className":16591},[251],[32,16593,16595],{"className":16594,"style":1441},[255],[32,16596],{},[32,16598],{"className":16599,"style":177},[105],[32,16601,12067],{"className":16602},[181],[32,16604],{"className":16605,"style":177},[105],[32,16607,16609,16612,16615,16618,16621,16624],{"className":16608},[82],[32,16610],{"className":16611,"style":87},[86],[32,16613,157],{"className":16614},[95],[32,16616,69],{"className":16617},[113],[32,16619],{"className":16620,"style":177},[105],[32,16622,368],{"className":16623},[181],[32,16625],{"className":16626,"style":177},[105],[32,16628,16630,16633,16668],{"className":16629},[82],[32,16631],{"className":16632,"style":6534},[86],[32,16634,16636,16639],{"className":16635},[95],[32,16637,5511],{"className":16638},[95],[32,16640,16642],{"className":16641},[242],[32,16643,16645],{"className":16644},[246],[32,16646,16648],{"className":16647},[251],[32,16649,16651],{"className":16650,"style":6534},[255],[32,16652,16653,16656],{"style":6555},[32,16654],{"className":16655,"style":264},[263],[32,16657,16659],{"className":16658},[268,269,270,271],[32,16660,16662,16665],{"className":16661},[95,271],[32,16663,1089],{"className":16664},[95,271],[32,16666,5322],{"className":16667,"style":1530},[95,96,271],[32,16669,6076],{"className":16670},[95],[11,16672,16673,16674,16702],{},"这已经在 ",[32,16675,16677,16690],{"className":16676,"translate":36},[35],[32,16678,16680],{"className":16679},[40],[42,16681,16682],{"xmlns":44},[46,16683,16684,16688],{},[49,16685,16686],{},[57,16687,5322],{},[71,16689,5322],{"encoding":73},[32,16691,16693],{"className":16692,"ariaHidden":62},[78],[32,16694,16696,16699],{"className":16695},[82],[32,16697],{"className":16698,"style":212},[86],[32,16700,5322],{"className":16701,"style":1530},[95,96]," 上指数衰减。独立性假设对对称随机矩阵的特征值并不成立——特征值彼此强烈排斥——但「正定很罕见」这一结论仍然正确，而且真正的随机矩阵估计还要更尖锐。",[16321,16704,16706],{"id":16705},"goe-hessian-与半圆律","GOE Hessian 与半圆律",[11,16708,16709,16712,16713,16763,16764,16816,16817,16863],{},[1364,16710,16711],{},"高斯正交系综（GOE）"," 是中心化随机实对称矩阵的标准模型。非正式地说，",[32,16714,16716,16733],{"className":16715,"translate":36},[35],[32,16717,16719],{"className":16718},[40],[42,16720,16721],{"xmlns":44},[46,16722,16723,16731],{},[49,16724,16725,16727,16729],{},[57,16726,5322],{},[52,16728,136],{},[57,16730,5322],{},[71,16732,6992],{"encoding":73},[32,16734,16736,16754],{"className":16735,"ariaHidden":62},[78],[32,16737,16739,16742,16745,16748,16751],{"className":16738},[82],[32,16740],{"className":16741,"style":191},[86],[32,16743,5322],{"className":16744,"style":1530},[95,96],[32,16746],{"className":16747,"style":198},[105],[32,16749,136],{"className":16750},[202],[32,16752],{"className":16753,"style":198},[105],[32,16755,16757,16760],{"className":16756},[82],[32,16758],{"className":16759,"style":212},[86],[32,16761,5322],{"className":16762,"style":1530},[95,96]," 的 GOE 矩阵是其独立元素为高斯的对称矩阵，归一化方式使得特征值在 ",[32,16765,16767,16786],{"className":16766,"translate":36},[35],[32,16768,16770],{"className":16769},[40],[42,16771,16772],{"xmlns":44},[46,16773,16774,16783],{},[49,16775,16776,16778,16780],{},[57,16777,5322],{},[52,16779,141],{},[57,16781,16782],{"mathvariant":1083},"∞",[71,16784,16785],{"encoding":73},"N\\to\\infty",[32,16787,16789,16807],{"className":16788,"ariaHidden":62},[78],[32,16790,16792,16795,16798,16801,16804],{"className":16791},[82],[32,16793],{"className":16794,"style":212},[86],[32,16796,5322],{"className":16797,"style":1530},[95,96],[32,16799],{"className":16800,"style":177},[105],[32,16802,141],{"className":16803},[181],[32,16805],{"className":16806,"style":177},[105],[32,16808,16810,16813],{"className":16809},[82],[32,16811],{"className":16812,"style":1261},[86],[32,16814,16782],{"className":16815},[95]," 时保持在 ",[32,16818,16820,16841],{"className":16819,"translate":36},[35],[32,16821,16823],{"className":16822},[40],[42,16824,16825],{"xmlns":44},[46,16826,16827,16838],{},[49,16828,16829,16832,16834,16836],{},[57,16830,16831],{},"O",[52,16833,55],{"stretchy":54},[155,16835,1732],{},[52,16837,69],{"stretchy":54},[71,16839,16840],{"encoding":73},"O(1)",[32,16842,16844],{"className":16843,"ariaHidden":62},[78],[32,16845,16847,16850,16854,16857,16860],{"className":16846},[82],[32,16848],{"className":16849,"style":87},[86],[32,16851,16831],{"className":16852,"style":16853},[95,96],"margin-right:0.0278em;",[32,16855,55],{"className":16856},[91],[32,16858,1732],{"className":16859},[95],[32,16861,69],{"className":16862},[113]," 量级。",[11,16865,16866,16869,16870,16920,16921,16958,16959,3568],{},[1364,16867,16868],{},"维格纳半圆律（Wigner's semicircle law）"," 表明，中心化 GOE 矩阵的经验特征值分布在 ",[32,16871,16873,16890],{"className":16872,"translate":36},[35],[32,16874,16876],{"className":16875},[40],[42,16877,16878],{"xmlns":44},[46,16879,16880,16888],{},[49,16881,16882,16884,16886],{},[57,16883,5322],{},[52,16885,141],{},[57,16887,16782],{"mathvariant":1083},[71,16889,16785],{"encoding":73},[32,16891,16893,16911],{"className":16892,"ariaHidden":62},[78],[32,16894,16896,16899,16902,16905,16908],{"className":16895},[82],[32,16897],{"className":16898,"style":212},[86],[32,16900,5322],{"className":16901,"style":1530},[95,96],[32,16903],{"className":16904,"style":177},[105],[32,16906,141],{"className":16907},[181],[32,16909],{"className":16910,"style":177},[105],[32,16912,16914,16917],{"className":16913},[82],[32,16915],{"className":16916,"style":1261},[86],[32,16918,16782],{"className":16919},[95]," 时收敛到关于零点对称的确定性半圆分布。因此，典型的大 GOE 矩阵有大量正特征值和大量负特征值，它的指数通常接近 ",[32,16922,16924,16942],{"className":16923,"translate":36},[35],[32,16925,16927],{"className":16926},[40],[42,16928,16929],{"xmlns":44},[46,16930,16931,16939],{},[49,16932,16933,16935,16937],{},[57,16934,5322],{},[57,16936,5099],{"mathvariant":1083},[155,16938,5511],{},[71,16940,16941],{"encoding":73},"N\u002F2",[32,16943,16945],{"className":16944,"ariaHidden":62},[78],[32,16946,16948,16951,16954],{"className":16947},[82],[32,16949],{"className":16950,"style":87},[86],[32,16952,5322],{"className":16953,"style":1530},[95,96],[32,16955,16957],{"className":16956},[95],"\u002F2","，而不是接近 0 或 ",[32,16960,16962,16975],{"className":16961,"translate":36},[35],[32,16963,16965],{"className":16964},[40],[42,16966,16967],{"xmlns":44},[46,16968,16969,16973],{},[49,16970,16971],{},[57,16972,5322],{},[71,16974,5322],{"encoding":73},[32,16976,16978],{"className":16977,"ariaHidden":62},[78],[32,16979,16981,16984],{"className":16980},[82],[32,16982],{"className":16983,"style":212},[86],[32,16985,5322],{"className":16986,"style":1530},[95,96],[11,16988,16989],{},"然而，一个临界点要成为非退化局部极小值，Hessian 必须正定。在 GOE 模型中，这就是事件",[32,16991,16993],{"className":16992,"translate":36},[299],[32,16994,16996,17027],{"className":16995,"translate":36},[35],[32,16997,16999],{"className":16998},[40],[42,17000,17001],{"xmlns":44,"display":308},[46,17002,17003,17024],{},[49,17004,17005,17014,17016,17018,17020,17022],{},[143,17006,17007,17009,17012],{},[57,17008,9849],{},[57,17010,17011],{},"min",[52,17013,11692],{},[52,17015,55],{"stretchy":54},[57,17017,7043],{},[52,17019,69],{"stretchy":54},[52,17021,12067],{},[155,17023,12105],{},[71,17025,17026],{"encoding":73},"\\lambda_{\\min}(H) > 0.",[32,17028,17030,17108],{"className":17029,"ariaHidden":62},[78],[32,17031,17033,17036,17090,17093,17096,17099,17102,17105],{"className":17032},[82],[32,17034],{"className":17035,"style":87},[86],[32,17037,17039,17042],{"className":17038},[95],[32,17040,9849],{"className":17041},[95,96],[32,17043,17045],{"className":17044},[242],[32,17046,17048,17082],{"className":17047},[246,247],[32,17049,17051,17079],{"className":17050},[251],[32,17052,17055],{"className":17053,"style":17054},[255],"height:0.3175em;",[32,17056,17057,17060],{"style":259},[32,17058],{"className":17059,"style":264},[263],[32,17061,17063],{"className":17062},[268,269,270,271],[32,17064,17066],{"className":17065},[95,271],[32,17067,17069,17073,17076],{"className":17068},[10058,271],[32,17070,17072],{"className":17071},[271],"m",[32,17074,2084],{"className":17075},[271],[32,17077,1746],{"className":17078},[271],[32,17080,285],{"className":17081},[284],[32,17083,17085],{"className":17084},[251],[32,17086,17088],{"className":17087,"style":1441},[255],[32,17089],{},[32,17091,55],{"className":17092},[91],[32,17094,7043],{"className":17095,"style":7154},[95,96],[32,17097,69],{"className":17098},[113],[32,17100],{"className":17101,"style":177},[105],[32,17103,12067],{"className":17104},[181],[32,17106],{"className":17107,"style":177},[105],[32,17109,17111,17114],{"className":17110},[82],[32,17112],{"className":17113,"style":11978},[86],[32,17115,12105],{"className":17116},[95],[11,17118,17119],{},"这个事件迫使整个特征值谱——其自然的极限形状以零为中心——平移到正的一侧。这是一个大偏差事件。",[3222,17121,17122,17351,17767],{},[11,17123,17124,10677,17127,1265,17197,17247,17248,17299,17300,17350],{},[1364,17125,17126],{},"定理 4.1（GOE 正定呈指数级罕见）",[32,17128,17130,17148],{"className":17129,"translate":36},[35],[32,17131,17133],{"className":17132},[40],[42,17134,17135],{"xmlns":44},[46,17136,17137,17145],{},[49,17138,17139],{},[143,17140,17141,17143],{},[57,17142,7043],{},[57,17144,5322],{},[71,17146,17147],{"encoding":73},"H_N",[32,17149,17151],{"className":17150,"ariaHidden":62},[78],[32,17152,17154,17157],{"className":17153},[82],[32,17155],{"className":17156,"style":2137},[86],[32,17158,17160,17163],{"className":17159},[95],[32,17161,7043],{"className":17162,"style":7154},[95,96],[32,17164,17166],{"className":17165},[242],[32,17167,17169,17189],{"className":17168},[246,247],[32,17170,17172,17186],{"className":17171},[251],[32,17173,17175],{"className":17174,"style":4238},[255],[32,17176,17177,17180],{"style":7169},[32,17178],{"className":17179,"style":264},[263],[32,17181,17183],{"className":17182},[268,269,270,271],[32,17184,5322],{"className":17185,"style":1530},[95,96,271],[32,17187,285],{"className":17188},[284],[32,17190,17192],{"className":17191},[251],[32,17193,17195],{"className":17194,"style":1441},[255],[32,17196],{},[32,17198,17200,17217],{"className":17199,"translate":36},[35],[32,17201,17203],{"className":17202},[40],[42,17204,17205],{"xmlns":44},[46,17206,17207,17215],{},[49,17208,17209,17211,17213],{},[57,17210,5322],{},[52,17212,136],{},[57,17214,5322],{},[71,17216,6992],{"encoding":73},[32,17218,17220,17238],{"className":17219,"ariaHidden":62},[78],[32,17221,17223,17226,17229,17232,17235],{"className":17222},[82],[32,17224],{"className":17225,"style":191},[86],[32,17227,5322],{"className":17228,"style":1530},[95,96],[32,17230],{"className":17231,"style":198},[105],[32,17233,136],{"className":17234},[202],[32,17236],{"className":17237,"style":198},[105],[32,17239,17241,17244],{"className":17240},[82],[32,17242],{"className":17243,"style":212},[86],[32,17245,5322],{"className":17246,"style":1530},[95,96]," 中心化 GOE 矩阵，采用通常的随机矩阵标度。则存在常数 ",[32,17249,17251,17269],{"className":17250,"translate":36},[35],[32,17252,17254],{"className":17253},[40],[42,17255,17256],{"xmlns":44},[46,17257,17258,17266],{},[49,17259,17260,17262,17264],{},[57,17261,6594],{},[52,17263,12067],{},[155,17265,157],{},[71,17267,17268],{"encoding":73},"c>0",[32,17270,17272,17290],{"className":17271,"ariaHidden":62},[78],[32,17273,17275,17278,17281,17284,17287],{"className":17274},[82],[32,17276],{"className":17277,"style":6923},[86],[32,17279,6594],{"className":17280},[95,96],[32,17282],{"className":17283,"style":177},[105],[32,17285,12067],{"className":17286},[181],[32,17288],{"className":17289,"style":177},[105],[32,17291,17293,17296],{"className":17292},[82],[32,17294],{"className":17295,"style":11978},[86],[32,17297,157],{"className":17298},[95],"，使得在 ",[32,17301,17303,17320],{"className":17302,"translate":36},[35],[32,17304,17306],{"className":17305},[40],[42,17307,17308],{"xmlns":44},[46,17309,17310,17318],{},[49,17311,17312,17314,17316],{},[57,17313,5322],{},[52,17315,141],{},[57,17317,16782],{"mathvariant":1083},[71,17319,16785],{"encoding":73},[32,17321,17323,17341],{"className":17322,"ariaHidden":62},[78],[32,17324,17326,17329,17332,17335,17338],{"className":17325},[82],[32,17327],{"className":17328,"style":212},[86],[32,17330,5322],{"className":17331,"style":1530},[95,96],[32,17333],{"className":17334,"style":177},[105],[32,17336,141],{"className":17337},[181],[32,17339],{"className":17340,"style":177},[105],[32,17342,17344,17347],{"className":17343},[82],[32,17345],{"className":17346,"style":1261},[86],[32,17348,16782],{"className":17349},[95]," 时",[32,17352,17354],{"className":17353,"translate":36},[299],[32,17355,17357,17450],{"className":17356,"translate":36},[35],[32,17358,17360],{"className":17359},[40],[42,17361,17362],{"xmlns":44,"display":308},[46,17363,17364,17447],{},[49,17365,17366,17368,17370,17376,17378,17380,17382,17384,17386,17394,17396,17402,17404,17406,17408,17410,17412,17415,17417,17419,17421,17423,17429,17431,17433,17435,17441,17443,17445],{},[57,17367,2101],{"mathvariant":147},[52,17369,55],{"stretchy":54},[143,17371,17372,17374],{},[57,17373,7043],{},[57,17375,5322],{},[372,17377,5291],{},[52,17379,69],{"stretchy":54},[52,17381,368],{},[57,17383,2101],{"mathvariant":147},[52,17385,55],{"stretchy":54},[143,17387,17388,17390,17392],{},[57,17389,9849],{},[57,17391,17011],{},[52,17393,11692],{},[52,17395,55],{"stretchy":54},[143,17397,17398,17400],{},[57,17399,7043],{},[57,17401,5322],{},[52,17403,69],{"stretchy":54},[52,17405,12067],{},[155,17407,157],{},[52,17409,69],{"stretchy":54},[52,17411,368],{},[57,17413,17414],{},"exp",[52,17416,11692],{},[52,17418,315],{"stretchy":54},[52,17420,1089],{},[57,17422,6594],{},[5397,17424,17425,17427],{},[57,17426,5322],{},[155,17428,5511],{},[52,17430,478],{},[57,17432,1512],{},[52,17434,55],{"stretchy":54},[5397,17436,17437,17439],{},[57,17438,5322],{},[155,17440,5511],{},[52,17442,69],{"stretchy":54},[52,17444,1749],{"stretchy":54},[57,17446,6076],{"mathvariant":1083},[71,17448,17449],{"encoding":73},"\\mathbb{P}(H_N \\text{正定}) = \\mathbb{P}(\\lambda_{\\min}(H_N)>0) = \\exp\\{-cN^2 + o(N^2)\\}.",[32,17451,17453,17523,17642,17663,17719],{"className":17452,"ariaHidden":62},[78],[32,17454,17456,17459,17462,17465,17505,17511,17514,17517,17520],{"className":17455},[82],[32,17457],{"className":17458,"style":87},[86],[32,17460,2101],{"className":17461},[95,238],[32,17463,55],{"className":17464},[91],[32,17466,17468,17471],{"className":17467},[95],[32,17469,7043],{"className":17470,"style":7154},[95,96],[32,17472,17474],{"className":17473},[242],[32,17475,17477,17497],{"className":17476},[246,247],[32,17478,17480,17494],{"className":17479},[251],[32,17481,17483],{"className":17482,"style":4238},[255],[32,17484,17485,17488],{"style":7169},[32,17486],{"className":17487,"style":264},[263],[32,17489,17491],{"className":17490},[268,269,270,271],[32,17492,5322],{"className":17493,"style":1530},[95,96,271],[32,17495,285],{"className":17496},[284],[32,17498,17500],{"className":17499},[251],[32,17501,17503],{"className":17502,"style":1441},[255],[32,17504],{},[32,17506,17508],{"className":17507},[95,4000],[32,17509,5291],{"className":17510},[95,4712],[32,17512,69],{"className":17513},[113],[32,17515],{"className":17516,"style":177},[105],[32,17518,368],{"className":17519},[181],[32,17521],{"className":17522,"style":177},[105],[32,17524,17526,17529,17532,17535,17587,17590,17630,17633,17636,17639],{"className":17525},[82],[32,17527],{"className":17528,"style":87},[86],[32,17530,2101],{"className":17531},[95,238],[32,17533,55],{"className":17534},[91],[32,17536,17538,17541],{"className":17537},[95],[32,17539,9849],{"className":17540},[95,96],[32,17542,17544],{"className":17543},[242],[32,17545,17547,17579],{"className":17546},[246,247],[32,17548,17550,17576],{"className":17549},[251],[32,17551,17553],{"className":17552,"style":17054},[255],[32,17554,17555,17558],{"style":259},[32,17556],{"className":17557,"style":264},[263],[32,17559,17561],{"className":17560},[268,269,270,271],[32,17562,17564],{"className":17563},[95,271],[32,17565,17567,17570,17573],{"className":17566},[10058,271],[32,17568,17072],{"className":17569},[271],[32,17571,2084],{"className":17572},[271],[32,17574,1746],{"className":17575},[271],[32,17577,285],{"className":17578},[284],[32,17580,17582],{"className":17581},[251],[32,17583,17585],{"className":17584,"style":1441},[255],[32,17586],{},[32,17588,55],{"className":17589},[91],[32,17591,17593,17596],{"className":17592},[95],[32,17594,7043],{"className":17595,"style":7154},[95,96],[32,17597,17599],{"className":17598},[242],[32,17600,17602,17622],{"className":17601},[246,247],[32,17603,17605,17619],{"className":17604},[251],[32,17606,17608],{"className":17607,"style":4238},[255],[32,17609,17610,17613],{"style":7169},[32,17611],{"className":17612,"style":264},[263],[32,17614,17616],{"className":17615},[268,269,270,271],[32,17617,5322],{"className":17618,"style":1530},[95,96,271],[32,17620,285],{"className":17621},[284],[32,17623,17625],{"className":17624},[251],[32,17626,17628],{"className":17627,"style":1441},[255],[32,17629],{},[32,17631,69],{"className":17632},[113],[32,17634],{"className":17635,"style":177},[105],[32,17637,12067],{"className":17638},[181],[32,17640],{"className":17641,"style":177},[105],[32,17643,17645,17648,17651,17654,17657,17660],{"className":17644},[82],[32,17646],{"className":17647,"style":87},[86],[32,17649,157],{"className":17650},[95],[32,17652,69],{"className":17653},[113],[32,17655],{"className":17656,"style":177},[105],[32,17658,368],{"className":17659},[181],[32,17661],{"className":17662,"style":177},[105],[32,17664,17666,17669,17672,17675,17678,17681,17710,17713,17716],{"className":17665},[82],[32,17667],{"className":17668,"style":7214},[86],[32,17670,17414],{"className":17671},[10058],[32,17673,315],{"className":17674},[91],[32,17676,1089],{"className":17677},[95],[32,17679,6594],{"className":17680},[95,96],[32,17682,17684,17687],{"className":17683},[95],[32,17685,5322],{"className":17686,"style":1530},[95,96],[32,17688,17690],{"className":17689},[242],[32,17691,17693],{"className":17692},[246],[32,17694,17696],{"className":17695},[251],[32,17697,17699],{"className":17698,"style":7233},[255],[32,17700,17701,17704],{"style":6555},[32,17702],{"className":17703,"style":264},[263],[32,17705,17707],{"className":17706},[268,269,270,271],[32,17708,5511],{"className":17709},[95,271],[32,17711],{"className":17712,"style":198},[105],[32,17714,478],{"className":17715},[202],[32,17717],{"className":17718,"style":198},[105],[32,17720,17722,17725,17728,17731,17760,17764],{"className":17721},[82],[32,17723],{"className":17724,"style":7214},[86],[32,17726,1512],{"className":17727},[95,96],[32,17729,55],{"className":17730},[91],[32,17732,17734,17737],{"className":17733},[95],[32,17735,5322],{"className":17736,"style":1530},[95,96],[32,17738,17740],{"className":17739},[242],[32,17741,17743],{"className":17742},[246],[32,17744,17746],{"className":17745},[251],[32,17747,17749],{"className":17748,"style":7233},[255],[32,17750,17751,17754],{"style":6555},[32,17752],{"className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",[32,17771,17773,17788],{"className":17772,"translate":36},[35],[32,17774,17776],{"className":17775},[40],[42,17777,17778],{"xmlns":44},[46,17779,17780,17785],{},[49,17781,17782],{},[57,17783,17784],{},"β",[71,17786,17787],{"encoding":73},"\\beta",[32,17789,17791],{"className":17790,"ariaHidden":62},[78],[32,17792,17794,17797],{"className":17793},[82],[32,17795],{"className":17796,"style":4025},[86],[32,17798,17784],{"className":17799,"style":17800},[95,96],"margin-right:0.0528em;"," 系综，指数具有 ",[32,17803,17805,17839],{"className":17804,"translate":36},[35],[32,17806,17808],{"className":17807},[40],[42,17809,17810],{"xmlns":44},[46,17811,17812,17836],{},[49,17813,17814,17816,17819,17822,17824,17826,17828,17830],{},[57,17815,17784],{},[372,17817,17818],{}," ",[57,17820,17821],{},"θ",[52,17823,55],{"stretchy":54},[155,17825,157],{},[52,17827,69],{"stretchy":54},[372,17829,17818],{},[5397,17831,17832,17834],{},[57,17833,5322],{},[155,17835,5511],{},[71,17837,17838],{"encoding":73},"\\beta\\,\\theta(0)\\,N^2",[32,17840,17842],{"className":17841,"ariaHidden":62},[78],[32,17843,17845,17849,17852,17855,17858,17861,17864,17867,17870],{"className":17844},[82],[32,17846],{"className":17847,"style":17848},[86],"height:1.0641em;vertical-align:-0.25em;",[32,17850,17784],{"className":17851,"style":17800},[95,96],[32,17853],{"className":17854,"style":106},[105],[32,17856,17821],{"className":17857,"style":16853},[95,96],[32,17859,55],{"className":17860},[91],[32,17862,157],{"className":17863},[95],[32,17865,69],{"className":17866},[113],[32,17868],{"className":17869,"style":106},[105],[32,17871,17873,17876],{"className":17872},[95],[32,17874,5322],{"className":17875,"style":1530},[95,96],[32,17877,17879],{"className":17878},[242],[32,17880,17882],{"className":17881},[246],[32,17883,17885],{"className":17884},[251],[32,17886,17888],{"className":17887,"style":5571},[255],[32,17889,17890,17893],{"style":5458},[32,17891],{"className":17892,"style":264},[263],[32,17894,17896],{"className":17895},[268,269,270,271],[32,17897,5511],{"className":17898},[95,271]," 的形式，其中 ",[32,17901,17903,17942],{"className":17902,"translate":36},[35],[32,17904,17906],{"className":17905},[40],[42,17907,17908],{"xmlns":44},[46,17909,17910,17939],{},[49,17911,17912,17914,17916,17918,17920,17922,17924,17927,17929,17932,17934,17936],{},[57,17913,17821],{},[52,17915,55],{"stretchy":54},[155,17917,157],{},[52,17919,69],{"stretchy":54},[52,17921,368],{},[52,17923,55],{"stretchy":54},[57,17925,17926],{},"log",[52,17928,11692],{},[155,17930,17931],{},"3",[52,17933,69],{"stretchy":54},[57,17935,5099],{"mathvariant":1083},[155,17937,17938],{},"4",[71,17940,17941],{"encoding":73},"\\theta(0) = (\\log 3)\u002F4",[32,17943,17945,17972],{"className":17944,"ariaHidden":62},[78],[32,17946,17948,17951,17954,17957,17960,17963,17966,17969],{"className":17947},[82],[32,17949],{"className":17950,"style":87},[86],[32,17952,17821],{"className":17953,"style":16853},[95,96],[32,17955,55],{"className":17956},[91],[32,17958,157],{"className":17959},[95],[32,17961,69],{"className":17962},[113],[32,17964],{"className":17965,"style":177},[105],[32,17967,368],{"className":17968},[181],[32,17970],{"className":17971,"style":177},[105],[32,17973,17975,17978,17981,17989,17992,17995,17998],{"className":17974},[82],[32,17976],{"className":17977,"style":87},[86],[32,17979,55],{"className":17980},[91],[32,17982,17984,17985],{"className":17983},[10058],"lo",[32,17986,17988],{"style":17987},"margin-right:0.0139em;","g",[32,17990],{"className":17991,"style":106},[105],[32,17993,17931],{"className":17994},[95],[32,17996,69],{"className":17997},[113],[32,17999,18001],{"className":18000},[95],"\u002F4","（Dean & Majumdar 的结果与约定）。",[11,18004,18005,18006,18065,18066,18069,18070,18139],{},"精确常数不如标度重要。要点在于：正定不仅是「不太可能」，对中心化 GOE 矩阵而言，它在 ",[32,18007,18009,18027],{"className":18008,"translate":36},[35],[32,18010,18012],{"className":18011},[40],[42,18013,18014],{"xmlns":44},[46,18015,18016,18024],{},[49,18017,18018],{},[5397,18019,18020,18022],{},[57,18021,5322],{},[155,18023,5511],{},[71,18025,18026],{"encoding":73},"N^2",[32,18028,18030],{"className":18029,"ariaHidden":62},[78],[32,18031,18033,18036],{"className":18032},[82],[32,18034],{"className":18035,"style":5571},[86],[32,18037,18039,18042],{"className":18038},[95],[32,18040,5322],{"className":18041,"style":1530},[95,96],[32,18043,18045],{"className":18044},[242],[32,18046,18048],{"className":18047},[246],[32,18049,18051],{"className":18050},[251],[32,18052,18054],{"className":18053,"style":5571},[255],[32,18055,18056,18059],{"style":5458},[32,18057],{"className":18058,"style":264},[263],[32,18060,18062],{"className":18061},[268,269,270,271],[32,18063,5511],{"className":18064},[95,271]," 上呈",[1364,18067,18068],{},"指数级","不可能——比前面粗糙估计里的 ",[32,18071,18073,18095],{"className":18072,"translate":36},[35],[32,18074,18076],{"className":18075},[40],[42,18077,18078],{"xmlns":44},[46,18079,18080,18092],{},[49,18081,18082],{},[5397,18083,18084,18086],{},[155,18085,5511],{},[49,18087,18088,18090],{},[52,18089,1089],{},[57,18091,5322],{},[71,18093,18094],{"encoding":73},"2^{-N}",[32,18096,18098],{"className":18097,"ariaHidden":62},[78],[32,18099,18101,18104],{"className":18100},[82],[32,18102],{"className":18103,"style":5437},[86],[32,18105,18107,18110],{"className":18106},[95],[32,18108,5511],{"className":18109},[95],[32,18111,18113],{"className":18112},[242],[32,18114,18116],{"className":18115},[246],[32,18117,18119],{"className":18118},[251],[32,18120,18122],{"className":18121,"style":5437},[255],[32,18123,18124,18127],{"style":5458},[32,18125],{"className":18126,"style":264},[263],[32,18128,18130],{"className":18129},[268,269,270,271],[32,18131,18133,18136],{"className":18132},[95,271],[32,18134,1089],{"className":18135},[95,271],[32,18137,5322],{"className":18138,"style":1530},[95,96,271]," 还要猛烈。于是，在随机 Hessian 模型中，",[32,18141,18143],{"className":18142,"translate":36},[299],[32,18144,18146,18189],{"className":18145,"translate":36},[35],[32,18147,18149],{"className":18148},[40],[42,18150,18151],{"xmlns":44,"display":308},[46,18152,18153,18186],{},[49,18154,18155,18157,18159,18162,18164,18167,18169,18171,18173,18175,18177,18180,18182,18184],{},[57,18156,2101],{"mathvariant":147},[52,18158,55],{"stretchy":54},[372,18160,18161],{},"临界点是鞍点",[52,18163,69],{"stretchy":54},[52,18165,18166],{},"⟶",[155,18168,1732],{},[52,18170,63],{"separator":62},[105,18172],{"width":14843},[57,18174,2101],{"mathvariant":147},[52,18176,55],{"stretchy":54},[372,18178,18179],{},"临界点是非退化局部极小值",[52,18181,69],{"stretchy":54},[52,18183,18166],{},[155,18185,12105],{},[71,18187,18188],{"encoding":73},"\\mathbb{P}(\\text{临界点是鞍点}) \\longrightarrow 1, \\qquad \\mathbb{P}(\\text{临界点是非退化局部极小值}) \\longrightarrow 0.",[32,18190,18192,18222,18264],{"className":18191,"ariaHidden":62},[78],[32,18193,18195,18198,18201,18204,18210,18213,18216,18219],{"className":18194},[82],[32,18196],{"className":18197,"style":87},[86],[32,18199,2101],{"className":18200},[95,238],[32,18202,55],{"className":18203},[91],[32,18205,18207],{"className":18206},[95,4000],[32,18208,18161],{"className":18209},[95,4712],[32,18211,69],{"className":18212},[113],[32,18214],{"className":18215,"style":177},[105],[32,18217,18166],{"className":18218},[181],[32,18220],{"className":18221,"style":177},[105],[32,18223,18225,18228,18231,18234,18237,18240,18243,18246,18252,18255,18258,18261],{"className":18224},[82],[32,18226],{"className":18227,"style":87},[86],[32,18229,1732],{"className":18230},[95],[32,18232,63],{"className":18233},[101],[32,18235],{"className":18236,"style":15029},[105],[32,18238],{"className":18239,"style":106},[105],[32,18241,2101],{"className":18242},[95,238],[32,18244,55],{"className":18245},[91],[32,18247,18249],{"className":18248},[95,4000],[32,18250,18179],{"className":18251},[95,4712],[32,18253,69],{"className":18254},[113],[32,18256],{"className":18257,"style":177},[105],[32,18259,18166],{"className":18260},[181],[32,18262],{"className":18263,"style":177},[105],[32,18265,18267,18270],{"className":18266},[82],[32,18268],{"className":18269,"style":11978},[86],[32,18271,12105],{"className":18272},[95],[16321,18274,18275],{"id":18275},"更明确的几何诠释",[11,18277,8991,18278,18347],{},[32,18279,18281,18298],{"className":18280,"translate":36},[35],[32,18282,18284],{"className":18283},[40],[42,18285,18286],{"xmlns":44},[46,18287,18288,18296],{},[49,18289,18290],{},[143,18291,18292,18294],{},[57,18293,358],{},[57,18295,6594],{},[71,18297,9012],{"encoding":73},[32,18299,18301],{"className":18300,"ariaHidden":62},[78],[32,18302,18304,18307],{"className":18303},[82],[32,18305],{"className":18306,"style":8289},[86],[32,18308,18310,18313],{"className":18309},[95],[32,18311,358],{"className":18312},[95,96],[32,18314,18316],{"className":18315},[242],[32,18317,18319,18339],{"className":18318},[246,247],[32,18320,18322,18336],{"className":18321},[251],[32,18323,18325],{"className":18324,"style":1860},[255],[32,18326,18327,18330],{"style":259},[32,18328],{"className":18329,"style":264},[263],[32,18331,18333],{"className":18332},[268,269,270,271],[32,18334,6594],{"className":18335},[95,96,271],[32,18337,285],{"className":18338},[284],[32,18340,18342],{"className":18341},[251],[32,18343,18345],{"className":18344,"style":1441},[255],[32,18346],{}," 处，二阶近似为",[32,18349,18351],{"className":18350,"translate":36},[299],[32,18352,18354,18431],{"className":18353,"translate":36},[35],[32,18355,18357],{"className":18356},[40],[42,18358,18359],{"xmlns":44,"display":308},[46,18360,18361,18428],{},[49,18362,18363,18365,18367,18373,18375,18377,18379,18381,18383,18385,18391,18393,18396,18402,18408,18414,18416,18422,18424,18426],{},[57,18364,1909],{},[52,18366,55],{"stretchy":54},[143,18368,18369,18371],{},[57,18370,358],{},[57,18372,6594],{},[52,18374,478],{},[57,18376,8227],{},[52,18378,69],{"stretchy":54},[52,18380,1089],{},[57,18382,1909],{},[52,18384,55],{"stretchy":54},[143,18386,18387,18389],{},[57,18388,358],{},[57,18390,6594],{},[52,18392,69],{"stretchy":54},[52,18394,18395],{},"≈",[5987,18397,18398,18400],{},[155,18399,1732],{},[155,18401,5511],{},[5397,18403,18404,18406],{},[57,18405,8227],{},[57,18407,1509],{},[143,18409,18410,18412],{},[57,18411,7043],{},[57,18413,1909],{},[52,18415,55],{"stretchy":54},[143,18417,18418,18420],{},[57,18419,358],{},[57,18421,6594],{},[52,18423,69],{"stretchy":54},[57,18425,8227],{},[57,18427,6076],{"mathvariant":1083},[71,18429,18430],{"encoding":73},"f(x_c+h) - f(x_c) \\approx \\frac{1}{2}h^T H_f(x_c) h.",[32,18432,18434,18495,18516,18580],{"className":18433,"ariaHidden":62},[78],[32,18435,18437,18440,18443,18446,18486,18489,18492],{"className":18436},[82],[32,18438],{"className":18439,"style":87},[86],[32,18441,1909],{"className":18442,"style":1965},[95,96],[32,18444,55],{"className":18445},[91],[32,18447,18449,18452],{"className":18448},[95],[32,18450,358],{"className":18451},[95,96],[32,18453,18455],{"className":18454},[242],[32,18456,18458,18478],{"className":18457},[246,247],[32,18459,18461,18475],{"className":18460},[251],[32,18462,18464],{"className":18463,"style":1860},[255],[32,18465,18466,18469],{"style":259},[32,18467],{"className":18468,"style":264},[263],[32,18470,18472],{"className":18471},[268,269,270,271],[32,18473,6594],{"className":18474},[95,96,271],[32,18476,285],{"className":18477},[284],[32,18479,18481],{"className":18480},[251],[32,18482,18484],{"className":18483,"style":1441},[255],[32,18485],{},[32,18487],{"className":18488,"style":198},[105],[32,18490,478],{"className":18491},[202],[32,18493],{"className":18494,"style":198},[105],[32,18496,18498,18501,18504,18507,18510,18513],{"className":18497},[82],[32,18499],{"className":18500,"style":87},[86],[32,18502,8227],{"className":18503},[95,96],[32,18505,69],{"className":18506},[113],[32,18508],{"className":18509,"style":198},[105],[32,18511,1089],{"className":18512},[202],[32,18514],{"className":18515,"style":198},[105],[32,18517,18519,18522,18525,18528,18568,18571,18574,18577],{"className":18518},[82],[32,18520],{"className":18521,"style":87},[86],[32,18523,1909],{"className":18524,"style":1965},[95,96],[32,18526,55],{"className":18527},[91],[32,18529,18531,18534],{"className":18530},[95],[32,18532,358],{"className":18533},[95,96],[32,18535,18537],{"className":18536},[242],[32,18538,18540,18560],{"className":18539},[246,247],[32,18541,18543,18557],{"className":18542},[251],[32,18544,18546],{"className":18545,"style":1860},[255],[32,18547,18548,18551],{"style":259},[32,18549],{"className":18550,"style":264},[263],[32,18552,18554],{"className":18553},[268,269,270,271],[32,18555,6594],{"className":18556},[95,96,271],[32,18558,285],{"className":18559},[284],[32,18561,18563],{"className":18562},[251],[32,18564,18566],{"className":18565,"style":1441},[255],[32,18567],{},[32,18569,69],{"className":18570},[113],[32,18572],{"className":18573,"style":177},[105],[32,18575,18395],{"className":18576},[181],[32,18578],{"className":18579,"style":177},[105],[32,18581,18583,18586,18648