[{"data":1,"prerenderedAt":72743},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0013-satisfiability-modulo-theories":3,"i-ci:tag":-1,"i-mingcute:time-line":-1},{"post":4,"nextPost":19959,"prevPost":72554},{"id":5,"title":6,"body":7,"description":13,"draft":19946,"enableCodeHighlight":19947,"enableCodeHightlight":19946,"enableComment":19947,"enableD2":19947,"enableKatex":19947,"extension":19948,"image":23,"important":19946,"location":19949,"meta":19950,"navigation":19947,"ogImage":19949,"path":19951,"seo":19952,"stem":19953,"summary":19954,"tags":19955,"time":19957,"__hash__":19958},"blog\u002Fblog\u002F2026\u002F0013-satisfiability-modulo-theories.md","可满足性模理论 (Satisfiability Modulo Theories)",{"type":8,"value":9,"toc":19907},"minimark",[10,14,17,171,186,189,374,377,385,407,413,418,423,461,464,467,616,680,851,855,858,1189,1752,1755,2479,2847,2906,2917,2924,2928,2931,3144,3150,3182,3185,3188,3192,3196,3203,3249,3379,4011,4429,4432,4435,4463,4571,4790,4793,4797,4800,4803,4998,5001,5387,5578,5581,5584,5588,5591,5594,5765,5768,6196,6457,6593,6600,6604,6610,6613,6650,6653,7269,7272,7275,7279,7282,7296,7302,7306,7310,7317,7320,7419,7454,7473,7476,7480,7486,7670,7673,7843,7846,8286,8386,8393,8525,8529,8535,8542,8545,8715,9160,9163,9166,9351,9452,9458,9648,9652,9658,9737,9743,9747,9754,9757,9770,9773,9777,9781,9792,9799,9802,9823,9826,9830,9833,9836,10052,10055,10309,10312,10408,10615,10622,10778,10858,11167,11171,11174,11182,11185,11205,11208,11212,11248,11251,11256,11259,11400,11403,11629,11850,11853,11856,11861,11997,11999,12196,12321,12726,12729,12731,12736,12739,12938,12941,12948,13212,13355,13357,13365,13376,13378,13679,13899,13902,13908,13912,13919,13922,14077,14080,14309,14312,14423,14484,14487,14491,14494,14709,15164,15292,15448,15543,15666,15669,15676,15679,15716,15720,15726,15754,15757,15760,15764,15768,15771,15777,15780,16210,16296,16305,16311,16315,16318,16407,16424,16435,16566,16569,16761,16764,16796,16799,16805,16941,16944,16984,16987,17180,17186,17200,17238,17255,17259,17352,17430,17433,17541,17544,17935,18000,18003,18045,18048,18054,18057,18065,18261,18347,18357,18361,18364,18429,18440,18531,18537,18684,18690,18693,18759,18849,18906,18916,18923,18927,19001,19004,19113,19123,19126,19166,19169,19332,19337,19348,19668,19759,19763,19766,19773,19779,19786,19790,19893,19903],[11,12,13],"p",{},"SMT 的出现，本质上是为了解决一个核心矛盾：很多实际问题用纯命题逻辑（SAT）表达时，要么根本无法表达，要么表达后规模爆炸；而用一般的一阶逻辑或高阶逻辑表达虽然自然，却无法自动高效求解。",[11,15,16],{},"看一段再普通不过的代码：",[18,19,24],"pre",{"className":20,"code":21,"language":22,"meta":23,"style":23},"language-c shiki shiki-themes github-light github-light github-dark","void f(int x, int y) {\n    int a[100] = {0};\n    a[0] = 5;\n    if (x + y > 10 && a[0] == 5) {\n        \u002F\u002F 目标分支\n    }\n}\n","c","",[25,26,27,62,93,113,152,159,165],"code",{"__ignoreMap":23},[28,29,32,36,40,44,47,51,54,56,59],"span",{"class":30,"line":31},"line",1,[28,33,35],{"class":34},"s8jYJ","void",[28,37,39],{"class":38},"snPdu"," f",[28,41,43],{"class":42},"sxrX7","(",[28,45,46],{"class":34},"int",[28,48,50],{"class":49},"sP4rz"," x",[28,52,53],{"class":42},", ",[28,55,46],{"class":34},[28,57,58],{"class":49}," y",[28,60,61],{"class":42},") {\n",[28,63,65,68,71,74,78,81,84,87,90],{"class":30,"line":64},2,[28,66,67],{"class":34},"    int",[28,69,70],{"class":49}," a",[28,72,73],{"class":42},"[",[28,75,77],{"class":76},"sBjJW","100",[28,79,80],{"class":42},"] ",[28,82,83],{"class":34},"=",[28,85,86],{"class":42}," {",[28,88,89],{"class":76},"0",[28,91,92],{"class":42},"};\n",[28,94,96,99,101,103,105,107,110],{"class":30,"line":95},3,[28,97,98],{"class":49},"    a",[28,100,73],{"class":42},[28,102,89],{"class":76},[28,104,80],{"class":42},[28,106,83],{"class":34},[28,108,109],{"class":76}," 5",[28,111,112],{"class":42},";\n",[28,114,116,119,122,125,128,131,134,137,139,141,143,145,148,150],{"class":30,"line":115},4,[28,117,118],{"class":34},"    if",[28,120,121],{"class":42}," (x ",[28,123,124],{"class":34},"+",[28,126,127],{"class":42}," y ",[28,129,130],{"class":34},">",[28,132,133],{"class":76}," 10",[28,135,136],{"class":34}," &&",[28,138,70],{"class":49},[28,140,73],{"class":42},[28,142,89],{"class":76},[28,144,80],{"class":42},[28,146,147],{"class":34},"==",[28,149,109],{"class":76},[28,151,61],{"class":42},[28,153,155],{"class":30,"line":154},5,[28,156,158],{"class":157},"sCsY4","        \u002F\u002F 目标分支\n",[28,160,162],{"class":30,"line":161},6,[28,163,164],{"class":42},"    }\n",[28,166,168],{"class":30,"line":167},7,[28,169,170],{"class":42},"}\n",[11,172,173,174,177,178,181,182],{},"假设 ",[25,175,176],{},"x","、",[25,179,180],{},"y"," 是来自外部的整数。我们想知道：",[183,184,185],"strong",{},"有没有一组输入，能走进这个分支？",[11,187,188],{},"沿着这条路径，把进入分支需要满足的条件收集起来：",[28,190,194],{"className":191,"translate":193},[192],"katex-display","no",[28,195,198,257],{"className":196,"translate":193},[197],"katex",[28,199,202],{"className":200},[201],"katex-mathml",[203,204,207],"math",{"xmlns":205,"display":206},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[208,209,210,252],"semantics",{},[211,212,213,216,219,221,223,227,231,234,236,239,242,244,247,249],"mrow",{},[214,215,176],"mi",{},[217,218,124],"mo",{},[214,220,180],{},[217,222,130],{},[224,225,226],"mn",{},"10",[228,229,230],"mtext",{},"  ",[217,232,233],{},"∧",[228,235,230],{},[214,237,238],{},"a",[217,240,73],{"stretchy":241},"false",[224,243,89],{},[217,245,246],{"stretchy":241},"]",[217,248,83],{},[224,250,251],{},"5",[253,254,256],"annotation",{"encoding":255},"application\u002Fx-tex","x + y > 10 \\;\\wedge\\; a[0] = 5",[28,258,262,288,310,335,365],{"className":259,"ariaHidden":261},[260],"katex-html","true",[28,263,266,271,276,281,285],{"className":264},[265],"base",[28,267],{"className":268,"style":270},[269],"strut","height:0.6667em;vertical-align:-0.0833em;",[28,272,176],{"className":273},[274,275],"mord","mathnormal",[28,277],{"className":278,"style":280},[279],"mspace","margin-right:0.2222em;",[28,282,124],{"className":283},[284],"mbin",[28,286],{"className":287,"style":280},[279],[28,289,291,295,299,303,307],{"className":290},[265],[28,292],{"className":293,"style":294},[269],"height:0.7335em;vertical-align:-0.1944em;",[28,296,180],{"className":297,"style":298},[274,275],"margin-right:0.0359em;",[28,300],{"className":301,"style":302},[279],"margin-right:0.2778em;",[28,304,130],{"className":305},[306],"mrel",[28,308],{"className":309,"style":302},[279],[28,311,313,317,320,323,326,329,332],{"className":312},[265],[28,314],{"className":315,"style":316},[269],"height:0.6444em;",[28,318,226],{"className":319},[274],[28,321],{"className":322,"style":302},[279],[28,324],{"className":325,"style":280},[279],[28,327,233],{"className":328},[284],[28,330],{"className":331,"style":302},[279],[28,333],{"className":334,"style":280},[279],[28,336,338,342,345,349,352,356,359,362],{"className":337},[265],[28,339],{"className":340,"style":341},[269],"height:1em;vertical-align:-0.25em;",[28,343,238],{"className":344},[274,275],[28,346,73],{"className":347},[348],"mopen",[28,350,89],{"className":351},[274],[28,353,246],{"className":354},[355],"mclose",[28,357],{"className":358,"style":302},[279],[28,360,83],{"className":361},[306],[28,363],{"className":364,"style":302},[279],[28,366,368,371],{"className":367},[265],[28,369],{"className":370,"style":316},[269],[28,372,251],{"className":373},[274],[11,375,376],{},"问题于是变成：这组约束有没有解？",[11,378,379,380,177,382,384],{},"纯 SAT 求解器也能做，但你得先把 ",[25,381,176],{},[25,383,180],{}," 各拆成 32 个布尔变量，把加法展开成加法器电路，把数组展开成 100 个元素，最后交给它几千条子句。它只认识真假，不认识整数，也不认识数组。",[11,386,387,388,391,392,395,396,399,400,402,403,406],{},"SMT 求解器则直接读原式：",[25,389,390],{},"x + y > 10"," 交给算术，",[25,393,394],{},"a[0] = 5"," 交给数组。算术说取 ",[25,397,398],{},"x = 6, y = 5"," 就行，数组说 ",[25,401,394],{}," 本来就成立。两边一对，返回 ",[183,404,405],{},"sat","，并给出这组解。",[11,408,409,410],{},"同一个问题，一个要把语义拆成门电路，一个能直接利用语义。",[183,411,412],{},"SMT 要做的，就是后者。",[414,415,417],"h2",{"id":416},"一smt-是什么","一、SMT 是什么",[419,420,422],"h3",{"id":421},"_11-定义","1.1 定义",[11,424,425,426],{},"把上面的过程抽象一下。",[183,427,428,429,460],{},"SMT 是「SAT + 背景理论」：在某个理论 ",[28,430,432,446],{"className":431,"translate":193},[197],[28,433,435],{"className":434},[201],[203,436,437],{"xmlns":205},[208,438,439,444],{},[211,440,441],{},[214,442,443],{},"T",[253,445,443],{"encoding":255},[28,447,449],{"className":448,"ariaHidden":261},[260],[28,450,452,456],{"className":451},[265],[28,453],{"className":454,"style":455},[269],"height:0.6833em;",[28,457,443],{"className":458,"style":459},[274,275],"margin-right:0.1389em;"," 规定的语义下，判定一个一阶逻辑公式是否存在满足它的赋值。",[11,462,463],{},"这里的关键词是「模理论」（modulo theories）。「模」可以理解为「相对于」：我们不再问公式在全部数学结构里是否成立，而是问它在某个特定理论——比如整数算术、位向量、数组——的解释下是否成立。",[11,465,466],{},"形式化一点，SMT 要回答的是：",[28,468,470],{"className":469,"translate":193},[192],[28,471,473,514],{"className":472,"translate":193},[197],[28,474,476],{"className":475},[201],[203,477,478],{"xmlns":205,"display":206},[208,479,480,511],{},[211,481,482,486,489,492,495,498,500,508],{},[214,483,485],{"mathvariant":484},"normal","∃",[228,487,488],{}," ",[214,490,491],{},"σ",[214,493,494],{"mathvariant":484},".",[279,496],{"width":497},"1em",[214,499,491],{},[501,502,503,506],"msub",{},[217,504,505],{},"⊨",[214,507,443],{},[214,509,510],{},"φ",[253,512,513],{"encoding":255},"\\exists\\, \\sigma.\\quad \\sigma \\models_T \\varphi",[28,515,517,606],{"className":516,"ariaHidden":261},[260],[28,518,520,524,527,531,534,537,541,544,547,603],{"className":519},[265],[28,521],{"className":522,"style":523},[269],"height:0.999em;vertical-align:-0.249em;",[28,525,485],{"className":526},[274],[28,528],{"className":529,"style":530},[279],"margin-right:0.1667em;",[28,532,491],{"className":533,"style":298},[274,275],[28,535,494],{"className":536},[274],[28,538],{"className":539,"style":540},[279],"margin-right:1em;",[28,542,491],{"className":543,"style":298},[274,275],[28,545],{"className":546,"style":302},[279],[28,548,550,553],{"className":549},[306],[28,551,505],{"className":552},[306],[28,554,557],{"className":555},[556],"msupsub",[28,558,562,594],{"className":559},[560,561],"vlist-t","vlist-t2",[28,563,566,589],{"className":564},[565],"vlist-r",[28,567,571],{"className":568,"style":570},[569],"vlist","height:0.3283em;",[28,572,574,579],{"style":573},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[28,575],{"className":576,"style":578},[577],"pstrut","height:2.7em;",[28,580,586],{"className":581},[582,583,584,585],"sizing","reset-size6","size3","mtight",[28,587,443],{"className":588,"style":459},[274,275,585],[28,590,593],{"className":591},[592],"vlist-s","​",[28,595,597],{"className":596},[565],[28,598,601],{"className":599,"style":600},[569],"height:0.15em;",[28,602],{},[28,604],{"className":605,"style":302},[279],[28,607,609,613],{"className":608},[265],[28,610],{"className":611,"style":612},[269],"height:0.625em;vertical-align:-0.1944em;",[28,614,510],{"className":615},[274,275],[11,617,618,619,649,650,679],{},"也就是：是否存在一个赋值 ",[28,620,622,636],{"className":621,"translate":193},[197],[28,623,625],{"className":624},[201],[203,626,627],{"xmlns":205},[208,628,629,633],{},[211,630,631],{},[214,632,491],{},[253,634,635],{"encoding":255},"\\sigma",[28,637,639],{"className":638,"ariaHidden":261},[260],[28,640,642,646],{"className":641},[265],[28,643],{"className":644,"style":645},[269],"height:0.4306em;",[28,647,491],{"className":648,"style":298},[274,275],"，把每个变量映射到理论域里的值，使得 ",[28,651,653,667],{"className":652,"translate":193},[197],[28,654,656],{"className":655},[201],[203,657,658],{"xmlns":205},[208,659,660,664],{},[211,661,662],{},[214,663,510],{},[253,665,666],{"encoding":255},"\\varphi",[28,668,670],{"className":669,"ariaHidden":261},[260],[28,671,673,676],{"className":672},[265],[28,674],{"className":675,"style":612},[269],[28,677,510],{"className":678},[274,275]," 在这个理论的所有公理下为真。",[11,681,682,683,755,756,821,822,850],{},"回到开头的例子：",[28,684,686,707],{"className":685,"translate":193},[197],[28,687,689],{"className":688},[201],[203,690,691],{"xmlns":205},[208,692,693,705],{},[211,694,695,697,699,701,703],{},[214,696,176],{},[217,698,124],{},[214,700,180],{},[217,702,130],{},[224,704,226],{},[253,706,390],{"encoding":255},[28,708,710,728,746],{"className":709,"ariaHidden":261},[260],[28,711,713,716,719,722,725],{"className":712},[265],[28,714],{"className":715,"style":270},[269],[28,717,176],{"className":718},[274,275],[28,720],{"className":721,"style":280},[279],[28,723,124],{"className":724},[284],[28,726],{"className":727,"style":280},[279],[28,729,731,734,737,740,743],{"className":730},[265],[28,732],{"className":733,"style":294},[269],[28,735,180],{"className":736,"style":298},[274,275],[28,738],{"className":739,"style":302},[279],[28,741,130],{"className":742},[306],[28,744],{"className":745,"style":302},[279],[28,747,749,752],{"className":748},[265],[28,750],{"className":751,"style":316},[269],[28,753,226],{"className":754},[274]," 是算术原子，",[28,757,759,782],{"className":758,"translate":193},[197],[28,760,762],{"className":761},[201],[203,763,764],{"xmlns":205},[208,765,766,780],{},[211,767,768,770,772,774,776,778],{},[214,769,238],{},[217,771,73],{"stretchy":241},[224,773,89],{},[217,775,246],{"stretchy":241},[217,777,83],{},[224,779,251],{},[253,781,394],{"encoding":255},[28,783,785,812],{"className":784,"ariaHidden":261},[260],[28,786,788,791,794,797,800,803,806,809],{"className":787},[265],[28,789],{"className":790,"style":341},[269],[28,792,238],{"className":793},[274,275],[28,795,73],{"className":796},[348],[28,798,89],{"className":799},[274],[28,801,246],{"className":802},[355],[28,804],{"className":805,"style":302},[279],[28,807,83],{"className":808},[306],[28,810],{"className":811,"style":302},[279],[28,813,815,818],{"className":814},[265],[28,816],{"className":817,"style":316},[269],[28,819,251],{"className":820},[274]," 是数组原子，理论 ",[28,823,825,838],{"className":824,"translate":193},[197],[28,826,828],{"className":827},[201],[203,829,830],{"xmlns":205},[208,831,832,836],{},[211,833,834],{},[214,835,443],{},[253,837,443],{"encoding":255},[28,839,841],{"className":840,"ariaHidden":261},[260],[28,842,844,847],{"className":843},[265],[28,845],{"className":846,"style":455},[269],[28,848,443],{"className":849,"style":459},[274,275]," 就是整数算术与数组的组合。SMT 要回答的，正是这组原子能不能同时为真。",[419,852,854],{"id":853},"_12-公式的样子","1.2 公式的样子",[11,856,857],{},"SMT 公式由布尔结构和理论原子组成：",[28,859,861],{"className":860,"translate":193},[192],[28,862,864,956],{"className":863,"translate":193},[197],[28,865,867],{"className":866},[201],[203,868,869],{"xmlns":205,"display":206},[208,870,871,953],{},[211,872,873,875,878,880,882,885,887,890,892,895,897,899,901,904,906,908,910,913,915,917,919,921,923,925,927,929,931,933,935,938,940,942,944,946,948,951],{},[214,874,510],{},[217,876,877],{},":",[217,879,877],{},[217,881,83],{},[214,883,884],{"mathvariant":484},"⊤",[228,886,230],{},[217,888,889],{},"∣",[228,891,230],{},[214,893,894],{"mathvariant":484},"⊥",[228,896,230],{},[217,898,889],{},[228,900,230],{},[214,902,903],{},"A",[228,905,230],{},[217,907,889],{},[228,909,230],{},[214,911,912],{"mathvariant":484},"¬",[214,914,510],{},[228,916,230],{},[217,918,889],{},[228,920,230],{},[214,922,510],{},[217,924,233],{},[214,926,510],{},[228,928,230],{},[217,930,889],{},[228,932,230],{},[214,934,510],{},[217,936,937],{},"∨",[214,939,510],{},[228,941,230],{},[217,943,889],{},[228,945,230],{},[214,947,510],{},[217,949,950],{},"→",[214,952,510],{},[253,954,955],{"encoding":255},"\\varphi ::= \\top \\;\\mid\\; \\bot \\;\\mid\\; A \\;\\mid\\; \\neg\\varphi \\;\\mid\\; \\varphi \\wedge \\varphi \\;\\mid\\; \\varphi \\vee \\varphi \\;\\mid\\; \\varphi \\rightarrow \\varphi",[28,957,959,978,1002,1026,1050,1077,1096,1120,1138,1162,1180],{"className":958,"ariaHidden":261},[260],[28,960,962,965,968,971,975],{"className":961},[265],[28,963],{"className":964,"style":612},[269],[28,966,510],{"className":967},[274,275],[28,969],{"className":970,"style":302},[279],[28,972,974],{"className":973},[306],"::=",[28,976],{"className":977,"style":302},[279],[28,979,981,984,987,990,993,996,999],{"className":980},[265],[28,982],{"className":983,"style":341},[269],[28,985,884],{"className":986},[274],[28,988],{"className":989,"style":302},[279],[28,991],{"className":992,"style":302},[279],[28,994,889],{"className":995},[306],[28,997],{"className":998,"style":302},[279],[28,1000],{"className":1001,"style":302},[279],[28,1003,1005,1008,1011,1014,1017,1020,1023],{"className":1004},[265],[28,1006],{"className":1007,"style":341},[269],[28,1009,894],{"className":1010},[274],[28,1012],{"className":1013,"style":302},[279],[28,1015],{"className":1016,"style":302},[279],[28,1018,889],{"className":1019},[306],[28,1021],{"className":1022,"style":302},[279],[28,1024],{"className":1025,"style":302},[279],[28,1027,1029,1032,1035,1038,1041,1044,1047],{"className":1028},[265],[28,1030],{"className":1031,"style":341},[269],[28,1033,903],{"className":1034},[274,275],[28,1036],{"className":1037,"style":302},[279],[28,1039],{"className":1040,"style":302},[279],[28,1042,889],{"className":1043},[306],[28,1045],{"className":1046,"style":302},[279],[28,1048],{"className":1049,"style":302},[279],[28,1051,1053,1056,1059,1062,1065,1068,1071,1074],{"className":1052},[265],[28,1054],{"className":1055,"style":341},[269],[28,1057,912],{"className":1058},[274],[28,1060,510],{"className":1061},[274,275],[28,1063],{"className":1064,"style":302},[279],[28,1066],{"className":1067,"style":302},[279],[28,1069,889],{"className":1070},[306],[28,1072],{"className":1073,"style":302},[279],[28,1075],{"className":1076,"style":302},[279],[28,1078,1080,1084,1087,1090,1093],{"className":1079},[265],[28,1081],{"className":1082,"style":1083},[269],"height:0.75em;vertical-align:-0.1944em;",[28,1085,510],{"className":1086},[274,275],[28,1088],{"className":1089,"style":280},[279],[28,1091,233],{"className":1092},[284],[28,1094],{"className":1095,"style":280},[279],[28,1097,1099,1102,1105,1108,1111,1114,1117],{"className":1098},[265],[28,1100],{"className":1101,"style":341},[269],[28,1103,510],{"className":1104},[274,275],[28,1106],{"className":1107,"style":302},[279],[28,1109],{"className":1110,"style":302},[279],[28,1112,889],{"className":1113},[306],[28,1115],{"className":1116,"style":302},[279],[28,1118],{"className":1119,"style":302},[279],[28,1121,1123,1126,1129,1132,1135],{"className":1122},[265],[28,1124],{"className":1125,"style":1083},[269],[28,1127,510],{"className":1128},[274,275],[28,1130],{"className":1131,"style":280},[279],[28,1133,937],{"className":1134},[284],[28,1136],{"className":1137,"style":280},[279],[28,1139,1141,1144,1147,1150,1153,1156,1159],{"className":1140},[265],[28,1142],{"className":1143,"style":341},[269],[28,1145,510],{"className":1146},[274,275],[28,1148],{"className":1149,"style":302},[279],[28,1151],{"className":1152,"style":302},[279],[28,1154,889],{"className":1155},[306],[28,1157],{"className":1158,"style":302},[279],[28,1160],{"className":1161,"style":302},[279],[28,1163,1165,1168,1171,1174,1177],{"className":1164},[265],[28,1166],{"className":1167,"style":612},[269],[28,1169,510],{"className":1170},[274,275],[28,1172],{"className":1173,"style":302},[279],[28,1175,950],{"className":1176},[306],[28,1178],{"className":1179,"style":302},[279],[28,1181,1183,1186],{"className":1182},[265],[28,1184],{"className":1185,"style":612},[269],[28,1187,510],{"className":1188},[274,275],[11,1190,1191,1192,1220,1221,1224,1225,177,1363,177,1497,177,1648,1751],{},"其中 ",[28,1193,1195,1208],{"className":1194,"translate":193},[197],[28,1196,1198],{"className":1197},[201],[203,1199,1200],{"xmlns":205},[208,1201,1202,1206],{},[211,1203,1204],{},[214,1205,903],{},[253,1207,903],{"encoding":255},[28,1209,1211],{"className":1210,"ariaHidden":261},[260],[28,1212,1214,1217],{"className":1213},[265],[28,1215],{"className":1216,"style":455},[269],[28,1218,903],{"className":1219},[274,275]," 是",[183,1222,1223],{},"理论原子","，形如 ",[28,1226,1228,1257],{"className":1227,"translate":193},[197],[28,1229,1231],{"className":1230},[201],[203,1232,1233],{"xmlns":205},[208,1234,1235,1254],{},[211,1236,1237,1245,1247],{},[501,1238,1239,1242],{},[214,1240,1241],{},"t",[224,1243,1244],{},"1",[217,1246,83],{},[501,1248,1249,1251],{},[214,1250,1241],{},[224,1252,1253],{},"2",[253,1255,1256],{"encoding":255},"t_1 = t_2",[28,1258,1260,1317],{"className":1259,"ariaHidden":261},[260],[28,1261,1263,1267,1308,1311,1314],{"className":1262},[265],[28,1264],{"className":1265,"style":1266},[269],"height:0.7651em;vertical-align:-0.15em;",[28,1268,1270,1273],{"className":1269},[274],[28,1271,1241],{"className":1272},[274,275],[28,1274,1276],{"className":1275},[556],[28,1277,1279,1300],{"className":1278},[560,561],[28,1280,1282,1297],{"className":1281},[565],[28,1283,1286],{"className":1284,"style":1285},[569],"height:0.3011em;",[28,1287,1288,1291],{"style":573},[28,1289],{"className":1290,"style":578},[577],[28,1292,1294],{"className":1293},[582,583,584,585],[28,1295,1244],{"className":1296},[274,585],[28,1298,593],{"className":1299},[592],[28,1301,1303],{"className":1302},[565],[28,1304,1306],{"className":1305,"style":600},[569],[28,1307],{},[28,1309],{"className":1310,"style":302},[279],[28,1312,83],{"className":1313},[306],[28,1315],{"className":1316,"style":302},[279],[28,1318,1320,1323],{"className":1319},[265],[28,1321],{"className":1322,"style":1266},[269],[28,1324,1326,1329],{"className":1325},[274],[28,1327,1241],{"className":1328},[274,275],[28,1330,1332],{"className":1331},[556],[28,1333,1335,1355],{"className":1334},[560,561],[28,1336,1338,1352],{"className":1337},[565],[28,1339,1341],{"className":1340,"style":1285},[569],[28,1342,1343,1346],{"style":573},[28,1344],{"className":1345,"style":578},[577],[28,1347,1349],{"className":1348},[582,583,584,585],[28,1350,1253],{"className":1351},[274,585],[28,1353,593],{"className":1354},[592],[28,1356,1358],{"className":1357},[565],[28,1359,1361],{"className":1360,"style":600},[569],[28,1362],{},[28,1364,1366,1393],{"className":1365,"translate":193},[197],[28,1367,1369],{"className":1368},[201],[203,1370,1371],{"xmlns":205},[208,1372,1373,1390],{},[211,1374,1375,1381,1384],{},[501,1376,1377,1379],{},[214,1378,1241],{},[224,1380,1244],{},[217,1382,1383],{},"\u003C",[501,1385,1386,1388],{},[214,1387,1241],{},[224,1389,1253],{},[253,1391,1392],{"encoding":255},"t_1 \u003C t_2",[28,1394,1396,1451],{"className":1395,"ariaHidden":261},[260],[28,1397,1399,1402,1442,1445,1448],{"className":1398},[265],[28,1400],{"className":1401,"style":1266},[269],[28,1403,1405,1408],{"className":1404},[274],[28,1406,1241],{"className":1407},[274,275],[28,1409,1411],{"className":1410},[556],[28,1412,1414,1434],{"className":1413},[560,561],[28,1415,1417,1431],{"className":1416},[565],[28,1418,1420],{"className":1419,"style":1285},[569],[28,1421,1422,1425],{"style":573},[28,1423],{"className":1424,"style":578},[577],[28,1426,1428],{"className":1427},[582,583,584,585],[28,1429,1244],{"className":1430},[274,585],[28,1432,593],{"className":1433},[592],[28,1435,1437],{"className":1436},[565],[28,1438,1440],{"className":1439,"style":600},[569],[28,1441],{},[28,1443],{"className":1444,"style":302},[279],[28,1446,1383],{"className":1447},[306],[28,1449],{"className":1450,"style":302},[279],[28,1452,1454,1457],{"className":1453},[265],[28,1455],{"className":1456,"style":1266},[269],[28,1458,1460,1463],{"className":1459},[274],[28,1461,1241],{"className":1462},[274,275],[28,1464,1466],{"className":1465},[556],[28,1467,1469,1489],{"className":1468},[560,561],[28,1470,1472,1486],{"className":1471},[565],[28,1473,1475],{"className":1474,"style":1285},[569],[28,1476,1477,1480],{"style":573},[28,1478],{"className":1479,"style":578},[577],[28,1481,1483],{"className":1482},[582,583,584,585],[28,1484,1253],{"className":1485},[274,585],[28,1487,593],{"className":1488},[592],[28,1490,1492],{"className":1491},[565],[28,1493,1495],{"className":1494,"style":600},[569],[28,1496],{},[28,1498,1500,1534],{"className":1499,"translate":193},[197],[28,1501,1503],{"className":1502},[201],[203,1504,1505],{"xmlns":205},[208,1506,1507,1531],{},[211,1508,1509,1512,1514,1520,1523,1525],{},[214,1510,1511],{},"f",[217,1513,43],{"stretchy":241},[501,1515,1516,1518],{},[214,1517,1241],{},[224,1519,1244],{},[217,1521,1522],{"stretchy":241},")",[217,1524,83],{},[501,1526,1527,1529],{},[214,1528,1241],{},[224,1530,1253],{},[253,1532,1533],{"encoding":255},"f(t_1) = t_2",[28,1535,1537,1602],{"className":1536,"ariaHidden":261},[260],[28,1538,1540,1543,1547,1550,1590,1593,1596,1599],{"className":1539},[265],[28,1541],{"className":1542,"style":341},[269],[28,1544,1511],{"className":1545,"style":1546},[274,275],"margin-right:0.1076em;",[28,1548,43],{"className":1549},[348],[28,1551,1553,1556],{"className":1552},[274],[28,1554,1241],{"className":1555},[274,275],[28,1557,1559],{"className":1558},[556],[28,1560,1562,1582],{"className":1561},[560,561],[28,1563,1565,1579],{"className":1564},[565],[28,1566,1568],{"className":1567,"style":1285},[569],[28,1569,1570,1573],{"style":573},[28,1571],{"className":1572,"style":578},[577],[28,1574,1576],{"className":1575},[582,583,584,585],[28,1577,1244],{"className":1578},[274,585],[28,1580,593],{"className":1581},[592],[28,1583,1585],{"className":1584},[565],[28,1586,1588],{"className":1587,"style":600},[569],[28,1589],{},[28,1591,1522],{"className":1592},[355],[28,1594],{"className":1595,"style":302},[279],[28,1597,83],{"className":1598},[306],[28,1600],{"className":1601,"style":302},[279],[28,1603,1605,1608],{"className":1604},[265],[28,1606],{"className":1607,"style":1266},[269],[28,1609,1611,1614],{"className":1610},[274],[28,1612,1241],{"className":1613},[274,275],[28,1615,1617],{"className":1616},[556],[28,1618,1620,1640],{"className":1619},[560,561],[28,1621,1623,1637],{"className":1622},[565],[28,1624,1626],{"className":1625,"style":1285},[569],[28,1627,1628,1631],{"style":573},[28,1629],{"className":1630,"style":578},[577],[28,1632,1634],{"className":1633},[582,583,584,585],[28,1635,1253],{"className":1636},[274,585],[28,1638,593],{"className":1639},[592],[28,1641,1643],{"className":1642},[565],[28,1644,1646],{"className":1645,"style":600},[569],[28,1647],{},[28,1649,1651,1697],{"className":1650,"translate":193},[197],[28,1652,1654],{"className":1653},[201],[203,1655,1656],{"xmlns":205},[208,1657,1658,1694],{},[211,1659,1660,1677,1679,1681,1684,1687,1689,1691],{},[211,1661,1662,1665,1668,1671,1673,1675],{},[214,1663,1664],{"mathvariant":484},"s",[214,1666,1667],{"mathvariant":484},"e",[214,1669,1670],{"mathvariant":484},"l",[214,1672,1667],{"mathvariant":484},[214,1674,22],{"mathvariant":484},[214,1676,1241],{"mathvariant":484},[217,1678,43],{"stretchy":241},[214,1680,238],{},[217,1682,1683],{"separator":261},",",[214,1685,1686],{},"i",[217,1688,1522],{"stretchy":241},[217,1690,83],{},[214,1692,1693],{},"v",[253,1695,1696],{"encoding":255},"\\mathrm{select}(a, i) = v",[28,1698,1700,1742],{"className":1699,"ariaHidden":261},[260],[28,1701,1703,1706,1714,1717,1720,1724,1727,1730,1733,1736,1739],{"className":1702},[265],[28,1704],{"className":1705,"style":341},[269],[28,1707,1709],{"className":1708},[274],[28,1710,1713],{"className":1711},[274,1712],"mathrm","select",[28,1715,43],{"className":1716},[348],[28,1718,238],{"className":1719},[274,275],[28,1721,1683],{"className":1722},[1723],"mpunct",[28,1725],{"className":1726,"style":530},[279],[28,1728,1686],{"className":1729},[274,275],[28,1731,1522],{"className":1732},[355],[28,1734],{"className":1735,"style":302},[279],[28,1737,83],{"className":1738},[306],[28,1740],{"className":1741,"style":302},[279],[28,1743,1745,1748],{"className":1744},[265],[28,1746],{"className":1747,"style":645},[269],[28,1749,1693],{"className":1750,"style":298},[274,275],"。原子内部的项由变量、常量和理论函数符号组成。",[11,1753,1754],{},"下面这个公式同时用到了三种理论：",[28,1756,1758],{"className":1757,"translate":193},[192],[28,1759,1761,1884],{"className":1760,"translate":193},[197],[28,1762,1764],{"className":1763},[201],[203,1765,1766],{"xmlns":205,"display":206},[208,1767,1768,1881],{},[211,1769,1770,1796,1798,1800,1802,1830,1832,1834,1836,1838,1868,1870,1872,1874,1877,1879],{},[1771,1772,1773,1794],"munder",{},[1771,1774,1775,1791],{},[211,1776,1777,1779,1781,1783,1785,1787,1789],{},[217,1778,43],{"stretchy":241},[214,1780,176],{},[217,1782,124],{},[214,1784,180],{},[217,1786,130],{},[224,1788,226],{},[217,1790,1522],{"stretchy":241},[217,1792,1793],{"stretchy":261},"⏟",[214,1795,11],{},[228,1797,230],{},[217,1799,233],{},[228,1801,230],{},[1771,1803,1804,1827],{},[1771,1805,1806,1825],{},[211,1807,1808,1811,1813,1815,1817,1819,1821,1823],{},[217,1809,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},"1.2em",[214,1812,238],{},[217,1814,73],{"stretchy":241},[224,1816,89],{},[217,1818,246],{"stretchy":241},[217,1820,83],{},[224,1822,251],{},[217,1824,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,1826,1793],{"stretchy":261},[214,1828,1829],{},"q",[228,1831,230],{},[217,1833,233],{},[228,1835,230],{},[217,1837,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[1771,1839,1840,1865],{},[1771,1841,1842,1863],{},[211,1843,1844,1846,1848,1850,1852,1855,1857,1859,1861],{},[214,1845,1511],{},[217,1847,43],{"stretchy":241},[214,1849,176],{},[217,1851,1522],{"stretchy":241},[217,1853,1854],{"mathvariant":484},"≠",[214,1856,1511],{},[217,1858,43],{"stretchy":241},[214,1860,180],{},[217,1862,1522],{"stretchy":241},[217,1864,1793],{"stretchy":261},[214,1866,1867],{},"r",[217,1869,937],{},[228,1871,488],{},[214,1873,180],{},[217,1875,1876],{},"≤",[224,1878,89],{},[217,1880,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[253,1882,1883],{"encoding":255},"\\underbrace{(x + y > 10)}_{p}\\;\\wedge\\;\\underbrace{\\bigl(a[0] = 5\\bigr)}_{q}\\;\\wedge\\;\\bigl(\\underbrace{f(x) \\ne f(y)}_{r} \\vee\\, y \\le 0\\bigr)",[28,1885,1887,2072,2240,2444,2463],{"className":1886,"ariaHidden":261},[260],[28,1888,1890,1894,2057,2060,2063,2066,2069],{"className":1889},[265],[28,1891],{"className":1892,"style":1893},[269],"height:2.2855em;vertical-align:-1.5355em;",[28,1895,1898],{"className":1896},[1897,1771],"minner",[28,1899,1901,2048],{"className":1900},[560,561],[28,1902,1904,2045],{"className":1903},[565],[28,1905,1908,1924],{"className":1906,"style":1907},[569],"height:0.75em;",[28,1909,1911,1915],{"style":1910},"top:-1.6006em;",[28,1912],{"className":1913,"style":1914},[577],"height:3em;",[28,1916,1918],{"className":1917},[582,583,584,585],[28,1919,1921],{"className":1920},[274,585],[28,1922,11],{"className":1923},[274,275,585],[28,1925,1927,1930],{"style":1926},"top:-3em;",[28,1928],{"className":1929,"style":1914},[577],[28,1931,1933],{"className":1932},[1897,1771],[28,1934,1936,2036],{"className":1935},[560,561],[28,1937,1939,2033],{"className":1938},[565],[28,1940,1942,1992],{"className":1941,"style":1907},[569],[28,1943,1947,1950],{"className":1944,"style":1946},[1945],"svg-align","top:-2.102em;",[28,1948],{"className":1949,"style":1914},[577],[28,1951,1955,1972,1982],{"className":1952,"style":1954},[1953],"stretchy","height:0.548em;min-width:1.6em;",[28,1956,1960],{"className":1957,"style":1959},[1958],"brace-left","height:0.548em;",[1961,1962,1968],"svg",{"xmlns":1963,"width":1964,"height":1965,"viewBox":1966,"preserveAspectRatio":1967},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","0.548em","0 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属于",[183,2559,2560],{},"线性整数算术","：它知道 ",[28,2563,2565,2578],{"className":2564,"translate":193},[197],[28,2566,2568],{"className":2567},[201],[203,2569,2570],{"xmlns":205},[208,2571,2572,2576],{},[211,2573,2574],{},[217,2575,124],{},[253,2577,124],{"encoding":255},[28,2579,2581],{"className":2580,"ariaHidden":261},[260],[28,2582,2584,2587],{"className":2583},[265],[28,2585],{"className":2586,"style":270},[269],[28,2588,124],{"className":2589},[274]," 是加法、",[28,2592,2594,2607],{"className":2593,"translate":193},[197],[28,2595,2597],{"className":2596},[201],[203,2598,2599],{"xmlns":205},[208,2600,2601,2605],{},[211,2602,2603],{},[217,2604,130],{},[253,2606,130],{"encoding":255},[28,2608,2610],{"className":2609,"ariaHidden":261},[260],[28,2611,2613,2617],{"className":2612},[265],[28,2614],{"className":2615,"style":2616},[269],"height:0.5782em;vertical-align:-0.0391em;",[28,2618,130],{"className":2619},[306]," 是全序；",[2483,2622,2623,2557,2688,2561,2691,2693,2694,2697],{},[28,2624,2626,2649],{"className":2625,"translate":193},[197],[28,2627,2629],{"className":2628},[201],[203,2630,2631],{"xmlns":205},[208,2632,2633,2647],{},[211,2634,2635,2637,2639,2641,2643,2645],{},[214,2636,238],{},[217,2638,73],{"stretchy":241},[224,2640,89],{},[217,2642,246],{"stretchy":241},[217,2644,83],{},[224,2646,251],{},[253,2648,394],{"encoding":255},[28,2650,2652,2679],{"className":2651,"ariaHidden":261},[260],[28,2653,2655,2658,2661,2664,2667,2670,2673,2676],{"className":2654},[265],[28,2656],{"className":2657,"style":341},[269],[28,2659,238],{"className":2660},[274,275],[28,2662,73],{"className":2663},[348],[28,2665,89],{"className":2666},[274],[28,2668,246],{"className":2669},[355],[28,2671],{"className":2672,"style":302},[279],[28,2674,83],{"className":2675},[306],[28,2677],{"className":2678,"style":302},[279],[28,2680,2682,2685],{"className":2681},[265],[28,2683],{"className":2684,"style":316},[269],[28,2686,251],{"className":2687},[274],[183,2689,2690],{},"数组理论",[25,2692,1713],{}," 和 ",[25,2695,2696],{},"store"," 的读写语义；",[2483,2699,2700,2557,2814,2817,2818,2846],{},[28,2701,2703,2733],{"className":2702,"translate":193},[197],[28,2704,2706],{"className":2705},[201],[203,2707,2708],{"xmlns":205},[208,2709,2710,2730],{},[211,2711,2712,2714,2716,2718,2720,2722,2724,2726,2728],{},[214,2713,1511],{},[217,2715,43],{"stretchy":241},[214,2717,176],{},[217,2719,1522],{"stretchy":241},[217,2721,1854],{"mathvariant":484},[214,2723,1511],{},[217,2725,43],{"stretchy":241},[214,2727,180],{},[217,2729,1522],{"stretchy":241},[253,2731,2732],{"encoding":255},"f(x) \\ne f(y)",[28,2734,2736,2796],{"className":2735,"ariaHidden":261},[260],[28,2737,2739,2742,2745,2748,2751,2754,2757,2793],{"className":2738},[265],[28,2740],{"className":2741,"style":341},[269],[28,2743,1511],{"className":2744,"style":1546},[274,275],[28,2746,43],{"className":2747},[348],[28,2749,176],{"className":2750},[274,275],[28,2752,1522],{"className":2753},[355],[28,2755],{"className":2756,"style":302},[279],[28,2758,2760,2787,2790],{"className":2759},[306],[28,2761,2763],{"className":2762},[306],[28,2764,2766],{"className":2765},[274,2359],[28,2767,2769],{"className":2768},[2363],[28,2770,2772,2775,2784],{"className":2771},[2367],[28,2773],{"className":2774,"style":2371},[269],[28,2776,2778],{"className":2777},[2375],[28,2779,2781],{"className":2780},[274],[28,2782,2382],{"className":2783},[306],[28,2785],{"className":2786},[2386],[28,2788],{"className":2789},[279,2390],[28,2791,83],{"className":2792},[306],[28,2794],{"className":2795,"style":302},[279],[28,2797,2799,2802,2805,2808,2811],{"className":2798},[265],[28,2800],{"className":2801,"style":341},[269],[28,2803,1511],{"className":2804,"style":1546},[274,275],[28,2806,43],{"className":2807},[348],[28,2809,180],{"className":2810,"style":298},[274,275],[28,2812,1522],{"className":2813},[355],[183,2815,2816],{},"未解释函数","：它不知道 ",[28,2819,2821,2834],{"className":2820,"translate":193},[197],[28,2822,2824],{"className":2823},[201],[203,2825,2826],{"xmlns":205},[208,2827,2828,2832],{},[211,2829,2830],{},[214,2831,1511],{},[253,2833,1511],{"encoding":255},[28,2835,2837],{"className":2836,"ariaHidden":261},[260],[28,2838,2840,2843],{"className":2839},[265],[28,2841],{"className":2842,"style":2371},[269],[28,2844,1511],{"className":2845,"style":1546},[274,275]," 具体算什么，只知道「同一个输入必须得到同一个输出」。",[11,2848,2849,2850,2905],{},"对 SAT 来说，这三个原子只是三个裸的布尔变量 ",[28,2851,2853,2875],{"className":2852,"translate":193},[197],[28,2854,2856],{"className":2855},[201],[203,2857,2858],{"xmlns":205},[208,2859,2860,2872],{},[211,2861,2862,2864,2866,2868,2870],{},[214,2863,11],{},[217,2865,1683],{"separator":261},[214,2867,1829],{},[217,2869,1683],{"separator":261},[214,2871,1867],{},[253,2873,2874],{"encoding":255},"p, q, r",[28,2876,2878],{"className":2877,"ariaHidden":261},[260],[28,2879,2881,2884,2887,2890,2893,2896,2899,2902],{"className":2880},[265],[28,2882],{"className":2883,"style":612},[269],[28,2885,11],{"className":2886},[274,275],[28,2888,1683],{"className":2889},[1723],[28,2891],{"className":2892,"style":530},[279],[28,2894,1829],{"className":2895,"style":298},[274,275],[28,2897,1683],{"className":2898},[1723],[28,2900],{"className":2901,"style":530},[279],[28,2903,1867],{"className":2904,"style":2279},[274,275],"。对 SMT 来说，它们各自带着一整套内部结构。",[2907,2908,2909],"div",{},[2910,2911],"img",{"height":2912,"width":2913,"style":2914,"data-d2-theme":2915,"alt":23,"src":2916},614,552,"max-width:100%;height:auto","default","data:image\u002Fsvg+xml;base64,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\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",[11,2918,2919,2920,2923],{},"这张图就是 SMT 的骨架：",[183,2921,2922],{},"布尔层负责「怎么组合」，理论层负责「每个原子是什么意思」","。",[419,2925,2927],{"id":2926},"_13-背景理论","1.3 背景理论",[11,2929,2930],{},"SMT 里常用的理论有这么几类：",[2480,2932,2933,2943,3020,3032,3105,3122,3131],{},[2483,2934,2935,2942],{},[183,2936,2937,2938,2941],{},"EUF（",[25,2939,2940],{},"QF_UF","）","：只处理相等和未解释函数，适合描述对象相等、指针别名。",[2483,2944,2945,2955,2956,177,2984,3014,3015,3017,3018,2923],{},[183,2946,2947,2948,2951,2952,2941],{},"线性算术（",[25,2949,2950],{},"QF_LRA"," \u002F ",[25,2953,2954],{},"QF_LIA","：处理 ",[28,2957,2959,2972],{"className":2958,"translate":193},[197],[28,2960,2962],{"className":2961},[201],[203,2963,2964],{"xmlns":205},[208,2965,2966,2970],{},[211,2967,2968],{},[217,2969,124],{},[253,2971,124],{"encoding":255},[28,2973,2975],{"className":2974,"ariaHidden":261},[260],[28,2976,2978,2981],{"className":2977},[265],[28,2979],{"className":2980,"style":270},[269],[28,2982,124],{"className":2983},[274],[28,2985,2987,3001],{"className":2986,"translate":193},[197],[28,2988,2990],{"className":2989},[201],[203,2991,2992],{"xmlns":205},[208,2993,2994,2998],{},[211,2995,2996],{},[217,2997,1876],{},[253,2999,3000],{"encoding":255},"\\le",[28,3002,3004],{"className":3003,"ariaHidden":261},[260],[28,3005,3007,3011],{"className":3006},[265],[28,3008],{"className":3009,"style":3010},[269],"height:0.7719em;vertical-align:-0.136em;",[28,3012,1876],{"className":3013},[306]," 和常数；实数版是 ",[25,3016,2950],{},"，整数版是 ",[25,3019,2954],{},[2483,3021,3022,3031],{},[183,3023,3024,3025,2951,3028,2941],{},"非线性算术（",[25,3026,3027],{},"QF_NRA",[25,3029,3030],{},"QF_NIA","：允许乘法；实数版可判定，整数版不可判定。",[2483,3033,3034,3040,3041,3104],{},[183,3035,3036,3037,2941],{},"位向量（",[25,3038,3039],{},"QF_BV","：描述机器整数和硬件电路，加法是模 ",[28,3042,3044,3064],{"className":3043,"translate":193},[197],[28,3045,3047],{"className":3046},[201],[203,3048,3049],{"xmlns":205},[208,3050,3051,3061],{},[211,3052,3053],{},[3054,3055,3056,3058],"msup",{},[224,3057,1253],{},[214,3059,3060],{},"n",[253,3062,3063],{"encoding":255},"2^n",[28,3065,3067],{"className":3066,"ariaHidden":261},[260],[28,3068,3070,3074],{"className":3069},[265],[28,3071],{"className":3072,"style":3073},[269],"height:0.6644em;",[28,3075,3077,3080],{"className":3076},[274],[28,3078,1253],{"className":3079},[274],[28,3081,3083],{"className":3082},[556],[28,3084,3086],{"className":3085},[560],[28,3087,3089],{"className":3088},[565],[28,3090,3092],{"className":3091,"style":3073},[569],[28,3093,3095,3098],{"style":3094},"top:-3.063em;margin-right:0.05em;",[28,3096],{"className":3097,"style":578},[577],[28,3099,3101],{"className":3100},[582,583,584,585],[28,3102,3060],{"className":3103},[274,275,585]," 的。",[2483,3106,3107,3116,3117,2693,3119,3121],{},[183,3108,3109,3110,2951,3113,2941],{},"数组（",[25,3111,3112],{},"QF_AX",[25,3114,3115],{},"QF_AUFBV","：用 ",[25,3118,1713],{},[25,3120,2696],{}," 描述内存读写。",[2483,3123,3124,3130],{},[183,3125,3126,3127,2941],{},"浮点（",[25,3128,3129],{},"QF_FP","：描述 IEEE 754 浮点运算。",[2483,3132,3133,3136,3137,177,3140,3143],{},[183,3134,3135],{},"字符串","：描述 ",[25,3138,3139],{},"concat",[25,3141,3142],{},"contains"," 等操作，工业实现常常不完整。",[11,3145,3146,3149],{},[25,3147,3148],{},"QF"," 是 quantifier-free 的缩写，意思是「无量词」。绝大多数 SMT 求解器面对的都是这个片段。",[11,3151,3152,3153,3181],{},"可以把 SAT 看成 SMT 的一个特例：当理论 ",[28,3154,3156,3169],{"className":3155,"translate":193},[197],[28,3157,3159],{"className":3158},[201],[203,3160,3161],{"xmlns":205},[208,3162,3163,3167],{},[211,3164,3165],{},[214,3166,443],{},[253,3168,443],{"encoding":255},[28,3170,3172],{"className":3171,"ariaHidden":261},[260],[28,3173,3175,3178],{"className":3174},[265],[28,3176],{"className":3177,"style":455},[269],[28,3179,443],{"className":3180,"style":459},[274,275]," 为空、所有原子都是裸命题时，SMT 就退化成 SAT。",[11,3183,3184],{},"反过来，SMT 也可以看成 SAT 的一次「语义升级」：布尔骨架不变，但每个命题不再是黑盒，而是有内部结构、可以被理论求解器检查和反驳的对象。",[11,3186,3187],{},"这个升级带来的好处，要放到具体的历史困境里才看得清楚。",[414,3189,3191],{"id":3190},"二smt-之前","二、SMT 之前",[419,3193,3195],{"id":3194},"_21-位爆破","2.1 位爆破",[11,3197,3198,3199,3202],{},"SAT 只认布尔变量。要用它处理整数，标准做法叫",[183,3200,3201],{},"位爆破","（bit-blasting）：把整数拆成一个个布尔变量，再把算术运算展开成逻辑门电路，最后用 Tseitin 变换把电路转成 CNF 子句。",[11,3204,3205,3206,3248],{},"看一个具体的小例子。取两个 4 位无符号整数 ",[28,3207,3209,3227],{"className":3208,"translate":193},[197],[28,3210,3212],{"className":3211},[201],[203,3213,3214],{"xmlns":205},[208,3215,3216,3224],{},[211,3217,3218,3220,3222],{},[214,3219,176],{},[217,3221,1683],{"separator":261},[214,3223,180],{},[253,3225,3226],{"encoding":255},"x, y",[28,3228,3230],{"className":3229,"ariaHidden":261},[260],[28,3231,3233,3236,3239,3242,3245],{"className":3232},[265],[28,3234],{"className":3235,"style":612},[269],[28,3237,176],{"className":3238},[274,275],[28,3240,1683],{"className":3241},[1723],[28,3243],{"className":3244,"style":530},[279],[28,3246,180],{"className":3247,"style":298},[274,275],"，约束是：",[28,3250,3252],{"className":3251,"translate":193},[192],[28,3253,3255,3289],{"className":3254,"translate":193},[197],[28,3256,3258],{"className":3257},[201],[203,3259,3260],{"xmlns":205,"display":206},[208,3261,3262,3286],{},[211,3263,3264,3266,3268,3270,3272,3274,3276,3278,3280,3282,3284],{},[214,3265,176],{},[217,3267,124],{},[214,3269,180],{},[217,3271,83],{},[224,3273,226],{},[228,3275,230],{},[217,3277,233],{},[228,3279,230],{},[214,3281,176],{},[217,3283,130],{},[214,3285,180],{},[253,3287,3288],{"encoding":255},"x + y = 10 \\;\\wedge\\; x > y",[28,3290,3292,3310,3328,3352,3370],{"className":3291,"ariaHidden":261},[260],[28,3293,3295,3298,3301,3304,3307],{"className":3294},[265],[28,3296],{"className":3297,"style":270},[269],[28,3299,176],{"className":3300},[274,275],[28,3302],{"className":3303,"style":280},[279],[28,3305,124],{"className":3306},[284],[28,3308],{"className":3309,"style":280},[279],[28,3311,3313,3316,3319,3322,3325],{"className":3312},[265],[28,3314],{"className":3315,"style":612},[269],[28,3317,180],{"className":3318,"style":298},[274,275],[28,3320],{"className":3321,"style":302},[279],[28,3323,83],{"className":3324},[306],[28,3326],{"className":3327,"style":302},[279],[28,3329,3331,3334,3337,3340,3343,3346,3349],{"className":3330},[265],[28,3332],{"className":3333,"style":316},[269],[28,3335,226],{"className":3336},[274],[28,3338],{"className":3339,"style":302},[279],[28,3341],{"className":3342,"style":280},[279],[28,3344,233],{"className":3345},[284],[28,3347],{"className":3348,"style":302},[279],[28,3350],{"className":3351,"style":280},[279],[28,3353,3355,3358,3361,3364,3367],{"className":3354},[265],[28,3356],{"className":3357,"style":2616},[269],[28,3359,176],{"className":3360},[274,275],[28,3362],{"className":3363,"style":302},[279],[28,3365,130],{"className":3366},[306],[28,3368],{"className":3369,"style":302},[279],[28,3371,3373,3376],{"className":3372},[265],[28,3374],{"className":3375,"style":612},[269],[28,3377,180],{"className":3378,"style":298},[274,275],[11,3380,3381,3382,2693,3592,3801,3802,3908,3909,4010],{},"SAT 看到的是 8 个布尔变量 ",[28,3383,3385,3422],{"className":3384,"translate":193},[197],[28,3386,3388],{"className":3387},[201],[203,3389,3390],{"xmlns":205},[208,3391,3392,3419],{},[211,3393,3394,3401,3407,3413],{},[501,3395,3396,3398],{},[214,3397,176],{},[224,3399,3400],{},"3",[501,3402,3403,3405],{},[214,3404,176],{},[224,3406,1253],{},[501,3408,3409,3411],{},[214,3410,176],{},[224,3412,1244],{},[501,3414,3415,3417],{},[214,3416,176],{},[224,3418,89],{},[253,3420,3421],{"encoding":255},"x_3 x_2 x_1 x_0",[28,3423,3425],{"className":3424,"ariaHidden":261},[260],[28,3426,3428,3432,3472,3512,3552],{"className":3427},[265],[28,3429],{"className":3430,"style":3431},[269],"height:0.5806em;vertical-align:-0.15em;",[28,3433,3435,3438],{"className":3434},[274],[28,3436,176],{"className":3437},[274,275],[28,3439,3441],{"className":3440},[556],[28,3442,3444,3464],{"className":3443},[560,561],[28,3445,3447,3461],{"className":3446},[565],[28,3448,3450],{"className":3449,"style":1285},[569],[28,3451,3452,3455],{"style":573},[28,3453],{"className":3454,"style":578},[577],[28,3456,3458],{"className":3457},[582,583,584,585],[28,3459,3400],{"className":3460},[274,585],[28,3462,593],{"className":3463},[592],[28,3465,3467],{"className":3466},[565],[28,3468,3470],{"className":3469,"style":600},[569],[28,3471],{},[28,3473,3475,3478],{"className":3474},[274],[28,3476,176],{"className":3477},[274,275],[28,3479,3481],{"className":3480},[556],[28,3482,3484,3504],{"className":3483},[560,561],[28,3485,3487,3501],{"className":3486},[565],[28,3488,3490],{"className":3489,"style":1285},[569],[28,3491,3492,3495],{"style":573},[28,3493],{"className":3494,"style":578},[577],[28,3496,3498],{"className":3497},[582,583,584,585],[28,3499,1253],{"className":3500},[274,585],[28,3502,593],{"className":3503},[592],[28,3505,3507],{"className":3506},[565],[28,3508,3510],{"className":3509,"style":600},[569],[28,3511],{},[28,3513,3515,3518],{"className":3514},[274],[28,3516,176],{"className":3517},[274,275],[28,3519,3521],{"className":3520},[556],[28,3522,3524,3544],{"className":3523},[560,561],[28,3525,3527,3541],{"className":3526},[565],[28,3528,3530],{"className":3529,"style":1285},[569],[28,3531,3532,3535],{"style":573},[28,3533],{"className":3534,"style":578},[577],[28,3536,3538],{"className":3537},[582,583,584,585],[28,3539,1244],{"className":3540},[274,585],[28,3542,593],{"className":3543},[592],[28,3545,3547],{"className":3546},[565],[28,3548,3550],{"className":3549,"style":600},[569],[28,3551],{},[28,3553,3555,3558],{"className":3554},[274],[28,3556,176],{"className":3557},[274,275],[28,3559,3561],{"className":3560},[556],[28,3562,3564,3584],{"className":3563},[560,561],[28,3565,3567,3581],{"className":3566},[565],[28,3568,3570],{"className":3569,"style":1285},[569],[28,3571,3572,3575],{"style":573},[28,3573],{"className":3574,"style":578},[577],[28,3576,3578],{"className":3577},[582,583,584,585],[28,3579,89],{"className":3580},[274,585],[28,3582,593],{"className":3583},[592],[28,3585,3587],{"className":3586},[565],[28,3588,3590],{"className":3589,"style":600},[569],[28,3591],{},[28,3593,3595,3631],{"className":3594,"translate":193},[197],[28,3596,3598],{"className":3597},[201],[203,3599,3600],{"xmlns":205},[208,3601,3602,3628],{},[211,3603,3604,3610,3616,3622],{},[501,3605,3606,3608],{},[214,3607,180],{},[224,3609,3400],{},[501,3611,3612,3614],{},[214,3613,180],{},[224,3615,1253],{},[501,3617,3618,3620],{},[214,3619,180],{},[224,3621,1244],{},[501,3623,3624,3626],{},[214,3625,180],{},[224,3627,89],{},[253,3629,3630],{"encoding":255},"y_3 y_2 y_1 y_0",[28,3632,3634],{"className":3633,"ariaHidden":261},[260],[28,3635,3637,3640,3681,3721,3761],{"className":3636},[265],[28,3638],{"className":3639,"style":612},[269],[28,3641,3643,3646],{"className":3642},[274],[28,3644,180],{"className":3645,"style":298},[274,275],[28,3647,3649],{"className":3648},[556],[28,3650,3652,3673],{"className":3651},[560,561],[28,3653,3655,3670],{"className":3654},[565],[28,3656,3658],{"className":3657,"style":1285},[569],[28,3659,3661,3664],{"style":3660},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[28,3662],{"className":3663,"style":578},[577],[28,3665,3667],{"className":3666},[582,583,584,585],[28,3668,3400],{"className":3669},[274,585],[28,3671,593],{"className":3672},[592],[28,3674,3676],{"className":3675},[565],[28,3677,3679],{"className":3678,"style":600},[569],[28,3680],{},[28,3682,3684,3687],{"className":3683},[274],[28,3685,180],{"className":3686,"style":298},[274,275],[28,3688,3690],{"className":3689},[556],[28,3691,3693,3713],{"className":3692},[560,561],[28,3694,3696,3710],{"className":3695},[565],[28,3697,3699],{"className":3698,"style":1285},[569],[28,3700,3701,3704],{"style":3660},[28,3702],{"className":3703,"style":578},[577],[28,3705,3707],{"className":3706},[582,583,584,585],[28,3708,1253],{"className":3709},[274,585],[28,3711,593],{"className":3712},[592],[28,3714,3716],{"className":3715},[565],[28,3717,3719],{"className":3718,"style":600},[569],[28,3720],{},[28,3722,3724,3727],{"className":3723},[274],[28,3725,180],{"className":3726,"style":298},[274,275],[28,3728,3730],{"className":3729},[556],[28,3731,3733,3753],{"className":3732},[560,561],[28,3734,3736,3750],{"className":3735},[565],[28,3737,3739],{"className":3738,"style":1285},[569],[28,3740,3741,3744],{"style":3660},[28,3742],{"className":3743,"style":578},[577],[28,3745,3747],{"className":3746},[582,583,584,585],[28,3748,1244],{"className":3749},[274,585],[28,3751,593],{"className":3752},[592],[28,3754,3756],{"className":3755},[565],[28,3757,3759],{"className":3758,"style":600},[569],[28,3760],{},[28,3762,3764,3767],{"className":3763},[274],[28,3765,180],{"className":3766,"style":298},[274,275],[28,3768,3770],{"className":3769},[556],[28,3771,3773,3793],{"className":3772},[560,561],[28,3774,3776,3790],{"className":3775},[565],[28,3777,3779],{"className":3778,"style":1285},[569],[28,3780,3781,3784],{"style":3660},[28,3782],{"className":3783,"style":578},[577],[28,3785,3787],{"className":3786},[582,583,584,585],[28,3788,89],{"className":3789},[274,585],[28,3791,593],{"className":3792},[592],[28,3794,3796],{"className":3795},[565],[28,3797,3799],{"className":3798,"style":600},[569],[28,3800],{},"。为了表达加法，需要 4 个全加器。每个全加器有三个输入 ",[28,3803,3805,3836],{"className":3804,"translate":193},[197],[28,3806,3808],{"className":3807},[201],[203,3809,3810],{"xmlns":205},[208,3811,3812,3833],{},[211,3813,3814,3816,3818,3821,3823],{},[214,3815,238],{},[217,3817,1683],{"separator":261},[214,3819,3820],{},"b",[217,3822,1683],{"separator":261},[501,3824,3825,3827],{},[214,3826,22],{},[211,3828,3829,3831],{},[214,3830,1686],{},[214,3832,3060],{},[253,3834,3835],{"encoding":255},"a, b, c_{in}",[28,3837,3839],{"className":3838,"ariaHidden":261},[260],[28,3840,3842,3845,3848,3851,3854,3857,3860,3863],{"className":3841},[265],[28,3843],{"className":3844,"style":2371},[269],[28,3846,238],{"className":3847},[274,275],[28,3849,1683],{"className":3850},[1723],[28,3852],{"className":3853,"style":530},[279],[28,3855,3820],{"className":3856},[274,275],[28,3858,1683],{"className":3859},[1723],[28,3861],{"className":3862,"style":530},[279],[28,3864,3866,3869],{"className":3865},[274],[28,3867,22],{"className":3868},[274,275],[28,3870,3872],{"className":3871},[556],[28,3873,3875,3900],{"className":3874},[560,561],[28,3876,3878,3897],{"className":3877},[565],[28,3879,3882],{"className":3880,"style":3881},[569],"height:0.3117em;",[28,3883,3884,3887],{"style":573},[28,3885],{"className":3886,"style":578},[577],[28,3888,3890],{"className":3889},[582,583,584,585],[28,3891,3893],{"className":3892},[274,585],[28,3894,3896],{"className":3895},[274,275,585],"in",[28,3898,593],{"className":3899},[592],[28,3901,3903],{"className":3902},[565],[28,3904,3906],{"className":3905,"style":600},[569],[28,3907],{},"、两个输出 ",[28,3910,3912,3942],{"className":3911,"translate":193},[197],[28,3913,3915],{"className":3914},[201],[203,3916,3917],{"xmlns":205},[208,3918,3919,3939],{},[211,3920,3921,3923,3925],{},[214,3922,1664],{},[217,3924,1683],{"separator":261},[501,3926,3927,3929],{},[214,3928,22],{},[211,3930,3931,3934,3937],{},[214,3932,3933],{},"o",[214,3935,3936],{},"u",[214,3938,1241],{},[253,3940,3941],{"encoding":255},"s, c_{out}",[28,3943,3945],{"className":3944,"ariaHidden":261},[260],[28,3946,3948,3951,3954,3957,3960],{"className":3947},[265],[28,3949],{"className":3950,"style":612},[269],[28,3952,1664],{"className":3953},[274,275],[28,3955,1683],{"className":3956},[1723],[28,3958],{"className":3959,"style":530},[279],[28,3961,3963,3966],{"className":3962},[274],[28,3964,22],{"className":3965},[274,275],[28,3967,3969],{"className":3968},[556],[28,3970,3972,4002],{"className":3971},[560,561],[28,3973,3975,3999],{"className":3974},[565],[28,3976,3979],{"className":3977,"style":3978},[569],"height:0.2806em;",[28,3980,3981,3984],{"style":573},[28,3982],{"className":3983,"style":578},[577],[28,3985,3987],{"className":3986},[582,583,584,585],[28,3988,3990,3993,3996],{"className":3989},[274,585],[28,3991,3933],{"className":3992},[274,275,585],[28,3994,3936],{"className":3995},[274,275,585],[28,3997,1241],{"className":3998},[274,275,585],[28,4000,593],{"className":4001},[592],[28,4003,4005],{"className":4004},[565],[28,4006,4008],{"className":4007,"style":600},[569],[28,4009],{},"，满足：",[28,4012,4014],{"className":4013,"translate":193},[192],[28,4015,4017,4109],{"className":4016,"translate":193},[197],[28,4018,4020],{"className":4019},[201],[203,4021,4022],{"xmlns":205,"display":206},[208,4023,4024,4106],{},[211,4025,4026,4028,4030,4032,4035,4037,4039,4049,4051,4054,4066,4068,4070,4072,4074,4076,4078,4080,4082,4092,4094,4096,4098,4100,4102,4104],{},[214,4027,1664],{},[217,4029,83],{},[214,4031,238],{},[217,4033,4034],{},"⊕",[214,4036,3820],{},[217,4038,4034],{},[501,4040,4041,4043],{},[214,4042,22],{},[211,4044,4045,4047],{},[214,4046,1686],{},[214,4048,3060],{},[217,4050,1683],{"separator":261},[279,4052],{"width":4053},"2em",[501,4055,4056,4058],{},[214,4057,22],{},[211,4059,4060,4062,4064],{},[214,4061,3933],{},[214,4063,3936],{},[214,4065,1241],{},[217,4067,83],{},[217,4069,43],{"stretchy":241},[214,4071,238],{},[217,4073,233],{},[214,4075,3820],{},[217,4077,1522],{"stretchy":241},[217,4079,937],{},[217,4081,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[501,4083,4084,4086],{},[214,4085,22],{},[211,4087,4088,4090],{},[214,4089,1686],{},[214,4091,3060],{},[217,4093,233],{},[217,4095,43],{"stretchy":241},[214,4097,238],{},[217,4099,4034],{},[214,4101,3820],{},[217,4103,1522],{"stretchy":241},[217,4105,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[253,4107,4108],{"encoding":255},"s = a \\oplus b \\oplus c_{in}, \\qquad c_{out} = (a \\wedge b) \\vee \\bigl(c_{in} \\wedge (a \\oplus b)\\bigr)",[28,4110,4112,4130,4148,4167,4284,4305,4326,4390,4411],{"className":4111,"ariaHidden":261},[260],[28,4113,4115,4118,4121,4124,4127],{"className":4114},[265],[28,4116],{"className":4117,"style":645},[269],[28,4119,1664],{"className":4120},[274,275],[28,4122],{"className":4123,"style":302},[279],[28,4125,83],{"className":4126},[306],[28,4128],{"className":4129,"style":302},[279],[28,4131,4133,4136,4139,4142,4145],{"className":4132},[265],[28,4134],{"className":4135,"style":270},[269],[28,4137,238],{"className":4138},[274,275],[28,4140],{"className":4141,"style":280},[279],[28,4143,4034],{"className":4144},[284],[28,4146],{"className":4147,"style":280},[279],[28,4149,4151,4155,4158,4161,4164],{"className":4150},[265],[28,4152],{"className":4153,"style":4154},[269],"height:0.7778em;vertical-align:-0.0833em;",[28,4156,3820],{"className":4157},[274,275],[28,4159],{"className":4160,"style":280},[279],[28,4162,4034],{"className":4163},[284],[28,4165],{"className":4166,"style":280},[279],[28,4168,4170,4173,4216,4219,4223,4226,4275,4278,4281],{"className":4169},[265],[28,4171],{"className":4172,"style":612},[269],[28,4174,4176,4179],{"className":4175},[274],[28,4177,22],{"className":4178},[274,275],[28,4180,4182],{"className":4181},[556],[28,4183,4185,4208],{"className":4184},[560,561],[28,4186,4188,4205],{"className":4187},[565],[28,4189,4191],{"className":4190,"style":3881},[569],[28,4192,4193,4196],{"style":573},[28,4194],{"className":4195,"style":578},[577],[28,4197,4199],{"className":4198},[582,583,584,585],[28,4200,4202],{"className":4201},[274,585],[28,4203,3896],{"className":4204},[274,275,585],[28,4206,593],{"className":4207},[592],[28,4209,4211],{"className":4210},[565],[28,4212,4214],{"className":4213,"style":600},[569],[28,4215],{},[28,4217,1683],{"className":4218},[1723],[28,4220],{"className":4221,"style":4222},[279],"margin-right:2em;",[28,4224],{"className":4225,"style":530},[279],[28,4227,4229,4232],{"className":4228},[274],[28,4230,22],{"className":4231},[274,275],[28,4233,4235],{"className":4234},[556],[28,4236,4238,4267],{"className":4237},[560,561],[28,4239,4241,4264],{"className":4240},[565],[28,4242,4244],{"className":4243,"style":3978},[569],[28,4245,4246,4249],{"style":573},[28,4247],{"className":4248,"style":578},[577],[28,4250,4252],{"className":4251},[582,583,584,585],[28,4253,4255,4258,4261],{"className":4254},[274,585],[28,4256,3933],{"className":4257},[274,275,585],[28,4259,3936],{"className":4260},[274,275,585],[28,4262,1241],{"className":4263},[274,275,585],[28,4265,593],{"className":4266},[592],[28,4268,4270],{"className":4269},[565],[28,4271,4273],{"className":4272,"style":600},[569],[28,4274],{},[28,4276],{"className":4277,"style":302},[279],[28,4279,83],{"className":4280},[306],[28,4282],{"className":4283,"style":302},[279],[28,4285,4287,4290,4293,4296,4299,4302],{"className":4286},[265],[28,4288],{"className":4289,"style":341},[269],[28,4291,43],{"className":4292},[348],[28,4294,238],{"className":4295},[274,275],[28,4297],{"className":4298,"style":280},[279],[28,4300,233],{"className":4301},[284],[28,4303],{"className":4304,"style":280},[279],[28,4306,4308,4311,4314,4317,4320,4323],{"className":4307},[265],[28,4309],{"className":4310,"style":341},[269],[28,4312,3820],{"className":4313},[274,275],[28,4315,1522],{"className":4316},[355],[28,4318],{"className":4319,"style":280},[279],[28,4321,937],{"className":4322},[284],[28,4324],{"className":4325,"style":280},[279],[28,4327,4329,4332,4338,4381,4384,4387],{"className":4328},[265],[28,4330],{"className":4331,"style":2469},[269],[28,4333,4335],{"className":4334},[348],[28,4336,43],{"className":4337},[2169,2170],[28,4339,4341,4344],{"className":4340},[274],[28,4342,22],{"className":4343},[274,275],[28,4345,4347],{"className":4346},[556],[28,4348,4350,4373],{"className":4349},[560,561],[28,4351,4353,4370],{"className":4352},[565],[28,4354,4356],{"className":4355,"style":3881},[569],[28,4357,4358,4361],{"style":573},[28,4359],{"className":4360,"style":578},[577],[28,4362,4364],{"className":4363},[582,583,584,585],[28,4365,4367],{"className":4366},[274,585],[28,4368,3896],{"className":4369},[274,275,585],[28,4371,593],{"className":4372},[592],[28,4374,4376],{"className":4375},[565],[28,4377,4379],{"className":4378,"style":600},[569],[28,4380],{},[28,4382],{"className":4383,"style":280},[279],[28,4385,233],{"className":4386},[284],[28,4388],{"className":4389,"style":280},[279],[28,4391,4393,4396,4399,4402,4405,4408],{"className":4392},[265],[28,4394],{"className":4395,"style":341},[269],[28,4397,43],{"className":4398},[348],[28,4400,238],{"className":4401},[274,275],[28,4403],{"className":4404,"style":280},[279],[28,4406,4034],{"className":4407},[284],[28,4409],{"className":4410,"style":280},[279],[28,4412,4414,4417,4420,4423],{"className":4413},[265],[28,4415],{"className":4416,"style":2469},[269],[28,4418,3820],{"className":4419},[274,275],[28,4421,1522],{"className":4422},[355],[28,4424,4426],{"className":4425},[355],[28,4427,1522],{"className":4428},[2169,2170],[11,4430,4431],{},"把这两个等式转成 CNF，一个全加器大约 17 条子句。4 位加法器就是 4 个全加器，约 70 条子句。再加上「和等于 1010、进位为 0」的 5 条单元子句，以及一个 4 位比较器的几十条子句，整个约束变成一百多条子句、几十个辅助变量。",[11,4433,4434],{},"而原始约束只有一行：",[18,4436,4438],{"className":20,"code":4437,"language":22,"meta":23,"style":23},"x + y == 10 && x > y\n",[25,4439,4440],{"__ignoreMap":23},[28,4441,4442,4445,4447,4449,4451,4453,4455,4458,4460],{"class":30,"line":31},[28,4443,4444],{"class":42},"x ",[28,4446,124],{"class":34},[28,4448,127],{"class":42},[28,4450,147],{"class":34},[28,4452,133],{"class":76},[28,4454,136],{"class":34},[28,4456,4457],{"class":42}," x ",[28,4459,130],{"class":34},[28,4461,4462],{"class":42}," y\n",[11,4464,4465,4466,177,4518,4570],{},"求解器可能给出 ",[28,4467,4469,4488],{"className":4468,"translate":193},[197],[28,4470,4472],{"className":4471},[201],[203,4473,4474],{"xmlns":205},[208,4475,4476,4485],{},[211,4477,4478,4480,4482],{},[214,4479,176],{},[217,4481,83],{},[224,4483,4484],{},"0110",[253,4486,4487],{"encoding":255},"x = 0110",[28,4489,4491,4509],{"className":4490,"ariaHidden":261},[260],[28,4492,4494,4497,4500,4503,4506],{"className":4493},[265],[28,4495],{"className":4496,"style":645},[269],[28,4498,176],{"className":4499},[274,275],[28,4501],{"className":4502,"style":302},[279],[28,4504,83],{"className":4505},[306],[28,4507],{"className":4508,"style":302},[279],[28,4510,4512,4515],{"className":4511},[265],[28,4513],{"className":4514,"style":316},[269],[28,4516,4484],{"className":4517},[274],[28,4519,4521,4540],{"className":4520,"translate":193},[197],[28,4522,4524],{"className":4523},[201],[203,4525,4526],{"xmlns":205},[208,4527,4528,4537],{},[211,4529,4530,4532,4534],{},[214,4531,180],{},[217,4533,83],{},[224,4535,4536],{},"0100",[253,4538,4539],{"encoding":255},"y = 0100",[28,4541,4543,4561],{"className":4542,"ariaHidden":261},[260],[28,4544,4546,4549,4552,4555,4558],{"className":4545},[265],[28,4547],{"className":4548,"style":612},[269],[28,4550,180],{"className":4551,"style":298},[274,275],[28,4553],{"className":4554,"style":302},[279],[28,4556,83],{"className":4557},[306],[28,4559],{"className":4560,"style":302},[279],[28,4562,4564,4567],{"className":4563},[265],[28,4565],{"className":4566,"style":316},[269],[28,4568,4536],{"className":4569},[274],"，但它并不知道这是 6 和 4。它只知道这一组真假值恰好让所有子句为真。",[11,4572,4573,4574,4577,4578,4674,4675,4703,4704,4732,4733,4703,4761,4789],{},"规模还不是最麻烦的，",[183,4575,4576],{},"语义丢失","才是。假设我们想证明 ",[28,4579,4581,4607],{"className":4580,"translate":193},[197],[28,4582,4584],{"className":4583},[201],[203,4585,4586],{"xmlns":205},[208,4587,4588,4604],{},[211,4589,4590,4592,4594,4596,4598,4600,4602],{},[214,4591,176],{},[217,4593,124],{},[214,4595,180],{},[217,4597,83],{},[214,4599,180],{},[217,4601,124],{},[214,4603,176],{},[253,4605,4606],{"encoding":255},"x + y = y + x",[28,4608,4610,4628,4646,4665],{"className":4609,"ariaHidden":261},[260],[28,4611,4613,4616,4619,4622,4625],{"className":4612},[265],[28,4614],{"className":4615,"style":270},[269],[28,4617,176],{"className":4618},[274,275],[28,4620],{"className":4621,"style":280},[279],[28,4623,124],{"className":4624},[284],[28,4626],{"className":4627,"style":280},[279],[28,4629,4631,4634,4637,4640,4643],{"className":4630},[265],[28,4632],{"className":4633,"style":612},[269],[28,4635,180],{"className":4636,"style":298},[274,275],[28,4638],{"className":4639,"style":302},[279],[28,4641,83],{"className":4642},[306],[28,4644],{"className":4645,"style":302},[279],[28,4647,4649,4653,4656,4659,4662],{"className":4648},[265],[28,4650],{"className":4651,"style":4652},[269],"height:0.7778em;vertical-align:-0.1944em;",[28,4654,180],{"className":4655,"style":298},[274,275],[28,4657],{"className":4658,"style":280},[279],[28,4660,124],{"className":4661},[284],[28,4663],{"className":4664,"style":280},[279],[28,4666,4668,4671],{"className":4667},[265],[28,4669],{"className":4670,"style":645},[269],[28,4672,176],{"className":4673},[274,275],"。在算术里这是一条公理，一秒都不用。但在 SAT 里，左边是一个「先加 ",[28,4676,4678,4691],{"className":4677,"translate":193},[197],[28,4679,4681],{"className":4680},[201],[203,4682,4683],{"xmlns":205},[208,4684,4685,4689],{},[211,4686,4687],{},[214,4688,176],{},[253,4690,176],{"encoding":255},[28,4692,4694],{"className":4693,"ariaHidden":261},[260],[28,4695,4697,4700],{"className":4696},[265],[28,4698],{"className":4699,"style":645},[269],[28,4701,176],{"className":4702},[274,275]," 再加 ",[28,4705,4707,4720],{"className":4706,"translate":193},[197],[28,4708,4710],{"className":4709},[201],[203,4711,4712],{"xmlns":205},[208,4713,4714,4718],{},[211,4715,4716],{},[214,4717,180],{},[253,4719,180],{"encoding":255},[28,4721,4723],{"className":4722,"ariaHidden":261},[260],[28,4724,4726,4729],{"className":4725},[265],[28,4727],{"className":4728,"style":612},[269],[28,4730,180],{"className":4731,"style":298},[274,275],"」的电路，右边是一个「先加 ",[28,4734,4736,4749],{"className":4735,"translate":193},[197],[28,4737,4739],{"className":4738},[201],[203,4740,4741],{"xmlns":205},[208,4742,4743,4747],{},[211,4744,4745],{},[214,4746,180],{},[253,4748,180],{"encoding":255},[28,4750,4752],{"className":4751,"ariaHidden":261},[260],[28,4753,4755,4758],{"className":4754},[265],[28,4756],{"className":4757,"style":612},[269],[28,4759,180],{"className":4760,"style":298},[274,275],[28,4762,4764,4777],{"className":4763,"translate":193},[197],[28,4765,4767],{"className":4766},[201],[203,4768,4769],{"xmlns":205},[208,4770,4771,4775],{},[211,4772,4773],{},[214,4774,176],{},[253,4776,176],{"encoding":255},[28,4778,4780],{"className":4779,"ariaHidden":261},[260],[28,4781,4783,4786],{"className":4782},[265],[28,4784],{"className":4785,"style":645},[269],[28,4787,176],{"className":4788},[274,275],"」的电路，两个电路内部结构不同，SAT 只能把两边都展开，再逐位证明输出相等。它不知道「加法可交换」，只知道一堆门。",[11,4791,4792],{},"打个比方：这就像为了判断一首诗是否押韵，先把它拆成一个个字母，再让机器从字母的排列里重新发现什么是韵脚。信息没有丢，但结构没了。",[419,4794,4796],{"id":4795},"_22-不可判定","2.2 不可判定",[11,4798,4799],{},"那干脆用表达力最强的一阶逻辑，让自动定理证明器去证，不行吗？",[11,4801,4802],{},"不行。