,18677,18717,18720,18760,18763,18766],{"className":18582},[82],[32,18584],{"className":18585,"style":8722},[86],[32,18587,18589,18592,18645],{"className":18588},[95],[32,18590],{"className":18591},[91,974],[32,18593,18595],{"className":18594},[5987],[32,18596,18598,18637],{"className":18597},[246,247],[32,18599,18601,18634],{"className":18600},[251],[32,18602,18604,18615,18623],{"className":18603,"style":8741},[255],[32,18605,18606,18609],{"style":6222},[32,18607],{"className":18608,"style":715},[263],[32,18610,18612],{"className":18611},[95],[32,18613,5511],{"className":18614},[95],[32,18616,18617,18620],{"style":6275},[32,18618],{"className":18619,"style":715},[263],[32,18621],{"className":18622,"style":6283},[6282],[32,18624,18625,18628],{"style":6286},[32,18626],{"className":18627,"style":715},[263],[32,18629,18631],{"className":18630},[95],[32,18632,1732],{"className":18633},[95],[32,18635,285],{"className":18636},[284],[32,18638,18640],{"className":18639},[251],[32,18641,18643],{"className":18642,"style":8781},[255],[32,18644],{},[32,18646],{"className":18647},[113,974],[32,18649,18651,18654],{"className":18650},[95],[32,18652,8227],{"className":18653},[95,96],[32,18655,18657],{"className":18656},[242],[32,18658,18660],{"className":18659},[246],[32,18661,18663],{"className":18662},[251],[32,18664,18666],{"className":18665,"style":6534},[255],[32,18667,18668,18671],{"style":6555},[32,18669],{"className":18670,"style":264},[263],[32,18672,18674],{"className":18673},[268,269,270,271],[32,18675,1509],{"className":18676,"style":1675},[95,96,271],[32,18678,18680,18683],{"className":18679},[95],[32,18681,7043],{"className":18682,"style":7154},[95,96],[32,18684,18686],{"className":18685},[242],[32,18687,18689,18709],{"className":18688},[246,247],[32,18690,18692,18706],{"className":18691},[251],[32,18693,18695],{"className":18694,"style":1419},[255],[32,18696,18697,18700],{"style":7169},[32,18698],{"className":18699,"style":264},[263],[32,18701,18703],{"className":18702},[268,269,270,271],[32,18704,1909],{"className":18705,"style":1965},[95,96,271],[32,18707,285],{"className":18708},[284],[32,18710,18712],{"className":18711},[251],[32,18713,18715],{"className":18714,"style":2519},[255],[32,18716],{},[32,18718,55],{"className":18719},[91],[32,18721,18723,18726],{"className":18722},[95],[32,18724,358],{"className":18725},[95,96],[32,18727,18729],{"className":18728},[242],[32,18730,18732,18752],{"className":18731},[246,247],[32,18733,18735,18749],{"className":18734},[251],[32,18736,18738],{"className":18737,"style":1860},[255],[32,18739,18740,18743],{"style":259},[32,18741],{"className":18742,"style":264},[263],[32,18744,18746],{"className":18745},[268,269,270,271],[32,18747,6594],{"className":18748},[95,96,271],[32,18750,285],{"className":18751},[284],[32,18753,18755],{"className":18754},[251],[32,18756,18758],{"className":18757,"style":1441},[255],[32,18759],{},[32,18761,69],{"className":18762},[113],[32,18764,8227],{"className":18765},[95,96],[32,18767,6076],{"className":18768},[95],[11,18770,3877,18771,18896,18897,18926,18927,18955],{},[32,18772,18774,18801],{"className":18773,"translate":36},[35],[32,18775,18777],{"className":18776},[40],[42,18778,18779],{"xmlns":44},[46,18780,18781,18799],{},[49,18782,18783,18789,18791,18797],{},[143,18784,18785,18787],{},[57,18786,7043],{},[57,18788,1909],{},[52,18790,55],{"stretchy":54},[143,18792,18793,18795],{},[57,18794,358],{},[57,18796,6594],{},[52,18798,69],{"stretchy":54},[71,18800,9592],{"encoding":73},[32,18802,18804],{"className":18803,"ariaHidden":62},[78],[32,18805,18807,18810,18850,18853,18893],{"className":18806},[82],[32,18808],{"className":18809,"style":7147},[86],[32,18811,18813,18816],{"className":18812},[95],[32,18814,7043],{"className":18815,"style":7154},[95,96],[32,18817,18819],{"className":18818},[242],[32,18820,18822,18842],{"className":18821},[246,247],[32,18823,18825,18839],{"className":18824},[251],[32,18826,18828],{"className":18827,"style":1419},[255],[32,18829,18830,18833],{"style":7169},[32,18831],{"className":18832,"style":264},[263],[32,18834,18836],{"className":18835},[268,269,270,271],[32,18837,1909],{"className":18838,"style":1965},[95,96,271],[32,18840,285],{"className":18841},[284],[32,18843,18845],{"className":18844},[251],[32,18846,18848],{"className":18847,"style":2519},[255],[32,18849],{},[32,18851,55],{"className":18852},[91],[32,18854,18856,18859],{"className":18855},[95],[32,18857,358],{"className":18858},[95,96],[32,18860,18862],{"className":18861},[242],[32,18863,18865,18885],{"className":18864},[246,247],[32,18866,18868,18882],{"className":18867},[251],[32,18869,18871],{"className":18870,"style":1860},[255],[32,18872,18873,18876],{"style":259},[32,18874],{"className":18875,"style":264},[263],[32,18877,18879],{"className":18878},[268,269,270,271],[32,18880,6594],{"className":18881},[95,96,271],[32,18883,285],{"className":18884},[284],[32,18886,18888],{"className":18887},[251],[32,18889,18891],{"className":18890,"style":1441},[255],[32,18892],{},[32,18894,69],{"className":18895},[113]," 只要有一个负特征值 ",[32,18898,18900,18914],{"className":18899,"translate":36},[35],[32,18901,18903],{"className":18902},[40],[42,18904,18905],{"xmlns":44},[46,18906,18907,18911],{},[49,18908,18909],{},[57,18910,9849],{},[71,18912,18913],{"encoding":73},"\\lambda",[32,18915,18917],{"className":18916,"ariaHidden":62},[78],[32,18918,18920,18923],{"className":18919},[82],[32,18921],{"className":18922,"style":170},[86],[32,18924,9849],{"className":18925},[95,96],"，对应特征向量 ",[32,18928,18930,18943],{"className":18929,"translate":36},[35],[32,18931,18933],{"className":18932},[40],[42,18934,18935],{"xmlns":44},[46,18936,18937,18941],{},[49,18938,18939],{},[57,18940,9705],{},[71,18942,9705],{"encoding":73},[32,18944,18946],{"className":18945,"ariaHidden":62},[78],[32,18947,18949,18952],{"className":18948},[82],[32,18950],{"className":18951,"style":1261},[86],[32,18953,9705],{"className":18954,"style":740},[95,96],"，那么",[32,18957,18959],{"className":18958,"translate":36},[299],[32,18960,18962,19004],{"className":18961,"translate":36},[35],[32,18963,18965],{"className":18964},[40],[42,18966,18967],{"xmlns":44,"display":308},[46,18968,18969,19001],{},[49,18970,18971,18977,18983,18985,18991,18993,18995,18997,18999],{},[5397,18972,18973,18975],{},[57,18974,9705],{},[57,18976,1509],{},[143,18978,18979,18981],{},[57,18980,7043],{},[57,18982,1909],{},[52,18984,55],{"stretchy":54},[143,18986,18987,18989],{},[57,18988,358],{},[57,18990,6594],{},[52,18992,69],{"stretchy":54},[57,18994,9705],{},[52,18996,11724],{},[155,18998,157],{},[52,19000,63],{"separator":62},[71,19002,19003],{"encoding":73},"v^T H_f(x_c) v \u003C 0,",[32,19005,19007,19140],{"className":19006,"ariaHidden":62},[78],[32,19008,19010,19013,19042,19082,19085,19125,19128,19131,19134,19137],{"className":19009},[82],[32,19011],{"className":19012,"style":10316},[86],[32,19014,19016,19019],{"className":19015},[95],[32,19017,9705],{"className":19018,"style":740},[95,96],[32,19020,19022],{"className":19021},[242],[32,19023,19025],{"className":19024},[246],[32,19026,19028],{"className":19027},[251],[32,19029,19031],{"className":19030,"style":6534},[255],[32,19032,19033,19036],{"style":6555},[32,19034],{"className":19035,"style":264},[263],[32,19037,19039],{"className":19038},[268,269,270,271],[32,19040,1509],{"className":19041,"style":1675},[95,96,271],[32,19043,19045,19048],{"className":19044},[95],[32,19046,7043],{"className":19047,"style":7154},[95,96],[32,19049,19051],{"className":19050},[242],[32,19052,19054,19074],{"className":19053},[246,247],[32,19055,19057,19071],{"className":19056},[251],[32,19058,19060],{"className":19059,"style":1419},[255],[32,19061,19062,19065],{"style":7169},[32,19063],{"className":19064,"style":264},[263],[32,19066,19068],{"className":19067},[268,269,270,271],[32,19069,1909],{"className":19070,"style":1965},[95,96,271],[32,19072,285],{"className":19073},[284],[32,19075,19077],{"className":19076},[251],[32,19078,19080],{"className":19079,"style":2519},[255],[32,19081],{},[32,19083,55],{"className":19084},[91],[32,19086,19088,19091],{"className":19087},[95],[32,19089,358],{"className":19090},[95,96],[32,19092,19094],{"className":19093},[242],[32,19095,19097,19117],{"className":19096},[246,247],[32,19098,19100,19114],{"className":19099},[251],[32,19101,19103],{"className":19102,"style":1860},[255],[32,19104,19105,19108],{"style":259},[32,19106],{"className":19107,"style":264},[263],[32,19109,19111],{"className":19110},[268,269,270,271],[32,19112,6594],{"className":19113},[95,96,271],[32,19115,285],{"className":19116},[284],[32,19118,19120],{"className":19119},[251],[32,19121,19123],{"className":19122,"style":1441},[255],[32,19124],{},[32,19126,69],{"className":19127},[113],[32,19129,9705],{"className":19130,"style":740},[95,96],[32,19132],{"className":19133,"style":177},[105],[32,19135,11724],{"className":19136},[181],[32,19138],{"className":19139,"style":177},[105],[32,19141,19143,19146,19149],{"className":19142},[82],[32,19144],{"className":19145,"style":6815},[86],[32,19147,157],{"className":19148},[95],[32,19150,63],{"className":19151},[101],[11,19153,19154,19155,19183,19184,19212],{},"沿 ",[32,19156,19158,19171],{"className":19157,"translate":36},[35],[32,19159,19161],{"className":19160},[40],[42,19162,19163],{"xmlns":44},[46,19164,19165,19169],{},[49,19166,19167],{},[57,19168,9705],{},[71,19170,9705],{"encoding":73},[32,19172,19174],{"className":19173,"ariaHidden":62},[78],[32,19175,19177,19180],{"className":19176},[82],[32,19178],{"className":19179,"style":1261},[86],[32,19181,9705],{"className":19182,"style":740},[95,96]," 移动一小段距离会使 ",[32,19185,19187,19200],{"className":19186,"translate":36},[35],[32,19188,19190],{"className":19189},[40],[42,19191,19192],{"xmlns":44},[46,19193,19194,19198],{},[49,19195,19196],{},[57,19197,1909],{},[71,19199,1909],{"encoding":73},[32,19201,19203],{"className":19202,"ariaHidden":62},[78],[32,19204,19206,19209],{"className":19205},[82],[32,19207],{"className":19208,"style":4025},[86],[32,19210,1909],{"className":19211,"style":1965},[95,96]," 在二阶意义上减小：",[32,19214,19216],{"className":19215,"translate":36},[299],[32,19217,19219,19307],{"className":19218,"translate":36},[35],[32,19220,19222],{"className":19221},[40],[42,19223,19224],{"xmlns":44,"display":308},[46,19225,19226,19304],{},[49,19227,19228,19230,19232,19238,19240,19242,19244,19246,19248,19250,19252,19258,19260,19262,19268,19274,19276,19282,19288,19290,19296,19298,19300,19302],{},[57,19229,1909],{},[52,19231,55],{"stretchy":54},[143,19233,19234,19236],{},[57,19235,358],{},[57,19237,6594],{},[52,19239,478],{},[57,19241,1502],{},[57,19243,9705],{},[52,19245,69],{"stretchy":54},[52,19247,1089],{},[57,19249,1909],{},[52,19251,55],{"stretchy":54},[143,19253,19254,19256],{},[57,19255,358],{},[57,19257,6594],{},[52,19259,69],{"stretchy":54},[52,19261,18395],{},[5987,19263,19264,19266],{},[155,19265,1732],{},[155,19267,5511],{},[5397,19269,19270,19272],{},[57,19271,1502],{},[155,19273,5511],{},[372,19275,17818],{},[5397,19277,19278,19280],{},[57,19279,9705],{},[57,19281,1509],{},[143,19283,19284,19286],{},[57,19285,7043],{},[57,19287,1909],{},[52,19289,55],{"stretchy":54},[143,19291,19292,19294],{},[57,19293,358],{},[57,19295,6594],{},[52,19297,69],{"stretchy":54},[57,19299,9705],{},[52,19301,11724],{},[155,19303,12105],{},[71,19305,19306],{"encoding":73},"f(x_c+tv) - f(x_c) \\approx \\frac{1}{2}t^2\\, v^T H_f(x_c) v \u003C 0.",[32,19308,19310,19371,19395,19459,19686],{"className":19309,"ariaHidden":62},[78],[32,19311,19313,19316,19319,19322,19362,19365,19368],{"className":19312},[82],[32,19314],{"className":19315,"style":87},[86],[32,19317,1909],{"className":19318,"style":1965},[95,96],[32,19320,55],{"className":19321},[91],[32,19323,19325,19328],{"className":19324},[95],[32,19326,358],{"className":19327},[95,96],[32,19329,19331],{"className":19330},[242],[32,19332,19334,19354],{"className":19333},[246,247],[32,19335,19337,19351],{"className":19336},[251],[32,19338,19340],{"className":19339,"style":1860},[255],[32,19341,19342,19345],{"style":259},[32,19343],{"className":19344,"style":264},[263],[32,19346,19348],{"className":19347},[268,269,270,271],[32,19349,6594],{"className":19350},[95,96,271],[32,19352,285],{"className":19353},[284],[32,19355,19357],{"className":19356},[251],[32,19358,19360],{"className":19359,"style":1441},[255],[32,19361],{},[32,19363],{"className":19364,"style":198},[105],[32,19366,478],{"className":19367},[202],[32,19369],{"className":19370,"style":198},[105],[32,19372,19374,19377,19380,19383,19386,19389,19392],{"className":19373},[82],[32,19375],{"className":19376,"style":87},[86],[32,19378,1502],{"className":19379},[95,96],[32,19381,9705],{"className":19382,"style":740},[95,96],[32,19384,69],{"className":19385},[113],[32,19387],{"className":19388,"style":198},[105],[32,19390,1089],{"className":19391},[202],[32,19393],{"className":19394,"style":198},[105],[32,19396,19398,19401,19404,19407,19447,19450,19453,19456],{"className":19397},[82],[32,19399],{"className":19400,"style":87},[86],[32,19402,1909],{"className":19403,"style":1965},[95,96],[32,19405,55],{"className":19406},[91],[32,19408,19410,19413],{"className":19409},[95],[32,19411,358],{"className":19412},[95,96],[32,19414,19416],{"className":19415},[242],[32,19417,19419,19439],{"className":19418},[246,247],[32,19420,19422,19436],{"className":19421},[251],[32,19423,19425],{"className":19424,"style":1860},[255],[32,19426,19427,19430],{"style":259},[32,19428],{"className":19429,"style":264},[263],[32,19431,19433],{"className":19432},[268,269,270,271],[32,19434,6594],{"className":19435},[95,96,271],[32,19437,285],{"className":19438},[284],[32,19440,19442],{"className":19441},[251],[32,19443,19445],{"className":19444,"style":1441},[255],[32,19446],{},[32,19448,69],{"className":19449},[113],[32,19451],{"className":19452,"style":177},[105],[32,19454,18395],{"className":19455},[181],[32,19457],{"className":19458,"style":177},[105],[32,19460,19462,19465,19527,19556,19559,19588,19628,19631,19671,19674,19677,19680,19683],{"className":19461},[82],[32,19463],{"className":19464,"style":8722},[86],[32,19466,19468,19471,19524],{"className":19467},[95],[32,19469],{"className":19470},[91,974],[32,19472,19474],{"className":19473},[5987],[32,19475,19477,19516],{"className":19476},[246,247],[32,19478,19480,19513],{"className":19479},[251],[32,19481,19483,19494,19502],{"className":19482,"style":8741},[255],[32,19484,19485,19488],{"style":6222},[32,19486],{"className":19487,"style":715},[263],[32,19489,19491],{"className":19490},[95],[32,19492,5511],{"className":19493},[95],[32,19495,19496,19499],{"style":6275},[32,19497],{"className":19498,"style":715},[263],[32,19500],{"className":19501,"style":6283},[6282],[32,19503,19504,19507],{"style":6286},[32,19505],{"className":19506,"style":715},[263],[32,19508,19510],{"className":19509},[95],[32,19511,1732],{"className":19512},[95],[32,19514,285],{"className":19515},[284],[32,19517,19519],{"className":19518},[251],[32,19520,19522],{"className":19521,"style":8781},[255],[32,19523],{},[32,19525],{"className":19526},[113,974],[32,19528,19530,19533],{"className":19529},[95],[32,19531,1502],{"className":19532},[95,96],[32,19534,19536],{"className":19535},[242],[32,19537,19539],{"className":19538},[246],[32,19540,19542],{"className":19541},[251],[32,19543,19545],{"className":19544,"style":7233},[255],[32,19546,19547,19550],{"style":6555},[32,19548],{"className":19549,"style":264},[263],[32,19551,19553],{"className":19552},[268,269,270,271],[32,19554,5511],{"className":19555},[95,271],[32,19557],{"className":19558,"style":106},[105],[32,19560,19562,19565],{"className":19561},[95],[32,19563,9705],{"className":19564,"style":740},[95,96],[32,19566,19568],{"className":19567},[242],[32,19569,19571],{"className":19570},[246],[32,19572,19574],{"className":19573},[251],[32,19575,19577],{"className":19576,"style":6534},[255],[32,19578,19579,19582],{"style":6555},[32,19580],{"className":19581,"style":264},[263],[32,19583,19585],{"className":19584},[268,269,270,271],[32,19586,1509],{"className":19587,"style":1675},[95,96,271],[32,19589,19591,19594],{"className":19590},[95],[32,19592,7043],{"className":19593,"style":7154},[95,96],[32,19595,19597],{"className":19596},[242],[32,19598,19600,19620],{"className":19599},[246,247],[32,19601,19603,19617],{"className":19602},[251],[32,19604,19606],{"className":19605,"style":1419},[255],[32,19607,19608,19611],{"style":7169},[32,19609],{"className":19610,"style":264},[263],[32,19612,19614],{"className":19613},[268,269,270,271],[32,19615,1909],{"className":19616,"style":1965},[95,96,271],[32,19618,285],{"className":19619},[284],[32,19621,19623],{"className":19622},[251],[32,19624,19626],{"className":19625,"style":2519},[255],[32,19627],{},[32,19629,55],{"className":19630},[91],[32,19632,19634,19637],{"className":19633},[95],[32,19635,358],{"className":19636},[95,96],[32,19638,19640],{"className":19639},[242],[32,19641,19643,19663],{"className":19642},[246,247],[32,19644,19646,19660],{"className":19645},[251],[32,19647,19649],{"className":19648,"style":1860},[255],[32,19650,19651,19654],{"style":259},[32,19652],{"className":19653,"style":264},[263],[32,19655,19657],{"className":19656},[268,269,270,271],[32,19658,6594],{"className":19659},[95,96,271],[32,19661,285],{"className":19662},[284],[32,19664,19666],{"className":19665},[251],[32,19667,19669],{"className":19668,"style":1441},[255],[32,19670],{},[32,19672,69],{"className":19673},[113],[32,19675,9705],{"className":19676,"style":740},[95,96],[32,19678],{"className":19679,"style":177},[105],[32,19681,11724],{"className":19682},[181],[32,19684],{"className":19685,"style":177},[105],[32,19687,19689,19692],{"className":19688},[82],[32,19690],{"className":19691,"style":11978},[86],[32,19693,12105],{"className":19694},[95],[11,19696,10636,19697,19700],{},[1364,19698,19699],{},"单个负特征值就摧毁了非退化局部极小值的可能性","。