看一个具体方程：",[28,4804,4806],{"className":4805,"translate":193},[192],[28,4807,4809,4849],{"className":4808,"translate":193},[197],[28,4810,4812],{"className":4811},[201],[203,4813,4814],{"xmlns":205,"display":206},[208,4815,4816,4846],{},[211,4817,4818,4824,4826,4832,4834,4841,4843],{},[3054,4819,4820,4822],{},[214,4821,176],{},[224,4823,3400],{},[217,4825,124],{},[3054,4827,4828,4830],{},[214,4829,180],{},[224,4831,3400],{},[217,4833,124],{},[3054,4835,4836,4839],{},[214,4837,4838],{},"z",[224,4840,3400],{},[217,4842,83],{},[224,4844,4845],{},"33",[253,4847,4848],{"encoding":255},"x^3 + y^3 + z^3 = 33",[28,4850,4852,4899,4944,4989],{"className":4851,"ariaHidden":261},[260],[28,4853,4855,4859,4890,4893,4896],{"className":4854},[265],[28,4856],{"className":4857,"style":4858},[269],"height:0.9474em;vertical-align:-0.0833em;",[28,4860,4862,4865],{"className":4861},[274],[28,4863,176],{"className":4864},[274,275],[28,4866,4868],{"className":4867},[556],[28,4869,4871],{"className":4870},[560],[28,4872,4874],{"className":4873},[565],[28,4875,4878],{"className":4876,"style":4877},[569],"height:0.8641em;",[28,4879,4881,4884],{"style":4880},"top:-3.113em;margin-right:0.05em;",[28,4882],{"className":4883,"style":578},[577],[28,4885,4887],{"className":4886},[582,583,584,585],[28,4888,3400],{"className":4889},[274,585],[28,4891],{"className":4892,"style":280},[279],[28,4894,124],{"className":4895},[284],[28,4897],{"className":4898,"style":280},[279],[28,4900,4902,4906,4935,4938,4941],{"className":4901},[265],[28,4903],{"className":4904,"style":4905},[269],"height:1.0585em;vertical-align:-0.1944em;",[28,4907,4909,4912],{"className":4908},[274],[28,4910,180],{"className":4911,"style":298},[274,275],[28,4913,4915],{"className":4914},[556],[28,4916,4918],{"className":4917},[560],[28,4919,4921],{"className":4920},[565],[28,4922,4924],{"className":4923,"style":4877},[569],[28,4925,4926,4929],{"style":4880},[28,4927],{"className":4928,"style":578},[577],[28,4930,4932],{"className":4931},[582,583,584,585],[28,4933,3400],{"className":4934},[274,585],[28,4936],{"className":4937,"style":280},[279],[28,4939,124],{"className":4940},[284],[28,4942],{"className":4943,"style":280},[279],[28,4945,4947,4950,4980,4983,4986],{"className":4946},[265],[28,4948],{"className":4949,"style":4877},[269],[28,4951,4953,4957],{"className":4952},[274],[28,4954,4838],{"className":4955,"style":4956},[274,275],"margin-right:0.044em;",[28,4958,4960],{"className":4959},[556],[28,4961,4963],{"className":4962},[560],[28,4964,4966],{"className":4965},[565],[28,4967,4969],{"className":4968,"style":4877},[569],[28,4970,4971,4974],{"style":4880},[28,4972],{"className":4973,"style":578},[577],[28,4975,4977],{"className":4976},[582,583,584,585],[28,4978,3400],{"className":4979},[274,585],[28,4981],{"className":4982,"style":302},[279],[28,4984,83],{"className":4985},[306],[28,4987],{"className":4988,"style":302},[279],[28,4990,4992,4995],{"className":4991},[265],[28,4993],{"className":4994,"style":316},[269],[28,4996,4845],{"className":4997},[274],[11,4999,5000],{},"它看起来很简单。它有整数解，一个已知解是：",[28,5002,5004],{"className":5003,"translate":193},[192],[28,5005,5007,5118],{"className":5006,"translate":193},[197],[28,5008,5010],{"className":5009},[201],[203,5011,5012],{"xmlns":205,"display":206},[208,5013,5014,5115],{},[5015,5016,5020,5052,5084],"mtable",{"rowspacing":5017,"columnalign":5018,"columnspacing":5019},"0.25em","right left","0em",[5021,5022,5023,5037],"mtr",{},[5024,5025,5026],"mtd",{},[5027,5028,5029],"mstyle",{"scriptlevel":89,"displaystyle":261},[211,5030,5031,5033,5035],{},[224,5032,4845],{},[217,5034,83],{},[211,5036],{},[5024,5038,5039],{},[5027,5040,5041],{"scriptlevel":89,"displaystyle":261},[211,5042,5043,5045],{},[211,5044],{},[3054,5046,5047,5050],{},[224,5048,5049],{},"8866128975287528",[224,5051,3400],{},[5021,5053,5054,5060],{},[5024,5055,5056],{},[5027,5057,5058],{"scriptlevel":89,"displaystyle":261},[211,5059],{},[5024,5061,5062],{},[5027,5063,5064],{"scriptlevel":89,"displaystyle":261},[211,5065,5066,5068,5070,5072,5075,5078],{},[211,5067],{},[217,5069,124],{},[217,5071,43],{"stretchy":241},[217,5073,5074],{},"−",[224,5076,5077],{},"8778405442862239",[3054,5079,5080,5082],{},[217,5081,1522],{"stretchy":241},[224,5083,3400],{},[5021,5085,5086,5092],{},[5024,5087,5088],{},[5027,5089,5090],{"scriptlevel":89,"displaystyle":261},[211,5091],{},[5024,5093,5094],{},[5027,5095,5096],{"scriptlevel":89,"displaystyle":261},[211,5097,5098,5100,5102,5104,5106,5109],{},[211,5099],{},[217,5101,124],{},[217,5103,43],{"stretchy":241},[217,5105,5074],{},[224,5107,5108],{},"2736111468807040",[3054,5110,5111,5113],{},[217,5112,1522],{"stretchy":241},[224,5114,3400],{},[253,5116,5117],{"encoding":255},"\\begin{aligned}\n33 ={}& 8866128975287528^3 \\\\\n      &+ (-8778405442862239)^3 \\\\\n      &+ (-2736111468807040)^3\n\\end{aligned}",[28,5119,5121],{"className":5120,"ariaHidden":261},[260],[28,5122,5124,5128],{"className":5123},[265],[28,5125],{"className":5126,"style":5127},[269],"height:4.2723em;vertical-align:-1.8862em;",[28,5129,5131],{"className":5130},[274],[28,5132,5134,5202],{"className":5133},[5015],[28,5135,5138],{"className":5136},[5137],"col-align-r",[28,5139,5141,5193],{"className":5140},[560,561],[28,5142,5144,5190],{"className":5143},[565],[28,5145,5148,5172,5181],{"className":5146,"style":5147},[569],"height:2.3862em;",[28,5149,5151,5154],{"style":5150},"top:-4.5221em;",[28,5152],{"className":5153,"style":1914},[577],[28,5155,5157,5160,5163,5166,5169],{"className":5156},[274],[28,5158,4845],{"className":5159},[274],[28,5161],{"className":5162,"style":302},[279],[28,5164,83],{"className":5165},[306],[28,5167],{"className":5168,"style":302},[279],[28,5170],{"className":5171},[274],[28,5173,5175,5178],{"style":5174},"top:-2.9979em;",[28,5176],{"className":5177,"style":1914},[577],[28,5179],{"className":5180},[274],[28,5182,5184,5187],{"style":5183},"top:-1.4738em;",[28,5185],{"className":5186,"style":1914},[577],[28,5188],{"className":5189},[274],[28,5191,593],{"className":5192},[592],[28,5194,5196],{"className":5195},[565],[28,5197,5200],{"className":5198,"style":5199},[569],"height:1.8862em;",[28,5201],{},[28,5203,5206],{"className":5204},[5205],"col-align-l",[28,5207,5209,5379],{"className":5208},[560,561],[28,5210,5212,5376],{"className":5211},[565],[28,5213,5215,5260,5318],{"className":5214,"style":5147},[569],[28,5216,5217,5220],{"style":5150},[28,5218],{"className":5219,"style":1914},[577],[28,5221,5223,5226,5230],{"className":5222},[274],[28,5224],{"className":5225},[274],[28,5227,5229],{"className":5228},[274],"886612897528752",[28,5231,5233,5237],{"className":5232},[274],[28,5234,5236],{"className":5235},[274],"8",[28,5238,5240],{"className":5239},[556],[28,5241,5243],{"className":5242},[560],[28,5244,5246],{"className":5245},[565],[28,5247,5249],{"className":5248,"style":4877},[569],[28,5250,5251,5254],{"style":4880},[28,5252],{"className":5253,"style":578},[577],[28,5255,5257],{"className":5256},[582,583,584,585],[28,5258,3400],{"className":5259},[274,585],[28,5261,5262,5265],{"style":5174},[28,5263],{"className":5264,"style":1914},[577],[28,5266,5268,5271,5274,5277,5280,5283,5286,5289],{"className":5267},[274],[28,5269],{"className":5270},[274],[28,5272],{"className":5273,"style":280},[279],[28,5275,124],{"className":5276},[284],[28,5278],{"className":5279,"style":280},[279],[28,5281,43],{"className":5282},[348],[28,5284,5074],{"className":5285},[274],[28,5287,5077],{"className":5288},[274],[28,5290,5292,5295],{"className":5291},[355],[28,5293,1522],{"className":5294},[355],[28,5296,5298],{"className":5297},[556],[28,5299,5301],{"className":5300},[560],[28,5302,5304],{"className":5303},[565],[28,5305,5307],{"className":5306,"style":4877},[569],[28,5308,5309,5312],{"style":4880},[28,5310],{"className":5311,"style":578},[577],[28,5313,5315],{"className":5314},[582,583,584,585],[28,5316,3400],{"className":5317},[274,585],[28,5319,5320,5323],{"style":5183},[28,5321],{"className":5322,"style":1914},[577],[28,5324,5326,5329,5332,5335,5338,5341,5344,5347],{"className":5325},[274],[28,5327],{"className":5328},[274],[28,5330],{"className":5331,"style":280},[279],[28,5333,124],{"className":5334},[284],[28,5336],{"className":5337,"style":280},[279],[28,5339,43],{"className":5340},[348],[28,5342,5074],{"className":5343},[274],[28,5345,5108],{"className":5346},[274],[28,5348,5350,5353],{"className":5349},[355],[28,5351,1522],{"className":5352},[355],[28,5354,5356],{"className":5355},[556],[28,5357,5359],{"className":5358},[560],[28,5360,5362],{"className":5361},[565],[28,5363,5365],{"className":5364,"style":4877},[569],[28,5366,5367,5370],{"style":4880},[28,5368],{"className":5369,"style":578},[577],[28,5371,5373],{"className":5372},[582,583,584,585],[28,5374,3400],{"className":5375},[274,585],[28,5377,593],{"className":5378},[592],[28,5380,5382],{"className":5381},[565],[28,5383,5385],{"className":5384,"style":5199},[569],[28,5386],{},[11,5388,5389,5390,5445,5446,5510,5511,5577],{},"这三个数都是 16 位数，直到 2019 年才被找到。如果把 ",[28,5391,5393,5415],{"className":5392,"translate":193},[197],[28,5394,5396],{"className":5395},[201],[203,5397,5398],{"xmlns":205},[208,5399,5400,5412],{},[211,5401,5402,5404,5406,5408,5410],{},[214,5403,176],{},[217,5405,1683],{"separator":261},[214,5407,180],{},[217,5409,1683],{"separator":261},[214,5411,4838],{},[253,5413,5414],{"encoding":255},"x, y, z",[28,5416,5418],{"className":5417,"ariaHidden":261},[260],[28,5419,5421,5424,5427,5430,5433,5436,5439,5442],{"className":5420},[265],[28,5422],{"className":5423,"style":612},[269],[28,5425,176],{"className":5426},[274,275],[28,5428,1683],{"className":5429},[1723],[28,5431],{"className":5432,"style":530},[279],[28,5434,180],{"className":5435,"style":298},[274,275],[28,5437,1683],{"className":5438},[1723],[28,5440],{"className":5441,"style":530},[279],[28,5443,4838],{"className":5444,"style":4956},[274,275]," 都编码成 64 位整数，搜索空间是 ",[28,5447,5449,5468],{"className":5448,"translate":193},[197],[28,5450,5452],{"className":5451},[201],[203,5453,5454],{"xmlns":205},[208,5455,5456,5465],{},[211,5457,5458],{},[3054,5459,5460,5462],{},[224,5461,1253],{},[224,5463,5464],{},"192",[253,5466,5467],{"encoding":255},"2^{192}",[28,5469,5471],{"className":5470,"ariaHidden":261},[260],[28,5472,5474,5478],{"className":5473},[265],[28,5475],{"className":5476,"style":5477},[269],"height:0.8141em;",[28,5479,5481,5484],{"className":5480},[274],[28,5482,1253],{"className":5483},[274],[28,5485,5487],{"className":5486},[556],[28,5488,5490],{"className":5489},[560],[28,5491,5493],{"className":5492},[565],[28,5494,5496],{"className":5495,"style":5477},[569],[28,5497,5498,5501],{"style":3094},[28,5499],{"className":5500,"style":578},[577],[28,5502,5504],{"className":5503},[582,583,584,585],[28,5505,5507],{"className":5506},[274,585],[28,5508,5464],{"className":5509},[274,585],"；按每秒试十亿个赋值来算，也要 ",[28,5512,5514,5533],{"className":5513,"translate":193},[197],[28,5515,5517],{"className":5516},[201],[203,5518,5519],{"xmlns":205},[208,5520,5521,5530],{},[211,5522,5523],{},[3054,5524,5525,5527],{},[224,5526,226],{},[224,5528,5529],{},"41",[253,5531,5532],{"encoding":255},"10^{41}",[28,5534,5536],{"className":5535,"ariaHidden":261},[260],[28,5537,5539,5542,5545],{"className":5538},[265],[28,5540],{"className":5541,"style":5477},[269],[28,5543,1244],{"className":5544},[274],[28,5546,5548,5551],{"className":5547},[274],[28,5549,89],{"className":5550},[274],[28,5552,5554],{"className":5553},[556],[28,5555,5557],{"className":5556},[560],[28,5558,5560],{"className":5559},[565],[28,5561,5563],{"className":5562,"style":5477},[569],[28,5564,5565,5568],{"style":3094},[28,5566],{"className":5567,"style":578},[577],[28,5569,5571],{"className":5570},[582,583,584,585],[28,5572,5574],{"className":5573},[274,585],[28,5575,5529],{"className":5576},[274,585]," 年量级。位爆破在这里基本不可行。",[11,5579,5580],{},"更根本的是，这不是「这一个方程特别难」的问题。希尔伯特第十问题问的是「是否存在算法判定整系数多项式方程有没有整数解」，这个问题在 1970 年被马蒂亚谢维奇证明为否。证明的思路很巧妙：任给一台图灵机和一段输入，都能构造出一个整系数多项式方程，使方程有解当且仅当图灵机停机。既然停机问题不可判定，一般的方程判定问题也不可判定。换句话说，想写一个能判定所有这类方程的通用算法，这件事本身就是不可能的。",[11,5582,5583],{},"工程上的后果很直接：自动定理证明器可能跑了一整天，最后告诉你「我不知道」。对于一个需要每秒判定成千上万条路径条件的程序分析工具来说，这是不可接受的。",[419,5585,5587],{"id":5586},"_23-难以组合","2.3 难以组合",[11,5589,5590],{},"另一条路是「一个问题配一个专用算法」：线性规划用单纯形法，只懂实数线性约束；并查集只懂「相等」；BDD 只懂布尔函数；自动机只懂正则语言和字符串匹配。",[11,5592,5593],{},"这些算法在各自领域都很好用，但真实的程序约束往往是混合的。看一个只有两个原子的例子：",[28,5595,5597],{"className":5596,"translate":193},[192],[28,5598,5600,5642],{"className":5599,"translate":193},[197],[28,5601,5603],{"className":5602},[201],[203,5604,5605],{"xmlns":205,"display":206},[208,5606,5607,5639],{},[211,5608,5609,5611,5613,5615,5617,5619,5621,5623,5625,5627,5629,5631,5633,5635,5637],{},[214,5610,176],{},[217,5612,83],{},[214,5614,180],{},[228,5616,230],{},[217,5618,233],{},[228,5620,230],{},[214,5622,1511],{},[217,5624,43],{"stretchy":241},[214,5626,176],{},[217,5628,1522],{"stretchy":241},[217,5630,1854],{"mathvariant":484},[214,5632,1511],{},[217,5634,43],{"stretchy":241},[214,5636,180],{},[217,5638,1522],{"stretchy":241},[253,5640,5641],{"encoding":255},"x = y \\;\\wedge\\; f(x) \\ne f(y)",[28,5643,5645,5663,5687,5747],{"className":5644,"ariaHidden":261},[260],[28,5646,5648,5651,5654,5657,5660],{"className":5647},[265],[28,5649],{"className":5650,"style":645},[269],[28,5652,176],{"className":5653},[274,275],[28,5655],{"className":5656,"style":302},[279],[28,5658,83],{"className":5659},[306],[28,5661],{"className":5662,"style":302},[279],[28,5664,5666,5669,5672,5675,5678,5681,5684],{"className":5665},[265],[28,5667],{"className":5668,"style":1083},[269],[28,5670,180],{"className":5671,"style":298},[274,275],[28,5673],{"className":5674,"style":302},[279],[28,5676],{"className":5677,"style":280},[279],[28,5679,233],{"className":5680},[284],[28,5682],{"className":5683,"style":302},[279],[28,5685],{"className":5686,"style":280},[279],[28,5688,5690,5693,5696,5699,5702,5705,5708,5744],{"className":5689},[265],[28,5691],{"className":5692,"style":341},[269],[28,5694,1511],{"className":5695,"style":1546},[274,275],[28,5697,43],{"className":5698},[348],[28,5700,176],{"className":5701},[274,275],[28,5703,1522],{"className":5704},[355],[28,5706],{"className":5707,"style":302},[279],[28,5709,5711,5738,5741],{"className":5710},[306],[28,5712,5714],{"className":5713},[306],[28,5715,5717],{"className":5716},[274,2359],[28,5718,5720],{"className":5719},[2363],[28,5721,5723,5726,5735],{"className":5722},[2367],[28,5724],{"className":5725,"style":2371},[269],[28,5727,5729],{"className":5728},[2375],[28,5730,5732],{"className":5731},[274],[28,5733,2382],{"className":5734},[306],[28,5736],{"className":5737},[2386],[28,5739],{"className":5740},[279,2390],[28,5742,83],{"className":5743},[306],[28,5745],{"className":5746,"style":302},[279],[28,5748,5750,5753,5756,5759,5762],{"className":5749},[265],[28,5751],{"className":5752,"style":341},[269],[28,5754,1511],{"className":5755,"style":1546},[274,275],[28,5757,43],{"className":5758},[348],[28,5760,180],{"className":5761,"style":298},[274,275],[28,5763,1522],{"className":5764},[355],[11,5766,5767],{},"假设我们分别去问两个求解器：",[2480,5769,5770,5902],{},[2483,5771,5772,5773,5824,5825,5827,5828,5901],{},"算术求解器只看 ",[28,5774,5776,5794],{"className":5775,"translate":193},[197],[28,5777,5779],{"className":5778},[201],[203,5780,5781],{"xmlns":205},[208,5782,5783,5791],{},[211,5784,5785,5787,5789],{},[214,5786,176],{},[217,5788,83],{},[214,5790,180],{},[253,5792,5793],{"encoding":255},"x = y",[28,5795,5797,5815],{"className":5796,"ariaHidden":261},[260],[28,5798,5800,5803,5806,5809,5812],{"className":5799},[265],[28,5801],{"className":5802,"style":645},[269],[28,5804,176],{"className":5805},[274,275],[28,5807],{"className":5808,"style":302},[279],[28,5810,83],{"className":5811},[306],[28,5813],{"className":5814,"style":302},[279],[28,5816,5818,5821],{"className":5817},[265],[28,5819],{"className":5820,"style":612},[269],[28,5822,180],{"className":5823,"style":298},[274,275],"，回答 ",[183,5826,405],{},"，比如 ",[28,5829,5831,5853],{"className":5830,"translate":193},[197],[28,5832,5834],{"className":5833},[201],[203,5835,5836],{"xmlns":205},[208,5837,5838,5850],{},[211,5839,5840,5842,5844,5846,5848],{},[214,5841,176],{},[217,5843,83],{},[214,5845,180],{},[217,5847,83],{},[224,5849,89],{},[253,5851,5852],{"encoding":255},"x = y = 0",[28,5854,5856,5874,5892],{"className":5855,"ariaHidden":261},[260],[28,5857,5859,5862,5865,5868,5871],{"className":5858},[265],[28,5860],{"className":5861,"style":645},[269],[28,5863,176],{"className":5864},[274,275],[28,5866],{"className":5867,"style":302},[279],[28,5869,83],{"className":5870},[306],[28,5872],{"className":5873,"style":302},[279],[28,5875,5877,5880,5883,5886,5889],{"className":5876},[265],[28,5878],{"className":5879,"style":612},[269],[28,5881,180],{"className":5882,"style":298},[274,275],[28,5884],{"className":5885,"style":302},[279],[28,5887,83],{"className":5888},[306],[28,5890],{"className":5891,"style":302},[279],[28,5893,5895,5898],{"className":5894},[265],[28,5896],{"className":5897,"style":316},[269],[28,5899,89],{"className":5900},[274],"；",[2483,5903,5904,5905,6018,6019,6021,6022,6063,6064,177,6130,2923],{},"EUF 求解器只看 ",[28,5906,5908,5937],{"className":5907,"translate":193},[197],[28,5909,5911],{"className":5910},[201],[203,5912,5913],{"xmlns":205},[208,5914,5915,5935],{},[211,5916,5917,5919,5921,5923,5925,5927,5929,5931,5933],{},[214,5918,1511],{},[217,5920,43],{"stretchy":241},[214,5922,176],{},[217,5924,1522],{"stretchy":241},[217,5926,1854],{"mathvariant":484},[214,5928,1511],{},[217,5930,43],{"stretchy":241},[214,5932,180],{},[217,5934,1522],{"stretchy":241},[253,5936,2732],{"encoding":255},[28,5938,5940,6000],{"className":5939,"ariaHidden":261},[260],[28,5941,5943,5946,5949,5952,5955,5958,5961,5997],{"className":5942},[265],[28,5944],{"className":5945,"style":341},[269],[28,5947,1511],{"className":5948,"style":1546},[274,275],[28,5950,43],{"className":5951},[348],[28,5953,176],{"className":5954},[274,275],[28,5956,1522],{"className":5957},[355],[28,5959],{"className":5960,"style":302},[279],[28,5962,5964,5991,5994],{"className":5963},[306],[28,5965,5967],{"className":5966},[306],[28,5968,5970],{"className":5969},[274,2359],[28,5971,5973],{"className":5972},[2363],[28,5974,5976,5979,5988],{"className":5975},[2367],[28,5977],{"className":5978,"style":2371},[269],[28,5980,5982],{"className":5981},[2375],[28,5983,5985],{"className":5984},[274],[28,5986,2382],{"className":5987},[306],[28,5989],{"className":5990},[2386],[28,5992],{"className":5993},[279,2390],[28,5995,83],{"className":5996},[306],[28,5998],{"className":5999,"style":302},[279],[28,6001,6003,6006,6009,6012,6015],{"className":6002},[265],[28,6004],{"className":6005,"style":341},[269],[28,6007,1511],{"className":6008,"style":1546},[274,275],[28,6010,43],{"className":6011},[348],[28,6013,180],{"className":6014,"style":298},[274,275],[28,6016,1522],{"className":6017},[355],"，也回答 ",[183,6020,405],{},"：它把 ",[28,6023,6025,6042],{"className":6024,"translate":193},[197],[28,6026,6028],{"className":6027},[201],[203,6029,6030],{"xmlns":205},[208,6031,6032,6040],{},[211,6033,6034,6036,6038],{},[214,6035,176],{},[217,6037,1683],{"separator":261},[214,6039,180],{},[253,6041,3226],{"encoding":255},[28,6043,6045],{"className":6044,"ariaHidden":261},[260],[28,6046,6048,6051,6054,6057,6060],{"className":6047},[265],[28,6049],{"className":6050,"style":612},[269],[28,6052,176],{"className":6053},[274,275],[28,6055,1683],{"className":6056},[1723],[28,6058],{"className":6059,"style":530},[279],[28,6061,180],{"className":6062,"style":298},[274,275]," 当成两个不同的常量，令 ",[28,6065,6067,6091],{"className":6066,"translate":193},[197],[28,6068,6070],{"className":6069},[201],[203,6071,6072],{"xmlns":205},[208,6073,6074,6088],{},[211,6075,6076,6078,6080,6082,6084,6086],{},[214,6077,1511],{},[217,6079,43],{"stretchy":241},[214,6081,176],{},[217,6083,1522],{"stretchy":241},[217,6085,83],{},[224,6087,89],{},[253,6089,6090],{"encoding":255},"f(x) = 0",[28,6092,6094,6121],{"className":6093,"ariaHidden":261},[260],[28,6095,6097,6100,6103,6106,6109,6112,6115,6118],{"className":6096},[265],[28,6098],{"className":6099,"style":341},[269],[28,6101,1511],{"className":6102,"style":1546},[274,275],[28,6104,43],{"className":6105},[348],[28,6107,176],{"className":6108},[274,275],[28,6110,1522],{"className":6111},[355],[28,6113],{"className":6114,"style":302},[279],[28,6116,83],{"className":6117},[306],[28,6119],{"className":6120,"style":302},[279],[28,6122,6124,6127],{"className":6123},[265],[28,6125],{"className":6126,"style":316},[269],[28,6128,89],{"className":6129},[274],[28,6131,6133,6157],{"className":6132,"translate":193},[197],[28,6134,6136],{"className":6135},[201],[203,6137,6138],{"xmlns":205},[208,6139,6140,6154],{},[211,6141,6142,6144,6146,6148,6150,6152],{},[214,6143,1511],{},[217,6145,43],{"stretchy":241},[214,6147,180],{},[217,6149,1522],{"stretchy":241},[217,6151,83],{},[224,6153,1244],{},[253,6155,6156],{"encoding":255},"f(y) = 1",[28,6158,6160,6187],{"className":6159,"ariaHidden":261},[260],[28,6161,6163,6166,6169,6172,6175,6178,6181,6184],{"className":6162},[265],[28,6164],{"className":6165,"style":341},[269],[28,6167,1511],{"className":6168,"style":1546},[274,275],[28,6170,43],{"className":6171},[348],[28,6173,180],{"className":6174,"style":298},[274,275],[28,6176,1522],{"className":6177},[355],[28,6179],{"className":6180,"style":302},[279],[28,6182,83],{"className":6183},[306],[28,6185],{"className":6186,"style":302},[279],[28,6188,6190,6193],{"className":6189},[265],[28,6191],{"className":6192,"style":316},[269],[28,6194,1244],{"className":6195},[274],[11,6197,6198,6199,6249,6250,6294,6295,6339,6340,6453,6454,2923],{},"如果把两个答案一拼，说「两边都可满足，所以整体可满足」，就错了。因为 ",[28,6200,6202,6219],{"className":6201,"translate":193},[197],[28,6203,6205],{"className":6204},[201],[203,6206,6207],{"xmlns":205},[208,6208,6209,6217],{},[211,6210,6211,6213,6215],{},[214,6212,176],{},[217,6214,83],{},[214,6216,180],{},[253,6218,5793],{"encoding":255},[28,6220,6222,6240],{"className":6221,"ariaHidden":261},[260],[28,6223,6225,6228,6231,6234,6237],{"className":6224},[265],[28,6226],{"className":6227,"style":645},[269],[28,6229,176],{"className":6230},[274,275],[28,6232],{"className":6233,"style":302},[279],[28,6235,83],{"className":6236},[306],[28,6238],{"className":6239,"style":302},[279],[28,6241,6243,6246],{"className":6242},[265],[28,6244],{"className":6245,"style":612},[269],[28,6247,180],{"className":6248,"style":298},[274,275]," 意味着 ",[28,6251,6253,6273],{"className":6252,"translate":193},[197],[28,6254,6256],{"className":6255},[201],[203,6257,6258],{"xmlns":205},[208,6259,6260,6270],{},[211,6261,6262,6264,6266,6268],{},[214,6263,1511],{},[217,6265,43],{"stretchy":241},[214,6267,176],{},[217,6269,1522],{"stretchy":241},[253,6271,6272],{"encoding":255},"f(x)",[28,6274,6276],{"className":6275,"ariaHidden":261},[260],[28,6277,6279,6282,6285,6288,6291],{"className":6278},[265],[28,6280],{"className":6281,"style":341},[269],[28,6283,1511],{"className":6284,"style":1546},[274,275],[28,6286,43],{"className":6287},[348],[28,6289,176],{"className":6290},[274,275],[28,6292,1522],{"className":6293},[355]," 必须等于 ",[28,6296,6298,6318],{"className":6297,"translate":193},[197],[28,6299,6301],{"className":6300},[201],[203,6302,6303],{"xmlns":205},[208,6304,6305,6315],{},[211,6306,6307,6309,6311,6313],{},[214,6308,1511],{},[217,6310,43],{"stretchy":241},[214,6312,180],{},[217,6314,1522],{"stretchy":241},[253,6316,6317],{"encoding":255},"f(y)",[28,6319,6321],{"className":6320,"ariaHidden":261},[260],[28,6322,6324,6327,6330,6333,6336],{"className":6323},[265],[28,6325],{"className":6326,"style":341},[269],[28,6328,1511],{"className":6329,"style":1546},[274,275],[28,6331,43],{"className":6332},[348],[28,6334,180],{"className":6335,"style":298},[274,275],[28,6337,1522],{"className":6338},[355],"，与 ",[28,6341,6343,6372],{"className":6342,"translate":193},[197],[28,6344,6346],{"className":6345},[201],[203,6347,6348],{"xmlns":205},[208,6349,6350,6370],{},[211,6351,6352,6354,6356,6358,6360,6362,6364,6366,6368],{},[214,6353,1511],{},[217,6355,43],{"stretchy":241},[214,6357,176],{},[217,6359,1522],{"stretchy":241},[217,6361,1854],{"mathvariant":484},[214,6363,1511],{},[217,6365,43],{"stretchy":241},[214,6367,180],{},[217,6369,1522],{"stretchy":241},[253,6371,2732],{"encoding":255},[28,6373,6375,6435],{"className":6374,"ariaHidden":261},[260],[28,6376,6378,6381,6384,6387,6390,6393,6396,6432],{"className":6377},[265],[28,6379],{"className":6380,"style":341},[269],[28,6382,1511],{"className":6383,"style":1546},[274,275],[28,6385,43],{"className":6386},[348],[28,6388,176],{"className":6389},[274,275],[28,6391,1522],{"className":6392},[355],[28,6394],{"className":6395,"style":302},[279],[28,6397,6399,6426,6429],{"className":6398},[306],[28,6400,6402],{"className":6401},[306],[28,6403,6405],{"className":6404},[274,2359],[28,6406,6408],{"className":6407},[2363],[28,6409,6411,6414,6423],{"className":6410},[2367],[28,6412],{"className":6413,"style":2371},[269],[28,6415,6417],{"className":6416},[2375],[28,6418,6420],{"className":6419},[274],[28,6421,2382],{"className":6422},[306],[28,6424],{"className":6425},[2386],[28,6427],{"className":6428},[279,2390],[28,6430,83],{"className":6431},[306],[28,6433],{"className":6434,"style":302},[279],[28,6436,6438,6441,6444,6447,6450],{"className":6437},[265],[28,6439],{"className":6440,"style":341},[269],[28,6442,1511],{"className":6443,"style":1546},[274,275],[28,6445,43],{"className":6446},[348],[28,6448,180],{"className":6449,"style":298},[274,275],[28,6451,1522],{"className":6452},[355]," 直接矛盾。整体其实是 ",[183,6455,6456],{},"unsat",[11,6458,6459,6460,6510,6511,6592],{},"要得到正确答案，两个求解器必须交换信息：算术把等式 ",[28,6461,6463,6480],{"className":6462,"translate":193},[197],[28,6464,6466],{"className":6465},[201],[203,6467,6468],{"xmlns":205},[208,6469,6470,6478],{},[211,6471,6472,6474,6476],{},[214,6473,176],{},[217,6475,83],{},[214,6477,180],{},[253,6479,5793],{"encoding":255},[28,6481,6483,6501],{"className":6482,"ariaHidden":261},[260],[28,6484,6486,6489,6492,6495,6498],{"className":6485},[265],[28,6487],{"className":6488,"style":645},[269],[28,6490,176],{"className":6491},[274,275],[28,6493],{"className":6494,"style":302},[279],[28,6496,83],{"className":6497},[306],[28,6499],{"className":6500,"style":302},[279],[28,6502,6504,6507],{"className":6503},[265],[28,6505],{"className":6506,"style":612},[269],[28,6508,180],{"className":6509,"style":298},[274,275]," 告诉 EUF，EUF 立刻推出 ",[28,6512,6514,6544],{"className":6513,"translate":193},[197],[28,6515,6517],{"className":6516},[201],[203,6518,6519],{"xmlns":205},[208,6520,6521,6541],{},[211,6522,6523,6525,6527,6529,6531,6533,6535,6537,6539],{},[214,6524,1511],{},[217,6526,43],{"stretchy":241},[214,6528,176],{},[217,6530,1522],{"stretchy":241},[217,6532,83],{},[214,6534,1511],{},[217,6536,43],{"stretchy":241},[214,6538,180],{},[217,6540,1522],{"stretchy":241},[253,6542,6543],{"encoding":255},"f(x) = f(y)",[28,6545,6547,6574],{"className":6546,"ariaHidden":261},[260],[28,6548,6550,6553,6556,6559,6562,6565,6568,6571],{"className":6549},[265],[28,6551],{"className":6552,"style":341},[269],[28,6554,1511],{"className":6555,"style":1546},[274,275],[28,6557,43],{"className":6558},[348],[28,6560,176],{"className":6561},[274,275],[28,6563,1522],{"className":6564},[355],[28,6566],{"className":6567,"style":302},[279],[28,6569,83],{"className":6570},[306],[28,6572],{"className":6573,"style":302},[279],[28,6575,6577,6580,6583,6586,6589],{"className":6576},[265],[28,6578],{"className":6579,"style":341},[269],[28,6581,1511],{"className":6582,"style":1546},[274,275],[28,6584,43],{"className":6585},[348],[28,6587,180],{"className":6588,"style":298},[274,275],[28,6590,1522],{"className":6591},[355],"，冲突。谁来负责交换？交换哪些事实？什么时候交换？在 SMT 之前，这些都是需要人工设计的事情。",[11,6594,6595,6596,6599],{},"数组和算术之间也有同样的坑。",[25,6597,6598],{},"i = j ∧ a[i] ≠ a[j]"," 单独看两边都可满足，合起来却不可满足，因为下标相等就意味着读到的值相等。没有任何一个专用算法能独自看出这一点。",[419,6601,6603],{"id":6602},"_24-门槛太高","2.4 门槛太高",[11,6605,6606,6607,2923],{},"还有一条路是 Coq、Isabelle、ACL2 这类交互式定理证明器。它们的表达力极强，能处理带量词、带归纳的复杂性质，但代价是",[183,6608,6609],{},"需要人写证明脚本",[11,6611,6612],{},"比如在 Coq 里证明「任何自然数加零都等于它自己」，要写：",[18,6614,6618],{"className":6615,"code":6616,"language":6617,"meta":23,"style":23},"language-coq shiki shiki-themes github-light github-light github-dark","Theorem plus_0_r : forall n : nat, n + 0 = n.\nProof.\n  induction n as [| n' IH].\n  - simpl. reflexivity.\n  - simpl. rewrite IH. reflexivity.\nQed.\n","coq",[25,6619,6620,6625,6630,6635,6640,6645],{"__ignoreMap":23},[28,6621,6622],{"class":30,"line":31},[28,6623,6624],{},"Theorem plus_0_r : forall n : nat, n + 0 = n.\n",[28,6626,6627],{"class":30,"line":64},[28,6628,6629],{},"Proof.\n",[28,6631,6632],{"class":30,"line":95},[28,6633,6634],{},"  induction n as [| n' IH].\n",[28,6636,6637],{"class":30,"line":115},[28,6638,6639],{},"  - simpl. reflexivity.\n",[28,6641,6642],{"class":30,"line":154},[28,6643,6644],{},"  - simpl. rewrite IH. reflexivity.\n",[28,6646,6647],{"class":30,"line":161},[28,6648,6649],{},"Qed.\n",[11,6651,6652],{},"这几行不是装饰：",[2480,6654,6655,6816,6909],{},[2483,6656,6657,6660,6661,2693,6712,5901],{},[25,6658,6659],{},"induction n"," 把目标拆成两种情况：",[28,6662,6664,6682],{"className":6663,"translate":193},[197],[28,6665,6667],{"className":6666},[201],[203,6668,6669],{"xmlns":205},[208,6670,6671,6679],{},[211,6672,6673,6675,6677],{},[214,6674,3060],{},[217,6676,83],{},[224,6678,89],{},[253,6680,6681],{"encoding":255},"n = 0",[28,6683,6685,6703],{"className":6684,"ariaHidden":261},[260],[28,6686,6688,6691,6694,6697,6700],{"className":6687},[265],[28,6689],{"className":6690,"style":645},[269],[28,6692,3060],{"className":6693},[274,275],[28,6695],{"className":6696,"style":302},[279],[28,6698,83],{"className":6699},[306],[28,6701],{"className":6702,"style":302},[279],[28,6704,6706,6709],{"className":6705},[265],[28,6707],{"className":6708,"style":316},[269],[28,6710,89],{"className":6711},[274],[28,6713,6715,6745],{"className":6714,"translate":193},[197],[28,6716,6718],{"className":6717},[201],[203,6719,6720],{"xmlns":205},[208,6721,6722,6742],{},[211,6723,6724,6726,6728,6731,6733,6740],{},[214,6725,3060],{},[217,6727,83],{},[214,6729,6730],{},"S",[217,6732,43],{"stretchy":241},[3054,6734,6735,6737],{},[214,6736,3060],{},[217,6738,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},"′",[217,6741,1522],{"stretchy":241},[253,6743,6744],{"encoding":255},"n = S(n')",[28,6746,6748,6766],{"className":6747,"ariaHidden":261},[260],[28,6749,6751,6754,6757,6760,6763],{"className":6750},[265],[28,6752],{"className":6753,"style":645},[269],[28,6755,3060],{"className":6756},[274,275],[28,6758],{"className":6759,"style":302},[279],[28,6761,83],{"className":6762},[306],[28,6764],{"className":6765,"style":302},[279],[28,6767,6769,6773,6777,6780,6813],{"className":6768},[265],[28,6770],{"className":6771,"style":6772},[269],"height:1.0019em;vertical-align:-0.25em;",[28,6774,6730],{"className":6775,"style":6776},[274,275],"margin-right:0.0576em;",[28,6778,43],{"className":6779},[348],[28,6781,6783,6786],{"className":6782},[274],[28,6784,3060],{"className":6785},[274,275],[28,6787,6789],{"className":6788},[556],[28,6790,6792],{"className":6791},[560],[28,6793,6795],{"className":6794},[565],[28,6796,6799],{"className":6797,"style":6798},[569],"height:0.7519em;",[28,6800,6801,6804],{"style":3094},[28,6802],{"className":6803,"style":578},[577],[28,6805,6807],{"className":6806},[582,583,584,585],[28,6808,6810],{"className":6809},[274,585],[28,6811,6739],{"className":6812},[274,585],[28,6814,1522],{"className":6815},[355],[2483,6817,6818,6819,6822,6823,6875,6876,6904,6905,6908],{},"第一种情况，",[25,6820,6821],{},"simpl"," 把 ",[28,6824,6826,6844],{"className":6825,"translate":193},[197],[28,6827,6829],{"className":6828},[201],[203,6830,6831],{"xmlns":205},[208,6832,6833,6841],{},[211,6834,6835,6837,6839],{},[224,6836,89],{},[217,6838,124],{},[224,6840,89],{},[253,6842,6843],{"encoding":255},"0 + 0",[28,6845,6847,6866],{"className":6846,"ariaHidden":261},[260],[28,6848,6850,6854,6857,6860,6863],{"className":6849},[265],[28,6851],{"className":6852,"style":6853},[269],"height:0.7278em;vertical-align:-0.0833em;",[28,6855,89],{"className":6856},[274],[28,6858],{"className":6859,"style":280},[279],[28,6861,124],{"className":6862},[284],[28,6864],{"className":6865,"style":280},[279],[28,6867,6869,6872],{"className":6868},[265],[28,6870],{"className":6871,"style":316},[269],[28,6873,89],{"className":6874},[274]," 化简成 ",[28,6877,6879,6892],{"className":6878,"translate":193},[197],[28,6880,6882],{"className":6881},[201],[203,6883,6884],{"xmlns":205},[208,6885,6886,6890],{},[211,6887,6888],{},[224,6889,89],{},[253,6891,89],{"encoding":255},[28,6893,6895],{"className":6894,"ariaHidden":261},[260],[28,6896,6898,6901],{"className":6897},[265],[28,6899],{"className":6900,"style":316},[269],[28,6902,89],{"className":6903},[274],"，",[25,6906,6907],{},"reflexivity"," 确认两边相等；",[2483,6910,6911,6912,6822,6914,7013,7014,7113,7114,6822,7117,7202,7203,7265,7266,7268],{},"第二种情况，",[25,6913,6821],{},[28,6915,6917,6945],{"className":6916,"translate":193},[197],[28,6918,6920],{"className":6919},[201],[203,6921,6922],{"xmlns":205},[208,6923,6924,6942],{},[211,6925,6926,6928,6930,6936,6938,6940],{},[214,6927,6730],{},[217,6929,43],{"stretchy":241},[3054,6931,6932,6934],{},[214,6933,3060],{},[217,6935,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[217,6937,1522],{"stretchy":241},[217,6939,124],{},[224,6941,89],{},[253,6943,6944],{"encoding":255},"S(n') + 0",[28,6946,6948,7004],{"className":6947,"ariaHidden":261},[260],[28,6949,6951,6954,6957,6960,6992,6995,6998,7001],{"className":6950},[265],[28,6952],{"className":6953,"style":6772},[269],[28,6955,6730],{"className":6956,"style":6776},[274,275],[28,6958,43],{"className":6959},[348],[28,6961,6963,6966],{"className":6962},[274],[28,6964,3060],{"className":6965},[274,275],[28,6967,6969],{"className":6968},[556],[28,6970,6972],{"className":6971},[560],[28,6973,6975],{"className":6974},[565],[28,6976,6978],{"className":6977,"style":6798},[569],[28,6979,6980,6983],{"style":3094},[28,6981],{"className":6982,"style":578},[577],[28,6984,6986],{"className":6985},[582,583,584,585],[28,6987,6989],{"className":6988},[274,585],[28,6990,6739],{"className":6991},[274,585],[28,6993,1522],{"className":6994},[355],[28,6996],{"className":6997,"style":280},[279],[28,6999,124],{"className":7000},[284],[28,7002],{"className":7003,"style":280},[279],[28,7005,7007,7010],{"className":7006},[265],[28,7008],{"className":7009,"style":316},[269],[28,7011,89],{"className":7012},[274]," 化成 ",[28,7015,7017,7045],{"className":7016,"translate":193},[197],[28,7018,7020],{"className":7019},[201],[203,7021,7022],{"xmlns":205},[208,7023,7024,7042],{},[211,7025,7026,7028,7030,7036,7038,7040],{},[214,7027,6730],{},[217,7029,43],{"stretchy":241},[3054,7031,7032,7034],{},[214,7033,3060],{},[217,7035,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[217,7037,124],{},[224,7039,89],{},[217,7041,1522],{"stretchy":241},[253,7043,7044],{"encoding":255},"S(n' + 0)",[28,7046,7048,7101],{"className":7047,"ariaHidden":261},[260],[28,7049,7051,7054,7057,7060,7092,7095,7098],{"className":7050},[265],[28,7052],{"className":7053,"style":6772},[269],[28,7055,6730],{"className":7056,"style":6776},[274,275],[28,7058,43],{"className":7059},[348],[28,7061,7063,7066],{"className":7062},[274],[28,7064,3060],{"className":7065},[274,275],[28,7067,7069],{"className":7068},[556],[28,7070,7072],{"className":7071},[560],[28,7073,7075],{"className":7074},[565],[28,7076,7078],{"className":7077,"style":6798},[569],[28,7079,7080,7083],{"style":3094},[28,7081],{"className":7082,"style":578},[577],[28,7084,7086],{"className":7085},[582,583,584,585],[28,7087,7089],{"className":7088},[274,585],[28,7090,6739],{"className":7091},[274,585],[28,7093],{"className":7094,"style":280},[279],[28,7096,124],{"className":7097},[284],[28,7099],{"className":7100,"style":280},[279],[28,7102,7104,7107,7110],{"className":7103},[265],[28,7105],{"className":7106,"style":341},[269],[28,7108,89],{"className":7109},[274],[28,7111,1522],{"className":7112},[355],"，再用归纳假设 ",[25,7115,7116],{},"IH",[28,7118,7120,7142],{"className":7119,"translate":193},[197],[28,7121,7123],{"className":7122},[201],[203,7124,7125],{"xmlns":205},[208,7126,7127,7139],{},[211,7128,7129,7135,7137],{},[3054,7130,7131,7133],{},[214,7132,3060],{},[217,7134,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[217,7136,124],{},[224,7138,89],{},[253,7140,7141],{"encoding":255},"n' + 0",[28,7143,7145,7193],{"className":7144,"ariaHidden":261},[260],[28,7146,7148,7152,7184,7187,7190],{"className":7147},[265],[28,7149],{"className":7150,"style":7151},[269],"height:0.8352em;vertical-align:-0.0833em;",[28,7153,7155,7158],{"className":7154},[274],[28,7156,3060],{"className":7157},[274,275],[28,7159,7161],{"className":7160},[556],[28,7162,7164],{"className":7163},[560],[28,7165,7167],{"className":7166},[565],[28,7168,7170],{"className":7169,"style":6798},[569],[28,7171,7172,7175],{"style":3094},[28,7173],{"className":7174,"style":578},[577],[28,7176,7178],{"className":7177},[582,583,584,585],[28,7179,7181],{"className":7180},[274,585],[28,7182,6739],{"className":7183},[274,585],[28,7185],{"className":7186,"style":280},[279],[28,7188,124],{"className":7189},[284],[28,7191],{"className":7192,"style":280},[279],[28,7194,7196,7199],{"className":7195},[265],[28,7197],{"className":7198,"style":316},[269],[28,7200,89],{"className":7201},[274]," 换成 ",[28,7204,7206,7224],{"className":7205,"translate":193},[197],[28,7207,7209],{"className":7208},[201],[203,7210,7211],{"xmlns":205},[208,7212,7213,7221],{},[211,7214,7215],{},[3054,7216,7217,7219],{},[214,7218,3060],{},[217,7220,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[253,7222,7223],{"encoding":255},"n'",[28,7225,7227],{"className":7226,"ariaHidden":261},[260],[28,7228,7230,7233],{"className":7229},[265],[28,7231],{"className":7232,"style":6798},[269],[28,7234,7236,7239],{"className":7235},[274],[28,7237,3060],{"className":7238},[274,275],[28,7240,7242],{"className":7241},[556],[28,7243,7245],{"className":7244},[560],[28,7246,7248],{"className":7247},[565],[28,7249,7251],{"className":7250,"style":6798},[569],[28,7252,7253,7256],{"style":3094},[28,7254],{"className":7255,"style":578},[577],[28,7257,7259],{"className":7258},[582,583,584,585],[28,7260,7262],{"className":7261},[274,585],[28,7263,6739],{"className":7264},[274,585],"，最后 ",[25,7267,6907],{}," 收尾。",[11,7270,7271],{},"每一步往哪走、用哪条归纳假设，都是人选的。这还只是自然数加零。真实的程序验证要证循环不变式、数组不变式、指针分离，脚本可能几百上千行。",[11,7273,7274],{},"程序分析场景里，工具每秒要生成并判定几千条路径条件。不可能每条都停下来等工程师写一段证明。自动化程度不够，就无法工业部署。",[419,7276,7278],{"id":7277},"_25-小结","2.5 小结",[11,7280,7281],{},"几条路，各有各的难处：",[2480,7283,7284,7287,7290,7293],{},[2483,7285,7286],{},"纯 SAT 快，但把高层语义压成门电路之后，规模爆炸、语义丢失；",[2483,7288,7289],{},"一阶逻辑表达力强，但一般情形不可判定，自动证明器可能永远给不出答案；",[2483,7291,7292],{},"专用算法在自己的理论里很好用，碰到混合约束就各自为政；",[2483,7294,7295],{},"交互式定理证明能处理复杂性质，但每条路径都等人写脚本，工业化无从谈起。",[11,7297,7298,7299],{},"SMT 要做的，就是在这几条路的夹缝里找一个中间地带：",[183,7300,7301],{},"表达力足够描述程序，判定过程保证自动，还能把多个理论组合起来。",[414,7303,7305],{"id":7304},"三smt-的解法","三、SMT 的解法",[419,7307,7309],{"id":7308},"_31-收缩片段","3.1 收缩片段",[11,7311,7312,7313,7316],{},"SMT 的第一个策略是",[183,7314,7315],{},"主动放弃一部分表达力","，只保留可判定的片段。",[11,7318,7319],{},"具体做法是：把主战场放在无量词片段，只允许特定理论的组合。这样既避开了 2.2 节里的不可判定性，又保留了描述程序行为所需的大部分能力。",[11,7321,7322,7323,6904,7352,7418],{},"为什么可以放弃量词？因为程序分析产生的验证条件，大多数天然就是无量词的。符号执行收集到的路径条件，是对「这一条路径上各个变量的具体取值」的约束，不需要「对所有 ",[28,7324,7326,7339],{"className":7325,"translate":193},[197],[28,7327,7329],{"className":7328},[201],[203,7330,7331],{"xmlns":205},[208,7332,7333,7337],{},[211,7334,7335],{},[214,7336,1686],{},[253,7338,1686],{"encoding":255},[28,7340,7342],{"className":7341,"ariaHidden":261},[260],[28,7343,7345,7349],{"className":7344},[265],[28,7346],{"className":7347,"style":7348},[269],"height:0.6595em;",[28,7350,1686],{"className":7351},[274,275],[28,7353,7355,7379],{"className":7354,"translate":193},[197],[28,7356,7358],{"className":7357},[201],[203,7359,7360],{"xmlns":205},[208,7361,7362,7376],{},[211,7363,7364,7366,7368,7370,7372,7374],{},[214,7365,238],{},[217,7367,73],{"stretchy":241},[214,7369,1686],{},[217,7371,246],{"stretchy":241},[217,7373,130],{},[224,7375,89],{},[253,7377,7378],{"encoding":255},"a[i] > 0",[28,7380,7382,7409],{"className":7381,"ariaHidden":261},[260],[28,7383,7385,7388,7391,7394,7397,7400,7403,7406],{"className":7384},[265],[28,7386],{"className":7387,"style":341},[269],[28,7389,238],{"className":7390},[274,275],[28,7392,73],{"className":7393},[348],[28,7395,1686],{"className":7396},[274,275],[28,7398,246],{"className":7399},[355],[28,7401],{"className":7402,"style":302},[279],[28,7404,130],{"className":7405},[306],[28,7407],{"className":7408,"style":302},[279],[28,7410,7412,7415],{"className":7411},[265],[28,7413],{"className":7414,"style":316},[269],[28,7416,89],{"className":7417},[274],"」这种全称陈述。",[11,7420,7421,7422,7453],{},"需要量词的地方——循环不变式、数组的整体性质、归纳证明——通常由上层工具负责：抽象解释负责生成不变式，",[28,7423,7425,7439],{"className":7424,"translate":193},[197],[28,7426,7428],{"className":7427},[201],[203,7429,7430],{"xmlns":205},[208,7431,7432,7437],{},[211,7433,7434],{},[214,7435,7436],{},"k",[253,7438,7436],{"encoding":255},[28,7440,7442],{"className":7441,"ariaHidden":261},[260],[28,7443,7445,7449],{"className":7444},[265],[28,7446],{"className":7447,"style":7448},[269],"height:0.6944em;",[28,7450,7436],{"className":7451,"style":7452},[274,275],"margin-right:0.0315em;","-induction 负责把归纳化归为有限次查询，SMT 只作为底层的判定引擎。这是一个分工，而不是缺陷。",[11,7455,7456,7457,177,7459,177,7461,7463,7464,7466,7467,7469,7470,7472],{},"当然，「可判定」不等于「容易」。",[25,7458,2940],{},[25,7460,2954],{},[25,7462,3039],{}," 都是 NP-complete；",[25,7465,2950],{}," 是多项式时间；",[25,7468,3027],{}," 可判定，但最坏情况是双指数；",[25,7471,3030],{}," 则直接不可判定。",[11,7474,7475],{},"SMT 在选定的片段内保证了「答案一定会来」，但不保证「答案来得快」。工程上的成败，很大程度上取决于启发式和问题编码。",[419,7477,7479],{"id":7478},"_32-直接表达","3.2 直接表达",[11,7481,7482,7483,2923],{},"第二个策略是",[183,7484,7485],{},"让公式直接说人话",[11,7487,7488,7489,7561,7562,7627,7628,7669],{},"同样是「",[28,7490,7492,7513],{"className":7491,"translate":193},[197],[28,7493,7495],{"className":7494},[201],[203,7496,7497],{"xmlns":205},[208,7498,7499,7511],{},[211,7500,7501,7503,7505,7507,7509],{},[214,7502,176],{},[217,7504,124],{},[214,7506,180],{},[217,7508,130],{},[224,7510,226],{},[253,7512,390],{"encoding":255},[28,7514,7516,7534,7552],{"className":7515,"ariaHidden":261},[260],[28,7517,7519,7522,7525,7528,7531],{"className":7518},[265],[28,7520],{"className":7521,"style":270},[269],[28,7523,176],{"className":7524},[274,275],[28,7526],{"className":7527,"style":280},[279],[28,7529,124],{"className":7530},[284],[28,7532],{"className":7533,"style":280},[279],[28,7535,7537,7540,7543,7546,7549],{"className":7536},[265],[28,7538],{"className":7539,"style":294},[269],[28,7541,180],{"className":7542,"style":298},[274,275],[28,7544],{"className":7545,"style":302},[279],[28,7547,130],{"className":7548},[306],[28,7550],{"className":7551,"style":302},[279],[28,7553,7555,7558],{"className":7554},[265],[28,7556],{"className":7557,"style":316},[269],[28,7559,226],{"className":7560},[274]," 且 ",[28,7563,7565,7588],{"className":7564,"translate":193},[197],[28,7566,7568],{"className":7567},[201],[203,7569,7570],{"xmlns":205},[208,7571,7572,7586],{},[211,7573,7574,7576,7578,7580,7582,7584],{},[214,7575,238],{},[217,7577,73],{"stretchy":241},[224,7579,89],{},[217,7581,246],{"stretchy":241},[217,7583,83],{},[224,7585,251],{},[253,7587,394],{"encoding":255},[28,7589,7591,7618],{"className":7590,"ariaHidden":261},[260],[28,7592,7594,7597,7600,7603,7606,7609,7612,7615],{"className":7593},[265],[28,7595],{"className":7596,"style":341},[269],[28,7598,238],{"className":7599},[274,275],[28,7601,73],{"className":7602},[348],[28,7604,89],{"className":7605},[274],[28,7607,246],{"className":7608},[355],[28,7610],{"className":7611,"style":302},[279],[28,7613,83],{"className":7614},[306],[28,7616],{"className":7617,"style":302},[279],[28,7619,7621,7624],{"className":7620},[265],[28,7622],{"className":7623,"style":316},[269],[28,7625,251],{"className":7626},[274],"」，SMT 不需要把 ",[28,7629,7631,7648],{"className":7630,"translate":193},[197],[28,7632,7634],{"className":7633},[201],[203,7635,7636],{"xmlns":205},[208,7637,7638,7646],{},[211,7639,7640,7642,7644],{},[214,7641,176],{},[217,7643,1683],{"separator":261},[214,7645,180],{},[253,7647,3226],{"encoding":255},[28,7649,7651],{"className":7650,"ariaHidden":261},[260],[28,7652,7654,7657,7660,7663,7666],{"className":7653},[265],[28,7655],{"className":7656,"style":612},[269],[28,7658,176],{"className":7659},[274,275],[28,7661,1683],{"className":7662},[1723],[28,7664],{"className":7665,"style":530},[279],[28,7667,180],{"className":7668,"style":298},[274,275]," 拆成 64 个布尔变量，也不需要把数组展开成 100 个元素。它把这两个条件原样保留为两个理论原子，交给对应的求解器。",[11,7671,7672],{},"关键在于「按需展开」：",[2480,7674,7675,7681,7687],{},[2483,7676,7677,7680],{},[183,7678,7679],{},"算术","：直接用单纯形法或分支定界，在整数\u002F实数域上推理，完全不经过布尔编码。",[2483,7682,7683,7686],{},[183,7684,7685],{},"数组","：用 read-over-write 公理，只在公式里真的出现某个下标时才实例化。",[2483,7688,7689,7692,7693,7696,7697,177,7769,7842],{},[183,7690,7691],{},"位向量","：这一条要诚实一点——大多数求解器内部",[183,7694,7695],{},"仍然会做位爆破","，因为位级编码对硬件电路太有效了。但这件事发生在求解器内部，只针对相关子表达式，而且会配合字级重写（比如 ",[28,7698,7700,7727],{"className":7699,"translate":193},[197],[28,7701,7703],{"className":7702},[201],[203,7704,7705],{"xmlns":205},[208,7706,7707,7724],{},[211,7708,7709,7711,7713,7716,7718,7720,7722],{},[214,7710,176],{},[228,7712,488],{},[214,7714,7715],{"mathvariant":484},"&",[228,7717,488],{},[224,7719,89],{},[217,7721,83],{},[224,7723,89],{},[253,7725,7726],{"encoding":255},"x \\,\\&\\, 0 = 0",[28,7728,7730,7760],{"className":7729,"ariaHidden":261},[260],[28,7731,7733,7736,7739,7742,7745,7748,7751,7754,7757],{"className":7732},[265],[28,7734],{"className":7735,"style":7448},[269],[28,7737,176],{"className":7738},[274,275],[28,7740],{"className":7741,"style":530},[279],[28,7743,7715],{"className":7744},[274],[28,7746],{"className":7747,"style":530},[279],[28,7749,89],{"className":7750},[274],[28,7752],{"className":7753,"style":302},[279],[28,7755,83],{"className":7756},[306],[28,7758],{"className":7759,"style":302},[279],[28,7761,7763,7766],{"className":7762},[265],[28,7764],{"className":7765,"style":316},[269],[28,7767,89],{"className":7768},[274],[28,7770,7772,7794],{"className":7771,"translate":193},[197],[28,7773,7775],{"className":7774},[201],[203,7776,7777],{"xmlns":205},[208,7778,7779,7791],{},[211,7780,7781,7783,7785,7787,7789],{},[214,7782,176],{},[217,7784,5074],{},[214,7786,176],{},[217,7788,83],{},[224,7790,89],{},[253,7792,7793],{"encoding":255},"x - x = 0",[28,7795,7797,7815,7833],{"className":7796,"ariaHidden":261},[260],[28,7798,7800,7803,7806,7809,7812],{"className":7799},[265],[28,7801],{"className":7802,"style":270},[269],[28,7804,176],{"className":7805},[274,275],[28,7807],{"className":7808,"style":280},[279],[28,7810,5074],{"className":7811},[284],[28,7813],{"className":7814,"style":280},[279],[28,7816,7818,7821,7824,7827,7830],{"className":7817},[265],[28,7819],{"className":7820,"style":645},[269],[28,7822,176],{"className":7823},[274,275],[28,7825],{"className":7826,"style":302},[279],[28,7828,83],{"className":7829},[306],[28,7831],{"className":7832,"style":302},[279],[28,7834,7836,7839],{"className":7835},[265],[28,7837],{"className":7838,"style":316},[269],[28,7840,89],{"className":7841},[274],"）和等价类合并。对使用者来说，它仍然是透明的。",[11,7844,7845],{},"数组的 read-over-write 公理长这样：",[28,7847,7849],{"className":7848,"translate":193},[192],[28,7850,7852,7994],{"className":7851,"translate":193},[197],[28,7853,7855],{"className":7854},[201],[203,7856,7857],{"xmlns":205,"display":206},[208,7858,7859,7991],{},[211,7860,7861,7875,7877,7889,7891,7893,7895,7897,7899,7901,7903,7905,7908,7910,7912],{},[211,7862,7863,7865,7867,7869,7871,7873],{},[214,7864,1664],{"mathvariant":484},[214,7866,1667],{"mathvariant":484},[214,7868,1670],{"mathvariant":484},[214,7870,1667],{"mathvariant":484},[214,7872,22],{"mathvariant":484},[214,7874,1241],{"mathvariant":484},[217,7876,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[211,7878,7879,7881,7883,7885,7887],{},[214,7880,1664],{"mathvariant":484},[214,7882,1241],{"mathvariant":484},[214,7884,3933],{"mathvariant":484},[214,7886,1867],{"mathvariant":484},[214,7888,1667],{"mathvariant":484},[217,7890,43],{"stretchy":241},[214,7892,238],{},[217,7894,1683],{"separator":261},[214,7896,1686],{},[217,7898,1683],{"separator":261},[214,7900,1693],{},[217,7902,1522],{"stretchy":241},[217,7904,1683],{"separator":261},[214,7906,7907],{},"j",[217,7909,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,7911,83],{},[211,7913,7914,7917],{},[217,7915,7916],{"fence":261},"{",[5015,7918,7921,7945],{"rowspacing":7919,"columnalign":7920,"columnspacing":497},"0.36em","left left",[5021,7922,7923,7933],{},[5024,7924,7925],{},[5027,7926,7927],{"scriptlevel":89,"displaystyle":241},[211,7928,7929,7931],{},[214,7930,1693],{},[217,7932,1683],{"separator":261},[5024,7934,7935],{},[5027,7936,7937],{"scriptlevel":89,"displaystyle":241},[211,7938,7939,7941,7943],{},[214,7940,1686],{},[217,7942,83],{},[214,7944,7907],{},[5021,7946,7947,7979],{},[5024,7948,7949],{},[5027,7950,7951],{"scriptlevel":89,"displaystyle":241},[211,7952,7953,7967,7969,7971,7973,7975,7977],{},[211,7954,7955,7957,7959,7961,7963,7965],{},[214,7956,1664],{"mathvariant":484},[214,7958,1667],{"mathvariant":484},[214,7960,1670],{"mathvariant":484},[214,7962,1667],{"mathvariant":484},[214,7964,22],{"mathvariant":484},[214,7966,1241],{"mathvariant":484},[217,7968,43],{"stretchy":241},[214,7970,238],{},[217,7972,1683],{"separator":261},[214,7974,7907],{},[217,7976,1522],{"stretchy":241},[217,7978,1683],{"separator":261},[5024,7980,7981],{},[5027,7982,7983],{"scriptlevel":89,"displaystyle":241},[211,7984,7985,7987,7989],{},[214,7986,1686],{},[217,7988,1854],{"mathvariant":484},[214,7990,7907],{},[253,7992,7993],{"encoding":255},"\\mathrm{select}\\bigl(\\mathrm{store}(a, i, v), j\\bigr) =\n\\begin{cases}\nv, & i = j \\\\\n\\mathrm{select}(a, j), & i \\ne j\n\\end{cases}",[28,7995,7997,8073],{"className":7996,"ariaHidden":261},[260],[28,7998,8000,8003,8009,8015,8021,8024,8027,8030,8033,8036,8039,8042,8045,8048,8051,8054,8058,8064,8067,8070],{"className":7999},[265],[28,8001],{"className":8002,"style":2469},[269],[28,8004,8006],{"className":8005},[274],[28,8007,1713],{"className":8008},[274,1712],[28,8010,8012],{"className":8011},[348],[28,8013,43],{"className":8014},[2169,2170],[28,8016,8018],{"className":8017},[274],[28,8019,2696],{"className":8020},[274,1712],[28,8022,43],{"className":8023},[348],[28,8025,238],{"className":8026},[274,275],[28,8028,1683],{"className":8029},[1723],[28,8031],{"className":8032,"style":530},[279],[28,8034,1686],{"className":8035},[274,275],[28,8037,1683],{"className":8038},[1723],[28,8040],{"className":8041,"style":530},[279],[28,8043,1693],{"className":8044,"style":298},[274,275],[28,8046,1522],{"className":8047},[355],[28,8049,1683],{"className":8050},[1723],[28,8052],{"className":8053,"style":530},[279],[28,8055,7907],{"className":8056,"style":8057},[274,275],"margin-right:0.0572em;",[28,8059,8061],{"className":8060},[355],[28,8062,1522],{"className":8063},[2169,2170],[28,8065],{"className":8066,"style":302},[279],[28,8068,83],{"className":8069},[306],[28,8071],{"className":8072,"style":302},[279],[28,8074,8076,8080],{"className":8075},[265],[28,8077],{"className":8078,"style":8079},[269],"height:3em;vertical-align:-1.25em;",[28,8081,8083,8092,8282],{"className":8082},[1897],[28,8084,8088],{"className":8085,"style":8087},[348,8086],"delimcenter","top:0em;",[28,8089,7916],{"className":8090},[2169,8091],"size4",[28,8093,8095],{"className":8094},[274],[28,8096,8098,8175,8180],{"className":8097},[5015],[28,8099,8101],{"className":8100},[5205],[28,8102,8104,8166],{"className":8103},[560,561],[28,8105,8107,8163],{"className":8106},[565],[28,8108,8111,8127],{"className":8109,"style":8110},[569],"height:1.69em;",[28,8112,8114,8118],{"style":8113},"top:-3.69em;",[28,8115],{"className":8116,"style":8117},[577],"height:3.008em;",[28,8119,8121,8124],{"className":8120},[274],[28,8122,1693],{"className":8123,"style":298},[274,275],[28,8125,1683],{"className":8126},[1723],[28,8128,8130,8133],{"style":8129},"top:-2.25em;",[28,8131],{"className":8132,"style":8117},[577],[28,8134,8136,8142,8145,8148,8151,8154,8157,8160],{"className":8135},[274],[28,8137,8139],{"className":8138},[274],[28,8140,1713],{"className":8141},[274,1712],[28,8143,43],{"className":8144},[348],[28,8146,238],{"className":8147},[274,275],[28,8149,1683],{"className":8150},[1723],[28,8152],{"className":8153,"style":530},[279],[28,8155,7907],{"className":8156,"style":8057},[274,275],[28,8158,1522],{"className":8159},[355],[28,8161,1683],{"className":8162},[1723],[28,8164,593],{"className":8165},[592],[28,8167,8169],{"className":8168},[565],[28,8170,8173],{"className":8171,"style":8172},[569],"height:1.19em;",[28,8174],{},[28,8176],{"className":8177,"style":8179},[8178],"arraycolsep","width:1em;",[28,8181,8183],{"className":8182},[5205],[28,8184,8186,8274],{"className":8185},[560,561],[28,8187,8189,8271],{"className":8188},[565],[28,8190,8192,8215],{"className":8191,"style":8110},[569],[28,8193,8194,8197],{"style":8113},[28,8195],{"className":8196,"style":8117},[577],[28,8198,8200,8203,8206,8209,8212],{"className":8199},[274],[28,8201,1686],{"className":8202},[274,275],[28,8204],{"className":8205,"style":302},[279],[28,8207,83],{"className":8208},[306],[28,8210],{"className":8211,"style":302},[279],[28,8213,7907],{"className":8214,"style":8057},[274,275],[28,8216,8217,8220],{"style":8129},[28,8218],{"className":8219,"style":8117},[577],[28,8221,8223,8226,8229,8265,8268],{"className":8222},[274],[28,8224,1686],{"className":8225},[274,275],[28,8227],{"className":8228,"style":302},[279],[28,8230,8232,8259,8262],{"className":8231},[306],[28,8233,8235],{"className":8234},[306],[28,8236,8238],{"className":8237},[274,2359],[28,8239,8241],{"className":8240},[2363],[28,8242,8244,8247,8256],{"className":8243},[2367],[28,8245],{"className":8246,"style":2371},[269],[28,8248,8250],{"className":8249},[2375],[28,8251,8253],{"className":8252},[274],[28,8254,2382],{"className":8255},[306],[28,8257],{"className":8258},[2386],[28,8260],{"className":8261},[279,2390],[28,8263,83],{"className":8264},[306],[28,8266],{"className":8267,"style":302},[279],[28,8269,7907],{"className":8270,"style":8057},[274,275],[28,8272,593],{"className":8273},[592],[28,8275,8277],{"className":8276},[565],[28,8278,8280],{"className":8279,"style":8172},[569],[28,8281],{},[28,8283],{"className":8284},[355,8285],"nulldelimiter",[11,8287,8288,8289,8333,8334,8385],{},"如果公式里只关心 ",[28,8290,8292,8312],{"className":8291,"translate":193},[197],[28,8293,8295],{"className":8294},[201],[203,8296,8297],{"xmlns":205},[208,8298,8299,8309],{},[211,8300,8301,8303,8305,8307],{},[214,8302,238],{},[217,8304,73],{"stretchy":241},[224,8306,89],{},[217,8308,246],{"stretchy":241},[253,8310,8311],{"encoding":255},"a[0]",[28,8313,8315],{"className":8314,"ariaHidden":261},[260],[28,8316,8318,8321,8324,8327,8330],{"className":8317},[265],[28,8319],{"className":8320,"style":341},[269],[28,8322,238],{"className":8323},[274,275],[28,8325,73],{"className":8326},[348],[28,8328,89],{"className":8329},[274],[28,8331,246],{"className":8332},[355],"，求解器就只实例化 ",[28,8335,8337,8355],{"className":8336,"translate":193},[197],[28,8338,8340],{"className":8339},[201],[203,8341,8342],{"xmlns":205},[208,8343,8344,8352],{},[211,8345,8346,8348,8350],{},[214,8347,1686],{},[217,8349,83],{},[224,8351,89],{},[253,8353,8354],{"encoding":255},"i = 0",[28,8356,8358,8376],{"className":8357,"ariaHidden":261},[260],[28,8359,8361,8364,8367,8370,8373],{"className":8360},[265],[28,8362],{"className":8363,"style":7348},[269],[28,8365,1686],{"className":8366},[274,275],[28,8368],{"className":8369,"style":302},[279],[28,8371,83],{"className":8372},[306],[28,8374],{"className":8375,"style":302},[279],[28,8377,8379,8382],{"className":8378},[265],[28,8380],{"className":8381,"style":316},[269],[28,8383,89],{"className":8384},[274]," 这一条，剩下的 99 个元素根本不会出现。这是「按需」和「全局」的区别，也是 SMT 能在数组上跑赢位爆破的原因。",[11,8387,8388,8389,8392],{},"所以更准确的说法不是「SMT 消灭了底层编码」，而是：",[183,8390,8391],{},"SMT 把「怎么编码」的决定权从用户手里交给了求解器","。求解器可以根据理论特性选择最合适的内部表示，而不是让用户先做一次不可逆的全局展开。",[11,8394,8395,8396,8437,8438,2923],{},"用同一个约束对比一下。纯 SAT 的版本要生成数百到上千个布尔变量、上千条子句；SMT 的版本只保留两个理论原子。SAT 只能做布尔剪枝，SMT 能利用算术和数组的语义；SAT 的答案是一组布尔赋值，还要反解出 ",[28,8397,8399,8416],{"className":8398,"translate":193},[197],[28,8400,8402],{"className":8401},[201],[203,8403,8404],{"xmlns":205},[208,8405,8406,8414],{},[211,8407,8408,8410,8412],{},[214,8409,176],{},[217,8411,1683],{"separator":261},[214,8413,180],{},[253,8415,3226],{"encoding":255},[28,8417,8419],{"className":8418,"ariaHidden":261},[260],[28,8420,8422,8425,8428,8431,8434],{"className":8421},[265],[28,8423],{"className":8424,"style":612},[269],[28,8426,176],{"className":8427},[274,275],[28,8429,1683],{"className":8430},[1723],[28,8432],{"className":8433,"style":530},[279],[28,8435,180],{"className":8436,"style":298},[274,275],"，SMT 直接给出 ",[28,8439,8441,8467],{"className":8440,"translate":193},[197],[28,8442,8444],{"className":8443},[201],[203,8445,8446],{"xmlns":205},[208,8447,8448,8465],{},[211,8449,8450,8452,8454,8457,8459,8461,8463],{},[214,8451,176],{},[217,8453,83],{},[224,8455,8456],{},"6",[217,8458,1683],{"separator":261},[214,8460,180],{},[217,8462,83],{},[224,8464,251],{},[253,8466,398],{"encoding":255},[28,8468,8470,8488,8516],{"className":8469,"ariaHidden":261},[260],[28,8471,8473,8476,8479,8482,8485],{"className":8472},[265],[28,8474],{"className":8475,"style":645},[269],[28,8477,176],{"className":8478},[274,275],[28,8480],{"className":8481,"style":302},[279],[28,8483,83],{"className":8484},[306],[28,8486],{"className":8487,"style":302},[279],[28,8489,8491,8495,8498,8501,8504,8507,8510,8513],{"className":8490},[265],[28,8492],{"className":8493,"style":8494},[269],"height:0.8389em;vertical-align:-0.1944em;",[28,8496,8456],{"className":8497},[274],[28,8499,1683],{"className":8500},[1723],[28,8502],{"className":8503,"style":530},[279],[28,8505,180],{"className":8506,"style":298},[274,275],[28,8508],{"className":8509,"style":302},[279],[28,8511,83],{"className":8512},[306],[28,8514],{"className":8515,"style":302},[279],[28,8517,8519,8522],{"className":8518},[265],[28,8520],{"className":8521,"style":316},[269],[28,8523,251],{"className":8524},[274],[419,8526,8528],{"id":8527},"_33-组合理论","3.3 组合理论",[11,8530,8531,8532,2923],{},"第三个策略是",[183,8533,8534],{},"让不同理论通过共享等式来协作",[11,8536,8537,8538,8541],{},"最经典的方法是 Nelson–Oppen 组合。它的前提是：两个理论的签名不相交（各自管各自的符号），并且都满足「稳定无限」性质（任何可满足的公式都有无穷模型）。在这个前提下，两个理论只需要交换",[183,8539,8540],{},"变量之间的等式","，就能组合成一个完整的判定过程。",[11,8543,8544],{},"看一个例子：",[28,8546,8548],{"className":8547,"translate":193},[192],[28,8549,8551,8592],{"className":8550,"translate":193},[197],[28,8552,8554],{"className":8553},[201],[203,8555,8556],{"xmlns":205,"display":206},[208,8557,8558,8590],{},[211,8559,8560,8562,8564,8566,8568,8570,8572,8574,8576,8578,8580,8582,8584,8586,8588],{},[214,8561,176],{},[217,8563,83],{},[214,8565,180],{},[228,8567,230],{},[217,8569,233],{},[228,8571,230],{},[214,8573,1511],{},[217,8575,43],{"stretchy":241},[214,8577,176],{},[217,8579,1522],{"stretchy":241},[217,8581,1854],{"mathvariant":484},[214,8583,1511],{},[217,8585,43],{"stretchy":241},[214,8587,180],{},[217,8589,1522],{"stretchy":241},[253,8591,5641],{"encoding":255},[28,8593,8595,8613,8637,8697],{"className":8594,"ariaHidden":261},[260],[28,8596,8598,8601,8604,8607,8610],{"className":8597},[265],[28,8599],{"className":8600,"style":645},[269],[28,8602,176],{"className":8603},[274,275],[28,8605],{"className":8606,"style":302},[279],[28,8608,83],{"className":8609},[306],[28,8611],{"className":8612,"style":302},[279],[28,8614,8616,8619,8622,8625,8628,8631,8634],{"className":8615},[265],[28,8617],{"className":8618,"style":1083},[269],[28,8620,180],{"className":8621,"style":298},[274,275],[28,8623],{"className":8624,"style":302},[279],[28,8626],{"className":8627,"style":280},[279],[28,8629,233],{"className":8630},[284],[28,8632],{"className":8633,"style":302},[279],[28,8635],{"className":8636,"style":280},[279],[28,8638,8640,8643,8646,8649,8652,8655,8658,8694],{"className":8639},[265],[28,8641],{"className":8642,"style":341},[269],[28,8644,1511],{"className":8645,"style":1546},[274,275],[28,8647,43],{"className":8648},[348],[28,8650,176],{"className":8651},[274,275],[28,8653,1522],{"className":8654},[355],[28,8656],{"className":8657,"style":302},[279],[28,8659,8661,8688,8691],{"className":8660},[306],[28,8662,8664],{"className":8663},[306],[28,8665,8667],{"className":8666},[274,2359],[28,8668,8670],{"className":8669},[2363],[28,8671,8673,8676,8685],{"className":8672},[2367],[28,8674],{"className":8675,"style":2371},[269],[28,8677,8679],{"className":8678},[2375],[28,8680,8682],{"className":8681},[274],[28,8683,2382],{"className":8684},[306],[28,8686],{"className":8687},[2386],[28,8689],{"className":8690},[279,2390],[28,8692,83],{"className":8693},[306],[28,8695],{"className":8696,"style":302},[279],[28,8698,8700,8703,8706,8709,8712],{"className":8699},[265],[28,8701],{"className":8702,"style":341},[269],[28,8704,1511],{"className":8705,"style":1546},[274,275],[28,8707,43],{"className":8708},[348],[28,8710,180],{"className":8711,"style":298},[274,275],[28,8713,1522],{"className":8714},[355],[2480,8716,8717,8771,8912],{},[2483,8718,8719,8720,8770],{},"算术部分看到 ",[28,8721,8723,8740],{"className":8722,"translate":193},[197],[28,8724,8726],{"className":8725},[201],[203,8727,8728],{"xmlns":205},[208,8729,8730,8738],{},[211,8731,8732,8734,8736],{},[214,8733,176],{},[217,8735,83],{},[214,8737,180],{},[253,8739,5793],{"encoding":255},[28,8741,8743,8761],{"className":8742,"ariaHidden":261},[260],[28,8744,8746,8749,8752,8755,8758],{"className":8745},[265],[28,8747],{"className":8748,"style":645},[269],[28,8750,176],{"className":8751},[274,275],[28,8753],{"className":8754,"style":302},[279],[28,8756,83],{"className":8757},[306],[28,8759],{"className":8760,"style":302},[279],[28,8762,8764,8767],{"className":8763},[265],[28,8765],{"className":8766,"style":612},[269],[28,8768,180],{"className":8769,"style":298},[274,275],"，本身没有矛盾；",[2483,8772,8773,8774,8824,8825,2693,8868,8911],{},"EUF 部分如果不知道 ",[28,8775,8777,8794],{"className":8776,"translate":193},[197],[28,8778,8780],{"className":8779},[201],[203,8781,8782],{"xmlns":205},[208,8783,8784,8792],{},[211,8785,8786,8788,8790],{},[214,8787,176],{},[217,8789,83],{},[214,8791,180],{},[253,8793,5793],{"encoding":255},[28,8795,8797,8815],{"className":8796,"ariaHidden":261},[260],[28,8798,8800,8803,8806,8809,8812],{"className":8799},[265],[28,8801],{"className":8802,"style":645},[269],[28,8804,176],{"className":8805},[274,275],[28,8807],{"className":8808,"style":302},[279],[28,8810,83],{"className":8811},[306],[28,8813],{"className":8814,"style":302},[279],[28,8816,8818,8821],{"className":8817},[265],[28,8819],{"className":8820,"style":612},[269],[28,8822,180],{"className":8823,"style":298},[274,275],"，就会认为 ",[28,8826,8828,8847],{"className":8827,"translate":193},[197],[28,8829,8831],{"className":8830},[201],[203,8832,8833],{"xmlns":205},[208,8834,8835,8845],{},[211,8836,8837,8839,8841,8843],{},[214,8838,1511],{},[217,8840,43],{"stretchy":241},[214,8842,176],{},[217,8844,1522],{"stretchy":241},[253,8846,6272],{"encoding":255},[28,8848,8850],{"className":8849,"ariaHidden":261},[260],[28,8851,8853,8856,8859,8862,8865],{"className":8852},[265],[28,8854],{"className":8855,"style":341},[269],[28,8857,1511],{"className":8858,"style":1546},[274,275],[28,8860,43],{"className":8861},[348],[28,8863,176],{"className":8864},[274,275],[28,8866,1522],{"className":8867},[355],[28,8869,8871,8890],{"className":8870,"translate":193},[197],[28,8872,8874],{"className":8873},[201],[203,8875,8876],{"xmlns":205},[208,8877,8878,8888],{},[211,8879,8880,8882,8884,8886],{},[214,8881,1511],{},[217,8883,43],{"stretchy":241},[214,8885,180],{},[217,8887,1522],{"stretchy":241},[253,8889,6317],{"encoding":255},[28,8891,8893],{"className":8892,"ariaHidden":261},[260],[28,8894,8896,8899,8902,8905,8908],{"className":8895},[265],[28,8897],{"className":8898,"style":341},[269],[28,8900,1511],{"className":8901,"style":1546},[274,275],[28,8903,43],{"className":8904},[348],[28,8906,180],{"className":8907,"style":298},[274,275],[28,8909,1522],{"className":8910},[355]," 可以不同，也不会报错；",[2483,8913,8914,8915,8965,8966,6339,9046,9159],{},"一旦算术把等式 ",[28,8916,8918,8935],{"className":8917,"translate":193},[197],[28,8919,8921],{"className":8920},[201],[203,8922,8923],{"xmlns":205},[208,8924,8925,8933],{},[211,8926,8927,8929,8931],{},[214,8928,176],{},[217,8930,83],{},[214,8932,180],{},[253,8934,5793],{"encoding":255},[28,8936,8938,8956],{"className":8937,"ariaHidden":261},[260],[28,8939,8941,8944,8947,8950,8953],{"className":8940},[265],[28,8942],{"className":8943,"style":645},[269],[28,8945,176],{"className":8946},[274,275],[28,8948],{"className":8949,"style":302},[279],[28,8951,83],{"className":8952},[306],[28,8954],{"className":8955,"style":302},[279],[28,8957,8959,8962],{"className":8958},[265],[28,8960],{"className":8961,"style":612},[269],[28,8963,180],{"className":8964,"style":298},[274,275]," 传给 EUF，EUF 立刻推出 ",[28,8967,8969,8998],{"className":8968,"translate":193},[197],[28,8970,8972],{"className":8971},[201],[203,8973,8974],{"xmlns":205},[208,8975,8976,8996],{},[211,8977,8978,8980,8982,8984,8986,8988,8990,8992,8994],{},[214,8979,1511],{},[217,8981,43],{"stretchy":241},[214,8983,176],{},[217,8985,1522],{"stretchy":241},[217,8987,83],{},[214,8989,1511],{},[217,8991,43],{"stretchy":241},[214,8993,180],{},[217,8995,1522],{"stretchy":241},[253,8997,6543],{"encoding":255},[28,8999,9001,9028],{"className":9000,"ariaHidden":261},[260],[28,9002,9004,9007,9010,9013,9016,9019,9022,9025],{"className":9003},[265],[28,9005],{"className":9006,"style":341},[269],[28,9008,1511],{"className":9009,"style":1546},[274,275],[28,9011,43],{"className":9012},[348],[28,9014,176],{"className":9015},[274,275],[28,9017,1522],{"className":9018},[355],[28,9020],{"className":9021,"style":302},[279],[28,9023,83],{"className":9024},[306],[28,9026],{"className":9027,"style":302},[279],[28,9029,9031,9034,9037,9040,9043],{"className":9030},[265],[28,9032],{"className":9033,"style":341},[269],[28,9035,1511],{"className":9036,"style":1546},[274,275],[28,9038,43],{"className":9039},[348],[28,9041,180],{"className":9042,"style":298},[274,275],[28,9044,1522],{"className":9045},[355],[28,9047,9049,9078],{"className":9048,"translate":193},[197],[28,9050,9052],{"className":9051},[201],[203,9053,9054],{"xmlns":205},[208,9055,9056,9076],{},[211,9057,9058,9060,9062,9064,9066,9068,9070,9072,9074],{},[214,9059,1511],{},[217,9061,43],{"stretchy":241},[214,9063,176],{},[217,9065,1522],{"stretchy":241},[217,9067,1854],{"mathvariant":484},[214,9069,1511],{},[217,9071,43],{"stretchy":241},[214,9073,180],{},[217,9075,1522],{"stretchy":241},[253,9077,2732],{"encoding":255},[28,9079,9081,9141],{"className":9080,"ariaHidden":261},[260],[28,9082,9084,9087,9090,9093,9096,9099,9102,9138],{"className":9083},[265],[28,9085],{"className":9086,"style":341},[269],[28,9088,1511],{"className":9089,"style":1546},[274,275],[28,9091,43],{"className":9092},[348],[28,9094,176],{"className":9095},[274,275],[28,9097,1522],{"className":9098},[355],[28,9100],{"className":9101,"style":302},[279],[28,9103,9105,9132,9135],{"className":9104},[306],[28,9106,9108],{"className":9107},[306],[28,9109,9111],{"className":9110},[274,2359],[28,9112,9114],{"className":9113},[2363],[28,9115,9117,9120,9129],{"className":9116},[2367],[28,9118],{"className":9119,"style":2371},[269],[28,9121,9123],{"className":9122},[2375],[28,9124,9126],{"className":9125},[274],[28,9127,2382],{"className":9128},[306],[28,9130],{"className":9131},[2386],[28,9133],{"className":9134},[279,2390],[28,9136,83],{"className":9137},[306],[28,9139],{"className":9140,"style":302},[279],[28,9142,9144,9147,9150,9153,9156],{"className":9143},[265],[28,9145],{"className":9146,"style":341},[269],[28,9148,1511],{"className":9149,"style":1546},[274,275],[28,9151,43],{"className":9152},[348],[28,9154,180],{"className":9155,"style":298},[274,275],[28,9157,1522],{"className":9158},[355]," 冲突。",[11,9161,9162],{},"于是整个公式被判为不可满足。