在高维中，随机 Hessian 有很多机会获得这样的负方向。只有当这些方向都不向下弯曲时，非退化局部极小值才能存活。",[21,19702,19704],{"id":19703},"kacrice-公式","Kac–Rice 公式",[11,19706,19707,19708,3568],{},"随机矩阵论证解释了为什么随机 Hessian 不太可能正定。要把这一论证与实际随机函数联系起来，需要一种计数临界点并按指数分类的方法。标准工具是 ",[1364,19709,19704],{},[11,19711,5379,19712,19740,19741,19825,19826,19901],{},[32,19713,19715,19728],{"className":19714,"translate":36},[35],[32,19716,19718],{"className":19717},[40],[42,19719,19720],{"xmlns":44},[46,19721,19722,19726],{},[49,19723,19724],{},[57,19725,1909],{},[71,19727,1909],{"encoding":73},[32,19729,19731],{"className":19730,"ariaHidden":62},[78],[32,19732,19734,19737],{"className":19733},[82],[32,19735],{"className":19736,"style":4025},[86],[32,19738,1909],{"className":19739,"style":1965},[95,96]," 是区域 ",[32,19742,19744,19768],{"className":19743,"translate":36},[35],[32,19745,19747],{"className":19746},[40],[42,19748,19749],{"xmlns":44},[46,19750,19751,19765],{},[49,19752,19753,19756,19759],{},[57,19754,19755],{},"D",[52,19757,19758],{},"⊂",[5397,19760,19761,19763],{},[57,19762,148],{"mathvariant":147},[57,19764,5322],{},[71,19766,19767],{"encoding":73},"D\\subset\\mathbb{R}^N",[32,19769,19771,19790],{"className":19770,"ariaHidden":62},[78],[32,19772,19774,19778,19781,19784,19787],{"className":19773},[82],[32,19775],{"className":19776,"style":19777},[86],"height:0.7224em;vertical-align:-0.0391em;",[32,19779,19755],{"className":19780,"style":16853},[95,96],[32,19782],{"className":19783,"style":177},[105],[32,19785,19758],{"className":19786},[181],[32,19788],{"className":19789,"style":177},[105],[32,19791,19793,19796],{"className":19792},[82],[32,19794],{"className":19795,"style":5437},[86],[32,19797,19799,19802],{"className":19798},[95],[32,19800,148],{"className":19801},[95,238],[32,19803,19805],{"className":19804},[242],[32,19806,19808],{"className":19807},[246],[32,19809,19811],{"className":19810},[251],[32,19812,19814],{"className":19813,"style":5437},[255],[32,19815,19816,19819],{"style":5458},[32,19817],{"className":19818,"style":264},[263],[32,19820,19822],{"className":19821},[268,269,270,271],[32,19823,5322],{"className":19824,"style":1530},[95,96,271]," 上足够正则的随机场。对 ",[32,19827,19829,19851],{"className":19828,"translate":36},[35],[32,19830,19832],{"className":19831},[40],[42,19833,19834],{"xmlns":44},[46,19835,19836,19848],{},[49,19837,19838,19840,19842,19844,19846],{},[155,19839,157],{},[52,19841,463],{},[57,19843,1387],{},[52,19845,463],{},[57,19847,5322],{},[71,19849,19850],{"encoding":73},"0\\le k\\le N",[32,19852,19854,19873,19892],{"className":19853,"ariaHidden":62},[78],[32,19855,19857,19861,19864,19867,19870],{"className":19856},[82],[32,19858],{"className":19859,"style":19860},[86],"height:0.7804em;vertical-align:-0.136em;",[32,19862,157],{"className":19863},[95],[32,19865],{"className":19866,"style":177},[105],[32,19868,463],{"className":19869},[181],[32,19871],{"className":19872,"style":177},[105],[32,19874,19876,19880,19883,19886,19889],{"className":19875},[82],[32,19877],{"className":19878,"style":19879},[86],"height:0.8304em;vertical-align:-0.136em;",[32,19881,1387],{"className":19882,"style":1431},[95,96],[32,19884],{"className":19885,"style":177},[105],[32,19887,463],{"className":19888},[181],[32,19890],{"className":19891,"style":177},[105],[32,19893,19895,19898],{"className":19894},[82],[32,19896],{"className":19897,"style":212},[86],[32,19899,5322],{"className":19900,"style":1530},[95,96],"，定义",[32,19903,19905],{"className":19904,"translate":36},[299],[32,19906,19908,19994],{"className":19907,"translate":36},[35],[32,19909,19911],{"className":19910},[40],[42,19912,19913],{"xmlns":44,"display":308},[46,19914,19915,19991],{},[49,19916,19917,19923,19925,19927,19929,19931,19933,19935,19937,19939,19941,19943,19945,19947,19949,19951,19953,19955,19957,19959,19961,19963,19965,19967,19969,19975,19977,19979,19981,19983,19985,19987,19989],{},[143,19918,19919,19921],{},[57,19920,5508],{},[57,19922,1387],{},[52,19924,55],{"stretchy":54},[57,19926,19755],{},[52,19928,69],{"stretchy":54},[52,19930,131],{},[52,19932,368],{},[57,19934,11709],{"mathvariant":1083},[52,19936,315],{"stretchy":54},[57,19938,358],{},[52,19940,1316],{},[57,19942,19755],{},[52,19944,131],{},[57,19946,5957],{"mathvariant":1083},[57,19948,1909],{},[52,19950,55],{"stretchy":54},[57,19952,358],{},[52,19954,69],{"stretchy":54},[52,19956,368],{},[155,19958,157],{},[52,19960,63],{"separator":62},[372,19962,12074],{},[57,19964,11689],{"mathvariant":1083},[52,19966,11692],{},[52,19968,55],{"stretchy":54},[143,19970,19971,19973],{},[57,19972,7043],{},[57,19974,1909],{},[52,19976,55],{"stretchy":54},[57,19978,358],{},[52,19980,69],{"stretchy":54},[52,19982,69],{"stretchy":54},[52,19984,368],{},[57,19986,1387],{},[52,19988,1749],{"stretchy":54},[52,19990,63],{"separator":62},[71,19992,19993],{"encoding":73},"C_k(D) := \\#\\{x\\in D : \\nabla f(x)=0,\\ 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",[32,20952,20954,20988],{"className":20953,"translate":36},[35],[32,20955,20957],{"className":20956},[40],[42,20958,20959],{"xmlns":44},[46,20960,20961,20985],{},[49,20962,20963,20979,20981,20983],{},[143,20964,20965,20967],{},[57,20966,11],{},[49,20968,20969,20971,20973,20975,20977],{},[57,20970,5957],{"mathvariant":1083},[57,20972,1909],{},[52,20974,55],{"stretchy":54},[57,20976,358],{},[52,20978,69],{"stretchy":54},[52,20980,55],{"stretchy":54},[155,20982,157],{},[52,20984,69],{"stretchy":54},[71,20986,20987],{"encoding":73},"p_{\\nabla f(x)}(0)",[32,20989,20991],{"className":20990,"ariaHidden":62},[78],[32,20992,20994,20998,21053,21056,21059],{"className":20993},[82],[32,20995],{"className":20996,"style":20997},[86],"height:1.1052em;vertical-align:-0.3552em;",[32,20999,21001,21004],{"className":21000},[95],[32,21002,11],{"className":21003},[95,96],[32,21005,21007],{"className":21006},[242],[32,21008,21010,21045],{"className":21009},[246,247],[32,21011,21013,21042],{"className":21012},[251],[32,21014,21016],{"className":21015,"style":20866},[255],[32,21017,21018,21021],{"style":20869},[32,21019],{"className":21020,"style":264},[263],[32,21022,21024],{"className":21023},[268,269,270,271],[32,21025,21027,21030,21033,21036,21039],{"className":21026},[95,271],[32,21028,5957],{"className":21029},[95,271],[32,21031,1909],{"className":21032,"style":1965},[95,96,271],[32,21034,55],{"className":21035},[91,271],[32,21037,358],{"className":21038},[95,96,271],[32,21040,69],{"className":21041},[113,271],[32,21043,285],{"className":21044},[284],[32,21046,21048],{"className":21047},[251],[32,21049,21051],{"className":21050,"style":20903},[255],[32,21052],{},[32,21054,55],{"className":21055},[91],[32,21057,157],{"className":21058},[95],[32,21060,69],{"className":21061},[113]," 是梯度在零处的密度。行列式因子的出现，是因为梯度映射的零点要按局部雅可比体积变化来计数。指示函数则选出所需的指数。",[11,21064,21065,21066,13749],{},"方程 (3) 使问题变得精确。计数非退化局部极小值就是取 ",[32,21067,21069,21087],{"className":21068,"translate":36},[35],[32,21070,21072],{"className":21071},[40],[42,21073,21074],{"xmlns":44},[46,21075,21076,21084],{},[49,21077,21078,21080,21082],{},[57,21079,1387],{},[52,21081,368],{},[155,21083,157],{},[71,21085,21086],{"encoding":73},"k=0",[32,21088,21090,21108],{"className":21089,"ariaHidden":62},[78],[32,21091,21093,21096,21099,21102,21105],{"className":21092},[82],[32,21094],{"className":21095,"style":170},[86],[32,21097,1387],{"className":21098,"style":1431},[95,96],[32,21100],{"className":21101,"style":177},[105],[32,21103,368],{"className":21104},[181],[32,21106],{"className":21107,"style":177},[105],[32,21109,21111,21114],{"className":21110},[82],[32,21112],{"className":21113,"style":11978},[86],[32,21115,157],{"className":21116},[95],[32,21118,21120],{"className":21119,"translate":36},[299],[32,21121,21123,21200],{"className":21122,"translate":36},[35],[32,21124,21126],{"className":21125},[40],[42,21127,21128],{"xmlns":44,"display":308},[46,21129,21130,21197],{},[49,21131,21132,21138,21140,21142,21144,21146,21148,21150,21152,21154,21156,21158,21160,21162,21164,21166,21168,21170,21172,21174,21176,21182,21184,21186,21188,21191,21193,21195],{},[143,21133,21134,21136],{},[57,21135,5508],{},[155,21137,157],{},[52,21139,55],{"stretchy":54},[57,21141,19755],{},[52,21143,69],{"stretchy":54},[52,21145,368],{},[57,21147,11709],{"mathvariant":1083},[52,21149,315],{"stretchy":54},[57,21151,358],{},[52,21153,1316],{},[57,21155,19755],{},[52,21157,131],{},[57,21159,5957],{"mathvariant":1083},[57,21161,1909],{},[52,21163,55],{"stretchy":54},[57,21165,358],{},[52,21167,69],{"stretchy":54},[52,21169,368],{},[155,21171,157],{},[52,21173,63],{"separator":62},[372,21175,12074],{},[143,21177,21178,21180],{},[57,21179,7043],{},[57,21181,1909],{},[52,21183,55],{"stretchy":54},[57,21185,358],{},[52,21187,69],{"stretchy":54},[52,21189,21190],{},"≻",[155,21192,157],{},[52,21194,1749],{"stretchy":54},[57,21196,6076],{"mathvariant":1083},[71,21198,21199],{"encoding":73},"C_0(D) = \\#\\{x\\in D : \\nabla f(x)=0,\\ H_f(x)\\succ 0\\}.",[32,21201,21203,21267,21291,21309,21339,21415],{"className":21202,"ariaHidden":62},[78],[32,21204,21206,21209,21249,21252,21255,21258,21261,21264],{"className":21205},[82],[32,21207],{"className":21208,"style":87},[86],[32,21210,21212,21215],{"className":21211},[95],[32,21213,5508],{"className":21214,"style":3492},[95,96],[32,21216,21218],{"className":21217},[242],[32,21219,21221,21241],{"className":21220},[246,247],[32,21222,21224,21238],{"className":21223},[251],[32,21225,21227],{"className":21226,"style":256},[255],[32,21228,21229,21232],{"style":20023},[32,21230],{"className":21231,"style":264},[263],[32,21233,21235],{"className":21234},[268,269,270,271],[32,21236,157],{"className":21237},[95,271],[32,21239,285],{"className":21240},[284],[32,21242,21244],{"className":21243},[251],[32,21245,21247],{"className":21246,"style":1441},[255],[32,21248],{},[32,21250,55],{"className":21251},[91],[32,21253,19755],{"className":21254,"style":16853},[95,96],[32,21256,69],{"className":21257},[113],[32,21259],{"className":21260,"style":177},[105],[32,21262,368],{"className":21263},[181],[32,21265],{"className":21266,"style":177},[105],[32,21268,21270,21273,21276,21279,21282,21285,21288],{"className":21269},[82],[32,21271],{"className":21272,"style":87},[86],[32,21274,11709],{"className":21275},[95],[32,21277,315],{"className":21278},[91],[32,21280,358],{"className":21281},[95,96],[32,21283],{"className":21284,"style":177},[105],[32,21286,1316],{"className":21287},[181],[32,21289],{"className":21290,"style":177},[105],[32,21292,21294,21297,21300,21303,21306],{"className":21293},[82],[32,21295],{"className":21296,"style":212},[86],[32,21298,19755],{"className":21299,"style":16853},[95,96],[32,21301],{"className":21302,"style":177},[105],[32,21304,131],{"className":21305},[181],[32,21307],{"className":21308,"style":177},[105],[32,21310,21312,21315,21318,21321,21324,21327,21330,21333,21336],{"className":21311},[82],[32,21313],{"className":21314,"style":87},[86],[32,21316,5957],{"className":21317},[95],[32,21319,1909],{"className":21320,"style":1965},[95,96],[32,21322,55],{"className":21323},[91],[32,21325,358],{"className":21326},[95,96],[32,21328,69],{"className":21329},[113],[32,21331],{"className":21332,"style":177},[105],[32,21334,368],{"className":21335},[181],[32,21337],{"className":21338,"style":177},[105],[32,21340,21342,21345,21348,21351,21354,21357,21397,21400,21403,21406,21409,21412],{"className":21341},[82],[32,21343],{"className":21344,"style":7147},[86],[32,21346,157],{"className":21347},[95],[32,21349,63],{"className":21350},[101],[32,21352,12074],{"className":21353},[105],[32,21355],{"className":21356,"style":106},[105],[32,21358,21360,21363],{"className":21359},[95],[32,21361,7043],{"className":21362,"style":7154},[95,96],[32,21364,21366],{"className":21365},[242],[32,21367,21369,21389],{"className":21368},[246,247],[32,21370,21372,21386],{"className":21371},[251],[32,21373,21375],{"className":21374,"style":1419},[255],[32,21376,21377,21380],{"style":7169},[32,21378],{"className":21379,"style":264},[263],[32,21381,21383],{"className":21382},[268,269,270,271],[32,21384,1909],{"className":21385,"style":1965},[95,96,271],[32,21387,285],{"className":21388},[284],[32,21390,21392],{"className":21391},[251],[32,21393,21395],{"className":21394,"style":2519},[255],[32,21396],{},[32,21398,55],{"className":21399},[91],[32,21401,358],{"className":21402},[95,96],[32,21404,69],{"className":21405},[113],[32,21407],{"className":21408,"style":177},[105],[32,21410,21190],{"className":21411},[181],[32,21413],{"className":21414,"style":177},[105],[32,21416,21418,21421,21424,21427],{"className":21417},[82],[32,21419],{"className":21420,"style":87},[86],[32,21422,157],{"className":21423},[95],[32,21425,1749],{"className":21426},[113],[32,21428,6076],{"className":21429},[95],[11,21431,21432,21433,21528],{},"计数鞍点则是对 ",[32,21434,21436,21462],{"className":21435,"translate":36},[35],[32,21437,21439],{"className":21438},[40],[42,21440,21441],{"xmlns":44},[46,21442,21443,21459],{},[49,21444,21445,21447,21449,21451,21453,21455,21457],{},[155,21446,1732],{},[52,21448,463],{},[57,21450,1387],{},[52,21452,463],{},[57,21454,5322],{},[52,21456,1089],{},[155,21458,1732],{},[71,21460,21461],{"encoding":73},"1\\le k\\le N-1",[32,21463,21465,21483,21501,21519],{"className":21464,"ariaHidden":62},[78],[32,21466,21468,21471,21474,21477,21480],{"className":21467},[82],[32,21469],{"className":21470,"style":19860},[86],[32,21472,1732],{"className":21473},[95],[32,21475],{"className":21476,"style":177},[105],[32,21478,463],{"className":21479},[181],[32,21481],{"className":21482,"style":177},[105],[32,21484,21486,21489,21492,21495,21498],{"className":21485},[82],[32,21487],{"className":21488,"style":19879},[86],[32,21490,1387],{"className":21491,"style":1431},[95,96],[32,21493],{"className":21494,"style":177},[105],[32,21496,463],{"className":21497},[181],[32,21499],{"className":21500,"style":177},[105],[32,21502,21504,21507,21510,21513,21516],{"className":21503},[82],[32,21505],{"className":21506,"style":191},[86],[32,21508,5322],{"className":21509,"style":1530},[95,96],[32,21511],{"className":21512,"style":198},[105],[32,21514,1089],{"className":21515},[202],[32,21517],{"className":21518,"style":198},[105],[32,21520,21522,21525],{"className":21521},[82],[32,21523],{"className":21524,"style":11978},[86],[32,21526,1732],{"className":21527},[95]," 求和。",[11,21530,21531,21532,21603],{},"对许多高斯随机场，在 ",[32,21533,21535,21561],{"className":21534,"translate":36},[35],[32,21536,21538],{"className":21537},[40],[42,21539,21540],{"xmlns":44},[46,21541,21542,21558],{},[49,21543,21544,21546,21548,21550,21552,21554,21556],{},[57,21545,5957],{"mathvariant":1083},[57,21547,1909],{},[52,21549,55],{"stretchy":54},[57,21551,358],{},[52,21553,69],{"stretchy":54},[52,21555,368],{},[155,21557,157],{},[71,21559,21560],{"encoding":73},"\\nabla f(x)=0",[32,21562,21564,21594],{"className":21563,"ariaHidden":62},[78],[32,21565,21567,21570,21573,21576,21579,21582,21585,21588,21591],{"className":21566},[82],[32,21568],{"className":21569,"style":87},[86],[32,21571,5957],{"className":21572},[95],[32,21574,1909],{"className":21575,"style":1965},[95,96],[32,21577,55],{"className":21578},[91],[32,21580,358],{"className":21581},[95,96],[32,21583,69],{"className":21584},[113],[32,21586],{"className":21587,"style":177},[105],[32,21589,368],{"className":21590},[181],[32,21592],{"className":21593,"style":177},[105],[32,21595,21597,21600],{"className":21596},[82],[32,21598],{"className":21599,"style":11978},[86],[32,21601,157],{"className":21602},[95]," 条件下，Hessian 的条件分布是一个正交不变的高斯矩阵，通常是 GOE 型矩阵加上一个依赖于场值的标量平移——这正是随机矩阵理论严格介入之处，Hessian 的指数分布控制着每一类临界点的期望个数。",[11,21605,21606,21607,21635],{},"在自旋玻璃模型中，这一方案已被详细执行。