这个结论不是某一个理论单独得出的，而是两个理论「对了一下账」才对出来的。",[11,9164,9165],{},"再看一个反方向的例子：",[28,9167,9169],{"className":9168,"translate":193},[192],[28,9170,9172,9219],{"className":9171,"translate":193},[197],[28,9173,9175],{"className":9174},[201],[203,9176,9177],{"xmlns":205,"display":206},[208,9178,9179,9216],{},[211,9180,9181,9183,9185,9187,9189,9191,9193,9195,9197,9199,9201,9203,9205,9207,9209,9211,9213],{},[214,9182,176],{},[217,9184,124],{},[214,9186,180],{},[217,9188,130],{},[224,9190,226],{},[228,9192,230],{},[217,9194,233],{},[228,9196,230],{},[214,9198,176],{},[217,9200,83],{},[224,9202,3400],{},[228,9204,230],{},[217,9206,233],{},[228,9208,230],{},[214,9210,180],{},[217,9212,83],{},[224,9214,9215],{},"4",[253,9217,9218],{"encoding":255},"x + y > 10 \\;\\wedge\\; x = 3 \\;\\wedge\\; y = 4",[28,9220,9222,9240,9258,9282,9300,9324,9342],{"className":9221,"ariaHidden":261},[260],[28,9223,9225,9228,9231,9234,9237],{"className":9224},[265],[28,9226],{"className":9227,"style":270},[269],[28,9229,176],{"className":9230},[274,275],[28,9232],{"className":9233,"style":280},[279],[28,9235,124],{"className":9236},[284],[28,9238],{"className":9239,"style":280},[279],[28,9241,9243,9246,9249,9252,9255],{"className":9242},[265],[28,9244],{"className":9245,"style":294},[269],[28,9247,180],{"className":9248,"style":298},[274,275],[28,9250],{"className":9251,"style":302},[279],[28,9253,130],{"className":9254},[306],[28,9256],{"className":9257,"style":302},[279],[28,9259,9261,9264,9267,9270,9273,9276,9279],{"className":9260},[265],[28,9262],{"className":9263,"style":316},[269],[28,9265,226],{"className":9266},[274],[28,9268],{"className":9269,"style":302},[279],[28,9271],{"className":9272,"style":280},[279],[28,9274,233],{"className":9275},[284],[28,9277],{"className":9278,"style":302},[279],[28,9280],{"className":9281,"style":280},[279],[28,9283,9285,9288,9291,9294,9297],{"className":9284},[265],[28,9286],{"className":9287,"style":645},[269],[28,9289,176],{"className":9290},[274,275],[28,9292],{"className":9293,"style":302},[279],[28,9295,83],{"className":9296},[306],[28,9298],{"className":9299,"style":302},[279],[28,9301,9303,9306,9309,9312,9315,9318,9321],{"className":9302},[265],[28,9304],{"className":9305,"style":316},[269],[28,9307,3400],{"className":9308},[274],[28,9310],{"className":9311,"style":302},[279],[28,9313],{"className":9314,"style":280},[279],[28,9316,233],{"className":9317},[284],[28,9319],{"className":9320,"style":302},[279],[28,9322],{"className":9323,"style":280},[279],[28,9325,9327,9330,9333,9336,9339],{"className":9326},[265],[28,9328],{"className":9329,"style":612},[269],[28,9331,180],{"className":9332,"style":298},[274,275],[28,9334],{"className":9335,"style":302},[279],[28,9337,83],{"className":9338},[306],[28,9340],{"className":9341,"style":302},[279],[28,9343,9345,9348],{"className":9344},[265],[28,9346],{"className":9347,"style":316},[269],[28,9349,9215],{"className":9350},[274],[11,9352,9353,9354,9451],{},"这里算术部分自己就能算出 ",[28,9355,9357,9384],{"className":9356,"translate":193},[197],[28,9358,9360],{"className":9359},[201],[203,9361,9362],{"xmlns":205},[208,9363,9364,9381],{},[211,9365,9366,9368,9370,9372,9374,9377,9379],{},[224,9367,3400],{},[217,9369,124],{},[224,9371,9215],{},[217,9373,83],{},[224,9375,9376],{},"7",[217,9378,1383],{},[224,9380,226],{},[253,9382,9383],{"encoding":255},"3 + 4 = 7 \u003C 10",[28,9385,9387,9405,9423,9442],{"className":9386,"ariaHidden":261},[260],[28,9388,9390,9393,9396,9399,9402],{"className":9389},[265],[28,9391],{"className":9392,"style":6853},[269],[28,9394,3400],{"className":9395},[274],[28,9397],{"className":9398,"style":280},[279],[28,9400,124],{"className":9401},[284],[28,9403],{"className":9404,"style":280},[279],[28,9406,9408,9411,9414,9417,9420],{"className":9407},[265],[28,9409],{"className":9410,"style":316},[269],[28,9412,9215],{"className":9413},[274],[28,9415],{"className":9416,"style":302},[279],[28,9418,83],{"className":9419},[306],[28,9421],{"className":9422,"style":302},[279],[28,9424,9426,9430,9433,9436,9439],{"className":9425},[265],[28,9427],{"className":9428,"style":9429},[269],"height:0.6835em;vertical-align:-0.0391em;",[28,9431,9376],{"className":9432},[274],[28,9434],{"className":9435,"style":302},[279],[28,9437,1383],{"className":9438},[306],[28,9440],{"className":9441,"style":302},[279],[28,9443,9445,9448],{"className":9444},[265],[28,9446],{"className":9447,"style":316},[269],[28,9449,226],{"className":9450},[274],"，直接判 unsat，根本不需要 EUF 参与。",[11,9453,9454,9455],{},"现代求解器（Z3、cvc5 等）通常不严格照搬经典 Nelson–Oppen，而是在 DPLL(T) 框架下做「理论传播 + 冲突解释」。但核心思想没变：",[183,9456,9457],{},"每个理论只懂自己那一摊，通过交换少量关键事实来达成全局一致。",[11,9459,9460,9461,9545,9546,9647],{},"有一个细节值得注意：如果理论是「非凸」的，光交换等式还不够。比如在整数算术里，",[28,9462,9464,9482],{"className":9463,"translate":193},[197],[28,9465,9467],{"className":9466},[201],[203,9468,9469],{"xmlns":205},[208,9470,9471,9479],{},[211,9472,9473,9475,9477],{},[214,9474,176],{},[217,9476,1854],{"mathvariant":484},[224,9478,89],{},[253,9480,9481],{"encoding":255},"x \\ne 0",[28,9483,9485,9536],{"className":9484,"ariaHidden":261},[260],[28,9486,9488,9491,9494,9497,9533],{"className":9487},[265],[28,9489],{"className":9490,"style":2371},[269],[28,9492,176],{"className":9493},[274,275],[28,9495],{"className":9496,"style":302},[279],[28,9498,9500,9527,9530],{"className":9499},[306],[28,9501,9503],{"className":9502},[306],[28,9504,9506],{"className":9505},[274,2359],[28,9507,9509],{"className":9508},[2363],[28,9510,9512,9515,9524],{"className":9511},[2367],[28,9513],{"className":9514,"style":2371},[269],[28,9516,9518],{"className":9517},[2375],[28,9519,9521],{"className":9520},[274],[28,9522,2382],{"className":9523},[306],[28,9525],{"className":9526},[2386],[28,9528],{"className":9529},[279,2390],[28,9531,83],{"className":9532},[306],[28,9534],{"className":9535,"style":302},[279],[28,9537,9539,9542],{"className":9538},[265],[28,9540],{"className":9541,"style":316},[269],[28,9543,89],{"className":9544},[274]," 等价于 ",[28,9547,9549,9578],{"className":9548,"translate":193},[197],[28,9550,9552],{"className":9551},[201],[203,9553,9554],{"xmlns":205},[208,9555,9556,9575],{},[211,9557,9558,9560,9562,9564,9566,9568,9570,9573],{},[214,9559,176],{},[217,9561,1876],{},[217,9563,5074],{},[224,9565,1244],{},[217,9567,937],{},[214,9569,176],{},[217,9571,9572],{},"≥",[224,9574,1244],{},[253,9576,9577],{"encoding":255},"x \\le -1 \\vee x \\ge 1",[28,9579,9581,9599,9620,9638],{"className":9580,"ariaHidden":261},[260],[28,9582,9584,9587,9590,9593,9596],{"className":9583},[265],[28,9585],{"className":9586,"style":3010},[269],[28,9588,176],{"className":9589},[274,275],[28,9591],{"className":9592,"style":302},[279],[28,9594,1876],{"className":9595},[306],[28,9597],{"className":9598,"style":302},[279],[28,9600,9602,9605,9608,9611,9614,9617],{"className":9601},[265],[28,9603],{"className":9604,"style":6853},[269],[28,9606,5074],{"className":9607},[274],[28,9609,1244],{"className":9610},[274],[28,9612],{"className":9613,"style":280},[279],[28,9615,937],{"className":9616},[284],[28,9618],{"className":9619,"style":280},[279],[28,9621,9623,9626,9629,9632,9635],{"className":9622},[265],[28,9624],{"className":9625,"style":3010},[269],[28,9627,176],{"className":9628},[274,275],[28,9630],{"className":9631,"style":302},[279],[28,9633,9572],{"className":9634},[306],[28,9636],{"className":9637,"style":302},[279],[28,9639,9641,9644],{"className":9640},[265],[28,9642],{"className":9643,"style":316},[269],[28,9645,1244],{"className":9646},[274],"，这是一个析取，没法用一组等式表达。上层只能做情况分裂（case split），分别假设每一支再检查。这会让组合复杂度上升，也是混合理论求解有时会突然变慢的原因之一。",[419,9649,9651],{"id":9650},"_34-给出解释","3.4 给出解释",[11,9653,9654,9655,2923],{},"第四个策略是",[183,9656,9657],{},"不只回答是或否，还给出可操作的结果",[2480,9659,9660,9665,9670,9731],{},[2483,9661,9662,9664],{},[183,9663,405],{},"：求解器返回一个模型，给出每个变量的具体取值。对符号执行来说，这就是一条能触发目标路径的测试输入。",[2483,9666,9667,9669],{},[183,9668,6456],{},"：求解器可以返回一个 unsat core，也就是一组不可满足的子约束，告诉你「是这几条打架了」。",[2483,9671,9672,9675,9676,9730],{},[183,9673,9674],{},"插值","：从 ",[28,9677,9679,9699],{"className":9678,"translate":193},[197],[28,9680,9682],{"className":9681},[201],[203,9683,9684],{"xmlns":205},[208,9685,9686,9696],{},[211,9687,9688,9690,9693],{},[214,9689,903],{},[217,9691,9692],{},"⇒",[214,9694,9695],{},"B",[253,9697,9698],{"encoding":255},"A \\Rightarrow B",[28,9700,9702,9720],{"className":9701,"ariaHidden":261},[260],[28,9703,9705,9708,9711,9714,9717],{"className":9704},[265],[28,9706],{"className":9707,"style":455},[269],[28,9709,903],{"className":9710},[274,275],[28,9712],{"className":9713,"style":302},[279],[28,9715,9692],{"className":9716},[306],[28,9718],{"className":9719,"style":302},[279],[28,9721,9723,9726],{"className":9722},[265],[28,9724],{"className":9725,"style":455},[269],[28,9727,9695],{"className":9728,"style":9729},[274,275],"margin-right:0.0502em;"," 的不可满足性中提取一个只含共同变量的中间公式（Craig interpolant），用于程序验证中的不变式合成和抽象精化。",[2483,9732,9733,9736],{},[183,9734,9735],{},"证明","：部分求解器能输出 resolution 证明，供独立检查，避免「求解器说有漏洞，但你不知道为什么」的黑盒困境。",[11,9738,9739,9740,9742],{},"这一点经常被忽略，但它决定了 SMT 能不能真正嵌进工具链。一个只会输出 ",[25,9741,6456],{}," 的求解器，在调试时几乎没用；一个能告诉你「第 3、7、9 条约束冲突」的求解器，才能帮工程师定位问题。",[419,9744,9746],{"id":9745},"_35-增量求解","3.5 增量求解",[11,9748,9749,9750,9753],{},"第五个策略是",[183,9751,9752],{},"增量求解","（incremental solving）。",[11,9755,9756],{},"符号执行的路径是一棵树：父路径的约束是子路径约束的前缀。如果每换一条路径都从头求解，大量工作会被重复浪费。",[11,9758,9759,9760,2951,9763,2693,9766,9769],{},"SMT-LIB 提供了 ",[25,9761,9762],{},"push",[25,9764,9765],{},"pop",[25,9767,9768],{},"check-sat-assuming","，允许在已有约束的基础上临时加入假设、求解、再回退。求解器会保留之前学到的子句和理论状态。一条路径判 unsat 后回退到分叉点，下一条路径可以接着用。",[11,9771,9772],{},"KLEE、angr 这类工具都重度依赖这个特性。没有增量求解，符号执行的性能会差一个数量级。",[414,9774,9776],{"id":9775},"四dpllt","四、DPLL(T)",[419,9778,9780],{"id":9779},"_41-总体架构","4.1 总体架构",[11,9782,9783,9784,9787,9788,9791],{},"现代 SMT 求解器的标准架构叫 ",[183,9785,9786],{},"DPLL(T)","，更准确的名字是 ",[183,9789,9790],{},"CDCL(T)","：外层是带冲突学习的 SAT 求解器，内层挂一个或多个理论求解器。",[2907,9793,9794],{},[2910,9795],{"height":9796,"width":9797,"style":2914,"data-d2-theme":2915,"alt":23,"src":9798},320,2578,"data:image\u002Fsvg+xml;base64,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\u002Fmu6HotrM\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",[11,9800,9801],{},"用一句话概括这个循环：",[9803,9804,9805,9808,9811,9814,9817,9820],"ol",{},[2483,9806,9807],{},"把公式里的每个理论原子替换成一个命题变量，得到一个纯布尔公式；",[2483,9809,9810],{},"让 SAT 求解器找一个布尔模型；",[2483,9812,9813],{},"把布尔模型翻译回理论原子的真假，交给理论求解器检查；",[2483,9815,9816],{},"如果理论层说「一致」，就得到 sat 和模型；",[2483,9818,9819],{},"如果理论层说「冲突」，就把冲突总结成一条新的布尔子句，塞回 SAT 层，让它换个方向再搜；",[2483,9821,9822],{},"如果 SAT 层把所有可能性都穷尽了，就得到 unsat。",[11,9824,9825],{},"可以把它想象成一个总承包商和几个专业工种的协作：总承包商（SAT）负责排期和组合，电工、水管工（理论求解器）负责检查自己那部分是否可行。电工说「这个排期下电线会短路」，总承包商就把这条经验记下来，换一个排期重新试。",[419,9827,9829],{"id":9828},"_42-布尔抽象","4.2 布尔抽象",[11,9831,9832],{},"布尔抽象（也叫 propositional abstraction）是两层之间的接口。",[11,9834,9835],{},"以这个公式为例：",[28,9837,9839],{"className":9838,"translate":193},[192],[28,9840,9842,9896],{"className":9841,"translate":193},[197],[28,9843,9845],{"className":9844},[201],[203,9846,9847],{"xmlns":205,"display":206},[208,9848,9849,9893],{},[211,9850,9851,9853,9855,9857,9859,9861,9863,9865,9867,9869,9871,9873,9875,9877,9879,9881,9883,9885,9887,9889,9891],{},[214,9852,510],{},[217,9854,83],{},[217,9856,43],{"stretchy":241},[214,9858,176],{},[217,9860,130],{},[224,9862,89],{},[217,9864,937],{},[214,9866,180],{},[217,9868,130],{},[224,9870,89],{},[217,9872,1522],{"stretchy":241},[228,9874,230],{},[217,9876,233],{},[228,9878,230],{},[217,9880,43],{"stretchy":241},[214,9882,176],{},[217,9884,124],{},[214,9886,180],{},[217,9888,1383],{},[224,9890,89],{},[217,9892,1522],{"stretchy":241},[253,9894,9895],{"encoding":255},"\\varphi = (x > 0 \\vee y > 0) \\;\\wedge\\; (x + y \u003C 0)",[28,9897,9899,9917,9938,9956,9974,10001,10022,10040],{"className":9898,"ariaHidden":261},[260],[28,9900,9902,9905,9908,9911,9914],{"className":9901},[265],[28,9903],{"className":9904,"style":612},[269],[28,9906,510],{"className":9907},[274,275],[28,9909],{"className":9910,"style":302},[279],[28,9912,83],{"className":9913},[306],[28,9915],{"className":9916,"style":302},[279],[28,9918,9920,9923,9926,9929,9932,9935],{"className":9919},[265],[28,9921],{"className":9922,"style":341},[269],[28,9924,43],{"className":9925},[348],[28,9927,176],{"className":9928},[274,275],[28,9930],{"className":9931,"style":302},[279],[28,9933,130],{"className":9934},[306],[28,9936],{"className":9937,"style":302},[279],[28,9939,9941,9944,9947,9950,9953],{"className":9940},[265],[28,9942],{"className":9943,"style":316},[269],[28,9945,89],{"className":9946},[274],[28,9948],{"className":9949,"style":280},[279],[28,9951,937],{"className":9952},[284],[28,9954],{"className":9955,"style":280},[279],[28,9957,9959,9962,9965,9968,9971],{"className":9958},[265],[28,9960],{"className":9961,"style":294},[269],[28,9963,180],{"className":9964,"style":298},[274,275],[28,9966],{"className":9967,"style":302},[279],[28,9969,130],{"className":9970},[306],[28,9972],{"className":9973,"style":302},[279],[28,9975,9977,9980,9983,9986,9989,9992,9995,9998],{"className":9976},[265],[28,9978],{"className":9979,"style":341},[269],[28,9981,89],{"className":9982},[274],[28,9984,1522],{"className":9985},[355],[28,9987],{"className":9988,"style":302},[279],[28,9990],{"className":9991,"style":280},[279],[28,9993,233],{"className":9994},[284],[28,9996],{"className":9997,"style":302},[279],[28,9999],{"className":10000,"style":280},[279],[28,10002,10004,10007,10010,10013,10016,10019],{"className":10003},[265],[28,10005],{"className":10006,"style":341},[269],[28,10008,43],{"className":10009},[348],[28,10011,176],{"className":10012},[274,275],[28,10014],{"className":10015,"style":280},[279],[28,10017,124],{"className":10018},[284],[28,10020],{"className":10021,"style":280},[279],[28,10023,10025,10028,10031,10034,10037],{"className":10024},[265],[28,10026],{"className":10027,"style":294},[269],[28,10029,180],{"className":10030,"style":298},[274,275],[28,10032],{"className":10033,"style":302},[279],[28,10035,1383],{"className":10036},[306],[28,10038],{"className":10039,"style":302},[279],[28,10041,10043,10046,10049],{"className":10042},[265],[28,10044],{"className":10045,"style":341},[269],[28,10047,89],{"className":10048},[274],[28,10050,1522],{"className":10051},[355],[11,10053,10054],{},"引入三个命题变量：",[28,10056,10058],{"className":10057,"translate":193},[192],[28,10059,10061,10128],{"className":10060,"translate":193},[197],[28,10062,10064],{"className":10063},[201],[203,10065,10066],{"xmlns":205,"display":206},[208,10067,10068,10125],{},[211,10069,10070,10072,10075,10077,10079,10081,10083,10085,10087,10089,10091,10093,10095,10097,10099,10101,10103,10105,10107,10109,10111,10113,10115,10117,10119,10121,10123],{},[214,10071,11],{},[217,10073,10074],{},"≡",[217,10076,43],{"stretchy":241},[214,10078,176],{},[217,10080,130],{},[224,10082,89],{},[217,10084,1522],{"stretchy":241},[217,10086,1683],{"separator":261},[279,10088],{"width":4053},[214,10090,1829],{},[217,10092,10074],{},[217,10094,43],{"stretchy":241},[214,10096,180],{},[217,10098,130],{},[224,10100,89],{},[217,10102,1522],{"stretchy":241},[217,10104,1683],{"separator":261},[279,10106],{"width":4053},[214,10108,1867],{},[217,10110,10074],{},[217,10112,43],{"stretchy":241},[214,10114,176],{},[217,10116,124],{},[214,10118,180],{},[217,10120,1383],{},[224,10122,89],{},[217,10124,1522],{"stretchy":241},[253,10126,10127],{"encoding":255},"p \\equiv (x > 0), \\qquad q \\equiv (y > 0), \\qquad r \\equiv (x + y \u003C 0)",[28,10129,10131,10150,10171,10204,10225,10258,10279,10297],{"className":10130,"ariaHidden":261},[260],[28,10132,10134,10138,10141,10144,10147],{"className":10133},[265],[28,10135],{"className":10136,"style":10137},[269],"height:0.6582em;vertical-align:-0.1944em;",[28,10139,11],{"className":10140},[274,275],[28,10142],{"className":10143,"style":302},[279],[28,10145,10074],{"className":10146},[306],[28,10148],{"className":10149,"style":302},[279],[28,10151,10153,10156,10159,10162,10165,10168],{"className":10152},[265],[28,10154],{"className":10155,"style":341},[269],[28,10157,43],{"className":10158},[348],[28,10160,176],{"className":10161},[274,275],[28,10163],{"className":10164,"style":302},[279],[28,10166,130],{"className":10167},[306],[28,10169],{"className":10170,"style":302},[279],[28,10172,10174,10177,10180,10183,10186,10189,10192,10195,10198,10201],{"className":10173},[265],[28,10175],{"className":10176,"style":341},[269],[28,10178,89],{"className":10179},[274],[28,10181,1522],{"className":10182},[355],[28,10184,1683],{"className":10185},[1723],[28,10187],{"className":10188,"style":4222},[279],[28,10190],{"className":10191,"style":530},[279],[28,10193,1829],{"className":10194,"style":298},[274,275],[28,10196],{"className":10197,"style":302},[279],[28,10199,10074],{"className":10200},[306],[28,10202],{"className":10203,"style":302},[279],[28,10205,10207,10210,10213,10216,10219,10222],{"className":10206},[265],[28,10208],{"className":10209,"style":341},[269],[28,10211,43],{"className":10212},[348],[28,10214,180],{"className":10215,"style":298},[274,275],[28,10217],{"className":10218,"style":302},[279],[28,10220,130],{"className":10221},[306],[28,10223],{"className":10224,"style":302},[279],[28,10226,10228,10231,10234,10237,10240,10243,10246,10249,10252,10255],{"className":10227},[265],[28,10229],{"className":10230,"style":341},[269],[28,10232,89],{"className":10233},[274],[28,10235,1522],{"className":10236},[355],[28,10238,1683],{"className":10239},[1723],[28,10241],{"className":10242,"style":4222},[279],[28,10244],{"className":10245,"style":530},[279],[28,10247,1867],{"className":10248,"style":2279},[274,275],[28,10250],{"className":10251,"style":302},[279],[28,10253,10074],{"className":10254},[306],[28,10256],{"className":10257,"style":302},[279],[28,10259,10261,10264,10267,10270,10273,10276],{"className":10260},[265],[28,10262],{"className":10263,"style":341},[269],[28,10265,43],{"className":10266},[348],[28,10268,176],{"className":10269},[274,275],[28,10271],{"className":10272,"style":280},[279],[28,10274,124],{"className":10275},[284],[28,10277],{"className":10278,"style":280},[279],[28,10280,10282,10285,10288,10291,10294],{"className":10281},[265],[28,10283],{"className":10284,"style":294},[269],[28,10286,180],{"className":10287,"style":298},[274,275],[28,10289],{"className":10290,"style":302},[279],[28,10292,1383],{"className":10293},[306],[28,10295],{"className":10296,"style":302},[279],[28,10298,10300,10303,10306],{"className":10299},[265],[28,10301],{"className":10302,"style":341},[269],[28,10304,89],{"className":10305},[274],[28,10307,1522],{"className":10308},[355],[11,10310,10311],{},"布尔骨架变成：",[28,10313,10315],{"className":10314,"translate":193},[192],[28,10316,10318,10348],{"className":10317,"translate":193},[197],[28,10319,10321],{"className":10320},[201],[203,10322,10323],{"xmlns":205,"display":206},[208,10324,10325,10345],{},[211,10326,10327,10329,10331,10333,10335,10337,10339,10341,10343],{},[217,10328,43],{"stretchy":241},[214,10330,11],{},[217,10332,937],{},[214,10334,1829],{},[217,10336,1522],{"stretchy":241},[228,10338,230],{},[217,10340,233],{},[228,10342,230],{},[214,10344,1867],{},[253,10346,10347],{"encoding":255},"(p \\vee q) \\;\\wedge\\; r",[28,10349,10351,10372,10399],{"className":10350,"ariaHidden":261},[260],[28,10352,10354,10357,10360,10363,10366,10369],{"className":10353},[265],[28,10355],{"className":10356,"style":341},[269],[28,10358,43],{"className":10359},[348],[28,10361,11],{"className":10362},[274,275],[28,10364],{"className":10365,"style":280},[279],[28,10367,937],{"className":10368},[284],[28,10370],{"className":10371,"style":280},[279],[28,10373,10375,10378,10381,10384,10387,10390,10393,10396],{"className":10374},[265],[28,10376],{"className":10377,"style":341},[269],[28,10379,1829],{"className":10380,"style":298},[274,275],[28,10382,1522],{"className":10383},[355],[28,10385],{"className":10386,"style":302},[279],[28,10388],{"className":10389,"style":280},[279],[28,10391,233],{"className":10392},[284],[28,10394],{"className":10395,"style":302},[279],[28,10397],{"className":10398,"style":280},[279],[28,10400,10402,10405],{"className":10401},[265],[28,10403],{"className":10404,"style":645},[269],[28,10406,1867],{"className":10407,"style":2279},[274,275],[11,10409,10410,10411,10492,10493,10614],{},"SAT 求解器在这个骨架上的搜索空间只有 ",[28,10412,10414,10436],{"className":10413,"translate":193},[197],[28,10415,10417],{"className":10416},[201],[203,10418,10419],{"xmlns":205},[208,10420,10421,10433],{},[211,10422,10423,10429,10431],{},[3054,10424,10425,10427],{},[224,10426,1253],{},[224,10428,3400],{},[217,10430,83],{},[224,10432,5236],{},[253,10434,10435],{"encoding":255},"2^3 = 8",[28,10437,10439,10483],{"className":10438,"ariaHidden":261},[260],[28,10440,10442,10445,10474,10477,10480],{"className":10441},[265],[28,10443],{"className":10444,"style":5477},[269],[28,10446,10448,10451],{"className":10447},[274],[28,10449,1253],{"className":10450},[274],[28,10452,10454],{"className":10453},[556],[28,10455,10457],{"className":10456},[560],[28,10458,10460],{"className":10459},[565],[28,10461,10463],{"className":10462,"style":5477},[569],[28,10464,10465,10468],{"style":3094},[28,10466],{"className":10467,"style":578},[577],[28,10469,10471],{"className":10470},[582,583,584,585],[28,10472,3400],{"className":10473},[274,585],[28,10475],{"className":10476,"style":302},[279],[28,10478,83],{"className":10479},[306],[28,10481],{"className":10482,"style":302},[279],[28,10484,10486,10489],{"className":10485},[265],[28,10487],{"className":10488,"style":316},[269],[28,10490,5236],{"className":10491},[274]," 种赋值。它先试 ",[28,10494,10496,10530],{"className":10495,"translate":193},[197],[28,10497,10499],{"className":10498},[201],[203,10500,10501],{"xmlns":205},[208,10502,10503,10527],{},[211,10504,10505,10507,10509,10511,10513,10515,10517,10519,10521,10523,10525],{},[214,10506,11],{},[217,10508,83],{},[214,10510,884],{"mathvariant":484},[217,10512,1683],{"separator":261},[214,10514,1829],{},[217,10516,83],{},[214,10518,884],{"mathvariant":484},[217,10520,1683],{"separator":261},[214,10522,1867],{},[217,10524,83],{},[214,10526,884],{"mathvariant":484},[253,10528,10529],{"encoding":255},"p = \\top, q = \\top, r = \\top",[28,10531,10533,10551,10578,10605],{"className":10532,"ariaHidden":261},[260],[28,10534,10536,10539,10542,10545,10548],{"className":10535},[265],[28,10537],{"className":10538,"style":612},[269],[28,10540,11],{"className":10541},[274,275],[28,10543],{"className":10544,"style":302},[279],[28,10546,83],{"className":10547},[306],[28,10549],{"className":10550,"style":302},[279],[28,10552,10554,10557,10560,10563,10566,10569,10572,10575],{"className":10553},[265],[28,10555],{"className":10556,"style":2371},[269],[28,10558,884],{"className":10559},[274],[28,10561,1683],{"className":10562},[1723],[28,10564],{"className":10565,"style":530},[279],[28,10567,1829],{"className":10568,"style":298},[274,275],[28,10570],{"className":10571,"style":302},[279],[28,10573,83],{"className":10574},[306],[28,10576],{"className":10577,"style":302},[279],[28,10579,10581,10584,10587,10590,10593,10596,10599,10602],{"className":10580},[265],[28,10582],{"className":10583,"style":2371},[269],[28,10585,884],{"className":10586},[274],[28,10588,1683],{"className":10589},[1723],[28,10591],{"className":10592,"style":530},[279],[28,10594,1867],{"className":10595,"style":2279},[274,275],[28,10597],{"className":10598,"style":302},[279],[28,10600,83],{"className":10601},[306],[28,10603],{"className":10604,"style":302},[279],[28,10606,10608,10611],{"className":10607},[265],[28,10609],{"className":10610,"style":7448},[269],[28,10612,884],{"className":10613},[274],"，把这三个真值交回理论层。",[11,10616,10617,10618,10621],{},"理论层看到的是三个",[183,10619,10620],{},"文字","（literal）：",[28,10623,10625],{"className":10624,"translate":193},[192],[28,10626,10628,10670],{"className":10627,"translate":193},[197],[28,10629,10631],{"className":10630},[201],[203,10632,10633],{"xmlns":205,"display":206},[208,10634,10635,10667],{},[211,10636,10637,10639,10641,10643,10645,10647,10649,10651,10653,10655,10657,10659,10661,10663,10665],{},[214,10638,176],{},[217,10640,130],{},[224,10642,89],{},[217,10644,1683],{"separator":261},[279,10646],{"width":4053},[214,10648,180],{},[217,10650,130],{},[224,10652,89],{},[217,10654,1683],{"separator":261},[279,10656],{"width":4053},[214,10658,176],{},[217,10660,124],{},[214,10662,180],{},[217,10664,1383],{},[224,10666,89],{},[253,10668,10669],{"encoding":255},"x > 0, \\qquad y > 0, \\qquad x + y \u003C 0",[28,10671,10673,10691,10721,10751,10769],{"className":10672,"ariaHidden":261},[260],[28,10674,10676,10679,10682,10685,10688],{"className":10675},[265],[28,10677],{"className":10678,"style":2616},[269],[28,10680,176],{"className":10681},[274,275],[28,10683],{"className":10684,"style":302},[279],[28,10686,130],{"className":10687},[306],[28,10689],{"className":10690,"style":302},[279],[28,10692,10694,10697,10700,10703,10706,10709,10712,10715,10718],{"className":10693},[265],[28,10695],{"className":10696,"style":8494},[269],[28,10698,89],{"className":10699},[274],[28,10701,1683],{"className":10702},[1723],[28,10704],{"className":10705,"style":4222},[279],[28,10707],{"className":10708,"style":530},[279],[28,10710,180],{"className":10711,"style":298},[274,275],[28,10713],{"className":10714,"style":302},[279],[28,10716,130],{"className":10717},[306],[28,10719],{"className":10720,"style":302},[279],[28,10722,10724,10727,10730,10733,10736,10739,10742,10745,10748],{"className":10723},[265],[28,10725],{"className":10726,"style":8494},[269],[28,10728,89],{"className":10729},[274],[28,10731,1683],{"className":10732},[1723],[28,10734],{"className":10735,"style":4222},[279],[28,10737],{"className":10738,"style":530},[279],[28,10740,176],{"className":10741},[274,275],[28,10743],{"className":10744,"style":280},[279],[28,10746,124],{"className":10747},[284],[28,10749],{"className":10750,"style":280},[279],[28,10752,10754,10757,10760,10763,10766],{"className":10753},[265],[28,10755],{"className":10756,"style":294},[269],[28,10758,180],{"className":10759,"style":298},[274,275],[28,10761],{"className":10762,"style":302},[279],[28,10764,1383],{"className":10765},[306],[28,10767],{"className":10768,"style":302},[279],[28,10770,10772,10775],{"className":10771},[265],[28,10773],{"className":10774,"style":316},[269],[28,10776,89],{"className":10777},[274],[11,10779,10780,10781,10854,10855,2923],{},"线性算术求解器立刻发现：前两个推出 ",[28,10782,10784,10806],{"className":10783,"translate":193},[197],[28,10785,10787],{"className":10786},[201],[203,10788,10789],{"xmlns":205},[208,10790,10791,10803],{},[211,10792,10793,10795,10797,10799,10801],{},[214,10794,176],{},[217,10796,124],{},[214,10798,180],{},[217,10800,130],{},[224,10802,89],{},[253,10804,10805],{"encoding":255},"x + y > 0",[28,10807,10809,10827,10845],{"className":10808,"ariaHidden":261},[260],[28,10810,10812,10815,10818,10821,10824],{"className":10811},[265],[28,10813],{"className":10814,"style":270},[269],[28,10816,176],{"className":10817},[274,275],[28,10819],{"className":10820,"style":280},[279],[28,10822,124],{"className":10823},[284],[28,10825],{"className":10826,"style":280},[279],[28,10828,10830,10833,10836,10839,10842],{"className":10829},[265],[28,10831],{"className":10832,"style":294},[269],[28,10834,180],{"className":10835,"style":298},[274,275],[28,10837],{"className":10838,"style":302},[279],[28,10840,130],{"className":10841},[306],[28,10843],{"className":10844,"style":302},[279],[28,10846,10848,10851],{"className":10847},[265],[28,10849],{"className":10850,"style":316},[269],[28,10852,89],{"className":10853},[274],"，与第三个矛盾。这是一次",[183,10856,10857],{},"理论冲突",[11,10859,10860,10861,10912,10913,10979,10980,11031,11032,11166],{},"注意，理论层收到的不是「原子为真」，而是「带正负号的文字」。",[28,10862,10864,10882],{"className":10863,"translate":193},[197],[28,10865,10867],{"className":10866},[201],[203,10868,10869],{"xmlns":205},[208,10870,10871,10879],{},[211,10872,10873,10875,10877],{},[214,10874,1829],{},[217,10876,83],{},[214,10878,894],{"mathvariant":484},[253,10880,10881],{"encoding":255},"q = \\bot",[28,10883,10885,10903],{"className":10884,"ariaHidden":261},[260],[28,10886,10888,10891,10894,10897,10900],{"className":10887},[265],[28,10889],{"className":10890,"style":612},[269],[28,10892,1829],{"className":10893,"style":298},[274,275],[28,10895],{"className":10896,"style":302},[279],[28,10898,83],{"className":10899},[306],[28,10901],{"className":10902,"style":302},[279],[28,10904,10906,10909],{"className":10905},[265],[28,10907],{"className":10908,"style":7448},[269],[28,10910,894],{"className":10911},[274]," 对应的是 ",[28,10914,10916,10940],{"className":10915,"translate":193},[197],[28,10917,10919],{"className":10918},[201],[203,10920,10921],{"xmlns":205},[208,10922,10923,10937],{},[211,10924,10925,10927,10929,10931,10933,10935],{},[214,10926,912],{"mathvariant":484},[217,10928,43],{"stretchy":241},[214,10930,180],{},[217,10932,130],{},[224,10934,89],{},[217,10936,1522],{"stretchy":241},[253,10938,10939],{"encoding":255},"\\neg(y > 0)",[28,10941,10943,10967],{"className":10942,"ariaHidden":261},[260],[28,10944,10946,10949,10952,10955,10958,10961,10964],{"className":10945},[265],[28,10947],{"className":10948,"style":341},[269],[28,10950,912],{"className":10951},[274],[28,10953,43],{"className":10954},[348],[28,10956,180],{"className":10957,"style":298},[274,275],[28,10959],{"className":10960,"style":302},[279],[28,10962,130],{"className":10963},[306],[28,10965],{"className":10966,"style":302},[279],[28,10968,10970,10973,10976],{"className":10969},[265],[28,10971],{"className":10972,"style":341},[269],[28,10974,89],{"className":10975},[274],[28,10977,1522],{"className":10978},[355],"，也就是 ",[28,10981,10983,11001],{"className":10982,"translate":193},[197],[28,10984,10986],{"className":10985},[201],[203,10987,10988],{"xmlns":205},[208,10989,10990,10998],{},[211,10991,10992,10994,10996],{},[214,10993,180],{},[217,10995,1876],{},[224,10997,89],{},[253,10999,11000],{"encoding":255},"y \\le 0",[28,11002,11004,11022],{"className":11003,"ariaHidden":261},[260],[28,11005,11007,11010,11013,11016,11019],{"className":11006},[265],[28,11008],{"className":11009,"style":2450},[269],[28,11011,180],{"className":11012,"style":298},[274,275],[28,11014],{"className":11015,"style":302},[279],[28,11017,1876],{"className":11018},[306],[28,11020],{"className":11021,"style":302},[279],[28,11023,11025,11028],{"className":11024},[265],[28,11026],{"className":11027,"style":316},[269],[28,11029,89],{"className":11030},[274],"。理论求解器必须能处理这种否定形式，这也是为什么它通常直接操作形如 ",[28,11033,11035,11061],{"className":11034,"translate":193},[197],[28,11036,11038],{"className":11037},[201],[203,11039,11040],{"xmlns":205},[208,11041,11042,11058],{},[211,11043,11044,11050,11052],{},[501,11045,11046,11048],{},[214,11047,1241],{},[224,11049,1244],{},[217,11051,1876],{},[501,11053,11054,11056],{},[214,11055,1241],{},[224,11057,1253],{},[253,11059,11060],{"encoding":255},"t_1 \\le t_2",[28,11062,11064,11120],{"className":11063,"ariaHidden":261},[260],[28,11065,11067,11071,11111,11114,11117],{"className":11066},[265],[28,11068],{"className":11069,"style":11070},[269],"height:0.786em;vertical-align:-0.15em;",[28,11072,11074,11077],{"className":11073},[274],[28,11075,1241],{"className":11076},[274,275],[28,11078,11080],{"className":11079},[556],[28,11081,11083,11103],{"className":11082},[560,561],[28,11084,11086,11100],{"className":11085},[565],[28,11087,11089],{"className":11088,"style":1285},[569],[28,11090,11091,11094],{"style":573},[28,11092],{"className":11093,"style":578},[577],[28,11095,11097],{"className":11096},[582,583,584,585],[28,11098,1244],{"className":11099},[274,585],[28,11101,593],{"className":11102},[592],[28,11104,11106],{"className":11105},[565],[28,11107,11109],{"className":11108,"style":600},[569],[28,11110],{},[28,11112],{"className":11113,"style":302},[279],[28,11115,1876],{"className":11116},[306],[28,11118],{"className":11119,"style":302},[279],[28,11121,11123,11126],{"className":11122},[265],[28,11124],{"className":11125,"style":1266},[269],[28,11127,11129,11132],{"className":11128},[274],[28,11130,1241],{"className":11131},[274,275],[28,11133,11135],{"className":11134},[556],[28,11136,11138,11158],{"className":11137},[560,561],[28,11139,11141,11155],{"className":11140},[565],[28,11142,11144],{"className":11143,"style":1285},[569],[28,11145,11146,11149],{"style":573},[28,11147],{"className":11148,"style":578},[577],[28,11150,11152],{"className":11151},[582,583,584,585],[28,11153,1253],{"className":11154},[274,585],[28,11156,593],{"className":11157},[592],[28,11159,11161],{"className":11160},[565],[28,11162,11164],{"className":11163,"style":600},[569],[28,11165],{}," 的不等式，而不是只处理等式。",[419,11168,11170],{"id":11169},"_43-cdcl","4.3 CDCL",[11,11172,11173],{},"SAT 层用的是 CDCL（冲突驱动子句学习）。它的核心循环大致如下：",[18,11175,11180],{"className":11176,"code":11178,"language":11179},[11177],"language-text","loop:\n    conflict = unit_propagate()      # 单元传播\n    if conflict:\n        if decision_level == 0: return UNSAT\n        lemma = analyze(conflict)    # 冲突分析\n        backtrack()                  # 非时序回跳\n        add_clause(lemma)            # 学习子句\n    else:\n        if all_variables_assigned(): return SAT\n        decide()                     # 选一个未赋值的变量\n","text",[25,11181,11178],{"__ignoreMap":23},[11,11183,11184],{},"三个关键机制：",[2480,11186,11187,11193,11199],{},[2483,11188,11189,11192],{},[183,11190,11191],{},"单元传播","：一条子句里只剩一个未赋值的文字时，强制它取能让子句为真的值。",[2483,11194,11195,11198],{},[183,11196,11197],{},"冲突分析","：从冲突出发，反向找出是哪几个决策导致的，生成一条新的子句。这条子句是原公式的逻辑推论，加进去不会改变解集。",[2483,11200,11201,11204],{},[183,11202,11203],{},"非时序回跳","：不退回上一层，而是直接跳回冲突的真正源头，跳过大量无用的搜索。",[11,11206,11207],{},"CDCL 是过去二十年 SAT 求解器性能提升的主要来源。SMT 做的事情，基本上是把这套成熟的工程能力「复用」到理论层。",[419,11209,11211],{"id":11210},"_44-理论求解器","4.4 理论求解器",[11,11213,11214,11215,11218,11219,11247],{},"理论层要做的事情叫 ",[183,11216,11217],{},"T-一致性检查","：给定一组带正负号的原子，判断它们在理论 ",[28,11220,11222,11235],{"className":11221,"translate":193},[197],[28,11223,11225],{"className":11224},[201],[203,11226,11227],{"xmlns":205},[208,11228,11229,11233],{},[211,11230,11231],{},[214,11232,443],{},[253,11234,443],{"encoding":255},[28,11236,11238],{"className":11237,"ariaHidden":261},[260],[28,11239,11241,11244],{"className":11240},[265],[28,11242],{"className":11243,"style":455},[269],[28,11245,443],{"className":11246,"style":459},[274,275]," 下是否可能同时为真。",[11,11249,11250],{},"不同理论有不同的算法。",[11252,11253,11255],"h4",{"id":11254},"euf","EUF",[11,11257,11258],{},"EUF（Equality with Uninterpreted Functions）只关心相等关系和函数的一致性。核心是两件事：",[9803,11260,11261,11264],{},[2483,11262,11263],{},"用并查集维护变量的等价类；",[2483,11265,11266,11267,11318,11319,2923],{},"用同余闭包（congruence closure）保证：如果 ",[28,11268,11270,11288],{"className":11269,"translate":193},[197],[28,11271,11273],{"className":11272},[201],[203,11274,11275],{"xmlns":205},[208,11276,11277,11285],{},[211,11278,11279,11281,11283],{},[214,11280,238],{},[217,11282,83],{},[214,11284,3820],{},[253,11286,11287],{"encoding":255},"a = b",[28,11289,11291,11309],{"className":11290,"ariaHidden":261},[260],[28,11292,11294,11297,11300,11303,11306],{"className":11293},[265],[28,11295],{"className":11296,"style":645},[269],[28,11298,238],{"className":11299},[274,275],[28,11301],{"className":11302,"style":302},[279],[28,11304,83],{"className":11305},[306],[28,11307],{"className":11308,"style":302},[279],[28,11310,11312,11315],{"className":11311},[265],[28,11313],{"className":11314,"style":7448},[269],[28,11316,3820],{"className":11317},[274,275],"，那么 ",[28,11320,11322,11352],{"className":11321,"translate":193},[197],[28,11323,11325],{"className":11324},[201],[203,11326,11327],{"xmlns":205},[208,11328,11329,11349],{},[211,11330,11331,11333,11335,11337,11339,11341,11343,11345,11347],{},[214,11332,1511],{},[217,11334,43],{"stretchy":241},[214,11336,238],{},[217,11338,1522],{"stretchy":241},[217,11340,83],{},[214,11342,1511],{},[217,11344,43],{"stretchy":241},[214,11346,3820],{},[217,11348,1522],{"stretchy":241},[253,11350,11351],{"encoding":255},"f(a) = f(b)",[28,11353,11355,11382],{"className":11354,"ariaHidden":261},[260],[28,11356,11358,11361,11364,11367,11370,11373,11376,11379],{"className":11357},[265],[28,11359],{"className":11360,"style":341},[269],[28,11362,1511],{"className":11363,"style":1546},[274,275],[28,11365,43],{"className":11366},[348],[28,11368,238],{"className":11369},[274,275],[28,11371,1522],{"className":11372},[355],[28,11374],{"className":11375,"style":302},[279],[28,11377,83],{"className":11378},[306],[28,11380],{"className":11381,"style":302},[279],[28,11383,11385,11388,11391,11394,11397],{"className":11384},[265],[28,11386],{"className":11387,"style":341},[269],[28,11389,1511],{"className":11390,"style":1546},[274,275],[28,11392,43],{"className":11393},[348],[28,11395,3820],{"className":11396},[274,275],[28,11398,1522],{"className":11399},[355],[11,11401,11402],{},"例子：",[28,11404,11406],{"className":11405,"translate":193},[192],[28,11407,11409,11463],{"className":11408,"translate":193},[197],[28,11410,11412],{"className":11411},[201],[203,11413,11414],{"xmlns":205,"display":206},[208,11415,11416,11460],{},[211,11417,11418,11420,11422,11424,11426,11428,11430,11432,11434,11436,11438,11440,11442,11444,11446,11448,11450,11452,11454,11456,11458],{},[214,11419,238],{},[217,11421,83],{},[214,11423,3820],{},[228,11425,230],{},[217,11427,233],{},[228,11429,230],{},[214,11431,3820],{},[217,11433,83],{},[214,11435,22],{},[228,11437,230],{},[217,11439,233],{},[228,11441,230],{},[214,11443,1511],{},[217,11445,43],{"stretchy":241},[214,11447,238],{},[217,11449,1522],{"stretchy":241},[217,11451,1854],{"mathvariant":484},[214,11453,1511],{},[217,11455,43],{"stretchy":241},[214,11457,22],{},[217,11459,1522],{"stretchy":241},[253,11461,11462],{"encoding":255},"a = b \\;\\wedge\\; b = c \\;\\wedge\\; f(a) \\ne f(c)",[28,11464,11466,11484,11508,11526,11551,11611],{"className":11465,"ariaHidden":261},[260],[28,11467,11469,11472,11475,11478,11481],{"className":11468},[265],[28,11470],{"className":11471,"style":645},[269],[28,11473,238],{"className":11474},[274,275],[28,11476],{"className":11477,"style":302},[279],[28,11479,83],{"className":11480},[306],[28,11482],{"className":11483,"style":302},[279],[28,11485,11487,11490,11493,11496,11499,11502,11505],{"className":11486},[265],[28,11488],{"className":11489,"style":7448},[269],[28,11491,3820],{"className":11492},[274,275],[28,11494],{"className":11495,"style":302},[279],[28,11497],{"className":11498,"style":280},[279],[28,11500,233],{"className":11501},[284],[28,11503],{"className":11504,"style":302},[279],[28,11506],{"className":11507,"style":280},[279],[28,11509,11511,11514,11517,11520,11523],{"className":11510},[265],[28,11512],{"className":11513,"style":7448},[269],[28,11515,3820],{"className":11516},[274,275],[28,11518],{"className":11519,"style":302},[279],[28,11521,83],{"className":11522},[306],[28,11524],{"className":11525,"style":302},[279],[28,11527,11529,11533,11536,11539,11542,11545,11548],{"className":11528},[265],[28,11530],{"className":11531,"style":11532},[269],"height:0.5556em;",[28,11534,22],{"className":11535},[274,275],[28,11537],{"className":11538,"style":302},[279],[28,11540],{"className":11541,"style":280},[279],[28,11543,233],{"className":11544},[284],[28,11546],{"className":11547,"style":302},[279],[28,11549],{"className":11550,"style":280},[279],[28,11552,11554,11557,11560,11563,11566,11569,11572,11608],{"className":11553},[265],[28,11555],{"className":11556,"style":341},[269],[28,11558,1511],{"className":11559,"style":1546},[274,275],[28,11561,43],{"className":11562},[348],[28,11564,238],{"className":11565},[274,275],[28,11567,1522],{"className":11568},[355],[28,11570],{"className":11571,"style":302},[279],[28,11573,11575,11602,11605],{"className":11574},[306],[28,11576,11578],{"className":11577},[306],[28,11579,11581],{"className":11580},[274,2359],[28,11582,11584],{"className":11583},[2363],[28,11585,11587,11590,11599],{"className":11586},[2367],[28,11588],{"className":11589,"style":2371},[269],[28,11591,11593],{"className":11592},[2375],[28,11594,11596],{"className":11595},[274],[28,11597,2382],{"className":11598},[306],[28,11600],{"className":11601},[2386],[28,11603],{"className":11604},[279,2390],[28,11606,83],{"className":11607},[306],[28,11609],{"className":11610,"style":302},[279],[28,11612,11614,11617,11620,11623,11626],{"className":11613},[265],[28,11615],{"className":11616,"style":341},[269],[28,11618,1511],{"className":11619,"style":1546},[274,275],[28,11621,43],{"className":11622},[348],[28,11624,22],{"className":11625},[274,275],[28,11627,1522],{"className":11628},[355],[11,11630,11631,11632,11687,11688,2693,11732,11776,11777,11849],{},"并查集把 ",[28,11633,11635,11657],{"className":11634,"translate":193},[197],[28,11636,11638],{"className":11637},[201],[203,11639,11640],{"xmlns":205},[208,11641,11642,11654],{},[211,11643,11644,11646,11648,11650,11652],{},[214,11645,238],{},[217,11647,1683],{"separator":261},[214,11649,3820],{},[217,11651,1683],{"separator":261},[214,11653,22],{},[253,11655,11656],{"encoding":255},"a, b, c",[28,11658,11660],{"className":11659,"ariaHidden":261},[260],[28,11661,11663,11666,11669,11672,11675,11678,11681,11684],{"className":11662},[265],[28,11664],{"className":11665,"style":2371},[269],[28,11667,238],{"className":11668},[274,275],[28,11670,1683],{"className":11671},[1723],[28,11673],{"className":11674,"style":530},[279],[28,11676,3820],{"className":11677},[274,275],[28,11679,1683],{"className":11680},[1723],[28,11682],{"className":11683,"style":530},[279],[28,11685,22],{"className":11686},[274,275]," 合并成一个类；同余闭包发现 ",[28,11689,11691,11711],{"className":11690,"translate":193},[197],[28,11692,11694],{"className":11693},[201],[203,11695,11696],{"xmlns":205},[208,11697,11698,11708],{},[211,11699,11700,11702,11704,11706],{},[214,11701,1511],{},[217,11703,43],{"stretchy":241},[214,11705,238],{},[217,11707,1522],{"stretchy":241},[253,11709,11710],{"encoding":255},"f(a)",[28,11712,11714],{"className":11713,"ariaHidden":261},[260],[28,11715,11717,11720,11723,11726,11729],{"className":11716},[265],[28,11718],{"className":11719,"style":341},[269],[28,11721,1511],{"className":11722,"style":1546},[274,275],[28,11724,43],{"className":11725},[348],[28,11727,238],{"className":11728},[274,275],[28,11730,1522],{"className":11731},[355],[28,11733,11735,11755],{"className":11734,"translate":193},[197],[28,11736,11738],{"className":11737},[201],[203,11739,11740],{"xmlns":205},[208,11741,11742,11752],{},[211,11743,11744,11746,11748,11750],{},[214,11745,1511],{},[217,11747,43],{"stretchy":241},[214,11749,22],{},[217,11751,1522],{"stretchy":241},[253,11753,11754],{"encoding":255},"f(c)",[28,11756,11758],{"className":11757,"ariaHidden":261},[260],[28,11759,11761,11764,11767,11770,11773],{"className":11760},[265],[28,11762],{"className":11763,"style":341},[269],[28,11765,1511],{"className":11766,"style":1546},[274,275],[28,11768,43],{"className":11769},[348],[28,11771,22],{"className":11772},[274,275],[28,11774,1522],{"className":11775},[355]," 的参数相等，于是把 ",[28,11778,11780,11810],{"className":11779,"translate":193},[197],[28,11781,11783],{"className":11782},[201],[203,11784,11785],{"xmlns":205},[208,11786,11787,11807],{},[211,11788,11789,11791,11793,11795,11797,11799,11801,11803,11805],{},[214,11790,1511],{},[217,11792,43],{"stretchy":241},[214,11794,238],{},[217,11796,1522],{"stretchy":241},[217,11798,1683],{"separator":261},[214,11800,1511],{},[217,11802,43],{"stretchy":241},[214,11804,22],{},[217,11806,1522],{"stretchy":241},[253,11808,11809],{"encoding":255},"f(a), f(c)",[28,11811,11813],{"className":11812,"ariaHidden":261},[260],[28,11814,11816,11819,11822,11825,11828,11831,11834,11837,11840,11843,11846],{"className":11815},[265],[28,11817],{"className":11818,"style":341},[269],[28,11820,1511],{"className":11821,"style":1546},[274,275],[28,11823,43],{"className":11824},[348],[28,11826,238],{"className":11827},[274,275],[28,11829,1522],{"className":11830},[355],[28,11832,1683],{"className":11833},[1723],[28,11835],{"className":11836,"style":530},[279],[28,11838,1511],{"className":11839,"style":1546},[274,275],[28,11841,43],{"className":11842},[348],[28,11844,22],{"className":11845},[274,275],[28,11847,1522],{"className":11848},[355]," 也合并；但公式又要求它们不等，冲突，判 unsat。",[11,11851,11852],{},"EUF 在程序分析里非常有用，因为指针别名、对象身份、类型相等，本质上都是「未解释函数的相等性」。",[11252,11854,11855],{"id":11855},"线性算术",[11,11857,11858,11860],{},[25,11859,2950],{}," 用单纯形法（更准确地说，是 SMT 里常用的 general simplex）。它先维护一组满足所有等式的赋值，再通过枢轴变换（pivot）调整变量的取值，直到所有上下界约束也被满足；如果怎么调都满足不了，就报告冲突。",[11,11862,11863,11865,11866,11869,11870,2693,11933,11996],{},[25,11864,2954],{}," 在此基础上加",[183,11867,11868],{},"分支定界","：先按实数求解，如果解里某个整数变量取了小数，就分裂成两支，比如 ",[28,11871,11873,11897],{"className":11872,"translate":193},[197],[28,11874,11876],{"className":11875},[201],[203,11877,11878],{"xmlns":205},[208,11879,11880,11894],{},[211,11881,11882,11884,11886,11889,11891],{},[214,11883,176],{},[217,11885,1876],{},[217,11887,11888],{"stretchy":241},"⌊",[214,11890,1693],{},[217,11892,11893],{"stretchy":241},"⌋",[253,11895,11896],{"encoding":255},"x \\le \\lfloor v \\rfloor",[28,11898,11900,11918],{"className":11899,"ariaHidden":261},[260],[28,11901,11903,11906,11909,11912,11915],{"className":11902},[265],[28,11904],{"className":11905,"style":3010},[269],[28,11907,176],{"className":11908},[274,275],[28,11910],{"className":11911,"style":302},[279],[28,11913,1876],{"className":11914},[306],[28,11916],{"className":11917,"style":302},[279],[28,11919,11921,11924,11927,11930],{"className":11920},[265],[28,11922],{"className":11923,"style":341},[269],[28,11925,11888],{"className":11926},[348],[28,11928,1693],{"className":11929,"style":298},[274,275],[28,11931,11893],{"className":11932},[355],[28,11934,11936,11960],{"className":11935,"translate":193},[197],[28,11937,11939],{"className":11938},[201],[203,11940,11941],{"xmlns":205},[208,11942,11943,11957],{},[211,11944,11945,11947,11949,11952,11954],{},[214,11946,176],{},[217,11948,9572],{},[217,11950,11951],{"stretchy":241},"⌈",[214,11953,1693],{},[217,11955,11956],{"stretchy":241},"⌉",[253,11958,11959],{"encoding":255},"x \\ge \\lceil v \\rceil",[28,11961,11963,11981],{"className":11962,"ariaHidden":261},[260],[28,11964,11966,11969,11972,11975,11978],{"className":11965},[265],[28,11967],{"className":11968,"style":3010},[269],[28,11970,176],{"className":11971},[274,275],[28,11973],{"className":11974,"style":302},[279],[28,11976,9572],{"className":11977},[306],[28,11979],{"className":11980,"style":302},[279],[28,11982,11984,11987,11990,11993],{"className":11983},[265],[28,11985],{"className":11986,"style":341},[269],[28,11988,11951],{"className":11989},[348],[28,11991,1693],{"className":11992,"style":298},[274,275],[28,11994,11956],{"className":11995},[355],"，分别再解。",[11,11998,11402],{},[28,12000,12002],{"className":12001,"translate":193},[192],[28,12003,12005,12062],{"className":12004,"translate":193},[197],[28,12006,12008],{"className":12007},[201],[203,12009,12010],{"xmlns":205,"display":206},[208,12011,12012,12059],{},[211,12013,12014,12016,12018,12020,12022,12024,12026,12028,12030,12032,12034,12036,12038,12040,12042,12044,12046,12048,12050,12052,12055],{},[224,12015,3400],{},[214,12017,176],{},[217,12019,124],{},[224,12021,1253],{},[214,12023,180],{},[217,12025,83],{},[224,12027,251],{},[217,12029,1683],{"separator":261},[279,12031],{"width":4053},[214,12033,176],{},[217,12035,1683],{"separator":261},[214,12037,180],{},[217,12039,9572],{},[224,12041,89],{},[217,12043,1683],{"separator":261},[279,12045],{"width":4053},[214,12047,176],{},[217,12049,1683],{"separator":261},[214,12051,180],{},[217,12053,12054],{},"∈",[214,12056,12058],{"mathvariant":12057},"double-struck","Z",[253,12060,12061],{"encoding":255},"3x + 2y = 5, \\qquad x, y \\ge 0, \\qquad x, y \\in \\mathbb{Z}",[28,12063,12065,12086,12107,12146,12185],{"className":12064,"ariaHidden":261},[260],[28,12066,12068,12071,12074,12077,12080,12083],{"className":12067},[265],[28,12069],{"className":12070,"style":6853},[269],[28,12072,3400],{"className":12073},[274],[28,12075,176],{"className":12076},[274,275],[28,12078],{"className":12079,"style":280},[279],[28,12081,124],{"className":12082},[284],[28,12084],{"className":12085,"style":280},[279],[28,12087,12089,12092,12095,12098,12101,12104],{"className":12088},[265],[28,12090],{"className":12091,"style":8494},[269],[28,12093,1253],{"className":12094},[274],[28,12096,180],{"className":12097,"style":298},[274,275],[28,12099],{"className":12100,"style":302},[279],[28,12102,83],{"className":12103},[306],[28,12105],{"className":12106,"style":302},[279],[28,12108,12110,12113,12116,12119,12122,12125,12128,12131,12134,12137,12140,12143],{"className":12109},[265],[28,12111],{"className":12112,"style":8494},[269],[28,12114,251],{"className":12115},[274],[28,12117,1683],{"className":12118},[1723],[28,12120],{"className":12121,"style":4222},[279],[28,12123],{"className":12124,"style":530},[279],[28,12126,176],{"className":12127},[274,275],[28,12129,1683],{"className":12130},[1723],[28,12132],{"className":12133,"style":530},[279],[28,12135,180],{"className":12136,"style":298},[274,275],[28,12138],{"className":12139,"style":302},[279],[28,12141,9572],{"className":12142},[306],[28,12144],{"className":12145,"style":302},[279],[28,12147,12149,12152,12155,12158,12161,12164,12167,12170,12173,12176,12179,12182],{"className":12148},[265],[28,12150],{"className":12151,"style":8494},[269],[28,12153,89],{"className":12154},[274],[28,12156,1683],{"className":12157},[1723],[28,12159],{"className":12160,"style":4222},[279],[28,12162],{"className":12163,"style":530},[279],[28,12165,176],{"className":12166},[274,275],[28,12168,1683],{"className":12169},[1723],[28,12171],{"className":12172,"style":530},[279],[28,12174,180],{"className":12175,"style":298},[274,275],[28,12177],{"className":12178,"style":302},[279],[28,12180,12054],{"className":12181},[306],[28,12183],{"className":12184,"style":302},[279],[28,12186,12188,12192],{"className":12187},[265],[28,12189],{"className":12190,"style":12191},[269],"height:0.6889em;",[28,12193,12058],{"className":12194},[274,12195],"mathbb",[11,12197,12198,12199,12291,12292,12320],{},"实数松弛下，",[28,12200,12202,12233],{"className":12201,"translate":193},[197],[28,12203,12205],{"className":12204},[201],[203,12206,12207],{"xmlns":205},[208,12208,12209,12230],{},[211,12210,12211,12213,12215,12217,12220,12222,12224,12226,12228],{},[214,12212,176],{},[217,12214,83],{},[224,12216,251],{},[214,12218,12219],{"mathvariant":484},"\u002F",[224,12221,3400],{},[217,12223,1683],{"separator":261},[214,12225,180],{},[217,12227,83],{},[224,12229,89],{},[253,12231,12232],{"encoding":255},"x = 5\u002F3, y = 0",[28,12234,12236,12254,12282],{"className":12235,"ariaHidden":261},[260],[28,12237,12239,12242,12245,12248,12251],{"className":12238},[265],[28,12240],{"className":12241,"style":645},[269],[28,12243,176],{"className":12244},[274,275],[28,12246],{"className":12247,"style":302},[279],[28,12249,83],{"className":12250},[306],[28,12252],{"className":12253,"style":302},[279],[28,12255,12257,12260,12264,12267,12270,12273,12276,12279],{"className":12256},[265],[28,12258],{"className":12259,"style":341},[269],[28,12261,12263],{"className":12262},[274],"5\u002F3",[28,12265,1683],{"className":12266},[1723],[28,12268],{"className":12269,"style":530},[279],[28,12271,180],{"className":12272,"style":298},[274,275],[28,12274],{"className":12275,"style":302},[279],[28,12277,83],{"className":12278},[306],[28,12280],{"className":12281,"style":302},[279],[28,12283,12285,12288],{"className":12284},[265],[28,12286],{"className":12287,"style":316},[269],[28,12289,89],{"className":12290},[274]," 是一个解，但 ",[28,12293,12295,12308],{"className":12294,"translate":193},[197],[28,12296,12298],{"className":12297},[201],[203,12299,12300],{"xmlns":205},[208,12301,12302,12306],{},[211,12303,12304],{},[214,12305,176],{},[253,12307,176],{"encoding":255},[28,12309,12311],{"className":12310,"ariaHidden":261},[260],[28,12312,12314,12317],{"className":12313},[265],[28,12315],{"className":12316,"style":645},[269],[28,12318,176],{"className":12319},[274,275]," 不是整数。分支：",[2480,12322,12323,12671],{},[2483,12324,173,12325,12376,12377,12405,12406,12434,12435,5901,12463,12514,12515,12567,12568,12514,12619,12670],{},[28,12326,12328,12346],{"className":12327,"translate":193},[197],[28,12329,12331],{"className":12330},[201],[203,12332,12333],{"xmlns":205},[208,12334,12335,12343],{},[211,12336,12337,12339,12341],{},[214,12338,176],{},[217,12340,1876],{},[224,12342,1244],{},[253,12344,12345],{"encoding":255},"x \\le 1",[28,12347,12349,12367],{"className":12348,"ariaHidden":261},[260],[28,12350,12352,12355,12358,12361,12364],{"className":12351},[265],[28,12353],{"className":12354,"style":3010},[269],[28,12356,176],{"className":12357},[274,275],[28,12359],{"className":12360,"style":302},[279],[28,12362,1876],{"className":12363},[306],[28,12365],{"className":12366,"style":302},[279],[28,12368,12370,12373],{"className":12369},[265],[28,12371],{"className":12372,"style":316},[269],[28,12374,1244],{"className":12375},[274],"：",[28,12378,12380,12393],{"className":12379,"translate":193},[197],[28,12381,12383],{"className":12382},[201],[203,12384,12385],{"xmlns":205},[208,12386,12387,12391],{},[211,12388,12389],{},[214,12390,176],{},[253,12392,176],{"encoding":255},[28,12394,12396],{"className":12395,"ariaHidden":261},[260],[28,12397,12399,12402],{"className":12398},[265],[28,12400],{"className":12401,"style":645},[269],[28,12403,176],{"className":12404},[274,275]," 只能取 ",[28,12407,12409,12422],{"className":12408,"translate":193},[197],[28,12410,12412],{"className":12411},[201],[203,12413,12414],{"xmlns":205},[208,12415,12416,12420],{},[211,12417,12418],{},[224,12419,89],{},[253,12421,89],{"encoding":255},[28,12423,12425],{"className":12424,"ariaHidden":261},[260],[28,12426,12428,12431],{"className":12427},[265],[28,12429],{"className":12430,"style":316},[269],[28,12432,89],{"className":12433},[274]," 或 ",[28,12436,12438,12451],{"className":12437,"translate":193},[197],[28,12439,12441],{"className":12440},[201],[203,12442,12443],{"xmlns":205},[208,12444,12445,12449],{},[211,12446,12447],{},[224,12448,1244],{},[253,12450,1244],{"encoding":255},[28,12452,12454],{"className":12453,"ariaHidden":261},[260],[28,12455,12457,12460],{"className":12456},[265],[28,12458],{"className":12459,"style":316},[269],[28,12461,1244],{"className":12462},[274],[28,12464,12466,12484],{"className":12465,"translate":193},[197],[28,12467,12469],{"className":12468},[201],[203,12470,12471],{"xmlns":205},[208,12472,12473,12481],{},[211,12474,12475,12477,12479],{},[214,12476,176],{},[217,12478,83],{},[224,12480,89],{},[253,12482,12483],{"encoding":255},"x = 0",[28,12485,12487,12505],{"className":12486,"ariaHidden":261},[260],[28,12488,12490,12493,12496,12499,12502],{"className":12489},[265],[28,12491],{"className":12492,"style":645},[269],[28,12494,176],{"className":12495},[274,275],[28,12497],{"className":12498,"style":302},[279],[28,12500,83],{"className":12501},[306],[28,12503],{"className":12504,"style":302},[279],[28,12506,12508,12511],{"className":12507},[265],[28,12509],{"className":12510,"style":316},[269],[28,12512,89],{"className":12513},[274]," 时 ",[28,12516,12518,12537],{"className":12517,"translate":193},[197],[28,12519,12521],{"className":12520},[201],[203,12522,12523],{"xmlns":205},[208,12524,12525,12534],{},[211,12526,12527,12529,12531],{},[214,12528,180],{},[217,12530,83],{},[224,12532,12533],{},"2.5",[253,12535,12536],{"encoding":255},"y = 2.5",[28,12538,12540,12558],{"className":12539,"ariaHidden":261},[260],[28,12541,12543,12546,12549,12552,12555],{"className":12542},[265],[28,12544],{"className":12545,"style":612},[269],[28,12547,180],{"className":12548,"style":298},[274,275],[28,12550],{"className":12551,"style":302},[279],[28,12553,83],{"className":12554},[306],[28,12556],{"className":12557,"style":302},[279],[28,12559,12561,12564],{"className":12560},[265],[28,12562],{"className":12563,"style":316},[269],[28,12565,12533],{"className":12566},[274]," 仍不是整数，",[28,12569,12571,12589],{"className":12570,"translate":193},[197],[28,12572,12574],{"className":12573},[201],[203,12575,12576],{"xmlns":205},[208,12577,12578,12586],{},[211,12579,12580,12582,12584],{},[214,12581,176],{},[217,12583,83],{},[224,12585,1244],{},[253,12587,12588],{"encoding":255},"x = 1",[28,12590,12592,12610],{"className":12591,"ariaHidden":261},[260],[28,12593,12595,12598,12601,12604,12607],{"className":12594},[265],[28,12596],{"className":12597,"style":645},[269],[28,12599,176],{"className":12600},[274,275],[28,12602],{"className":12603,"style":302},[279],[28,12605,83],{"className":12606},[306],[28,12608],{"className":12609,"style":302},[279],[28,12611,12613,12616],{"className":12612},[265],[28,12614],{"className":12615,"style":316},[269],[28,12617,1244],{"className":12618},[274],[28,12620,12622,12640],{"className":12621,"translate":193},[197],[28,12623,12625],{"className":12624},[201],[203,12626,12627],{"xmlns":205},[208,12628,12629,12637],{},[211,12630,12631,12633,12635],{},[214,12632,180],{},[217,12634,83],{},[224,12636,1244],{},[253,12638,12639],{"encoding":255},"y = 1",[28,12641,12643,12661],{"className":12642,"ariaHidden":261},[260],[28,12644,12646,12649,12652,12655,12658],{"className":12645},[265],[28,12647],{"className":12648,"style":612},[269],[28,12650,180],{"className":12651,"style":298},[274,275],[28,12653],{"className":12654,"style":302},[279],[28,12656,83],{"className":12657},[306],[28,12659],{"className":12660,"style":302},[279],[28,12662,12664,12667],{"className":12663},[265],[28,12665],{"className":12666,"style":316},[269],[28,12668,1244],{"className":12669},[274],"，得到整数解，sat；",[2483,12672,12673,12674,12725],{},"另一支 ",[28,12675,12677,12695],{"className":12676,"translate":193},[197],[28,12678,12680],{"className":12679},[201],[203,12681,12682],{"xmlns":205},[208,12683,12684,12692],{},[211,12685,12686,12688,12690],{},[214,12687,176],{},[217,12689,9572],{},[224,12691,1253],{},[253,12693,12694],{"encoding":255},"x \\ge 2",[28,12696,12698,12716],{"className":12697,"ariaHidden":261},[260],[28,12699,12701,12704,12707,12710,12713],{"className":12700},[265],[28,12702],{"className":12703,"style":3010},[269],[28,12705,176],{"className":12706},[274,275],[28,12708],{"className":12709,"style":302},[279],[28,12711,9572],{"className":12712},[306],[28,12714],{"className":12715,"style":302},[279],[28,12717,12719,12722],{"className":12718},[265],[28,12720],{"className":12721,"style":316},[269],[28,12723,1253],{"className":12724},[274]," 就无须再看了。",[11,12727,12728],{},"如果两支都不可行，就判 unsat。实际求解器还会加上割平面（cut）和更强的预处理，避免分支树爆炸。",[11252,12730,7691],{"id":7691},[11,12732,12733,12735],{},[25,12734,3039],{}," 最常用的方法还是位爆破：把位向量运算展开成门电路，再交给 SAT 层。但因为位向量有固定宽度，这种展开是有限且可预测的。",[11,12737,12738],{},"现代求解器会先做字级重写，把明显恒真的部分消掉，比如：",[28,12740,12742],{"className":12741,"translate":193},[192],[28,12743,12745,12800],{"className":12744,"translate":193},[197],[28,12746,12748],{"className":12747},[201],[203,12749,12750],{"xmlns":205,"display":206},[208,12751,12752,12797],{},[211,12753,12754,12756,12758,12760,12762,12764,12766,12768,12770,12772,12774,12776,12778,12780,12782,12784,12786,12788,12791,12793,12795],{},[214,12755,176],{},[228,12757,488],{},[214,12759,7715],{"mathvariant":484},[228,12761,488],{},[224,12763,89],{},[217,12765,83],{},[224,12767,89],{},[217,12769,1683],{"separator":261},[279,12771],{"width":4053},[214,12773,176],{},[217,12775,5074],{},[214,12777,176],{},[217,12779,83],{},[224,12781,89],{},[217,12783,1683],{"separator":261},[279,12785],{"width":4053},[214,12787,176],{},[217,12789,12790],{},"≪",[224,12792,89],{},[217,12794,83],{},[214,12796,176],{},[253,12798,12799],{"encoding":255},"x \\,\\&\\, 0 = 0, \\qquad x - x = 0, \\qquad x \\ll 0 = x",[28,12801,12803,12833,12863,12881,12911,12929],{"className":12802,"ariaHidden":261},[260],[28,12804,12806,12809,12812,12815,12818,12821,12824,12827,12830],{"className":12805},[265],[28,12807],{"className":12808,"style":7448},[269],[28,12810,176],{"className":12811},[274,275],[28,12813],{"className":12814,"style":530},[279],[28,12816,7715],{"className":12817},[274],[28,12819],{"className":12820,"style":530},[279],[28,12822,89],{"className":12823},[274],[28,12825],{"className":12826,"style":302},[279],[28,12828,83],{"className":12829},[306],[28,12831],{"className":12832,"style":302},[279],[28,12834,12836,12839,12842,12845,12848,12851,12854,12857,12860],{"className":12835},[265],[28,12837],{"className":12838,"style":8494},[269],[28,12840,89],{"className":12841},[274],[28,12843,1683],{"className":12844},[1723],[28,12846],{"className":12847,"style":4222},[279],[28,12849],{"className":12850,"style":530},[279],[28,12852,176],{"className":12853},[274,275],[28,12855],{"className":12856,"style":280},[279],[28,12858,5074],{"className":12859},[284],[28,12861],{"className":12862,"style":280},[279],[28,12864,12866,12869,12872,12875,12878],{"className":12865},[265],[28,12867],{"className":12868,"style":645},[269],[28,12870,176],{"className":12871},[274,275],[28,12873],{"className":12874,"style":302},[279],[28,12876,83],{"className":12877},[306],[28,12879],{"className":12880,"style":302},[279],[28,12882,12884,12887,12890,12893,12896,12899,12902,12905,12908],{"className":12883},[265],[28,12885],{"className":12886,"style":8494},[269],[28,12888,89],{"className":12889},[274],[28,12891,1683],{"className":12892},[1723],[28,12894],{"className":12895,"style":4222},[279],[28,12897],{"className":12898,"style":530},[279],[28,12900,176],{"className":12901},[274,275],[28,12903],{"className":12904,"style":302},[279],[28,12906,12790],{"className":12907},[306],[28,12909],{"className":12910,"style":302},[279],[28,12912,12914,12917,12920,12923,12926],{"className":12913},[265],[28,12915],{"className":12916,"style":316},[269],[28,12918,89],{"className":12919},[274],[28,12921],{"className":12922,"style":302},[279],[28,12924,83],{"className":12925},[306],[28,12927],{"className":12928,"style":302},[279],[28,12930,12932,12935],{"className":12931},[265],[28,12933],{"className":12934,"style":645},[269],[28,12936,176],{"className":12937},[274,275],[11,12939,12940],{},"然后再决定哪些子表达式值得展开。乘法是位向量的难点，因为乘法器电路规模大，求解器通常会配合专门的乘法推理和等价类传播。",[11,12942,12943,12944,12947],{},"位向量的语义是",[183,12945,12946],{},"模运算","，这一点非常重要：",[28,12949,12951],{"className":12950,"translate":193},[192],[28,12952,12954,13018],{"className":12953,"translate":193},[197],[28,12955,12957],{"className":12956},[201],[203,12958,12959],{"xmlns":205,"display":206},[208,12960,12961,13015],{},[211,12962,12963,12965,12967,12969,12971,12973,12975,12977,12979,12989,12992,12999,13001,13003,13005,13007,13009,13011],{},[214,12964,176],{},[217,12966,124],{},[224,12968,1244],{},[217,12970,10074],{},[224,12972,89],{},[279,12974],{},[279,12976],{"width":497},[217,12978,43],{"stretchy":241},[211,12980,12981,12984,12986],{},[214,12982,12983],{"mathvariant":484},"m",[214,12985,3933],{"mathvariant":484},[214,12987,12988],{"mathvariant":484},"d",[279,12990],{"width":12991},"0.3333em",[3054,12993,12994,12996],{},[224,12995,1253],{},[224,12997,12998],{},"32",[217,13000,1522],{"stretchy":241},[228,13002,230],{},[217,13004,233],{},[228,13006,230],{},[214,13008,176],{},[217,13010,1854],{"mathvariant":484},[228,13012,13014],{"mathvariant":13013},"monospace","0xFFFFFFFF",[253,13016,13017],{"encoding":255},"x + 1 \\equiv 0 \\pmod{2^{32}} \\;\\wedge\\; x \\ne \\texttt{0xFFFFFFFF}",[28,13019,13021,13039,13057,13073,13147,13198],{"className":13020,"ariaHidden":261},[260],[28,13022,13024,13027,13030,13033,13036],{"className":13023},[265],[28,13025],{"className":13026,"style":270},[269],[28,13028,176],{"className":13029},[274,275],[28,13031],{"className":13032,"style":280},[279],[28,13034,124],{"className":13035},[284],[28,13037],{"className":13038,"style":280},[279],[28,13040,13042,13045,13048,13051,13054],{"className":13041},[265],[28,13043],{"className":13044,"style":316},[269],[28,13046,1244],{"className":13047},[274],[28,13049],{"className":13050,"style":302},[279],[28,13052,10074],{"className":13053},[306],[28,13055],{"className":13056,"style":302},[279],[28,13058,13060,13063,13066,13070],{"className":13059},[265],[28,13061],{"className":13062,"style":316},[269],[28,13064,89],{"className":13065},[274],[28,13067],{"className":13068},[279,13069],"allowbreak",[28,13071],{"className":13072,"style":540},[279],[28,13074,13076,13080,13083,13093,13097,13129,13132,13135,13138,13141,13144],{"className":13075},[265],[28,13077],{"className":13078,"style":13079},[269],"height:1.1141em;vertical-align:-0.25em;",[28,13081,43],{"className":13082},[348],[28,13084,13086],{"className":13085},[274],[28,13087,13089],{"className":13088},[274],[28,13090,13092],{"className":13091},[274,1712],"mod",[28,13094],{"className":13095,"style":13096},[279],"margin-right:0.3333em;",[28,13098,13100,13103],{"className":13099},[274],[28,13101,1253],{"className":13102},[274],[28,13104,13106],{"className":13105},[556],[28,13107,13109],{"className":13108},[560],[28,13110,13112],{"className":13111},[565],[28,13113,13115],{"className":13114,"style":4877},[569],[28,13116,13117,13120],{"style":4880},[28,13118],{"className":13119,"style":578},[577],[28,13121,13123],{"className":13122},[582,583,584,585],[28,13124,13126],{"className":13125},[274,585],[28,13127,12998],{"className":13128},[274,585],[28,13130,1522],{"className":13131},[355],[28,13133],{"className":13134,"style":302},[279],[28,13136],{"className":13137,"style":280},[279],[28,13139,233],{"className":13140},[284],[28,13142],{"className":13143,"style":302},[279],[28,13145],{"className":13146,"style":280},[279],[28,13148,13150,13153,13156,13159,13195],{"className":13149},[265],[28,13151],{"className":13152,"style":2371},[269],[28,13154,176],{"className":13155},[274,275],[28,13157],{"className":13158,"style":302},[279],[28,13160,13162,13189,13192],{"className":13161},[306],[28,13163,13165],{"className":13164},[306],[28,13166,13168],{"className":13167},[274,2359],[28,13169,13171],{"className":13170},[2363],[28,13172,13174,13177,13186],{"className":13173},[2367],[28,13175],{"className":13176,"style":2371},[269],[28,13178,13180],{"className":13179},[2375],[28,13181,13183],{"className":13182},[274],[28,13184,2382],{"className":13185},[306],[28,13187],{"className":13188},[2386],[28,13190],{"className":13191},[279,2390],[28,13193,83],{"className":13194},[306],[28,13196],{"className":13197,"style":302},[279],[28,13199,13201,13205],{"className":13200},[265],[28,13202],{"className":13203,"style":13204},[269],"height:0.6111em;",[28,13206,13208],{"className":13207},[274,11179],[28,13209,13014],{"className":13210},[274,13211],"texttt",[11,13213,13214,13215,13288,13289,13291,13292,13348,13349,13351,13352],{},"在 32 位无符号整数里，唯一满足 ",[28,13216,13218,13240],{"className":13217,"translate":193},[197],[28,13219,13221],{"className":13220},[201],[203,13222,13223],{"xmlns":205},[208,13224,13225,13237],{},[211,13226,13227,13229,13231,13233,13235],{},[214,13228,176],{},[217,13230,124],{},[224,13232,1244],{},[217,13234,10074],{},[224,13236,89],{},[253,13238,13239],{"encoding":255},"x + 1 \\equiv 0",[28,13241,13243,13261,13279],{"className":13242,"ariaHidden":261},[260],[28,13244,13246,13249,13252,13255,13258],{"className":13245},[265],[28,13247],{"className":13248,"style":270},[269],[28,13250,176],{"className":13251},[274,275],[28,13253],{"className":13254,"style":280},[279],[28,13256,124],{"className":13257},[284],[28,13259],{"className":13260,"style":280},[279],[28,13262,13264,13267,13270,13273,13276],{"className":13263},[265],[28,13265],{"className":13266,"style":316},[269],[28,13268,1244],{"className":13269},[274],[28,13271],{"className":13272,"style":302},[279],[28,13274,10074],{"className":13275},[306],[28,13277],{"className":13278,"style":302},[279],[28,13280,13282,13285],{"className":13281},[265],[28,13283],{"className":13284,"style":316},[269],[28,13286,89],{"className":13287},[274]," 的值就是 ",[25,13290,13014],{},"，所以这个公式是 unsat。如果把它当成数学整数，",[28,13293,13295,13315],{"className":13294,"translate":193},[197],[28,13296,13298],{"className":13297},[201],[203,13299,13300],{"xmlns":205},[208,13301,13302,13312],{},[211,13303,13304,13306,13308,13310],{},[214,13305,176],{},[217,13307,83],{},[217,13309,5074],{},[224,13311,1244],{},[253,13313,13314],{"encoding":255},"x = -1",[28,13316,13318,13336],{"className":13317,"ariaHidden":261},[260],[28,13319,13321,13324,13327,13330,13333],{"className":13320},[265],[28,13322],{"className":13323,"style":645},[269],[28,13325,176],{"className":13326},[274,275],[28,13328],{"className":13329,"style":302},[279],[28,13331,83],{"className":13332},[306],[28,13334],{"className":13335,"style":302},[279],[28,13337,13339,13342,13345],{"className":13338},[265],[28,13340],{"className":13341,"style":6853},[269],[28,13343,5074],{"className":13344},[274],[28,13346,1244],{"className":13347},[274]," 又不等于 ",[25,13350,13014],{},"，结论就完全不同。",[183,13353,13354],{},"选错理论，漏洞检测就会漏报或误报。",[11252,13356,7685],{"id":7685},[11,13358,13359,13360,2693,13362,13364],{},"数组理论的核心是 ",[25,13361,1713],{},[25,13363,2696],{}," 两条操作，以及前面提到的 read-over-write 公理。",[11,13366,13367,13368,13371,13372,13375],{},"求解器不会一次性展开所有下标，而是采用",[183,13369,13370],{},"惰性公理实例化","（lazy axiom instantiation）：先假设数组是一致的，当检查中发现某个 ",[25,13373,13374],{},"select(store(...))"," 需要用到公理时，才生成对应的引理加入公式。这叫「按需生成」，也叫 lemma-on-demand。",[11,13377,11402],{},[28,13379,13381],{"className":13380,"translate":193},[192],[28,13382,13384,13473],{"className":13383,"translate":193},[197],[28,13385,13387],{"className":13386},[201],[203,13388,13389],{"xmlns":205,"display":206},[208,13390,13391,13470],{},[211,13392,13393,13407,13409,13421,13423,13425,13427,13429,13431,13433,13435,13437,13439,13441,13443,13446,13448,13450,13452,13454,13456,13458,13460,13462,13464,13466,13468],{},[211,13394,13395,13397,13399,13401,13403,13405],{},[214,13396,1664],{"mathvariant":484},[214,13398,1667],{"mathvariant":484},[214,13400,1670],{"mathvariant":484},[214,13402,1667],{"mathvariant":484},[214,13404,22],{"mathvariant":484},[214,13406,1241],{"mathvariant":484},[217,13408,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[211,13410,13411,13413,13415,13417,13419],{},[214,13412,1664],{"mathvariant":484},[214,13414,1241],{"mathvariant":484},[214,13416,3933],{"mathvariant":484},[214,13418,1867],{"mathvariant":484},[214,13420,1667],{"mathvariant":484},[217,13422,43],{"stretchy":241},[214,13424,238],{},[217,13426,1683],{"separator":261},[214,13428,1686],{},[217,13430,1683],{"separator":261},[214,13432,1693],{},[217,13434,1522],{"stretchy":241},[217,13436,1683],{"separator":261},[214,13438,7907],{},[217,13440,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,13442,83],{},[214,13444,13445],{},"w",[228,13447,230],{},[217,13449,233],{},[228,13451,230],{},[214,13453,1686],{},[217,13455,83],{},[214,13457,7907],{},[228,13459,230],{},[217,13461,233],{},[228,13463,230],{},[214,13465,13445],{},[217,13467,1854],{"mathvariant":484},[214,13469,1693],{},[253,13471,13472],{"encoding":255},"\\mathrm{select}\\bigl(\\mathrm{store}(a, i, v), j\\bigr) = w \\;\\wedge\\; i = j \\;\\wedge\\; w \\ne v",[28,13474,13476,13551,13576,13594,13619,13670],{"className":13475,"ariaHidden":261},[260],[28,13477,13479,13482,13488,13494,13500,13503,13506,13509,13512,13515,13518,13521,13524,13527,13530,13533,13536,13542,13545,13548],{"className":13478},[265],[28,13480],{"className":13481,"style":2469},[269],[28,13483,13485],{"className":13484},[274],[28,13486,1713],{"className":13487},[274,1712],[28,13489,13491],{"className":13490},[348],[28,13492,43],{"className":13493},[2169,2170],[28,13495,13497],{"className":13496},[274],[28,13498,2696],{"className":13499},[274,1712],[28,13501,43],{"className":13502},[348],[28,13504,238],{"className":13505},[274,275],[28,13507,1683],{"className":13508},[1723],[28,13510],{"className":13511,"style":530},[279],[28,13513,1686],{"className":13514},[274,275],[28,13516,1683],{"className":13517},[1723],[28,13519],{"className":13520,"style":530},[279],[28,13522,1693],{"className":13523,"style":298},[274,275],[28,13525,1522],{"className":13526},[355],[28,13528,1683],{"className":13529},[1723],[28,13531],{"className":13532,"style":530},[279],[28,13534,7907],{"className":13535,"style":8057},[274,275],[28,13537,13539],{"className":13538},[355],[28,13540,1522],{"className":13541},[2169,2170],[28,13543],{"className":13544,"style":302},[279],[28,13546,83],{"className":13547},[306],[28,13549],{"className":13550,"style":302},[279],[28,13552,13554,13557,13561,13564,13567,13570,13573],{"className":13553},[265],[28,13555],{"className":13556,"style":11532},[269],[28,13558,13445],{"className":13559,"style":13560},[274,275],"margin-right:0.0269em;",[28,13562],{"className":13563,"style":302},[279],[28,13565],{"className":13566,"style":280},[279],[28,13568,233],{"className":13569},[284],[28,13571],{"className":13572,"style":302},[279],[28,13574],{"className":13575,"style":280},[279],[28,13577,13579,13582,13585,13588,13591],{"className":13578},[265],[28,13580],{"className":13581,"style":7348},[269],[28,13583,1686],{"className":13584},[274,275],[28,13586],{"className":13587,"style":302},[279],[28,13589,83],{"className":13590},[306],[28,13592],{"className":13593,"style":302},[279],[28,13595,13597,13601,13604,13607,13610,13613,13616],{"className":13596},[265],[28,13598],{"className":13599,"style":13600},[269],"height:0.854em;vertical-align:-0.1944em;",[28,13602,7907],{"className":13603,"style":8057},[274,275],[28,13605],{"className":13606,"style":302},[279],[28,13608],{"className":13609,"style":280},[279],[28,13611,233],{"className":13612},[284],[28,13614],{"className":13615,"style":302},[279],[28,13617],{"className":13618,"style":280},[279],[28,13620,13622,13625,13628,13631,13667],{"className":13621},[265],[28,13623],{"className":13624,"style":2371},[269],[28,13626,13445],{"className":13627,"style":13560},[274,275],[28,13629],{"className":13630,"style":302},[279],[28,13632,13634,13661,13664],{"className":13633},[306],[28,13635,13637],{"className":13636},[306],[28,13638,13640],{"className":13639},[274,2359],[28,13641,13643],{"className":13642},[2363],[28,13644,13646,13649,13658],{"className":13645},[2367],[28,13647],{"className":13648,"style":2371},[269],[28,13650,13652],{"className":13651},[2375],[28,13653,13655],{"className":13654},[274],[28,13656,2382],{"className":13657},[306],[28,13659],{"className":13660},[2386],[28,13662],{"className":13663},[279,2390],[28,13665,83],{"className":13666},[306],[28,13668],{"className":13669,"style":302},[279],[28,13671,13673,13676],{"className":13672},[265],[28,13674],{"className":13675,"style":645},[269],[28,13677,1693],{"className":13678,"style":298},[274,275],[11,13680,13681,13682,13733,13734,13762,13763,6339,13814,13898],{},"由 ",[28,13683,13685,13703],{"className":13684,"translate":193},[197],[28,13686,13688],{"className":13687},[201],[203,13689,13690],{"xmlns":205},[208,13691,13692,13700],{},[211,13693,13694,13696,13698],{},[214,13695,1686],{},[217,13697,83],{},[214,13699,7907],{},[253,13701,13702],{"encoding":255},"i = j",[28,13704,13706,13724],{"className":13705,"ariaHidden":261},[260],[28,13707,13709,13712,13715,13718,13721],{"className":13708},[265],[28,13710],{"className":13711,"style":7348},[269],[28,13713,1686],{"className":13714},[274,275],[28,13716],{"className":13717,"style":302},[279],[28,13719,83],{"className":13720},[306],[28,13722],{"className":13723,"style":302},[279],[28,13725,13727,13730],{"className":13726},[265],[28,13728],{"className":13729,"style":13600},[269],[28,13731,7907],{"className":13732,"style":8057},[274,275]," 和 read-over-write 公理，左边等于 ",[28,13735,13737,13750],{"className":13736,"translate":193},[197],[28,13738,13740],{"className":13739},[201],[203,13741,13742],{"xmlns":205},[208,13743,13744,13748],{},[211,13745,13746],{},[214,13747,1693],{},[253,13749,1693],{"encoding":255},[28,13751,13753],{"className":13752,"ariaHidden":261},[260],[28,13754,13756,13759],{"className":13755},[265],[28,13757],{"className":13758,"style":645},[269],[28,13760,1693],{"className":13761,"style":298},[274,275],"；于是 ",[28,13764,13766,13784],{"className":13765,"translate":193},[197],[28,13767,13769],{"className":13768},[201],[203,13770,13771],{"xmlns":205},[208,13772,13773,13781],{},[211,13774,13775,13777,13779],{},[214,13776,1693],{},[217,13778,83],{},[214,13780,13445],{},[253,13782,13783],{"encoding":255},"v = w",[28,13785,13787,13805],{"className":13786,"ariaHidden":261},[260],[28,13788,13790,13793,13796,13799,13802],{"className":13789},[265],[28,13791],{"className":13792,"style":645},[269],[28,13794,1693],{"className":13795,"style":298},[274,275],[28,13797],{"className":13798,"style":302},[279],[28,13800,83],{"className":13801},[306],[28,13803],{"className":13804,"style":302},[279],[28,13806,13808,13811],{"className":13807},[265],[28,13809],{"className":13810,"style":645},[269],[28,13812,13445],{"className":13813,"style":13560},[274,275],[28,13815,13817,13835],{"className":13816,"translate":193},[197],[28,13818,13820],{"className":13819},[201],[203,13821,13822],{"xmlns":205},[208,13823,13824,13832],{},[211,13825,13826,13828,13830],{},[214,13827,13445],{},[217,13829,1854],{"mathvariant":484},[214,13831,1693],{},[253,13833,13834],{"encoding":255},"w \\ne v",[28,13836,13838,13889],{"className":13837,"ariaHidden":261},[260],[28,13839,13841,13844,13847,13850,13886],{"className":13840},[265],[28,13842],{"className":13843,"style":2371},[269],[28,13845,13445],{"className":13846,"style":13560},[274,275],[28,13848],{"className":13849,"style":302},[279],[28,13851,13853,13880,13883],{"className":13852},[306],[28,13854,13856],{"className":13855},[306],[28,13857,13859],{"className":13858},[274,2359],[28,13860,13862],{"className":13861},[2363],[28,13863,13865,13868,13877],{"className":13864},[2367],[28,13866],{"className":13867,"style":2371},[269],[28,13869,13871],{"className":13870},[2375],[28,13872,13874],{"className":13873},[274],[28,13875,2382],{"className":13876},[306],[28,13878],{"className":13879},[2386],[28,13881],{"className":13882},[279,2390],[28,13884,83],{"className":13885},[306],[28,13887],{"className":13888,"style":302},[279],[28,13890,13892,13895],{"className":13891},[265],[28,13893],{"className":13894,"style":645},[269],[28,13896,1693],{"className":13897,"style":298},[274,275]," 冲突，unsat。整个过程中，数组的长度、其他下标的值，全都不需要知道。",[11252,13900,13901],{"id":13901},"其他理论",[11,13903,13904,13905,13907],{},"字符串理论通常用自动机或序列理论处理，word equation 本身可判定，但加上长度约束、正则约束后很多片段会变得不可判定，工业求解器往往采用不完整但实用的策略。浮点理论通常把 IEEE 754 的位级语义编码成位向量，再交给 ",[25,13906,3039],{}," 后端。",[419,13909,13911],{"id":13910},"_45-冲突学习","4.5 冲突学习",[11,13913,13914,13915,13918],{},"理论冲突不会只让当前搜索失败一次，它会变成一条",[183,13916,13917],{},"理论引理","（theory lemma），永久加入布尔公式。",[11,13920,13921],{},"前面 4.2 节的例子：",[28,13923,13925],{"className":13924,"translate":193},[192],[28,13926,13928,13969],{"className":13927,"translate":193},[197],[28,13929,13931],{"className":13930},[201],[203,13932,13933],{"xmlns":205,"display":206},[208,13934,13935,13967],{},[211,13936,13937,13939,13941,13943,13945,13947,13949,13951,13953,13955,13957,13959,13961,13963,13965],{},[214,13938,176],{},[217,13940,130],{},[224,13942,89],{},[217,13944,1683],{"separator":261},[279,13946],{"width":4053},[214,13948,180],{},[217,13950,130],{},[224,13952,89],{},[217,13954,1683],{"separator":261},[279,13956],{"width":4053},[214,13958,176],{},[217,13960,124],{},[214,13962,180],{},[217,13964,1383],{},[224,13966,89],{},[253,13968,10669],{"encoding":255},[28,13970,13972,13990,14020,14050,14068],{"className":13971,"ariaHidden":261},[260],[28,13973,13975,13978,13981,13984,13987],{"className":13974},[265],[28,13976],{"className":13977,"style":2616},[269],[28,13979,176],{"className":13980},[274,275],[28,13982],{"className":13983,"style":302},[279],[28,13985,130],{"className":13986},[306],[28,13988],{"className":13989,"style":302},[279],[28,13991,13993,13996,13999,14002,14005,14008,14011,14014,14017],{"className":13992},[265],[28,13994],{"className":13995,"style":8494},[269],[28,13997,89],{"className":13998},[274],[28,14000,1683],{"className":14001},[1723],[28,14003],{"className":14004,"style":4222},[279],[28,14006],{"className":14007,"style":530},[279],[28,14009,180],{"className":14010,"style":298},[274,275],[28,14012],{"className":14013,"style":302},[279],[28,14015,130],{"className":14016},[306],[28,14018],{"className":14019,"style":302},[279],[28,14021,14023,14026,14029,14032,14035,14038,14041,14044,14047],{"className":14022},[265],[28,14024],{"className":14025,"style":8494},[269],[28,14027,89],{"className":14028},[274],[28,14030,1683],{"className":14031},[1723],[28,14033],{"className":14034,"style":4222},[279],[28,14036],{"className":14037,"style":530},[279],[28,14039,176],{"className":14040},[274,275],[28,14042],{"className":14043,"style":280},[279],[28,14045,124],{"className":14046},[284],[28,14048],{"className":14049,"style":280},[279],[28,14051,14053,14056,14059,14062,14065],{"className":14052},[265],[28,14054],{"className":14055,"style":294},[269],[28,14057,180],{"className":14058,"style":298},[274,275],[28,14060],{"className":14061,"style":302},[279],[28,14063,1383],{"className":14064},[306],[28,14066],{"className":14067,"style":302},[279],[28,14069,14071,14074],{"className":14070},[265],[28,14072],{"className":14073,"style":316},[269],[28,14075,89],{"className":14076},[274],[11,14078,14079],{},"这三条文字不可同时为真，所以下面这条子句是理论有效的：",[28,14081,14083],{"className":14082,"translate":193},[192],[28,14084,14086,14150],{"className":14085,"translate":193},[197],[28,14087,14089],{"className":14088},[201],[203,14090,14091],{"xmlns":205,"display":206},[208,14092,14093,14147],{},[211,14094,14095,14097,14099,14101,14103,14105,14107,14109,14111,14113,14115,14117,14119,14121,14123,14125,14127,14129,14131,14133,14135,14137,14139,14141,14143,14145],{},[214,14096,912],{"mathvariant":484},[217,14098,43],{"stretchy":241},[214,14100,176],{},[217,14102,130],{},[224,14104,89],{},[217,14106,1522],{"stretchy":241},[228,14108,230],{},[217,14110,937],{},[228,14112,230],{},[214,14114,912],{"mathvariant":484},[217,14116,43],{"stretchy":241},[214,14118,180],{},[217,14120,130],{},[224,14122,89],{},[217,14124,1522],{"stretchy":241},[228,14126,230],{},[217,14128,937],{},[228,14130,230],{},[214,14132,912],{"mathvariant":484},[217,14134,43],{"stretchy":241},[214,14136,176],{},[217,14138,124],{},[214,14140,180],{},[217,14142,1383],{},[224,14144,89],{},[217,14146,1522],{"stretchy":241},[253,14148,14149],{"encoding":255},"\\neg(x > 0) \\;\\vee\\; \\neg(y > 0) \\;\\vee\\; \\neg(x + y \u003C 0)",[28,14151,14153,14177,14204,14228,14255,14279,14297],{"className":14152,"ariaHidden":261},[260],[28,14154,14156,14159,14162,14165,14168,14171,14174],{"className":14155},[265],[28,14157],{"className":14158,"style":341},[269],[28,14160,912],{"className":14161},[274],[28,14163,43],{"className":14164},[348],[28,14166,176],{"className":14167},[274,275],[28,14169],{"className":14170,"style":302},[279],[28,14172,130],{"className":14173},[306],[28,14175],{"className":14176,"style":302},[279],[28,14178,14180,14183,14186,14189,14192,14195,14198,14201],{"className":14179},[265],[28,14181],{"className":14182,"style":341},[269],[28,14184,89],{"className":14185},[274],[28,14187,1522],{"className":14188},[355],[28,14190],{"className":14191,"style":302},[279],[28,14193],{"className":14194,"style":280},[279],[28,14196,937],{"className":14197},[284],[28,14199],{"className":14200,"style":302},[279],[28,14202],{"className":14203,"style":280},[279],[28,14205,14207,14210,14213,14216,14219,14222,14225],{"className":14206},[265],[28,14208],{"className":14209,"style":341},[269],[28,14211,912],{"className":14212},[274],[28,14214,43],{"className":14215},[348],[28,14217,180],{"className":14218,"style":298},[274,275],[28,14220],{"className":14221,"style":302},[279],[28,14223,130],{"className":14224},[306],[28,14226],{"className":14227,"style":302},[279],[28,14229,14231,14234,14237,14240,14243,14246,14249,14252],{"className":14230},[265],[28,14232],{"className":14233,"style":341},[269],[28,14235,89],{"className":14236},[274],[28,14238,1522],{"className":14239},[355],[28,14241],{"className":14242,"style":302},[279],[28,14244],{"className":14245,"style":280},[279],[28,14247,937],{"className":14248},[284],[28,14250],{"className":14251,"style":302},[279],[28,14253],{"className":14254,"style":280},[279],[28,14256,14258,14261,14264,14267,14270,14273,14276],{"className":14257},[265],[28,14259],{"className":14260,"style":341},[269],[28,14262,912],{"className":14263},[274],[28,14265,43],{"className":14266},[348],[28,14268,176],{"className":14269},[274,275],[28,14271],{"className":14272,"style":280},[279],[28,14274,124],{"className":14275},[284],[28,14277],{"className":14278,"style":280},[279],[28,14280,14282,14285,14288,14291,14294],{"className":14281},[265],[28,14283],{"className":14284,"style":294},[269],[28,14286,180],{"className":14287,"style":298},[274,275],[28,14289],{"className":14290,"style":302},[279],[28,14292,1383],{"className":14293},[306],[28,14295],{"className":14296,"style":302},[279],[28,14298,14300,14303,14306],{"className":14299},[265],[28,14301],{"className":14302,"style":341},[269],[28,14304,89],{"className":14305},[274],[28,14307,1522],{"className":14308},[355],[11,14310,14311],{},"把它翻译回命题变量，就是：",[28,14313,14315],{"className":14314,"translate":193},[192],[28,14316,14318,14354],{"className":14317,"translate":193},[197],[28,14319,14321],{"className":14320},[201],[203,14322,14323],{"xmlns":205,"display":206},[208,14324,14325,14351],{},[211,14326,14327,14329,14331,14333,14335,14337,14339,14341,14343,14345,14347,14349],{},[214,14328,912],{"mathvariant":484},[214,14330,11],{},[228,14332,230],{},[217,14334,937],{},[228,14336,230],{},[214,14338,912],{"mathvariant":484},[214,14340,1829],{},[228,14342,230],{},[217,14344,937],{},[228,14346,230],{},[214,14348,912],{"mathvariant":484},[214,14350,1867],{},[253,14352,14353],{"encoding":255},"\\neg p \\;\\vee\\; \\neg q \\;\\vee\\; \\neg r",[28,14355,14357,14384,14411],{"className":14356,"ariaHidden":261},[260],[28,14358,14360,14363,14366,14369,14372,14375,14378,14381],{"className":14359},[265],[28,14361],{"className":14362,"style":1083},[269],[28,14364,912],{"className":14365},[274],[28,14367,11],{"className":14368},[274,275],[28,14370],{"className":14371,"style":302},[279],[28,14373],{"className":14374,"style":280},[279],[28,14376,937],{"className":14377},[284],[28,14379],{"className":14380,"style":302},[279],[28,14382],{"className":14383,"style":280},[279],[28,14385,14387,14390,14393,14396,14399,14402,14405,14408],{"className":14386},[265],[28,14388],{"className":14389,"style":1083},[269],[28,14391,912],{"className":14392},[274],[28,14394,1829],{"className":14395,"style":298},[274,275],[28,14397],{"className":14398,"style":302},[279],[28,14400],{"className":14401,"style":280},[279],[28,14403,937],{"className":14404},[284],[28,14406],{"className":14407,"style":302},[279],[28,14409],{"className":14410,"style":280},[279],[28,14412,14414,14417,14420],{"className":14413},[265],[28,14415],{"className":14416,"style":645},[269],[28,14418,912],{"className":14419},[274],[28,14421,1867],{"className":14422,"style":2279},[274,275],[11,14424,14425,14426,14480,14481,2923],{},"SAT 层学到这条子句之后，就再也不会同时选 ",[28,14427,14429,14450],{"className":14428,"translate":193},[197],[28,14430,14432],{"className":14431},[201],[203,14433,14434],{"xmlns":205},[208,14435,14436,14448],{},[211,14437,14438,14440,14442,14444,14446],{},[214,14439,11],{},[217,14441,1683],{"separator":261},[214,14443,1829],{},[217,14445,1683],{"separator":261},[214,14447,1867],{},[253,14449,2874],{"encoding":255},[28,14451,14453],{"className":14452,"ariaHidden":261},[260],[28,14454,14456,14459,14462,14465,14468,14471,14474,14477],{"className":14455},[265],[28,14457],{"className":14458,"style":612},[269],[28,14460,11],{"className":14461},[274,275],[28,14463,1683],{"className":14464},[1723],[28,14466],{"className":14467,"style":530},[279],[28,14469,1829],{"className":14470,"style":298},[274,275],[28,14472,1683],{"className":14473},[1723],[28,14475],{"className":14476,"style":530},[279],[28,14478,1867],{"className":14479,"style":2279},[274,275]," 都为真了。这一步是 DPLL(T) 效率的关键：",[183,14482,14483],{},"理论层的「一次性检查」变成了布尔层的「永久知识」",[11,14485,14486],{},"如果没有冲突学习，SAT 层会在同一个理论冲突上反复撞墙；有了它，搜索空间被真正地剪掉了。",[419,14488,14490],{"id":14489},"_46-完整例子","4.6 完整例子",[11,14492,14493],{},"还是用这个公式：",[28,14495,14497],{"className":14496,"translate":193},[192],[28,14498,14500,14553],{"className":14499,"translate":193},[197],[28,14501,14503],{"className":14502},[201],[203,14504,14505],{"xmlns":205,"display":206},[208,14506,14507,14551],{},[211,14508,14509,14511,14513,14515,14517,14519,14521,14523,14525,14527,14529,14531,14533,14535,14537,14539,14541,14543,14545,14547,14549],{},[214,14510,510],{},[217,14512,83],{},[217,14514,43],{"stretchy":241},[214,14516,176],{},[217,14518,130],{},[224,14520,89],{},[217,14522,937],{},[214,14524,180],{},[217,14526,130],{},[224,14528,89],{},[217,14530,1522],{"stretchy":241},[228,14532,230],{},[217,14534,233],{},[228,14536,230],{},[217,14538,43],{"stretchy":241},[214,14540,176],{},[217,14542,124],{},[214,14544,180],{},[217,14546,1383],{},[224,14548,89],{},[217,14550,1522],{"stretchy":241},[253,14552,9895],{"encoding":255},[28,14554,14556,14574,14595,14613,14631,14658,14679,14697],{"className":14555,"ariaHidden":261},[260],[28,14557,14559,14562,14565,14568,14571],{"className":14558},[265],[28,14560],{"className":14561,"style":612},[269],[28,14563,510],{"className":14564},[274,275],[28,14566],{"className":14567,"style":302},[279],[28,14569,83],{"className":14570},[306],[28,14572],{"className":14573,"style":302},[279],[28,14575,14577,14580,14583,14586,14589,14592],{"className":14576},[265],[28,14578],{"className":14579,"style":341},[269],[28,14581,43],{"className":14582},[348],[28,14584,176],{"className":14585},[274,275],[28,14587],{"className":14588,"style":302},[279],[28,14590,130],{"className":14591},[306],[28,14593],{"className":14594,"style":302},[279],[28,14596,14598,14601,14604,14607,14610],{"className":14597},[265],[28,14599],{"className":14600,"style":316},[269],[28,14602,89],{"className":14603},[274],[28,14605],{"className":14606,"style":280},[279],[28,14608,937],{"className":14609},[284],[28,14611],{"className":14612,"style":280},[279],[28,14614,14616,14619,14622,14625,14628],{"className":14615},[265],[28,14617],{"className":14618,"style":294},[269],[28,14620,180],{"className":14621,"style":298},[274,275],[28,14623],{"className":14624,"style":302},[279],[28,14626,130],{"className":14627},[306],[28,14629],{"className":14630,"style":302},[279],[28,14632,14634,14637,14640,14643,14646,14649,14652,14655],{"className":14633},[265],[28,14635],{"className":14636,"style":341},[269],[28,14638,89],{"className":14639},[274],[28,14641,1522],{"className":14642},[355],[28,14644],{"className":14645,"style":302},[279],[28,14647],{"className":14648,"style":280},[279],[28,14650,233],{"className":14651},[284],[28,14653],{"className":14654,"style":302},[279],[28,14656],{"className":14657,"style":280},[279],[28,14659,14661,14664,14667,14670,14673,14676],{"className":14660},[265],[28,14662],{"className":14663,"style":341},[269],[28,14665,43],{"className":14666},[348],[28,14668,176],{"className":14669},[274,275],[28,14671],{"className":14672,"style":280},[279],[28,14674,124],{"className":14675},[284],[28,14677],{"className":14678,"style":280},[279],[28,14680,14682,14685,14688,14691,14694],{"className":14681},[265],[28,14683],{"className":14684,"style":294},[269],[28,14686,180],{"className":14687,"style":298},[274,275],[28,14689],{"className":14690,"style":302},[279],[28,14692,1383],{"className":14693},[306],[28,14695],{"className":14696,"style":302},[279],[28,14698,14700,14703,14706],{"className":14699},[265],[28,14701],{"className":14702,"style":341},[269],[28,14704,89],{"className":14705},[274],[28,14707,1522],{"className":14708},[355],[11,14710,14711,14714,14715,14835,14836,15001,15002,15074,15075,15163],{},[183,14712,14713],{},"第 1 轮","：SAT 层选择 ",[28,14716,14718,14751],{"className":14717,"translate":193},[197],[28,14719,14721],{"className":14720},[201],[203,14722,14723],{"xmlns":205},[208,14724,14725,14749],{},[211,14726,14727,14729,14731,14733,14735,14737,14739,14741,14743,14745,14747],{},[214,14728,11],{},[217,14730,83],{},[214,14732,884],{"mathvariant":484},[217,14734,1683],{"separator":261},[214,14736,1829],{},[217,14738,83],{},[214,14740,884],{"mathvariant":484},[217,14742,1683],{"separator":261},[214,14744,1867],{},[217,14746,83],{},[214,14748,884],{"mathvariant":484},[253,14750,10529],{"encoding":255},[28,14752,14754,14772,14799,14826],{"className":14753,"ariaHidden":261},[260],[28,14755,14757,14760,14763,14766,14769],{"className":14756},[265],[28,14758],{"className":14759,"style":612},[269],[28,14761,11],{"className":14762},[274,275],[28,14764],{"className":14765,"style":302},[279],[28,14767,83],{"className":14768},[306],[28,14770],{"className":14771,"style":302},[279],[28,14773,14775,14778,14781,14784,14787,14790,14793,14796],{"className":14774},[265],[28,14776],{"className":14777,"style":2371},[269],[28,14779,884],{"className":14780},[274],[28,14782,1683],{"className":14783},[1723],[28,14785],{"className":14786,"style":530},[279],[28,14788,1829],{"className":14789,"style":298},[274,275],[28,14791],{"className":14792,"style":302},[279],[28,14794,83],{"className":14795},[306],[28,14797],{"className":14798,"style":302},[279],[28,14800,14802,14805,14808,14811,14814,14817,14820,14823],{"className":14801},[265],[28,14803],{"className":14804,"style":2371},[269],[28,14806,884],{"className":14807},[274],[28,14809,1683],{"className":14810},[1723],[28,14812],{"className":14813,"style":530},[279],[28,14815,1867],{"className":14816,"style":2279},[274,275],[28,14818],{"className":14819,"style":302},[279],[28,14821,83],{"className":14822},[306],[28,14824],{"className":14825,"style":302},[279],[28,14827,14829,14832],{"className":14828},[265],[28,14830],{"className":14831,"style":7448},[269],[28,14833,884],{"className":14834},[274],"。理论层检查 ",[28,14837,14839,14887],{"className":14838,"translate":193},[197],[28,14840,14842],{"className":14841},[201],[203,14843,14844],{"xmlns":205},[208,14845,14846,14884],{},[211,14847,14848,14850,14852,14854,14856,14858,14861,14863,14865,14867,14869,14871,14873,14875,14877,14879,14881],{},[217,14849,7916],{"stretchy":241},[214,14851,176],{},[217,14853,130],{},[224,14855,89],{},[217,14857,1683],{"separator":261},[228,14859,14860],{}," ",[214,14862,180],{},[217,14864,130],{},[224,14866,89],{},[217,14868,1683],{"separator":261},[228,14870,14860],{},[214,14872,176],{},[217,14874,124],{},[214,14876,180],{},[217,14878,1383],{},[224,14880,89],{},[217,14882,14883],{"stretchy":241},"}",[253,14885,14886],{"encoding":255},"\\{x > 0,\\ y > 0,\\ x + y \u003C 0\\}",[28,14888,14890,14911,14941,14971,14989],{"className":14889,"ariaHidden":261},[260],[28,14891,14893,14896,14899,14902,14905,14908],{"className":14892},[265],[28,14894],{"className":14895,"style":341},[269],[28,14897,7916],{"className":14898},[348],[28,14900,176],{"className":14901},[274,275],[28,14903],{"className":14904,"style":302},[279],[28,14906,130],{"className":14907},[306],[28,14909],{"className":14910,"style":302},[279],[28,14912,14914,14917,14920,14923,14926,14929,14932,14935,14938],{"className":14913},[265],[28,14915],{"className":14916,"style":8494},[269],[28,14918,89],{"className":14919},[274],[28,14921,1683],{"className":14922},[1723],[28,14924,14860],{"className":14925},[279],[28,14927],{"className":14928,"style":530},[279],[28,14930,180],{"className":14931,"style":298},[274,275],[28,14933],{"className":14934,"style":302},[279],[28,14936,130],{"className":14937},[306],[28,14939],{"className":14940,"style":302},[279],[28,14942,14944,14947,14950,14953,14956,14959,14962,14965,14968],{"className":14943},[265],[28,14945],{"className":14946,"style":8494},[269],[28,14948,89],{"className":14949},[274],[28,14951,1683],{"className":14952},[1723],[28,14954,14860],{"className":14955},[279],[28,14957],{"className":14958,"style":530},[279],[28,14960,176],{"className":14961},[274,275],[28,14963],{"className":14964,"style":280},[279],[28,14966,124],{"className":14967},[284],[28,14969],{"className":14970,"style":280},[279],[28,14972,14974,14977,14980,14983,14986],{"className":14973},[265],[28,14975],{"className":14976,"style":294},[269],[28,14978,180],{"className":14979,"style":298},[274,275],[28,14981],{"className":14982,"style":302},[279],[28,14984,1383],{"className":14985},[306],[28,14987],{"className":14988,"style":302},[279],[28,14990,14992,14995,14998],{"className":14991},[265],[28,14993],{"className":14994,"style":341},[269],[28,14996,89],{"className":14997},[274],[28,14999,14883],{"className":15000},[355],"，发现 ",[28,15003,15005,15026],{"className":15004,"translate":193},[197],[28,15006,15008],{"className":15007},[201],[203,15009,15010],{"xmlns":205},[208,15011,15012,15024],{},[211,15013,15014,15016,15018,15020,15022],{},[214,15015,176],{},[217,15017,124],{},[214,15019,180],{},[217,15021,130],{},[224,15023,89],{},[253,15025,10805],{"encoding":255},[28,15027,15029,15047,15065],{"className":15028,"ariaHidden":261},[260],[28,15030,15032,15035,15038,15041,15044],{"className":15031},[265],[28,15033],{"className":15034,"style":270},[269],[28,15036,176],{"className":15037},[274,275],[28,15039],{"className":15040,"style":280},[279],[28,15042,124],{"className":15043},[284],[28,15045],{"className":15046,"style":280},[279],[28,15048,15050,15053,15056,15059,15062],{"className":15049},[265],[28,15051],{"className":15052,"style":294},[269],[28,15054,180],{"className":15055,"style":298},[274,275],[28,15057],{"className":15058,"style":302},[279],[28,15060,130],{"className":15061},[306],[28,15063],{"className":15064,"style":302},[279],[28,15066,15068,15071],{"className":15067},[265],[28,15069],{"className":15070,"style":316},[269],[28,15072,89],{"className":15073},[274],"，冲突。生成引理 ",[28,15076,15078,15106],{"className":15077,"translate":193},[197],[28,15079,15081],{"className":15080},[201],[203,15082,15083],{"xmlns":205},[208,15084,15085,15103],{},[211,15086,15087,15089,15091,15093,15095,15097,15099,15101],{},[214,15088,912],{"mathvariant":484},[214,15090,11],{},[217,15092,937],{},[214,15094,912],{"mathvariant":484},[214,15096,1829],{},[217,15098,937],{},[214,15100,912],{"mathvariant":484},[214,15102,1867],{},[253,15104,15105],{"encoding":255},"\\neg p \\vee \\neg q \\vee \\neg r",[28,15107,15109,15130,15151],{"className":15108,"ariaHidden":261},[260],[28,15110,15112,15115,15118,15121,15124,15127],{"className":15111},[265],[28,15113],{"className":15114,"style":1083},[269],[28,15116,912],{"className":15117},[274],[28,15119,11],{"className":15120},[274,275],[28,15122],{"className":15123,"style":280},[279],[28,15125,937],{"className":15126},[284],[28,15128],{"className":15129,"style":280},[279],[28,15131,15133,15136,15139,15142,15145,15148],{"className":15132},[265],[28,15134],{"className":15135,"style":1083},[269],[28,15137,912],{"className":15138},[274],[28,15140,1829],{"className":15141,"style":298},[274,275],[28,15143],{"className":15144,"style":280},[279],[28,15146,937],{"className":15147},[284],[28,15149],{"className":15150,"style":280},[279],[28,15152,15154,15157,15160],{"className":15153},[265],[28,15155],{"className":15156,"style":645},[269],[28,15158,912],{"className":15159},[274],[28,15161,1867],{"className":15162,"style":2279},[274,275]," 并回跳。",[11,15165,15166,15169,15170,15291],{},[183,15167,15168],{},"第 2 轮","：SAT 层在引理约束下重新搜索，可能选择 ",[28,15171,15173,15207],{"className":15172,"translate":193},[197],[28,15174,15176],{"className":15175},[201],[203,15177,15178],{"xmlns":205},[208,15179,15180,15204],{},[211,15181,15182,15184,15186,15188,15190,15192,15194,15196,15198,15200,15202],{},[214,15183,11],{},[217,15185,83],{},[214,15187,884],{"mathvariant":484},[217,15189,1683],{"separator":261},[214,15191,1829],{},[217,15193,83],{},[214,15195,894],{"mathvariant":484},[217,15197,1683],{"separator":261},[214,15199,1867],{},[217,15201,83],{},[214,15203,884],{"mathvariant":484},[253,15205,15206],{"encoding":255},"p = \\top, q = \\bot, r = \\top",[28,15208,15210,15228,15255,15282],{"className":15209,"ariaHidden":261},[260],[28,15211,15213,15216,15219,15222,15225],{"className":15212},[265],[28,15214],{"className":15215,"style":612},[269],[28,15217,11],{"className":15218},[274,275],[28,15220],{"className":15221,"style":302},[279],[28,15223,83],{"className":15224},[306],[28,15226],{"className":15227,"style":302},[279],[28,15229,15231,15234,15237,15240,15243,15246,15249,15252],{"className":15230},[265],[28,15232],{"className":15233,"style":2371},[269],[28,15235,884],{"className":15236},[274],[28,15238,1683],{"className":15239},[1723],[28,15241],{"className":15242,"style":530},[279],[28,15244,1829],{"className":15245,"style":298},[274,275],[28,15247],{"className":15248,"style":302},[279],[28,15250,83],{"className":15251},[306],[28,15253],{"className":15254,"style":302},[279],[28,15256,15258,15261,15264,15267,15270,15273,15276,15279],{"className":15257},[265],[28,15259],{"className":15260,"style":2371},[269],[28,15262,894],{"className":15263},[274],[28,15265,1683],{"className":15266},[1723],[28,15268],{"className":15269,"style":530},[279],[28,15271,1867],{"className":15272,"style":2279},[274,275],[28,15274],{"className":15275,"style":302},[279],[28,15277,83],{"className":15278},[306],[28,15280],{"className":15281,"style":302},[279],[28,15283,15285,15288],{"className":15284},[265],[28,15286],{"className":15287,"style":7448},[269],[28,15289,884],{"className":15290},[274],"。理论层检查：",[28,15293,15295],{"className":15294,"translate":193},[192],[28,15296,15298,15340],{"className":15297,"translate":193},[197],[28,15299,15301],{"className":15300},[201],[203,15302,15303],{"xmlns":205,"display":206},[208,15304,15305,15337],{},[211,15306,15307,15309,15311,15313,15315,15317,15319,15321,15323,15325,15327,15329,15331,15333,15335],{},[214,15308,176],{},[217,15310,130],{},[224,15312,89],{},[217,15314,1683],{"separator":261},[279,15316],{"width":4053},[214,15318,180],{},[217,15320,1876],{},[224,15322,89],{},[217,15324,1683],{"separator":261},[279,15326],{"width":4053},[214,15328,176],{},[217,15330,124],{},[214,15332,180],{},[217,15334,1383],{},[224,15336,89],{},[253,15338,15339],{"encoding":255},"x > 0, \\qquad y \\le 0, \\qquad x + y \u003C 0",[28,15341,15343,15361,15391,15421,15439],{"className":15342,"ariaHidden":261},[260],[28,15344,15346,15349,15352,15355,15358],{"className":15345},[265],[28,15347],{"className":15348,"style":2616},[269],[28,15350,176],{"className":15351},[274,275],[28,15353],{"className":15354,"style":302},[279],[28,15356,130],{"className":15357},[306],[28,15359],{"className":15360,"style":302},[279],[28,15362,15364,15367,15370,15373,15376,15379,15382,15385,15388],{"className":15363},[265],[28,15365],{"className":15366,"style":8494},[269],[28,15368,89],{"className":15369},[274],[28,15371,1683],{"className":15372},[1723],[28,15374],{"className":15375,"style":4222},[279],[28,15377],{"className":15378,"style":530},[279],[28,15380,180],{"className":15381,"style":298},[274,275],[28,15383],{"className":15384,"style":302},[279],[28,15386,1876],{"className":15387},[306],[28,15389],{"className":15390,"style":302},[279],[28,15392,15394,15397,15400,15403,15406,15409,15412,15415,15418],{"className":15393},[265],[28,15395],{"className":15396,"style":8494},[269],[28,15398,89],{"className":15399},[274],[28,15401,1683],{"className":15402},[1723],[28,15404],{"className":15405,"style":4222},[279],[28,15407],{"className":15408,"style":530},[279],[28,15410,176],{"className":15411},[274,275],[28,15413],{"className":15414,"style":280},[279],[28,15416,124],{"className":15417},[284],[28,15419],{"className":15420,"style":280},[279],[28,15422,15424,15427,15430,15433,15436],{"className":15423},[265],[28,15425],{"className":15426,"style":294},[269],[28,15428,180],{"className":15429,"style":298},[274,275],[28,15431],{"className":15432,"style":302},[279],[28,15434,1383],{"className":15435},[306],[28,15437],{"className":15438,"style":302},[279],[28,15440,15442,15445],{"className":15441},[265],[28,15443],{"className":15444,"style":316},[269],[28,15446,89],{"className":15447},[274],[11,15449,15450,15451,15542],{},"取 ",[28,15452,15454,15482],{"className":15453,"translate":193},[197],[28,15455,15457],{"className":15456},[201],[203,15458,15459],{"xmlns":205},[208,15460,15461,15479],{},[211,15462,15463,15465,15467,15469,15471,15473,15475,15477],{},[214,15464,176],{},[217,15466,83],{},[224,15468,1244],{},[217,15470,1683],{"separator":261},[214,15472,180],{},[217,15474,83],{},[217,15476,5074],{},[224,15478,1253],{},[253,15480,15481],{"encoding":255},"x = 1, y = -2",[28,15483,15485,15503,15530],{"className":15484,"ariaHidden":261},[260],[28,15486,15488,15491,15494,15497,15500],{"className":15487},[265],[28,15489],{"className":15490,"style":645},[269],[28,15492,176],{"className":15493},[274,275],[28,15495],{"className":15496,"style":302},[279],[28,15498,83],{"className":15499},[306],[28,15501],{"className":15502,"style":302},[279],[28,15504,15506,15509,15512,15515,15518,15521,15524,15527],{"className":15505},[265],[28,15507],{"className":15508,"style":8494},[269],[28,15510,1244],{"className":15511},[274],[28,15513,1683],{"className":15514},[1723],[28,15516],{"className":15517,"style":530},[279],[28,15519,180],{"className":15520,"style":298},[274,275],[28,15522],{"className":15523,"style":302},[279],[28,15525,83],{"className":15526},[306],[28,15528],{"className":15529,"style":302},[279],[28,15531,15533,15536,15539],{"className":15532},[265],[28,15534],{"className":15535,"style":6853},[269],[28,15537,5074],{"className":15538},[274],[28,15540,1253],{"className":15541},[274],"，三条同时成立。理论层返回一致，求解器输出：",[28,15544,15546],{"className":15545,"translate":193},[192],[28,15547,15549,15586],{"className":15548,"translate":193},[197],[28,15550,15552],{"className":15551},[201],[203,15553,15554],{"xmlns":205,"display":206},[208,15555,15556,15583],{},[211,15557,15558,15561,15563,15565,15567,15569,15571,15573,15575,15577,15579,15581],{},[228,15559,405],{"mathvariant":15560},"bold",[217,15562,1683],{"separator":261},[279,15564],{"width":4053},[214,15566,176],{},[217,15568,83],{},[224,15570,1244],{},[217,15572,1683],{"separator":261},[228,15574,14860],{},[214,15576,180],{},[217,15578,83],{},[217,15580,5074],{},[224,15582,1253],{},[253,15584,15585],{"encoding":255},"\\textbf{sat}, \\qquad x = 1,\\ y = -2",[28,15587,15589,15624,15654],{"className":15588,"ariaHidden":261},[260],[28,15590,15592,15596,15603,15606,15609,15612,15615,15618,15621],{"className":15591},[265],[28,15593],{"className":15594,"style":15595},[269],"height:0.8294em;vertical-align:-0.1944em;",[28,15597,15599],{"className":15598},[274,11179],[28,15600,405],{"className":15601},[274,15602],"textbf",[28,15604,1683],{"className":15605},[1723],[28,15607],{"className":15608,"style":4222},[279],[28,15610],{"className":15611,"style":530},[279],[28,15613,176],{"className":15614},[274,275],[28,15616],{"className":15617,"style":302},[279],[28,15619,83],{"className":15620},[306],[28,15622],{"className":15623,"style":302},[279],[28,15625,15627,15630,15633,15636,15639,15642,15645,15648,15651],{"className":15626},[265],[28,15628],{"className":15629,"style":8494},[269],[28,15631,1244],{"className":15632},[274],[28,15634,1683],{"className":15635},[1723],[28,15637,14860],{"className":15638},[279],[28,15640],{"className":15641,"style":530},[279],[28,15643,180],{"className":15644,"style":298},[274,275],[28,15646],{"className":15647,"style":302},[279],[28,15649,83],{"className":15650},[306],[28,15652],{"className":15653,"style":302},[279],[28,15655,15657,15660,15663],{"className":15656},[265],[28,15658],{"className":15659,"style":6853},[269],[28,15661,5074],{"className":15662},[274],[28,15664,1253],{"className":15665},[274],[11,15667,15668],{},"这就是一个完整的 DPLL(T) 循环：一次冲突、一次学习、一次成功。",[11,15670,15671,15672,15675],{},"在真实求解器里，理论层还会做",[183,15673,15674],{},"理论传播","：不等 SAT 层给出完整赋值，就提前推出某些文字必须为真或为假，从而更早剪枝。这让循环通常只需要很少几轮。",[11,15677,15678],{},"理论正确不代表跑得快。工业级求解器还依赖大量工程优化：",[2480,15680,15681,15687,15692,15698,15704,15710],{},[2483,15682,15683,15686],{},[183,15684,15685],{},"预处理","：化简公式、消除冗余变量、把公式规范化成求解器偏好的形式。",[2483,15688,15689,15691],{},[183,15690,15674],{},"：理论层主动推导文字，而不是被动等待完整赋值。",[2483,15693,15694,15697],{},[183,15695,15696],{},"重启策略","：按 Luby 序列或指数策略重启，避免陷入坏的搜索区域。",[2483,15699,15700,15703],{},[183,15701,15702],{},"子句删除","：学习子句太多会拖慢传播，需要定期清理。",[2483,15705,15706,15709],{},[183,15707,15708],{},"多引擎组合","：同一个问题用不同策略并行求解（portfolio），谁先出结果用谁。",[2483,15711,15712,15715],{},[183,15713,15714],{},"专用推理","：针对乘法、数组、字符串等难点，加入专门的推理规则。",[419,15717,15719],{"id":15718},"_47-复杂度边界","4.7 复杂度边界",[11,15721,15722,15723,2923],{},"必须强调：",[183,15724,15725],{},"可判定不等于可解",[2480,15727,15728,15731,15740,15745,15749],{},[2483,15729,15730],{},"SAT 是 NP-complete；",[2483,15732,15733,177,15735,177,15737,15739],{},[25,15734,3039],{},[25,15736,2954],{},[25,15738,2940],{}," 也都是 NP-complete；",[2483,15741,15742,15744],{},[25,15743,2950],{}," 虽然是多项式时间，但实际常数很大；",[2483,15746,15747,7469],{},[25,15748,3027],{},[2483,15750,15751,15753],{},[25,15752,3030],{}," 直接不可判定。",[11,15755,15756],{},"这意味着，工业问题的最坏情况仍然是指数级的。求解器的成功，很大程度上依赖问题结构、编码方式和启发式。遇到超时，工具通常只能返回「未知」，而不是「安全」。",[11,15758,15759],{},"这也是为什么实际系统里，SMT 很少单独使用，而是和抽象解释、fuzzing、测试生成等方法组合。",[414,15761,15763],{"id":15762},"五漏洞检测","五、漏洞检测",[419,15765,15767],{"id":15766},"_51-问题转化","5.1 问题转化",[11,15769,15770],{},"软件漏洞检测的核心问题可以写成一句话：",[15772,15773,15774],"blockquote",{},[11,15775,15776],{},"是否存在一个输入，让程序走到某个危险位置，并且违反某条安全性质？",[11,15778,15779],{},"形式化之后，它就是一个存在性查询：",[28,15781,15783],{"className":15782,"translate":193},[192],[28,15784,15786,15903],{"className":15785,"translate":193},[197],[28,15787,15789],{"className":15788},[201],[203,15790,15791],{"xmlns":205,"display":206},[208,15792,15793,15900],{},[211,15794,15795,15797,15799,15811,15813,15815,15850,15852,15854,15856],{},[214,15796,485],{"mathvariant":484},[228,15798,488],{},[211,15800,15801,15803,15805,15807,15809],{},[214,15802,1686],{},[214,15804,3060],{},[214,15806,11],{},[214,15808,3936],{},[214,15810,1241],{},[214,15812,494],{"mathvariant":484},[279,15814],{"width":497},[1771,15816,15817,15847],{},[1771,15818,15819,15845],{},[211,15820,15821,15829,15831,15843],{},[211,15822,15823,15826],{},[214,15824,15825],{"mathvariant":484},"P",[214,15827,15828],{"mathvariant":484},"C",[217,15830,43],{"stretchy":241},[211,15832,15833,15835,15837,15839,15841],{},[214,15834,1686],{},[214,15836,3060],{},[214,15838,11],{},[214,15840,3936],{},[214,15842,1241],{},[217,15844,1522],{"stretchy":241},[217,15846,1793],{"stretchy":261},[228,15848,15849],{},"路径条件",[228,15851,230],{},[217,15853,233],{},[228,15855,230],{},[1771,15857,15858,15897],{},[1771,15859,15860,15895],{},[211,15861,15862,15879,15881,15893],{},[211,15863,15864,15867,15869,15871,15873,15875,15877],{},[214,15865,15866],{"mathvariant":484},"V",[214,15868,1686],{"mathvariant":484},[214,15870,3933],{"mathvariant":484},[214,15872,1670],{"mathvariant":484},[214,15874,238],{"mathvariant":484},[214,15876,1241],{"mathvariant":484},[214,15878,1667],{"mathvariant":484},[217,15880,43],{"stretchy":241},[211,15882,15883,15885,15887,15889,15891],{},[214,15884,1686],{},[214,15886,3060],{},[214,15888,11],{},[214,15890,3936],{},[214,15892,1241],{},[217,15894,1522],{"stretchy":241},[217,15896,1793],{"stretchy":261},[228,15898,15899],{},"安全性质取反",[253,15901,15902],{"encoding":255},"\\exists\\, \\mathit{input}.\\quad \\underbrace{\\mathrm{PC}(\\mathit{input})}_{\\text{路径条件}} \\;\\wedge\\; \\underbrace{\\mathrm{Violate}(\\mathit{input})}_{\\text{安全性质取反}}",[28,15904,15906,16079],{"className":15905,"ariaHidden":261},[260],[28,15907,15909,15913,15916,15919,15927,15930,15933,15936,16064,16067,16070,16073,16076],{"className":15908},[265],[28,15910],{"className":15911,"style":15912},[269],"height:2.3263em;vertical-align:-1.5763em;",[28,15914,485],{"className":15915},[274],[28,15917],{"className":15918,"style":530},[279],[28,15920,15922],{"className":15921},[274],[28,15923,15926],{"className":15924},[274,15925],"mathit","input",[28,15928,494],{"className":15929},[274],[28,15931],{"className":15932,"style":540},[279],[28,15934],{"className":15935,"style":530},[279],[28,15937,15939],{"className":15938},[1897,1771],[28,15940,15942,16055],{"className":15941},[560,561],[28,15943,15945,16052],{"className":15944},[565],[28,15946,15948,15967],{"className":15947,"style":1907},[569],[28,15949,15951,15954],{"style":15950},"top:-1.4237em;",[28,15952],{"className":15953,"style":1914},[577],[28,15955,15957],{"className":15956},[582,583,584,585],[28,15958,15960],{"className":15959},[274,585],[28,15961,15963],{"className":15962},[274,11179,585],[28,15964,15849],{"className":15965},[274,15966,585],"cjk_fallback",[28,15968,15969,15972],{"style":1926},[28,15970],{"className":15971,"style":1914},[577],[28,15973,15975],{"className":15974},[1897,1771],[28,15976,15978,16044],{"className":15977},[560,561],[28,15979,15981,16041],{"className":15980},[565],[28,15982,15984,16014],{"className":15983,"style":1907},[569],[28,15985,15987,15990],{"className":15986,"style":1946},[1945],[28,15988],{"className":15989,"style":1914},[577],[28,15991,15993,16000,16007],{"className":15992,"style":1954},[1953],[28,15994,15996],{"className":15995,"style":1959},[1958],[1961,15997,15998],{"xmlns":1963,"width":1964,"height":1965,"viewBox":1966,"preserveAspectRatio":1967},[1969,15999],{"d":1971},[28,16001,16003],{"className":16002,"style":1959},[1975],[1961,16004,16005],{"xmlns":1963,"width":1964,"height":1965,"viewBox":1966,"preserveAspectRatio":1978},[1969,16006],{"d":1981},[28,16008,16010],{"className":16009,"style":1959},[1985],[1961,16011,16012],{"xmlns":1963,"width":1964,"height":1965,"viewBox":1966,"preserveAspectRatio":1988},[1969,16013],{"d":1991},[28,16015,16016,16019],{"style":1926},[28,16017],{"className":16018,"style":1914},[577],[28,16020,16022,16029,16032,16038],{"className":16021},[274],[28,16023,16025],{"className":16024},[274],[28,16026,16028],{"className":16027},[274,1712],"PC",[28,16030,43],{"className":16031},[348],[28,16033,16035],{"className":16034},[274],[28,16036,15926],{"className":16037},[274,15925],[28,16039,1522],{"className":16040},[355],[28,16042,593],{"className":16043},[592],[28,16045,16047],{"className":16046},[565],[28,16048,16050],{"className":16049,"style":2042},[569],[28,16051],{},[28,16053,593],{"className":16054},[592],[28,16056,16058],{"className":16057},[565],[28,16059,16062],{"className":16060,"style":16061},[569],"height:1.5763em;",[28,16063],{},[28,16065],{"className":16066,"style":302},[279],[28,16068],{"className":16069,"style":280},[279],[28,16071,233],{"className":16072},[284],[28,16074],{"className":16075,"style":302},[279],[28,16077],{"className":16078,"style":280},[279],[28,16080,16082,16085],{"className":16081},[265],[28,16083],{"className":16084,"style":15912},[269],[28,16086,16088],{"className":16087},[1897,1771],[28,16089,16091,16202],{"className":16090},[560,561],[28,16092,16094,16199],{"className":16093},[565],[28,16095,16097,16114],{"className":16096,"style":1907},[569],[28,16098,16099,16102],{"style":15950},[28,16100],{"className":16101,"style":1914},[577],[28,16103,16105],{"className":16104},[582,583,584,585],[28,16106,16108],{"className":16107},[274,585],[28,16109,16111],{"className":16110},[274,11179,585],[28,16112,15899],{"className":16113},[274,15966,585],[28,16115,16116,16119],{"style":1926},[28,16117],{"className":16118,"style":1914},[577],[28,16120,16122],{"className":16121},[1897,1771],[28,16123,16125,16191],{"className":16124},[560,561],[28,16126,16128,16188],{"className":16127},[565],[28,16129,16131,16161],{"className":16130,"style":1907},[569],[28,16132,16134,16137],{"className":16133,"style":1946},[1945],[28,16135],{"className":16136,"style":1914},[577],[28,16138,16140,16147,16154],{"className":16139,"style":1954},[1953],[28,16141,16143],{"className":16142,"style":1959},[1958],[1961,16144,16145],{"xmlns":1963,"width":1964,"height":1965,"viewBox":1966,"preserveAspectRatio":1967},[1969,16146],{"d":1971},[28,16148,16150],{"className":16149,"style":1959},[1975],[1961,16151,16152],{"xmlns":1963,"width":1964,"height":1965,"viewBox":1966,"preserveAspectRatio":1978},[1969,16153],{"d":1981},[28,16155,16157],{"className":16156,"style":1959},[1985],[1961,16158,16159],{"xmlns":1963,"width":1964,"height":1965,"viewBox":1966,"preserveAspectRatio":1988},[1969,16160],{"d":1991},[28,16162,16163,16166],{"style":1926},[28,16164],{"className":16165,"style":1914},[577],[28,16167,16169,16176,16179,16185],{"className":16168},[274],[28,16170,16172],{"className":16171},[274],[28,16173,16175],{"className":16174},[274,1712],"Violate",[28,16177,43],{"className":16178},[348],[28,16180,16182],{"className":16181},[274],[28,16183,15926],{"className":16184},[274,15925],[28,16186,1522],{"className":16187},[355],[28,16189,593],{"className":16190},[592],[28,16192,16194],{"className":16193},[565],[28,16195,16197],{"className":16196,"style":2042},[569],[28,16198],{},[28,16200,593],{"className":16201},[592],[28,16203,16205],{"className":16204},[565],[28,16206,16208],{"className":16207,"style":16061},[569],[28,16209],{},[2480,16211,16212,16249],{},[2483,16213,16214,16248],{},[28,16215,16217,16233],{"className":16216,"translate":193},[197],[28,16218,16220],{"className":16219},[201],[203,16221,16222],{"xmlns":205},[208,16223,16224,16230],{},[211,16225,16226,16228],{},[214,16227,15825],{"mathvariant":484},[214,16229,15828],{"mathvariant":484},[253,16231,16232],{"encoding":255},"\\mathrm{PC}",[28,16234,16236],{"className":16235,"ariaHidden":261},[260],[28,16237,16239,16242],{"className":16238},[265],[28,16240],{"className":16241,"style":455},[269],[28,16243,16245],{"className":16244},[274],[28,16246,16028],{"className":16247},[274,1712]," 是路径条件，描述「走到这里需要满足哪些约束」；",[2483,16250,16251,16295],{},[28,16252,16254,16280],{"className":16253,"translate":193},[197],[28,16255,16257],{"className":16256},[201],[203,16258,16259],{"xmlns":205},[208,16260,16261,16277],{},[211,16262,16263,16265,16267,16269,16271,16273,16275],{},[214,16264,15866],{"mathvariant":484},[214,16266,1686],{"mathvariant":484},[214,16268,3933],{"mathvariant":484},[214,16270,1670],{"mathvariant":484},[214,16272,238],{"mathvariant":484},[214,16274,1241],{"mathvariant":484},[214,16276,1667],{"mathvariant":484},[253,16278,16279],{"encoding":255},"\\mathrm{Violate}",[28,16281,16283],{"className":16282,"ariaHidden":261},[260],[28,16284,16286,16289],{"className":16285},[265],[28,16287],{"className":16288,"style":7448},[269],[28,16290,16292],{"className":16291},[274],[28,16293,16175],{"className":16294},[274,1712]," 是安全性质的否定，描述「怎样才算越界\u002F溢出\u002F空指针解引用」。",[11,16297,16298,16299,16301,16302,16304],{},"如果这个公式 ",[183,16300,405],{},"，求解器给出的模型就是一条能触发漏洞的输入，漏洞可达；如果 ",[183,16303,6456],{},"，说明在这条路径上不可能违反性质，可以安全剪枝。",[11,16306,16307,16308,2923],{},"这个转化是整个应用的基石：",[183,16309,16310],{},"「找漏洞」变成了「解约束」",[419,16312,16314],{"id":16313},"_52-缓冲区溢出","5.2 缓冲区溢出",[11,16316,16317],{},"先看一个最直观的例子：",[18,16319,16321],{"className":20,"code":16320,"language":22,"meta":23,"style":23},"void process(char *input, int len) {\n    char buf[8];\n    if (len > 4 && len \u003C 12) {\n        memcpy(buf, input, len);   \u002F\u002F 可疑点\n    }\n}\n",[25,16322,16323,16349,16364,16388,16399,16403],{"__ignoreMap":23},[28,16324,16325,16327,16330,16332,16335,16338,16340,16342,16344,16347],{"class":30,"line":31},[28,16326,35],{"class":34},[28,16328,16329],{"class":38}," process",[28,16331,43],{"class":42},[28,16333,16334],{"class":34},"char",[28,16336,16337],{"class":34}," *",[28,16339,15926],{"class":49},[28,16341,53],{"class":42},[28,16343,46],{"class":34},[28,16345,16346],{"class":49}," len",[28,16348,61],{"class":42},[28,16350,16351,16354,16357,16359,16361],{"class":30,"line":64},[28,16352,16353],{"class":34},"    char",[28,16355,16356],{"class":49}," buf",[28,16358,73],{"class":42},[28,16360,5236],{"class":76},[28,16362,16363],{"class":42},"];\n",[28,16365,16366,16368,16371,16373,16376,16378,16381,16383,16386],{"class":30,"line":95},[28,16367,118],{"class":34},[28,16369,16370],{"class":42}," (len ",[28,16372,130],{"class":34},[28,16374,16375],{"class":76}," 4",[28,16377,136],{"class":34},[28,16379,16380],{"class":42}," len ",[28,16382,1383],{"class":34},[28,16384,16385],{"class":76}," 12",[28,16387,61],{"class":42},[28,16389,16390,16393,16396],{"class":30,"line":115},[28,16391,16392],{"class":38},"        memcpy",[28,16394,16395],{"class":42},"(buf, input, len);",[28,16397,16398],{"class":157},"   \u002F\u002F 可疑点\n",[28,16400,16401],{"class":30,"line":154},[28,16402,164],{"class":42},[28,16404,16405],{"class":30,"line":161},[28,16406,170],{"class":42},[11,16408,16409,16412,16413,16416,16417,16420,16421,16423],{},[25,16410,16411],{},"buf"," 只有 8 字节，而 ",[25,16414,16415],{},"memcpy"," 拷贝 ",[25,16418,16419],{},"len"," 字节。如果 ",[25,16422,16419],{}," 能大于 8，就会发生栈溢出。",[11,16425,16426,16427,2693,16429,16431,16432,16434],{},"符号执行引擎把 ",[25,16428,15926],{},[25,16430,16419],{}," 当作符号变量，沿路径走到 ",[25,16433,16415],{}," 时，收集到的路径条件是：",[28,16436,16438],{"className":16437,"translate":193},[192],[28,16439,16441,16480],{"className":16440,"translate":193},[197],[28,16442,16444],{"className":16443},[201],[203,16445,16446],{"xmlns":205,"display":206},[208,16447,16448,16477],{},[211,16449,16450,16452,16454,16456,16458,16460,16462,16464,16466,16468,16470,16472,16474],{},[214,16451,1670],{},[214,16453,1667],{},[214,16455,3060],{},[217,16457,130],{},[224,16459,9215],{},[228,16461,230],{},[217,16463,233],{},[228,16465,230],{},[214,16467,1670],{},[214,16469,1667],{},[214,16471,3060],{},[217,16473,1383],{},[224,16475,16476],{},"12",[253,16478,16479],{"encoding":255},"len > 4 \\;\\wedge\\; len \u003C 12",[28,16481,16483,16509,16533,16557],{"className":16482,"ariaHidden":261},[260],[28,16484,16486,16490,16494,16497,16500,16503,16506],{"className":16485},[265],[28,16487],{"className":16488,"style":16489},[269],"height:0.7335em;vertical-align:-0.0391em;",[28,16491,1670],{"className":16492,"style":16493},[274,275],"margin-right:0.0197em;",[28,16495,1667],{"className":16496},[274,275],[28,16498,3060],{"className":16499},[274,275],[28,16501],{"className":16502,"style":302},[279],[28,16504,130],{"className":16505},[306],[28,16507],{"className":16508,"style":302},[279],[28,16510,16512,16515,16518,16521,16524,16527,16530],{"className":16511},[265],[28,16513],{"className":16514,"style":316},[269],[28,16516,9215],{"className":16517},[274],[28,16519],{"className":16520,"style":302},[279],[28,16522],{"className":16523,"style":280},[279],[28,16525,233],{"className":16526},[284],[28,16528],{"className":16529,"style":302},[279],[28,16531],{"className":16532,"style":280},[279],[28,16534,16536,16539,16542,16545,16548,16551,16554],{"className":16535},[265],[28,16537],{"className":16538,"style":16489},[269],[28,16540,1670],{"className":16541,"style":16493},[274,275],[28,16543,1667],{"className":16544},[274,275],[28,16546,3060],{"className":16547},[274,275],[28,16549],{"className":16550,"style":302},[279],[28,16552,1383],{"className":16553},[306],[28,16555],{"className":16556,"style":302},[279],[28,16558,16560,16563],{"className":16559},[265],[28,16561],{"className":16562,"style":316},[269],[28,16564,16476],{"className":16565},[274],[11,16567,16568],{},"要判断溢出是否可达，就在路径条件上再加一条「写入长度超过缓冲区大小」：",[28,16570,16572],{"className":16571,"translate":193},[192],[28,16573,16575,16629],{"className":16574,"translate":193},[197],[28,16576,16578],{"className":16577},[201],[203,16579,16580],{"xmlns":205,"display":206},[208,16581,16582,16626],{},[211,16583,16584,16586,16588,16590,16592,16594,16596,16598,16600,16602,16604,16606,16608,16610,16612,16614,16616,16618,16620,16622,16624],{},[214,16585,1670],{},[214,16587,1667],{},[214,16589,3060],{},[217,16591,130],{},[224,16593,9215],{},[228,16595,230],{},[217,16597,233],{},[228,16599,230],{},[214,16601,1670],{},[214,16603,1667],{},[214,16605,3060],{},[217,16607,1383],{},[224,16609,16476],{},[228,16611,230],{},[217,16613,233],{},[228,16615,230],{},[214,16617,1670],{},[214,16619,1667],{},[214,16621,3060],{},[217,16623,130],{},[224,16625,5236],{},[253,16627,16628],{"encoding":255},"len > 4 \\;\\wedge\\; len \u003C 12 \\;\\wedge\\; len > 8",[28,16630,16632,16656,16680,16704,16728,16752],{"className":16631,"ariaHidden":261},[260],[28,16633,16635,16638,16641,16644,16647,16650,16653],{"className":16634},[265],[28,16636],{"className":16637,"style":16489},[269],[28,16639,1670],{"className":16640,"style":16493},[274,275],[28,16642,1667],{"className":16643},[274,275],[28,16645,3060],{"className":16646},[274,275],[28,16648],{"className":16649,"style":302},[279],[28,16651,130],{"className":16652},[306],[28,16654],{"className":16655,"style":302},[279],[28,16657,16659,16662,16665,16668,16671,16674,16677],{"className":16658},[265],[28,16660],{"className":16661,"style":316},[269],[28,16663,9215],{"className":16664},[274],[28,16666],{"className":16667,"style":302},[279],[28,16669],{"className":16670,"style":280},[279],[28,16672,233],{"className":16673},[284],[28,16675],{"className":16676,"style":302},[279],[28,16678],{"className":16679,"style":280},[279],[28,16681,16683,16686,16689,16692,16695,16698,16701],{"className":16682},[265],[28,16684],{"className":16685,"style":16489},[269],[28,16687,1670],{"className":16688,"style":16493},[274,275],[28,16690,1667],{"className":16691},[274,275],[28,16693,3060],{"className":16694},[274,275],[28,16696],{"className":16697,"style":302},[279],[28,16699,1383],{"className":16700},[306],[28,16702],{"className":16703,"style":302},[279],[28,16705,16707,16710,16713,16716,16719,16722,16725],{"className":16706},[265],[28,16708],{"className":16709,"style":316},[269],[28,16711,16476],{"className":16712},[274],[28,16714],{"className":16715,"style":302},[279],[28,16717],{"className":16718,"style":280},[279],[28,16720,233],{"className":16721},[284],[28,16723],{"className":16724,"style":302},[279],[28,16726],{"className":16727,"style":280},[279],[28,16729,16731,16734,16737,16740,16743,16746,16749],{"className":16730},[265],[28,16732],{"className":16733,"style":16489},[269],[28,16735,1670],{"className":16736,"style":16493},[274,275],[28,16738,1667],{"className":16739},[274,275],[28,16741,3060],{"className":16742},[274,275],[28,16744],{"className":16745,"style":302},[279],[28,16747,130],{"className":16748},[306],[28,16750],{"className":16751,"style":302},[279],[28,16753,16755,16758],{"className":16754},[265],[28,16756],{"className":16757,"style":316},[269],[28,16759,5236],{"className":16760},[274],[11,16762,16763],{},"写成 SMT-LIB：",[18,16765,16769],{"className":16766,"code":16767,"language":16768,"meta":23,"style":23},"language-smt shiki shiki-themes github-light github-light github-dark","(set-logic QF_LIA)\n(declare-const len Int)\n(assert (and (> len 4) (\u003C len 12) (> len 8)))\n(check-sat)\n(get-model)\n","smt",[25,16770,16771,16776,16781,16786,16791],{"__ignoreMap":23},[28,16772,16773],{"class":30,"line":31},[28,16774,16775],{},"(set-logic QF_LIA)\n",[28,16777,16778],{"class":30,"line":64},[28,16779,16780],{},"(declare-const len Int)\n",[28,16782,16783],{"class":30,"line":95},[28,16784,16785],{},"(assert (and (> len 4) (\u003C len 12) (> len 8)))\n",[28,16787,16788],{"class":30,"line":115},[28,16789,16790],{},"(check-sat)\n",[28,16792,16793],{"class":30,"line":154},[28,16794,16795],{},"(get-model)\n",[11,16797,16798],{},"求解器返回：",[18,16800,16803],{"className":16801,"code":16802,"language":11179},[11177],"sat\n(model\n  (define-fun len () Int\n    9)\n)\n",[25,16804,16802],{"__ignoreMap":23},[11,16806,16807,16810,16811,16885,16886,16937,16938,16940],{},[25,16808,16809],{},"len = 9"," 是一个合法解：它满足 ",[28,16812,16814,16837],{"className":16813,"translate":193},[197],[28,16815,16817],{"className":16816},[201],[203,16818,16819],{"xmlns":205},[208,16820,16821,16834],{},[211,16822,16823,16825,16827,16830,16832],{},[224,16824,9215],{},[217,16826,1383],{},[224,16828,16829],{},"9",[217,16831,1383],{},[224,16833,16476],{},[253,16835,16836],{"encoding":255},"4 \u003C 9 \u003C 12",[28,16838,16840,16858,16876],{"className":16839,"ariaHidden":261},[260],[28,16841,16843,16846,16849,16852,16855],{"className":16842},[265],[28,16844],{"className":16845,"style":9429},[269],[28,16847,9215],{"className":16848},[274],[28,16850],{"className":16851,"style":302},[279],[28,16853,1383],{"className":16854},[306],[28,16856],{"className":16857,"style":302},[279],[28,16859,16861,16864,16867,16870,16873],{"className":16860},[265],[28,16862],{"className":16863,"style":9429},[269],[28,16865,16829],{"className":16866},[274],[28,16868],{"className":16869,"style":302},[279],[28,16871,1383],{"className":16872},[306],[28,16874],{"className":16875,"style":302},[279],[28,16877,16879,16882],{"className":16878},[265],[28,16880],{"className":16881,"style":316},[269],[28,16883,16476],{"className":16884},[274],"，同时 ",[28,16887,16889,16907],{"className":16888,"translate":193},[197],[28,16890,16892],{"className":16891},[201],[203,16893,16894],{"xmlns":205},[208,16895,16896,16904],{},[211,16897,16898,16900,16902],{},[224,16899,16829],{},[217,16901,130],{},[224,16903,5236],{},[253,16905,16906],{"encoding":255},"9 > 8",[28,16908,16910,16928],{"className":16909,"ariaHidden":261},[260],[28,16911,16913,16916,16919,16922,16925],{"className":16912},[265],[28,16914],{"className":16915,"style":9429},[269],[28,16917,16829],{"className":16918},[274],[28,16920],{"className":16921,"style":302},[279],[28,16923,130],{"className":16924},[306],[28,16926],{"className":16927,"style":302},[279],[28,16929,16931,16934],{"className":16930},[265],[28,16932],{"className":16933,"style":316},[269],[28,16935,5236],{"className":16936},[274],"。