Bray 与 Dean 计算了大维空间上高斯场的平均临界点个数，把它作为能量与指数的函数。Aufinger、Ben Arous 与 Černý 利用 Kac–Rice 与 GOE 特征值渐近，对球面 ",[32,21608,21610,21623],{"className":21609,"translate":36},[35],[32,21611,21613],{"className":21612},[40],[42,21614,21615],{"xmlns":44},[46,21616,21617,21621],{},[49,21618,21619],{},[57,21620,11],{},[71,21622,11],{"encoding":73},[32,21624,21626],{"className":21625,"ariaHidden":62},[78],[32,21627,21629,21632],{"className":21628},[82],[32,21630],{"className":21631,"style":9732},[86],[32,21633,11],{"className":21634},[95,96],"-自旋自旋玻璃哈密顿量的复杂度给出了严格的渐近分析。这些工作展示了一种典型模式：临界点可能是指数级众多的，但主导指数通常不是 0——相反，在景观的大部分区域，占主导的临界点具有正比例的负特征值，即它们是鞍点。",[11,21637,21638],{},"因此，严格的概率陈述不是「每个高维函数都只有很少的局部极小值」，而是更接近：",[3222,21640,21641],{},[11,21642,21643],{},"在广泛的高维随机景观模型中，期望的临界点计数以鞍点为主，而指数为零的临界点只构成一个趋于消失或指数级更小的比例，除非处于特殊的低能区域。",[21,21645,21647],{"id":21646},"morse-理论","Morse 理论",[11,21649,21650],{},"Morse 理论为指数提供了拓扑诠释。它本身并不证明极小值是罕见的，而是解释为什么指数是应当追踪的正确不变量。",[11,21652,5379,21653,21726,21727,21755,21756,21784],{},[32,21654,21656,21678],{"className":21655,"translate":36},[35],[32,21657,21659],{"className":21658},[40],[42,21660,21661],{"xmlns":44},[46,21662,21663,21675],{},[49,21664,21665,21667,21669,21671,21673],{},[57,21666,1909],{},[52,21668,131],{},[57,21670,1496],{},[52,21672,141],{},[57,21674,148],{"mathvariant":147},[71,21676,21677],{"encoding":73},"f: M\\to\\mathbb{R}",[32,21679,21681,21699,21717],{"className":21680,"ariaHidden":62},[78],[32,21682,21684,21687,21690,21693,21696],{"className":21683},[82],[32,21685],{"className":21686,"style":4025},[86],[32,21688,1909],{"className":21689,"style":1965},[95,96],[32,21691],{"className":21692,"style":177},[105],[32,21694,131],{"className":21695},[181],[32,21697],{"className":21698,"style":177},[105],[32,21700,21702,21705,21708,21711,21714],{"className":21701},[82],[32,21703],{"className":21704,"style":212},[86],[32,21706,1496],{"className":21707,"style":1530},[95,96],[32,21709],{"className":21710,"style":177},[105],[32,21712,141],{"className":21713},[181],[32,21715],{"className":21716,"style":177},[105],[32,21718,21720,21723],{"className":21719},[82],[32,21721],{"className":21722,"style":5483},[86],[32,21724,148],{"className":21725},[95,238]," 是紧光滑流形 ",[32,21728,21730,21743],{"className":21729,"translate":36},[35],[32,21731,21733],{"className":21732},[40],[42,21734,21735],{"xmlns":44},[46,21736,21737,21741],{},[49,21738,21739],{},[57,21740,1496],{},[71,21742,1496],{"encoding":73},[32,21744,21746],{"className":21745,"ariaHidden":62},[78],[32,21747,21749,21752],{"className":21748},[82],[32,21750],{"className":21751,"style":212},[86],[32,21753,1496],{"className":21754,"style":1530},[95,96]," 上的 Morse 函数。当抬高阈值 ",[32,21757,21759,21772],{"className":21758,"translate":36},[35],[32,21760,21762],{"className":21761},[40],[42,21763,21764],{"xmlns":44},[46,21765,21766,21770],{},[49,21767,21768],{},[57,21769,2104],{},[71,21771,2104],{"encoding":73},[32,21773,21775],{"className":21774,"ariaHidden":62},[78],[32,21776,21778,21781],{"className":21777},[82],[32,21779],{"className":21780,"style":1261},[86],[32,21782,2104],{"className":21783},[95,96]," 并考察下水平集",[32,21786,21788],{"className":21787,"translate":36},[299],[32,21789,21791,21837],{"className":21790,"translate":36},[35],[32,21792,21794],{"className":21793},[40],[42,21795,21796],{"xmlns":44,"display":308},[46,21797,21798,21834],{},[49,21799,21800,21806,21808,21810,21812,21814,21816,21818,21820,21822,21824,21826,21828,21830,21832],{},[143,21801,21802,21804],{},[57,21803,1496],{},[57,21805,2104],{},[52,21807,131],{},[52,21809,368],{},[52,21811,315],{"stretchy":54},[57,21813,358],{},[52,21815,1316],{},[57,21817,1496],{},[52,21819,131],{},[57,21821,1909],{},[52,21823,55],{"stretchy":54},[57,21825,358],{},[52,21827,69],{"stretchy":54},[52,21829,463],{},[57,21831,2104],{},[52,21833,1749],{"stretchy":54},[71,21835,21836],{"encoding":73},"M_a := \\{x\\in M : f(x)\\le a\\}",[32,21838,21840,21895,21916,21934,21961],{"className":21839,"ariaHidden":62},[78],[32,21841,21843,21846,21886,21889,21892],{"className":21842},[82],[32,21844],{"className":21845,"style":2137},[86],[32,21847,21849,21852],{"className":21848},[95],[32,21850,1496],{"className":21851,"style":1530},[95,96],[32,21853,21855],{"className":21854},[242],[32,21856,21858,21878],{"className":21857},[246,247],[32,21859,21861,21875],{"className":21860},[251],[32,21862,21864],{"className":21863,"style":1860},[255],[32,21865,21866,21869],{"style":2318},[32,21867],{"className":21868,"style":264},[263],[32,21870,21872],{"className":21871},[268,269,270,271],[32,21873,2104],{"className":21874},[95,96,271],[32,21876,285],{"className":21877},[284],[32,21879,21881],{"className":21880},[251],[32,21882,21884],{"className":21883,"style":1441},[255],[32,21885],{},[32,21887],{"className":21888,"style":177},[105],[32,21890,2185],{"className":21891},[181],[32,21893],{"className":21894,"style":177},[105],[32,21896,21898,21901,21904,21907,21910,21913],{"className":21897},[82],[32,21899],{"className":21900,"style":87},[86],[32,21902,315],{"className":21903},[91],[32,21905,358],{"className":21906},[95,96],[32,21908],{"className":21909,"style":177},[105],[32,21911,1316],{"className":21912},[181],[32,21914],{"className":21915,"style":177},[105],[32,21917,21919,21922,21925,21928,21931],{"className":21918},[82],[32,21920],{"className":21921,"style":212},[86],[32,21923,1496],{"className":21924,"style":1530},[95,96],[32,21926],{"className":21927,"style":177},[105],[32,21929,131],{"className":21930},[181],[32,21932],{"className":21933,"style":177},[105],[32,21935,21937,21940,21943,21946,21949,21952,21955,21958],{"className":21936},[82],[32,21938],{"className":21939,"style":87},[86],[32,21941,1909],{"className":21942,"style":1965},[95,96],[32,21944,55],{"className":21945},[91],[32,21947,358],{"className":21948},[95,96],[32,21950,69],{"className":21951},[113],[32,21953],{"className":21954,"style":177},[105],[32,21956,463],{"className":21957},[181],[32,21959],{"className":21960,"style":177},[105],[32,21962,21964,21967,21970],{"className":21963},[82],[32,21965],{"className":21966,"style":87},[86],[32,21968,2104],{"className":21969},[95,96],[32,21971,1749],{"className":21972},[113],[11,21974,21975,21976,22046,22047,22075,22076,22104,22105,22133],{},"时，",[32,21977,21979,21997],{"className":21978,"translate":36},[35],[32,21980,21982],{"className":21981},[40],[42,21983,21984],{"xmlns":44},[46,21985,21986,21994],{},[49,21987,21988],{},[143,21989,21990,21992],{},[57,21991,1496],{},[57,21993,2104],{},[71,21995,21996],{"encoding":73},"M_a",[32,21998,22000],{"className":21999,"ariaHidden":62},[78],[32,22001,22003,22006],{"className":22002},[82],[32,22004],{"className":22005,"style":2137},[86],[32,22007,22009,22012],{"className":22008},[95],[32,22010,1496],{"className":22011,"style":1530},[95,96],[32,22013,22015],{"className":22014},[242],[32,22016,22018,22038],{"className":22017},[246,247],[32,22019,22021,22035],{"className":22020},[251],[32,22022,22024],{"className":22023,"style":1860},[255],[32,22025,22026,22029],{"style":2318},[32,22027],{"className":22028,"style":264},[263],[32,22030,22032],{"className":22031},[268,269,270,271],[32,22033,2104],{"className":22034},[95,96,271],[32,22036,285],{"className":22037},[284],[32,22039,22041],{"className":22040},[251],[32,22042,22044],{"className":22043,"style":1441},[255],[32,22045],{}," 的拓扑只在 ",[32,22048,22050,22063],{"className":22049,"translate":36},[35],[32,22051,22053],{"className":22052},[40],[42,22054,22055],{"xmlns":44},[46,22056,22057,22061],{},[49,22058,22059],{},[57,22060,2104],{},[71,22062,2104],{"encoding":73},[32,22064,22066],{"className":22065,"ariaHidden":62},[78],[32,22067,22069,22072],{"className":22068},[82],[32,22070],{"className":22071,"style":1261},[86],[32,22073,2104],{"className":22074},[95,96]," 经过临界值时改变。若临界点的指数为 ",[32,22077,22079,22092],{"className":22078,"translate":36},[35],[32,22080,22082],{"className":22081},[40],[42,22083,22084],{"xmlns":44},[46,22085,22086,22090],{},[49,22087,22088],{},[57,22089,1387],{},[71,22091,1387],{"encoding":73},[32,22093,22095],{"className":22094,"ariaHidden":62},[78],[32,22096,22098,22101],{"className":22097},[82],[32,22099],{"className":22100,"style":170},[86],[32,22102,1387],{"className":22103,"style":1431},[95,96],"，那么粗略地说，穿过该水平会连接一个 ",[32,22106,22108,22121],{"className":22107,"translate":36},[35],[32,22109,22111],{"className":22110},[40],[42,22112,22113],{"xmlns":44},[46,22114,22115,22119],{},[49,22116,22117],{},[57,22118,1387],{},[71,22120,1387],{"encoding":73},[32,22122,22124],{"className":22123,"ariaHidden":62},[78],[32,22125,22127,22130],{"className":22126},[82],[32,22128],{"className":22129,"style":170},[86],[32,22131,1387],{"className":22132,"style":1431},[95,96]," 维手柄：局部极小值的指数为 0，因此它会为下水平集创造出新的连通分支。指数为 1 的鞍点可以把分支合并。更高指数的鞍点则创造出更高维的隧道与空洞。",[11,22135,22136,22137,13749],{},"一个推论是欧拉示性数的 ",[1364,22138,22139],{},"Morse 关系",[32,22141,22143],{"className":22142,"translate":36},[299],[32,22144,22146,22218],{"className":22145,"translate":36},[35],[32,22147,22149],{"className":22148},[40],[42,22150,22151],{"xmlns":44,"display":308},[46,22152,22153,22215],{},[317,22154,22155],{"width":8346},[323,22156,22157,22159,22208,22210],{},[326,22158],{"width":8351},[326,22160,22161],{},[49,22162,22163,22177,22179,22181,22183,22189,22195,22197,22200,22202,22204,22206],{},[9995,22164,22165,22167,22175],{},[52,22166,9999],{},[49,22168,22169,22171,22173],{},[57,22170,1387],{},[52,22172,368],{},[155,22174,157],{},[57,22176,5322],{},[52,22178,55],{"stretchy":54},[52,22180,1089],{},[155,22182,1732],{},[5397,22184,22185,22187],{},[52,22186,69],{"stretchy":54},[57,22188,1387],{},[143,22190,22191,22193],{},[57,22192,5508],{},[57,22194,1387],{},[52,22196,368],{},[57,22198,22199],{},"χ",[52,22201,55],{"stretchy":54},[57,22203,1496],{},[52,22205,69],{"stretchy":54},[52,22207,63],{"separator":62},[326,22209],{"width":8351},[326,22211,22212],{},[372,22213,22214],{},"(4)",[71,22216,22217],{"encoding":73},"\\sum_{k=0}^{N} (-1)^k C_k = \\chi(M), \\tag{4}",[32,22219,22221,22385,22406],{"className":22220,"ariaHidden":62},[78],[32,22222,22224,22228,22297,22300,22303,22306,22336,22376,22379,22382],{"className":22223},[82],[32,22225],{"className":22226,"style":22227},[86],"height:3.1304em;vertical-align:-1.3021em;",[32,22229,22231],{"className":22230},[10058,10059],[32,22232,22234,22288],{"className":22233},[246,247],[32,22235,22237,22285],{"className":22236},[251],[32,22238,22240,22261,22271],{"className":22239,"style":10069},[255],[32,22241,22243,22246],{"style":22242},"top:-1.8479em;margin-left:0em;",[32,22244],{"className":22245,"style":10076},[263],[32,22247,22249],{"className":22248},[268,269,270,271],[32,22250,22252,22255,22258],{"className":22251},[95,271],[32,22253,1387],{"className":22254,"style":1431},[95,96,271],[32,22256,368],{"className":22257},[181,271],[32,22259,157],{"className":22260},[95,271],[32,22262,22263,22266],{"style":10094},[32,22264],{"className":22265,"style":10076},[263],[32,22267,22268],{},[32,22269,9999],{"className":22270},[10058,10103,10104],[32,22272,22273,22276],{"style":10107},[32,22274],{"className":22275,"style":10076},[263],[32,22277,22279],{"className":22278},[268,269,270,271],[32,22280,22282],{"className":22281},[95,271],[32,22283,5322],{"className":22284,"style":1530},[95,96,271],[32,22286,285],{"className":22287},[284],[32,22289,22291],{"className":22290},[251],[32,22292,22295],{"className":22293,"style":22294},[255],"height:1.3021em;",[32,22296],{},[32,22298,55],{"className":22299},[91],[32,22301,1089],{"className":22302},[95],[32,22304,1732],{"className":22305},[95],[32,22307,22309,22312],{"className":22308},[113],[32,22310,69],{"className":22311},[113],[32,22313,22315],{"className":22314},[242],[32,22316,22318],{"className":22317},[246],[32,22319,22321],{"className":22320},[251],[32,22322,22325],{"className":22323,"style":22324},[255],"height:0.8991em;",[32,22326,22327,22330],{"style":6555},[32,22328],{"className":22329,"style":264},[263],[32,22331,22333],{"className":22332},[268,269,270,271],[32,22334,1387],{"className":22335,"style":1431},[95,96,271],[32,22337,22339,22342],{"className":22338},[95],[32,22340,5508],{"className":22341,"style":3492},[95,96],[32,22343,22345],{"className":22344},[242],[32,22346,22348,22368],{"className":22347},[246,247],[32,22349,22351,22365],{"className":22350},[251],[32,22352,22354],{"className":22353,"style":1419},[255],[32,22355,22356,22359],{"style":20023},[32,22357],{"className":22358,"style":264},[263],[32,22360,22362],{"className":22361},[268,269,270,271],[32,22363,1387],{"className":22364,"style":1431},[95,96,271],[32,22366,285],{"className":22367},[284],[32,22369,22371],{"className":22370},[251],[32,22372,22374],{"className":22373,"style":1441},[255],[32,22375],{},[32,22377],{"className":22378,"style":177},[105],[32,22380,368],{"className":22381},[181],[32,22383],{"className":22384,"style":177},[105],[32,22386,22388,22391,22394,22397,22400,22403],{"className":22387},[82],[32,22389],{"className":22390,"style":87},[86],[32,22392,22199],{"className":22393},[95,96],[32,22395,55],{"className":22396},[91],[32,22398,1496],{"className":22399,"style":1530},[95,96],[32,22401,69],{"className":22402},[113],[32,22404,63],{"className":22405},[101],[32,22407,22409,22412],{"className":22408},[8970],[32,22410],{"className":22411,"style":22227},[86],[32,22413,22415,22418,22424],{"className":22414},[95,4000],[32,22416,55],{"className":22417},[95],[32,22419,22421],{"className":22420},[95],[32,22422,17938],{"className":22423},[95],[32,22425,69],{"className":22426},[95],[11,22428,22429,22430,22500,22501,22529,22530,1265,22574,22602,22603,22646],{},"其中 ",[32,22431,22433,22451],{"className":22432,"translate":36},[35],[32,22434,22436],{"className":22435},[40],[42,22437,22438],{"xmlns":44},[46,22439,22440,22448],{},[49,22441,22442],{},[143,22443,22444,22446],{},[57,22445,5508],{},[57,22447,1387],{},[71,22449,22450],{"encoding":73},"C_k",[32,22452,22454],{"className":22453,"ariaHidden":62},[78],[32,22455,22457,22460],{"className":22456},[82],[32,22458],{"className":22459,"style":2137},[86],[32,22461,22463,22466],{"className":22462},[95],[32,22464,5508],{"className":22465,"style":3492},[95,96],[32,22467,22469],{"className":22468},[242],[32,22470,22472,22492],{"className":22471},[246,247],[32,22473,22475,22489],{"className":22474},[251],[32,22476,22478],{"className":22477,"style":1419},[255],[32,22479,22480,22483],{"style":20023},[32,22481],{"className":22482,"style":264},[263],[32,22484,22486],{"className":22485},[268,269,270,271],[32,22487,1387],{"className":22488,"style":1431},[95,96,271],[32,22490,285],{"className":22491},[284],[32,22493,22495],{"className":22494},[251],[32,22496,22498],{"className":22497,"style":1441},[255],[32,22499],{}," 是指数为 ",[32,22502,22504,22517],{"className":22503,"translate":36},[35],[32,22505,22507],{"className":22506},[40],[42,22508,22509],{"xmlns":44},[46,22510,22511,22515],{},[49,22512,22513],{},[57,22514,1387],{},[71,22516,1387],{"encoding":73},[32,22518,22520],{"className":22519,"ariaHidden":62},[78],[32,22521,22523,22526],{"className":22522},[82],[32,22524],{"className":22525,"style":170},[86],[32,22527,1387],{"className":22528,"style":1431},[95,96]," 的临界点个数，",[32,22531,22533,22553],{"className":22532,"translate":36},[35],[32,22534,22536],{"className":22535},[40],[42,22537,22538],{"xmlns":44},[46,22539,22540,22550],{},[49,22541,22542,22544,22546,22548],{},[57,22543,22199],{},[52,22545,55],{"stretchy":54},[57,22547,1496],{},[52,22549,69],{"stretchy":54},[71,22551,22552],{"encoding":73},"\\chi(M)",[32,22554,22556],{"className":22555,"ariaHidden":62},[78],[32,22557,22559,22562,22565,22568,22571],{"className":22558},[82],[32,22560],{"className":22561,"style":87},[86],[32,22563,22199],{"className":22564},[95,96],[32,22566,55],{"className":22567},[91],[32,22569,1496],{"className":22570,"style":1530},[95,96],[32,22572,69],{"className":22573},[113],[32,22575,22577,22590],{"className":22576,"translate":36},[35],[32,22578,22580],{"className":22579},[40],[42,22581,22582],{"xmlns":44},[46,22583,22584,22588],{},[49,22585,22586],{},[57,22587,1496],{},[71,22589,1496],{"encoding":73},[32,22591,22593],{"className":22592,"ariaHidden":62},[78],[32,22594,22596,22599],{"className":22595},[82],[32,22597],{"className":22598,"style":212},[86],[32,22600,1496],{"className":22601,"style":1530},[95,96]," 的欧拉示性数。由于在复杂的随机景观中，",[32,22604,22606,22625],{"className":22605,"translate":36},[35],[32,22607,22609],{"className":22608},[40],[42,22610,22611],{"xmlns":44},[46,22612,22613,22623],{},[49,22614,22615,22617,22619,22621],{},[57,22616,22199],{},[52,22618,55],{"stretchy":54},[57,22620,1496],{},[52,22622,69],{"stretchy":54},[71,22624,22552],{"encoding":73},[32,22626,22628],{"className":22627,"ariaHidden":62},[78],[32,22629,22631,22634,22637,22640,22643],{"className":22630},[82],[32,22632],{"className":22633,"style":87},[86],[32,22635,22199],{"className":22636},[95,96],[32,22638,55],{"className":22639},[91],[32,22641,1496],{"className":22642,"style":1530},[95,96],[32,22644,69],{"className":22645},[113]," 通常相对于临界点总数很小，许多临界点必须在这个交错求和中相互抵消——相邻指数的鞍点天然适合这种抵消。",[11,22648,22649],{},"这一拓扑图景与概率图景一致：随机高维景观常常含有大量散布在各个指数上的鞍点。但重要的是不要夸大拓扑论证：球面上的函数可以只有一个极小值和一个极大值。Morse 理论约束着可能的临界点计数。随机矩阵理论与 Kac–Rice 则解释了为什么在随机高维模型中，计数的绝大部分通常是鞍点式的。",[16321,22651,22653],{"id":22652},"坐标不变性morse-引理","坐标不变性：Morse 引理",[11,22655,22656,22657,22660],{},"在非退化临界点附近，经过光滑坐标变换后，每个光滑函数都看起来像它的 Hessian 标准形——这正是 ",[1364,22658,22659],{},"Morse 引理"," 的内容。",[3222,22662,22663,23002],{},[11,22664,22665,10677,22668,22740,22741,22500,22793,22821,22822,22850,22851,23001],{},[1364,22666,22667],{},"定理 B.1（Morse 引理）",[32,22669,22671,22692],{"className":22670,"translate":36},[35],[32,22672,22674],{"className":22673},[40],[42,22675,22676],{"xmlns":44},[46,22677,22678,22690],{},[49,22679,22680,22682,22684,22686,22688],{},[57,22681,1909],{},[52,22683,131],{},[57,22685,1496],{},[52,22687,141],{},[57,22689,148],{"mathvariant":147},[71,22691,21677],{"encoding":73},[32,22693,22695,22713,22731],{"className":22694,"ariaHidden":62},[78],[32,22696,22698,22701,22704,22707,22710],{"className":22697},[82],[32,22699],{"className":22700,"style":4025},[86],[32,22702,1909],{"className":22703,"style":1965},[95,96],[32,22705],{"className":22706,"style":177},[105],[32,22708,131],{"className":22709},[181],[32,22711],{"className":22712,"style":177},[105],[32,22714,22716,22719,22722,22725,22728],{"className":22715},[82],[32,22717],{"className":22718,"style":212},[86],[32,22720,1496],{"className":22721,"style":1530},[95,96],[32,22723],{"className":22724,"style":177},[105],[32,22726,141],{"className":22727},[181],[32,22729],{"className":22730,"style":177},[105],[32,22732,22734,22737],{"className":22733},[82],[32,22735],{"className":22736,"style":5483},[86],[32,22738,148],{"className":22739},[95,238]," 光滑，",[32,22742,22744,22762],{"className":22743,"translate":36},[35],[32,22745,22747],{"className":22746},[40],[42,22748,22749],{"xmlns":44},[46,22750,22751,22759],{},[49,22752,22753,22755,22757],{},[57,22754,11],{},[52,22756,1316],{},[57,22758,1496],{},[71,22760,22761],{"encoding":73},"p\\in M",[32,22763,22765,22784],{"className":22764,"ariaHidden":62},[78],[32,22766,22768,22772,22775,22778,22781],{"className":22767},[82],[32,22769],{"className":22770,"style":22771},[86],"height:0.7335em;vertical-align:-0.1944em;",[32,22773,11],{"className":22774},[95,96],[32,22776],{"className":22777,"style":177},[105],[32,22779,1316],{"className":22780},[181],[32,22782],{"className":22783,"style":177},[105],[32,22785,22787,22790],{"className":22786},[82],[32,22788],{"className":22789,"style":212},[86],[32,22791,1496],{"className":22792,"style":1530},[95,96],[32,22794,22796,22809],{"className":22795,"translate":36},[35],[32,22797,22799],{"className":22798},[40],[42,22800,22801],{"xmlns":44},[46,22802,22803,22807],{},[49,22804,22805],{},[57,22806,1387],{},[71,22808,1387],{"encoding":73},[32,22810,22812],{"className":22811,"ariaHidden":62},[78],[32,22813,22815,22818],{"className":22814},[82],[32,22816],{"className":22817,"style":170},[86],[32,22819,1387],{"className":22820,"style":1431},[95,96]," 的非退化临界点。则以 ",[32,22823,22825,22838],{"className":22824,"translate":36},[35],[32,22826,22828],{"className":22827},[40],[42,22829,22830],{"xmlns":44},[46,22831,22832,22836],{},[49,22833,22834],{},[57,22835,11],{},[71,22837,11],{"encoding":73},[32,22839,22841],{"className":22840,"ariaHidden":62},[78],[32,22842,22844,22847],{"className":22843},[82],[32,22845],{"className":22846,"style":9732},[86],[32,22848,11],{"className":22849},[95,96]," 为中心存在局部坐标 ",[32,22852,22854,22888],{"className":22853,"translate":36},[35],[32,22855,22857],{"className":22856},[40],[42,22858,22859],{"xmlns":44},[46,22860,22861,22885],{},[49,22862,22863,22865,22871,22873,22875,22877,22883],{},[52,22864,55],{"stretchy":54},[143,22866,22867,22869],{},[57,22868,14821],{},[155,22870,1732],{},[52,22872,63],{"separator":62},[52,22874,1737],{},[52,22876,63],{"separator":62},[143,22878,22879,22881],{},[57,22880,14821],{},[57,22882,5322],{},[52,22884,69],{"stretchy":54},[71,22886,22887],{"encoding":73},"(u_1,\\dots,u_N)",[32,22889,22891],{"className":22890,"ariaHidden":62},[78],[32,22892,22894,22897,22900,22940,22943,22946,22949,22952,22955,22958,22998],{"className":22893},[82],[32,22895],{"className":22896,"style":87},[86],[32,22898,55],{"className":22899},[91],[32,22901,22903,22906],{"className":22902},[95],[32,22904,14821],{"className":22905},[95,96],[32,22907,22909],{"className":22908},[242],[32,22910,22912,22932],{"className":22911},[246,247],[32,22913,22915,22929],{"className":22914},[251],[32,22916,22918],{"className":22917,"style":256},[255],[32,22919,22920,22923],{"style":259},[32,22921],{"className":22922,"style":264},[263],[32,22924,22926],{"className":22925},[268,269,270,271],[32,22927,1732],{"className":22928},[95,271],[32,22930,285],{"className":22931},[284],[32,22933,22935],{"className":22934},[251],[32,22936,22938],{"className":22937,"style":1441},[255],[32,22939],{},[32,22941,63],{"className":22942},[101],[32,22944],{"className":22945,"style":106},[105],[32,22947,1737],{"className":22948},[507],[32,22950],{"className":22951,"style":106},[105],[32,22953,63],{"className":22954},[101],[32,22956],{"className":22957,"style":106},[105],[32,22959,22961,22964],{"className":22960},[95],[32,22962,14821],{"className":22963},[95,96],[32,22965,22967],{"className":22966},[242],[32,22968,22970,22990],{"className":22969},[246,247],[32,22971,22973,22987],{"className":22972},[251],[32,22974,22976],{"className":22975,"style":4238},[255],[32,22977,22978,22981],{"style":259},[32,22979],{"className":22980,"style":264},[263],[32,22982,22984],{"className":22983},[268,269,270,271],[32,22985,5322],{"className":22986,"style":1530},[95,96,271],[32,22988,285],{"className":22989},[284],[32,22991,22993],{"className":22992},[251],[32,22994,22996],{"className":22995,"style":1441},[255],[32,22997],{},[32,22999,69],{"className":23000},[113],"，使得",[32,23003,23005],{"className":23004,"translate":36},[299],[32,23006,23008,23095],{"className":23007,"translate":36},[35],[32,23009,23011],{"className":23010},[40],[42,23012,23013],{"xmlns":44,"display":308},[46,23014,23015,23092],{},[49,23016,23017,23019,23021,23023,23025,23027,23029,23031,23033,23035,23037,23045,23047,23050,23052,23060,23062,23076,23078,23080,23082,23090],{},[57,23018,1909],{},[52,23020,55],{"stretchy":54},[57,23022,14821],{},[52,23024,69],{"stretchy":54},[52,23026,368],{},[57,23028,1909],{},[52,23030,55],{"stretchy":54},[57,23032,11],{},[52,23034,69],{"stretchy":54},[52,23036,1089],{},[7073,23038,23039,23041,23043],{},[57,23040,14821],{},[155,23042,1732],{},[155,23044,5511],{},[52,23046,1089],{},[52,23048,23049],{},"⋯",[52,23051,1089],{},[7073,23053,23054,23056,23058],{},[57,23055,14821],{},[57,23057,1387],{},[155,23059,5511],{},[52,23061,478],{},[7073,23063,23064,23066,23074],{},[57,23065,14821],{},[49,23067,23068,23070,23072],{},[57,23069,1387],{},[52,23071,478],{},[155,23073,1732],{},[155,23075,5511],{},[52,23077,478],{},[52,23079,23049],{},[52,23081,478],{},[7073,23083,23084,23086,23088],{},[57,23085,14821],{},[57,23087,5322],{},[155,23089,5511],{},[57,23091,6076],{"mathvariant":1083},[71,23093,23094],{"encoding":73},"f(u) = f(p) - u_1^2 - \\dots - u_k^2 + u_{k+1}^2 + \\dots + u_N^2.",[32,23096,23098,23125,23152,23219,23237,23303,23380,23398],{"className":23097,"ariaHidden":62},[78],[32,23099,23101,23104,23107,23110,23113,23116,23119,23122],{"className":23100},[82],[32,23102],{"className":23103,"style":87},[86],[32,23105,1909],{"className":23106,"style":1965},[95,96],[32,23108,55],{"className":23109},[91],[32,23111,14821],{"className":23112},[95,96],[32,23114,69],{"className":23115},[113],[32,23117],{"className":23118,"style":177},[105],[32,23120,368],{"className":23121},[181],[32,23123],{"className":23124,"style":177},[105],[32,23126,23128,23131,23134,23137,23140,23143,23146,23149],{"className":23127},[82],[32,23129],{"className":23130,"style":87},[86],[32,23132,1909],{"className":23133,"style":1965},[95,96],[32,23135,55],{"className":23136},[91],[32,23138,11],{"className":23139},[95,96],[32,23141,69],{"className":23142},[113],[32,23144],{"className":23145,"style":198},[105],[32,23147,1089],{"className":23148},[202],[32,23150],{"className":23151,"style":198},[105],[32,23153,23155,23159,23210,23213,23216],{"className":23154},[82],[32,23156],{"className":23157,"style":23158},[86],"height:1.1111em;vertical-align:-0.247em;",[32,23160,23162,23165],{"className":23161},[95],[32,23163,14821],{"className":23164},[95,96],[32,23166,23168],{"className":23167},[242],[32,23169,23171,23202],{"className":23170},[246,247],[32,23172,23174,23199],{"className":23173},[251],[32,23175,23177,23188],{"className":23176,"style":7233},[255],[32,23178,23179,23182],{"style":10577},[32,23180],{"className":23181,"style":264},[263],[32,23183,23185],{"className":23184},[268,269,270,271],[32,23186,1732],{"className":23187},[95,271],[32,23189,23190,23193],{"style":6555},[32,23191],{"className":23192,"style":264},[263],[32,23194,23196],{"className":23195},[268,269,270,271],[32,23197,5511],{"className":23198},[95,271],[32,23200,285],{"className":23201},[284],[32,23203,23205],{"className":23204},[251],[32,23206,23208],{"className":23207,"style":10607},[255],[32,23209],{},[32,23211],{"className":23212,"style":198},[105],[32,23214,1089],{"className":23215},[202],[32,23217],{"className":23218,"style":198},[105],[32,23220,23222,23225,23228,23231,23234],{"className":23221},[82],[32,23223],{"className":23224,"style":8270},[86],[32,23226,23049],{"className":23227},[507],[32,23229],{"className":23230,"style":198},[105],[32,23232,1089],{"className":23233},[202],[32,23235],{"className":23236,"style":198},[105],[32,23238,23240,23243,23294,23297,23300],{"className":23239},[82],[32,23241],{"className":23242,"style":23158},[86],[32,23244,23246,23249],{"className":23245},[95],[32,23247,14821],{"className":23248},[95,96],[32,23250,23252],{"className":23251},[242],[32,23253,23255,23286],{"className":23254},[246,247],[32,23256,23258,23283],{"className":23257},[251],[32,23259,23261,23272],{"className":23260,"style":7233},[255],[32,23262,23263,23266],{"style":10577},[32,23264],{"className":23265,"style":264},[263],[32,23267,23269],{"className":23268},[268,269,270,271],[32,23270,1387],{"className":23271,"style":1431},[95,96,271],[32,23273,23274,23277],{"style":6555},[32,23275],{"className":23276,"style":264},[263],[32,23278,23280],{"className":23279},[268,269,270,271],[32,23281,5511],{"className":23282},[95,271],[32,23284,285],{"className":23285},[284],[32,23287,23289],{"className":23288},[251],[32,23290,23292],{"className":23291,"style":10607},[255],[32,23293],{},[32,23295],{"className":23296,"style":198},[105],[32,23298,478],{"className":23299},[202],[32,23301],{"className":23302,"style":198},[105],[32,23304,23306,23310,23371,23374,23377],{"className":23305},[82],[32,23307],{"className":23308,"style":23309},[86],"height:1.1694em;vertical-align:-0.3053em;",[32,23311,23313,23316],{"className":23312},[95],[32,23314,14821],{"className":23315},[95,96],[32,23317,23319],{"className":23318},[242],[32,23320,23322,23362],{"className":23321},[246,247],[32,23323,23325,23359],{"className":23324},[251],[32,23326,23328,23348],{"className":23327,"style":7233},[255],[32,23329,23330,23333],{"style":10577},[32,23331],{"className":23332,"style":264},[263],[32,23334,23336],{"className":23335},[268,269,270,271],[32,23337,23339,23342,23345],{"className":23338},[95,271],[32,23340,1387],{"className":23341,"style":1431},[95,96,271],[32,23343,478],{"className":23344},[202,271],[32,23346,1732],{"className":23347},[95,271],[32,23349,23350,23353],{"style":6555},[32,23351],{"className":23352,"style":264},[263],[32,23354,23356],{"className":23355},[268,269,270,271],[32,23357,5511],{"className":23358},[95,271],[32,23360,285],{"className":23361},[284],[32,23363,23365],{"className":23364},[251],[32,23366,23369],{"className":23367,"style":23368},[255],"height:0.3053em;",[32,23370],{},[32,23372],{"className":23373,"style":198},[105],[32,23375,478],{"className":23376},[202],[32,23378],{"className":23379,"style":198},[105],[32,23381,23383,23386,23389,23392,23395],{"className":23382},[82],[32,23384],{"className":23385,"style":8270},[86],[32,23387,23049],{"className":23388},[507],[32,23390],{"className":23391,"style":198},[105],[32,23393,478],{"className":23394},[202],[32,23396],{"className":23397,"style":198},[105],[32,23399,23401,23404,23455],{"className":23400},[82],[32,23402],{"className":23403,"style":23158},[86],[32,23405,23407,23410],{"className":23406},[95],[32,23408,14821],{"className":23409},[95,96],[32,23411,23413],{"className":23412},[242],[32,23414,23416,23447],{"className":23415},[246,247],[32,23417,23419,23444],{"className":23418},[251],[32,23420,23422,23433],{"className":23421,"style":7233},[255],[32,23423,23424,23427],{"style":10577},[32,23425],{"className":23426,"style":264},[263],[32,23428,23430],{"className":23429},[268,269,270,271],[32,23431,5322],{"className":23432,"style":1530},[95,96,271],[32,23434,23435,23438],{"style":6555},[32,23436],{"className":23437,"style":264},[263],[32,23439,23441],{"className":23440},[268,269,270,271],[32,23442,5511],{"className":23443},[95,271],[32,23445,285],{"className":23446},[284],[32,23448,23450],{"className":23449},[251],[32,23451,23453],{"className":23452,"style":10607},[255],[32,23454],{},[32,23456,6076],{"className":23457},[95],[11,23459,23460],{},"因此，在光滑局部坐标变换意义下，非退化临界点完全由其指数分类。