这条路径可达，",[25,16939,16415],{}," 会往 8 字节的缓冲区里写 9 字节，溢出 1 字节。是否真的覆盖到返回地址取决于栈布局，但「越界写」这个事实本身已经成立。",[11,16942,16943],{},"如果把代码改成：",[18,16945,16947],{"className":20,"code":16946,"language":22,"meta":23,"style":23},"if (len > 4 && len \u003C= 8) {\n    memcpy(buf, input, len);\n}\n",[25,16948,16949,16972,16980],{"__ignoreMap":23},[28,16950,16951,16954,16956,16958,16960,16962,16964,16967,16970],{"class":30,"line":31},[28,16952,16953],{"class":34},"if",[28,16955,16370],{"class":42},[28,16957,130],{"class":34},[28,16959,16375],{"class":76},[28,16961,136],{"class":34},[28,16963,16380],{"class":42},[28,16965,16966],{"class":34},"\u003C=",[28,16968,16969],{"class":76}," 8",[28,16971,61],{"class":42},[28,16973,16974,16977],{"class":30,"line":64},[28,16975,16976],{"class":38},"    memcpy",[28,16978,16979],{"class":42},"(buf, input, len);\n",[28,16981,16982],{"class":30,"line":95},[28,16983,170],{"class":42},[11,16985,16986],{},"约束就变成：",[28,16988,16990],{"className":16989,"translate":193},[192],[28,16991,16993,17047],{"className":16992,"translate":193},[197],[28,16994,16996],{"className":16995},[201],[203,16997,16998],{"xmlns":205,"display":206},[208,16999,17000,17044],{},[211,17001,17002,17004,17006,17008,17010,17012,17014,17016,17018,17020,17022,17024,17026,17028,17030,17032,17034,17036,17038,17040,17042],{},[214,17003,1670],{},[214,17005,1667],{},[214,17007,3060],{},[217,17009,130],{},[224,17011,9215],{},[228,17013,230],{},[217,17015,233],{},[228,17017,230],{},[214,17019,1670],{},[214,17021,1667],{},[214,17023,3060],{},[217,17025,1876],{},[224,17027,5236],{},[228,17029,230],{},[217,17031,233],{},[228,17033,230],{},[214,17035,1670],{},[214,17037,1667],{},[214,17039,3060],{},[217,17041,130],{},[224,17043,5236],{},[253,17045,17046],{"encoding":255},"len > 4 \\;\\wedge\\; len \\le 8 \\;\\wedge\\; len > 8",[28,17048,17050,17074,17098,17123,17147,17171],{"className":17049,"ariaHidden":261},[260],[28,17051,17053,17056,17059,17062,17065,17068,17071],{"className":17052},[265],[28,17054],{"className":17055,"style":16489},[269],[28,17057,1670],{"className":17058,"style":16493},[274,275],[28,17060,1667],{"className":17061},[274,275],[28,17063,3060],{"className":17064},[274,275],[28,17066],{"className":17067,"style":302},[279],[28,17069,130],{"className":17070},[306],[28,17072],{"className":17073,"style":302},[279],[28,17075,17077,17080,17083,17086,17089,17092,17095],{"className":17076},[265],[28,17078],{"className":17079,"style":316},[269],[28,17081,9215],{"className":17082},[274],[28,17084],{"className":17085,"style":302},[279],[28,17087],{"className":17088,"style":280},[279],[28,17090,233],{"className":17091},[284],[28,17093],{"className":17094,"style":302},[279],[28,17096],{"className":17097,"style":280},[279],[28,17099,17101,17105,17108,17111,17114,17117,17120],{"className":17100},[265],[28,17102],{"className":17103,"style":17104},[269],"height:0.8304em;vertical-align:-0.136em;",[28,17106,1670],{"className":17107,"style":16493},[274,275],[28,17109,1667],{"className":17110},[274,275],[28,17112,3060],{"className":17113},[274,275],[28,17115],{"className":17116,"style":302},[279],[28,17118,1876],{"className":17119},[306],[28,17121],{"className":17122,"style":302},[279],[28,17124,17126,17129,17132,17135,17138,17141,17144],{"className":17125},[265],[28,17127],{"className":17128,"style":316},[269],[28,17130,5236],{"className":17131},[274],[28,17133],{"className":17134,"style":302},[279],[28,17136],{"className":17137,"style":280},[279],[28,17139,233],{"className":17140},[284],[28,17142],{"className":17143,"style":302},[279],[28,17145],{"className":17146,"style":280},[279],[28,17148,17150,17153,17156,17159,17162,17165,17168],{"className":17149},[265],[28,17151],{"className":17152,"style":16489},[269],[28,17154,1670],{"className":17155,"style":16493},[274,275],[28,17157,1667],{"className":17158},[274,275],[28,17160,3060],{"className":17161},[274,275],[28,17163],{"className":17164,"style":302},[279],[28,17166,130],{"className":17167},[306],[28,17169],{"className":17170,"style":302},[279],[28,17172,17174,17177],{"className":17173},[265],[28,17175],{"className":17176,"style":316},[269],[28,17178,5236],{"className":17179},[274],[11,17181,17182,17183,17185],{},"求解器返回 ",[183,17184,6456],{},"。这条溢出路径不可达，工具不会报告漏洞。",[11,17187,17188,17189,17191,17192,17194,17195,17197,17198,12376],{},"注意这里用的是 ",[25,17190,2954],{},"（整数线性算术）。真实的 C 代码里 ",[25,17193,16419],{}," 是 32 位 ",[25,17196,46],{},"，更精确的建模应该用 ",[25,17199,3039],{},[18,17201,17203],{"className":16766,"code":17202,"language":16768,"meta":23,"style":23},"(set-logic QF_BV)\n(declare-const len (_ BitVec 32))\n(assert (and (bvsgt len (_ bv4 32))\n             (bvslt len (_ bv12 32))\n             (bvsgt len (_ bv8 32))))\n(check-sat)\n(get-model)\n",[25,17204,17205,17210,17215,17220,17225,17230,17234],{"__ignoreMap":23},[28,17206,17207],{"class":30,"line":31},[28,17208,17209],{},"(set-logic QF_BV)\n",[28,17211,17212],{"class":30,"line":64},[28,17213,17214],{},"(declare-const len (_ BitVec 32))\n",[28,17216,17217],{"class":30,"line":95},[28,17218,17219],{},"(assert (and (bvsgt len (_ bv4 32))\n",[28,17221,17222],{"class":30,"line":115},[28,17223,17224],{},"             (bvslt len (_ bv12 32))\n",[28,17226,17227],{"class":30,"line":154},[28,17228,17229],{},"             (bvsgt len (_ bv8 32))))\n",[28,17231,17232],{"class":30,"line":161},[28,17233,16790],{},[28,17235,17236],{"class":30,"line":167},[28,17237,16795],{},[11,17239,17240,17241,2951,17244,17247,17248,17250,17251,17254],{},"注意这里用的是有符号比较（",[25,17242,17243],{},"bvsgt",[25,17245,17246],{},"bvslt","），因为 C 里的 ",[25,17249,46],{}," 是有符号的。一个可能的模型是 ",[25,17252,17253],{},"len = #x00000009","。对这个例子来说，两种建模的结论一样；但下一个例子就会看到，理论选错，结论可能完全相反。",[419,17256,17258],{"id":17257},"_53-整数溢出","5.3 整数溢出",[18,17260,17262],{"className":20,"code":17261,"language":22,"meta":23,"style":23},"void copy(char *src, unsigned int off, unsigned int len) {\n    char buf[64];\n    if (off + len \u003C= 64) {\n        memcpy(buf + off, src, len);\n    }\n}\n",[25,17263,17264,17301,17314,17332,17344,17348],{"__ignoreMap":23},[28,17265,17266,17268,17271,17273,17275,17277,17280,17282,17285,17288,17291,17293,17295,17297,17299],{"class":30,"line":31},[28,17267,35],{"class":34},[28,17269,17270],{"class":38}," copy",[28,17272,43],{"class":42},[28,17274,16334],{"class":34},[28,17276,16337],{"class":34},[28,17278,17279],{"class":49},"src",[28,17281,53],{"class":42},[28,17283,17284],{"class":34},"unsigned",[28,17286,17287],{"class":34}," int",[28,17289,17290],{"class":49}," off",[28,17292,53],{"class":42},[28,17294,17284],{"class":34},[28,17296,17287],{"class":34},[28,17298,16346],{"class":49},[28,17300,61],{"class":42},[28,17302,17303,17305,17307,17309,17312],{"class":30,"line":64},[28,17304,16353],{"class":34},[28,17306,16356],{"class":49},[28,17308,73],{"class":42},[28,17310,17311],{"class":76},"64",[28,17313,16363],{"class":42},[28,17315,17316,17318,17321,17323,17325,17327,17330],{"class":30,"line":95},[28,17317,118],{"class":34},[28,17319,17320],{"class":42}," (off ",[28,17322,124],{"class":34},[28,17324,16380],{"class":42},[28,17326,16966],{"class":34},[28,17328,17329],{"class":76}," 64",[28,17331,61],{"class":42},[28,17333,17334,17336,17339,17341],{"class":30,"line":115},[28,17335,16392],{"class":38},[28,17337,17338],{"class":42},"(buf ",[28,17340,124],{"class":34},[28,17342,17343],{"class":42}," off, src, len);\n",[28,17345,17346],{"class":30,"line":154},[28,17347,164],{"class":42},[28,17349,17350],{"class":30,"line":161},[28,17351,170],{"class":42},[11,17353,17354,17355,17358,17359,2693,17362,17364,17365,3104],{},"看起来检查很严：只有 ",[25,17356,17357],{},"off + len"," 不超过 64 才拷贝。但 ",[25,17360,17361],{},"off",[25,17363,16419],{}," 都是 32 位无符号整数，加法是",[183,17366,17367,17368],{},"模 ",[28,17369,17371,17389],{"className":17370,"translate":193},[197],[28,17372,17374],{"className":17373},[201],[203,17375,17376],{"xmlns":205},[208,17377,17378,17386],{},[211,17379,17380],{},[3054,17381,17382,17384],{},[224,17383,1253],{},[224,17385,12998],{},[253,17387,17388],{"encoding":255},"2^{32}",[28,17390,17392],{"className":17391,"ariaHidden":261},[260],[28,17393,17395,17398],{"className":17394},[265],[28,17396],{"className":17397,"style":5477},[269],[28,17399,17401,17404],{"className":17400},[274],[28,17402,1253],{"className":17403},[274],[28,17405,17407],{"className":17406},[556],[28,17408,17410],{"className":17409},[560],[28,17411,17413],{"className":17412},[565],[28,17414,17416],{"className":17415,"style":5477},[569],[28,17417,17418,17421],{"style":3094},[28,17419],{"className":17420,"style":578},[577],[28,17422,17424],{"className":17423},[582,583,584,585],[28,17425,17427],{"className":17426},[274,585],[28,17428,12998],{"className":17429},[274,585],[11,17431,17432],{},"攻击者可以取：",[28,17434,17436],{"className":17435,"translate":193},[192],[28,17437,17439,17469],{"className":17438,"translate":193},[197],[28,17440,17442],{"className":17441},[201],[203,17443,17444],{"xmlns":205,"display":206},[208,17445,17446,17466],{},[211,17447,17448,17450,17452,17455,17457,17459,17461,17463],{},[228,17449,17361],{},[217,17451,83],{},[228,17453,17454],{"mathvariant":13013},"0xFFFFFFF0",[217,17456,1683],{"separator":261},[279,17458],{"width":4053},[228,17460,16419],{},[217,17462,83],{},[228,17464,17465],{"mathvariant":13013},"0x20",[253,17467,17468],{"encoding":255},"\\text{off} = \\texttt{0xFFFFFFF0}, \\qquad \\text{len} = \\texttt{0x20}",[28,17470,17472,17493,17529],{"className":17471,"ariaHidden":261},[260],[28,17473,17475,17478,17484,17487,17490],{"className":17474},[265],[28,17476],{"className":17477,"style":7448},[269],[28,17479,17481],{"className":17480},[274,11179],[28,17482,17361],{"className":17483},[274],[28,17485],{"className":17486,"style":302},[279],[28,17488,83],{"className":17489},[306],[28,17491],{"className":17492,"style":302},[279],[28,17494,17496,17499,17505,17508,17511,17514,17520,17523,17526],{"className":17495},[265],[28,17497],{"className":17498,"style":2371},[269],[28,17500,17502],{"className":17501},[274,11179],[28,17503,17454],{"className":17504},[274,13211],[28,17506,1683],{"className":17507},[1723],[28,17509],{"className":17510,"style":4222},[279],[28,17512],{"className":17513,"style":530},[279],[28,17515,17517],{"className":17516},[274,11179],[28,17518,16419],{"className":17519},[274],[28,17521],{"className":17522,"style":302},[279],[28,17524,83],{"className":17525},[306],[28,17527],{"className":17528,"style":302},[279],[28,17530,17532,17535],{"className":17531},[265],[28,17533],{"className":17534,"style":13204},[269],[28,17536,17538],{"className":17537},[274,11179],[28,17539,17465],{"className":17540},[274,13211],[11,17542,17543],{},"计算一下：",[28,17545,17547],{"className":17546,"translate":193},[192],[28,17548,17550,17663],{"className":17549,"translate":193},[197],[28,17551,17553],{"className":17552},[201],[203,17554,17555],{"xmlns":205,"display":206},[208,17556,17557,17660],{},[5015,17558,17559,17589,17610],{"rowspacing":5017,"columnalign":5018,"columnspacing":5019},[5021,17560,17561,17573],{},[5024,17562,17563],{},[5027,17564,17565],{"scriptlevel":89,"displaystyle":261},[211,17566,17567,17569,17571],{},[228,17568,17361],{},[217,17570,124],{},[228,17572,16419],{},[5024,17574,17575],{},[5027,17576,17577],{"scriptlevel":89,"displaystyle":261},[211,17578,17579,17581,17583,17585,17587],{},[211,17580],{},[217,17582,83],{},[228,17584,17454],{"mathvariant":13013},[217,17586,124],{},[228,17588,17465],{"mathvariant":13013},[5021,17590,17591,17597],{},[5024,17592,17593],{},[5027,17594,17595],{"scriptlevel":89,"displaystyle":261},[211,17596],{},[5024,17598,17599],{},[5027,17600,17601],{"scriptlevel":89,"displaystyle":261},[211,17602,17603,17605,17607],{},[211,17604],{},[217,17606,83],{},[228,17608,17609],{"mathvariant":13013},"0x100000010",[5021,17611,17612,17618],{},[5024,17613,17614],{},[5027,17615,17616],{"scriptlevel":89,"displaystyle":261},[211,17617],{},[5024,17619,17620],{},[5027,17621,17622],{"scriptlevel":89,"displaystyle":261},[211,17623,17624,17626,17628,17631,17633,17636,17638,17640,17642,17650,17652,17658],{},[211,17625],{},[217,17627,10074],{},[228,17629,17630],{"mathvariant":13013},"0x10",[217,17632,83],{},[224,17634,17635],{},"16",[279,17637],{},[279,17639],{"width":497},[217,17641,43],{"stretchy":241},[211,17643,17644,17646,17648],{},[214,17645,12983],{"mathvariant":484},[214,17647,3933],{"mathvariant":484},[214,17649,12988],{"mathvariant":484},[279,17651],{"width":12991},[3054,17653,17654,17656],{},[224,17655,1253],{},[224,17657,12998],{},[217,17659,1522],{"stretchy":241},[253,17661,17662],{"encoding":255},"\\begin{aligned}\n\\text{off} + \\text{len}\n&= \\texttt{0xFFFFFFF0} + \\texttt{0x20} \\\\\n&= \\texttt{0x100000010} \\\\\n&\\equiv \\texttt{0x10} = 16 \\pmod{2^{32}}\n\\end{aligned}",[28,17664,17666],{"className":17665,"ariaHidden":261},[260],[28,17667,17669,17673],{"className":17668},[265],[28,17670],{"className":17671,"style":17672},[269],"height:4.2241em;vertical-align:-1.8621em;",[28,17674,17676],{"className":17675},[274],[28,17677,17679,17751],{"className":17678},[5015],[28,17680,17682],{"className":17681},[5137],[28,17683,17685,17742],{"className":17684},[560,561],[28,17686,17688,17739],{"className":17687},[565],[28,17689,17692,17721,17730],{"className":17690,"style":17691},[569],"height:2.3621em;",[28,17693,17694,17697],{"style":5150},[28,17695],{"className":17696,"style":1914},[577],[28,17698,17700,17706,17709,17712,17715],{"className":17699},[274],[28,17701,17703],{"className":17702},[274,11179],[28,17704,17361],{"className":17705},[274],[28,17707],{"className":17708,"style":280},[279],[28,17710,124],{"className":17711},[284],[28,17713],{"className":17714,"style":280},[279],[28,17716,17718],{"className":17717},[274,11179],[28,17719,16419],{"className":17720},[274],[28,17722,17724,17727],{"style":17723},"top:-3.0221em;",[28,17725],{"className":17726,"style":1914},[577],[28,17728],{"className":17729},[274],[28,17731,17733,17736],{"style":17732},"top:-1.4979em;",[28,17734],{"className":17735,"style":1914},[577],[28,17737],{"className":17738},[274],[28,17740,593],{"className":17741},[592],[28,17743,17745],{"className":17744},[565],[28,17746,17749],{"className":17747,"style":17748},[569],"height:1.8621em;",[28,17750],{},[28,17752,17754],{"className":17753},[5205],[28,17755,17757,17927],{"className":17756},[560,561],[28,17758,17760,17924],{"className":17759},[565],[28,17761,17763,17804,17830],{"className":17762,"style":17691},[569],[28,17764,17765,17768],{"style":5150},[28,17766],{"className":17767,"style":1914},[577],[28,17769,17771,17774,17777,17780,17783,17789,17792,17795,17798],{"className":17770},[274],[28,17772],{"className":17773},[274],[28,17775],{"className":17776,"style":302},[279],[28,17778,83],{"className":17779},[306],[28,17781],{"className":17782,"style":302},[279],[28,17784,17786],{"className":17785},[274,11179],[28,17787,17454],{"className":17788},[274,13211],[28,17790],{"className":17791,"style":280},[279],[28,17793,124],{"className":17794},[284],[28,17796],{"className":17797,"style":280},[279],[28,17799,17801],{"className":17800},[274,11179],[28,17802,17465],{"className":17803},[274,13211],[28,17805,17806,17809],{"style":17723},[28,17807],{"className":17808,"style":1914},[577],[28,17810,17812,17815,17818,17821,17824],{"className":17811},[274],[28,17813],{"className":17814},[274],[28,17816],{"className":17817,"style":302},[279],[28,17819,83],{"className":17820},[306],[28,17822],{"className":17823,"style":302},[279],[28,17825,17827],{"className":17826},[274,11179],[28,17828,17609],{"className":17829},[274,13211],[28,17831,17832,17835],{"style":17732},[28,17833],{"className":17834,"style":1914},[577],[28,17836,17838,17841,17844,17847,17850,17856,17859,17862,17865,17868,17871,17874,17877,17886,17889,17921],{"className":17837},[274],[28,17839],{"className":17840},[274],[28,17842],{"className":17843,"style":302},[279],[28,17845,10074],{"className":17846},[306],[28,17848],{"className":17849,"style":302},[279],[28,17851,17853],{"className":17852},[274,11179],[28,17854,17630],{"className":17855},[274,13211],[28,17857],{"className":17858,"style":302},[279],[28,17860,83],{"className":17861},[306],[28,17863],{"className":17864,"style":302},[279],[28,17866,17635],{"className":17867},[274],[28,17869],{"className":17870},[279,13069],[28,17872],{"className":17873,"style":540},[279],[28,17875,43],{"className":17876},[348],[28,17878,17880],{"className":17879},[274],[28,17881,17883],{"className":17882},[274],[28,17884,13092],{"className":17885},[274,1712],[28,17887],{"className":17888,"style":13096},[279],[28,17890,17892,17895],{"className":17891},[274],[28,17893,1253],{"className":17894},[274],[28,17896,17898],{"className":17897},[556],[28,17899,17901],{"className":17900},[560],[28,17902,17904],{"className":17903},[565],[28,17905,17907],{"className":17906,"style":4877},[569],[28,17908,17909,17912],{"style":4880},[28,17910],{"className":17911,"style":578},[577],[28,17913,17915],{"className":17914},[582,583,584,585],[28,17916,17918],{"className":17917},[274,585],[28,17919,12998],{"className":17920},[274,585],[28,17922,1522],{"className":17923},[355],[28,17925,593],{"className":17926},[592],[28,17928,17930],{"className":17929},[565],[28,17931,17933],{"className":17932,"style":17748},[569],[28,17934],{},[11,17936,17937,17989,17990,17993,17994,17996,17997,2923],{},[28,17938,17940,17958],{"className":17939,"translate":193},[197],[28,17941,17943],{"className":17942},[201],[203,17944,17945],{"xmlns":205},[208,17946,17947,17955],{},[211,17948,17949,17951,17953],{},[224,17950,17635],{},[217,17952,1876],{},[224,17954,17311],{},[253,17956,17957],{"encoding":255},"16 \\le 64",[28,17959,17961,17980],{"className":17960,"ariaHidden":261},[260],[28,17962,17964,17968,17971,17974,17977],{"className":17963},[265],[28,17965],{"className":17966,"style":17967},[269],"height:0.7804em;vertical-align:-0.136em;",[28,17969,17635],{"className":17970},[274],[28,17972],{"className":17973,"style":302},[279],[28,17975,1876],{"className":17976},[306],[28,17978],{"className":17979,"style":302},[279],[28,17981,17983,17986],{"className":17982},[265],[28,17984],{"className":17985,"style":316},[269],[28,17987,17311],{"className":17988},[274],"，检查通过。但 ",[25,17991,17992],{},"buf + off"," 指向的是缓冲区外大约 4GB 的位置，",[25,17995,16415],{}," 会往一个完全无关的地址写 32 字节。这是一个典型的",[183,17998,17999],{},"整数溢出导致的任意地址写",[11,18001,18002],{},"用位向量理论来问：",[18,18004,18006],{"className":16766,"code":18005,"language":16768,"meta":23,"style":23},"(set-logic QF_BV)\n(declare-const off (_ BitVec 32))\n(declare-const len (_ BitVec 32))\n(assert (bvule (bvadd off len) (_ bv64 32)))\n(assert (bvugt off (_ bv64 32)))\n(assert (bvule len (_ bv64 32)))\n(check-sat)\n(get-model)\n",[25,18007,18008,18012,18017,18021,18026,18031,18036,18040],{"__ignoreMap":23},[28,18009,18010],{"class":30,"line":31},[28,18011,17209],{},[28,18013,18014],{"class":30,"line":64},[28,18015,18016],{},"(declare-const off (_ BitVec 32))\n",[28,18018,18019],{"class":30,"line":95},[28,18020,17214],{},[28,18022,18023],{"class":30,"line":115},[28,18024,18025],{},"(assert (bvule (bvadd off len) (_ bv64 32)))\n",[28,18027,18028],{"class":30,"line":154},[28,18029,18030],{},"(assert (bvugt off (_ bv64 32)))\n",[28,18032,18033],{"class":30,"line":161},[28,18034,18035],{},"(assert (bvule len (_ bv64 32)))\n",[28,18037,18038],{"class":30,"line":167},[28,18039,16790],{},[28,18041,18043],{"class":30,"line":18042},8,[28,18044,16795],{},[11,18046,18047],{},"求解器返回 sat，模型可能是：",[18,18049,18052],{"className":18050,"code":18051,"language":11179},[11177],"off = #xfffffff0\nlen = #x00000020\n",[25,18053,18051],{"__ignoreMap":23},[11,18055,18056],{},"漏洞可达。",[11,18058,18059,18060,2693,18062,18064],{},"如果换一种建模，把 ",[25,18061,17361],{},[25,18063,16419],{}," 当成数学整数：",[28,18066,18068],{"className":18067,"translate":193},[192],[28,18069,18071,18117],{"className":18070,"translate":193},[197],[28,18072,18074],{"className":18073},[201],[203,18075,18076],{"xmlns":205,"display":206},[208,18077,18078,18114],{},[211,18079,18080,18082,18084,18086,18088,18090,18092,18094,18096,18098,18100,18102,18104,18106,18108,18110,18112],{},[228,18081,17361],{},[217,18083,124],{},[228,18085,16419],{},[217,18087,1876],{},[224,18089,17311],{},[228,18091,230],{},[217,18093,233],{},[228,18095,230],{},[228,18097,17361],{},[217,18099,130],{},[224,18101,17311],{},[228,18103,230],{},[217,18105,233],{},[228,18107,230],{},[228,18109,16419],{},[217,18111,9572],{},[224,18113,89],{},[253,18115,18116],{"encoding":255},"\\text{off} + \\text{len} \\le 64 \\;\\wedge\\; \\text{off} > 64 \\;\\wedge\\; \\text{len} \\ge 0",[28,18118,18120,18141,18162,18186,18207,18231,18252],{"className":18119,"ariaHidden":261},[260],[28,18121,18123,18126,18132,18135,18138],{"className":18122},[265],[28,18124],{"className":18125,"style":4154},[269],[28,18127,18129],{"className":18128},[274,11179],[28,18130,17361],{"className":18131},[274],[28,18133],{"className":18134,"style":280},[279],[28,18136,124],{"className":18137},[284],[28,18139],{"className":18140,"style":280},[279],[28,18142,18144,18147,18153,18156,18159],{"className":18143},[265],[28,18145],{"className":18146,"style":17104},[269],[28,18148,18150],{"className":18149},[274,11179],[28,18151,16419],{"className":18152},[274],[28,18154],{"className":18155,"style":302},[279],[28,18157,1876],{"className":18158},[306],[28,18160],{"className":18161,"style":302},[279],[28,18163,18165,18168,18171,18174,18177,18180,18183],{"className":18164},[265],[28,18166],{"className":18167,"style":316},[269],[28,18169,17311],{"className":18170},[274],[28,18172],{"className":18173,"style":302},[279],[28,18175],{"className":18176,"style":280},[279],[28,18178,233],{"className":18179},[284],[28,18181],{"className":18182,"style":302},[279],[28,18184],{"className":18185,"style":280},[279],[28,18187,18189,18192,18198,18201,18204],{"className":18188},[265],[28,18190],{"className":18191,"style":16489},[269],[28,18193,18195],{"className":18194},[274,11179],[28,18196,17361],{"className":18197},[274],[28,18199],{"className":18200,"style":302},[279],[28,18202,130],{"className":18203},[306],[28,18205],{"className":18206,"style":302},[279],[28,18208,18210,18213,18216,18219,18222,18225,18228],{"className":18209},[265],[28,18211],{"className":18212,"style":316},[269],[28,18214,17311],{"className":18215},[274],[28,18217],{"className":18218,"style":302},[279],[28,18220],{"className":18221,"style":280},[279],[28,18223,233],{"className":18224},[284],[28,18226],{"className":18227,"style":302},[279],[28,18229],{"className":18230,"style":280},[279],[28,18232,18234,18237,18243,18246,18249],{"className":18233},[265],[28,18235],{"className":18236,"style":17104},[269],[28,18238,18240],{"className":18239},[274,11179],[28,18241,16419],{"className":18242},[274],[28,18244],{"className":18245,"style":302},[279],[28,18247,9572],{"className":18248},[306],[28,18250],{"className":18251,"style":302},[279],[28,18253,18255,18258],{"className":18254},[265],[28,18256],{"className":18257,"style":316},[269],[28,18259,89],{"className":18260},[274],[11,18262,18263,18264,18266,18267,18346],{},"这个公式是 ",[183,18265,6456],{},"，因为 ",[28,18268,18270,18292],{"className":18269,"translate":193},[197],[28,18271,18273],{"className":18272},[201],[203,18274,18275],{"xmlns":205},[208,18276,18277,18289],{},[211,18278,18279,18281,18283,18285,18287],{},[228,18280,17361],{},[217,18282,124],{},[228,18284,16419],{},[217,18286,130],{},[224,18288,17311],{},[253,18290,18291],{"encoding":255},"\\text{off} + \\text{len} > 64",[28,18293,18295,18316,18337],{"className":18294,"ariaHidden":261},[260],[28,18296,18298,18301,18307,18310,18313],{"className":18297},[265],[28,18299],{"className":18300,"style":4154},[269],[28,18302,18304],{"className":18303},[274,11179],[28,18305,17361],{"className":18306},[274],[28,18308],{"className":18309,"style":280},[279],[28,18311,124],{"className":18312},[284],[28,18314],{"className":18315,"style":280},[279],[28,18317,18319,18322,18328,18331,18334],{"className":18318},[265],[28,18320],{"className":18321,"style":16489},[269],[28,18323,18325],{"className":18324},[274,11179],[28,18326,16419],{"className":18327},[274],[28,18329],{"className":18330,"style":302},[279],[28,18332,130],{"className":18333},[306],[28,18335],{"className":18336,"style":302},[279],[28,18338,18340,18343],{"className":18339},[265],[28,18341],{"className":18342,"style":316},[269],[28,18344,17311],{"className":18345},[274]," 直接和第一条矛盾。工具会得出结论「没有漏洞」，而实际上漏洞就在那里。",[11,18348,18349,18352,18353,18356],{},[183,18350,18351],{},"同一个程序，两种理论，两个相反的答案。"," 这就是为什么漏洞检测工具必须精确建模目标语言的整数语义：C 的 ",[25,18354,18355],{},"unsigned int"," 是模运算，不是数学整数。",[419,18358,18360],{"id":18359},"_54-空指针","5.4 空指针",[11,18362,18363],{},"静态分析经常报「可能的空指针解引用」。这类告警里有很多是误报，SMT 可以用来过滤。",[18,18365,18367],{"className":20,"code":18366,"language":22,"meta":23,"style":23},"int read_flag(struct node *p) {\n    if (p == NULL) return 0;\n    return p->flag;     \u002F\u002F 静态分析：p 可能为空？\n}\n",[25,18368,18369,18391,18414,18425],{"__ignoreMap":23},[28,18370,18371,18373,18376,18378,18381,18384,18387,18389],{"class":30,"line":31},[28,18372,46],{"class":34},[28,18374,18375],{"class":38}," read_flag",[28,18377,43],{"class":42},[28,18379,18380],{"class":34},"struct",[28,18382,18383],{"class":42}," node ",[28,18385,18386],{"class":34},"*",[28,18388,11],{"class":49},[28,18390,61],{"class":42},[28,18392,18393,18395,18398,18400,18403,18406,18409,18412],{"class":30,"line":64},[28,18394,118],{"class":34},[28,18396,18397],{"class":42}," (p ",[28,18399,147],{"class":34},[28,18401,18402],{"class":76}," NULL",[28,18404,18405],{"class":42},") ",[28,18407,18408],{"class":34},"return",[28,18410,18411],{"class":76}," 0",[28,18413,112],{"class":42},[28,18415,18416,18419,18422],{"class":30,"line":95},[28,18417,18418],{"class":34},"    return",[28,18420,18421],{"class":42}," p->flag;",[28,18423,18424],{"class":157},"     \u002F\u002F 静态分析：p 可能为空？\n",[28,18426,18427],{"class":30,"line":115},[28,18428,170],{"class":42},[11,18430,18431,18432,18435,18436,18439],{},"工具在 ",[25,18433,18434],{},"p->flag"," 处收集路径条件。走到这一行，必然经过了 ",[25,18437,18438],{},"p == NULL"," 为假的分支，所以路径条件里包含：",[28,18441,18443],{"className":18442,"translate":193},[192],[28,18444,18446,18465],{"className":18445,"translate":193},[197],[28,18447,18449],{"className":18448},[201],[203,18450,18451],{"xmlns":205,"display":206},[208,18452,18453,18462],{},[211,18454,18455,18457,18459],{},[214,18456,11],{},[217,18458,1854],{"mathvariant":484},[228,18460,18461],{"mathvariant":13013},"NULL",[253,18463,18464],{"encoding":255},"p \\ne \\texttt{NULL}",[28,18466,18468,18519],{"className":18467,"ariaHidden":261},[260],[28,18469,18471,18474,18477,18480,18516],{"className":18470},[265],[28,18472],{"className":18473,"style":2371},[269],[28,18475,11],{"className":18476},[274,275],[28,18478],{"className":18479,"style":302},[279],[28,18481,18483,18510,18513],{"className":18482},[306],[28,18484,18486],{"className":18485},[306],[28,18487,18489],{"className":18488},[274,2359],[28,18490,18492],{"className":18491},[2363],[28,18493,18495,18498,18507],{"className":18494},[2367],[28,18496],{"className":18497,"style":2371},[269],[28,18499,18501],{"className":18500},[2375],[28,18502,18504],{"className":18503},[274],[28,18505,2382],{"className":18506},[306],[28,18508],{"className":18509},[2386],[28,18511],{"className":18512},[279,2390],[28,18514,83],{"className":18515},[306],[28,18517],{"className":18518,"style":302},[279],[28,18520,18522,18525],{"className":18521},[265],[28,18523],{"className":18524,"style":13204},[269],[28,18526,18528],{"className":18527},[274,11179],[28,18529,18461],{"className":18530},[274,13211],[11,18532,18533,18534,18536],{},"静态分析如果只做局部启发式，可能会说「",[25,18535,11],{}," 可能是空指针」。这时问 SMT：",[28,18538,18540],{"className":18539,"translate":193},[192],[28,18541,18543,18573],{"className":18542,"translate":193},[197],[28,18544,18546],{"className":18545},[201],[203,18547,18548],{"xmlns":205,"display":206},[208,18549,18550,18570],{},[211,18551,18552,18554,18556,18558,18560,18562,18564,18566,18568],{},[214,18553,11],{},[217,18555,83],{},[228,18557,18461],{"mathvariant":13013},[228,18559,230],{},[217,18561,233],{},[228,18563,230],{},[214,18565,11],{},[217,18567,1854],{"mathvariant":484},[228,18569,18461],{"mathvariant":13013},[253,18571,18572],{"encoding":255},"p = \\texttt{NULL} \\;\\wedge\\; p \\ne \\texttt{NULL}",[28,18574,18576,18594,18621,18672],{"className":18575,"ariaHidden":261},[260],[28,18577,18579,18582,18585,18588,18591],{"className":18578},[265],[28,18580],{"className":18581,"style":612},[269],[28,18583,11],{"className":18584},[274,275],[28,18586],{"className":18587,"style":302},[279],[28,18589,83],{"className":18590},[306],[28,18592],{"className":18593,"style":302},[279],[28,18595,18597,18600,18606,18609,18612,18615,18618],{"className":18596},[265],[28,18598],{"className":18599,"style":13204},[269],[28,18601,18603],{"className":18602},[274,11179],[28,18604,18461],{"className":18605},[274,13211],[28,18607],{"className":18608,"style":302},[279],[28,18610],{"className":18611,"style":280},[279],[28,18613,233],{"className":18614},[284],[28,18616],{"className":18617,"style":302},[279],[28,18619],{"className":18620,"style":280},[279],[28,18622,18624,18627,18630,18633,18669],{"className":18623},[265],[28,18625],{"className":18626,"style":2371},[269],[28,18628,11],{"className":18629},[274,275],[28,18631],{"className":18632,"style":302},[279],[28,18634,18636,18663,18666],{"className":18635},[306],[28,18637,18639],{"className":18638},[306],[28,18640,18642],{"className":18641},[274,2359],[28,18643,18645],{"className":18644},[2363],[28,18646,18648,18651,18660],{"className":18647},[2367],[28,18649],{"className":18650,"style":2371},[269],[28,18652,18654],{"className":18653},[2375],[28,18655,18657],{"className":18656},[274],[28,18658,2382],{"className":18659},[306],[28,18661],{"className":18662},[2386],[28,18664],{"className":18665},[279,2390],[28,18667,83],{"className":18668},[306],[28,18670],{"className":18671,"style":302},[279],[28,18673,18675,18678],{"className":18674},[265],[28,18676],{"className":18677,"style":13204},[269],[28,18679,18681],{"className":18680},[274,11179],[28,18682,18461],{"className":18683},[274,13211],[11,18685,18686,18687,18689],{},"这是明显的矛盾，返回 ",[183,18688,6456],{},"，告警被排除。",[11,18691,18692],{},"反过来，如果代码是：",[18,18694,18696],{"className":20,"code":18695,"language":22,"meta":23,"style":23},"int read_flag(struct node *p) {\n    if (p->flag == 1) {   \u002F\u002F 这里没有检查 p\n        return 1;\n    }\n    return 0;\n}\n",[25,18697,18698,18716,18734,18743,18747,18755],{"__ignoreMap":23},[28,18699,18700,18702,18704,18706,18708,18710,18712,18714],{"class":30,"line":31},[28,18701,46],{"class":34},[28,18703,18375],{"class":38},[28,18705,43],{"class":42},[28,18707,18380],{"class":34},[28,18709,18383],{"class":42},[28,18711,18386],{"class":34},[28,18713,11],{"class":49},[28,18715,61],{"class":42},[28,18717,18718,18720,18723,18725,18728,18731],{"class":30,"line":64},[28,18719,118],{"class":34},[28,18721,18722],{"class":42}," (p->flag ",[28,18724,147],{"class":34},[28,18726,18727],{"class":76}," 1",[28,18729,18730],{"class":42},") {",[28,18732,18733],{"class":157},"   \u002F\u002F 这里没有检查 p\n",[28,18735,18736,18739,18741],{"class":30,"line":95},[28,18737,18738],{"class":34},"        return",[28,18740,18727],{"class":76},[28,18742,112],{"class":42},[28,18744,18745],{"class":30,"line":115},[28,18746,164],{"class":42},[28,18748,18749,18751,18753],{"class":30,"line":154},[28,18750,18418],{"class":34},[28,18752,18411],{"class":76},[28,18754,112],{"class":42},[28,18756,18757],{"class":30,"line":161},[28,18758,170],{"class":42},[11,18760,18761,18762,18848],{},"路径条件里没有 ",[28,18763,18765,18782],{"className":18764,"translate":193},[197],[28,18766,18768],{"className":18767},[201],[203,18769,18770],{"xmlns":205},[208,18771,18772,18780],{},[211,18773,18774,18776,18778],{},[214,18775,11],{},[217,18777,1854],{"mathvariant":484},[228,18779,18461],{"mathvariant":13013},[253,18781,18464],{"encoding":255},[28,18783,18785,18836],{"className":18784,"ariaHidden":261},[260],[28,18786,18788,18791,18794,18797,18833],{"className":18787},[265],[28,18789],{"className":18790,"style":2371},[269],[28,18792,11],{"className":18793},[274,275],[28,18795],{"className":18796,"style":302},[279],[28,18798,18800,18827,18830],{"className":18799},[306],[28,18801,18803],{"className":18802},[306],[28,18804,18806],{"className":18805},[274,2359],[28,18807,18809],{"className":18808},[2363],[28,18810,18812,18815,18824],{"className":18811},[2367],[28,18813],{"className":18814,"style":2371},[269],[28,18816,18818],{"className":18817},[2375],[28,18819,18821],{"className":18820},[274],[28,18822,2382],{"className":18823},[306],[28,18825],{"className":18826},[2386],[28,18828],{"className":18829},[279,2390],[28,18831,83],{"className":18832},[306],[28,18834],{"className":18835,"style":302},[279],[28,18837,18839,18842],{"className":18838},[265],[28,18840],{"className":18841,"style":13204},[269],[28,18843,18845],{"className":18844},[274,11179],[28,18846,18461],{"className":18847},[274,13211],"，查询：",[28,18850,18852],{"className":18851,"translate":193},[192],[28,18853,18855,18873],{"className":18854,"translate":193},[197],[28,18856,18858],{"className":18857},[201],[203,18859,18860],{"xmlns":205,"display":206},[208,18861,18862,18870],{},[211,18863,18864,18866,18868],{},[214,18865,11],{},[217,18867,83],{},[228,18869,18461],{"mathvariant":13013},[253,18871,18872],{"encoding":255},"p = \\texttt{NULL}",[28,18874,18876,18894],{"className":18875,"ariaHidden":261},[260],[28,18877,18879,18882,18885,18888,18891],{"className":18878},[265],[28,18880],{"className":18881,"style":612},[269],[28,18883,11],{"className":18884},[274,275],[28,18886],{"className":18887,"style":302},[279],[28,18889,83],{"className":18890},[306],[28,18892],{"className":18893,"style":302},[279],[28,18895,18897,18900],{"className":18896},[265],[28,18898],{"className":18899,"style":13204},[269],[28,18901,18903],{"className":18902},[274,11179],[28,18904,18461],{"className":18905},[274,13211],[11,18907,18908,18909,18911,18912,18915],{},"返回 ",[183,18910,405],{},"，模型给出 ",[25,18913,18914],{},"p = NULL","。这是一条真实可达的空指针路径，告警保留。",[11,18917,18918,18919,18922],{},"这个用法体现了一个重要原则：",[183,18920,18921],{},"SMT 不负责发现漏洞，它负责判断「这条路径到底可不可能」","。发现可疑点靠前端分析，排除误报靠 SMT。",[419,18924,18926],{"id":18925},"_55-数组越界","5.5 数组越界",[18,18928,18930],{"className":20,"code":18929,"language":22,"meta":23,"style":23},"int get(int *a, int i) {\n    if (i \u003C 100) {\n        return a[i];     \u002F\u002F i 可能为负\n    }\n    return -1;\n}\n",[25,18931,18932,18956,18970,18982,18986,18997],{"__ignoreMap":23},[28,18933,18934,18936,18939,18941,18943,18945,18947,18949,18951,18954],{"class":30,"line":31},[28,18935,46],{"class":34},[28,18937,18938],{"class":38}," get",[28,18940,43],{"class":42},[28,18942,46],{"class":34},[28,18944,16337],{"class":34},[28,18946,238],{"class":49},[28,18948,53],{"class":42},[28,18950,46],{"class":34},[28,18952,18953],{"class":49}," i",[28,18955,61],{"class":42},[28,18957,18958,18960,18963,18965,18968],{"class":30,"line":64},[28,18959,118],{"class":34},[28,18961,18962],{"class":42}," (i ",[28,18964,1383],{"class":34},[28,18966,18967],{"class":76}," 100",[28,18969,61],{"class":42},[28,18971,18972,18974,18976,18979],{"class":30,"line":95},[28,18973,18738],{"class":34},[28,18975,70],{"class":49},[28,18977,18978],{"class":42},"[i];",[28,18980,18981],{"class":157},"     \u002F\u002F i 可能为负\n",[28,18983,18984],{"class":30,"line":115},[28,18985,164],{"class":42},[28,18987,18988,18990,18993,18995],{"class":30,"line":154},[28,18989,18418],{"class":34},[28,18991,18992],{"class":34}," -",[28,18994,1244],{"class":76},[28,18996,112],{"class":42},[28,18998,18999],{"class":30,"line":161},[28,19000,170],{"class":42},[11,19002,19003],{},"只检查了上界，没检查下界。问 SMT：",[28,19005,19007],{"className":19006,"translate":193},[192],[28,19008,19010,19040],{"className":19009,"translate":193},[197],[28,19011,19013],{"className":19012},[201],[203,19014,19015],{"xmlns":205,"display":206},[208,19016,19017,19037],{},[211,19018,19019,19021,19023,19025,19027,19029,19031,19033,19035],{},[214,19020,1686],{},[217,19022,1383],{},[224,19024,77],{},[228,19026,230],{},[217,19028,233],{},[228,19030,230],{},[214,19032,1686],{},[217,19034,1383],{},[224,19036,89],{},[253,19038,19039],{"encoding":255},"i \u003C 100 \\;\\wedge\\; i \u003C 0",[28,19041,19043,19062,19086,19104],{"className":19042,"ariaHidden":261},[260],[28,19044,19046,19050,19053,19056,19059],{"className":19045},[265],[28,19047],{"className":19048,"style":19049},[269],"height:0.6986em;vertical-align:-0.0391em;",[28,19051,1686],{"className":19052},[274,275],[28,19054],{"className":19055,"style":302},[279],[28,19057,1383],{"className":19058},[306],[28,19060],{"className":19061,"style":302},[279],[28,19063,19065,19068,19071,19074,19077,19080,19083],{"className":19064},[265],[28,19066],{"className":19067,"style":316},[269],[28,19069,77],{"className":19070},[274],[28,19072],{"className":19073,"style":302},[279],[28,19075],{"className":19076,"style":280},[279],[28,19078,233],{"className":19079},[284],[28,19081],{"className":19082,"style":302},[279],[28,19084],{"className":19085,"style":280},[279],[28,19087,19089,19092,19095,19098,19101],{"className":19088},[265],[28,19090],{"className":19091,"style":19049},[269],[28,19093,1686],{"className":19094},[274,275],[28,19096],{"className":19097,"style":302},[279],[28,19099,1383],{"className":19100},[306],[28,19102],{"className":19103,"style":302},[279],[28,19105,19107,19110],{"className":19106},[265],[28,19108],{"className":19109,"style":316},[269],[28,19111,89],{"className":19112},[274],[11,19114,19115,19116,2923,19119,19122],{},"返回 sat，例如 ",[25,19117,19118],{},"i = -1",[25,19120,19121],{},"a[-1]"," 是一次越界读。",[11,19124,19125],{},"如果加上下界检查：",[18,19127,19129],{"className":20,"code":19128,"language":22,"meta":23,"style":23},"if (i >= 0 && i \u003C 100) {\n    return a[i];\n}\n",[25,19130,19131,19153,19162],{"__ignoreMap":23},[28,19132,19133,19135,19137,19140,19142,19144,19147,19149,19151],{"class":30,"line":31},[28,19134,16953],{"class":34},[28,19136,18962],{"class":42},[28,19138,19139],{"class":34},">=",[28,19141,18411],{"class":76},[28,19143,136],{"class":34},[28,19145,19146],{"class":42}," i ",[28,19148,1383],{"class":34},[28,19150,18967],{"class":76},[28,19152,61],{"class":42},[28,19154,19155,19157,19159],{"class":30,"line":64},[28,19156,18418],{"class":34},[28,19158,70],{"class":49},[28,19160,19161],{"class":42},"[i];\n",[28,19163,19164],{"class":30,"line":95},[28,19165,170],{"class":42},[11,19167,19168],{},"约束变成：",[28,19170,19172],{"className":19171,"translate":193},[192],[28,19173,19175,19217],{"className":19174,"translate":193},[197],[28,19176,19178],{"className":19177},[201],[203,19179,19180],{"xmlns":205,"display":206},[208,19181,19182,19214],{},[211,19183,19184,19186,19188,19190,19192,19194,19196,19198,19200,19202,19204,19206,19208,19210,19212],{},[214,19185,1686],{},[217,19187,9572],{},[224,19189,89],{},[228,19191,230],{},[217,19193,233],{},[228,19195,230],{},[214,19197,1686],{},[217,19199,1383],{},[224,19201,77],{},[228,19203,230],{},[217,19205,233],{},[228,19207,230],{},[214,19209,1686],{},[217,19211,1383],{},[224,19213,89],{},[253,19215,19216],{"encoding":255},"i \\ge 0 \\;\\wedge\\; i \u003C 100 \\;\\wedge\\; i \u003C 0",[28,19218,19220,19239,19263,19281,19305,19323],{"className":19219,"ariaHidden":261},[260],[28,19221,19223,19227,19230,19233,19236],{"className":19222},[265],[28,19224],{"className":19225,"style":19226},[269],"height:0.7955em;vertical-align:-0.136em;",[28,19228,1686],{"className":19229},[274,275],[28,19231],{"className":19232,"style":302},[279],[28,19234,9572],{"className":19235},[306],[28,19237],{"className":19238,"style":302},[279],[28,19240,19242,19245,19248,19251,19254,19257,19260],{"className":19241},[265],[28,19243],{"className":19244,"style":316},[269],[28,19246,89],{"className":19247},[274],[28,19249],{"className":19250,"style":302},[279],[28,19252],{"className":19253,"style":280},[279],[28,19255,233],{"className":19256},[284],[28,19258],{"className":19259,"style":302},[279],[28,19261],{"className":19262,"style":280},[279],[28,19264,19266,19269,19272,19275,19278],{"className":19265},[265],[28,19267],{"className":19268,"style":19049},[269],[28,19270,1686],{"className":19271},[274,275],[28,19273],{"className":19274,"style":302},[279],[28,19276,1383],{"className":19277},[306],[28,19279],{"className":19280,"style":302},[279],[28,19282,19284,19287,19290,19293,19296,19299,19302],{"className":19283},[265],[28,19285],{"className":19286,"style":316},[269],[28,19288,77],{"className":19289},[274],[28,19291],{"className":19292,"style":302},[279],[28,19294],{"className":19295,"style":280},[279],[28,19297,233],{"className":19298},[284],[28,19300],{"className":19301,"style":302},[279],[28,19303],{"className":19304,"style":280},[279],[28,19306,19308,19311,19314,19317,19320],{"className":19307},[265],[28,19309],{"className":19310,"style":19049},[269],[28,19312,1686],{"className":19313},[274,275],[28,19315],{"className":19316,"style":302},[279],[28,19318,1383],{"className":19319},[306],[28,19321],{"className":19322,"style":302},[279],[28,19324,19326,19329],{"className":19325},[265],[28,19327],{"className":19328,"style":316},[269],[28,19330,89],{"className":19331},[274],[11,19333,18908,19334,19336],{},[183,19335,6456],{},"，越界路径被排除。",[11,19338,19339,19340,19343,19344,19347],{},"这个例子还可以扩展成带数组理论的版本。比如要验证「下标不同时，写入 ",[25,19341,19342],{},"a[j]"," 不会影响对 ",[25,19345,19346],{},"a[i]"," 的读取」，可以把「影响了」写成公式，再问它是否可满足：",[28,19349,19351],{"className":19350,"translate":193},[192],[28,19352,19354,19452],{"className":19353,"translate":193},[197],[28,19355,19357],{"className":19356},[201],[203,19358,19359],{"xmlns":205,"display":206},[208,19360,19361,19449],{},[211,19362,19363,19365,19367,19369,19371,19373,19375,19389,19391,19403,19405,19407,19409,19411,19413,19415,19417,19419,19421,19423,19425,19439,19441,19443,19445,19447],{},[214,19364,1686],{},[217,19366,1854],{"mathvariant":484},[214,19368,7907],{},[228,19370,230],{},[217,19372,233],{},[228,19374,230],{},[211,19376,19377,19379,19381,19383,19385,19387],{},[214,19378,1664],{"mathvariant":484},[214,19380,1667],{"mathvariant":484},[214,19382,1670],{"mathvariant":484},[214,19384,1667],{"mathvariant":484},[214,19386,22],{"mathvariant":484},[214,19388,1241],{"mathvariant":484},[217,19390,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[211,19392,19393,19395,19397,19399,19401],{},[214,19394,1664],{"mathvariant":484},[214,19396,1241],{"mathvariant":484},[214,19398,3933],{"mathvariant":484},[214,19400,1867],{"mathvariant":484},[214,19402,1667],{"mathvariant":484},[217,19404,43],{"stretchy":241},[214,19406,238],{},[217,19408,1683],{"separator":261},[214,19410,7907],{},[217,19412,1683],{"separator":261},[214,19414,1693],{},[217,19416,1522],{"stretchy":241},[217,19418,1683],{"separator":261},[214,19420,1686],{},[217,19422,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,19424,1854],{"mathvariant":484},[211,19426,19427,19429,19431,19433,19435,19437],{},[214,19428,1664],{"mathvariant":484},[214,19430,1667],{"mathvariant":484},[214,19432,1670],{"mathvariant":484},[214,19434,1667],{"mathvariant":484},[214,19436,22],{"mathvariant":484},[214,19438,1241],{"mathvariant":484},[217,19440,43],{"stretchy":241},[214,19442,238],{},[217,19444,1683],{"separator":261},[214,19446,1686],{},[217,19448,1522],{"stretchy":241},[253,19450,19451],{"encoding":255},"i \\ne j \\;\\wedge\\; \\mathrm{select}\\bigl(\\mathrm{store}(a, j, v), i\\bigr) \\ne \\mathrm{select}(a, i)",[28,19453,19455,19506,19530,19638],{"className":19454,"ariaHidden":261},[260],[28,19456,19458,19461,19464,19467,19503],{"className":19457},[265],[28,19459],{"className":19460,"style":2371},[269],[28,19462,1686],{"className":19463},[274,275],[28,19465],{"className":19466,"style":302},[279],[28,19468,19470,19497,19500],{"className":19469},[306],[28,19471,19473],{"className":19472},[306],[28,19474,19476],{"className":19475},[274,2359],[28,19477,19479],{"className":19478},[2363],[28,19480,19482,19485,19494],{"className":19481},[2367],[28,19483],{"className":19484,"style":2371},[269],[28,19486,19488],{"className":19487},[2375],[28,19489,19491],{"className":19490},[274],[28,19492,2382],{"className":19493},[306],[28,19495],{"className":19496},[2386],[28,19498],{"className":19499},[279,2390],[28,19501,83],{"className":19502},[306],[28,19504],{"className":19505,"style":302},[279],[28,19507,19509,19512,19515,19518,19521,19524,19527],{"className":19508},[265],[28,19510],{"className":19511,"style":13600},[269],[28,19513,7907],{"className":19514,"style":8057},[274,275],[28,19516],{"className":19517,"style":302},[279],[28,19519],{"className":19520,"style":280},[279],[28,19522,233],{"className":19523},[284],[28,19525],{"className":19526,"style":302},[279],[28,19528],{"className":19529,"style":280},[279],[28,19531,19533,19536,19542,19548,19554,19557,19560,19563,19566,19569,19572,19575,19578,19581,19584,19587,19590,19596,19599,19635],{"className":19532},[265],[28,19534],{"className":19535,"style":2469},[269],[28,19537,19539],{"className":19538},[274],[28,19540,1713],{"className":19541},[274,1712],[28,19543,19545],{"className":19544},[348],[28,19546,43],{"className":19547},[2169,2170],[28,19549,19551],{"className":19550},[274],[28,19552,2696],{"className":19553},[274,1712],[28,19555,43],{"className":19556},[348],[28,19558,238],{"className":19559},[274,275],[28,19561,1683],{"className":19562},[1723],[28,19564],{"className":19565,"style":530},[279],[28,19567,7907],{"className":19568,"style":8057},[274,275],[28,19570,1683],{"className":19571},[1723],[28,19573],{"className":19574,"style":530},[279],[28,19576,1693],{"className":19577,"style":298},[274,275],[28,19579,1522],{"className":19580},[355],[28,19582,1683],{"className":19583},[1723],[28,19585],{"className":19586,"style":530},[279],[28,19588,1686],{"className":19589},[274,275],[28,19591,19593],{"className":19592},[355],[28,19594,1522],{"className":19595},[2169,2170],[28,19597],{"className":19598,"style":302},[279],[28,19600,19602,19629,19632],{"className":19601},[306],[28,19603,19605],{"className":19604},[306],[28,19606,19608],{"className":19607},[274,2359],[28,19609,19611],{"className":19610},[2363],[28,19612,19614,19617,19626],{"className":19613},[2367],[28,19615],{"className":19616,"style":2371},[269],[28,19618,19620],{"className":19619},[2375],[28,19621,19623],{"className":19622},[274],[28,19624,2382],{"className":19625},[306],[28,19627],{"className":19628},[2386],[28,19630],{"className":19631},[279,2390],[28,19633,83],{"className":19634},[306],[28,19636],{"className":19637,"style":302},[279],[28,19639,19641,19644,19650,19653,19656,19659,19662,19665],{"className":19640},[265],[28,19642],{"className":19643,"style":341},[269],[28,19645,19647],{"className":19646},[274],[28,19648,1713],{"className":19649},[274,1712],[28,19651,43],{"className":19652},[348],[28,19654,238],{"className":19655},[274,275],[28,19657,1683],{"className":19658},[1723],[28,19660],{"className":19661,"style":530},[279],[28,19663,1686],{"className":19664},[274,275],[28,19666,1522],{"className":19667},[355],[11,19669,19670,19671,19755,19756,19758],{},"数组理论会展开 read-over-write 公理，算术理论会处理 ",[28,19672,19674,19692],{"className":19673,"translate":193},[197],[28,19675,19677],{"className":19676},[201],[203,19678,19679],{"xmlns":205},[208,19680,19681,19689],{},[211,19682,19683,19685,19687],{},[214,19684,1686],{},[217,19686,1854],{"mathvariant":484},[214,19688,7907],{},[253,19690,19691],{"encoding":255},"i \\ne j",[28,19693,19695,19746],{"className":19694,"ariaHidden":261},[260],[28,19696,19698,19701,19704,19707,19743],{"className":19697},[265],[28,19699],{"className":19700,"style":2371},[269],[28,19702,1686],{"className":19703},[274,275],[28,19705],{"className":19706,"style":302},[279],[28,19708,19710,19737,19740],{"className":19709},[306],[28,19711,19713],{"className":19712},[306],[28,19714,19716],{"className":19715},[274,2359],[28,19717,19719],{"className":19718},[2363],[28,19720,19722,19725,19734],{"className":19721},[2367],[28,19723],{"className":19724,"style":2371},[269],[28,19726,19728],{"className":19727},[2375],[28,19729,19731],{"className":19730},[274],[28,19732,2382],{"className":19733},[306],[28,19735],{"className":19736},[2386],[28,19738],{"className":19739},[279,2390],[28,19741,83],{"className":19742},[306],[28,19744],{"className":19745,"style":302},[279],[28,19747,19749,19752],{"className":19748},[265],[28,19750],{"className":19751,"style":13600},[269],[28,19753,7907],{"className":19754,"style":8057},[274,275],"，两个理论协作给出 ",[183,19757,6456],{},"——也就是说，「写入影响了另一个下标」的情况不可能发生。这正是 3.3 节里理论组合的实际用武之地。",[419,19760,19762],{"id":19761},"_56-完整流程","5.6 完整流程",[11,19764,19765],{},"把上面的步骤串起来，一个典型的漏洞检测流程是这样的：",[2907,19767,19768],{},[2910,19769],{"height":19770,"width":19771,"style":2914,"data-d2-theme":2915,"alt":23,"src":19772},240,2053,"data:image\u002Fsvg+xml;base64,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\u002Fm+WItjwvdGV4dD48L2c+PGcgY2xhc3M9ImMzbHQgc3RlcCI+PHBhdGggZmlsbD0iI2VlZjRmZiIgc3Ryb2tlPSIjNGE2ZmE1IiBkPSJNMjk0IDgxaDE3NnY4MkgyOTR6IiBjbGFzcz0ic2hhcGUiIHN0eWxlPSJzdHJva2Utd2lkdGg6MiIvPjx0ZXh0IHg9IjM4MiIgeT0iMTE5LjUiIGZpbGw9IiMwYTBmMjUiIGNsYXNzPSJ0ZXh0LWJvbGQgZmlsbC1OMSIgc3R5bGU9InRleHQtYW5jaG9yOm1pZGRsZTtmb250LX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只负责中间那个方框。前面「怎么把程序变成公式」是符号执行引擎的事，后面「怎么确认漏洞」是测试和人工的事。",[183,19777,19778],{},"公式建得对不对，直接决定 SMT 的结论有没有意义。",[11,19780,19781,19782,19785],{},"所以，SMT 在漏洞检测里的正确定位是：",[183,19783,19784],{},"一个强大但需要被正确使用的判定引擎","。它能把「这条路径可不可能」回答得很准，但「该问哪些问题」「怎么把程序翻译成问题」，仍然依赖前端工具和工程师的判断。",[414,19787,19789],{"id":19788},"六小结","六、小结",[11,19791,19792,19793,19888,19889,19892],{},"回到开头那道小学题。SMT 的价值，不是把 ",[28,19794,19796,19822],{"className":19795,"translate":193},[197],[28,19797,19799],{"className":19798},[201],[203,19800,19801],{"xmlns":205},[208,19802,19803,19819],{},[211,19804,19805,19807,19809,19811,19813,19815,19817],{},[214,19806,176],{},[217,19808,130],{},[224,19810,3400],{},[217,19812,233],{},[214,19814,176],{},[217,19816,1383],{},[224,19818,251],{},[253,19820,19821],{"encoding":255},"x > 3 \\wedge x \u003C 5",[28,19823,19825,19843,19861,19879],{"className":19824,"ariaHidden":261},[260],[28,19826,19828,19831,19834,19837,19840],{"className":19827},[265],[28,19829],{"className":19830,"style":2616},[269],[28,19832,176],{"className":19833},[274,275],[28,19835],{"className":19836,"style":302},[279],[28,19838,130],{"className":19839},[306],[28,19841],{"className":19842,"style":302},[279],[28,19844,19846,19849,19852,19855,19858],{"className":19845},[265],[28,19847],{"className":19848,"style":316},[269],[28,19850,3400],{"className":19851},[274],[28,19853],{"className":19854,"style":280},[279],[28,19856,233],{"className":19857},[284],[28,19859],{"className":19860,"style":280},[279],[28,19862,19864,19867,19870,19873,19876],{"className":19863},[265],[28,19865],{"className":19866,"style":2616},[269],[28,19868,176],{"className":19869},[274,275],[28,19871],{"className":19872,"style":302},[279],[28,19874,1383],{"className":19875},[306],[28,19877],{"className":19878,"style":302},[279],[28,19880,19882,19885],{"className":19881},[265],[28,19883],{"className":19884,"style":316},[269],[28,19886,251],{"className":19887},[274]," 算得更快，而是让机器",[183,19890,19891],{},"在保留「整数」「数组」「位向量」这些概念的前提下","，仍然能自动地回答「有没有解」。",[11,19894,19895,19898,19899,19902],{},[183,19896,19897],{},"SMT = SAT + 背景理论。"," 布尔层负责组合，理论层负责语义，两者通过「原子真值」这一层接口通信。 ",[183,19900,19901],{},"它的核心取舍是用表达力换可判定性。"," 把主战场放在无量词的可判定片段，换来一个在片段内保证终止、且能处理工业规模混合约束的自动判定过程。",[19904,19905,19906],"style",{},"html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sP4rz, html code.shiki .sP4rz{--shiki-light:#E36209;--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sCsY4, html code.shiki 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var(--shiki-dark-text-decoration);}",{"title":23,"searchDepth":64,"depth":64,"links":19908},[19909,19914,19921,19928,19937,19945],{"id":416,"depth":64,"text":417,"children":19910},[19911,19912,19913],{"id":421,"depth":95,"text":422},{"id":853,"depth":95,"text":854},{"id":2926,"depth":95,"text":2927},{"id":3190,"depth":64,"text":3191,"children":19915},[19916,19917,19918,19919,19920],{"id":3194,"depth":95,"text":3195},{"id":4795,"depth":95,"text":4796},{"id":5586,"depth":95,"text":5587},{"id":6602,"depth":95,"text":6603},{"id":7277,"depth":95,"text":7278},{"id":7304,"depth":64,"text":7305,"children":19922},[19923,19924,19925,19926,19927],{"id":7308,"depth":95,"text":7309},{"id":7478,"depth":95,"text":7479},{"id":8527,"depth":95,"text":8528},{"id":9650,"depth":95,"text":9651},{"id":9745,"depth":95,"text":9746},{"id":9775,"depth":64,"text":9776,"children":19929},[19930,19931,19932,19933,19934,19935,19936],{"id":9779,"depth":95,"text":9780},{"id":9828,"depth":95,"text":9829},{"id":11169,"depth":95,"text":11170},{"id":11210,"depth":95,"text":11211},{"id":13910,"depth":95,"text":13911},{"id":14489,"depth":95,"text":14490},{"id":15718,"depth":95,"text":15719},{"id":15762,"depth":64,"text":15763,"children":19938},[19939,19940,19941,19942,19943,19944],{"id":15766,"depth":95,"text":15767},{"id":16313,"depth":95,"text":16314},{"id":17257,"depth":95,"text":17258},{"id":18359,"depth":95,"text":18360},{"id":18925,"depth":95,"text":18926},{"id":19761,"depth":95,"text":19762},{"id":19788,"depth":64,"text":19789},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0013-satisfiability-modulo-theories",{"title":6,"description":13},"blog\u002F2026\u002F0013-satisfiability-modulo-theories","很多实际问题用纯命题逻辑（SAT）表达时，要么根本无法表达。而用一般的一阶逻辑或高阶逻辑表达虽然自然，却无法自动高效求解。人们要么用专用算法解决特定问题（如线性规划、等价性检查），难以组合多种约束。SMT 把 SAT 的布尔搜索与多个理论求解器组合起来，使得混合理论约束可以被统一、自动地判定。",[19956],"logic","2026-10-29T09:03:03+08:00","1C-2B31uaFHKLW1GJ3_3j1Cbaf3yP10u6PKSP7bBHZ8",{"id":19960,"title":19961,"body":19962,"description":19966,"draft":19946,"enableCodeHighlight":19946,"enableCodeHightlight":19946,"enableComment":19947,"enableD2":19946,"enableKatex":19947,"extension":19948,"image":23,"important":19946,"location":19949,"meta":72543,"navigation":19947,"ogImage":19949,"path":72544,"seo":72545,"stem":72546,"summary":72547,"tags":72548,"time":72552,"__hash__":72553},"blog\u002Fblog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun.md","最优传输几何概念简论",{"type":8,"value":19963,"toc":72530},[19964,19967,19970,19973,19976,19979,20317,20656,20739,20805,21020,21265,21383,21722,22327,22330,22820,22938,23306,23339,23508,23879,24579,24741,24994,25026,25130,25133,25136,25197,25412,25415,26797,27059,27831,27834,29416,29937,30555,30558,30561,30916,31347,31674,31677,32050,32053,32056,32182,32406,32600,32791,32864,33383,33561,34147,34540,34946,35660,36143,36146,37078,38191,38402,38840,38843,39216,39219,39250,39492,39629,39632,40005,40008,40424,40654,40852,41325,41880,41935,42067,42355,42544,42651,42654,42872,42905,43130,43133,43136,43529,43532,43617,44357,44732,44735,44872,45818,45821,45824,45827,46030,46203,46370,46844,47284,47387,47750,48332,48335,49151,49154,49157,49218,49221,49453,49894,50275,50770,50773,52130,52253,52411,52887,53656,53659,53663,53816,54017,54222,54345,54912,54915,54975,55262,55433,56426,56429,56915,56991,56994,57457,57690,58091,59896,59971,59974,60202,60692,60695,61146,61650,61653,61791,62021,62111,62114,62719,63123,63854,64195,64536,65317,65320,65323,65326,65329,65393,65560,65710,66154,66157,66320,66775,66778,67151,67213,67840,68789,68892,69250,69451,69522,70319,70526,70529,70532,70535,70650,71138,71170,71173,71749,71752,71755,71907,72159,72162,72165,72168,72171,72203,72206,72524,72527],[11,19965,19966],{},"我从一个很具体的困扰开始。模型对一张从没见过的图片给出的 softmax 最大值是 0.99，可我没有任何依据说它“99% 属于猫”。softmax 只做了一件事：把 logits 归一化成和为 1 的数。它回答的问题是“在已经假定的这几个类里，哪个最像”，而不是“这个样本是否属于我见过的任何一类”。",[11,19968,19969],{},"要判断一个样本有没有越界，我需要知道的不是分类器给它打了多少分，而是已知类别的数据在空间里占据了多少概率质量、这个样本离那片质量有多远。前者是密度，后者是距离，两者都是分布的几何性质，而不是分类器的性质。",[414,19971,19972],{"id":19972},"搬土",[11,19974,19975],{},"先算一道运输题。",[11,19977,19978],{},"甲矿有 3 吨煤，乙矿有 1 吨；A 厂要 2 吨，B 厂要 2 吨。运价是：甲到 A 每吨 1 块，甲到 B 每吨 2 块，乙到 A 每吨 3 块，乙到 B 每吨 1 块。怎么运最省？",[11,19980,19981,19982,20010,20011,20062,20063,20114,20115,20242,20243,20316],{},"设甲运往 A 的吨数是 ",[28,19983,19985,19998],{"className":19984,"translate":193},[197],[28,19986,19988],{"className":19987},[201],[203,19989,19990],{"xmlns":205},[208,19991,19992,19996],{},[211,19993,19994],{},[214,19995,176],{},[253,19997,176],{"encoding":255},[28,19999,20001],{"className":20000,"ariaHidden":261},[260],[28,20002,20004,20007],{"className":20003},[265],[28,20005],{"className":20006,"style":645},[269],[28,20008,176],{"className":20009},[274,275],"。甲的 3 吨要分完，所以甲到 B 是 ",[28,20012,20014,20032],{"className":20013,"translate":193},[197],[28,20015,20017],{"className":20016},[201],[203,20018,20019],{"xmlns":205},[208,20020,20021,20029],{},[211,20022,20023,20025,20027],{},[224,20024,3400],{},[217,20026,5074],{},[214,20028,176],{},[253,20030,20031],{"encoding":255},"3-x",[28,20033,20035,20053],{"className":20034,"ariaHidden":261},[260],[28,20036,20038,20041,20044,20047,20050],{"className":20037},[265],[28,20039],{"className":20040,"style":6853},[269],[28,20042,3400],{"className":20043},[274],[28,20045],{"className":20046,"style":280},[279],[28,20048,5074],{"className":20049},[284],[28,20051],{"className":20052,"style":280},[279],[28,20054,20056,20059],{"className":20055},[265],[28,20057],{"className":20058,"style":645},[269],[28,20060,176],{"className":20061},[274,275],"；A 要 2 吨，所以乙到 A 是 ",[28,20064,20066,20084],{"className":20065,"translate":193},[197],[28,20067,20069],{"className":20068},[201],[203,20070,20071],{"xmlns":205},[208,20072,20073,20081],{},[211,20074,20075,20077,20079],{},[224,20076,1253],{},[217,20078,5074],{},[214,20080,176],{},[253,20082,20083],{"encoding":255},"2-x",[28,20085,20087,20105],{"className":20086,"ariaHidden":261},[260],[28,20088,20090,20093,20096,20099,20102],{"className":20089},[265],[28,20091],{"className":20092,"style":6853},[269],[28,20094,1253],{"className":20095},[274],[28,20097],{"className":20098,"style":280},[279],[28,20100,5074],{"className":20101},[284],[28,20103],{"className":20104,"style":280},[279],[28,20106,20108,20111],{"className":20107},[265],[28,20109],{"className":20110,"style":645},[269],[28,20112,176],{"className":20113},[274,275],"；乙只有 1 吨，所以乙到 B 是 ",[28,20116,20118,20152],{"className":20117,"translate":193},[197],[28,20119,20121],{"className":20120},[201],[203,20122,20123],{"xmlns":205},[208,20124,20125,20149],{},[211,20126,20127,20129,20131,20133,20135,20137,20139,20141,20143,20145,20147],{},[224,20128,1244],{},[217,20130,5074],{},[217,20132,43],{"stretchy":241},[224,20134,1253],{},[217,20136,5074],{},[214,20138,176],{},[217,20140,1522],{"stretchy":241},[217,20142,83],{},[214,20144,176],{},[217,20146,5074],{},[224,20148,1244],{},[253,20150,20151],{"encoding":255},"1-(2-x)=x-1",[28,20153,20155,20173,20194,20215,20233],{"className":20154,"ariaHidden":261},[260],[28,20156,20158,20161,20164,20167,20170],{"className":20157},[265],[28,20159],{"className":20160,"style":6853},[269],[28,20162,1244],{"className":20163},[274],[28,20165],{"className":20166,"style":280},[279],[28,20168,5074],{"className":20169},[284],[28,20171],{"className":20172,"style":280},[279],[28,20174,20176,20179,20182,20185,20188,20191],{"className":20175},[265],[28,20177],{"className":20178,"style":341},[269],[28,20180,43],{"className":20181},[348],[28,20183,1253],{"className":20184},[274],[28,20186],{"className":20187,"style":280},[279],[28,20189,5074],{"className":20190},[284],[28,20192],{"className":20193,"style":280},[279],[28,20195,20197,20200,20203,20206,20209,20212],{"className":20196},[265],[28,20198],{"className":20199,"style":341},[269],[28,20201,176],{"className":20202},[274,275],[28,20204,1522],{"className":20205},[355],[28,20207],{"className":20208,"style":302},[279],[28,20210,83],{"className":20211},[306],[28,20213],{"className":20214,"style":302},[279],[28,20216,20218,20221,20224,20227,20230],{"className":20217},[265],[28,20219],{"className":20220,"style":270},[269],[28,20222,176],{"className":20223},[274,275],[28,20225],{"className":20226,"style":280},[279],[28,20228,5074],{"className":20229},[284],[28,20231],{"className":20232,"style":280},[279],[28,20234,20236,20239],{"className":20235},[265],[28,20237],{"className":20238,"style":316},[269],[28,20240,1244],{"className":20241},[274],"。四个数都得非负，于是 ",[28,20244,20246,20268],{"className":20245,"translate":193},[197],[28,20247,20249],{"className":20248},[201],[203,20250,20251],{"xmlns":205},[208,20252,20253,20265],{},[211,20254,20255,20257,20259,20261,20263],{},[224,20256,1244],{},[217,20258,1876],{},[214,20260,176],{},[217,20262,1876],{},[224,20264,1253],{},[253,20266,20267],{"encoding":255},"1\\le x\\le 2",[28,20269,20271,20289,20307],{"className":20270,"ariaHidden":261},[260],[28,20272,20274,20277,20280,20283,20286],{"className":20273},[265],[28,20275],{"className":20276,"style":17967},[269],[28,20278,1244],{"className":20279},[274],[28,20281],{"className":20282,"style":302},[279],[28,20284,1876],{"className":20285},[306],[28,20287],{"className":20288,"style":302},[279],[28,20290,20292,20295,20298,20301,20304],{"className":20291},[265],[28,20293],{"className":20294,"style":3010},[269],[28,20296,176],{"className":20297},[274,275],[28,20299],{"className":20300,"style":302},[279],[28,20302,1876],{"className":20303},[306],[28,20305],{"className":20306,"style":302},[279],[28,20308,20310,20313],{"className":20309},[265],[28,20311],{"className":20312,"style":316},[269],[28,20314,1253],{"className":20315},[274],"。总运费",[28,20318,20320],{"className":20319,"translate":193},[192],[28,20321,20323,20403],{"className":20322,"translate":193},[197],[28,20324,20326],{"className":20325},[201],[203,20327,20328],{"xmlns":205,"display":206},[208,20329,20330,20400],{},[211,20331,20332,20334,20337,20339,20341,20343,20345,20347,20349,20351,20353,20355,20357,20359,20361,20363,20365,20367,20369,20371,20373,20375,20377,20379,20381,20383,20385,20387,20389,20392,20394,20396,20398],{},[214,20333,176],{},[217,20335,20336],{},"⋅",[224,20338,1244],{},[217,20340,124],{},[217,20342,43],{"stretchy":241},[224,20344,3400],{},[217,20346,5074],{},[214,20348,176],{},[217,20350,1522],{"stretchy":241},[217,20352,20336],{},[224,20354,1253],{},[217,20356,124],{},[217,20358,43],{"stretchy":241},[224,20360,1253],{},[217,20362,5074],{},[214,20364,176],{},[217,20366,1522],{"stretchy":241},[217,20368,20336],{},[224,20370,3400],{},[217,20372,124],{},[217,20374,43],{"stretchy":241},[214,20376,176],{},[217,20378,5074],{},[224,20380,1244],{},[217,20382,1522],{"stretchy":241},[217,20384,20336],{},[224,20386,1244],{},[217,20388,83],{},[224,20390,20391],{},"11",[217,20393,5074],{},[224,20395,3400],{},[214,20397,176],{},[214,20399,494],{"mathvariant":484},[253,20401,20402],{"encoding":255},"x\\cdot1+(3-x)\\cdot2+(2-x)\\cdot3+(x-1)\\cdot1=11-3x .",[28,20404,20406,20425,20443,20464,20485,20503,20524,20545,20563,20584,20605,20623,20641],{"className":20405,"ariaHidden":261},[260],[28,20407,20409,20413,20416,20419,20422],{"className":20408},[265],[28,20410],{"className":20411,"style":20412},[269],"height:0.4445em;",[28,20414,176],{"className":20415},[274,275],[28,20417],{"className":20418,"style":280},[279],[28,20420,20336],{"className":20421},[284],[28,20423],{"className":20424,"style":280},[279],[28,20426,20428,20431,20434,20437,20440],{"className":20427},[265],[28,20429],{"className":20430,"style":6853},[269],[28,20432,1244],{"className":20433},[274],[28,20435],{"className":20436,"style":280},[279],[28,20438,124],{"className":20439},[284],[28,20441],{"className":20442,"style":280},[279],[28,20444,20446,20449,20452,20455,20458,20461],{"className":20445},[265],[28,20447],{"className":20448,"style":341},[269],[28,20450,43],{"className":20451},[348],[28,20453,3400],{"className":20454},[274],[28,20456],{"className":20457,"style":280},[279],[28,20459,5074],{"className":20460},[284],[28,20462],{"className":20463,"style":280},[279],[28,20465,20467,20470,20473,20476,20479,20482],{"className":20466},[265],[28,20468],{"className":20469,"style":341},[269],[28,20471,176],{"className":20472},[274,275],[28,20474,1522],{"className":20475},[355],[28,20477],{"className":20478,"style":280},[279],[28,20480,20336],{"className":20481},[284],[28,20483],{"className":20484,"style":280},[279],[28,20486,20488,20491,20494,20497,20500],{"className":20487},[265],[28,20489],{"className":20490,"style":6853},[269],[28,20492,1253],{"className":20493},[274],[28,20495],{"className":20496,"style":280},[279],[28,20498,124],{"className":20499},[284],[28,20501],{"className":20502,"style":280},[279],[28,20504,20506,20509,20512,20515,20518,20521],{"className":20505},[265],[28,20507],{"className":20508,"style":341},[269],[28,20510,43],{"className":20511},[348],[28,20513,1253],{"className":20514},[274],[28,20516],{"className":20517,"style":280},[279],[28,20519,5074],{"className":20520},[284],[28,20522],{"className":20523,"style":280},[279],[28,20525,20527,20530,20533,20536,20539,20542],{"className":20526},[265],[28,20528],{"className":20529,"style":341},[269],[28,20531,176],{"className":20532},[274,275],[28,20534,1522],{"className":20535},[355],[28,20537],{"className":20538,"style":280},[279],[28,20540,20336],{"className":20541},[284],[28,20543],{"className":20544,"style":280},[279],[28,20546,20548,20551,20554,20557,20560],{"className":20547},[265],[28,20549],{"className":20550,"style":6853},[269],[28,20552,3400],{"className":20553},[274],[28,20555],{"className":20556,"style":280},[279],[28,20558,124],{"className":20559},[284],[28,20561],{"className":20562,"style":280},[279],[28,20564,20566,20569,20572,20575,20578,20581],{"className":20565},[265],[28,20567],{"className":20568,"style":341},[269],[28,20570,43],{"className":20571},[348],[28,20573,176],{"className":20574},[274,275],[28,20576],{"className":20577,"style":280},[279],[28,20579,5074],{"className":20580},[284],[28,20582],{"className":20583,"style":280},[279],[28,20585,20587,20590,20593,20596,20599,20602],{"className":20586},[265],[28,20588],{"className":20589,"style":341},[269],[28,20591,1244],{"className":20592},[274],[28,20594,1522],{"className":20595},[355],[28,20597],{"className":20598,"style":280},[279],[28,20600,20336],{"className":20601},[284],[28,20603],{"className":20604,"style":280},[279],[28,20606,20608,20611,20614,20617,20620],{"className":20607},[265],[28,20609],{"className":20610,"style":316},[269],[28,20612,1244],{"className":20613},[274],[28,20615],{"className":20616,"style":302},[279],[28,20618,83],{"className":20619},[306],[28,20621],{"className":20622,"style":302},[279],[28,20624,20626,20629,20632,20635,20638],{"className":20625},[265],[28,20627],{"className":20628,"style":6853},[269],[28,20630,20391],{"className":20631},[274],[28,20633],{"className":20634,"style":280},[279],[28,20636,5074],{"className":20637},[284],[28,20639],{"className":20640,"style":280},[279],[28,20642,20644,20647,20650,20653],{"className":20643},[265],[28,20645],{"className":20646,"style":316},[269],[28,20648,3400],{"className":20649},[274],[28,20651,176],{"className":20652},[274,275],[28,20654,494],{"className":20655},[274],[11,20657,20658,20686,20687,20738],{},[28,20659,20661,20674],{"className":20660,"translate":193},[197],[28,20662,20664],{"className":20663},[201],[203,20665,20666],{"xmlns":205},[208,20667,20668,20672],{},[211,20669,20670],{},[214,20671,176],{},[253,20673,176],{"encoding":255},[28,20675,20677],{"className":20676,"ariaHidden":261},[260],[28,20678,20680,20683],{"className":20679},[265],[28,20681],{"className":20682,"style":645},[269],[28,20684,176],{"className":20685},[274,275]," 越大越省，取 ",[28,20688,20690,20708],{"className":20689,"translate":193},[197],[28,20691,20693],{"className":20692},[201],[203,20694,20695],{"xmlns":205},[208,20696,20697,20705],{},[211,20698,20699,20701,20703],{},[214,20700,176],{},[217,20702,83],{},[224,20704,1253],{},[253,20706,20707],{"encoding":255},"x=2",[28,20709,20711,20729],{"className":20710,"ariaHidden":261},[260],[28,20712,20714,20717,20720,20723,20726],{"className":20713},[265],[28,20715],{"className":20716,"style":645},[269],[28,20718,176],{"className":20719},[274,275],[28,20721],{"className":20722,"style":302},[279],[28,20724,83],{"className":20725},[306],[28,20727],{"className":20728,"style":302},[279],[28,20730,20732,20735],{"className":20731},[265],[28,20733],{"className":20734,"style":316},[269],[28,20736,1253],{"className":20737},[274],"，总运费 5 块：甲到 A 运 2 吨，甲到 B 运 1 吨，乙到 B 运 1 吨。",[11,20740,20741,20742,20772,20773,20804],{},"这张“每个产地往每个销地运多少”的表，就是运输方案。Monge 在 1781 年问的是一般情形：给定一堆土 ",[28,20743,20745,20760],{"className":20744,"translate":193},[197],[28,20746,20748],{"className":20747},[201],[203,20749,20750],{"xmlns":205},[208,20751,20752,20757],{},[211,20753,20754],{},[214,20755,20756],{},"μ",[253,20758,20759],{"encoding":255},"\\mu",[28,20761,20763],{"className":20762,"ariaHidden":261},[260],[28,20764,20766,20769],{"className":20765},[265],[28,20767],{"className":20768,"style":612},[269],[28,20770,20756],{"className":20771},[274,275]," 和一个坑 ",[28,20774,20776,20791],{"className":20775,"translate":193},[197],[28,20777,20779],{"className":20778},[201],[203,20780,20781],{"xmlns":205},[208,20782,20783,20788],{},[211,20784,20785],{},[214,20786,20787],{},"ν",[253,20789,20790],{"encoding":255},"\\nu",[28,20792,20794],{"className":20793,"ariaHidden":261},[260],[28,20795,20797,20800],{"className":20796},[265],[28,20798],{"className":20799,"style":645},[269],[28,20801,20787],{"className":20802,"style":20803},[274,275],"margin-right:0.0637em;","，怎么搬最省。",[11,20806,20807,20808,20888,20889,20991,20992,6904],{},"他的写法是找一个映射 ",[28,20809,20811,20836],{"className":20810,"translate":193},[197],[28,20812,20814],{"className":20813},[201],[203,20815,20816],{"xmlns":205},[208,20817,20818,20833],{},[211,20819,20820,20822,20824,20828,20830],{},[214,20821,443],{},[217,20823,877],{},[214,20825,20827],{"mathvariant":20826},"script","X",[217,20829,950],{},[214,20831,20832],{"mathvariant":20826},"Y",[253,20834,20835],{"encoding":255},"T:\\mathcal X\\to\\mathcal Y",[28,20837,20839,20857,20877],{"className":20838,"ariaHidden":261},[260],[28,20840,20842,20845,20848,20851,20854],{"className":20841},[265],[28,20843],{"className":20844,"style":455},[269],[28,20846,443],{"className":20847,"style":459},[274,275],[28,20849],{"className":20850,"style":302},[279],[28,20852,877],{"className":20853},[306],[28,20855],{"className":20856,"style":302},[279],[28,20858,20860,20863,20868,20871,20874],{"className":20859},[265],[28,20861],{"className":20862,"style":455},[269],[28,20864,20827],{"className":20865,"style":20867},[274,20866],"mathcal","margin-right:0.1464em;",[28,20869],{"className":20870,"style":302},[279],[28,20872,950],{"className":20873},[306],[28,20875],{"className":20876,"style":302},[279],[28,20878,20880,20884],{"className":20879},[265],[28,20881],{"className":20882,"style":20883},[269],"height:0.7805em;vertical-align:-0.0972em;",[28,20885,20832],{"className":20886,"style":20887},[274,20866],"margin-right:0.0822em;","，要求“搬完之后每个地方刚好装满”，也就是推前测度 ",[28,20890,20892,20917],{"className":20891,"translate":193},[197],[28,20893,20895],{"className":20894},[201],[203,20896,20897],{"xmlns":205},[208,20898,20899,20914],{},[211,20900,20901,20908,20910,20912],{},[501,20902,20903,20905],{},[214,20904,443],{},[214,20906,20907],{"mathvariant":484},"#",[214,20909,20756],{},[217,20911,83],{},[214,20913,20787],{},[253,20915,20916],{"encoding":255},"T_\\#\\mu=\\nu",[28,20918,20920,20982],{"className":20919,"ariaHidden":261},[260],[28,20921,20923,20927,20970,20973,20976,20979],{"className":20922},[265],[28,20924],{"className":20925,"style":20926},[269],"height:0.9694em;vertical-align:-0.2861em;",[28,20928,20930,20933],{"className":20929},[274],[28,20931,443],{"className":20932,"style":459},[274,275],[28,20934,20936],{"className":20935},[556],[28,20937,20939,20961],{"className":20938},[560,561],[28,20940,20942,20958],{"className":20941},[565],[28,20943,20946],{"className":20944,"style":20945},[569],"height:0.3361em;",[28,20947,20949,20952],{"style":20948},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[28,20950],{"className":20951,"style":578},[577],[28,20953,20955],{"className":20954},[582,583,584,585],[28,20956,20907],{"className":20957},[274,585],[28,20959,593],{"className":20960},[592],[28,20962,20964],{"className":20963},[565],[28,20965,20968],{"className":20966,"style":20967},[569],"height:0.2861em;",[28,20969],{},[28,20971,20756],{"className":20972},[274,275],[28,20974],{"className":20975,"style":302},[279],[28,20977,83],{"className":20978},[306],[28,20980],{"className":20981,"style":302},[279],[28,20983,20985,20988],{"className":20984},[265],[28,20986],{"className":20987,"style":645},[269],[28,20989,20787],{"className":20990,"style":20803},[274,275],"，即对任意可测集 ",[28,20993,20995,21008],{"className":20994,"translate":193},[197],[28,20996,20998],{"className":20997},[201],[203,20999,21000],{"xmlns":205},[208,21001,21002,21006],{},[211,21003,21004],{},[214,21005,9695],{},[253,21007,9695],{"encoding":255},[28,21009,21011],{"className":21010,"ariaHidden":261},[260],[28,21012,21014,21017],{"className":21013},[265],[28,21015],{"className":21016,"style":455},[269],[28,21018,9695],{"className":21019,"style":9729},[274,275],[28,21021,21023],{"className":21022,"translate":193},[192],[28,21024,21026,21092],{"className":21025,"translate":193},[197],[28,21027,21029],{"className":21028},[201],[203,21030,21031],{"xmlns":205,"display":206},[208,21032,21033,21089],{},[211,21034,21035,21037,21043,21045,21047,21049,21051,21053,21055,21057,21059,21069,21071,21073,21075,21077,21079,21081,21083,21085,21087],{},[217,21036,43],{"stretchy":241},[501,21038,21039,21041],{},[214,21040,443],{},[214,21042,20907],{"mathvariant":484},[214,21044,20756],{},[217,21046,1522],{"stretchy":241},[217,21048,43],{"stretchy":241},[214,21050,9695],{},[217,21052,1522],{"stretchy":241},[217,21054,83],{},[214,21056,20756],{},[217,21058,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[3054,21060,21061,21063],{},[214,21062,443],{},[211,21064,21065,21067],{},[217,21066,5074],{},[224,21068,1244],{},[217,21070,43],{"stretchy":241},[214,21072,9695],{},[217,21074,1522],{"stretchy":241},[217,21076,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,21078,83],{},[214,21080,20787],{},[217,21082,43],{"stretchy":241},[214,21084,9695],{},[217,21086,1522],{"stretchy":241},[217,21088,1683],{"separator":261},[253,21090,21091],{"encoding":255},"(T_\\#\\mu)(B)=\\mu\\bigl(T^{-1}(B)\\bigr)=\\nu(B),",[28,21093,21095,21169,21244],{"className":21094,"ariaHidden":261},[260],[28,21096,21098,21102,21105,21145,21148,21151,21154,21157,21160,21163,21166],{"className":21097},[265],[28,21099],{"className":21100,"style":21101},[269],"height:1.0361em;vertical-align:-0.2861em;",[28,21103,43],{"className":21104},[348],[28,21106,21108,21111],{"className":21107},[274],[28,21109,443],{"className":21110,"style":459},[274,275],[28,21112,21114],{"className":21113},[556],[28,21115,21117,21137],{"className":21116},[560,561],[28,21118,21120,21134],{"className":21119},[565],[28,21121,21123],{"className":21122,"style":20945},[569],[28,21124,21125,21128],{"style":20948},[28,21126],{"className":21127,"style":578},[577],[28,21129,21131],{"className":21130},[582,583,584,585],[28,21132,20907],{"className":21133},[274,585],[28,21135,593],{"className":21136},[592],[28,21138,21140],{"className":21139},[565],[28,21141,21143],{"className":21142,"style":20967},[569],[28,21144],{},[28,21146,20756],{"className":21147},[274,275],[28,21149,1522],{"className":21150},[355],[28,21152,43],{"className":21153},[348],[28,21155,9695],{"className":21156,"style":9729},[274,275],[28,21158,1522],{"className":21159},[355],[28,21161],{"className":21162,"style":302},[279],[28,21164,83],{"className":21165},[306],[28,21167],{"className":21168,"style":302},[279],[28,21170,21172,21176,21179,21185,21220,21223,21226,21229,21235,21238,21241],{"className":21171},[265],[28,21173],{"className":21174,"style":21175},[269],"height:1.2141em;vertical-align:-0.35em;",[28,21177,20756],{"className":21178},[274,275],[28,21180,21182],{"className":21181},[348],[28,21183,43],{"className":21184},[2169,2170],[28,21186,21188,21191],{"className":21187},[274],[28,21189,443],{"className":21190,"style":459},[274,275],[28,21192,21194],{"className":21193},[556],[28,21195,21197],{"className":21196},[560],[28,21198,21200],{"className":21199},[565],[28,21201,21203],{"className":21202,"style":4877},[569],[28,21204,21205,21208],{"style":4880},[28,21206],{"className":21207,"style":578},[577],[28,21209,21211],{"className":21210},[582,583,584,585],[28,21212,21214,21217],{"className":21213},[274,585],[28,21215,5074],{"className":21216},[274,585],[28,21218,1244],{"className":21219},[274,585],[28,21221,43],{"className":21222},[348],[28,21224,9695],{"className":21225,"style":9729},[274,275],[28,21227,1522],{"className":21228},[355],[28,21230,21232],{"className":21231},[355],[28,21233,1522],{"className":21234},[2169,2170],[28,21236],{"className":21237,"style":302},[279],[28,21239,83],{"className":21240},[306],[28,21242],{"className":21243,"style":302},[279],[28,21245,21247,21250,21253,21256,21259,21262],{"className":21246},[265],[28,21248],{"className":21249,"style":341},[269],[28,21251,20787],{"className":21252,"style":20803},[274,275],[28,21254,43],{"className":21255},[348],[28,21257,9695],{"className":21258,"style":9729},[274,275],[28,21260,1522],{"className":21261},[355],[28,21263,1683],{"className":21264},[1723],[11,21266,21267,21268,21382],{},"代价是 ",[28,21269,21271,21316],{"className":21270,"translate":193},[197],[28,21272,21274],{"className":21273},[201],[203,21275,21276],{"xmlns":205},[208,21277,21278,21313],{},[211,21279,21280,21283,21285,21287,21289,21291,21293,21295,21297,21299,21301,21303,21305,21307,21309,21311],{},[217,21281,21282],{},"∫",[214,21284,22],{},[217,21286,43],{"stretchy":241},[214,21288,176],{},[217,21290,1683],{"separator":261},[214,21292,443],{},[217,21294,43],{"stretchy":241},[214,21296,176],{},[217,21298,1522],{"stretchy":241},[217,21300,1522],{"stretchy":241},[228,21302,488],{},[214,21304,12988],{},[214,21306,20756],{},[217,21308,43],{"stretchy":241},[214,21310,176],{},[217,21312,1522],{"stretchy":241},[253,21314,21315],{"encoding":255},"\\int c(x,T(x))\\,d\\mu(x)",[28,21317,21319],{"className":21318,"ariaHidden":261},[260],[28,21320,21322,21326,21333,21336,21339,21342,21345,21348,21351,21354,21357,21360,21364,21367,21370,21373,21376,21379],{"className":21321},[265],[28,21323],{"className":21324,"style":21325},[269],"height:1.1111em;vertical-align:-0.3061em;",[28,21327,21282],{"className":21328,"style":21332},[21329,21330,21331],"mop","op-symbol","small-op","margin-right:0.1945em;position:relative;top:-0.0006em;",[28,21334],{"className":21335,"style":530},[279],[28,21337,22],{"className":21338},[274,275],[28,21340,43],{"className":21341},[348],[28,21343,176],{"className":21344},[274,275],[28,21346,1683],{"className":21347},[1723],[28,21349],{"className":21350,"style":530},[279],[28,21352,443],{"className":21353,"style":459},[274,275],[28,21355,43],{"className":21356},[348],[28,21358,176],{"className":21359},[274,275],[28,21361,21363],{"className":21362},[355],"))",[28,21365],{"className":21366,"style":530},[279],[28,21368,12988],{"className":21369},[274,275],[28,21371,20756],{"className":21372},[274,275],[28,21374,43],{"className":21375},[348],[28,21377,176],{"className":21378},[274,275],[28,21380,1522],{"className":21381},[355],"。于是 Monge 问题就是",[28,21384,21386],{"className":21385,"translate":193},[192],[28,21387,21389,21472],{"className":21388,"translate":193},[197],[28,21390,21392],{"className":21391},[201],[203,21393,21394],{"xmlns":205,"display":206},[208,21395,21396,21469],{},[211,21397,21398,21410,21412,21418,21420,21422,21424,21426,21428,21430,21432,21434,21436,21438,21440,21442,21444,21446,21448,21450,21453,21455,21461,21463,21465,21467],{},[1771,21399,21400,21408],{},[211,21401,21402,21405],{},[214,21403,21404],{},"inf",[217,21406,21407],{},"⁡",[214,21409,443],{},[228,21411,14860],{},[501,21413,21414,21416],{},[217,21415,21282],{},[214,21417,20827],{"mathvariant":20826},[214,21419,22],{},[217,21421,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,21423,176],{},[217,21425,1683],{"separator":261},[214,21427,443],{},[217,21429,43],{"stretchy":241},[214,21431,176],{},[217,21433,1522],{"stretchy":241},[217,21435,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[228,21437,488],{},[214,21439,12988],{},[214,21441,20756],{},[217,21443,43],{"stretchy":241},[214,21445,176],{},[217,21447,1522],{"stretchy":241},[279,21449],{"width":497},[228,21451,21452],{},"s.t.",[279,21454],{"width":497},[501,21456,21457,21459],{},[214,21458,443],{},[214,21460,20907],{"mathvariant":484},[214,21462,20756],{},[217,21464,83],{},[214,21466,20787],{},[214,21468,494],{"mathvariant":484},[253,21470,21471],{"encoding":255},"\\inf_{T}\\ \\int_{\\mathcal X} c\\bigl(x,T(x)\\bigr)\\,d\\mu(x)\n\\quad\\text{s.t.}\\quad T_\\#\\mu=\\nu .",[28,21473,21475,21710],{"className":21474,"ariaHidden":261},[260],[28,21476,21478,21482,21535,21538,21541,21589,21592,21595,21601,21604,21607,21610,21613,21616,21619,21622,21628,21631,21634,21637,21640,21643,21646,21649,21655,21658,21698,21701,21704,21707],{"className":21477},[265],[28,21479],{"className":21480,"style":21481},[269],"height:2.2719em;vertical-align:-0.9119em;",[28,21483,21486],{"className":21484},[21329,21485],"op-limits",[28,21487,21489,21526],{"className":21488},[560,561],[28,21490,21492,21523],{"className":21491},[565],[28,21493,21495,21510],{"className":21494,"style":7448},[569],[28,21496,21498,21501],{"style":21497},"top:-2.3557em;margin-left:0em;",[28,21499],{"className":21500,"style":1914},[577],[28,21502,21504],{"className":21503},[582,583,584,585],[28,21505,21507],{"className":21506},[274,585],[28,21508,443],{"className":21509,"style":459},[274,275,585],[28,21511,21512,21515],{"style":1926},[28,21513],{"className":21514,"style":1914},[577],[28,21516,21517],{},[28,21518,3896,21520],{"className":21519},[21329],[28,21521,1511],{"style":21522},"margin-right:0.0778em;",[28,21524,593],{"className":21525},[592],[28,21527,21529],{"className":21528},[565],[28,21530,21533],{"className":21531,"style":21532},[569],"height:0.7443em;",[28,21534],{},[28,21536,14860],{"className":21537},[279],[28,21539],{"className":21540,"style":530},[279],[28,21542,21544,21549],{"className":21543},[21329],[28,21545,21282],{"className":21546,"style":21548},[21329,21330,21547],"large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[28,21550,21552],{"className":21551},[556],[28,21553,21555,21580],{"className":21554},[560,561],[28,21556,21558,21577],{"className":21557},[565],[28,21559,21562],{"className":21560,"style":21561},[569],"height:-0.4336em;",[28,21563,21565,21568],{"style":21564},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[28,21566],{"className":21567,"style":578},[577],[28,21569,21571],{"className":21570},[582,583,584,585],[28,21572,21574],{"className":21573},[274,585],[28,21575,20827],{"className":21576,"style":20867},[274,20866,585],[28,21578,593],{"className":21579},[592],[28,21581,21583],{"className":21582},[565],[28,21584,21587],{"className":21585,"style":21586},[569],"height:0.9119em;",[28,21588],{},[28,21590],{"className":21591,"style":530},[279],[28,21593,22],{"className":21594},[274,275],[28,21596,21598],{"className":21597},[348],[28,21599,43],{"className":21600},[2169,2170],[28,21602,176],{"className":21603},[274,275],[28,21605,1683],{"className":21606},[1723],[28,21608],{"className":21609,"style":530},[279],[28,21611,443],{"className":21612,"style":459},[274,275],[28,21614,43],{"className":21615},[348],[28,21617,176],{"className":21618},[274,275],[28,21620,1522],{"className":21621},[355],[28,21623,21625],{"className":21624},[355],[28,21626,1522],{"className":21627},[2169,2170],[28,21629],{"className":21630,"style":530},[279],[28,21632,12988],{"className":21633},[274,275],[28,21635,20756],{"className":21636},[274,275],[28,21638,43],{"className":21639},[348],[28,21641,176],{"className":21642},[274,275],[28,21644,1522],{"className":21645},[355],[28,21647],{"className":21648,"style":540},[279],[28,21650,21652],{"className":21651},[274,11179],[28,21653,21452],{"className":21654},[274],[28,21656],{"className":21657,"style":540},[279],[28,21659,21661,21664],{"className":21660},[274],[28,21662,443],{"className":21663,"style":459},[274,275],[28,21665,21667],{"className":21666},[556],[28,21668,21670,21690],{"className":21669},[560,561],[28,21671,21673,21687],{"className":21672},[565],[28,21674,21676],{"className":21675,"style":20945},[569],[28,21677,21678,21681],{"style":20948},[28,21679],{"className":21680,"style":578},[577],[28,21682,21684],{"className":21683},[582,583,584,585],[28,21685,20907],{"className":21686},[274,585],[28,21688,593],{"className":21689},[592],[28,21691,21693],{"className":21692},[565],[28,21694,21696],{"className":21695,"style":20967},[569],[28,21697],{},[28,21699,20756],{"className":21700},[274,275],[28,21702],{"className":21703,"style":302},[279],[28,21705,83],{"className":21706},[306],[28,21708],{"className":21709,"style":302},[279],[28,21711,21713,21716,21719],{"className":21712},[265],[28,21714],{"className":21715,"style":645},[269],[28,21717,20787],{"className":21718,"style":20803},[274,275],[28,21720,494],{"className":21721},[274],[11,21723,21724,21725,21753,21754,21782,21783,21811,21812,21908,21909,2923,22225,22269,22270,2693,22298,22326],{},"这个提法有一个硬伤：",[28,21726,21728,21741],{"className":21727,"translate":193},[197],[28,21729,21731],{"className":21730},[201],[203,21732,21733],{"xmlns":205},[208,21734,21735,21739],{},[211,21736,21737],{},[214,21738,443],{},[253,21740,443],{"encoding":255},[28,21742,21744],{"className":21743,"ariaHidden":261},[260],[28,21745,21747,21750],{"className":21746},[265],[28,21748],{"className":21749,"style":455},[269],[28,21751,443],{"className":21752,"style":459},[274,275]," 必须是确定性的，一个产地只能有一条固定去向。如果 ",[28,21755,21757,21770],{"className":21756,"translate":193},[197],[28,21758,21760],{"className":21759},[201],[203,21761,21762],{"xmlns":205},[208,21763,21764,21768],{},[211,21765,21766],{},[214,21767,20756],{},[253,21769,20759],{"encoding":255},[28,21771,21773],{"className":21772,"ariaHidden":261},[260],[28,21774,21776,21779],{"className":21775},[265],[28,21777],{"className":21778,"style":612},[269],[28,21780,20756],{"className":21781},[274,275]," 在某个点上有原子，而 ",[28,21784,21786,21799],{"className":21785,"translate":193},[197],[28,21787,21789],{"className":21788},[201],[203,21790,21791],{"xmlns":205},[208,21792,21793,21797],{},[211,21794,21795],{},[214,21796,20787],{},[253,21798,20790],{"encoding":255},[28,21800,21802],{"className":21801,"ariaHidden":261},[260],[28,21803,21805,21808],{"className":21804},[265],[28,21806],{"className":21807,"style":645},[269],[28,21809,20787],{"className":21810,"style":20803},[274,275]," 在那里要求把质量分到两个地方，就无解。举个最小的例子：全部质量集中在甲一个点上，",[28,21813,21815,21838],{"className":21814,"translate":193},[197],[28,21816,21818],{"className":21817},[201],[203,21819,21820],{"xmlns":205},[208,21821,21822,21835],{},[211,21823,21824,21826,21828],{},[214,21825,20756],{},[217,21827,83],{},[501,21829,21830,21833],{},[214,21831,21832],{},"δ",[224,21834,89],{},[253,21836,21837],{"encoding":255},"\\mu=\\delta_0",[28,21839,21841,21859],{"className":21840,"ariaHidden":261},[260],[28,21842,21844,21847,21850,21853,21856],{"className":21843},[265],[28,21845],{"className":21846,"style":612},[269],[28,21848,20756],{"className":21849},[274,275],[28,21851],{"className":21852,"style":302},[279],[28,21854,83],{"className":21855},[306],[28,21857],{"className":21858,"style":302},[279],[28,21860,21862,21866],{"className":21861},[265],[28,21863],{"className":21864,"style":21865},[269],"height:0.8444em;vertical-align:-0.15em;",[28,21867,21869,21873],{"className":21868},[274],[28,21870,21832],{"className":21871,"style":21872},[274,275],"margin-right:0.0379em;",[28,21874,21876],{"className":21875},[556],[28,21877,21879,21900],{"className":21878},[560,561],[28,21880,21882,21897],{"className":21881},[565],[28,21883,21885],{"className":21884,"style":1285},[569],[28,21886,21888,21891],{"style":21887},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[28,21889],{"className":21890,"style":578},[577],[28,21892,21894],{"className":21893},[582,583,584,585],[28,21895,89],{"className":21896},[274,585],[28,21898,593],{"className":21899},[592],[28,21901,21903],{"className":21902},[565],[28,21904,21906],{"className":21905,"style":600},[569],[28,21907],{},"；而两个厂各要一半，",[28,21910,21912,21959],{"className":21911,"translate":193},[197],[28,21913,21915],{"className":21914},[201],[203,21916,21917],{"xmlns":205},[208,21918,21919,21956],{},[211,21920,21921,21923,21925,21934,21940,21942,21950],{},[214,21922,20787],{},[217,21924,83],{},[5027,21926,21927],{"scriptlevel":89,"displaystyle":241},[21928,21929,21930,21932],"mfrac",{},[224,21931,1244],{},[224,21933,1253],{},[501,21935,21936,21938],{},[214,21937,21832],{},[224,21939,1244],{},[217,21941,124],{},[5027,21943,21944],{"scriptlevel":89,"displaystyle":241},[21928,21945,21946,21948],{},[224,21947,1244],{},[224,21949,1253],{},[501,21951,21952,21954],{},[214,21953,21832],{},[224,21955,1253],{},[253,21957,21958],{"encoding":255},"\\nu=\\tfrac12\\delta_1+\\tfrac12\\delta_2",[28,21960,21962,21980,22111],{"className":21961,"ariaHidden":261},[260],[28,21963,21965,21968,21971,21974,21977],{"className":21964},[265],[28,21966],{"className":21967,"style":645},[269],[28,21969,20787],{"className":21970,"style":20803},[274,275],[28,21972],{"className":21973,"style":302},[279],[28,21975,83],{"className":21976},[306],[28,21978],{"className":21979,"style":302},[279],[28,21981,21983,21987,22062,22102,22105,22108],{"className":21982},[265],[28,21984],{"className":21985,"style":21986},[269],"height:1.1901em;vertical-align:-0.345em;",[28,21988,21990,21993,22059],{"className":21989},[274],[28,21991],{"className":21992},[348,8285],[28,21994,21996],{"className":21995},[21928],[28,21997,21999,22050],{"className":21998},[560,561],[28,22000,22002,22047],{"className":22001},[565],[28,22003,22006,22021,22032],{"className":22004,"style":22005},[569],"height:0.8451em;",[28,22007,22009,22012],{"style":22008},"top:-2.655em;",[28,22010],{"className":22011,"style":1914},[577],[28,22013,22015],{"className":22014},[582,583,584,585],[28,22016,22018],{"className":22017},[274,585],[28,22019,1253],{"className":22020},[274,585],[28,22022,22024,22027],{"style":22023},"top:-3.23em;",[28,22025],{"className":22026,"style":1914},[577],[28,22028],{"className":22029,"style":22031},[22030],"frac-line","border-bottom-width:0.04em;",[28,22033,22035,22038],{"style":22034},"top:-3.394em;",[28,22036],{"className":22037,"style":1914},[577],[28,22039,22041],{"className":22040},[582,583,584,585],[28,22042,22044],{"className":22043},[274,585],[28,22045,1244],{"className":22046},[274,585],[28,22048,593],{"className":22049},[592],[28,22051,22053],{"className":22052},[565],[28,22054,22057],{"className":22055,"style":22056},[569],"height:0.345em;",[28,22058],{},[28,22060],{"className":22061},[355,8285],[28,22063,22065,22068],{"className":22064},[274],[28,22066,21832],{"className":22067,"style":21872},[274,275],[28,22069,22071],{"className":22070},[556],[28,22072,22074,22094],{"className":22073},[560,561],[28,22075,22077,22091],{"className":22076},[565],[28,22078,22080],{"className":22079,"style":1285},[569],[28,22081,22082,22085],{"style":21887},[28,22083],{"className":22084,"style":578},[577],[28,22086,22088],{"className":22087},[582,583,584,585],[28,22089,1244],{"className":22090},[274,585],[28,22092,593],{"className":22093},[592],[28,22095,22097],{"className":22096},[565],[28,22098,22100],{"className":22099,"style":600},[569],[28,22101],{},[28,22103],{"className":22104,"style":280},[279],[28,22106,124],{"className":22107},[284],[28,22109],{"className":22110,"style":280},[279],[28,22112,22114,22117,22185],{"className":22113},[265],[28,22115],{"className":22116,"style":21986},[269],[28,22118,22120,22123,22182],{"className":22119},[274],[28,22121],{"className":22122},[348,8285],[28,22124,22126],{"className":22125},[21928],[28,22127,22129,22174],{"className":22128},[560,561],[28,22130,22132,22171],{"className":22131},[565],[28,22133,22135,22149,22157],{"className":22134,"style":22005},[569],[28,22136,22137,22140],{"style":22008},[28,22138],{"className":22139,"style":1914},[577],[28,22141,22143],{"className":22142},[582,583,584,585],[28,22144,22146],{"className":22145},[274,585],[28,22147,1253],{"className":22148},[274,585],[28,22150,22151,22154],{"style":22023},[28,22152],{"className":22153,"style":1914},[577],[28,22155],{"className":22156,"style":22031},[22030],[28,22158,22159,22162],{"style":22034},[28,22160],{"className":22161,"style":1914},[577],[28,22163,22165],{"className":22164},[582,583,584,585],[28,22166,22168],{"className":22167},[274,585],[28,22169,1244],{"className":22170},[274,585],[28,22172,593],{"className":22173},[592],[28,22175,22177],{"className":22176},[565],[28,22178,22180],{"className":22179,"style":22056},[569],[28,22181],{},[28,22183],{"className":22184},[355,8285],[28,22186,22188,22191],{"className":22187},[274],[28,22189,21832],{"className":22190,"style":21872},[274,275],[28,22192,22194],{"className":22193},[556],[28,22195,22197,22217],{"className":22196},[560,561],[28,22198,22200,22214],{"className":22199},[565],[28,22201,22203],{"className":22202,"style":1285},[569],[28,22204,22205,22208],{"style":21887},[28,22206],{"className":22207,"style":578},[577],[28,22209,22211],{"className":22210},[582,583,584,585],[28,22212,1253],{"className":22213},[274,585],[28,22215,593],{"className":22216},[592],[28,22218,22220],{"className":22219},[565],[28,22221,22223],{"className":22222,"style":600},[569],[28,22224],{},[28,22226,22228,22248],{"className":22227,"translate":193},[197],[28,22229,22231],{"className":22230},[201],[203,22232,22233],{"xmlns":205},[208,22234,22235,22245],{},[211,22236,22237,22239,22241,22243],{},[214,22238,443],{},[217,22240,43],{"stretchy":241},[224,22242,89],{},[217,22244,1522],{"stretchy":241},[253,22246,22247],{"encoding":255},"T(0)",[28,22249,22251],{"className":22250,"ariaHidden":261},[260],[28,22252,22254,22257,22260,22263,22266],{"className":22253},[265],[28,22255],{"className":22256,"style":341},[269],[28,22258,443],{"className":22259,"style":459},[274,275],[28,22261,43],{"className":22262},[348],[28,22264,89],{"className":22265},[274],[28,22267,1522],{"className":22268},[355]," 必须同时等于 ",[28,22271,22273,22286],{"className":22272,"translate":193},[197],[28,22274,22276],{"className":22275},[201],[203,22277,22278],{"xmlns":205},[208,22279,22280,22284],{},[211,22281,22282],{},[224,22283,1244],{},[253,22285,1244],{"encoding":255},[28,22287,22289],{"className":22288,"ariaHidden":261},[260],[28,22290,22292,22295],{"className":22291},[265],[28,22293],{"className":22294,"style":316},[269],[28,22296,1244],{"className":22297},[274],[28,22299,22301,22314],{"className":22300,"translate":193},[197],[28,22302,22304],{"className":22303},[201],[203,22305,22306],{"xmlns":205},[208,22307,22308,22312],{},[211,22309,22310],{},[224,22311,1253],{},[253,22313,1253],{"encoding":255},[28,22315,22317],{"className":22316,"ariaHidden":261},[260],[28,22318,22320,22323],{"className":22319},[265],[28,22321],{"className":22322,"style":316},[269],[28,22324,1253],{"className":22325},[274],"，函数做不到。",[11,22328,22329],{},"可这件事在现实中天天发生：一辆车分两个厂送货而已。Kantorovich 在 1940 年代的办法是把映射换成联合分布。令",[28,22331,22333],{"className":22332,"translate":193},[192],[28,22334,22336,22454],{"className":22335,"translate":193},[197],[28,22337,22339],{"className":22338},[201],[203,22340,22341],{"xmlns":205,"display":206},[208,22342,22343,22451],{},[211,22344,22345,22348,22350,22352,22354,22356,22358,22360,22363,22366,22368,22370,22372,22374,22377,22379,22381,22383,22385,22387,22389,22403,22409,22411,22413,22415,22417,22419,22421,22435,22441,22443,22445,22447,22449],{},[214,22346,22347],{"mathvariant":484},"Π",[217,22349,43],{"stretchy":241},[214,22351,20756],{},[217,22353,1683],{"separator":261},[214,22355,20787],{},[217,22357,1522],{"stretchy":241},[217,22359,83],{},[217,22361,7916],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},"1.8em",[214,22364,22365],{},"π",[217,22367,12054],{},[214,22369,15825],{"mathvariant":20826},[217,22371,43],{"stretchy":241},[214,22373,20827],{"mathvariant":20826},[217,22375,22376],{},"×",[214,22378,20832],{"mathvariant":20826},[217,22380,1522],{"stretchy":241},[228,22382,14860],{},[217,22384,877],{},[228,22386,14860],{},[217,22388,43],{"stretchy":241},[501,22390,22391,22401],{},[211,22392,22393,22395,22397,22399],{},[214,22394,11],{"mathvariant":484},[214,22396,1867],{"mathvariant":484},[214,22398,3933],{"mathvariant":484},[214,22400,7907],{"mathvariant":484},[214,22402,20827],{"mathvariant":20826},[501,22404,22405,22407],{},[217,22406,1522],{"stretchy":241},[214,22408,20907],{"mathvariant":484},[214,22410,22365],{},[217,22412,83],{},[214,22414,20756],{},[217,22416,1683],{"separator":261},[228,22418,14860],{},[217,22420,43],{"stretchy":241},[501,22422,22423,22433],{},[211,22424,22425,22427,22429,22431],{},[214,22426,11],{"mathvariant":484},[214,22428,1867],{"mathvariant":484},[214,22430,3933],{"mathvariant":484},[214,22432,7907],{"mathvariant":484},[214,22434,20832],{"mathvariant":20826},[501,22436,22437,22439],{},[217,22438,1522],{"stretchy":241},[214,22440,20907],{"mathvariant":484},[214,22442,22365],{},[217,22444,83],{},[214,22446,20787],{},[217,22448,14883],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[217,22450,1683],{"separator":261},[253,22452,22453],{"encoding":255},"\\Pi(\\mu,\\nu)=\\Bigl\\{\\pi\\in\\mathcal P(\\mathcal X\\times\\mathcal Y)\\ :\\ (\\mathrm{proj}_{\\mathcal X})_\\#\\pi=\\mu,\\ (\\mathrm{proj}_{\\mathcal Y})_\\#\\pi=\\nu\\Bigr\\},",[28,22455,22457,22493,22519,22543,22570,22681,22802],{"className":22456,"ariaHidden":261},[260],[28,22458,22460,22463,22466,22469,22472,22475,22478,22481,22484,22487,22490],{"className":22459},[265],[28,22461],{"className":22462,"style":341},[269],[28,22464,22347],{"className":22465},[274],[28,22467,43],{"className":22468},[348],[28,22470,20756],{"className":22471},[274,275],[28,22473,1683],{"className":22474},[1723],[28,22476],{"className":22477,"style":530},[279],[28,22479,20787],{"className":22480,"style":20803},[274,275],[28,22482,1522],{"className":22483},[355],[28,22485],{"className":22486,"style":302},[279],[28,22488,83],{"className":22489},[306],[28,22491],{"className":22492,"style":302},[279],[28,22494,22496,22500,22507,22510,22513,22516],{"className":22495},[265],[28,22497],{"className":22498,"style":22499},[269],"height:1.8em;vertical-align:-0.65em;",[28,22501,22503],{"className":22502},[348],[28,22504,7916],{"className":22505},[2169,22506],"size2",[28,22508,22365],{"className":22509,"style":298},[274,275],[28,22511],{"className":22512,"style":302},[279],[28,22514,12054],{"className":22515},[306],[28,22517],{"className":22518,"style":302},[279],[28,22520,22522,22525,22528,22531,22534,22537,22540],{"className":22521},[265],[28,22523],{"className":22524,"style":341},[269],[28,22526,15825],{"className":22527,"style":20887},[274,20866],[28,22529,43],{"className":22530},[348],[28,22532,20827],{"className":22533,"style":20867},[274,20866],[28,22535],{"className":22536,"style":280},[279],[28,22538,22376],{"className":22539},[284],[28,22541],{"className":22542,"style":280},[279],[28,22544,22546,22549,22552,22555,22558,22561,22564,22567],{"className":22545},[265],[28,22547],{"className":22548,"style":341},[269],[28,22550,20832],{"className":22551,"style":20887},[274,20866],[28,22553,1522],{"className":22554},[355],[28,22556,14860],{"className":22557},[279],[28,22559],{"className":22560,"style":302},[279],[28,22562,877],{"className":22563},[306],[28,22565,14860],{"className":22566},[279],[28,22568],{"className":22569,"style":302},[279],[28,22571,22573,22576,22579,22629,22669,22672,22675,22678],{"className":22572},[265],[28,22574],{"className":22575,"style":21101},[269],[28,22577,43],{"className":22578},[348],[28,22580,22582,22589],{"className":22581},[274],[28,22583,22585],{"className":22584},[274],[28,22586,22588],{"className":22587},[274,1712],"proj",[28,22590,22592],{"className":22591},[556],[28,22593,22595,22620],{"className":22594},[560,561],[28,22596,22598,22617],{"className":22597},[565],[28,22599,22602],{"className":22600,"style":22601},[569],"height:0.2342em;",[28,22603,22605,22608],{"style":22604},"top:-2.4559em;margin-right:0.05em;",[28,22606],{"className":22607,"style":578},[577],[28,22609,22611],{"className":22610},[582,583,584,585],[28,22612,22614],{"className":22613},[274,585],[28,22615,20827],{"className":22616,"style":20867},[274,20866,585],[28,22618,593],{"className":22619},[592],[28,22621,22623],{"className":22622},[565],[28,22624,22627],{"className":22625,"style":22626},[569],"height:0.2441em;",[28,22628],{},[28,22630,22632,22635],{"className":22631},[355],[28,22633,1522],{"className":22634},[355],[28,22636,22638],{"className":22637},[556],[28,22639,22641,22661],{"className":22640},[560,561],[28,22642,22644,22658],{"className":22643},[565],[28,22645,22647],{"className":22646,"style":20945},[569],[28,22648,22649,22652],{"style":573},[28,22650],{"className":22651,"style":578},[577],[28,22653,22655],{"className":22654},[582,583,584,585],[28,22656,20907],{"className":22657},[274,585],[28,22659,593],{"className":22660},[592],[28,22662,22664],{"className":22663},[565],[28,22665,22667],{"className":22666,"style":20967},[569],[28,22668],{},[28,22670,22365],{"className":22671,"style":298},[274,275],[28,22673],{"className":22674,"style":302},[279],[28,22676,83],{"className":22677},[306],[28,22679],{"className":22680,"style":302},[279],[28,22682,22684,22688,22691,22694,22697,22700,22703,22750,22790,22793,22796,22799],{"className":22683},[265],[28,22685],{"className":22686,"style":22687},[269],"height:1.0622em;vertical-align:-0.3122em;",[28,22689,20756],{"className":22690},[274,275],[28,22692,1683],{"className":22693},[1723],[28,22695,14860],{"className":22696},[279],[28,22698],{"className":22699,"style":530},[279],[28,22701,43],{"className":22702},[348],[28,22704,22706,22712],{"className":22705},[274],[28,22707,22709],{"className":22708},[274],[28,22710,22588],{"className":22711},[274,1712],[28,22713,22715],{"className":22714},[556],[28,22716,22718,22741],{"className":22717},[560,561],[28,22719,22721,22738],{"className":22720},[565],[28,22722,22724],{"className":22723,"style":22601},[569],[28,22725,22726,22729],{"style":22604},[28,22727],{"className":22728,"style":578},[577],[28,22730,22732],{"className":22731},[582,583,584,585],[28,22733,22735],{"className":22734},[274,585],[28,22736,20832],{"className":22737,"style":20887},[274,20866,585],[28,22739,593],{"className":2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读作“从 ",[28,22881,22883,22896],{"className":22882,"translate":193},[197],[28,22884,22886],{"className":22885},[201],[203,22887,22888],{"xmlns":205},[208,22889,22890,22894],{},[211,22891,22892],{},[214,22893,176],{},[253,22895,176],{"encoding":255},[28,22897,22899],{"className":22898,"ariaHidden":261},[260],[28,22900,22902,22905],{"className":22901},[265],[28,22903],{"className":22904,"style":645},[269],[28,22906,176],{"className":22907},[274,275]," 搬到 ",[28,22910,22912,22925],{"className":22911,"translate":193},[197],[28,22913,22915],{"className":22914},[201],[203,22916,22917],{"xmlns":205},[208,22918,22919,22923],{},[211,22920,22921],{},[214,22922,180],{},[253,22924,180],{"encoding":255},[28,22926,22928],{"className":22927,"ariaHidden":261},[260],[28,22929,22931,22934],{"className":22930},[265],[28,22932],{"className":22933,"style":612},[269],[28,22935,180],{"className":22936,"style":298},[274,275]," 的那部分质量”，于是",[28,22939,22941],{"className":22940,"translate":193},[192],[28,22942,22944,23043],{"className":22943,"translate":193},[197],[28,22945,22947],{"className":22946},[201],[203,22948,22949],{"xmlns":205,"display":206},[208,22950,22951,23040],{},[211,22952,22953,22960,22962,22964,22966,22968,22970,22972,22998,23010,23012,23014,23016,23018,23020,23022,23024,23026,23028,23030,23032,23034,23036,23038],{},[501,22954,22955,22958],{},[214,22956,22957],{},"W",[214,22959,22],{},[217,22961,43],{"stretchy":241},[214,22963,20756],{},[217,22965,1683],{"separator":261},[214,22967,20787],{},[217,22969,1522],{"stretchy":241},[217,22971,83],{},[1771,22973,22974,22980],{},[211,22975,22976,22978],{},[214,22977,21404],{},[217,22979,21407],{},[211,22981,22982,22984,22986,22988,22990,22992,22994,22996],{},[214,22983,22365],{},[217,22985,12054],{},[214,22987,22347],{"mathvariant":484},[217,22989,43],{"stretchy":241},[214,22991,20756],{},[217,22993,1683],{"separator":261},[214,22995,20787],{},[217,22997,1522],{"stretchy":241},[501,22999,23000,23002],{},[217,23001,21282],{},[211,23003,23004,23006,23008],{},[214,23005,20827],{"mathvariant":20826},[217,23007,22376],{},[214,23009,20832],{"mathvariant":20826},[214,23011,22],{},[217,23013,43],{"stretchy":241},[214,23015,176],{},[217,23017,1683],{"separator":261},[214,23019,180],{},[217,23021,1522],{"stretchy":241},[228,23023,488],{},[214,23025,12988],{},[214,23027,22365],{},[217,23029,43],{"stretchy":241},[214,23031,176],{},[217,23033,1683],{"separator":261},[214,23035,180],{},[217,23037,1522],{"stretchy":241},[214,23039,494],{"mathvariant":484},[253,23041,23042],{"encoding":255},"W_c(\\mu,\\nu)=\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int_{\\mathcal X\\times\\mathcal Y}c(x,y)\\,d\\pi(x,y).",[28,23044,23046,23120],{"className":23045,"ariaHidden":261},[260],[28,23047,23049,23052,23093,23096,23099,23102,23105,23108,23111,23114,23117],{"className":23048},[265],[28,23050],{"className":23051,"style":341},[269],[28,23053,23055,23058],{"className":23054},[274],[28,23056,22957],{"className":23057,"style":459},[274,275],[28,23059,23061],{"className":23060},[556],[28,23062,23064,23085],{"className":23063},[560,561],[28,23065,23067,23082],{"className":23066},[565],[28,23068,23071],{"className":23069,"style":23070},[569],"height:0.1514em;",[28,23072,23073,23076],{"style":20948},[28,23074],{"className":23075,"style":578},[577],[28,23077,23079],{"className":23078},[582,583,584,585],[28,23080,22],{"className":23081},[274,275,585],[28,23083,593],{"className":23084},[592],[28,23086,23088],{"className":23087},[565],[28,23089,23091],{"className":23090,"style":600},[569],[28,23092],{},[28,23094,43],{"className":23095},[348],[28,23097,20756],{"className":23098},[274,275],[28,23100,1683],{"className":23101},[1723],[28,23103],{"className":23104,"style":530},[279],[28,23106,20787],{"className":23107,"style":20803},[274,275],[28,23109,1522],{"className":23110},[355],[28,23112],{"className":23113,"style":302},[279],[28,23115,83],{"className":23116},[306],[28,23118],{"className":23119,"style":302},[279],[28,23121,23123,23127,23199,23202,23252,23255,23258,23261,23264,23267,23270,23273,23276,23279,23282,23285,23288,23291,23294,23297,23300,23303],{"className":23122},[265],[28,23124],{"className":23125,"style":23126},[269],"height:2.34em;vertical-align:-0.98em;",[28,23128,23130],{"className":23129},[21329,21485],[28,23131,23133,23190],{"className":23132},[560,561],[28,23134,23136,23187],{"className":23135},[565],[28,23137,23139,23175],{"className":23138,"style":7448},[569],[28,23140,23142,23145],{"style":23141},"top:-2.309em;margin-left:0em;",[28,23143],{"className":23144,"style":1914},[577],[28,23146,23148],{"className":23147},[582,583,584,585],[28,23149,23151,23154,23157,23160,23163,23166,23169,23172],{"className":23150},[274,585],[28,23152,22365],{"className":23153,"style":298},[274,275,585],[28,23155,12054],{"className":23156},[306,585],[28,23158,22347],{"className":23159},[274,585],[28,23161,43],{"className":23162},[348,585],[28,23164,20756],{"className":23165},[274,275,585],[28,23167,1683],{"className":23168},[1723,585],[28,23170,20787],{"className":23171,"style":20803},[274,275,585],[28,23173,1522],{"className":23174},[355,585],[28,23176,23177,23180],{"style":1926},[28,23178],{"className":23179,"style":1914},[577],[28,23181,23182],{},[28,23183,3896,23185],{"className":23184},[21329],[28,23186,1511],{"style":21522},[28,23188,593],{"className":23189},[592],[28,23191,23193],{"className":23192},[565],[28,23194,23197],{"className":23195,"style":23196},[569],"height:0.966em;",[28,23198],{},[28,23200],{"className":23201,"style":530},[279],[28,23203,23205,23208],{"className":23204},[21329],[28,23206,21282],{"className":23207,"style":21548},[21329,21330,21547],[28,23209,23211],{"className":23210},[556],[28,23212,23214,23243],{"className":23213},[560,561],[28,23215,23217,23240],{"className":23216},[565],[28,23218,23220],{"className":23219,"style":21561},[569],[28,23221,23222,23225],{"style":21564},[28,23223],{"className":23224,"style":578},[577],[28,23226,23228],{"className":23227},[582,583,584,585],[28,23229,23231,23234,23237],{"className":23230},[274,585],[28,23232,20827],{"className":23233,"style":20867},[274,20866,585],[28,23235,22376],{"className":23236},[284,585],[28,23238,20832],{"className":23239,"style":20887},[274,20866,585],[28,23241,593],{"className":23242},[592],[28,23244,23246],{"className":23245},[565],[28,23247,23250],{"className":23248,"style":23249},[569],"height:0.98em;",[28,23251],{},[28,23253],{"className":23254,"style":530},[279],[28,23256,22],{"className":23257},[274,275],[28,23259,43],{"className":23260},[348],[28,23262,176],{"className":23263},[274,275],[28,23265,1683],{"className":23266},[1723],[28,23268],{"className":23269,"style":530},[279],[28,23271,180],{"className":23272,"style":298},[274,275],[28,23274,1522],{"className":23275},[355],[28,23277],{"className":23278,"style":530},[279],[28,23280,12988],{"className":23281},[274,275],[28,23283,22365],{"className":23284,"style":298},[274,275],[28,23286,43],{"className":23287},[348],[28,23289,176],{"className":23290},[274,275],[28,23292,1683],{"className":23293},[1723],[28,23295],{"className":23296,"style":530},[279],[28,23298,180],{"className":23299,"style":298},[274,275],[28,23301,1522],{"className":23302},[355],[28,23304,494],{"className":23305},[274],[11,23307,23308,23309,23338],{},"允许分裂之后一定有解，而且原问题变成一个线性规划：",[28,23310,23312,23326],{"className":23311,"translate":193},[197],[28,23313,23315],{"className":23314},[201],[203,23316,23317],{"xmlns":205},[208,23318,23319,23323],{},[211,23320,23321],{},[214,23322,22365],{},[253,23324,23325],{"encoding":255},"\\pi",[28,23327,23329],{"className":23328,"ariaHidden":261},[260],[28,23330,23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就是那张运输表。",[11,23340,15450,23341,23478,23479,23507],{},[28,23342,23344,23386],{"className":23343,"translate":193},[197],[28,23345,23347],{"className":23346},[201],[203,23348,23349],{"xmlns":205},[208,23350,23351,23383],{},[211,23352,23353,23355,23357,23359,23361,23363,23365,23367,23369,23371,23373,23375,23377],{},[214,23354,22],{},[217,23356,43],{"stretchy":241},[214,23358,176],{},[217,23360,1683],{"separator":261},[214,23362,180],{},[217,23364,1522],{"stretchy":241},[217,23366,83],{},[214,23368,12988],{},[217,23370,43],{"stretchy":241},[214,23372,176],{},[217,23374,1683],{"separator":261},[214,23376,180],{},[3054,23378,23379,23381],{},[217,23380,1522],{"stretchy":241},[214,23382,11],{},[253,23384,23385],{"encoding":255},"c(x,y)=d(x,y)^p",[28,23387,23389,23425],{"className":23388,"ariaHidden":261},[260],[28,23390,23392,23395,23398,23401,23404,23407,23410,23413,23416,23419,23422],{"className":23391},[265],[28,23393],{"className":23394,"style":341},[269],[28,23396,22],{"className":23397},[274,275],[28,23399,43],{"className":23400},[348],[28,23402,176],{"className":23403},[274,275],[28,23405,1683],{"className":23406},[1723],[28,23408],{"className":23409,"style":530},[279],[28,23411,180],{"className":23412,"style":298},[274,275],[28,23414,1522],{"className":23415},[355],[28,23417],{"className":23418,"style":302},[279],[28,23420,83],{"className":23421},[306],[28,23423],{"className":23424,"style":302},[279],[28,23426,23428,23431,23434,23437,23440,23443,23446,23449],{"className":23427},[265],[28,23429],{"className":23430,"style":341},[269],[28,23432,12988],{"className":23433},[274,275],[28,23435,43],{"className":23436},[348],[28,23438,176],{"className":23439},[274,275],[28,23441,1683],{"className":23442},[1723],[28,23444],{"className":23445,"style":530},[279],[28,23447,180],{"className":23448,"style":298},[274,275],[28,23450,23452,23455],{"className":23451},[355],[28,23453,1522],{"className":23454},[355],[28,23456,23458],{"className":23457},[556],[28,23459,23461],{"className":23460},[560],[28,23462,23464],{"className":23463},[565],[28,23465,23467],{"className":23466,"style":3073},[569],[28,23468,23469,23472],{"style":3094},[28,23470],{"className":23471,"style":578},[577],[28,23473,23475],{"className":23474},[582,583,584,585],[28,23476,11],{"className":23477},[274,275,585],"（",[28,23480,23482,23495],{"className":23481,"translate":193},[197],[28,23483,23485],{"className":23484},[201],[203,23486,23487],{"xmlns":205},[208,23488,23489,23493],{},[211,23490,23491],{},[214,23492,12988],{},[253,23494,12988],{"encoding":255},[28,23496,23498],{"className":23497,"ariaHidden":261},[260],[28,23499,23501,23504],{"className":23500},[265],[28,23502],{"className":23503,"style":7448},[269],[28,23505,12988],{"className":23506},[274,275]," 是度量），则",[28,23509,23511],{"className":23510,"translate":193},[192],[28,23512,23514,23610],{"className":23513,"translate":193},[197],[28,23515,23517],{"className":23516},[201],[203,23518,23519],{"xmlns":205,"display":206},[208,23520,23521,23607],{},[211,23522,23523,23529,23531,23533,23535,23537,23539,23541],{},[501,23524,23525,23527],{},[214,23526,22957],{},[214,23528,11],{},[217,23530,43],{"stretchy":241},[214,23532,20756],{},[217,23534,1683],{"separator":261},[214,23536,20787],{},[217,23538,1522],{"stretchy":241},[217,23540,83],{},[3054,23542,23543,23599],{},[211,23544,23545,23547,23573,23575,23577,23579,23581,23583,23585,23591,23593,23595,23597],{},[217,23546,43],{"fence":261},[1771,23548,23549,23555],{},[211,23550,23551,23553],{},[214,23552,21404],{},[217,23554,21407],{},[211,23556,23557,23559,23561,23563,23565,23567,23569,23571],{},[214,23558,22365],{},[217,23560,12054],{},[214,23562,22347],{"mathvariant":484},[217,23564,43],{"stretchy":241},[214,23566,20756],{},[217,23568,1683],{"separator":261},[214,23570,20787],{},[217,23572,1522],{"stretchy":241},[217,23574,21282],{},[214,23576,12988],{},[217,23578,43],{"stretchy":241},[214,23580,176],{},[217,23582,1683],{"separator":261},[214,23584,180],{},[3054,23586,23587,23589],{},[217,23588,1522],{"stretchy":241},[214,23590,11],{},[228,23592,488],{},[214,23594,12988],{},[214,23596,22365],{},[217,23598,1522],{"fence":261},[211,23600,23601,23603,23605],{},[224,23602,1244],{},[214,23604,12219],{"mathvariant":484},[214,23606,11],{},[253,23608,23609],{"encoding":255},"W_p(\\mu,\\nu)=\\left(\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int d(x,y)^p\\,d\\pi\\right)^{1\u002Fp}",[28,23611,23613,23686],{"className":23612,"ariaHidden":261},[260],[28,23614,23616,23619,23659,23662,23665,23668,23671,23674,23677,23680,23683],{"className":23615},[265],[28,23617],{"className":23618,"style":21101},[269],[28,23620,23622,23625],{"className":23621},[274],[28,23623,22957],{"className":23624,"style":459},[274,275],[28,23626,23628],{"className":23627},[556],[28,23629,23631,23651],{"className":23630},[560,561],[28,23632,23634,23648],{"className":23633},[565],[28,23635,23637],{"className":23636,"style":23070},[569],[28,23638,23639,23642],{"style":20948},[28,23640],{"className":23641,"style":578},[577],[28,23643,23645],{"className":23644},[582,583,584,585],[28,23646,11],{"className":23647},[274,275,585],[28,23649,593],{"className":23650},[592],[28,23652,23654],{"className":23653},[565],[28,23655,23657],{"className":23656,"style":20967},[569],[28,23658],{},[28,23660,43],{"className":23661},[348],[28,23663,20756],{"className":23664},[274,275],[28,23666,1683],{"className":23667},[1723],[28,23669],{"className":23670,"style":530},[279],[28,23672,20787],{"className":23673,"style":20803},[274,275],[28,23675,1522],{"className":23676},[355],[28,23678],{"className":23679,"style":302},[279],[28,23681,83],{"className":23682},[306],[28,23684],{"className":23685,"style":302},[279],[28,23687,23689,23693],{"className":23688},[265],[28,23690],{"className":23691,"style":23692},[269],"height:2.6939em;vertical-align:-0.966em;",[28,23694,23696,23847],{"className":23695},[1897],[28,23697,23699,23705,23775,23778,23781,23784,23787,23790,23793,23796,23799,23802,23832,23835,23838,23841],{"className":23698},[1897],[28,23700,23702],{"className":23701,"style":8087},[348,8086],[28,23703,43],{"className":23704},[2169,584],[28,23706,23708],{"className":23707},[21329,21485],[28,23709,23711,23767],{"className":23710},[560,561],[28,23712,23714,23764],{"className":23713},[565],[28,23715,23717,23752],{"className":23716,"style":7448},[569],[28,23718,23719,23722],{"style":23141},[28,23720],{"className":23721,"style":1914},[577],[28,23723,23725],{"className":23724},[582,583,584,585],[28,23726,23728,23731,23734,23737,23740,23743,23746,23749],{"className":23727},[274,585],[28,23729,22365],{"className":23730,"style":298},[274,275,585],[28,23732,12054],{"className":23733},[306,585],[28,23735,22347],{"className":23736},[274,585],[28,23738,43],{"className":23739},[348,585],[28,23741,20756],{"className":23742},[274,275,585],[28,23744,1683],{"className":23745},[1723,585],[28,23747,20787],{"className":23748,"style":20803},[274,275,585],[28,23750,1522],{"className":23751},[355,585],[28,23753,23754,23757],{"style":1926},[28,23755],{"className":23756,"style":1914},[577],[28,23758,23759],{},[28,23760,3896,23762],{"className":23761},[21329],[28,23763,1511],{"style":21522},[28,23765,593],{"className":23766},[592],[28,23768,23770],{"className":23769},[565],[28,23771,23773],{"className":23772,"style":23196},[569],[28,23774],{},[28,23776],{"className":23777,"style":530},[279],[28,23779,21282],{"className":23780,"style":21548},[21329,21330,21547],[28,23782],{"className":23783,"style":530},[279],[28,23785,12988],{"className":23786},[274,275],[28,23788,43],{"className":23789},[348],[28,23791,176],{"className":23792},[274,275],[28,23794,1683],{"className":23795},[1723],[28,23797],{"className":23798,"style":530},[279],[28,23800,180],{"className":23801,"style":298},[274,275],[28,23803,23805,23808],{"className":23804},[355],[28,23806,1522],{"className":23807},[355],[28,23809,23811],{"className":23810},[556],[28,23812,23814],{"className":23813},[560],[28,23815,23817],{"className":23816},[565],[28,23818,23821],{"className":23819,"style":23820},[569],"height:0.7144em;",[28,23822,23823,23826],{"style":4880},[28,23824],{"className":23825,"style":578},[577],[28,23827,23829],{"className":23828},[582,583,584,585],[28,23830,11],{"className":23831},[274,275,585],[28,23833],{"className":23834,"style":530},[279],[28,23836,12988],{"className":23837},[274,275],[28,23839,22365],{"className":23840,"style":298},[274,275],[28,23842,23844],{"className":23843,"style":8087},[355,8086],[28,23845,1522],{"className":23846},[2169,584],[28,23848,23850],{"className":23849},[556],[28,23851,23853],{"className":23852},[560],[28,23854,23856],{"className":23855},[565],[28,23857,23860],{"className":23858,"style":23859},[569],"height:1.7279em;",[28,23861,23863,23866],{"style":23862},"top:-3.9029em;margin-right:0.05em;",[28,23864],{"className":23865,"style":578},[577],[28,23867,23869],{"className":23868},[582,583,584,585],[28,23870,23872,23876],{"className":23871},[274,585],[28,23873,23875],{"className":23874},[274,585],"1\u002F",[28,23877,11],{"className":23878},[274,275,585],[11,23880,23881,23882,23968,23969,6904,24143,24373,24374,24549,24550,24578],{},"是 ",[28,23883,23885,23909],{"className":23884,"translate":193},[197],[28,23886,23888],{"className":23887},[201],[203,23889,23890],{"xmlns":205},[208,23891,23892,23906],{},[211,23893,23894,23900,23902,23904],{},[501,23895,23896,23898],{},[214,23897,15825],{"mathvariant":20826},[214,23899,11],{},[217,23901,43],{"stretchy":241},[214,23903,20827],{"mathvariant":20826},[217,23905,1522],{"stretchy":241},[253,23907,23908],{"encoding":255},"\\mathcal P_p(\\mathcal X)",[28,23910,23912],{"className":23911,"ariaHidden":261},[260],[28,23913,23915,23918,23959,23962,23965],{"className":23914},[265],[28,23916],{"className":23917,"style":21101},[269],[28,23919,23921,23924],{"className":23920},[274],[28,23922,15825],{"className":23923,"style":20887},[274,20866],[28,23925,23927],{"className":23926},[556],[28,23928,23930,23951],{"className":23929},[560,561],[28,23931,23933,23948],{"className":23932},[565],[28,23934,23936],{"className":23935,"style":23070},[569],[28,23937,23939,23942],{"style":23938},"top:-2.55em;margin-left:-0.0822em;margin-right:0.05em;",[28,23940],{"className":23941,"style":578},[577],[28,23943,23945],{"className":23944},[582,583,584,585],[28,23946,11],{"className":23947},[274,275,585],[28,23949,593],{"className":23950},[592],[28,23952,23954],{"className":23953},[565],[28,23955,23957],{"className":23956,"style":20967},[569],[28,23958],{},[28,23960,43],{"className":23961},[348],[28,23963,20827],{"className":23964,"style":20867},[274,20866],[28,23966,1522],{"className":23967},[355]," 上的一个真度量。这一点比看起来重要，因为 KL 不是。取 ",[28,23970,23972,24008],{"className":23971,"translate":193},[197],[28,23973,23975],{"className":23974},[201],[203,23976,23977],{"xmlns":205},[208,23978,23979,24005],{},[211,23980,23981,23983,23985,23991,23993,23995,23997,23999],{},[214,23982,20756],{},[217,23984,83],{},[501,23986,23987,23989],{},[214,23988,21832],{},[214,23990,176],{},[217,23992,1683],{"separator":261},[228,23994,14860],{},[214,23996,20787],{},[217,23998,83],{},[501,24000,24001,24003],{},[214,24002,21832],{},[214,24004,180],{},[253,24006,24007],{"encoding":255},"\\mu=\\delta_x,\\ 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305},[560,561],[28,24307,24309,24323],{"className":24308},[565],[28,24310,24312],{"className":24311,"style":23070},[569],[28,24313,24314,24317],{"style":21887},[28,24315],{"className":24316,"style":578},[577],[28,24318,24320],{"className":24319},[582,583,584,585],[28,24321,180],{"className":24322,"style":298},[274,275,585],[28,24324,593],{"className":24325},[592],[28,24327,24329],{"className":24328},[565],[28,24330,24332],{"className":24331,"style":20967},[569],[28,24333],{},[28,24335,1522],{"className":24336},[355],[28,24338],{"className":24339,"style":302},[279],[28,24341,83],{"className":24342},[306],[28,24344],{"className":24345,"style":302},[279],[28,24347,24349,24352,24355,24358,24361,24364,24367,24370],{"className":24348},[265],[28,24350],{"className":24351,"style":341},[269],[28,24353,12988],{"className":24354},[274,275],[28,24356,43],{"className":24357},[348],[28,24359,176],{"className":24360},[274,275],[28,24362,1683],{"className":24363},[1723],[28,24365],{"className":24366,"style":530},[279],[28,24368,180],{"className":24369,"style":298},[274,275],[28,24371,1522],{"className":24372},[355]," 是个有限的数，而 ",[28,24375,24377,24423],{"className":24376,"translate":193},[197],[28,24378,24380],{"className":24379},[201],[203,24381,24382],{"xmlns":205},[208,24383,24384,24420],{},[211,24385,24386,24394,24396,24402,24405,24411,24413,24415,24417],{},[211,24387,24388,24391],{},[214,24389,24390],{"mathvariant":484},"K",[214,24392,24393],{"mathvariant":484},"L",[217,24395,43],{"stretchy":241},[501,24397,24398,24400],{},[214,24399,21832],{},[214,24401,176],{},[214,24403,24404],{"mathvariant":484},"∥",[501,24406,24407,24409],{},[214,24408,21832],{},[214,24410,180],{},[217,24412,1522],{"stretchy":241},[217,24414,83],{},[217,24416,124],{},[214,24418,24419],{"mathvariant":484},"∞",[253,24421,24422],{"encoding":255},"\\mathrm{KL}(\\delta_x\\|\\delta_y)=+\\infty",[28,24424,24426,24537],{"className":24425,"ariaHidden":261},[260],[28,24427,24429,24432,24439,24442,24482,24485,24525,24528,24531,24534],{"className":24428},[265],[28,24430],{"className":24431,"style":21101},[269],[28,24433,24435],{"className":24434},[274],[28,24436,24438],{"className":24437},[274,1712],"KL",[28,24440,43],{"className":24441},[348],[28,24443,24445,24448],{"className":24444},[274],[28,24446,21832],{"className":24447,"style":21872},[274,275],[28,24449,24451],{"className":24450},[556],[28,24452,24454,24474],{"className":24453},[560,561],[28,24455,24457,24471],{"className":24456},[565],[28,24458,24460],{"className":24459,"style":23070},[569],[28,24461,24462,24465],{"style":21887},[28,24463],{"className":24464,"style":578},[577],[28,24466,24468],{"className":24467},[582,583,584,585],[28,24469,176],{"className":24470},[274,275,585],[28,24472,593],{"className":24473},[592],[28,24475,24477],{"className":24476},[565],[28,24478,24480],{"className":24479,"style":600},[569],[28,24481],{},[28,24483,24404],{"className":24484},[274],[28,24486,24488,24491],{"className":24487},[274],[28,24489,21832],{"className":24490,"style":21872},[274,275],[28,24492,24494],{"className":24493},[556],[28,24495,24497,24517],{"className":24496},[560,561],[28,24498,24500,24514],{"className":24499},[565],[28,24501,24503],{"className":24502,"style":23070},[569],[28,24504,24505,24508],{"style":21887},[28,24506],{"className":24507,"style":578},[577],[28,24509,24511],{"className":24510},[582,583,584,585],[28,24512,180],{"className":24513,"style":298},[274,275,585],[28,24515,593],{"className":24516},[592],[28,24518,24520],{"className":24519},[565],[28,24521,24523],{"className":24522,"style":20967},[569],[28,24524],{},[28,24526,1522],{"className":24527},[355],[28,24529],{"className":24530,"style":302},[279],[28,24532,83],{"className":24533},[306],[28,24535],{"className":24536,"style":302},[279],[28,24538,24540,24543,24546],{"className":24539},[265],[28,24541],{"className":24542,"style":270},[269],[28,24544,124],{"className":24545},[274],[28,24547,24419],{"className":24548},[274],"。支撑不交的时候，KL 要么发散要么给出零梯度，",[28,24551,24553,24566],{"className":24552,"translate":193},[197],[28,24554,24556],{"className":24555},[201],[203,24557,24558],{"xmlns":205},[208,24559,24560,24564],{},[211,24561,24562],{},[214,24563,22957],{},[253,24565,22957],{"encoding":255},[28,24567,24569],{"className":24568,"ariaHidden":261},[260],[28,24570,24572,24575],{"className":24571},[265],[28,24573],{"className":24574,"style":455},[269],[28,24576,22957],{"className":24577,"style":459},[274,275]," 却仍然告诉你“隔了多远”。",[11,24580,24581,24582,24610,24611,12376,24641,24711,24712,24740],{},"平移的例子更直白。把 ",[28,24583,24585,24598],{"className":24584,"translate":193},[197],[28,24586,24588],{"className":24587},[201],[203,24589,24590],{"xmlns":205},[208,24591,24592,24596],{},[211,24593,24594],{},[214,24595,20756],{},[253,24597,20759],{"encoding":255},[28,24599,24601],{"className":24600,"ariaHidden":261},[260],[28,24602,24604,24607],{"className":24603},[265],[28,24605],{"className":24606,"style":612},[269],[28,24608,20756],{"className":24609},[274,275]," 整体挪 ",[28,24612,24614,24629],{"className":24613,"translate":193},[197],[28,24615,24617],{"className":24616},[201],[203,24618,24619],{"xmlns":205},[208,24620,24621,24626],{},[211,24622,24623],{},[214,24624,24625],{},"ε",[253,24627,24628],{"encoding":255},"\\varepsilon",[28,24630,24632],{"className":24631,"ariaHidden":261},[260],[28,24633,24635,24638],{"className":24634},[265],[28,24636],{"className":24637,"style":645},[269],[28,24639,24625],{"className":24640},[274,275],[28,24642,24644,24662],{"className":24643,"translate":193},[197],[28,24645,24647],{"className":24646},[201],[203,24648,24649],{"xmlns":205},[208,24650,24651,24659],{},[211,24652,24653],{},[501,24654,24655,24657],{},[214,24656,22957],{},[214,24658,11],{},[253,24660,24661],{"encoding":255},"W_p",[28,24663,24665],{"className":24664,"ariaHidden":261},[260],[28,24666,24668,24671],{"className":24667},[265],[28,24669],{"className":24670,"style":20926},[269],[28,24672,24674,24677],{"className":24673},[274],[28,24675,22957],{"className":24676,"style":459},[274,275],[28,24678,24680],{"className":24679},[556],[28,24681,24683,24703],{"className":24682},[560,561],[28,24684,24686,24700],{"className":24685},[565],[28,24687,24689],{"className":24688,"style":23070},[569],[28,24690,24691,24694],{"style":20948},[28,24692],{"className":24693,"style":578},[577],[28,24695,24697],{"className":24696},[582,583,584,585],[28,24698,11],{"className":24699},[274,275,585],[28,24701,593],{"className":24702},[592],[28,24704,24706],{"className":24705},[565],[28,24707,24709],{"className":24708,"style":20967},[569],[28,24710],{}," 是 ",[28,24713,24715,24728],{"className":24714,"translate":193},[197],[28,24716,24718],{"className":24717},[201],[203,24719,24720],{"xmlns":205},[208,24721,24722,24726],{},[211,24723,24724],{},[214,24725,24625],{},[253,24727,24628],{"encoding":255},[28,24729,24731],{"className":24730,"ariaHidden":261},[260],[28,24732,24734,24737],{"className":24733},[265],[28,24735],{"className":24736,"style":645},[269],[28,24738,24625],{"className":24739},[274,275]," 量级，而",[28,24742,24744],{"className":24743,"translate":193},[192],[28,24745,24747,24810],{"className":24746,"translate":193},[197],[28,24748,24750],{"className":24749},[201],[203,24751,24752],{"xmlns":205,"display":206},[208,24753,24754,24807],{},[211,24755,24756,24762,24764,24767,24769,24771,24773,24775,24777,24779,24781,24783,24785,24787,24789,24791,24793,24795,24805],{},[211,24757,24758,24760],{},[214,24759,24390],{"mathvariant":484},[214,24761,24393],{"mathvariant":484},[217,24763,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,24765,24766],{"mathvariant":20826},"N",[217,24768,43],{"stretchy":241},[224,24770,89],{},[217,24772,1683],{"separator":261},[224,24774,1244],{},[217,24776,1522],{"stretchy":241},[214,24778,24404],{"mathvariant":484},[214,24780,24766],{"mathvariant":20826},[217,24782,43],{"stretchy":241},[214,24784,24625],{},[217,24786,1683],{"separator":261},[224,24788,1244],{},[217,24790,1522],{"stretchy":241},[217,24792,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,24794,83],{},[21928,24796,24797,24803],{},[3054,24798,24799,24801],{},[214,24800,24625],{},[224,24802,1253],{},[224,24804,1253],{},[217,24806,1683],{"separator":261},[253,24808,24809],{"encoding":255},"\\mathrm{KL}\\bigl(\\mathcal N(0,1)\\|\\mathcal N(\\varepsilon,1)\\bigr)=\\frac{\\varepsilon^2}{2},",[28,24811,24813,24892],{"className":24812,"ariaHidden":261},[260],[28,24814,24816,24819,24825,24831,24835,24838,24841,24844,24847,24850,24853,24856,24859,24862,24865,24868,24871,24874,24877,24883,24886,24889],{"className":24815},[265],[28,24817],{"className":24818,"style":2469},[269],[28,24820,24822],{"className":24821},[274],[28,24823,24438],{"className":24824},[274,1712],[28,24826,24828],{"className":24827},[348],[28,24829,43],{"className":24830},[2169,2170],[28,24832,24766],{"className":24833,"style":24834},[274,20866],"margin-right:0.1474em;",[28,24836,43],{"className":24837},[348],[28,24839,89],{"className":24840},[274],[28,24842,1683],{"className":24843},[1723],[28,24845],{"className":24846,"style":530},[279],[28,24848,1244],{"className":24849},[274],[28,24851,1522],{"className":24852},[355],[28,24854,24404],{"className":24855},[274],[28,24857,24766],{"className":24858,"style":24834},[274,20866],[28,24860,43],{"className":24861},[348],[28,24863,24625],{"className":24864},[274,275],[28,24866,1683],{"className":24867},[1723],[28,24869],{"className":24870,"style":530},[279],[28,24872,1244],{"className":24873},[274],[28,24875,1522],{"className":24876},[355],[28,24878,24880],{"className":24879},[355],[28,24881,1522],{"className":24882},[2169,2170],[28,24884],{"className":24885,"style":302},[279],[28,24887,83],{"className":24888},[306],[28,24890],{"className":24891,"style":302},[279],[28,24893,24895,24899,24991],{"className":24894},[265],[28,24896],{"className":24897,"style":24898},[269],"height:2.1771em;vertical-align:-0.686em;",[28,24900,24902,24905,24988],{"className":24901},[274],[28,24903],{"className":24904},[348,8285],[28,24906,24908],{"className":24907},[21928],[28,24909,24911,24979],{"className":24910},[560,561],[28,24912,24914,24976],{"className":24913},[565],[28,24915,24918,24930,24938],{"className":24916,"style":24917},[569],"height:1.4911em;",[28,24919,24921,24924],{"style":24920},"top:-2.314em;",[28,24922],{"className":24923,"style":1914},[577],[28,24925,24927],{"className":24926},[274],[28,24928,1253],{"className":24929},[274],[28,24931,24932,24935],{"style":22023},[28,24933],{"className":24934,"style":1914},[577],[28,24936],{"className":24937,"style":22031},[22030],[28,24939,24941,24944],{"style":24940},"top:-3.677em;",[28,24942],{"className":24943,"style":1914},[577],[28,24945,24947],{"className":24946},[274],[28,24948,24950,24953],{"className":24949},[274],[28,24951,24625],{"className":24952},[274,275],[28,24954,24956],{"className":24955},[556],[28,24957,24959],{"className":24958},[560],[28,24960,24962],{"className":24961},[565],[28,24963,24965],{"className":24964,"style":5477},[569],[28,24966,24967,24970],{"style":3094},[28,24968],{"className":24969,"style":578},[577],[28,24971,24973],{"className":24972},[582,583,584,585],[28,24974,1253],{"className":24975},[274,585],[28,24977,593],{"className":24978},[592],[28,24980,24982],{"className":24981},[565],[28,24983,24986],{"className":24984,"style":24985},[569],"height:0.686em;",[28,24987],{},[28,24989],{"className":24990},[355,8285],[28,24992,1683],{"className":24993},[1723],[11,24995,24996,24997,25025],{},"是二阶的。