非退化局部极小值的标准形为",[32,23462,23464],{"className":23463,"translate":36},[299],[32,23465,23467,23513],{"className":23466,"translate":36},[35],[32,23468,23470],{"className":23469},[40],[42,23471,23472],{"xmlns":44,"display":308},[46,23473,23474,23510],{},[49,23475,23476,23478,23480,23482,23484,23486,23494,23496,23498,23500,23508],{},[57,23477,1909],{},[52,23479,55],{"stretchy":54},[57,23481,11],{},[52,23483,69],{"stretchy":54},[52,23485,478],{},[7073,23487,23488,23490,23492],{},[57,23489,14821],{},[155,23491,1732],{},[155,23493,5511],{},[52,23495,478],{},[52,23497,23049],{},[52,23499,478],{},[7073,23501,23502,23504,23506],{},[57,23503,14821],{},[57,23505,5322],{},[155,23507,5511],{},[52,23509,63],{"separator":62},[71,23511,23512],{"encoding":73},"f(p) + u_1^2 + \\dots + u_N^2,",[32,23514,23516,23543,23609,23627],{"className":23515,"ariaHidden":62},[78],[32,23517,23519,23522,23525,23528,23531,23534,23537,23540],{"className":23518},[82],[32,23520],{"className":23521,"style":87},[86],[32,23523,1909],{"className":23524,"style":1965},[95,96],[32,23526,55],{"className":23527},[91],[32,23529,11],{"className":23530},[95,96],[32,23532,69],{"className":23533},[113],[32,23535],{"className":23536,"style":198},[105],[32,23538,478],{"className":23539},[202],[32,23541],{"className":23542,"style":198},[105],[32,23544,23546,23549,23600,23603,23606],{"className":23545},[82],[32,23547],{"className":23548,"style":23158},[86],[32,23550,23552,23555],{"className":23551},[95],[32,23553,14821],{"className":23554},[95,96],[32,23556,23558],{"className":23557},[242],[32,23559,23561,23592],{"className":23560},[246,247],[32,23562,23564,23589],{"className":23563},[251],[32,23565,23567,23578],{"className":23566,"style":7233},[255],[32,23568,23569,23572],{"style":10577},[32,23570],{"className":23571,"style":264},[263],[32,23573,23575],{"className":23574},[268,269,270,271],[32,23576,1732],{"className":23577},[95,271],[32,23579,23580,23583],{"style":6555},[32,23581],{"className":23582,"style":264},[263],[32,23584,23586],{"className":23585},[268,269,270,271],[32,23587,5511],{"className":23588},[95,271],[32,23590,285],{"className":23591},[284],[32,23593,23595],{"className":23594},[251],[32,23596,23598],{"className":23597,"style":10607},[255],[32,23599],{},[32,23601],{"className":23602,"style":198},[105],[32,23604,478],{"className":23605},[202],[32,23607],{"className":23608,"style":198},[105],[32,23610,23612,23615,23618,23621,23624],{"className":23611},[82],[32,23613],{"className":23614,"style":8270},[86],[32,23616,23049],{"className":23617},[507],[32,23619],{"className":23620,"style":198},[105],[32,23622,478],{"className":23623},[202],[32,23625],{"className":23626,"style":198},[105],[32,23628,23630,23633,23684],{"className":23629},[82],[32,23631],{"className":23632,"style":23158},[86],[32,23634,23636,23639],{"className":23635},[95],[32,23637,14821],{"className":23638},[95,96],[32,23640,23642],{"className":23641},[242],[32,23643,23645,23676],{"className":23644},[246,247],[32,23646,23648,23673],{"className":23647},[251],[32,23649,23651,23662],{"className":23650,"style":7233},[255],[32,23652,23653,23656],{"style":10577},[32,23654],{"className":23655,"style":264},[263],[32,23657,23659],{"className":23658},[268,269,270,271],[32,23660,5322],{"className":23661,"style":1530},[95,96,271],[32,23663,23664,23667],{"style":6555},[32,23665],{"className":23666,"style":264},[263],[32,23668,23670],{"className":23669},[268,269,270,271],[32,23671,5511],{"className":23672},[95,271],[32,23674,285],{"className":23675},[284],[32,23677,23679],{"className":23678},[251],[32,23680,23682],{"className":23681,"style":10607},[255],[32,23683],{},[32,23685,63],{"className":23686},[101],[11,23688,23689],{},"非退化局部极大值的标准形为",[32,23691,23693],{"className":23692,"translate":36},[299],[32,23694,23696,23742],{"className":23695,"translate":36},[35],[32,23697,23699],{"className":23698},[40],[42,23700,23701],{"xmlns":44,"display":308},[46,23702,23703,23739],{},[49,23704,23705,23707,23709,23711,23713,23715,23723,23725,23727,23729,23737],{},[57,23706,1909],{},[52,23708,55],{"stretchy":54},[57,23710,11],{},[52,23712,69],{"stretchy":54},[52,23714,1089],{},[7073,23716,23717,23719,23721],{},[57,23718,14821],{},[155,23720,1732],{},[155,23722,5511],{},[52,23724,1089],{},[52,23726,23049],{},[52,23728,1089],{},[7073,23730,23731,23733,23735],{},[57,23732,14821],{},[57,23734,5322],{},[155,23736,5511],{},[52,23738,63],{"separator":62},[71,23740,23741],{"encoding":73},"f(p) - u_1^2 - \\dots - u_N^2,",[32,23743,23745,23772,23838,23856],{"className":23744,"ariaHidden":62},[78],[32,23746,23748,23751,23754,23757,23760,23763,23766,23769],{"className":23747},[82],[32,23749],{"className":23750,"style":87},[86],[32,23752,1909],{"className":23753,"style":1965},[95,96],[32,23755,55],{"className":23756},[91],[32,23758,11],{"className":23759},[95,96],[32,23761,69],{"className":23762},[113],[32,23764],{"className":23765,"style":198},[105],[32,23767,1089],{"className":23768},[202],[32,23770],{"className":23771,"style":198},[105],[32,23773,23775,23778,23829,23832,23835],{"className":23774},[82],[32,23776],{"className":23777,"style":23158},[86],[32,23779,23781,23784],{"className":23780},[95],[32,23782,14821],{"className":23783},[95,96],[32,23785,23787],{"className":23786},[242],[32,23788,23790,23821],{"className":23789},[246,247],[32,23791,23793,23818],{"className":23792},[251],[32,23794,23796,23807],{"className":23795,"style":7233},[255],[32,23797,23798,23801],{"style":10577},[32,23799],{"className":23800,"style":264},[263],[32,23802,23804],{"className":23803},[268,269,270,271],[32,23805,1732],{"className":23806},[95,271],[32,23808,23809,23812],{"style":6555},[32,23810],{"className":23811,"style":264},[263],[32,23813,23815],{"className":23814},[268,269,270,271],[32,23816,5511],{"className":23817},[95,271],[32,23819,285],{"className":23820},[284],[32,23822,23824],{"className":23823},[251],[32,23825,23827],{"className":23826,"style":10607},[255],[32,23828],{},[32,23830],{"className":23831,"style":198},[105],[32,23833,1089],{"className":23834},[202],[32,23836],{"className":23837,"style":198},[105],[32,23839,23841,23844,23847,23850,23853],{"className":23840},[82],[32,23842],{"className":23843,"style":8270},[86],[32,23845,23049],{"className":23846},[507],[32,23848],{"className":23849,"style":198},[105],[32,23851,1089],{"className":23852},[202],[32,23854],{"className":23855,"style":198},[105],[32,23857,23859,23862,23913],{"className":23858},[82],[32,23860],{"className":23861,"style":23158},[86],[32,23863,23865,23868],{"className":23864},[95],[32,23866,14821],{"className":23867},[95,96],[32,23869,23871],{"className":23870},[242],[32,23872,23874,23905],{"className":23873},[246,247],[32,23875,23877,23902],{"className":23876},[251],[32,23878,23880,23891],{"className":23879,"style":7233},[255],[32,23881,23882,23885],{"style":10577},[32,23883],{"className":23884,"style":264},[263],[32,23886,23888],{"className":23887},[268,269,270,271],[32,23889,5322],{"className":23890,"style":1530},[95,96,271],[32,23892,23893,23896],{"style":6555},[32,23894],{"className":23895,"style":264},[263],[32,23897,23899],{"className":23898},[268,269,270,271],[32,23900,5511],{"className":23901},[95,271],[32,23903,285],{"className":23904},[284],[32,23906,23908],{"className":23907},[251],[32,23909,23911],{"className":23910,"style":10607},[255],[32,23912],{},[32,23914,63],{"className":23915},[101],[11,23917,23918,23919,23947,23948,23976,23977,24028],{},"而指数为 ",[32,23920,23922,23935],{"className":23921,"translate":36},[35],[32,23923,23925],{"className":23924},[40],[42,23926,23927],{"xmlns":44},[46,23928,23929,23933],{},[49,23930,23931],{},[57,23932,1387],{},[71,23934,1387],{"encoding":73},[32,23936,23938],{"className":23937,"ariaHidden":62},[78],[32,23939,23941,23944],{"className":23940},[82],[32,23942],{"className":23943,"style":170},[86],[32,23945,1387],{"className":23946,"style":1431},[95,96]," 的鞍点有 ",[32,23949,23951,23964],{"className":23950,"translate":36},[35],[32,23952,23954],{"className":23953},[40],[42,23955,23956],{"xmlns":44},[46,23957,23958,23962],{},[49,23959,23960],{},[57,23961,1387],{},[71,23963,1387],{"encoding":73},[32,23965,23967],{"className":23966,"ariaHidden":62},[78],[32,23968,23970,23973],{"className":23969},[82],[32,23971],{"className":23972,"style":170},[86],[32,23974,1387],{"className":23975,"style":1431},[95,96]," 个向下的方向和 ",[32,23978,23980,23998],{"className":23979,"translate":36},[35],[32,23981,23983],{"className":23982},[40],[42,23984,23985],{"xmlns":44},[46,23986,23987,23995],{},[49,23988,23989,23991,23993],{},[57,23990,5322],{},[52,23992,1089],{},[57,23994,1387],{},[71,23996,23997],{"encoding":73},"N-k",[32,23999,24001,24019],{"className":24000,"ariaHidden":62},[78],[32,24002,24004,24007,24010,24013,24016],{"className":24003},[82],[32,24005],{"className":24006,"style":191},[86],[32,24008,5322],{"className":24009,"style":1530},[95,96],[32,24011],{"className":24012,"style":198},[105],[32,24014,1089],{"className":24015},[202],[32,24017],{"className":24018,"style":198},[105],[32,24020,24022,24025],{"className":24021},[82],[32,24023],{"className":24024,"style":170},[86],[32,24026,1387],{"className":24027,"style":1431},[95,96]," 个向上的方向。这正是「指数」成为高维临界点自然语言的原因。",[21,24030,24031],{"id":24031},"局限性",[11,24033,24034],{},"任何听起来很「美」的结论都需要打上补丁，这个也不例外。",[5281,24036,24037,24209,24363,24369],{},[4063,24038,24039,24042,24043,24101,24102,24208],{},[1364,24040,24041],{},"不是所有函数都适用。"," ",[32,24044,24046,24063],{"className":24045,"translate":36},[35],[32,24047,24049],{"className":24048},[40],[42,24050,24051],{"xmlns":44},[46,24052,24053,24061],{},[49,24054,24055],{},[5397,24056,24057,24059],{},[57,24058,148],{"mathvariant":147},[57,24060,5322],{},[71,24062,12868],{"encoding":73},[32,24064,24066],{"className":24065,"ariaHidden":62},[78],[32,24067,24069,24072],{"className":24068},[82],[32,24070],{"className":24071,"style":5437},[86],[32,24073,24075,24078],{"className":24074},[95],[32,24076,148],{"className":24077},[95,238],[32,24079,24081],{"className":24080},[242],[32,24082,24084],{"className":24083},[246],[32,24085,24087],{"className":24086},[251],[32,24088,24090],{"className":24089,"style":5437},[255],[32,24091,24092,24095],{"style":5458},[32,24093],{"className":24094,"style":264},[263],[32,24096,24098],{"className":24097},[268,269,270,271],[32,24099,5322],{"className":24100,"style":1530},[95,96,271]," 上的强凸函数，例如 ",[32,24103,24105,24137],{"className":24104,"translate":36},[35],[32,24106,24108],{"className":24107},[40],[42,24109,24110],{"xmlns":44},[46,24111,24112,24134],{},[49,24113,24114,24116,24118,24120,24122,24124,24126,24128],{},[57,24115,1909],{},[52,24117,55],{"stretchy":54},[57,24119,358],{},[52,24121,69],{"stretchy":54},[52,24123,368],{},[57,24125,1084],{"mathvariant":1083},[57,24127,358],{},[5397,24129,24130,24132],{},[57,24131,1084],{"mathvariant":1083},[155,24133,5511],{},[71,24135,24136],{"encoding":73},"f(x)=\\|x\\|^2",[32,24138,24140,24167],{"className":24139,"ariaHidden":62},[78],[32,24141,24143,24146,24149,24152,24155,24158,24161,24164],{"className":24142},[82],[32,24144],{"className":24145,"style":87},[86],[32,24147,1909],{"className":24148,"style":1965},[95,96],[32,24150,55],{"className":24151},[91],[32,24153,358],{"className":24154},[95,96],[32,24156,69],{"className":24157},[113],[32,24159],{"className":24160,"style":177},[105],[32,24162,368],{"className":24163},[181],[32,24165],{"className":24166,"style":177},[105],[32,24168,24170,24173,24176,24179],{"className":24169},[82],[32,24171],{"className":24172,"style":17848},[86],[32,24174,1084],{"className":24175},[95],[32,24177,358],{"className":24178},[95,96],[32,24180,24182,24185],{"className":24181},[95],[32,24183,1084],{"className":24184},[95],[32,24186,24188],{"className":24187},[242],[32,24189,24191],{"className":24190},[246],[32,24192,24194],{"className":24193},[251],[32,24195,24197],{"className":24196,"style":5571},[255],[32,24198,24199,24202],{"style":5458},[32,24200],{"className":24201,"style":264},[263],[32,24203,24205],{"className":24204},[268,269,270,271],[32,24206,5511],{"className":24207},[95,271],"，只有一个临界点，而且它是严格的全局极小值，没有鞍点。因此，仅仅高维并不意味着极小值稀缺——起决定作用的是 Hessian 有没有系统性地偏向正定。",[4063,24210,24211,24214,24215,5714,24311,24362],{},[1364,24212,24213],{},"正定是严格极小值的充分条件，而非必要条件。"," 退化的严格极小值是存在的：一维函数 ",[32,24216,24218,24246],{"className":24217,"translate":36},[35],[32,24219,24221],{"className":24220},[40],[42,24222,24223],{"xmlns":44},[46,24224,24225,24243],{},[49,24226,24227,24229,24231,24233,24235,24237],{},[57,24228,1909],{},[52,24230,55],{"stretchy":54},[57,24232,358],{},[52,24234,69],{"stretchy":54},[52,24236,368],{},[5397,24238,24239,24241],{},[57,24240,358],{},[155,24242,17938],{},[71,24244,24245],{"encoding":73},"f(x)=x^4",[32,24247,24249,24276],{"className":24248,"ariaHidden":62},[78],[32,24250,24252,24255,24258,24261,24264,24267,24270,24273],{"className":24251},[82],[32,24253],{"className":24254,"style":87},[86],[32,24256,1909],{"className":24257,"style":1965},[95,96],[32,24259,55],{"className":24260},[91],[32,24262,358],{"className":24263},[95,96],[32,24265,69],{"className":24266},[113],[32,24268],{"className":24269,"style":177},[105],[32,24271,368],{"className":24272},[181],[32,24274],{"className":24275,"style":177},[105],[32,24277,24279,24282],{"className":24278},[82],[32,24280],{"className":24281,"style":5571},[86],[32,24283,24285,24288],{"className":24284},[95],[32,24286,358],{"className":24287},[95,96],[32,24289,24291],{"className":24290},[242],[32,24292,24294],{"className":24293},[246],[32,24295,24297],{"className":24296},[251],[32,24298,24300],{"className":24299,"style":5571},[255],[32,24301,24302,24305],{"style":5458},[32,24303],{"className":24304,"style":264},[263],[32,24306,24308],{"className":24307},[268,269,270,271],[32,24309,17938],{"className":24310},[95,271],[32,24312,24314,24332],{"className":24313,"translate":36},[35],[32,24315,24317],{"className":24316},[40],[42,24318,24319],{"xmlns":44},[46,24320,24321,24329],{},[49,24322,24323,24325,24327],{},[57,24324,358],{},[52,24326,368],{},[155,24328,157],{},[71,24330,24331],{"encoding":73},"x=0",[32,24333,24335,24353],{"className":24334,"ariaHidden":62},[78],[32,24336,24338,24341,24344,24347,24350],{"className":24337},[82],[32,24339],{"className":24340,"style":1261},[86],[32,24342,358],{"className":24343},[95,96],[32,24345],{"className":24346,"style":177},[105],[32,24348,368],{"className":24349},[181],[32,24351],{"className":24352,"style":177},[105],[32,24354,24356,24359],{"className":24355},[82],[32,24357],{"className":24358,"style":11978},[86],[32,24360,157],{"className":24361},[95]," 处有严格局部极小值，但其 Hessian 在那里为零。因此，Hessian 符号论证针对的是非退化的、即 Morse 的临界点。",[4063,24364,24365,24368],{},[1364,24366,24367],{},"结论依赖于随机景观系综。"," 中心化、类 GOE 的 Hessian 导致鞍点主导。Hessian 强烈向正方向平移的模型可以有许多极小值。在高斯场模型中，以极低函数值为条件，可以把 Hessian 谱向上平移，使极小值在景观底部附近更可能出现。",[4063,24370,24371,24374,24375,24403],{},[1364,24372,24373],{},"罕见并不意味着不存在。"," 即使极小值在所有临界点中的占比趋于零，在某些模型中极小值的绝对数目仍可能随 ",[32,24376,24378,24391],{"className":24377,"translate":36},[35],[32,24379,24381],{"className":24380},[40],[42,24382,24383],{"xmlns":44},[46,24384,24385,24389],{},[49,24386,24387],{},[57,24388,5322],{},[71,24390,5322],{"encoding":73},[32,24392,24394],{"className":24393,"ariaHidden":62},[78],[32,24395,24397,24400],{"className":24396},[82],[32,24398],{"className":24399,"style":212},[86],[32,24401,5322],{"className":24402,"style":1530},[95,96]," 增长。变得罕见的只是「随机选出的临界点是极小值」这一概率。",[11,24405,24406,24407,24410],{},"有了这些限定，可以更准确地表述为：",[1364,24408,24409],{},"在许多 Hessian 谱近似无偏的高维随机景观中，一个典型的非退化临界点压倒性地倾向于成为鞍点。"," Morse 临界点中的局部极小值要求所有 Hessian 特征值为正，而这一正定事件随维度增长呈指数级罕见。",[21,24412,24413],{"id":24413},"优化实践的启发",[11,24415,24416],{},"这有助于解释为什么高维优化常被描述为「鞍点逃逸问题」，而非单纯的「坏局部极小值问题」。这一陈述不应被过度解读：它并不证明梯度下降总是有效，也不是说所有实际目标都表现得像 GOE 随机矩阵——真实的损失函数具有架构、数据、对称性、退化与约束，远比理想化模型复杂。",[11,24418,24419],{},"尽管如此，几何上的教训是有用的：",[5281,24421,24422,24425,24428],{},[4063,24423,24424],{},"高维中，绝大多数临界点是鞍点，其负特征值方向提供了逃离的「出口」（负曲率方向）。",[4063,24426,24427],{},"因此，优化卡住的主因不是「掉进局部极小值的陷阱」，而是在高原或鞍点附近「停滞」。",[4063,24429,24430],{},"算法如 SGD（随机梯度下降）自带的噪声，恰好能高效地把优化过程推离这些鞍点。