所以小位移时 KL 的梯度趋近于零，",[28,24998,25000,25013],{"className":24999,"translate":193},[197],[28,25001,25003],{"className":25002},[201],[203,25004,25005],{"xmlns":205},[208,25006,25007,25011],{},[211,25008,25009],{},[214,25010,22957],{},[253,25012,22957],{"encoding":255},[28,25014,25016],{"className":25015,"ariaHidden":261},[260],[28,25017,25019,25022],{"className":25018},[265],[28,25020],{"className":25021,"style":455},[269],[28,25023,22957],{"className":25024,"style":459},[274,275]," 的梯度是常数。生成模型里那句“Wasserstein 提供不饱和梯度”，准确的数学内容就是这一行。",[11,25027,25028,25029,25100,25101,25129],{},"顺带说一句，计算机视觉里管它叫 Earth Mover's Distance，数学里叫 Wasserstein 距离，是同一个东西；EMD 通常特指 ",[28,25030,25032,25050],{"className":25031,"translate":193},[197],[28,25033,25035],{"className":25034},[201],[203,25036,25037],{"xmlns":205},[208,25038,25039,25047],{},[211,25040,25041],{},[501,25042,25043,25045],{},[214,25044,22957],{},[224,25046,1244],{},[253,25048,25049],{"encoding":255},"W_1",[28,25051,25053],{"className":25052,"ariaHidden":261},[260],[28,25054,25056,25060],{"className":25055},[265],[28,25057],{"className":25058,"style":25059},[269],"height:0.8333em;vertical-align:-0.15em;",[28,25061,25063,25066],{"className":25062},[274],[28,25064,22957],{"className":25065,"style":459},[274,275],[28,25067,25069],{"className":25068},[556],[28,25070,25072,25092],{"className":25071},[560,561],[28,25073,25075,25089],{"className":25074},[565],[28,25076,25078],{"className":25077,"style":1285},[569],[28,25079,25080,25083],{"style":20948},[28,25081],{"className":25082,"style":578},[577],[28,25084,25086],{"className":25085},[582,583,584,585],[28,25087,1244],{"className":25088},[274,585],[28,25090,593],{"className":25091},[592],[28,25093,25095],{"className":25094},[565],[28,25096,25098],{"className":25097,"style":600},[569],[28,25099],{},"，或者指带整数质量的离散版本。把一张 28×28 的手写数字的灰度值当成质量，",[28,25102,25104,25117],{"className":25103,"translate":193},[197],[28,25105,25107],{"className":25106},[201],[203,25108,25109],{"xmlns":205},[208,25110,25111,25115],{},[211,25112,25113],{},[214,25114,22957],{},[253,25116,22957],{"encoding":255},[28,25118,25120],{"className":25119,"ariaHidden":261},[260],[28,25121,25123,25126],{"className":25122},[265],[28,25124],{"className":25125,"style":455},[269],[28,25127,22957],{"className":25128,"style":459},[274,275]," 就是“把这张画改写成那张画的最小工作量”，我最初就是靠这个画面理解它的。",[414,25131,25132],{"id":25132},"对偶",[11,25134,25135],{},"回到那道运输题。我算出最低运费是 5 块。如果有人说他能做到 4 块，我怎么证明他做不到？",[11,25137,25138,25139,25167,25168,25196],{},"不去枚举所有运输表。我给每个矿定一个“提货加价” ",[28,25140,25142,25155],{"className":25141,"translate":193},[197],[28,25143,25145],{"className":25144},[201],[203,25146,25147],{"xmlns":205},[208,25148,25149,25153],{},[211,25150,25151],{},[214,25152,3936],{},[253,25154,3936],{"encoding":255},[28,25156,25158],{"className":25157,"ariaHidden":261},[260],[28,25159,25161,25164],{"className":25160},[265],[28,25162],{"className":25163,"style":645},[269],[28,25165,3936],{"className":25166},[274,275],"，每个厂定一个“到货减价” ",[28,25169,25171,25184],{"className":25170,"translate":193},[197],[28,25172,25174],{"className":25173},[201],[203,25175,25176],{"xmlns":205},[208,25177,25178,25182],{},[211,25179,25180],{},[214,25181,1693],{},[253,25183,1693],{"encoding":255},[28,25185,25187],{"className":25186,"ariaHidden":261},[260],[28,25188,25190,25193],{"className":25189},[265],[28,25191],{"className":25192,"style":645},[269],[28,25194,1693],{"className":25195,"style":298},[274,275],"，只要求一条规矩：对任意一条路线，两个价加起来不超过运费，",[28,25198,25200],{"className":25199,"translate":193},[192],[28,25201,25203,25243],{"className":25202,"translate":193},[197],[28,25204,25206],{"className":25205},[201],[203,25207,25208],{"xmlns":205,"display":206},[208,25209,25210,25240],{},[211,25211,25212,25218,25220,25226,25228,25238],{},[501,25213,25214,25216],{},[214,25215,3936],{},[214,25217,1686],{},[217,25219,124],{},[501,25221,25222,25224],{},[214,25223,1693],{},[214,25225,7907],{},[217,25227,1876],{},[501,25229,25230,25232],{},[214,25231,22],{},[211,25233,25234,25236],{},[214,25235,1686],{},[214,25237,7907],{},[214,25239,494],{"mathvariant":484},[253,25241,25242],{"encoding":255},"u_i+v_j\\le c_{ij}.",[28,25244,25246,25302,25358],{"className":25245,"ariaHidden":261},[260],[28,25247,25249,25253,25293,25296,25299],{"className":25248},[265],[28,25250],{"className":25251,"style":25252},[269],"height:0.7333em;vertical-align:-0.15em;",[28,25254,25256,25259],{"className":25255},[274],[28,25257,3936],{"className":25258},[274,275],[28,25260,25262],{"className":25261},[556],[28,25263,25265,25285],{"className":25264},[560,561],[28,25266,25268,25282],{"className":25267},[565],[28,25269,25271],{"className":25270,"style":3881},[569],[28,25272,25273,25276],{"style":573},[28,25274],{"className":25275,"style":578},[577],[28,25277,25279],{"className":25278},[582,583,584,585],[28,25280,1686],{"className":25281},[274,275,585],[28,25283,593],{"className":25284},[592],[28,25286,25288],{"className":25287},[565],[28,25289,25291],{"className":25290,"style":600},[569],[28,25292],{},[28,25294],{"className":25295,"style":280},[279],[28,25297,124],{"className":25298},[284],[28,25300],{"className":25301,"style":280},[279],[28,25303,25305,25309,25349,25352,25355],{"className":25304},[265],[28,25306],{"className":25307,"style":25308},[269],"height:0.9221em;vertical-align:-0.2861em;",[28,25310,25312,25315],{"className":25311},[274],[28,25313,1693],{"className":25314,"style":298},[274,275],[28,25316,25318],{"className":25317},[556],[28,25319,25321,25341],{"className":25320},[560,561],[28,25322,25324,25338],{"className":25323},[565],[28,25325,25327],{"className":25326,"style":3881},[569],[28,25328,25329,25332],{"style":3660},[28,25330],{"className":25331,"style":578},[577],[28,25333,25335],{"className":25334},[582,583,584,585],[28,25336,7907],{"className":25337,"style":8057},[274,275,585],[28,25339,593],{"className":25340},[592],[28,25342,25344],{"className":25343},[565],[28,25345,25347],{"className":25346,"style":20967},[569],[28,25348],{},[28,25350],{"className":25351,"style":302},[279],[28,25353,1876],{"className":25354},[306],[28,25356],{"className":25357,"style":302},[279],[28,25359,25361,25365,25409],{"className":25360},[265],[28,25362],{"className":25363,"style":25364},[269],"height:0.7167em;vertical-align:-0.2861em;",[28,25366,25368,25371],{"className":25367},[274],[28,25369,22],{"className":25370},[274,275],[28,25372,25374],{"className":25373},[556],[28,25375,25377,25401],{"className":25376},[560,561],[28,25378,25380,25398],{"className":25379},[565],[28,25381,25383],{"className":25382,"style":3881},[569],[28,25384,25385,25388],{"style":573},[28,25386],{"className":25387,"style":578},[577],[28,25389,25391],{"className":25390},[582,583,584,585],[28,25392,25394],{"className":25393},[274,585],[28,25395,25397],{"className":25396,"style":8057},[274,275,585],"ij",[28,25399,593],{"className":25400},[592],[28,25402,25404],{"className":25403},[565],[28,25405,25407],{"className":25406,"style":20967},[569],[28,25408],{},[28,25410,494],{"className":25411},[274],[11,25413,25414],{},"那么任何一张运单的总价都满足",[28,25416,25418],{"className":25417,"translate":193},[192],[28,25419,25421,25662],{"className":25420,"translate":193},[197],[28,25422,25424],{"className":25423},[201],[203,25425,25426],{"xmlns":205,"display":206},[208,25427,25428,25659],{},[5015,25429,25430,25517,25601],{"rowspacing":5017,"columnalign":5018,"columnspacing":5019},[5021,25431,25432,25469],{},[5024,25433,25434],{},[5027,25435,25436],{"scriptlevel":89,"displaystyle":261},[211,25437,25438,25449,25459],{},[1771,25439,25440,25443],{},[217,25441,25442],{},"∑",[211,25444,25445,25447],{},[214,25446,1686],{},[214,25448,7907],{},[501,25450,25451,25453],{},[214,25452,22365],{},[211,25454,25455,25457],{},[214,25456,1686],{},[214,25458,7907],{},[501,25460,25461,25463],{},[214,25462,22],{},[211,25464,25465,25467],{},[214,25466,1686],{},[214,25468,7907],{},[5024,25470,25471],{},[5027,25472,25473],{"scriptlevel":89,"displaystyle":261},[211,25474,25475,25477,25479,25489,25499,25501,25507,25509,25515],{},[211,25476],{},[217,25478,9572],{},[1771,25480,25481,25483],{},[217,25482,25442],{},[211,25484,25485,25487],{},[214,25486,1686],{},[214,25488,7907],{},[501,25490,25491,25493],{},[214,25492,22365],{},[211,25494,25495,25497],{},[214,25496,1686],{},[214,25498,7907],{},[217,25500,43],{"stretchy":241},[501,25502,25503,25505],{},[214,25504,3936],{},[214,25506,1686],{},[217,25508,124],{},[501,25510,25511,25513],{},[214,25512,1693],{},[214,25514,7907],{},[217,25516,1522],{"stretchy":241},[5021,25518,25519,25525],{},[5024,25520,25521],{},[5027,25522,25523],{"scriptlevel":89,"displaystyle":261},[211,25524],{},[5024,25526,25527],{},[5027,25528,25529],{"scriptlevel":89,"displaystyle":261},[211,25530,25531,25533,25535,25541,25547,25549,25555,25565,25567,25569,25575,25581,25583,25589,25599],{},[211,25532],{},[217,25534,83],{},[1771,25536,25537,25539],{},[217,25538,25442],{},[214,25540,1686],{},[501,25542,25543,25545],{},[214,25544,3936],{},[214,25546,1686],{},[217,25548,43],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[1771,25550,25551,25553],{},[217,25552,25442],{},[214,25554,7907],{},[501,25556,25557,25559],{},[214,25558,22365],{},[211,25560,25561,25563],{},[214,25562,1686],{},[214,25564,7907],{},[217,25566,1522],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[217,25568,124],{},[1771,25570,25571,25573],{},[217,25572,25442],{},[214,25574,7907],{},[501,25576,25577,25579],{},[214,25578,1693],{},[214,25580,7907],{},[217,25582,43],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[1771,25584,25585,25587],{},[217,25586,25442],{},[214,25588,1686],{},[501,25590,25591,25593],{},[214,25592,22365],{},[211,25594,25595,25597],{},[214,25596,1686],{},[214,25598,7907],{},[217,25600,1522],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[5021,25602,25603,25609],{},[5024,25604,25605],{},[5027,25606,25607],{"scriptlevel":89,"displaystyle":261},[211,25608],{},[5024,25610,25611],{},[5027,25612,25613],{"scriptlevel":89,"displaystyle":261},[211,25614,25615,25617,25619,25625,25631,25637,25639,25645,25651,25657],{},[211,25616],{},[217,25618,83],{},[1771,25620,25621,25623],{},[217,25622,25442],{},[214,25624,1686],{},[501,25626,25627,25629],{},[214,25628,238],{},[214,25630,1686],{},[501,25632,25633,25635],{},[214,25634,3936],{},[214,25636,1686],{},[217,25638,124],{},[1771,25640,25641,25643],{},[217,25642,25442],{},[214,25644,7907],{},[501,25646,25647,25649],{},[214,25648,3820],{},[214,25650,7907],{},[501,25652,25653,25655],{},[214,25654,1693],{},[214,25656,7907],{},[214,25658,494],{"mathvariant":484},[253,25660,25661],{"encoding":255},"\\begin{aligned}\n\\sum_{ij}\\pi_{ij}c_{ij}\n&\\ge\\sum_{ij}\\pi_{ij}(u_i+v_j)\\\\\n&=\\sum_i u_i\\Bigl(\\sum_j\\pi_{ij}\\Bigr)+\\sum_j v_j\\Bigl(\\sum_i\\pi_{ij}\\Bigr)\\\\\n&=\\sum_i a_iu_i+\\sum_j b_jv_j .\n\\end{aligned}",[28,25663,25665],{"className":25664,"ariaHidden":261},[260],[28,25666,25668,25672],{"className":25667},[265],[28,25669],{"className":25670,"style":25671},[269],"height:8.0913em;vertical-align:-3.7957em;",[28,25673,25675],{"className":25674},[274],[28,25676,25678,25872],{"className":25677},[5015],[28,25679,25681],{"className":25680},[5137],[28,25682,25684,25863],{"className":25683},[560,561],[28,25685,25687,25860],{"className":25686},[565],[28,25688,25691,25842,25851],{"className":25689,"style":25690},[569],"height:4.2957em;",[28,25692,25694,25698],{"style":25693},"top:-6.3957em;",[28,25695],{"className":25696,"style":25697},[577],"height:3.15em;",[28,25699,25701,25753,25756,25799],{"className":25700},[274],[28,25702,25704],{"className":25703},[21329,21485],[28,25705,25707,25744],{"className":25706},[560,561],[28,25708,25710,25741],{"className":25709},[565],[28,25711,25714,25730],{"className":25712,"style":25713},[569],"height:1.05em;",[28,25715,25717,25721],{"style":25716},"top:-1.8723em;margin-left:0em;",[28,25718],{"className":25719,"style":25720},[577],"height:3.05em;",[28,25722,25724],{"className":25723},[582,583,584,585],[28,25725,25727],{"className":25726},[274,585],[28,25728,25397],{"className":25729,"style":8057},[274,275,585],[28,25731,25733,25736],{"style":25732},"top:-3.05em;",[28,25734],{"className":25735,"style":25720},[577],[28,25737,25738],{},[28,25739,25442],{"className":25740},[21329,21330,21547],[28,25742,593],{"className":25743},[592],[28,25745,25747],{"className":25746},[565],[28,25748,25751],{"className":25749,"style":25750},[569],"height:1.4138em;",[28,25752],{},[28,25754],{"className":25755,"style":530},[279],[28,25757,25759,25762],{"className":25758},[274],[28,25760,22365],{"className":25761,"style":298},[274,275],[28,25763,25765],{"className":25764},[556],[28,25766,25768,25791],{"className":25767},[560,561],[28,25769,25771,25788],{"className":25770},[565],[28,25772,25774],{"className":25773,"style":3881},[569],[28,25775,25776,25779],{"style":3660},[28,25777],{"className":25778,"style":578},[577],[28,25780,25782],{"className":25781},[582,583,584,585],[28,25783,25785],{"className":25784},[274,585],[28,25786,25397],{"className":25787,"style":8057},[274,275,585],[28,25789,593],{"className":25790},[592],[28,25792,25794],{"className":25793},[565],[28,25795,25797],{"className":25796,"style":20967},[569],[28,25798],{},[28,25800,25802,25805],{"className":25801},[274],[28,25803,22],{"className":25804},[274,275],[28,25806,25808],{"className":25807},[556],[28,25809,25811,25834],{"className":25810},[560,561],[28,25812,25814,25831],{"className":25813},[565],[28,25815,25817],{"className":25816,"style":3881},[569],[28,25818,25819,25822],{"style":573},[28,25820],{"className":25821,"style":578},[577],[28,25823,25825],{"className":25824},[582,583,584,585],[28,25826,25828],{"className":25827},[274,585],[28,25829,25397],{"className":25830,"style":8057},[274,275,585],[28,25832,593],{"className":25833},[592],[28,25835,25837],{"className":25836},[565],[28,25838,25840],{"className":25839,"style":20967},[569],[28,25841],{},[28,25843,25845,25848],{"style":25844},"top:-3.5319em;",[28,25846],{"className":25847,"style":25697},[577],[28,25849],{"className":25850},[274],[28,25852,25854,25857],{"style":25853},"top:-0.7681em;",[28,25855],{"className":25856,"style":25697},[577],[28,25858],{"className":25859},[274],[28,25861,593],{"className":25862},[592],[28,25864,25866],{"className":25865},[565],[28,25867,25870],{"className":25868,"style":25869},[569],"height:3.7957em;",[28,25871],{},[28,25873,25875],{"className":25874},[5205],[28,25876,25878,26789],{"className":25877},[560,561],[28,25879,25881,26786],{"className":25880},[565],[28,25882,25884,26092,26500],{"className":25883,"style":25690},[569],[28,25885,25886,25889],{"style":25693},[28,25887],{"className":25888,"style":25697},[577],[28,25890,25892,25895,25898,25901,25904,25951,25954,25997,26000,26040,26043,26046,26049,26089],{"className":25891},[274],[28,25893],{"className":25894},[274],[28,25896],{"className":25897,"style":302},[279],[28,25899,9572],{"className":25900},[306],[28,25902],{"className":25903,"style":302},[279],[28,25905,25907],{"className":25906},[21329,21485],[28,25908,25910,25943],{"className":25909},[560,561],[28,25911,25913,25940],{"className":25912},[565],[28,25914,25916,25930],{"className":25915,"style":25713},[569],[28,25917,25918,25921],{"style":25716},[28,25919],{"className":25920,"style":25720},[577],[28,25922,25924],{"className":25923},[582,583,584,585],[28,25925,25927],{"className":25926},[274,585],[28,25928,25397],{"className":25929,"style":8057},[274,275,585],[28,25931,25932,25935],{"style":25732},[28,25933],{"className":25934,"style":25720},[577],[28,25936,25937],{},[28,25938,25442],{"className":25939},[21329,21330,21547],[28,25941,593],{"className":25942},[592],[28,25944,25946],{"className":25945},[565],[28,25947,25949],{"className":25948,"style":25750},[569],[28,25950],{},[28,25952],{"className":25953,"style":530},[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},[577],[28,26549,26550],{},[28,26551,25442],{"className":26552},[21329,21330,21547],[28,26554,593],{"className":26555},[592],[28,26557,26559],{"className":26558},[565],[28,26560,26562],{"className":26561,"style":26154},[569],[28,26563],{},[28,26565],{"className":26566,"style":530},[279],[28,26568,26570,26573],{"className":26569},[274],[28,26571,238],{"className":26572},[274,275],[28,26574,26576],{"className":26575},[556],[28,26577,26579,26599],{"className":26578},[560,561],[28,26580,26582,26596],{"className":26581},[565],[28,26583,26585],{"className":26584,"style":3881},[569],[28,26586,26587,26590],{"style":573},[28,26588],{"className":26589,"style":578},[577],[28,26591,26593],{"className":26592},[582,583,584,585],[28,26594,1686],{"className":26595},[274,275,585],[28,26597,593],{"className":26598},[592],[28,26600,26602],{"className":26601},[565],[28,26603,26605],{"className":26604,"style":600},[569],[28,26606],{},[28,26608,26610,26613],{"className":26609},[274],[28,26611,3936],{"className":26612},[274,275],[28,26614,26616],{"className":26615},[556],[28,26617,26619,26639],{"className":26618},[560,561],[28,26620,26622,26636],{"className":26621},[565],[28,26623,26625],{"className":26624,"style":3881},[569],[28,26626,26627,26630],{"style":573},[28,26628],{"className":26629,"style":578},[577],[28,26631,26633],{"className":26632},[582,583,584,585],[28,26634,1686],{"className":26635},[274,275,585],[28,26637,593],{"className":26638},[592],[28,26640,26642],{"className":26641},[565],[28,26643,26645],{"className":26644,"style":600},[569],[28,26646],{},[28,26648],{"className":26649,"style":280},[279],[28,26651,124],{"className":26652},[284],[28,26654],{"className":26655,"style":280},[279],[28,26657,26659],{"className":26658},[21329,21485],[28,26660,26662,26692],{"className":26661},[560,561],[28,26663,26665,26689],{"className":26664},[565],[28,26666,26668,26679],{"className":26667,"style":25713},[569],[28,26669,26670,26673],{"style":25716},[28,26671],{"className":26672,"style":25720},[577],[28,26674,26676],{"className":26675},[582,583,584,585],[28,26677,7907],{"className":26678,"style":8057},[274,275,585],[28,26680,26681,26684],{"style":25732},[28,26682],{"className":26683,"style":25720},[577],[28,26685,26686],{},[28,26687,25442],{"className":26688},[21329,21330,21547],[28,26690,593],{"className":26691},[592],[28,26693,26695],{"className":26694},[565],[28,26696,26698],{"className":26697,"style":25750},[569],[28,26699],{},[28,26701],{"className":26702,"style":530},[279],[28,26704,26706,26709],{"className":26705},[274],[28,26707,3820],{"className":26708},[274,275],[28,26710,26712],{"className":26711},[556],[28,26713,26715,26735],{"className":26714},[560,561],[28,26716,26718,26732],{"className":26717},[565],[28,26719,26721],{"className":26720,"style":3881},[569],[28,26722,26723,26726],{"style":573},[28,26724],{"className":26725,"style":578},[577],[28,26727,26729],{"className":26728},[582,583,584,585],[28,26730,7907],{"className":26731,"style":8057},[274,275,585],[28,26733,593],{"className":26734},[592],[28,26736,26738],{"className":26737},[565],[28,26739,26741],{"className":26740,"style":20967},[569],[28,26742],{},[28,26744,26746,26749],{"className":26745},[274],[28,26747,1693],{"className":26748,"style":298},[274,275],[28,26750,26752],{"className":26751},[556],[28,26753,26755,26775],{"className":26754},[560,561],[28,26756,26758,26772],{"className":26757},[565],[28,26759,26761],{"className":26760,"style":3881},[569],[28,26762,26763,26766],{"style":3660},[28,26764],{"className":26765,"style":578},[577],[28,26767,26769],{"className":26768},[582,583,584,585],[28,26770,7907],{"className":26771,"style":8057},[274,275,585],[28,26773,593],{"className":26774},[592],[28,26776,26778],{"className":26777},[565],[28,26779,26781],{"className":26780,"style":20967},[569],[28,26782],{},[28,26784,494],{"className":26785},[274],[28,26787,593],{"className":26788},[592],[28,26790,26792],{"className":26791},[565],[28,26793,26795],{"className":26794,"style":25869},[569],[28,26796],{},[11,26798,26799,26800,26842,26843,26871,26872,26942,26943,27013,27014,27055,27056,27058],{},"最后这个数只跟 ",[28,26801,26803,26821],{"className":26802,"translate":193},[197],[28,26804,26806],{"className":26805},[201],[203,26807,26808],{"xmlns":205},[208,26809,26810,26818],{},[211,26811,26812,26814,26816],{},[214,26813,3936],{},[217,26815,1683],{"separator":261},[214,26817,1693],{},[253,26819,26820],{"encoding":255},"u,v",[28,26822,26824],{"className":26823,"ariaHidden":261},[260],[28,26825,26827,26830,26833,26836,26839],{"className":26826},[265],[28,26828],{"className":26829,"style":612},[269],[28,26831,3936],{"className":26832},[274,275],[28,26834,1683],{"className":26835},[1723],[28,26837],{"className":26838,"style":530},[279],[28,26840,1693],{"className":26841,"style":298},[274,275]," 有关，跟运单无关；而中间那步把 ",[28,26844,26846,26859],{"className":26845,"translate":193},[197],[28,26847,26849],{"className":26848},[201],[203,26850,26851],{"xmlns":205},[208,26852,26853,26857],{},[211,26854,26855],{},[214,26856,22365],{},[253,26858,23325],{"encoding":255},[28,26860,26862],{"className":26861,"ariaHidden":261},[260],[28,26863,26865,26868],{"className":26864},[265],[28,26866],{"className":26867,"style":645},[269],[28,26869,22365],{"className":26870,"style":298},[274,275]," 的边缘和换成了产量 ",[28,26873,26875,26893],{"className":26874,"translate":193},[197],[28,26876,26878],{"className":26877},[201],[203,26879,26880],{"xmlns":205},[208,26881,26882,26890],{},[211,26883,26884],{},[501,26885,26886,26888],{},[214,26887,238],{},[214,26889,1686],{},[253,26891,26892],{"encoding":255},"a_i",[28,26894,26896],{"className":26895,"ariaHidden":261},[260],[28,26897,26899,26902],{"className":26898},[265],[28,26900],{"className":26901,"style":3431},[269],[28,26903,26905,26908],{"className":26904},[274],[28,26906,238],{"className":26907},[274,275],[28,26909,26911],{"className":26910},[556],[28,26912,26914,26934],{"className":26913},[560,561],[28,26915,26917,26931],{"className":26916},[565],[28,26918,26920],{"className":26919,"style":3881},[569],[28,26921,26922,26925],{"style":573},[28,26923],{"className":26924,"style":578},[577],[28,26926,26928],{"className":26927},[582,583,584,585],[28,26929,1686],{"className":26930},[274,275,585],[28,26932,593],{"className":26933},[592],[28,26935,26937],{"className":26936},[565],[28,26938,26940],{"className":26939,"style":600},[569],[28,26941],{},"、需求 ",[28,26944,26946,26964],{"className":26945,"translate":193},[197],[28,26947,26949],{"className":26948},[201],[203,26950,26951],{"xmlns":205},[208,26952,26953,26961],{},[211,26954,26955],{},[501,26956,26957,26959],{},[214,26958,3820],{},[214,26960,7907],{},[253,26962,26963],{"encoding":255},"b_j",[28,26965,26967],{"className":26966,"ariaHidden":261},[260],[28,26968,26970,26973],{"className":26969},[265],[28,26971],{"className":26972,"style":24102},[269],[28,26974,26976,26979],{"className":26975},[274],[28,26977,3820],{"className":26978},[274,275],[28,26980,26982],{"className":26981},[556],[28,26983,26985,27005],{"className":26984},[560,561],[28,26986,26988,27002],{"className":26987},[565],[28,26989,26991],{"className":26990,"style":3881},[569],[28,26992,26993,26996],{"style":573},[28,26994],{"className":26995,"style":578},[577],[28,26997,26999],{"className":26998},[582,583,584,585],[28,27000,7907],{"className":27001,"style":8057},[274,275,585],[28,27003,593],{"className":27004},[592],[28,27006,27008],{"className":27007},[565],[28,27009,27011],{"className":27010,"style":20967},[569],[28,27012],{},"。所以只要我找到一组 ",[28,27015,27017,27034],{"className":27016,"translate":193},[197],[28,27018,27020],{"className":27019},[201],[203,27021,27022],{"xmlns":205},[208,27023,27024,27032],{},[211,27025,27026,27028,27030],{},[214,27027,3936],{},[217,27029,1683],{"separator":261},[214,27031,1693],{},[253,27033,26820],{"encoding":255},[28,27035,27037],{"className":27036,"ariaHidden":261},[260],[28,27038,27040,27043,27046,27049,27052],{"className":27039},[265],[28,27041],{"className":27042,"style":612},[269],[28,27044,3936],{"className":27045},[274,275],[28,27047,1683],{"className":27048},[1723],[28,27050],{"className":27051,"style":530},[279],[28,27053,1693],{"className":27054,"style":298},[274,275],"，使这个和等于 5，我就",[183,27057,9735],{},"了运费不可能低于 5 块。",[11,27060,27061,27062,6904,27141,27215,27216,27311,27312,27407,27408,27508,27509,27609,27610,27830],{},"试一试：取 ",[28,27063,27065,27093],{"className":27064,"translate":193},[197],[28,27066,27068],{"className":27067},[201],[203,27069,27070],{"xmlns":205},[208,27071,27072,27090],{},[211,27073,27074,27076,27078,27080,27082,27084,27086,27088],{},[214,27075,3936],{},[217,27077,83],{},[217,27079,43],{"stretchy":241},[224,27081,89],{},[217,27083,1683],{"separator":261},[217,27085,5074],{},[224,27087,1244],{},[217,27089,1522],{"stretchy":241},[253,27091,27092],{"encoding":255},"u=(0,-1)",[28,27094,27096,27114],{"className":27095,"ariaHidden":261},[260],[28,27097,27099,27102,27105,27108,27111],{"className":27098},[265],[28,27100],{"className":27101,"style":645},[269],[28,27103,3936],{"className":27104},[274,275],[28,27106],{"className":27107,"style":302},[279],[28,27109,83],{"className":27110},[306],[28,27112],{"className":27113,"style":302},[279],[28,27115,27117,27120,27123,27126,27129,27132,27135,27138],{"className":27116},[265],[28,27118],{"className":27119,"style":341},[269],[28,27121,43],{"className":27122},[348],[28,27124,89],{"className":27125},[274],[28,27127,1683],{"className":27128},[1723],[28,27130],{"className":27131,"style":530},[279],[28,27133,5074],{"className":27134},[274],[28,27136,1244],{"className":27137},[274],[28,27139,1522],{"className":27140},[355],[28,27142,27144,27170],{"className":27143,"translate":193},[197],[28,27145,27147],{"className":27146},[201],[203,27148,27149],{"xmlns":205},[208,27150,27151,27167],{},[211,27152,27153,27155,27157,27159,27161,27163,27165],{},[214,27154,1693],{},[217,27156,83],{},[217,27158,43],{"stretchy":241},[224,27160,1244],{},[217,27162,1683],{"separator":261},[224,27164,1253],{},[217,27166,1522],{"stretchy":241},[253,27168,27169],{"encoding":255},"v=(1,2)",[28,27171,27173,27191],{"className":27172,"ariaHidden":261},[260],[28,27174,27176,27179,27182,27185,27188],{"className":27175},[265],[28,27177],{"className":27178,"style":645},[269],[28,27180,1693],{"className":27181,"style":298},[274,275],[28,27183],{"className":27184,"style":302},[279],[28,27186,83],{"className":27187},[306],[28,27189],{"className":27190,"style":302},[279],[28,27192,27194,27197,27200,27203,27206,27209,27212],{"className":27193},[265],[28,27195],{"className":27196,"style":341},[269],[28,27198,43],{"className":27199},[348],[28,27201,1244],{"className":27202},[274],[28,27204,1683],{"className":27205},[1723],[28,27207],{"className":27208,"style":530},[279],[28,27210,1253],{"className":27211},[274],[28,27213,1522],{"className":27214},[355],"。四条路线逐一检查：甲到 A 是 ",[28,27217,27219,27245],{"className":27218,"translate":193},[197],[28,27220,27222],{"className":27221},[201],[203,27223,27224],{"xmlns":205},[208,27225,27226,27242],{},[211,27227,27228,27230,27232,27234,27236,27238,27240],{},[224,27229,89],{},[217,27231,124],{},[224,27233,1244],{},[217,27235,83],{},[224,27237,1244],{},[217,27239,1876],{},[224,27241,1244],{},[253,27243,27244],{"encoding":255},"0+1=1\\le 1",[28,27246,27248,27266,27284,27302],{"className":27247,"ariaHidden":261},[260],[28,27249,27251,27254,27257,27260,27263],{"className":27250},[265],[28,27252],{"className":27253,"style":6853},[269],[28,27255,89],{"className":27256},[274],[28,27258],{"className":27259,"style":280},[279],[28,27261,124],{"className":27262},[284],[28,27264],{"className":27265,"style":280},[279],[28,27267,27269,27272,27275,27278,27281],{"className":27268},[265],[28,27270],{"className":27271,"style":316},[269],[28,27273,1244],{"className":27274},[274],[28,27276],{"className":27277,"style":302},[279],[28,27279,83],{"className":27280},[306],[28,27282],{"className":27283,"style":302},[279],[28,27285,27287,27290,27293,27296,27299],{"className":27286},[265],[28,27288],{"className":27289,"style":17967},[269],[28,27291,1244],{"className":27292},[274],[28,27294],{"className":27295,"style":302},[279],[28,27297,1876],{"className":27298},[306],[28,27300],{"className":27301,"style":302},[279],[28,27303,27305,27308],{"className":27304},[265],[28,27306],{"className":27307,"style":316},[269],[28,27309,1244],{"className":27310},[274],"，甲到 B 是 ",[28,27313,27315,27341],{"className":27314,"translate":193},[197],[28,27316,27318],{"className":27317},[201],[203,27319,27320],{"xmlns":205},[208,27321,27322,27338],{},[211,27323,27324,27326,27328,27330,27332,27334,27336],{},[224,27325,89],{},[217,27327,124],{},[224,27329,1253],{},[217,27331,83],{},[224,27333,1253],{},[217,27335,1876],{},[224,27337,1253],{},[253,27339,27340],{"encoding":255},"0+2=2\\le 2",[28,27342,27344,27362,27380,27398],{"className":27343,"ariaHidden":261},[260],[28,27345,27347,27350,27353,27356,27359],{"className":27346},[265],[28,27348],{"className":27349,"style":6853},[269],[28,27351,89],{"className":27352},[274],[28,27354],{"className":27355,"style":280},[279],[28,27357,124],{"className":27358},[284],[28,27360],{"className":27361,"style":280},[279],[28,27363,27365,27368,27371,27374,27377],{"className":27364},[265],[28,27366],{"className":27367,"style":316},[269],[28,27369,1253],{"className":27370},[274],[28,27372],{"className":27373,"style":302},[279],[28,27375,83],{"className":27376},[306],[28,27378],{"className":27379,"style":302},[279],[28,27381,27383,27386,27389,27392,27395],{"className":27382},[265],[28,27384],{"className":27385,"style":17967},[269],[28,27387,1253],{"className":27388},[274],[28,27390],{"className":27391,"style":302},[279],[28,27393,1876],{"className":27394},[306],[28,27396],{"className":27397,"style":302},[279],[28,27399,27401,27404],{"className":27400},[265],[28,27402],{"className":27403,"style":316},[269],[28,27405,1253],{"className":27406},[274],"，乙到 A 是 ",[28,27409,27411,27439],{"className":27410,"translate":193},[197],[28,27412,27414],{"className":27413},[201],[203,27415,27416],{"xmlns":205},[208,27417,27418,27436],{},[211,27419,27420,27422,27424,27426,27428,27430,27432,27434],{},[217,27421,5074],{},[224,27423,1244],{},[217,27425,124],{},[224,27427,1244],{},[217,27429,83],{},[224,27431,89],{},[217,27433,1876],{},[224,27435,3400],{},[253,27437,27438],{"encoding":255},"-1+1=0\\le 3",[28,27440,27442,27463,27481,27499],{"className":27441,"ariaHidden":261},[260],[28,27443,27445,27448,27451,27454,27457,27460],{"className":27444},[265],[28,27446],{"className":27447,"style":6853},[269],[28,27449,5074],{"className":27450},[274],[28,27452,1244],{"className":27453},[274],[28,27455],{"className":27456,"style":280},[279],[28,27458,124],{"className":27459},[284],[28,27461],{"className":27462,"style":280},[279],[28,27464,27466,27469,27472,27475,27478],{"className":27465},[265],[28,27467],{"className":27468,"style":316},[269],[28,27470,1244],{"className":27471},[274],[28,27473],{"className":27474,"style":302},[279],[28,27476,83],{"className":27477},[306],[28,27479],{"className":27480,"style":302},[279],[28,27482,27484,27487,27490,27493,27496],{"className":27483},[265],[28,27485],{"className":27486,"style":17967},[269],[28,27488,89],{"className":27489},[274],[28,27491],{"className":27492,"style":302},[279],[28,27494,1876],{"className":27495},[306],[28,27497],{"className":27498,"style":302},[279],[28,27500,27502,27505],{"className":27501},[265],[28,27503],{"className":27504,"style":316},[269],[28,27506,3400],{"className":27507},[274],"，乙到 B 是 ",[28,27510,27512,27540],{"className":27511,"translate":193},[197],[28,27513,27515],{"className":27514},[201],[203,27516,27517],{"xmlns":205},[208,27518,27519,27537],{},[211,27520,27521,27523,27525,27527,27529,27531,27533,27535],{},[217,27522,5074],{},[224,27524,1244],{},[217,27526,124],{},[224,27528,1253],{},[217,27530,83],{},[224,27532,1244],{},[217,27534,1876],{},[224,27536,1244],{},[253,27538,27539],{"encoding":255},"-1+2=1\\le 1",[28,27541,27543,27564,27582,27600],{"className":27542,"ariaHidden":261},[260],[28,27544,27546,27549,27552,27555,27558,27561],{"className":27545},[265],[28,27547],{"className":27548,"style":6853},[269],[28,27550,5074],{"className":27551},[274],[28,27553,1244],{"className":27554},[274],[28,27556],{"className":27557,"style":280},[279],[28,27559,124],{"className":27560},[284],[28,27562],{"className":27563,"style":280},[279],[28,27565,27567,27570,27573,27576,27579],{"className":27566},[265],[28,27568],{"className":27569,"style":316},[269],[28,27571,1253],{"className":27572},[274],[28,27574],{"className":27575,"style":302},[279],[28,27577,83],{"className":27578},[306],[28,27580],{"className":27581,"style":302},[279],[28,27583,27585,27588,27591,27594,27597],{"className":27584},[265],[28,27586],{"className":27587,"style":17967},[269],[28,27589,1244],{"className":27590},[274],[28,27592],{"className":27593,"style":302},[279],[28,27595,1876],{"className":27596},[306],[28,27598],{"className":27599,"style":302},[279],[28,27601,27603,27606],{"className":27602},[265],[28,27604],{"className":27605,"style":316},[269],[28,27607,1244],{"className":27608},[274],"，全部成立。而 ",[28,27611,27613,27665],{"className":27612,"translate":193},[197],[28,27614,27616],{"className":27615},[201],[203,27617,27618],{"xmlns":205},[208,27619,27620,27662],{},[211,27621,27622,27624,27626,27628,27630,27632,27634,27636,27638,27640,27642,27644,27646,27648,27650,27652,27654,27656,27658,27660],{},[224,27623,3400],{},[217,27625,20336],{},[224,27627,89],{},[217,27629,124],{},[224,27631,1244],{},[217,27633,20336],{},[217,27635,43],{"stretchy":241},[217,27637,5074],{},[224,27639,1244],{},[217,27641,1522],{"stretchy":241},[217,27643,124],{},[224,27645,1253],{},[217,27647,20336],{},[224,27649,1244],{},[217,27651,124],{},[224,27653,1253],{},[217,27655,20336],{},[224,27657,1253],{},[217,27659,83],{},[224,27661,251],{},[253,27663,27664],{"encoding":255},"3\\cdot0+1\\cdot(-1)+2\\cdot1+2\\cdot2=5",[28,27666,27668,27686,27704,27722,27749,27767,27785,27803,27821],{"className":27667,"ariaHidden":261},[260],[28,27669,27671,27674,27677,27680,27683],{"className":27670},[265],[28,27672],{"className":27673,"style":316},[269],[28,27675,3400],{"className":27676},[274],[28,27678],{"className":27679,"style":280},[279],[28,27681,20336],{"className":27682},[284],[28,27684],{"className":27685,"style":280},[279],[28,27687,27689,27692,27695,27698,27701],{"className":27688},[265],[28,27690],{"className":27691,"style":6853},[269],[28,27693,89],{"className":27694},[274],[28,27696],{"className":27697,"style":280},[279],[28,27699,124],{"className":27700},[284],[28,27702],{"className":27703,"style":280},[279],[28,27705,27707,27710,27713,27716,27719],{"className":27706},[265],[28,27708],{"className":27709,"style":316},[269],[28,27711,1244],{"className":27712},[274],[28,27714],{"className":27715,"style":280},[279],[28,27717,20336],{"className":27718},[284],[28,27720],{"className":27721,"style":280},[279],[28,27723,27725,27728,27731,27734,27737,27740,27743,27746],{"className":27724},[265],[28,27726],{"className":27727,"style":341},[269],[28,27729,43],{"className":27730},[348],[28,27732,5074],{"className":27733},[274],[28,27735,1244],{"className":27736},[274],[28,27738,1522],{"className":27739},[355],[28,27741],{"className":27742,"style":280},[279],[28,27744,124],{"className":27745},[284],[28,27747],{"className":27748,"style":280},[279],[28,27750,27752,27755,27758,27761,27764],{"className":27751},[265],[28,27753],{"className":27754,"style":316},[269],[28,27756,1253],{"className":27757},[274],[28,27759],{"className":27760,"style":280},[279],[28,27762,20336],{"className":27763},[284],[28,27765],{"className":27766,"style":280},[279],[28,27768,27770,27773,27776,27779,27782],{"className":27769},[265],[28,27771],{"className":27772,"style":6853},[269],[28,27774,1244],{"className":27775},[274],[28,27777],{"className":27778,"style":280},[279],[28,27780,124],{"className":27781},[284],[28,27783],{"className":27784,"style":280},[279],[28,27786,27788,27791,27794,27797,27800],{"className":27787},[265],[28,27789],{"className":27790,"style":316},[269],[28,27792,1253],{"className":27793},[274],[28,27795],{"className":27796,"style":280},[279],[28,27798,20336],{"className":27799},[284],[28,27801],{"className":27802,"style":280},[279],[28,27804,27806,27809,27812,27815,27818],{"className":27805},[265],[28,27807],{"className":27808,"style":316},[269],[28,27810,1253],{"className":27811},[274],[28,27813],{"className":27814,"style":302},[279],[28,27816,83],{"className":27817},[306],[28,27819],{"className":27820,"style":302},[279],[28,27822,27824,27827],{"className":27823},[265],[28,27825],{"className":27826,"style":316},[269],[28,27828,251],{"className":27829},[274],"。证书找到了。",[11,27832,27833],{},"这就是对偶的思想：不去找运单，而是找一张“价格表”，让价格表卡住运费的下界。最优的价格表能卡到最紧，那个值就等于最低运费。离散情形下，这件事可以写成拉格朗日形式，把两组等式约束配乘子：",[28,27835,27837],{"className":27836,"translate":193},[192],[28,27838,27840,28111],{"className":27839,"translate":193},[197],[28,27841,27843],{"className":27842},[201],[203,27844,27845],{"xmlns":205,"display":206},[208,27846,27847,28108],{},[5015,27848,27849,27998],{"rowspacing":5017,"columnalign":5018,"columnspacing":5019},[5021,27850,27851,27874],{},[5024,27852,27853],{},[5027,27854,27855],{"scriptlevel":89,"displaystyle":261},[211,27856,27857,27859,27861,27863,27865,27867,27869,27872],{},[214,27858,24393],{},[217,27860,43],{"stretchy":241},[214,27862,22365],{},[217,27864,1683],{"separator":261},[214,27866,510],{},[217,27868,1683],{"separator":261},[214,27870,27871],{},"ψ",[217,27873,1522],{"stretchy":241},[5024,27875,27876],{},[5027,27877,27878],{"scriptlevel":89,"displaystyle":261},[211,27879,27880,27882,27884,27894,27904,27914,27916,27922,27928,27930,27936,27938,27944,27954,27956,27958,27964,27970,27972,27978,27980,27986,27996],{},[211,27881],{},[217,27883,83],{},[1771,27885,27886,27888],{},[217,27887,25442],{},[211,27889,27890,27892],{},[214,27891,1686],{},[214,27893,7907],{},[501,27895,27896,27898],{},[214,27897,22365],{},[211,27899,27900,27902],{},[214,27901,1686],{},[214,27903,7907],{},[501,27905,27906,27908],{},[214,27907,15828],{},[211,27909,27910,27912],{},[214,27911,1686],{},[214,27913,7907],{},[217,27915,124],{},[1771,27917,27918,27920],{},[217,27919,25442],{},[214,27921,1686],{},[501,27923,27924,27926],{},[214,27925,510],{},[214,27927,1686],{},[217,27929,43],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[501,27931,27932,27934],{},[214,27933,238],{},[214,27935,1686],{},[217,27937,5074],{},[1771,27939,27940,27942],{},[217,27941,25442],{},[214,27943,7907],{},[501,27945,27946,27948],{},[214,27947,22365],{},[211,27949,27950,27952],{},[214,27951,1686],{},[214,27953,7907],{},[217,27955,1522],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[217,27957,124],{},[1771,27959,27960,27962],{},[217,27961,25442],{},[214,27963,7907],{},[501,27965,27966,27968],{},[214,27967,27871],{},[214,27969,7907],{},[217,27971,43],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[501,27973,27974,27976],{},[214,27975,3820],{},[214,27977,7907],{},[217,27979,5074],{},[1771,27981,27982,27984],{},[217,27983,25442],{},[214,27985,1686],{},[501,27987,27988,27990],{},[214,27989,22365],{},[211,27991,27992,27994],{},[214,27993,1686],{},[214,27995,7907],{},[217,27997,1522],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[5021,27999,28000,28006],{},[5024,28001,28002],{},[5027,28003,28004],{"scriptlevel":89,"displaystyle":261},[211,28005],{},[5024,28007,28008],{},[5027,28009,28010],{"scriptlevel":89,"displaystyle":261},[211,28011,28012,28014,28016,28022,28028,28034,28036,28042,28048,28054,28056,28066,28076,28078,28088,28090,28096,28098,28104,28106],{},[211,28013],{},[217,28015,83],{},[1771,28017,28018,28020],{},[217,28019,25442],{},[214,28021,1686],{},[501,28023,28024,28026],{},[214,28025,238],{},[214,28027,1686],{},[501,28029,28030,28032],{},[214,28031,510],{},[214,28033,1686],{},[217,28035,124],{},[1771,28037,28038,28040],{},[217,28039,25442],{},[214,28041,7907],{},[501,28043,28044,28046],{},[214,28045,3820],{},[214,28047,7907],{},[501,28049,28050,28052],{},[214,28051,27871],{},[214,28053,7907],{},[217,28055,124],{},[1771,28057,28058,28060],{},[217,28059,25442],{},[211,28061,28062,28064],{},[214,28063,1686],{},[214,28065,7907],{},[501,28067,28068,28070],{},[214,28069,22365],{},[211,28071,28072,28074],{},[214,28073,1686],{},[214,28075,7907],{},[217,28077,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[501,28079,28080,28082],{},[214,28081,15828],{},[211,28083,28084,28086],{},[214,28085,1686],{},[214,28087,7907],{},[217,28089,5074],{},[501,28091,28092,28094],{},[214,28093,510],{},[214,28095,1686],{},[217,28097,5074],{},[501,28099,28100,28102],{},[214,28101,27871],{},[214,28103,7907],{},[217,28105,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,28107,494],{"mathvariant":484},[253,28109,28110],{"encoding":255},"\\begin{aligned}\nL(\\pi,\\varphi,\\psi)\n&=\\sum_{ij}\\pi_{ij}C_{ij}+\\sum_i\\varphi_i\\Bigl(a_i-\\sum_j\\pi_{ij}\\Bigr)+\\sum_j\\psi_j\\Bigl(b_j-\\sum_i\\pi_{ij}\\Bigr)\\\\\n&=\\sum_i a_i\\varphi_i+\\sum_j b_j\\psi_j+\\sum_{ij}\\pi_{ij}\\bigl(C_{ij}-\\varphi_i-\\psi_j\\bigr).\n\\end{aligned}",[28,28112,28114],{"className":28113,"ariaHidden":261},[260],[28,28115,28117,28121],{"className":28116},[265],[28,28118],{"className":28119,"style":28120},[269],"height:5.3276em;vertical-align:-2.4138em;",[28,28122,28124],{"className":28123},[274],[28,28125,28127,28200],{"className":28126},[5015],[28,28128,28130],{"className":28129},[5137],[28,28131,28133,28191],{"className":28132},[560,561],[28,28134,28136,28188],{"className":28135},[565],[28,28137,28140,28179],{"className":28138,"style":28139},[569],"height:2.9138em;",[28,28141,28143,28146],{"style":28142},"top:-4.9138em;",[28,28144],{"className":28145,"style":25697},[577],[28,28147,28149,28152,28155,28158,28161,28164,28167,28170,28173,28176],{"className":28148},[274],[28,28150,24393],{"className":28151},[274,275],[28,28153,43],{"className":28154},[348],[28,28156,22365],{"className":28157,"style":298},[274,275],[28,28159,1683],{"className":28160},[1723],[28,28162],{"className":28163,"style":530},[279],[28,28165,510],{"className":28166},[274,275],[28,28168,1683],{"className":28169},[1723],[28,28171],{"className":28172,"style":530},[279],[28,28174,27871],{"className":28175,"style":298},[274,275],[28,28177,1522],{"className":28178},[355],[28,28180,28182,28185],{"style":28181},"top:-2.15em;",[28,28183],{"className":28184,"style":25697},[577],[28,28186],{"className":28187},[274],[28,28189,593],{"className":28190},[592],[28,28192,28194],{"className":28193},[565],[28,28195,28198],{"className":28196,"style":28197},[569],"height:2.4138em;",[28,28199],{},[28,28201,28203],{"className":28202},[5205],[28,28204,28206,29408],{"className":28205},[560,561],[28,28207,28209,29405],{"className":28208},[565],[28,28210,28212,28864],{"className":28211,"style":28139},[569],[28,28213,28214,28217],{"style":28142},[28,28215],{"className":28216,"style":25697},[577],[28,28218,28220,28223,28226,28229,28232,28279,28282,28325,28370,28373,28376,28379,28423,28426,28466,28472,28512,28515,28518,28521,28565,28568,28611,28617,28620,28623,28626,28670,28673,28713,28719,28759,28762,28765,28768,28812,28815,28858],{"className":28219},[274],[28,28221],{"className":28222},[274],[28,28224],{"className":28225,"style":302},[279],[28,28227,83],{"className":28228},[306],[28,28230],{"className":28231,"style":302},[279],[28,28233,28235],{"className":28234},[21329,21485],[28,28236,28238,28271],{"className":28237},[560,561],[28,28239,28241,28268],{"className":28240},[565],[28,28242,28244,28258],{"className":28243,"style":25713},[569],[28,28245,28246,28249],{"style":25716},[28,28247],{"className":28248,"style":25720},[577],[28,28250,28252],{"className":28251},[582,583,584,585],[28,28253,28255],{"className":28254},[274,585],[28,28256,25397],{"className":28257,"style":8057},[274,275,585],[28,28259,28260,28263],{"style":25732},[28,28261],{"className":28262,"style":25720},[577],[28,28264,28265],{},[28,28266,25442],{"className":28267},[21329,21330,21547],[28,28269,593],{"className":28270},[592],[28,28272,28274],{"className":28273},[565],[28,28275,28277],{"className":28276,"style":25750},[569],[28,28278],{},[28,28280],{"className":28281,"style":530},[279],[28,28283,28285,28288],{"className":28284},[274],[28,28286,22365],{"className":28287,"style":298},[274,275],[28,28289,28291],{"className":28290},[556],[28,28292,28294,28317],{"className":28293},[560,561],[28,28295,28297,28314],{"className":28296},[565],[28,28298,28300],{"className":28299,"style":3881},[569],[28,28301,28302,28305],{"style":3660},[28,28303],{"className":28304,"style":578},[577],[28,28306,28308],{"className":28307},[582,583,584,585],[28,28309,28311],{"className":28310},[274,585],[28,28312,25397],{"className":28313,"style":8057},[274,275,585],[28,28315,593],{"className":28316},[592],[28,28318,28320],{"className":28319},[565],[28,28321,28323],{"className":28322,"style":20967},[569],[28,28324],{},[28,28326,28328,28332],{"className":28327},[274],[28,28329,15828],{"className":28330,"style":28331},[274,275],"margin-right:0.0715em;",[28,28333,28335],{"className":28334},[556],[28,28336,28338,28362],{"className":28337},[560,561],[28,28339,28341,28359],{"className":28340},[565],[28,28342,28344],{"className":28343,"style":3881},[569],[28,28345,28347,28350],{"style":28346},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[28,28348],{"className":28349,"style":578},[577],[28,28351,28353],{"className":28352},[582,583,584,585],[28,28354,28356],{"className":28355},[274,585],[28,28357,25397],{"className":28358,"style":8057},[274,275,585],[28,28360,593],{"className":28361},[592],[28,28363,28365],{"className":28364},[565],[28,28366,28368],{"className":28367,"style":20967},[569],[28,28369],{},[28,28371],{"className":28372,"style":280},[279],[28,28374,124],{"className":28375},[284],[28,28377],{"className":28378,"style":280},[279],[28,28380,28382],{"className":28381},[21329,21485],[28,28383,28385,28415],{"className":28384},[560,561],[28,28386,28388,28412],{"className":28387},[565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",[28,29525,29527,29569],{"className":29526,"translate":193},[197],[28,29528,29530],{"className":29529},[201],[203,29531,29532],{"xmlns":205},[208,29533,29534,29566],{},[211,29535,29536,29546,29548,29554,29556,29562,29564],{},[501,29537,29538,29540],{},[214,29539,15828],{},[211,29541,29542,29544],{},[214,29543,1686],{},[214,29545,7907],{},[217,29547,5074],{},[501,29549,29550,29552],{},[214,29551,510],{},[214,29553,1686],{},[217,29555,5074],{},[501,29557,29558,29560],{},[214,29559,27871],{},[214,29561,7907],{},[217,29563,1383],{},[224,29565,89],{},[253,29567,29568],{"encoding":255},"C_{ij}-\\varphi_i-\\psi_j\u003C0",[28,29570,29572,29630,29685,29740],{"className":29571,"ariaHidden":261},[260],[28,29573,29575,29578,29621,29624,29627],{"className":29574},[265],[28,29576],{"className":29577,"style":20926},[269],[28,29579,29581,29584],{"className":29580},[274],[28,29582,15828],{"className":29583,"style":28331},[274,275],[28,29585,29587],{"className":29586},[556],[28,29588,29590,29613],{"className":29589},[560,561],[28,29591,29593,29610],{"className":29592},[565],[28,29594,29596],{"className":29595,"style":3881},[569],[28,29597,29598,29601],{"style":28346},[28,29599],{"className":29600,"style":578},[577],[28,29602,29604],{"className":29603},[582,583,584,585],[28,29605,29607],{"className":29606},[274,585],[28,29608,25397],{"className":29609,"style":8057},[274,275,585],[28,29611,593],{"className":29612},[592],[28,29614,29616],{"className":29615},[565],[28,29617,29619],{"className":29618,"style":20967},[569],[28,29620],{},[28,29622],{"className":29623,"style":280},[279],[28,29625,5074],{"className":29626},[284],[28,29628],{"className":29629,"style":280},[279],[28,29631,29633,29636,29676,29679,29682],{"className":29632},[265],[28,29634],{"className":29635,"style":4652},[269],[28,29637,29639,29642],{"className":29638},[274],[28,29640,510],{"className":29641},[274,275],[28,29643,29645],{"className":29644},[556],[28,29646,29648,29668],{"className":29647},[560,561],[28,29649,29651,29665],{"className":29650},[565],[28,29652,29654],{"className":29653,"style":3881},[569],[28,29655,29656,29659],{"style":573},[28,29657],{"className":29658,"style":578},[577],[28,29660,29662],{"className":29661},[582,583,584,585],[28,29663,1686],{"className":29664},[274,275,585],[28,29666,593],{"className":29667},[592],[28,29669,29671],{"className":29670},[565],[28,29672,29674],{"className":29673,"style":600},[569],[28,29675],{},[28,29677],{"className":29678,"style":280},[279],[28,29680,5074],{"className":29681},[284],[28,29683],{"className":29684,"style":280},[279],[28,29686,29688,29691,29731,29734,29737],{"className":29687},[265],[28,29689],{"className":29690,"style":24102},[269],[28,29692,29694,29697],{"className":29693},[274],[28,29695,27871],{"className":29696,"style":298},[274,275],[28,29698,29700],{"className":29699},[556],[28,29701,29703,29723],{"className":29702},[560,561],[28,29704,29706,29720],{"className":29705},[565],[28,29707,29709],{"className":29708,"style":3881},[569],[28,29710,29711,29714],{"style":3660},[28,29712],{"className":29713,"style":578},[577],[28,29715,29717],{"className":29716},[582,583,584,585],[28,29718,7907],{"className":29719,"style":8057},[274,275,585],[28,29721,593],{"className":29722},[592],[28,29724,29726],{"className":29725},[565],[28,29727,29729],{"className":29728,"style":20967},[569],[28,29730],{},[28,29732],{"className":29733,"style":302},[279],[28,29735,1383],{"className":29736},[306],[28,29738],{"className":29739,"style":302},[279],[28,29741,29743,29746],{"className":29742},[265],[28,29744],{"className":29745,"style":316},[269],[28,29747,89],{"className":29748},[274],"，令 ",[28,29751,29753,29779],{"className":29752,"translate":193},[197],[28,29754,29756],{"className":29755},[201],[203,29757,29758],{"xmlns":205},[208,29759,29760,29776],{},[211,29761,29762,29772,29774],{},[501,29763,29764,29766],{},[214,29765,22365],{},[211,29767,29768,29770],{},[214,29769,1686],{},[214,29771,7907],{},[217,29773,950],{},[214,29775,24419],{"mathvariant":484},[253,29777,29778],{"encoding":255},"\\pi_{ij}\\to\\infty",[28,29780,29782,29840],{"className":29781,"ariaHidden":261},[260],[28,29783,29785,29788,29831,29834,29837],{"className":29784},[265],[28,29786],{"className":29787,"style":25364},[269],[28,29789,29791,29794],{"className":29790},[274],[28,29792,22365],{"className":29793,"style":298},[274,275],[28,29795,29797],{"className":29796},[556],[28,29798,29800,29823],{"className":29799},[560,561],[28,29801,29803,29820],{"className":29802},[565],[28,29804,29806],{"className":29805,"style":3881},[569],[28,29807,29808,29811],{"style":3660},[28,29809],{"className":29810,"style":578},[577],[28,29812,29814],{"className":29813},[582,583,584,585],[28,29815,29817],{"className":29816},[274,585],[28,29818,25397],{"className":29819,"style":8057},[274,275,585],[28,29821,593],{"className":29822},[592],[28,29824,29826],{"className":29825},[565],[28,29827,29829],{"className":29828,"style":20967},[569],[28,29830],{},[28,29832],{"className":29833,"style":302},[279],[28,29835,950],{"className":29836},[306],[28,29838],{"className":29839,"style":302},[279],[28,29841,29843,29846],{"className":29842},[265],[28,29844],{"className":29845,"style":645},[269],[28,29847,24419],{"className":29848},[274]," 就得到 ",[28,29851,29853,29869],{"className":29852,"translate":193},[197],[28,29854,29856],{"className":29855},[201],[203,29857,29858],{"xmlns":205},[208,29859,29860,29866],{},[211,29861,29862,29864],{},[217,29863,5074],{},[214,29865,24419],{"mathvariant":484},[253,29867,29868],{"encoding":255},"-\\infty",[28,29870,29872],{"className":29871,"ariaHidden":261},[260],[28,29873,29875,29878,29881],{"className":29874},[265],[28,29876],{"className":29877,"style":270},[269],[28,29879,5074],{"className":29880},[274],[28,29882,24419],{"className":29883},[274],"；否则下确界在 ",[28,29886,29888,29906],{"className":29887,"translate":193},[197],[28,29889,29891],{"className":29890},[201],[203,29892,29893],{"xmlns":205},[208,29894,29895,29903],{},[211,29896,29897,29899,29901],{},[214,29898,22365],{},[217,29900,83],{},[224,29902,89],{},[253,29904,29905],{"encoding":255},"\\pi=0",[28,29907,29909,29927],{"className":29908,"ariaHidden":261},[260],[28,29910,29912,29915,29918,29921,29924],{"className":29911},[265],[28,29913],{"className":29914,"style":645},[269],[28,29916,22365],{"className":29917,"style":298},[274,275],[28,29919],{"className":29920,"style":302},[279],[28,29922,83],{"className":29923},[306],[28,29925],{"className":29926,"style":302},[279],[28,29928,29930,29933],{"className":29929},[265],[28,29931],{"className":29932,"style":316},[269],[28,29934,89],{"className":29935},[274]," 处取到。于是对偶问题就是",[28,29938,29940],{"className":29939,"translate":193},[192],[28,29941,29943,30046],{"className":29942,"translate":193},[197],[28,29944,29946],{"className":29945},[201],[203,29947,29948],{"xmlns":205,"display":206},[208,29949,29950,30043],{},[211,29951,29952,29969,29971,29977,29983,29989,29991,29997,30003,30009,30011,30013,30015,30021,30023,30029,30031,30041],{},[1771,29953,29954,29961],{},[211,29955,29956,29959],{},[214,29957,29958],{},"max",[217,29960,21407],{},[211,29962,29963,29965,29967],{},[214,29964,510],{},[217,29966,1683],{"separator":261},[214,29968,27871],{},[228,29970,14860],{},[1771,29972,29973,29975],{},[217,29974,25442],{},[214,29976,1686],{},[501,29978,29979,29981],{},[214,29980,238],{},[214,29982,1686],{},[501,29984,29985,29987],{},[214,29986,510],{},[214,29988,1686],{},[217,29990,124],{},[1771,29992,29993,29995],{},[217,29994,25442],{},[214,29996,7907],{},[501,29998,29999,30001],{},[214,30000,3820],{},[214,30002,7907],{},[501,30004,30005,30007],{},[214,30006,27871],{},[214,30008,7907],{},[279,30010],{"width":497},[228,30012,21452],{},[279,30014],{"width":497},[501,30016,30017,30019],{},[214,30018,510],{},[214,30020,1686],{},[217,30022,124],{},[501,30024,30025,30027],{},[214,30026,27871],{},[214,30028,7907],{},[217,30030,1876],{},[501,30032,30033,30035],{},[214,30034,15828],{},[211,30036,30037,30039],{},[214,30038,1686],{},[214,30040,7907],{},[217,30042,1683],{"separator":261},[253,30044,30045],{"encoding":255},"\\max_{\\varphi,\\psi}\\ \\sum_i a_i\\varphi_i+\\sum_j b_j\\psi_j\n\\quad\\text{s.t.}\\quad \\varphi_i+\\psi_j\\le C_{ij},",[28,30047,30049,30253,30448,30503],{"className":30048,"ariaHidden":261},[260],[28,30050,30052,30056,30111,30114,30117,30161,30164,30204,30244,30247,30250],{"className":30051},[265],[28,30053],{"className":30054,"style":30055},[269],"height:2.3277em;vertical-align:-1.2777em;",[28,30057,30059],{"className":30058},[21329,21485],[28,30060,30062,30102],{"className":30061},[560,561],[28,30063,30065,30099],{"className":30064},[565],[28,30066,30068,30089],{"className":30067,"style":645},[569],[28,30069,30071,30074],{"style":30070},"top:-2.3479em;margin-left:0em;",[28,30072],{"className":30073,"style":1914},[577],[28,30075,30077],{"className":30076},[582,583,584,585],[28,30078,30080,30083,30086],{"className":30079},[274,585],[28,30081,510],{"className":30082},[274,275,585],[28,30084,1683],{"className":30085},[1723,585],[28,30087,27871],{"className":30088,"style":298},[274,275,585],[28,30090,30091,30094],{"style":1926},[28,30092],{"className":30093,"style":1914},[577],[28,30095,30096],{},[28,30097,29958],{"className":30098},[21329],[28,30100,593],{"className":30101},[592],[28,30103,30105],{"className":30104},[565],[28,30106,30109],{"className":30107,"style":30108},[569],"height:0.8882em;",[28,30110],{},[28,30112,14860],{"className":30113},[279],[28,30115],{"className":30116,"style":530},[279],[28,30118,30120],{"className":30119},[21329,21485],[28,30121,30123,30153],{"className":30122},[560,561],[28,30124,30126,30150],{"className":30125},[565],[28,30127,30129,30140],{"className":30128,"style":25713},[569],[28,30130,30131,30134],{"style":25716},[28,30132],{"className":30133,"style":25720},[577],[28,30135,30137],{"className":30136},[582,583,584,585],[28,30138,1686],{"className":30139},[274,275,585],[28,30141,30142,30145],{"style":25732},[28,30143],{"className":30144,"style":25720},[577],[28,30146,30147],{},[28,30148,25442],{"className":30149},[21329,21330,21547],[28,30151,593],{"className":30152},[592],[28,30154,30156],{"className":30155},[565],[28,30157,30159],{"className":30158,"style":26154},[569],[28,30160],{},[28,30162],{"className":30163,"style":530},[279],[28,30165,30167,30170],{"className":30166},[274],[28,30168,238],{"className":30169},[274,275],[28,30171,30173],{"className":30172},[556],[28,30174,30176,30196],{"className":30175},[560,561],[28,30177,30179,30193],{"className":30178},[565],[28,30180,30182],{"className":30181,"style":3881},[569],[28,30183,30184,30187],{"style":573},[28,30185],{"className":30186,"style":578},[577],[28,30188,30190],{"className":30189},[582,583,584,585],[28,30191,1686],{"className":30192},[274,275,585],[28,30194,593],{"className":30195},[592],[28,30197,30199],{"className":30198},[565],[28,30200,30202],{"className":30201,"style":600},[569],[28,30203],{},[28,30205,30207,30210],{"className":30206},[274],[28,30208,510],{"className":30209},[274,275],[28,30211,30213],{"className":30212},[556],[28,30214,30216,30236],{"className":30215},[560,561],[28,30217,30219,30233],{"className":30218},[565],[28,30220,30222],{"className":30221,"style":3881},[569],[28,30223,30224,30227],{"style":573},[28,30225],{"className":30226,"style":578},[577],[28,30228,30230],{"className":30229},[582,583,584,585],[28,30231,1686],{"className":30232},[274,275,585],[28,30234,593],{"className":30235},[592],[28,30237,30239],{"className":30238},[565],[28,30240,30242],{"className":30241,"style":600},[569],[28,30243],{},[28,30245],{"className":30246,"style":280},[279],[28,30248,124],{"className":30249},[284],[28,30251],{"className":30252,"style":280},[279],[28,30254,30256,30260,30304,30307,30347,30387,30390,30396,30399,30439,30442,30445],{"className":30255},[265],[28,30257],{"className":30258,"style":30259},[269],"height:2.4638em;vertical-align:-1.4138em;",[28,30261,30263],{"className":30262},[21329,21485],[28,30264,30266,30296],{"className":30265},[560,561],[28,30267,30269,30293],{"className":30268},[565],[28,30270,30272,30283],{"className":30271,"style":25713},[569],[28,30273,30274,30277],{"style":25716},[28,30275],{"className":30276,"style":25720},[577],[28,30278,30280],{"className":30279},[582,583,584,585],[28,30281,7907],{"className":30282,"style":8057},[274,275,585],[28,30284,30285,30288],{"style":25732},[28,30286],{"className":30287,"style":25720},[577],[28,30289,30290],{},[28,30291,25442],{"className":30292},[21329,21330,21547],[28,30294,593],{"className":30295},[592],[28,30297,30299],{"className":30298},[565],[28,30300,30302],{"className":30301,"style":25750},[569],[28,30303],{},[28,30305],{"className":30306,"style":530},[279],[28,30308,30310,30313],{"className":30309},[274],[28,30311,3820],{"className":30312},[274,275],[28,30314,30316],{"className":30315},[556],[28,30317,30319,30339],{"className":30318},[560,561],[28,30320,30322,30336],{"className":30321},[565],[28,30323,30325],{"className":30324,"style":3881},[569],[28,30326,30327,30330],{"style":573},[28,30328],{"className":30329,"style":578},[577],[28,30331,30333],{"className":30332},[582,583,584,585],[28,30334,7907],{"className":30335,"style":8057},[27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:\\ \\varphi(x)+\\psi(y)\\le c(x,y)\\Bigr\\}.",[28,30669,30671,30744,30787,30826,30853,30880],{"className":30670,"ariaHidden":261},[260],[28,30672,30674,30677,30717,30720,30723,30726,30729,30732,30735,30738,30741],{"className":30673},[265],[28,30675],{"className":30676,"style":341},[269],[28,30678,30680,30683],{"className":30679},[274],[28,30681,22957],{"className":30682,"style":459},[274,275],[28,30684,30686],{"className":30685},[556],[28,30687,30689,30709],{"className":30688},[560,561],[28,30690,30692,30706],{"className":30691},[565],[28,30693,30695],{"className":30694,"style":23070},[569],[28,30696,30697,30700],{"style":20948},[28,30698],{"className":30699,"style":578},[577],[28,30701,30703],{"className":30702},[582,583,584,585],[28,30704,22],{"className":30705},[274,275,585],[28,30707,593],{"className":30708},[592],[28,30710,30712],{"className":30711},[565],[28,30713,30715],{"className":30714,"style":600},[569],[28,30716],{},[28,30718,43],{"className":30719},[348],[28,30721,20756],{"className":30722},[274,275],[28,30724,1683],{"className":30725},[1723],[28,30727],{"className":30728,"style":530},[279],[28,30730,20787],{"className":30731,"style":20803},[274,275],[28,30733,1522],{"className":30734},[355],[28,30736],{"className":30737,"style":302},[279],[28,30739,83],{"className":30740},[306],[28,30742],{"className":30743,"style":302},[279],[28,30745,30747,30751,30754,30760,30763,30766,30769,30772,30775,30778,30781,30784],{"className":30746},[265],[28,30748],{"className":30749,"style":30750},[269],"height:2.2222em;vertical-align:-0.8622em;",[28,30752,30596],{"className":30753},[21329],[28,30755,30757],{"className":30756},[348],[28,30758,7916],{"className":30759},[2169,22506],[28,30761,21282],{"className":30762,"style":21548},[21329,21330,21547],[28,30764],{"className":30765,"style":530},[279],[28,30767,510],{"className":30768},[274,275],[28,30770],{"className":30771,"style":530},[279],[28,30773,12988],{"className":30774},[274,275],[28,30776,20756],{"className":30777},[274,275],[28,30779],{"className":30780,"style":280},[279],[28,30782,124],{"className":30783},[284],[28,30785],{"className":30786,"style":280},[279],[28,30788,30790,30793,30796,30799,30802,30805,30808,30811,30814,30817,30820,30823],{"className":30789},[265],[28,30791],{"className":30792,"style":30750},[269],[28,30794,21282],{"className":30795,"style":21548},[21329,21330,21547],[28,30797],{"className":30798,"style":530},[279],[28,30800,27871],{"className":30801,"style":298},[274,275],[28,30803],{"className":30804,"style":530},[279],[28,30806,12988],{"className":30807},[274,275],[28,30809,20787],{"className":30810,"style":20803},[274,275],[28,30812,14860],{"className":30813},[279],[28,30815],{"className":30816,"style":302},[279],[28,30818,877],{"className":30819},[306],[28,30821,14860],{"className":30822},[279],[28,30824],{"className":30825,"style":302},[279],[28,30827,30829,30832,30835,30838,30841,30844,30847,30850],{"className":30828},[265],[28,30830],{"className":30831,"style":341},[269],[28,30833,510],{"className":30834},[274,275],[28,30836,43],{"className":30837},[348],[28,30839,176],{"className":30840},[274,275],[28,30842,1522],{"className":30843},[355],[28,30845],{"className":30846,"style":280},[279],[28,30848,124],{"className":30849},[284],[28,30851],{"className":30852,"style":280},[279],[28,30854,30856,30859,30862,30865,30868,30871,30874,30877],{"className":30855},[265],[28,30857],{"className":30858,"style":341},[269],[28,30860,27871],{"className":30861,"style":298},[274,275],[28,30863,43],{"className":30864},[348],[28,30866,180],{"className":30867,"style":298},[274,275],[28,30869,1522],{"className":30870},[355],[28,30872],{"className":30873,"style":302},[279],[28,30875,1876],{"className":30876},[306],[28,30878],{"className":30879,"style":302},[279],[28,30881,30883,30886,30889,30892,30895,30898,30901,30904,30907,30913],{"className":30882},[265],[28,30884],{"className":30885,"style":22499},[269],[28,30887,22],{"className":30888},[274,275],[28,30890,43],{"className":30891},[348],[28,30893,176],{"className":30894},[274,275],[28,30896,1683],{"className":30897},[1723],[28,30899],{"className":30900,"style":530},[279],[28,30902,180],{"className":30903,"style":298},[274,275],[28,30905,1522],{"className":30906},[355],[28,30908,30910],{"className":30909},[355],[28,30911,14883],{"className":30912},[2169,22506],[28,30914,494],{"className":30915},[274],[11,30917,15450,30918,30969,30970,31101,31102,31158,31159,10979,31290,24711,31318,31346],{},[28,30919,30921,30939],{"className":30920,"translate":193},[197],[28,30922,30924],{"className":30923},[201],[203,30925,30926],{"xmlns":205},[208,30927,30928,30936],{},[211,30929,30930,30932,30934],{},[214,30931,22],{},[217,30933,83],{},[214,30935,12988],{},[253,30937,30938],{"encoding":255},"c=d",[28,30940,30942,30960],{"className":30941,"ariaHidden":261},[260],[28,30943,30945,30948,30951,30954,30957],{"className":30944},[265],[28,30946],{"className":30947,"style":645},[269],[28,30949,22],{"className":30950},[274,275],[28,30952],{"className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",[28,30971,30973,31017],{"className":30972,"translate":193},[197],[28,30974,30976],{"className":30975},[201],[203,30977,30978],{"xmlns":205},[208,30979,30980,31014],{},[211,30981,30982,30984,30986,30988,30990,30992,30994,30996,30998,31000,31002,31004,31006,31008,31010,31012],{},[214,30983,510],{},[217,30985,43],{"stretchy":241},[214,30987,176],{},[217,30989,1522],{"stretchy":241},[217,30991,124],{},[214,30993,27871],{},[217,30995,43],{"stretchy":241},[214,30997,180],{},[217,30999,1522],{"stretchy":241},[217,31001,1876],{},[214,31003,12988],{},[217,31005,43],{"stretchy":241},[214,31007,176],{},[217,31009,1683],{"separator":261},[214,31011,180],{},[217,31013,1522],{"stretchy":241},[253,31015,31016],{"encoding":255},"\\varphi(x)+\\psi(y)\\le d(x,y)",[28,31018,31020,31047,31074],{"className":31019,"ariaHidden":261},[260],[28,31021,31023,31026,31029,31032,31035,31038,31041,31044],{"className":31022},[265],[28,31024],{"className":31025,"style":341},[269],[28,31027,510],{"className":31028},[274,275],[28,31030,43],{"className":31031},[348],[28,31033,176],{"className":31034},[274,275],[28,31036,1522],{"className":31037},[355],[28,31039],{"className":31040,"style":280},[279],[28,31042,124],{"className":31043},[284],[28,31045],{"className":31046,"style":280},[279],[28,31048,31050,31053,31056,31059,31062,31065,31068,31071],{"className":31049},[265],[28,31051],{"className":31052,"style":341},[269],[28,31054,27871],{"className":31055,"style":298},[274,275],[28,31057,43],{"className":31058},[348],[28,31060,180],{"className":31061,"style":298},[274,275],[28,31063,1522],{"className":31064},[355],[28,31066],{"className":31067,"style":302},[279],[28,31069,1876],{"className":31070},[306],[28,31072],{"className":31073,"style":302},[279],[28,31075,31077,31080,31083,31086,31089,31092,31095,31098],{"className":31076},[265],[28,31078],{"className":31079,"style":341},[269],[28,31081,12988],{"className":31082},[274,275],[28,31084,43],{"className":31085},[348],[28,31087,176],{"className":31088},[274,275],[28,31090,1683],{"className":31091},[1723],[28,31093],{"className":31094,"style":530},[279],[28,31096,180],{"className":31097,"style":298},[274,275],[28,31099,1522],{"className":31100},[355]," 允许我取 ",[28,31103,31105,31125],{"className":31104,"translate":193},[197],[28,31106,31108],{"className":31107},[201],[203,31109,31110],{"xmlns":205},[208,31111,31112,31122],{},[211,31113,31114,31116,31118,31120],{},[214,31115,27871],{},[217,31117,83],{},[217,31119,5074],{},[214,31121,510],{},[253,31123,31124],{"encoding":255},"\\psi=-\\varphi",[28,31126,31128,31146],{"className":31127,"ariaHidden":261},[260],[28,31129,31131,31134,31137,31140,31143],{"className":31130},[265],[28,31132],{"className":31133,"style":2371},[269],[28,31135,27871],{"className":31136,"style":298},[274,275],[28,31138],{"className":31139,"style":302},[279],[28,31141,83],{"className":31142},[306],[28,31144],{"className":31145,"style":302},[279],[28,31147,31149,31152,31155],{"className":31148},[265],[28,31150],{"className":31151,"style":4652},[269],[28,31153,5074],{"className":31154},[274],[28,31156,510],{"className":31157},[274,275],"，此时它变成 ",[28,31160,31162,31206],{"className":31161,"translate":193},[197],[28,31163,31165],{"className":31164},[201],[203,31166,31167],{"xmlns":205},[208,31168,31169,31203],{},[211,31170,31171,31173,31175,31177,31179,31181,31183,31185,31187,31189,31191,31193,31195,31197,31199,31201],{},[214,31172,510],{},[217,31174,43],{"stretchy":241},[214,31176,176],{},[217,31178,1522],{"stretchy":241},[217,31180,5074],{},[214,31182,510],{},[217,31184,43],{"stretchy":241},[214,31186,180],{},[217,31188,1522],{"stretchy":241},[217,31190,1876],{},[214,31192,12988],{},[217,31194,43],{"stretchy":241},[214,31196,176],{},[217,31198,1683],{"separator":261},[214,31200,180],{},[217,31202,1522],{"stretchy":241},[253,31204,31205],{"encoding":255},"\\varphi(x)-\\varphi(y)\\le d(x,y)",[28,31207,31209,31236,31263],{"className":31208,"ariaHidden":261},[260],[28,31210,31212,31215,31218,31221,31224,31227,31230,31233],{"className":31211},[265],[28,31213],{"className":31214,"style":341},[269],[28,31216,510],{"className":31217},[274,275],[28,31219,43],{"className":31220},[348],[28,31222,176],{"className":31223},[274,275],[28,31225,1522],{"className":31226},[355],[28,31228],{"className":31229,"style":280},[279],[28,31231,5074],{"className":31232},[284],[28,31234],{"className":31235,"style":280},[279],[28,31237,31239,31242,31245,31248,31251,31254,31257,31260],{"className":31238},[265],[28,31240],{"className":31241,"style":341},[269],[28,31243,510],{"className":31244},[274,275],[28,31246,43],{"className":31247},[348],[28,31249,180],{"className":31250,"style":298},[274,275],[28,31252,1522],{"className":31253},[355],[28,31255],{"className":31256,"style":302},[279],[28,31258,1876],{"className":31259},[306],[28,31261],{"className":31262,"style":302},[279],[28,31264,31266,31269,31272,31275,31278,31281,31284,31287],{"className":31265},[265],[28,31267],{"className":31268,"style":341},[269],[28,31270,12988],{"className":31271},[274,275],[28,31273,43],{"className":31274},[348],[28,31276,176],{"className":31277},[274,275],[28,31279,1683],{"className":31280},[1723],[28,31282],{"className":31283,"style":530},[279],[28,31285,180],{"className":31286,"style":298},[274,275],[28,31288,1522],{"className":31289},[355],[28,31291,31293,31306],{"className":31292,"translate":193},[197],[28,31294,31296],{"className":31295},[201],[203,31297,31298],{"xmlns":205},[208,31299,31300,31304],{},[211,31301,31302],{},[214,31303,510],{},[253,31305,666],{"encoding":255},[28,31307,31309],{"className":31308,"ariaHidden":261},[260],[28,31310,31312,31315],{"className":31311},[265],[28,31313],{"className":31314,"style":612},[269],[28,31316,510],{"className":31317},[274,275],[28,31319,31321,31334],{"className":31320,"translate":193},[197],[28,31322,31324],{"className":31323},[201],[203,31325,31326],{"xmlns":205},[208,31327,31328,31332],{},[211,31329,31330],{},[224,31331,1244],{},[253,31333,1244],{"encoding":255},[28,31335,31337],{"className":31336,"ariaHidden":261},[260],[28,31338,31340,31343],{"className":31339},[265],[28,31341],{"className":31342,"style":316},[269],[28,31344,1244],{"className":31345},[274],"-Lipschitz 的。