而二阶算法（如牛顿法）反而需要谨慎处理负曲率方向，否则容易被带偏。",[11,24432,24433],{},"最后提醒一点：上述结论适用于随机、无结构化的景观。在深度学习中，由于对称性（如神经元置换）和平坦极小值的存在，景观结构更加复杂，但「鞍点主导」这一高维几何直觉依然成立。",[21,24435,24436],{"id":24436},"结论",[11,24438,24439,24467,24468,24496],{},[32,24440,24442,24455],{"className":24441,"translate":36},[35],[32,24443,24445],{"className":24444},[40],[42,24446,24447],{"xmlns":44},[46,24448,24449,24453],{},[49,24450,24451],{},[57,24452,5322],{},[71,24454,5322],{"encoding":73},[32,24456,24458],{"className":24457,"ariaHidden":62},[78],[32,24459,24461,24464],{"className":24460},[82],[32,24462],{"className":24463,"style":212},[86],[32,24465,5322],{"className":24466,"style":1530},[95,96]," 维中的非退化局部极小值必须通过 ",[32,24469,24471,24484],{"className":24470,"translate":36},[35],[32,24472,24474],{"className":24473},[40],[42,24475,24476],{"xmlns":44},[46,24477,24478,24482],{},[49,24479,24480],{},[57,24481,5322],{},[71,24483,5322],{"encoding":73},[32,24485,24487],{"className":24486,"ariaHidden":62},[78],[32,24488,24490,24493],{"className":24489},[82],[32,24491],{"className":24492,"style":212},[86],[32,24494,5322],{"className":24495,"style":1530},[95,96]," 个同时成立的曲率检验——在临界点处，每个 Hessian 特征值都必须为正。鞍点只需要正负号的混合。随机矩阵理论表明，大型中心化对称矩阵天然具有这样的混合：它们的谱散布在零附近，有大量正、负特征值。「所有特征值都落在正侧」这一事件是一个大偏差。",[11,24498,24499,24500,24528,24529,24557],{},"这解释了高维的直觉。临界点本身是光滑景观中体积为零的特征，但在临界点之中，决定性的问题是 Hessian 的符号。在随机模型中，典型的符号其指数靠近中间，而不是极端：非退化局部极小值指数为 0，非退化局部极大值指数为 ",[32,24501,24503,24516],{"className":24502,"translate":36},[35],[32,24504,24506],{"className":24505},[40],[42,24507,24508],{"xmlns":44},[46,24509,24510,24514],{},[49,24511,24512],{},[57,24513,5322],{},[71,24515,5322],{"encoding":73},[32,24517,24519],{"className":24518,"ariaHidden":62},[78],[32,24520,24522,24525],{"className":24521},[82],[32,24523],{"className":24524,"style":212},[86],[32,24526,5322],{"className":24527,"style":1530},[95,96],"，鞍点占据两者之间的所有指数。随着 ",[32,24530,24532,24545],{"className":24531,"translate":36},[35],[32,24533,24535],{"className":24534},[40],[42,24536,24537],{"xmlns":44},[46,24538,24539,24543],{},[49,24540,24541],{},[57,24542,5322],{},[71,24544,5322],{"encoding":73},[32,24546,24548],{"className":24547,"ariaHidden":62},[78],[32,24549,24551,24554],{"className":24550},[82],[32,24552],{"className":24553,"style":212},[86],[32,24555,5322],{"className":24556,"style":1530},[95,96]," 增长，中间压倒了极端。",[11,24559,24560],{},"因此，最好把这一观点理解为一条稳健的概率原则，而非关于每个高维函数的定理：",[3222,24562,24563],{},[11,24564,24565],{},"在高维随机景观中，指数为零的 Morse 临界点是例外，而鞍点才是默认。",[11,24567,24568],{},"这正是深度学习等现代优化能够成功的理论基石之一：绝大多数临界点是鞍点，其负曲率方向提供了逃离的出口。带噪声的一阶方法能高效地把优化过程推开这些鞍点。",[24570,24571,24572],"h4",{"id":24572},"参考",[11,24574,24575,24576,24582],{},"参考来源：",[2104,24577,24581],{"href":24578,"rel":24579},"https:\u002F\u002Fwww.grantstenger.com\u002Flocal-minima",[24580],"nofollow","《Why Local Minima Are Rare in High-Dimensional Landscapes》","，Grant Stenger.",{"title":5247,"searchDepth":5248,"depth":5248,"links":24584},[24585,24586,24587,24588,24589,24595,24596,24599,24600,24601],{"id":5301,"depth":5248,"text":5301},{"id":5375,"depth":5248,"text":5376},{"id":12330,"depth":5248,"text":12330},{"id":13760,"depth":5248,"text":13760},{"id":16313,"depth":5248,"text":16313,"children":24590},[24591,24593,24594],{"id":16323,"depth":24592,"text":16323},3,{"id":16705,"depth":24592,"text":16706},{"id":18275,"depth":24592,"text":18275},{"id":19703,"depth":5248,"text":19704},{"id":21646,"depth":5248,"text":21647,"children":24597},[24598],{"id":22652,"depth":24592,"text":22653},{"id":24031,"depth":5248,"text":24031},{"id":24413,"depth":5248,"text":24413},{"id":24436,"depth":5248,"text":24436},"兰州",{},"\u002Fblog\u002F2026\u002F2026-08-22-are-local-minima-rare-in-high-dimensions",{"title":5271,"description":5276},"blog\u002F2026\u002F2026-08-22-are-local-minima-rare-in-high-dimensions","高维空间中的局部极小值极为罕见——因为一个真正的局部极小值要求所有方向都向上弯曲，也就是 Hessian 矩阵的所有特征值都必须为正。而高维空间里，随机的临界点几乎总有某个方向可以向下走，也就是鞍点。这正是为什么神经网络训练更像是「逃离鞍点」的游戏，而不是「跳出局部最优」的游戏。",[5266],"2026-08-22T14:32:17+08:00","T8eGeIi0CA1m4XaOFzLI17zaHqy_AQWibRj-YAZ432c",{"id":24612,"title":24613,"body":24614,"description":24618,"draft":5256,"enableComment":5257,"extension":5258,"image":5247,"important":5256,"location":24870,"meta":24871,"navigation":5257,"ogImage":5259,"path":24872,"seo":24873,"stem":24874,"summary":24875,"tags":24876,"time":24878,"__hash__":24879},"blog\u002Fblog\u002F2026\u002F2026-08-14-demystification-research.md","祛魅科研，每个研究生的必修课",{"type":8,"value":24615,"toc":24862},[24616,24619,24622,24625,24628,24631,24634,24643,24646,24649,24652,24655,24658,24661,24664,24667,24670,24673,24677,24680,24683,24686,24693,24700,24703,24706,24709,24712,24715,24721,24724,24727,24730,24733,24738,24741,24744,24747,24750,24755,24758,24761,24768,24771,24774,24777,24780,24783,24786,24792,24795,24798,24801,24804,24807,24810,24813,24816,24819,24826,24830,24836,24841,24844,24847,24850,24853,24856,24859],[11,24617,24618],{},"从 2025 年开始入学读研，到现在已经有一个年头了，不知不觉已经研二了。在过去的一年的研究生学习中，我发现我的心态、世界观经历了一次完整的、打击性的重塑。如果说之前我的心态是中规中矩，那么现在则是满心失落。自从经历过完整的科研训练，调研课题、找 idea，做实验、处理数据、撰写论文、修改文章、等投稿、写 Rebuttal、拒稿再转投、论文改版等一系列流程完整经历过后，只剩下满心苦涩。一度怀疑，我们搞的这个所谓的「科研」的价值和意义。😇",[11,24620,24621],{},"因为笔者读的研究生是计算机工程类专业，对于其他理工科领域、文科领域，笔者不是很熟悉，所以以下观点体验仅适用于该专业。",[21,24623,24624],{"id":24624},"草台班子",[11,24626,24627],{},"第一个对我的改变就是对学术成果的怀疑。学术圈的科研，其实并没有那么高端，可以说是草台班子。",[11,24629,24630],{},"在读研之前，我一直认为学术论文其实是非常高端的东西，至少是可信度很高的参考。但是读研之后，这个想法被改变了：绝大多数论文，他们的观点、数据、实验，看看就好，不必认真。",[11,24632,24633],{},"一篇论文的结论，往往只有作者自己的机器上成立。审稿人不会复现，编辑不会复现，引用者也不会复现。大家都默认了这个不可验证的游戏潜规则。论文并不是经过验证的知识，而是一个经过美化的故事。",[11,24635,24636,24637,24642],{},"有人专门对被 ICML 收录的 Oral 论文进行了复现，并给出了",[2104,24638,24641],{"href":24639,"rel":24640},"https:\u002F\u002Fx.com\u002FChenhaoTan\u002Fstatus\u002F2079969545629118737",[24580],"论文的报告信息","。结果发现，在 105 篇完整的复现论文中，只有 27 篇复现了超过 40% 的声称效果，剩余的几乎无法复现，要么效果不达标。原因包括代码无法运行、文件缺失以及模型已弃用等等。",[11,24644,24645],{},"复现困难确实是机器学习领域长期存在的问题。因为，CUDA 版本、显卡型号、CPU 型号、PyTorch 框架版本，可能都有大量隐蔽的细节差异，如果再组合起来，那么效果就完全无法验证。实验中的很多关键 Hack（比如特定的随机种子、数据预处理的微妙顺序、超参数的精细调节）可能只存在于作者的实验笔记或潜意识里。这些隐性经验很难通过文字传递，导致复现者像在黑暗中摸索。",[11,24647,24648],{},"甚至如果作者的代码里有 Bug，也很可能会造成完全错误的结论。我曾经跑过前人的基线实验，发现一篇论文，他的某个指标又非常虚高。最后才发现，作者对数据集根本没有清洗干净，也就是说，这份成绩是假的。",[11,24650,24651],{},"但问题是，这个论文已经被大量后人引用且作为基线了，后人想要再投稿，那么他的成绩必须要比这个假成绩更高，但是你根本比不过这假成绩，怎么办呢？你只能偷偷摸摸用点魔法手段，万不可认真，否则反而会显得像是你做错了，审稿人不接受。后来的研究者反而成了受害者。这不就是难为后人么？",[11,24653,24654],{},"所以，后人也只能不得不逼着也参与这种数字游戏，偷偷使用手段。结果就是，学术论文逐渐沦为一场“数字通胀”。每一篇新论文都在前人的数字泡沫上再吹一层泡沫，直到某个数据集上的准确率达到 99.9%，然后这个方向就“死掉”了——因为没人能再超越了，而大家都知道那个 99.9% 是假的，但谁都不想捅破。",[11,24656,24657],{},"这已经超出了学术不端的范畴，演变成了一种系统性的「囚徒困境」，很遗憾，这种困境，在学术圈里几乎无解。",[11,24659,24660],{},"无法复现不一定是作者故意有造假意向，而是因为深度学习本身就是一个巨大的黑盒，可验证性、可解释性非常差。很多错误，作者不一定能及时发现，审稿人也不一定能发现，于是，“只要能 work 就能发文”。当审稿人面对一篇充满“惊艳”实验数据的论文时，他既没有算力去复现，也没有理论工具去证伪其内部的混沌过程。于是，审稿的标准退化为：“只要故事逻辑自洽，且结果看起来比 SOTA（当前最优）高，我就信。”这就给出了巨大的浑水摸鱼的空间。",[11,24662,24663],{},"这种无法复现的现象，也会给自己带来很多的麻烦：如果你的文章需要依赖前人的工作，但是前人的工作本身就有问题，那么你的工作就几乎不可能顺利下去。你只能花大把时间来反思、调试，最后还很可能一无所获。",[11,24665,24666],{},"最后被浪费大把的时间和精力。这种消耗是精神凌迟。后续工作往往不得不替它买单。",[11,24668,24669],{},"有时候，我也甚至怀疑作者是故意在开源里缺斤少两，删减关键代码和文件或者埋入 Bug，或者论文被 Accept 后立即下架数据集，极力避免复现。😇要知道，审稿人通常只有 2-3 位，他们在几周内免费审稿，不可能复现你的实验，也无法验证你的原始数据。他们主要检查的是逻辑是否顺畅和方法是否看起来合理，而不是结论是否绝对正确。",[11,24671,24672],{},"学术论文它的首要目标是发表，而不是传世。为了发表，作者必须讲一个完整且自洽的故事。他们会突出最漂亮的数据，而弱化不支持的“噪音”结果。所以，他们会在讨论部分夹带私货，把故事讲得更有吸引力。它也确实是痛苦的。因为这意味着你失去了一个可以无条件信任的知识权威。",[21,24674,24676],{"id":24675},"ai-审稿危机","AI 审稿危机",[11,24678,24679],{},"读研起到现在，我已经投稿了三篇文章，这点我是有亲身体会的。",[11,24681,24682],{},"另一种尴尬是 AI 时代的审稿危机。AI 时代的恶果就是，论文灌水越来越容易。往常，写一篇论文很可能需要一学期、大半年的时间，要辛辛苦苦做实验、分析结果、画图表、斟酌写作。但是现在不一样了，利用自动化学术 Agent 如 AutoResearch，只要输入合适的提示词和模糊的想法再交给 Agent，它两天时间就可以编出一篇像模像样的论文，自动编写程序做实验，自动画图表，一个人一个月就能编写出 10 篇论文，直接投到顶会轰炸。",[11,24684,24685],{},"AAAI 在 2026 年收到了五万多份取号，几乎是 2025 年的两倍。而前年 2024 年才不过九千多份，短短两年时间就增长了五倍！其实不止 AAAI，其他的计算机会议的投稿量也是几乎翻倍增长。假设每篇论文标准配备 3 名审稿人，组委会将需要管理 120,000 份独立审稿意见：",[11,24687,24688],{},[24689,24690],"img",{"alt":24691,"src":24692},"计算机顶会近五年投稿量。自从 2025 年后，大多数投稿量都是翻倍式增长","https:\u002F\u002Fimage-assets.dreams.plus\u002F202608111219721.jpg",[11,24694,24695,24696,24699],{},"这就会导致一个后果：劣币驱逐良币。审稿体系正在加速崩溃。就算你辛辛苦苦完成了心血，设计实验，写好文章，",[1364,24697,24698],{},"审稿人也几乎不会认真看你的文章，而是直接交给 LLM 敷衍。"," 因为审稿人压根无法区分哪篇论文是 AI 写得，哪篇不是。更糟糕的是，学术领域是高度分化的，审稿人极有可能也不熟悉、也不理解你的研究领域。所以，他也只能同样一股脑交给 AI 审稿。",[11,24701,24702],{},"在这种情况下，灌水零成本，中间大量灌水的作者和敷衍的审稿人占多数。认真的审稿人和认真的作者只能被动深受其害。受害的永远只是认真搞学术的你。在这种情况下，审稿工作还能正常进行下去吗？",[11,24704,24705],{},"而 LLM 审稿本身就有很大的问题，AI 并不会理解你的文章，LLM 的幻觉问题众所周知。它会用非常刁钻的角度在你的文章里挑刺，哪怕是没有问题也会制造出一些问题。这是因为 LLM 在训练、SFT 偏好微调的时候，天生就倾向于给你打低分。LLM 并不真正拥有论文作者的研究上下文。它尤其容易犯一种很危险的错误：把“我没理解”转换成“作者的方法存在问题”。",[11,24707,24708],{},"LLM 模型学习到的统计规律是：“审稿”这个动作的语义空间，几乎完全由“批评性词汇”构成。因此，即使一篇论文毫无瑕疵，模型根据概率分布生成的“审稿风格”文本，天然就带有负面倾向。SFT 微调的时候，宽松的论文意见并不受欢迎，为了要尽可能压榨 LLM 能力，在微调的时候会特意设计出非常严苛的训练案例。这种奖惩机制直接塑造了模型的“人格”：它必须“生产批评”来证明自己的价值。哪怕没有真实缺陷，它也会启动“防御性挑刺”模式，利用模式匹配强行构造出看似合理的问题。",[11,24710,24711],{},"我的文章被拒稿过一次，三个 Reviewer 里两人给出的意见，明显就是 AI 复制粘贴的。😅 但是你不能抗议，你还要捏着鼻子忍着恶心，在 rebuttal 里面假模假样地感谢审稿人，再假模假样地写 rebuttal，跪求他赏赐一口饭吃。😅",[11,24713,24714],{},"审稿体系存在一个非常明显的权力不对称。Reviewer 可以随意评价“The novelty is insufficient.”而作者很难说：“你根本没读懂我的论文。”因为作者没有证据。另外，Reviewer 可以洋洋洒洒写出几千字的审稿意见，没有字数限制，而作者给的 Rebuttal 却严格限制在几百字以内。这也是最恶心的一点😅",[11,24716,24717],{},[24689,24718],{"alt":24719,"src":24720},"如图，是几个 LLM 在审稿 ArXiv 计算机学科论文时给出的平均得分（10 分制）","https:\u002F\u002Fimage-assets.dreams.plus\u002F202608111210591.png",[11,24722,24723],{},"试过把 21-23 年，GPT 出来以前的三大顶会里，已经发表收录的论文随机爬下来，投给 AI 审，50 篇里 30 多个 weak reject，10 多个 weak accept，剩下的全部都是 reject。一篇 accept 都没有。后 AI 时代所谓的顶会论文已经变成笑话哩。",[11,24725,24726],{},"审稿人因为反驳文太长而疲惫，作者因为审稿人固执而绝望。当作者知道审稿人是 AI，审稿人知道作者用了 AI 时，作者 - 审稿人 - 编辑三方之间的这场学术对话，就彻底沦为了一场荒诞剧。这就造成了非常滑稽的景象，AI 写 AI 审：",[11,24728,24729],{},"用 AI 写论文、写代码，再用 AI 初审，根据 AI 的意见修改，完成初稿。审稿人拿到初稿，再交给 AI 审稿，用 AI 的意见给出 review 意见，作者拿到 AI 写的意见后再交给 AI 写 rebuttal。审稿人再用 AI 根据 rebuttal 做出决定。堪称学术出版领域正在逼近的赛博朋克式奇观。",[11,24731,24732],{},"在三方中，编辑\u002F主席的地位最为尴尬。当所有文字意见如审稿、反驳都由 AI 生成时，编辑失去了判断学术创新性的任何抓手。他唯一能做的，就是检查流程是否走完：AI 是否提了 3 个问题？作者是否逐条回复？回复长度是否达标？只要格式合规，就可以按下接受键。学术判断的权力，在此刻已完全让渡给了硅基算法。",[11,24734,24735],{},[1364,24736,24737],{},"所以，能否发顶会、顶刊，实际上已经越来越像摸彩票中奖的运气、概率问题，跟你的文章质量、工作效果已经几乎没有什么关系了。",[21,24739,24740],{"id":24740},"手艺人",[11,24742,24743],{},"硕博生这个身份曾经是不少人引以为傲的根本。但是真正体验过他们的生活，也会发现他们本质上和流水线的螺丝工人相差无几，这种生活状态其实是非常压抑的，如果过这种日子，那只能用「熬」来形容。",[11,24745,24746],{},"硕博生本质上也是出卖高强度脑力劳动换取生存资格的劳动者。跟工地上抗水泥的农民工、顶着大太阳湿透衣服的清洁工没有本质区别。",[11,24748,24749],{},"对于理工科研究生，996 是常态，实验室的灯永远亮着。每天睁眼闭眼就是要面对屏幕上密密麻麻的实验数据、代码和仪器。高强度脑力劳动后，带给身心的除了疲劳还是疲劳。日子久了，精气神会被消磨掉，慢慢丧失对这个世界的好奇心和一切欲望，不想谈恋爱，不想出去旅游，哪怕是手里的游戏，日子久了玩起来也没意思。",[3222,24751,24752],{},[11,24753,24754],{},"到周末后，只想在床上躺着，啥也不干。脑一旦被单一的高强度任务长期占据，负责发散思维、感受情绪、产生欲望的脑区就会被持续抑制。只像一个漂浮在数据海洋里的意识，拖着一具沉重而麻木的肉体，犹如冢中枯骨而已。",[11,24756,24757],{},"这是最致命的。当你看清你所做的研究可能只是学术游戏里的一个废棋，对外部世界毫无影响时，熬就变成了一种精神上的凌迟。不禁会问，「我做的这一切，受了那么多折磨，到底有什么意义？」",[11,24759,24760],{},"工人进厂时，起码能自知之明、清醒地知道这是出卖身体，用劳动换生存。但硕博生被社会、被家人、被曾经的自己赋予了天子骄子、知识精英的光环。当现实变成日复一日地跑数据、伺候仪器时，日子久了你会有这样一种感觉：自己并不是一个人，而是一个庞大机器中的一颗零件。",[11,24762,24763,24764,24767],{},"硕士生（Master）也不是大师。博士生也不博学。他的知识广度，甚至还可能不如一个高中生。",[1364,24765,24766],{},"读研后，你的视角只会被限制在高度狭窄且专业的小领域里。"," 在自己的学术孤岛里自娱自乐。",[11,24769,24770],{},"中世纪的经院哲学家热衷于争论「一个针尖上能站几个天使」，而今天的学术圈，大量精力被消耗在维护主流范式上。正如前文所提到的，学术论文本身也是高度固定化的、范式的。有时候你不会感觉自己「在创造一个想法」，而是「完成一个八股文」。",[11,24772,24773],{},"如果要比喻硕博生这个群体的身份，它更像是一个高度程序化的手艺人、螺丝工。搞科研的流程本身就是高度固定化、流水线化的。调研、idea、实验、写作、改稿、Rebuttal。这一圈下来，恭喜你，你已经是一名合格的劳工！",[21,24775,24776],{"id":24776},"学术圈",[11,24778,24779],{},"曾经对学术圈的浪漫想象，至少代表了知识的前沿、先进的生产力和思想文化，具有进步性。我其实对科研领域内的学者，高校里的教授、教师等群体，长期是存在敬仰的，认为他们或多或少都是代表了人类开拓认知知识、征服星辰大海的一批人，在心里也会敬三分。",[11,24781,24782],{},"现在才知道，学术圈其实是高度封闭的。因为知识本身就有很强的入门壁垒，当人类认知突破到一定边界时，工具、术语和范式的复杂度必然形成门槛，学术方向往往会走向高度分化、隔行如隔山的细碎分支，各个分支又会高度壁垒。这就造成了学术圈的高度封闭性。",[11,24784,24785],{},"其实，越是高度封闭的圈子，越容易产生高度固化的权力结构，行事作风越是封建化、越是讲政治。😅其实现在的学术圈其实跟欧洲中世纪的经学院教派之争、西藏喇嘛们的辩经并没有什么区别。那些专家，领域学者，头衔看得是挺唬人，但做的无非在极度封闭的圈子里，用只有内部人能懂的黑话，争论着对外部世界影响甚微的问题，而决定胜负的常常不是真理，而是资历、人脉和对经典的诠释权。",[11,24787,24788,24791],{},[1364,24789,24790],{},"学术圈，其实比大多数人想象的还要小。"," 因为现代学科已经进入高度分化、高度专业化的时代了。如果细分下去，全中国十四亿人里，同一个领域的研究同行很可能不超过百人，甚至十几人。在一个村落里，任何小动作，全落在这几十个低头不见抬头见的人手里。这里没法对事不对人，因为在结构上，所有的事，最终都是人的事。",[11,24793,24794],{},"然而，现在的学术圈是零和博弈，资源是极其有限的。这个在申请基金、文章版面、学术交流等等活动中，只要有人胜出，必然会有人落选。这也就意味着，你的小圈子里，可能到处都在「树敌」。这并不意味着本人有错，而是你的存在，本身就是威胁。",[11,24796,24797],{},"所以遇到同行暗中使绊子，也是常见的事。你的审稿人很可能跟你的导师有竞争或者过节，就直接轻松 Reject 你的心血。即使现在的审稿制度大多是双盲制，在一个领域只有几十上百人的圈子里，根本不存在真正的双向匿名。看研究问题、看方法、看引用的文献，审稿人闭着眼都能猜到这篇稿子出自哪个课题组。",[11,24799,24800],{},"对一个埋头苦干的学生来说，这是最深的打击。你相信公正，相信学术质量至上。然后，一堵由学派、人情、资源争夺构成的墙，悄无声息地挡在你面前，将你的心血轻松驳回。你甚至没有一个明确的敌人去质问抗争，你甚至不知道你的敌人是谁，又得罪了谁，只剩下无尽的无力感和被戏弄的愤怒。这种被暗算的体验会深刻腐蚀对学术共同体的信任。",[11,24802,24803],{},"像中世纪的领主分封土地一样，大牛导师和顶尖实验室把持着顶级期刊的版面、重大项目的经费和学阀圈子的话语权。你想在他的领地上发文章，就得遵循他的范式、引他的文章、甚至拜他的码头。学术圈的游戏规则是，正确不等于接受。投稿像一场赌博，审稿人的口味、当期版面、甚至运气，都比你那篇精心打磨的论文权重更高。",[21,24805,24806],{"id":24806},"事业意义",[11,24808,24809],{},"虽然小时候有「长大要当科学家」这种理想，但是，个人认为「学术」这条路并不适合像我这样的平民子弟。没有充裕的家底和财力作为后盾，吃学术这碗饭，也是一种高风险职业。",[11,24811,24812],{},"我觉得那些在学术圈里搞研究的人，其实也挺可悲的。自己把大量的青春，时间，精力投入到自己的课题里，勉强能讨得经费，靠这个饭碗。因为，选择某个研究领域，在初期往往带有偶然性。但一旦投入，就成了无法回头的豪赌。赌的是这个方向在几十年内不被证伪、不被超越、不被认为是死胡同、不会没落。他用的是几十年的青春和精力做赌注，用最严谨的头脑，从事着一项本质上充满不确定性的高风险事业。",[11,24814,24815],{},"如果有人突然跳出来用新理论、新范式挑战他，或者推翻了他的观点课题，这无异于把他降维打击成了一块废品，弃之如敝履，这不是嘲讽，而是真实的悲剧。在高度职业化的学术圈里，一个人的身份、地位、自尊，都深深扎根于他那一亩三分地的研究课题。",[11,24817,24818],{},"当他的理论被推翻，在外人看来不过是一个观点被证伪。但对他而言，无异于整个学术人格被判处了死刑，在这个圈子里，会被迅速边缘化，从而判了死刑。他毕生构建的意义大厦，瞬间崩塌为一座废墟。这就是一种存在主义危机。",[11,24820,24821,24822,24825],{},"大多数普通人活下去本身就很难。因为 ",[1364,24823,24824],{},"他们的人生，还有其他责任"," 。若压制住七情六欲，寒窗几十年，去碰学术圈，也未免太委屈了。如果一个人 25 岁读博士，30 岁左右博士毕业，然后经历博士后、非升即走、青年项目竞争，他可能在四十岁前都处于高度竞争状态。而同期进入工业界的人，可能已经积累了财富、住房和职业资本。",[21,24827,24829],{"id":24828},"破局功利化读研","破局：功利化读研",[11,24831,24832,24833,3568],{},"在中国，虽然知识分子往往被冠以社会期待的光环，但是这也是一种负担和枷锁。请记住：我们是普通人尤其是出身平民家庭，我们首要目标是生存。在生存生计成为问题的时候，我们 ",[1364,24834,24835],{},"没有义务背负太多期待",[3222,24837,24838],{},[11,24839,24840],{},"沧浪之水清兮，可以濯吾缨；沧浪之水浊兮，可以濯吾足。",[11,24842,24843],{},"请卸下你的「学术羞耻感」。把研究生学历视为一份职业准入资格证，而非学术朝圣。学术职业是一种高风险选择，而不是所有人都必须承担的使命。",[11,24845,24846],{},"如何破局——功利化读研不失为一种出路。研究生必须思考：我读研的目的是为了什么？获得更好的就业门槛，暂时避开竞争激烈的就业市场，获得更多选择权。这完全是一种合理的人生规划。",[11,24848,24849],{},"首要的当然是毕业、混文凭，获得硕博的身份————这也是最现实、也最重要的一条。所以，在读研前期，你的一切目标是，必须以尽可能短的时间，完成最低毕业要求。大量水论文、蹭项目就足矣，不需要尽善尽美。科研嘛，也就那样，随便搞搞就行。",[11,24851,24852],{},"这是功利化读研真正的溢价所在。既然科研只求及格，那你必须把多出来的精力毫无愧疚地投入生存技能的构建。",[11,24854,24855],{},"在当前的环境下，「包装」比「做事」更重要，「数量」比「质量」更重要。不要把自己的全部人生价值绑定在学术成果上。科研如此，创业如此，艺术如此，很多长期主义事业都是如此。我不需要证明自己是英雄，我只需要把自己的人生过好。知识值得敬畏，但人的生命也值得敬畏。学术可以是人生的一部分，但不必成为人生的全部。",[11,24857,24858],{},"希望研究生们，不必神化科研、不必自我内耗、不必绑定学术理想，认清行业真相后，依然可以清醒活着、务实成长。",[11,24860,24861],{},"到最后，这种「虽千万人，吾往矣」，本身就有壮士断腕的秋风式悲凉，不是吗？😮‍💨",{"title":5247,"searchDepth":5248,"depth":5248,"links":24863},[24864,24865,24866,24867,24868,24869],{"id":24624,"depth":5248,"text":24624},{"id":24675,"depth":5248,"text":24676},{"id":24740,"depth":5248,"text":24740},{"id":24776,"depth":5248,"text":24776},{"id":24806,"depth":5248,"text":24806},{"id":24828,"depth":5248,"text":24829},"河南郑州",{},"\u002Fblog\u002F2026\u002F2026-08-14-demystification-research",{"title":24613,"description":24618},"blog\u002F2026\u002F2026-08-14-demystification-research","自 2025 年入学至今，一年光阴悄然而逝，我已步入研二。回望这一年的研究生生活，我的心态与世界观经历了一场彻底而沉重的重塑——曾经的从容平实，如今已被挥之不去的失落感取代。",[24877],"thoughts","2026-08-14T00:00:00+08:00","utt_zVmdYpEoW-2HkgNeH6tsTcym6PstUAxslVH-wUE",{"left":24881,"top":24881,"width":24882,"height":24882,"rotate":24881,"vFlip":5256,"hFlip":5256,"body":24883},0,24,"\u003Cpath fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\" d=\"M12 7v5l3 3m6-3a9 9 0 1 1-18 0a9 9 0 0 1 18 0\"\u002F>",{"left":24881,"top":24881,"width":24882,"height":24882,"rotate":24881,"vFlip":5256,"hFlip":5256,"body":24885},"\u003Cg fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\">\u003Cpath d=\"m4.748 7.645l-.331 3.644c-.05.54-.074.811-.03 1.07a2 2 0 0 0 .238.655c.131.228.325.422.711.808l5.176 5.176c.787.787 1.18 1.18 1.636 1.328c.402.131.835.131 1.237 0c.456-.148.853-.544 1.645-1.336l3.96-3.96c.792-.792 1.187-1.188 1.336-1.644a2 2 0 0 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