所以",[28,31348,31350],{"className":31349,"translate":193},[192],[28,31351,31353,31433],{"className":31352,"translate":193},[197],[28,31354,31356],{"className":31355},[201],[203,31357,31358],{"xmlns":205,"display":206},[208,31359,31360,31430],{},[211,31361,31362,31368,31370,31372,31374,31376,31378,31380,31410,31412,31414,31416,31418,31420,31422,31424,31426,31428],{},[501,31363,31364,31366],{},[214,31365,22957],{},[224,31367,1244],{},[217,31369,43],{"stretchy":241},[214,31371,20756],{},[217,31373,1683],{"separator":261},[214,31375,20787],{},[217,31377,1522],{"stretchy":241},[217,31379,83],{},[1771,31381,31382,31388],{},[211,31383,31384,31386],{},[214,31385,30596],{},[217,31387,21407],{},[211,31389,31390,31392,31394,31406,31408],{},[214,31391,24404],{"mathvariant":484},[214,31393,510],{},[501,31395,31396,31398],{},[214,31397,24404],{"mathvariant":484},[211,31399,31400,31402,31404],{},[214,31401,24393],{"mathvariant":484},[214,31403,1686],{"mathvariant":484},[214,31405,11],{"mathvariant":484},[217,31407,1876],{},[224,31409,1244],{},[217,31411,21282],{},[214,31413,510],{},[228,31415,488],{},[214,31417,12988],{},[217,31419,43],{"stretchy":241},[214,31421,20756],{},[217,31423,5074],{},[214,31425,20787],{},[217,31427,1522],{"stretchy":241},[214,31429,494],{"mathvariant":484},[253,31431,31432],{"encoding":255},"W_1(\\mu,\\nu)=\\sup_{\\|\\varphi\\|_{\\mathrm{Lip}}\\le 1}\\int\\varphi\\,d(\\mu-\\nu).",[28,31434,31436,31509,31659],{"className":31435,"ariaHidden":261},[260],[28,31437,31439,31442,31482,31485,31488,31491,31494,31497,31500,31503,31506],{"className":31438},[265],[28,31440],{"className":31441,"style":341},[269],[28,31443,31445,31448],{"className":31444},[274],[28,31446,22957],{"className":31447,"style":459},[274,275],[28,31449,31451],{"className":31450},[556],[28,31452,31454,31474],{"className":31453},[560,561],[28,31455,31457,31471],{"className":31456},[565],[28,31458,31460],{"className":31459,"style":1285},[569],[28,31461,31462,31465],{"style":20948},[28,31463],{"className":31464,"style":578},[577],[28,31466,31468],{"className":31467},[582,583,584,585],[28,31469,1244],{"className":31470},[274,585],[28,31472,593],{"className":31473},[592],[28,31475,31477],{"className":31476},[565],[28,31478,31480],{"className":31479,"style":600},[569],[28,31481],{},[28,31483,43],{"className":31484},[348],[28,31486,20756],{"className":31487},[274,275],[28,31489,1683],{"className":31490},[1723],[28,31492],{"className":31493,"style":530},[279],[28,31495,20787],{"className":31496,"style":20803},[274,275],[28,31498,1522],{"className":31499},[355],[28,31501],{"className":31502,"style":302},[279],[28,31504,83],{"className":31505},[306],[28,31507],{"className":31508,"style":302},[279],[28,31510,31512,31516,31626,31629,31632,31635,31638,31641,31644,31647,31650,31653,31656],{"className":31511},[265],[28,31513],{"className":31514,"style":31515},[269],"height:2.543em;vertical-align:-1.183em;",[28,31517,31519],{"className":31518},[21329,21485],[28,31520,31522,31617],{"className":31521},[560,561],[28,31523,31525,31614],{"className":31524},[565],[28,31526,31528,31604],{"className":31527,"style":645},[569],[28,31529,31531,31534],{"style":31530},"top:-2.1146em;margin-left:0em;",[28,31532],{"className":31533,"style":1914},[577],[28,31535,31537],{"className":31536},[582,583,584,585],[28,31538,31540,31543,31546,31598,31601],{"className":31539},[274,585],[28,31541,24404],{"className":31542},[274,585],[28,31544,510],{"className":31545},[274,275,585],[28,31547,31549,31552],{"className":31548},[274,585],[28,31550,24404],{"className":31551},[274,585],[28,31553,31555],{"className":31554},[556],[28,31556,31558,31589],{"className":31557},[560,561],[28,31559,31561,31586],{"className":31560},[565],[28,31562,31565],{"className":31563,"style":31564},[569],"height:0.3448em;",[28,31566,31568,31572],{"style":31567},"top:-2.3567em;margin-left:0em;margin-right:0.0714em;",[28,31569],{"className":31570,"style":31571},[577],"height:2.5em;",[28,31573,31576],{"className":31574},[582,31575,2170,585],"reset-size3",[28,31577,31579],{"className":31578},[274,585],[28,31580,31582],{"className":31581},[274,585],[28,31583,31585],{"className":31584},[274,1712,585],"Lip",[28,31587,593],{"className":31588},[592],[28,31590,31592],{"className":31591},[565],[28,31593,31596],{"className":31594,"style":31595},[569],"height:0.2822em;",[28,31597],{},[28,31599,1876],{"className":31600},[306,585],[28,31602,1244],{"className":31603},[274,585],[28,31605,31606,31609],{"style":1926},[28,31607],{"className":31608,"style":1914},[577],[28,31610,31611],{},[28,31612,30596],{"className":31613},[21329],[28,31615,593],{"className":31616},[592],[28,31618,31620],{"className":31619},[565],[28,31621,31624],{"className":31622,"style":31623},[569],"height:1.183em;",[28,31625],{},[28,31627],{"className":31628,"style":530},[279],[28,31630,21282],{"className":31631,"style":21548},[21329,21330,21547],[28,31633],{"className":31634,"style":530},[279],[28,31636,510],{"className":31637},[274,275],[28,31639],{"className":31640,"style":530},[279],[28,31642,12988],{"className":31643},[274,275],[28,31645,43],{"className":31646},[348],[28,31648,20756],{"className":31649},[274,275],[28,31651],{"className":31652,"style":280},[279],[28,31654,5074],{"className":31655},[284],[28,31657],{"className":31658,"style":280},[279],[28,31660,31662,31665,31668,31671],{"className":31661},[265],[28,31663],{"className":31664,"style":341},[269],[28,31666,20787],{"className":31667,"style":20803},[274,275],[28,31669,1522],{"className":31670},[355],[28,31672,494],{"className":31673},[274],[11,31675,31676],{},"这个式子的分量很重：左边是关于分布的优化问题，右边是关于函数的优化问题，而函数可以用神经网络参数化。WGAN 干的就是这件事，只差把 Lipschitz 约束用 weight clipping 或 gradient penalty 近似实现。",[11,31678,31679,31680,31870,31871,31899,31900,31928,31929,31986,31987,32015,32016,2923],{},"二次代价 ",[28,31681,31683,31721],{"className":31682,"translate":193},[197],[28,31684,31686],{"className":31685},[201],[203,31687,31688],{"xmlns":205},[208,31689,31690,31718],{},[211,31691,31692,31694,31696,31704,31706,31708,31710,31712],{},[214,31693,22],{},[217,31695,83],{},[5027,31697,31698],{"scriptlevel":89,"displaystyle":241},[21928,31699,31700,31702],{},[224,31701,1244],{},[224,31703,1253],{},[214,31705,24404],{"mathvariant":484},[214,31707,176],{},[217,31709,5074],{},[214,31711,180],{},[3054,31713,31714,31716],{},[214,31715,24404],{"mathvariant":484},[224,31717,1253],{},[253,31719,31720],{"encoding":255},"c=\\tfrac12\\|x-y\\|^2",[28,31722,31724,31742,31831],{"className":31723,"ariaHidden":261},[260],[28,31725,31727,31730,31733,31736,31739],{"className":31726},[265],[28,31728],{"className":31729,"style":645},[269],[28,31731,22],{"className":31732},[274,275],[28,31734],{"className":31735,"style":302},[279],[28,31737,83],{"className":31738},[306],[28,31740],{"className":31741,"style":302},[279],[28,31743,31745,31748,31816,31819,31822,31825,31828],{"className":31744},[265],[28,31746],{"className":31747,"style":21986},[269],[28,31749,31751,31754,31813],{"className":31750},[274],[28,31752],{"className":31753},[348,8285],[28,31755,31757],{"className":31756},[21928],[28,31758,31760,31805],{"className":31759},[560,561],[28,31761,31763,31802],{"className":31762},[565],[28,31764,31766,31780,31788],{"className":31765,"style":22005},[569],[28,31767,31768,31771],{"style":22008},[28,31769],{"className":31770,"style":1914},[577],[28,31772,31774],{"className":31773},[582,583,584,585],[28,31775,31777],{"className":31776},[274,585],[28,31778,1253],{"className":31779},[274,585],[28,31781,31782,31785],{"style":22023},[28,31783],{"className":31784,"style":1914},[577],[28,31786],{"className":31787,"style":22031},[22030],[28,31789,31790,31793],{"style":22034},[28,31791],{"className":31792,"style":1914},[577],[28,31794,31796],{"className":31795},[582,583,584,585],[28,31797,31799],{"className":31798},[274,585],[28,31800,1244],{"className":31801},[274,585],[28,31803,593],{"className":31804},[592],[28,31806,31808],{"className":31807},[565],[28,31809,31811],{"className":31810,"style":22056},[569],[28,31812],{},[28,31814],{"className":31815},[355,8285],[28,31817,24404],{"className":31818},[274],[28,31820,176],{"className":31821},[274,275],[28,31823],{"className":31824,"style":280},[279],[28,31826,5074],{"className":31827},[284],[28,31829],{"className":31830,"style":280},[279],[28,31832,31834,31838,31841],{"className":31833},[265],[28,31835],{"className":31836,"style":31837},[269],"height:1.0641em;vertical-align:-0.25em;",[28,31839,180],{"className":31840,"style":298},[274,275],[28,31842,31844,31847],{"className":31843},[274],[28,31845,24404],{"className":31846},[274],[28,31848,31850],{"className":31849},[556],[28,31851,31853],{"className":31852},[560],[28,31854,31856],{"className":31855},[565],[28,31857,31859],{"className":31858,"style":5477},[569],[28,31860,31861,31864],{"style":3094},[28,31862],{"className":31863,"style":578},[577],[28,31865,31867],{"className":31866},[582,583,584,585],[28,31868,1253],{"className":31869},[274,585]," 时还有一个 Brenier 定理：若 ",[28,31872,31874,31887],{"className":31873,"translate":193},[197],[28,31875,31877],{"className":31876},[201],[203,31878,31879],{"xmlns":205},[208,31880,31881,31885],{},[211,31882,31883],{},[214,31884,20756],{},[253,31886,20759],{"encoding":255},[28,31888,31890],{"className":31889,"ariaHidden":261},[260],[28,31891,31893,31896],{"className":31892},[265],[28,31894],{"className":31895,"style":612},[269],[28,31897,20756],{"className":31898},[274,275]," 关于 Lebesgue 测度绝对连续，最优的 ",[28,31901,31903,31916],{"className":31902,"translate":193},[197],[28,31904,31906],{"className":31905},[201],[203,31907,31908],{"xmlns":205},[208,31909,31910,31914],{},[211,31911,31912],{},[214,31913,22365],{},[253,31915,23325],{"encoding":255},[28,31917,31919],{"className":31918,"ariaHidden":261},[260],[28,31920,31922,31925],{"className":31921},[265],[28,31923],{"className":31924,"style":645},[269],[28,31926,22365],{"className":31927,"style":298},[274,275]," 由映射 ",[28,31930,31932,31953],{"className":31931,"translate":193},[197],[28,31933,31935],{"className":31934},[201],[203,31936,31937],{"xmlns":205},[208,31938,31939,31950],{},[211,31940,31941,31943,31945,31948],{},[214,31942,443],{},[217,31944,83],{},[214,31946,31947],{"mathvariant":484},"∇",[214,31949,3936],{},[253,31951,31952],{"encoding":255},"T=\\nabla u",[28,31954,31956,31974],{"className":31955,"ariaHidden":261},[260],[28,31957,31959,31962,31965,31968,31971],{"className":31958},[265],[28,31960],{"className":31961,"style":455},[269],[28,31963,443],{"className":31964,"style":459},[274,275],[28,31966],{"className":31967,"style":302},[279],[28,31969,83],{"className":31970},[306],[28,31972],{"className":31973,"style":302},[279],[28,31975,31977,31980,31983],{"className":31976},[265],[28,31978],{"className":31979,"style":455},[269],[28,31981,31947],{"className":31982},[274],[28,31984,3936],{"className":31985},[274,275]," 给出，其中 ",[28,31988,31990,32003],{"className":31989,"translate":193},[197],[28,31991,31993],{"className":31992},[201],[203,31994,31995],{"xmlns":205},[208,31996,31997,32001],{},[211,31998,31999],{},[214,32000,3936],{},[253,32002,3936],{"encoding":255},[28,32004,32006],{"className":32005,"ariaHidden":261},[260],[28,32007,32009,32012],{"className":32008},[265],[28,32010],{"className":32011,"style":645},[269],[28,32013,3936],{"className":32014},[274,275]," 是凸函数。换句话说，最优搬运方案是某个凸势函数的梯度，是一个“单调”的变换。这句话是后半段所有流模型的伏笔：如果我要把一个分布搬到另一个分布，最优的那个可逆映射长得像 ",[28,32017,32019,32035],{"className":32018,"translate":193},[197],[28,32020,32022],{"className":32021},[201],[203,32023,32024],{"xmlns":205},[208,32025,32026,32032],{},[211,32027,32028,32030],{},[214,32029,31947],{"mathvariant":484},[214,32031,3936],{},[253,32033,32034],{"encoding":255},"\\nabla u",[28,32036,32038],{"className":32037,"ariaHidden":261},[260],[28,32039,32041,32044,32047],{"className":32040},[265],[28,32042],{"className":32043,"style":455},[269],[28,32045,31947],{"className":32046},[274],[28,32048,3936],{"className":32049},[274,275],[414,32051,32052],{"id":32052},"一维与高斯",[11,32054,32055],{},"A 公司的薪资在 0 到 1 万之间均匀分布，B 公司在 0 到 2 万之间均匀分布。一个人从 A 跳到 B，平均涨多少？",[11,32057,32058,32059,32088,32089,32117,32118,32152,32153,32181],{},"“最省力”的搬法是按百分位对齐：A 的第 30 百分位是 0.3 万，B 的第 30 百分位是 0.6 万，所以 A 里处在 30 分位的那个人涨 0.3 万。一般地，分位 ",[28,32060,32062,32075],{"className":32061,"translate":193},[197],[28,32063,32065],{"className":32064},[201],[203,32066,32067],{"xmlns":205},[208,32068,32069,32073],{},[211,32070,32071],{},[214,32072,1241],{},[253,32074,1241],{"encoding":255},[28,32076,32078],{"className":32077,"ariaHidden":261},[260],[28,32079,32081,32085],{"className":32080},[265],[28,32082],{"className":32083,"style":32084},[269],"height:0.6151em;",[28,32086,1241],{"className":32087},[274,275]," 的人从 ",[28,32090,32092,32105],{"className":32091,"translate":193},[197],[28,32093,32095],{"className":32094},[201],[203,32096,32097],{"xmlns":205},[208,32098,32099,32103],{},[211,32100,32101],{},[214,32102,1241],{},[253,32104,1241],{"encoding":255},[28,32106,32108],{"className":32107,"ariaHidden":261},[260],[28,32109,32111,32114],{"className":32110},[265],[28,32112],{"className":32113,"style":32084},[269],[28,32115,1241],{"className":32116},[274,275]," 万变成 ",[28,32119,32121,32137],{"className":32120,"translate":193},[197],[28,32122,32124],{"className":32123},[201],[203,32125,32126],{"xmlns":205},[208,32127,32128,32134],{},[211,32129,32130,32132],{},[224,32131,1253],{},[214,32133,1241],{},[253,32135,32136],{"encoding":255},"2t",[28,32138,32140],{"className":32139,"ariaHidden":261},[260],[28,32141,32143,32146,32149],{"className":32142},[265],[28,32144],{"className":32145,"style":316},[269],[28,32147,1253],{"className":32148},[274],[28,32150,1241],{"className":32151},[274,275]," 万，涨幅是 ",[28,32154,32156,32169],{"className":32155,"translate":193},[197],[28,32157,32159],{"className":32158},[201],[203,32160,32161],{"xmlns":205},[208,32162,32163,32167],{},[211,32164,32165],{},[214,32166,1241],{},[253,32168,1241],{"encoding":255},[28,32170,32172],{"className":32171,"ariaHidden":261},[260],[28,32173,32175,32178],{"className":32174},[265],[28,32176],{"className":32177,"style":32084},[269],[28,32179,1241],{"className":32180},[274,275]," 万，平均",[28,32183,32185],{"className":32184,"translate":193},[192],[28,32186,32188,32232],{"className":32187,"translate":193},[197],[28,32189,32191],{"className":32190},[201],[203,32192,32193],{"xmlns":205,"display":206},[208,32194,32195,32229],{},[211,32196,32197,32206,32208,32210,32212,32214,32216,32224,32227],{},[32198,32199,32200,32202,32204],"msubsup",{},[217,32201,21282],{},[224,32203,89],{},[224,32205,1244],{},[214,32207,1241],{},[228,32209,488],{},[214,32211,12988],{},[214,32213,1241],{},[217,32215,83],{},[5027,32217,32218],{"scriptlevel":89,"displaystyle":241},[21928,32219,32220,32222],{},[224,32221,1244],{},[224,32223,1253],{},[228,32225,32226],{}," 万",[214,32228,494],{"mathvariant":484},[253,32230,32231],{"encoding":255},"\\int_0^1 t\\,dt=\\tfrac12\\ \\text{万}.",[28,32233,32235,32319],{"className":32234,"ariaHidden":261},[260],[28,32236,32238,32242,32295,32298,32301,32304,32307,32310,32313,32316],{"className":32237},[265],[28,32239],{"className":32240,"style":32241},[269],"height:2.476em;vertical-align:-0.9119em;",[28,32243,32245,32248],{"className":32244},[21329],[28,32246,21282],{"className":32247,"style":21548},[21329,21330,21547],[28,32249,32251],{"className":32250},[556],[28,32252,32254,32287],{"className":32253},[560,561],[28,32255,32257,32284],{"className":32256},[565],[28,32258,32261,32272],{"className":32259,"style":32260},[569],"height:1.564em;",[28,32262,32263,32266],{"style":21564},[28,32264],{"className":32265,"style":578},[577],[28,32267,32269],{"className":32268},[582,583,584,585],[28,32270,89],{"className":32271},[274,585],[28,32273,32275,32278],{"style":32274},"top:-3.8129em;margin-right:0.05em;",[28,32276],{"className":32277,"style":578},[577],[28,32279,32281],{"className":32280},[582,583,584,585],[28,32282,1244],{"className":32283},[274,585],[28,32285,593],{"className":32286},[592],[28,32288,32290],{"className":32289},[565],[28,32291,32293],{"className":32292,"style":21586},[569],[28,32294],{},[28,32296],{"className":32297,"style":530},[279],[28,32299,1241],{"className":32300},[274,275],[28,32302],{"className":32303,"style":530},[279],[28,32305,12988],{"className":32306},[274,275],[28,32308,1241],{"className":32309},[274,275],[28,32311],{"className":32312,"style":302},[279],[28,32314,83],{"className":32315},[306],[28,32317],{"className":32318,"style":302},[279],[28,32320,32322,32325,32393,32396,32403],{"className":32321},[265],[28,32323],{"className":32324,"style":21986},[269],[28,32326,32328,32331,32390],{"className":32327},[274],[28,32329],{"className":32330},[348,8285],[28,32332,32334],{"className":32333},[21928],[28,32335,32337,32382],{"className":32336},[560,561],[28,32338,32340,32379],{"className":32339},[565],[28,32341,32343,32357,32365],{"className":32342,"style":22005},[569],[28,32344,32345,32348],{"style":22008},[28,32346],{"className":32347,"style":1914},[577],[28,32349,32351],{"className":32350},[582,583,584,585],[28,32352,32354],{"className":32353},[274,585],[28,32355,1253],{"className":32356},[274,585],[28,32358,32359,32362],{"style":22023},[28,32360],{"className":32361,"style":1914},[577],[28,32363],{"className":32364,"style":22031},[22030],[28,32366,32367,32370],{"style":22034},[28,32368],{"className":32369,"style":1914},[577],[28,32371,32373],{"className":32372},[582,583,584,585],[28,32374,32376],{"className":32375},[274,585],[28,32377,1244],{"className":32378},[274,585],[28,32380,593],{"className":32381},[592],[28,32383,32385],{"className":32384},[565],[28,32386,32388],{"className":32387,"style":22056},[569],[28,32389],{},[28,32391],{"className":32392},[355,8285],[28,32394,14860],{"className":32395},[279],[28,32397,32399],{"className":32398},[274,11179],[28,32400,32402],{"className":32401},[274,15966],"万",[28,32404,494],{"className":32405},[274],[11,32407,32408,32409,32465,32466,32530,32531,2923],{},"这里可以直接用 ",[28,32410,32412,32432],{"className":32411,"translate":193},[197],[28,32413,32415],{"className":32414},[201],[203,32416,32417],{"xmlns":205},[208,32418,32419,32429],{},[211,32420,32421,32423,32425,32427],{},[224,32422,1253],{},[214,32424,1241],{},[217,32426,5074],{},[214,32428,1241],{},[253,32430,32431],{"encoding":255},"2t-t",[28,32433,32435,32456],{"className":32434,"ariaHidden":261},[260],[28,32436,32438,32441,32444,32447,32450,32453],{"className":32437},[265],[28,32439],{"className":32440,"style":6853},[269],[28,32442,1253],{"className":32443},[274],[28,32445,1241],{"className":32446},[274,275],[28,32448],{"className":32449,"style":280},[279],[28,32451,5074],{"className":32452},[284],[28,32454],{"className":32455,"style":280},[279],[28,32457,32459,32462],{"className":32458},[265],[28,32460],{"className":32461,"style":32084},[269],[28,32463,1241],{"className":32464},[274,275]," 而不是 ",[28,32467,32469,32493],{"className":32468,"translate":193},[197],[28,32470,32472],{"className":32471},[201],[203,32473,32474],{"xmlns":205},[208,32475,32476,32490],{},[211,32477,32478,32480,32482,32484,32486,32488],{},[214,32479,889],{"mathvariant":484},[224,32481,1253],{},[214,32483,1241],{},[217,32485,5074],{},[214,32487,1241],{},[214,32489,889],{"mathvariant":484},[253,32491,32492],{"encoding":255},"|2t-t|",[28,32494,32496,32518],{"className":32495,"ariaHidden":261},[260],[28,32497,32499,32502,32506,32509,32512,32515],{"className":32498},[265],[28,32500],{"className":32501,"style":341},[269],[28,32503,32505],{"className":32504},[274],"∣2",[28,32507,1241],{"className":32508},[274,275],[28,32510],{"className":32511,"style":280},[279],[28,32513,5074],{"className":32514},[284],[28,32516],{"className":32517,"style":280},[279],[28,32519,32521,32524,32527],{"className":32520},[265],[28,32522],{"className":32523,"style":341},[269],[28,32525,1241],{"className":32526},[274,275],[28,32528,889],{"className":32529},[274],"，因为所有人都在往同一个方向涨。而这个“平均涨幅”正是 ",[28,32532,32534,32551],{"className":32533,"translate":193},[197],[28,32535,32537],{"className":32536},[201],[203,32538,32539],{"xmlns":205},[208,32540,32541,32549],{},[211,32542,32543],{},[501,32544,32545,32547],{},[214,32546,22957],{},[224,32548,1244],{},[253,32550,25049],{"encoding":255},[28,32552,32554],{"className":32553,"ariaHidden":261},[260],[28,32555,32557,32560],{"className":32556},[265],[28,32558],{"className":32559,"style":25059},[269],[28,32561,32563,32566],{"className":32562},[274],[28,32564,22957],{"className":32565,"style":459},[274,275],[28,32567,32569],{"className":32568},[556],[28,32570,32572,32592],{"className":32571},[560,561],[28,32573,32575,32589],{"className":32574},[565],[28,32576,32578],{"className":32577,"style":1285},[569],[28,32579,32580,32583],{"style":20948},[28,32581],{"className":32582,"style":578},[577],[28,32584,32586],{"className":32585},[582,583,584,585],[28,32587,1244],{"className":32588},[274,585],[28,32590,593],{"className":32591},[592],[28,32593,32595],{"className":32594},[565],[28,32596,32598],{"className":32597,"style":600},[569],[28,32599],{},[11,32601,32602,32603,32662,32663,32720,32721,32790],{},"如果反过来，A 是 ",[28,32604,32606,32631],{"className":32605,"translate":193},[197],[28,32607,32609],{"className":32608},[201],[203,32610,32611],{"xmlns":205},[208,32612,32613,32628],{},[211,32614,32615,32618,32620,32622,32624,32626],{},[214,32616,32617],{},"U",[217,32619,43],{"stretchy":241},[224,32621,89],{},[217,32623,1683],{"separator":261},[224,32625,1253],{},[217,32627,1522],{"stretchy":241},[253,32629,32630],{"encoding":255},"U(0,2)",[28,32632,32634],{"className":32633,"ariaHidden":261},[260],[28,32635,32637,32640,32644,32647,32650,32653,32656,32659],{"className":32636},[265],[28,32638],{"className":32639,"style":341},[269],[28,32641,32617],{"className":32642,"style":32643},[274,275],"margin-right:0.109em;",[28,32645,43],{"className":32646},[348],[28,32648,89],{"className":32649},[274],[28,32651,1683],{"className":32652},[1723],[28,32654],{"className":32655,"style":530},[279],[28,32657,1253],{"className":32658},[274],[28,32660,1522],{"className":32661},[355],"、B 是 ",[28,32664,32666,32690],{"className":32665,"translate":193},[197],[28,32667,32669],{"className":32668},[201],[203,32670,32671],{"xmlns":205},[208,32672,32673,32687],{},[211,32674,32675,32677,32679,32681,32683,32685],{},[214,32676,32617],{},[217,32678,43],{"stretchy":241},[224,32680,89],{},[217,32682,1683],{"separator":261},[224,32684,1244],{},[217,32686,1522],{"stretchy":241},[253,32688,32689],{"encoding":255},"U(0,1)",[28,32691,32693],{"className":32692,"ariaHidden":261},[260],[28,32694,32696,32699,32702,32705,32708,32711,32714,32717],{"className":32695},[265],[28,32697],{"className":32698,"style":341},[269],[28,32700,32617],{"className":32701,"style":32643},[274,275],[28,32703,43],{"className":32704},[348],[28,32706,89],{"className":32707},[274],[28,32709,1683],{"className":32710},[1723],[28,32712],{"className":32713,"style":530},[279],[28,32715,1244],{"className":32716},[274],[28,32718,1522],{"className":32719},[355],"，那就是平均降 0.5 万，",[28,32722,32724,32741],{"className":32723,"translate":193},[197],[28,32725,32727],{"className":32726},[201],[203,32728,32729],{"xmlns":205},[208,32730,32731,32739],{},[211,32732,32733],{},[501,32734,32735,32737],{},[214,32736,22957],{},[224,32738,1244],{},[253,32740,25049],{"encoding":255},[28,32742,32744],{"className":32743,"ariaHidden":261},[260],[28,32745,32747,32750],{"className":32746},[265],[28,32748],{"className":32749,"style":25059},[269],[28,32751,32753,32756],{"className":32752},[274],[28,32754,22957],{"className":32755,"style":459},[274,275],[28,32757,32759],{"className":32758},[556],[28,32760,32762,32782],{"className":32761},[560,561],[28,32763,32765,32779],{"className":32764},[565],[28,32766,32768],{"className":32767,"style":1285},[569],[28,32769,32770,32773],{"style":20948},[28,32771],{"className":32772,"style":578},[577],[28,32774,32776],{"className":32775},[582,583,584,585],[28,32777,1244],{"className":32778},[274,585],[28,32780,593],{"className":32781},[592],[28,32783,32785],{"className":32784},[565],[28,32786,32788],{"className":32787,"style":600},[569],[28,32789],{}," 仍然是 0.5——它量的是“搬了多少”，不管涨还是跌。",[11,32792,32793,32794,32863],{},"一般地，一维的 ",[28,32795,32797,32814],{"className":32796,"translate":193},[197],[28,32798,32800],{"className":32799},[201],[203,32801,32802],{"xmlns":205},[208,32803,32804,32812],{},[211,32805,32806],{},[501,32807,32808,32810],{},[214,32809,22957],{},[214,32811,11],{},[253,32813,24661],{"encoding":255},[28,32815,32817],{"className":32816,"ariaHidden":261},[260],[28,32818,32820,32823],{"className":32819},[265],[28,32821],{"className":32822,"style":20926},[269],[28,32824,32826,32829],{"className":32825},[274],[28,32827,22957],{"className":32828,"style":459},[274,275],[28,32830,32832],{"className":32831},[556],[28,32833,32835,32855],{"className":32834},[560,561],[28,32836,32838,32852],{"className":32837},[565],[28,32839,32841],{"className":32840,"style":23070},[569],[28,32842,32843,32846],{"style":20948},[28,32844],{"className":32845,"style":578},[577],[28,32847,32849],{"className":32848},[582,583,584,585],[28,32850,11],{"className":32851},[274,275,585],[28,32853,593],{"className":32854},[592],[28,32856,32858],{"className":32857},[565],[28,32859,32861],{"className":32860,"style":20967},[569],[28,32862],{}," 有闭式解：",[28,32865,32867],{"className":32866,"translate":193},[192],[28,32868,32870,32965],{"className":32869,"translate":193},[197],[28,32871,32873],{"className":32872},[201],[203,32874,32875],{"xmlns":205,"display":206},[208,32876,32877,32962],{},[211,32878,32879,32887,32889,32891,32893,32895,32897,32899,32907,32909,32922,32924,32926,32928,32930,32942,32944,32946,32948,32954,32956,32958,32960],{},[32198,32880,32881,32883,32885],{},[214,32882,22957],{},[214,32884,11],{},[214,32886,11],{},[217,32888,43],{"stretchy":241},[214,32890,20756],{},[217,32892,1683],{"separator":261},[214,32894,20787],{},[217,32896,1522],{"stretchy":241},[217,32898,83],{},[32198,32900,32901,32903,32905],{},[217,32902,21282],{},[224,32904,89],{},[224,32906,1244],{},[217,32908,889],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[32198,32910,32911,32914,32916],{},[214,32912,32913],{},"F",[214,32915,20756],{},[211,32917,32918,32920],{},[217,32919,5074],{},[224,32921,1244],{},[217,32923,43],{"stretchy":241},[214,32925,1241],{},[217,32927,1522],{"stretchy":241},[217,32929,5074],{},[32198,32931,32932,32934,32936],{},[214,32933,32913],{},[214,32935,20787],{},[211,32937,32938,32940],{},[217,32939,5074],{},[224,32941,1244],{},[217,32943,43],{"stretchy":241},[214,32945,1241],{},[217,32947,1522],{"stretchy":241},[3054,32949,32950,32952],{},[217,32951,889],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,32953,11],{},[228,32955,488],{},[214,32957,12988],{},[214,32959,1241],{},[217,32961,1683],{"separator":261},[253,32963,32964],{"encoding":255},"W_p^p(\\mu,\\nu)=\\int_0^1\\bigl|F_\\mu^{-1}(t)-F_\\nu^{-1}(t)\\bigr|^p\\,dt ,",[28,32966,32968,33055,33232],{"className":32967,"ariaHidden":261},[260],[28,32969,32971,32975,33028,33031,33034,33037,33040,33043,33046,33049,33052],{"className":32970},[265],[28,32972],{"className":32973,"style":32974},[269],"height:1.1331em;vertical-align:-0.3831em;",[28,32976,32978,32981],{"className":32977},[274],[28,32979,22957],{"className":32980,"style":459},[274,275],[28,32982,32984],{"className":32983},[556],[28,32985,32987,33019],{"className":32986},[560,561],[28,32988,32990,33016],{"className":32989},[565],[28,32991,32993,33005],{"className":32992,"style":23820},[569],[28,32994,32996,32999],{"style":32995},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[28,32997],{"className":32998,"style":578},[577],[28,33000,33002],{"className":33001},[582,583,584,585],[28,33003,11],{"className":33004},[274,275,585],[28,33006,33007,33010],{"style":4880},[28,33008],{"className":33009,"style":578},[577],[28,33011,33013],{"className":33012},[582,583,584,585],[28,33014,11],{"className":33015},[274,275,585],[28,33017,593],{"className":33018},[592],[28,33020,33022],{"className":33021},[565],[28,33023,33026],{"className":33024,"style":33025},[569],"height:0.3831em;",[28,33027],{},[28,33029,43],{"className":33030},[348],[28,33032,20756],{"className":33033},[274,275],[28,33035,1683],{"className":33036},[1723],[28,33038],{"className":33039,"style":530},[279],[28,33041,20787],{"className":33042,"style":20803},[274,275],[28,33044,1522],{"className":33045},[355],[28,33047],{"className":33048,"style":302},[279],[28,33050,83],{"className":33051},[306],[28,33053],{"className":33054,"style":302},[279],[28,33056,33058,33061,33112,33157,33214,33217,33220,33223,33226,33229],{"className":33057},[265],[28,33059],{"className":33060,"style":32241},[269],[28,33062,33064,33067],{"className":33063},[21329],[28,33065,21282],{"className":33066,"style":21548},[21329,21330,21547],[28,33068,33070],{"className":33069},[556],[28,33071,33073,33104],{"className":33072},[560,561],[28,33074,33076,33101],{"className":33075},[565],[28,33077,33079,33090],{"className":33078,"style":32260},[569],[28,33080,33081,33084],{"style":21564},[28,33082],{"className":33083,"style":578},[577],[28,33085,33087],{"className":33086},[582,583,584,585],[28,33088,89],{"className":33089},[274,585],[28,33091,33092,33095],{"style":32274},[28,33093],{"className":33094,"style":578},[577],[28,33096,33098],{"className":33097},[582,583,584,585],[28,33099,1244],{"className":33100},[274,585],[28,33102,593],{"className":33103},[592],[28,33105,33107],{"className":33106},[565],[28,33108,33110],{"className":33109,"style":21586},[569],[28,33111],{},[28,33113,33115],{"className":33114},[348],[28,33116,33119],{"className":33117},[2169,33118],"mult",[28,33120,33122,33148],{"className":33121},[560,561],[28,33123,33125,33145],{"className":33124},[565],[28,33126,33128],{"className":33127,"style":2091},[569],[28,33129,33131,33135],{"style":33130},"top:-2.85em;",[28,33132],{"className":33133,"style":33134},[577],"height:3.2em;",[28,33136,33138],{"style":33137},"width:0.333em;height:1.2em;",[1961,33139,33142],{"xmlns":1963,"width":33140,"height":1810,"viewBox":33141},"0.333em","0 0 333 1200",[1969,33143],{"d":33144},"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 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时有更漂亮的形式。用上一节的对偶，对绝对连续的 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",[28,34318,34320,34371],{"className":34319,"translate":193},[197],[28,34321,34323],{"className":34322},[201],[203,34324,34325],{"xmlns":205},[208,34326,34327,34368],{},[211,34328,34329,34335,34337,34339,34350,34352,34358,34360,34366],{},[3054,34330,34331,34333],{},[214,34332,510],{},[217,34334,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[217,34336,83],{},[217,34338,5074],{},[211,34340,34341,34343,34345,34348],{},[214,34342,1664],{"mathvariant":484},[214,34344,1686],{"mathvariant":484},[214,34346,34347],{"mathvariant":484},"g",[214,34349,3060],{"mathvariant":484},[217,34351,43],{"stretchy":241},[501,34353,34354,34356],{},[214,34355,32913],{},[214,34357,20756],{},[217,34359,5074],{},[501,34361,34362,34364],{},[214,34363,32913],{},[214,34365,20787],{},[217,34367,1522],{"stretchy":241},[253,34369,34370],{"encoding":255},"\\varphi'=-\\mathrm{sign}(F_\\mu-F_\\nu)",[28,34372,34374,34422,34490],{"className":34373,"ariaHidden":261},[260],[28,34375,34377,34381,34413,34416,34419],{"className":34376},[265],[28,34378],{"className":34379,"style":34380},[269],"height:0.9463em;vertical-align:-0.1944em;",[28,34382,34384,34387],{"className":34383},[274],[28,34385,510],{"className":34386},[274,275],[28,34388,34390],{"className":34389},[556],[28,34391,34393],{"className":34392},[560],[28,34394,34396],{"className":34395},[565],[28,34397,34399],{"className":34398,"style":6798},[569],[28,34400,34401,34404],{"style":3094},[28,34402],{"className":34403,"style":578},[577],[28,34405,34407],{"className":34406},[582,583,584,585],[28,34408,34410],{"className":34409},[274,585],[28,34411,6739],{"className":34412},[274,585],[28,34414],{"className":34415,"style":302},[279],[28,34417,83],{"className":34418},[306],[28,34420],{"className":34421,"style":302},[279],[28,34423,34425,34428,34431,34438,34441,34481,34484,34487],{"className":34424},[265],[28,34426],{"className":34427,"style":21101},[269],[28,34429,5074],{"className":34430},[274],[28,34432,34434],{"className":34433},[274],[28,34435,34437],{"className":34436},[274,1712],"sign",[28,34439,43],{"className":34440},[348],[28,34442,34444,34447],{"className":34443},[274],[28,34445,32913],{"className":34446,"style":459},[274,275],[28,34448,34450],{"className":34449},[556],[28,34451,34453,34473],{"className":34452},[560,561],[28,34454,34456,34470],{"className":34455},[565],[28,34457,34459],{"className":34458,"style":23070},[569],[28,34460,34461,34464],{"style":20948},[28,34462],{"className":34463,"style":578},[577],[28,34465,34467],{"className":34466},[582,583,584,585],[28,34468,20756],{"className":34469},[274,275,585],[28,34471,593],{"className":34472},[592],[28,34474,34476],{"className":34475},[565],[28,34477,34479],{"className":34478,"style":20967},[569],[28,34480],{},[28,34482],{"className":34483,"style":280},[279],[28,34485,5074],{"className":34486},[284],[28,34488],{"className":34489,"style":280},[279],[28,34491,34493,34496,34536],{"className":34492},[265],[28,34494],{"className":34495,"style":341},[269],[28,34497,34499,34502],{"className":34498},[274],[28,34500,32913],{"className":34501,"style":459},[274,275],[28,34503,34505],{"className":34504},[556],[28,34506,34508,34528],{"className":34507},[560,561],[28,34509,34511,34525],{"className":34510},[565],[28,34512,34514],{"className":34513,"style":23070},[569],[28,34515,34516,34519],{"style":20948},[28,34517],{"className":34518,"style":578},[577],[28,34520,34522],{"className":34521},[582,583,584,585],[28,34523,20787],{"className":34524,"style":20803},[274,275,585],[28,34526,593],{"className":34527},[592],[28,34529,34531],{"className":34530},[565],[28,34532,34534],{"className":34533,"style":600},[569],[28,34535],{},[28,34537,1522],{"className":34538},[355]," 时等号成立。于是",[28,34541,34543],{"className":34542,"translate":193},[192],[28,34544,34546,34621],{"className":34545,"translate":193},[197],[28,34547,34549],{"className":34548},[201],[203,34550,34551],{"xmlns":205,"display":206},[208,34552,34553,34618],{},[211,34554,34555,34561,34563,34565,34567,34569,34571,34573,34580,34582,34588,34590,34592,34594,34596,34602,34604,34606,34608,34610,34612,34614,34616],{},[501,34556,34557,34559],{},[214,34558,22957],{},[224,34560,1244],{},[217,34562,43],{"stretchy":241},[214,34564,20756],{},[217,34566,1683],{"separator":261},[214,34568,20787],{},[217,34570,1522],{"stretchy":241},[217,34572,83],{},[501,34574,34575,34577],{},[217,34576,21282],{},[214,34578,34579],{"mathvariant":12057},"R",[217,34581,889],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[501,34583,34584,34586],{},[214,34585,32913],{},[214,34587,20756],{},[217,34589,43],{"stretchy":241},[214,34591,176],{},[217,34593,1522],{"stretchy":241},[217,34595,5074],{},[501,34597,34598,34600],{},[214,34599,32913],{},[214,34601,20787],{},[217,34603,43],{"stretchy":241},[214,34605,176],{},[217,34607,1522],{"stretchy":241},[217,34609,889],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[228,34611,488],{},[214,34613,12988],{},[214,34615,176],{},[214,34617,494],{"mathvariant":484},[253,34619,34620],{"encoding":255},"W_1(\\mu,\\nu)=\\int_{\\mathbb R}\\bigl|F_\\mu(x)-F_\\nu(x)\\bigr|\\,dx .",[28,34622,34624,34697,34842],{"className":34623,"ariaHidden":261},[260],[28,34625,34627,34630,34670,34673,34676,34679,34682,34685,34688,34691,34694],{"className":34626},[265],[28,34628],{"className":34629,"style":341},[269],[28,34631,34633,34636],{"className":34632},[274],[28,34634,22957],{"className":34635,"style":459},[274,275],[28,34637,34639],{"className":34638},[556],[28,34640,34642,34662],{"className":34641},[560,561],[28,34643,34645,34659],{"className":34644},[565],[28,34646,34648],{"className":34647,"style":1285},[569],[28,34649,34650,34653],{"style":20948},[28,34651],{"className":34652,"style":578},[577],[28,34654,34656],{"className":34655},[582,583,584,585],[28,34657,1244],{"className":34658},[274,585],[28,34660,593],{"className":34661},[592],[28,34663,34665],{"className":34664},[565],[28,34666,34668],{"className":34667,"style":600},[569],[28,34669],{},[28,34671,43],{"className":34672},[348],[28,34674,20756],{"className":34675},[274,275],[28,34677,1683],{"className":34678},[1723],[28,34680],{"className":34681,"style":530},[279],[28,34683,20787],{"className":34684,"style":20803},[274,275],[28,34686,1522],{"className":34687},[355],[28,34689],{"className":34690,"style":302},[279],[28,34692,83],{"className":34693},[306],[28,34695],{"className":34696,"style":302},[279],[28,34698,34700,34703,34747,34784,34824,34827,34830,34833,34836,34839],{"className":34699},[265],[28,34701],{"className":34702,"style":21481},[269],[28,34704,34706,34709],{"className":34705},[21329],[28,34707,21282],{"className":34708,"style":21548},[21329,21330,21547],[28,34710,34712],{"className":34711},[556],[28,34713,34715,34739],{"className":34714},[560,561],[28,34716,34718,34736],{"className":34717},[565],[28,34719,34722],{"className":34720,"style":34721},[569],"height:-0.4297em;",[28,34723,34724,34727],{"style":21564},[28,34725],{"className":34726,"style":578},[577],[28,34728,34730],{"className":34729},[582,583,584,585],[28,34731,34733],{"className":34732},[274,585],[28,34734,34579],{"className":34735},[274,12195,585],[28,34737,593],{"className":34738},[592],[28,34740,34742],{"className":34741},[565],[28,34743,34745],{"className":34744,"style":21586},[569],[28,34746],{},[28,34748,34750],{"className":34749},[348],[28,34751,34753],{"className":34752},[2169,33118],[28,34754,34756,34776],{"className":34755},[560,561],[28,34757,34759,34773],{"className":34758},[565],[28,34760,34762],{"className":34761,"style":2091},[569],[28,34763,34764,34767],{"style":33130},[28,34765],{"className":34766,"style":33134},[577],[28,34768,34769],{"style":33137},[1961,34770,34771],{"xmlns":1963,"width":33140,"height":1810,"viewBox":33141},[1969,34772],{"d":33144},[28,34774,593],{"className":34775},[592],[28,34777,34779],{"className":34778},[565],[28,34780,34782],{"className":34781,"style":33154},[569],[28,34783],{},[28,34785,34787,34790],{"className":34786},[274],[28,34788,32913],{"className":34789,"style":459},[274,275],[28,34791,34793],{"className":34792},[556],[28,34794,34796,34816],{"className":34795},[560,561],[28,34797,34799,34813],{"className":34798},[565],[28,34800,34802],{"className":34801,"style":23070},[569],[28,34803,34804,34807],{"style":20948},[28,34805],{"className":34806,"style":578},[577],[28,34808,34810],{"className":34809},[582,583,584,585],[28,34811,20756],{"className":34812},[274,275,585],[28,34814,593],{"className":34815},[592],[28,34817,34819],{"className":34818},[565],[28,34820,34822],{"className":34821,"style":20967},[569],[28,34823],{},[28,34825,43],{"className":34826},[348],[28,34828,176],{"className":34829},[274,275],[28,34831,1522],{"className":34832},[355],[28,34834],{"className":34835,"style":280},[279],[28,34837,5074],{"className":34838},[284],[28,34840],{"className":34841,"style":280},[279],[28,34843,34845,34848,34888,34891,34894,34897,34934,34937,34940,34943],{"className":34844},[265],[28,34846],{"className":34847,"style":2469},[269],[28,34849,34851,34854],{"className":34850},[274],[28,34852,32913],{"className":34853,"style":459},[274,275],[28,34855,34857],{"className":34856},[556],[28,34858,34860,34880],{"className":34859},[560,561],[28,34861,34863,34877],{"className":34862},[565],[28,34864,34866],{"className":34865,"style":23070},[569],[28,34867,34868,34871],{"style":20948},[28,34869],{"className":34870,"style":578},[577],[28,34872,34874],{"className":34873},[582,583,584,585],[28,34875,20787],{"className":34876,"style":20803},[274,275,585],[28,34878,593],{"className":34879},[592],[28,34881,34883],{"className":34882},[565],[28,34884,34886],{"className":34885,"style":600},[569],[28,34887],{},[28,34889,43],{"className":34890},[348],[28,34892,176],{"className":34893},[274,275],[28,34895,1522],{"className":34896},[355],[28,34898,34900],{"className":34899},[355],[28,34901,34903],{"className":34902},[2169,33118],[28,34904,34906,34926],{"className":34905},[560,561],[28,34907,34909,34923],{"className":34908},[565],[28,34910,34912],{"className":34911,"style":2091},[569],[28,34913,34914,34917],{"style":33130},[28,34915],{"className":34916,"style":33134},[577],[28,34918,34919],{"style":33137},[1961,34920,34921],{"xmlns":1963,"width":33140,"height":1810,"viewBox":33141},[1969,34922],{"d":33144},[28,34924,593],{"className":34925},[592],[28,34927,34929],{"className":34928},[565],[28,34930,34932],{"className":34931,"style":33154},[569],[28,34933],{},[28,34935],{"className":34936,"style":530},[279],[28,34938,12988],{"className":34939},[274,275],[28,34941,176],{"className":34942},[274,275],[28,34944,494],{"className":34945},[274],[11,34947,34948,34949,35001,35002,177,35068,35143,35144,35180,35181,35281,35282,35001,35334,35143,35400,35458,35459,35558,35559,35659],{},"两条 CDF 曲线之间的面积，就是搬运的总量。拿刚才的薪资例验一下：在 ",[28,34950,34952,34974],{"className":34951,"translate":193},[197],[28,34953,34955],{"className":34954},[201],[203,34956,34957],{"xmlns":205},[208,34958,34959,34971],{},[211,34960,34961,34963,34965,34967,34969],{},[217,34962,73],{"stretchy":241},[224,34964,89],{},[217,34966,1683],{"separator":261},[224,34968,1244],{},[217,34970,246],{"stretchy":241},[253,34972,34973],{"encoding":255},"[0,1]",[28,34975,34977],{"className":34976,"ariaHidden":261},[260],[28,34978,34980,34983,34986,34989,34992,34995,34998],{"className":34979},[265],[28,34981],{"className":34982,"style":341},[269],[28,34984,73],{"className":34985},[348],[28,34987,89],{"className":34988},[274],[28,34990,1683],{"className":34991},[1723],[28,34993],{"className":34994,"style":530},[279],[28,34996,1244],{"className":34997},[274],[28,34999,246],{"className":35000},[355]," 上 ",[28,35003,35005,35029],{"className":35004,"translate":193},[197],[28,35006,35008],{"className":35007},[201],[203,35009,35010],{"xmlns":205},[208,35011,35012,35026],{},[211,35013,35014,35016,35018,35020,35022,35024],{},[214,35015,32913],{},[217,35017,43],{"stretchy":241},[214,35019,176],{},[217,35021,1522],{"stretchy":241},[217,35023,83],{},[214,35025,176],{},[253,35027,35028],{"encoding":255},"F(x)=x",[28,35030,35032,35059],{"className":35031,"ariaHidden":261},[260],[28,35033,35035,35038,35041,35044,35047,35050,35053,35056],{"className":35034},[265],[28,35036],{"className":35037,"style":341},[269],[28,35039,32913],{"className":35040,"style":459},[274,275],[28,35042,43],{"className":35043},[348],[28,35045,176],{"className":35046},[274,275],[28,35048,1522],{"className":35049},[355],[28,35051],{"className":35052,"style":302},[279],[28,35054,83],{"className":35055},[306],[28,35057],{"className":35058,"style":302},[279],[28,35060,35062,35065],{"className":35061},[265],[28,35063],{"className":35064,"style":645},[269],[28,35066,176],{"className":35067},[274,275],[28,35069,35071,35100],{"className":35070,"translate":193},[197],[28,35072,35074],{"className":35073},[201],[203,35075,35076],{"xmlns":205},[208,35077,35078,35097],{},[211,35079,35080,35083,35085,35087,35089,35091,35093,35095],{},[214,35081,35082],{},"G",[217,35084,43],{"stretchy":241},[214,35086,176],{},[217,35088,1522],{"stretchy":241},[217,35090,83],{},[214,35092,176],{},[214,35094,12219],{"mathvariant":484},[224,35096,1253],{},[253,35098,35099],{"encoding":255},"G(x)=x\u002F2",[28,35101,35103,35130],{"className":35102,"ariaHidden":261},[260],[28,35104,35106,35109,35112,35115,35118,35121,35124,35127],{"className":35105},[265],[28,35107],{"className":35108,"style":341},[269],[28,35110,35082],{"className":35111},[274,275],[28,35113,43],{"className":35114},[348],[28,35116,176],{"className":35117},[274,275],[28,35119,1522],{"className":35120},[355],[28,35122],{"className":35123,"style":302},[279],[28,35125,83],{"className":35126},[306],[28,35128],{"className":35129,"style":302},[279],[28,35131,35133,35136,35139],{"className":35132},[265],[28,35134],{"className":35135,"style":341},[269],[28,35137,176],{"className":35138},[274,275],[28,35140,35142],{"className":35141},[274],"\u002F2","，差是 ",[28,35145,35147,35165],{"className":35146,"translate":193},[197],[28,35148,35150],{"className":35149},[201],[203,35151,35152],{"xmlns":205},[208,35153,35154,35162],{},[211,35155,35156,35158,35160],{},[214,35157,176],{},[214,35159,12219],{"mathvariant":484},[224,35161,1253],{},[253,35163,35164],{"encoding":255},"x\u002F2",[28,35166,35168],{"className":35167,"ariaHidden":261},[260],[28,35169,35171,35174,35177],{"className":35170},[265],[28,35172],{"className":35173,"style":341},[269],[28,35175,176],{"className":35176},[274,275],[28,35178,35142],{"className":35179},[274],"，积出 ",[28,35182,35184,35204],{"className":35183,"translate":193},[197],[28,35185,35187],{"className":35186},[201],[203,35188,35189],{"xmlns":205},[208,35190,35191,35201],{},[211,35192,35193],{},[5027,35194,35195],{"scriptlevel":89,"displaystyle":241},[21928,35196,35197,35199],{},[224,35198,1244],{},[224,35200,9215],{},[253,35202,35203],{"encoding":255},"\\tfrac14",[28,35205,35207],{"className":35206,"ariaHidden":261},[260],[28,35208,35210,35213],{"className":35209},[265],[28,35211],{"className":35212,"style":21986},[269],[28,35214,35216,35219,35278],{"className":35215},[274],[28,35217],{"className":35218},[348,8285],[28,35220,35222],{"className":35221},[21928],[28,35223,35225,35270],{"className":35224},[560,561],[28,35226,35228,35267],{"className":35227},[565],[28,35229,35231,35245,35253],{"className":35230,"style":22005},[569],[28,35232,35233,35236],{"style":22008},[28,35234],{"className":35235,"style":1914},[577],[28,35237,35239],{"className":35238},[582,583,584,585],[28,35240,35242],{"className":35241},[274,585],[28,35243,9215],{"className":35244},[274,585],[28,35246,35247,35250],{"style":22023},[28,35248],{"className":35249,"style":1914},[577],[28,35251],{"className":35252,"style":22031},[22030],[28,35254,35255,35258],{"style":22034},[28,35256],{"className":35257,"style":1914},[577],[28,35259,35261],{"className":35260},[582,583,584,585],[28,35262,35264],{"className":35263},[274,585],[28,35265,1244],{"className":35266},[274,585],[28,35268,593],{"className":35269},[592],[28,35271,35273],{"className":35272},[565],[28,35274,35276],{"className":35275,"style":22056},[569],[28,35277],{},[28,35279],{"className":35280},[355,8285],"；在 ",[28,35283,35285,35307],{"className":35284,"translate":193},[197],[28,35286,35288],{"className":35287},[201],[203,35289,35290],{"xmlns":205},[208,35291,35292,35304],{},[211,35293,35294,35296,35298,35300,35302],{},[217,35295,73],{"stretchy":241},[224,35297,1244],{},[217,35299,1683],{"separator":261},[224,35301,1253],{},[217,35303,246],{"stretchy":241},[253,35305,35306],{"encoding":255},"[1,2]",[28,35308,35310],{"className":35309,"ariaHidden":261},[260],[28,35311,35313,35316,35319,35322,35325,35328,35331],{"className":35312},[265],[28,35314],{"className":35315,"style":341},[269],[28,35317,73],{"className":35318},[348],[28,35320,1244],{"className":35321},[274],[28,35323,1683],{"className":35324},[1723],[28,35326],{"className":35327,"style":530},[279],[28,35329,1253],{"className":35330},[274],[28,35332,246],{"className":35333},[355],[28,35335,35337,35361],{"className":35336,"translate":193},[197],[28,35338,35340],{"className":35339},[201],[203,35341,35342],{"xmlns":205},[208,35343,35344,35358],{},[211,35345,35346,35348,35350,35352,35354,35356],{},[214,35347,32913],{},[217,35349,43],{"stretchy":241},[214,35351,176],{},[217,35353,1522],{"stretchy":241},[217,35355,83],{},[224,35357,1244],{},[253,35359,35360],{"encoding":255},"F(x)=1",[28,35362,35364,35391],{"className":35363,"ariaHidden":261},[260],[28,35365,35367,35370,35373,35376,35379,35382,35385,35388],{"className":35366},[265],[28,35368],{"className":35369,"style":341},[269],[28,35371,32913],{"className":35372,"style":459},[274,275],[28,35374,43],{"className":35375},[348],[28,35377,176],{"className":35378},[274,275],[28,35380,1522],{"className":35381},[355],[28,35383],{"className":35384,"style":302},[279],[28,35386,83],{"className":35387},[306],[28,35389],{"className":35390,"style":302},[279],[28,35392,35394,35397],{"className":35393},[265],[28,35395],{"className":35396,"style":316},[269],[28,35398,1244],{"className":35399},[274],[28,35401,35403,35425],{"className":35402,"translate":193},[197],[28,35404,35406],{"className":35405},[201],[203,35407,35408],{"xmlns":205},[208,35409,35410,35422],{},[211,35411,35412,35414,35416,35418,35420],{},[224,35413,1244],{},[217,35415,5074],{},[214,35417,176],{},[214,35419,12219],{"mathvariant":484},[224,35421,1253],{},[253,35423,35424],{"encoding":255},"1-x\u002F2",[28,35426,35428,35446],{"className":35427,"ariaHidden":261},[260],[28,35429,35431,35434,35437,35440,35443],{"className":35430},[265],[28,35432],{"className":35433,"style":6853},[269],[28,35435,1244],{"className":35436},[274],[28,35438],{"className":35439,"style":280},[279],[28,35441,5074],{"className":35442},[284],[28,35444],{"className":35445,"style":280},[279],[28,35447,35449,35452,35455],{"className":35448},[265],[28,35450],{"className":35451,"style":341},[269],[28,35453,176],{"className":35454},[274,275],[28,35456,35142],{"className":35457},[274],"，也积出 ",[28,35460,35462,35481],{"className":35461,"translate":193},[197],[28,35463,35465],{"className":35464},[201],[203,35466,35467],{"xmlns":205},[208,35468,35469,35479],{},[211,35470,35471],{},[5027,35472,35473],{"scriptlevel":89,"displaystyle":241},[21928,35474,35475,35477],{},[224,35476,1244],{},[224,35478,9215],{},[253,35480,35203],{"encoding":255},[28,35482,35484],{"className":35483,"ariaHidden":261},[260],[28,35485,35487,35490],{"className":35486},[265],[28,35488],{"className":35489,"style":21986},[269],[28,35491,35493,35496,35555],{"className":35492},[274],[28,35494],{"className":35495},[348,8285],[28,35497,35499],{"className":35498},[21928],[28,35500,35502,35547],{"className":35501},[560,561],[28,35503,35505,35544],{"className":35504},[565],[28,35506,35508,35522,35530],{"className":35507,"style":22005},[569],[28,35509,35510,35513],{"style":22008},[28,35511],{"className":35512,"style":1914},[577],[28,35514,35516],{"className":35515},[582,583,584,585],[28,35517,35519],{"className":35518},[274,585],[28,35520,9215],{"className":35521},[274,585],[28,35523,35524,35527],{"style":22023},[28,35525],{"className":35526,"style":1914},[577],[28,35528],{"className":35529,"style":22031},[22030],[28,35531,35532,35535],{"style":22034},[28,35533],{"className":35534,"style":1914},[577],[28,35536,35538],{"className":35537},[582,583,584,585],[28,35539,35541],{"className":35540},[274,585],[28,35542,1244],{"className":35543},[274,585],[28,35545,593],{"className":35546},[592],[28,35548,35550],{"className":35549},[565],[28,35551,35553],{"className":35552,"style":22056},[569],[28,35554],{},[28,35556],{"className":35557},[355,8285],"。合计 ",[28,35560,35562,35582],{"className":35561,"translate":193},[197],[28,35563,35565],{"className":35564},[201],[203,35566,35567],{"xmlns":205},[208,35568,35569,35579],{},[211,35570,35571],{},[5027,35572,35573],{"scriptlevel":89,"displaystyle":241},[21928,35574,35575,35577],{},[224,35576,1244],{},[224,35578,1253],{},[253,35580,35581],{"encoding":255},"\\tfrac12",[28,35583,35585],{"className":35584,"ariaHidden":261},[260],[28,35586,35588,35591],{"className":35587},[265],[28,35589],{"className":35590,"style":21986},[269],[28,35592,35594,35597,35656],{"className":35593},[274],[28,35595],{"className":35596},[348,8285],[28,35598,35600],{"className":35599},[21928],[28,35601,35603,35648],{"className":35602},[560,561],[28,35604,35606,35645],{"className":35605},[565],[28,35607,35609,35623,35631],{"className":35608,"style":22005},[569],[28,35610,35611,35614],{"style":22008},[28,35612],{"className":35613,"style":1914},[577],[28,35615,35617],{"className":35616},[582,583,584,585],[28,35618,35620],{"className":35619},[274,585],[28,35621,1253],{"className":35622},[274,585],[28,35624,35625,35628],{"style":22023},[28,35626],{"className":35627,"style":1914},[577],[28,35629],{"className":35630,"style":22031},[22030],[28,35632,35633,35636],{"style":22034},[28,35634],{"className":35635,"style":1914},[577],[28,35637,35639],{"className":35638},[582,583,584,585],[28,35640,35642],{"className":35641},[274,585],[28,35643,1244],{"className":35644},[274,585],[28,35646,593],{"className":35647},[592],[28,35649,35651],{"className":35650},[565],[28,35652,35654],{"className":35653,"style":22056},[569],[28,35655],{},[28,35657],{"className":35658},[355,8285],"，和分位数算法的结果一致。",[11,35661,35662,35663,35797,35798,35961,35962,36142],{},"一维之所以这么简单，是因为凸代价下最优耦合是“单调”的：不会出现 ",[28,35664,35666,35692],{"className":35665,"translate":193},[197],[28,35667,35669],{"className":35668},[201],[203,35670,35671],{"xmlns":205},[208,35672,35673,35689],{},[211,35674,35675,35681,35683],{},[501,35676,35677,35679],{},[214,35678,176],{},[224,35680,1244],{},[217,35682,1383],{},[501,35684,35685,35687],{},[214,35686,176],{},[224,35688,1253],{},[253,35690,35691],{"encoding":255},"x_1\u003Cx_2",[28,35693,35695,35751],{"className":35694,"ariaHidden":261},[260],[28,35696,35698,35702,35742,35745,35748],{"className":35697},[265],[28,35699],{"className":35700,"style":35701},[269],"height:0.6891em;vertical-align:-0.15em;",[28,35703,35705,35708],{"className":35704},[274],[28,35706,176],{"className":35707},[274,275],[28,35709,35711],{"className":35710},[556],[28,35712,35714,35734],{"className":35713},[560,561],[28,35715,35717,35731],{"className":35716},[565],[28,35718,35720],{"className":35719,"style":1285},[569],[28,35721,35722,35725],{"style":573},[28,35723],{"className":35724,"style":578},[577],[28,35726,35728],{"className":35727},[582,583,584,585],[28,35729,1244],{"className":35730},[274,585],[28,35732,593],{"className":35733},[592],[28,35735,35737],{"className":35736},[565],[28,35738,35740],{"className":35739,"style":600},[569],[28,35741],{},[28,35743],{"className":35744,"style":302},[279],[28,35746,1383],{"className":35747},[306],[28,35749],{"className":35750,"style":302},[279],[28,35752,35754,35757],{"className":35753},[265],[28,35755],{"className":35756,"style":3431},[269],[28,35758,35760,35763],{"className":35759},[274],[28,35761,176],{"className":35762},[274,275],[28,35764,35766],{"className":35765},[556],[28,35767,35769,35789],{"className":35768},[560,561],[28,35770,35772,35786],{"className":35771},[565],[28,35773,35775],{"className":35774,"style":1285},[569],[28,35776,35777,35780],{"style":573},[28,35778],{"className":35779,"style":578},[577],[28,35781,35783],{"className":35782},[582,583,584,585],[28,35784,1253],{"className":35785},[274,585],[28,35787,593],{"className":35788},[592],[28,35790,35792],{"className":35791},[565],[28,35793,35795],{"className":35794,"style":600},[569],[28,35796],{}," 却 ",[28,35799,35801,35839],{"className":35800,"translate":193},[197],[28,35802,35804],{"className":35803},[201],[203,35805,35806],{"xmlns":205},[208,35807,35808,35836],{},[211,35809,35810,35812,35814,35820,35822,35824,35826,35828,35834],{},[214,35811,443],{},[217,35813,43],{"stretchy":241},[501,35815,35816,35818],{},[214,35817,176],{},[224,35819,1244],{},[217,35821,1522],{"stretchy":241},[217,35823,130],{},[214,35825,443],{},[217,35827,43],{"stretchy":241},[501,35829,35830,35832],{},[214,35831,176],{},[224,35833,1253],{},[217,35835,1522],{"stretchy":241},[253,35837,35838],{"encoding":255},"T(x_1)>T(x_2)",[28,35840,35842,35906],{"className":35841,"ariaHidden":261},[260],[28,35843,35845,35848,35851,35854,35894,35897,35900,35903],{"className":35844},[265],[28,35846],{"className":35847,"style":341},[269],[28,35849,443],{"className":35850,"style":459},[274,275],[28,35852,43],{"className":35853},[348],[28,35855,35857,35860],{"className":35856},[274],[28,35858,176],{"className":35859},[274,275],[28,35861,35863],{"className":35862},[556],[28,35864,35866,35886],{"className":35865},[560,561],[28,35867,35869,35883],{"className":35868},[565],[28,35870,35872],{"className":35871,"style":1285},[569],[28,35873,35874,35877],{"style":573},[28,35875],{"className":35876,"style":578},[577],[28,35878,35880],{"className":35879},[582,583,584,585],[28,35881,1244],{"className":35882},[274,585],[28,35884,593],{"className":35885},[592],[28,35887,35889],{"className":35888},[565],[28,35890,35892],{"className":35891,"style":600},[569],[28,35893],{},[28,35895,1522],{"className":35896},[355],[28,35898],{"className":35899,"style":302},[279],[28,35901,130],{"className":35902},[306],[28,35904],{"className":35905,"style":302},[279],[28,35907,35909,35912,35915,35918,35958],{"className":35908},[265],[28,35910],{"className":35911,"style":341},[269],[28,35913,443],{"className":35914,"style":459},[274,275],[28,35916,43],{"className":35917},[348],[28,35919,35921,35924],{"className":35920},[274],[28,35922,176],{"className":35923},[274,275],[28,35925,35927],{"className":35926},[556],[28,35928,35930,35950],{"className":35929},[560,561],[28,35931,35933,35947],{"className":35932},[565],[28,35934,35936],{"className":35935,"style":1285},[569],[28,35937,35938,35941],{"style":573},[28,35939],{"className":35940,"style":578},[577],[28,35942,35944],{"className":35943},[582,583,584,585],[28,35945,1253],{"className":35946},[274,585],[28,35948,593],{"className":35949},[592],[28,35951,35953],{"className":35952},[565],[28,35954,35956],{"className":35955,"style":600},[569],[28,35957],{},[28,35959,1522],{"className":35960},[355],"，否则交换一下就更省。所以 ",[28,35963,35965,36002],{"className":35964,"translate":193},[197],[28,35966,35968],{"className":35967},[201],[203,35969,35970],{"xmlns":205},[208,35971,35972,35999],{},[211,35973,35974,35976,35978,35990,35993],{},[214,35975,443],{},[217,35977,83],{},[32198,35979,35980,35982,35984],{},[214,35981,32913],{},[214,35983,20787],{},[211,35985,35986,35988],{},[217,35987,5074],{},[224,35989,1244],{},[217,35991,35992],{},"∘",[501,35994,35995,35997],{},[214,35996,32913],{},[214,35998,20756],{},[253,36000,36001],{"encoding":255},"T=F_\\nu^{-1}\\circ F_\\mu",[28,36003,36005,36023,36096],{"className":36004,"ariaHidden":261},[260],[28,36006,36008,36011,36014,36017,36020],{"className":36007},[265],[28,36009],{"className":36010,"style":455},[269],[28,36012,443],{"className":36013,"style":459},[274,275],[28,36015],{"className":36016,"style":302},[279],[28,36018,83],{"className":36019},[306],[28,36021],{"className":36022,"style":302},[279],[28,36024,36026,36030,36087,36090,36093],{"className":36025},[265],[28,36027],{"className":36028,"style":36029},[269],"height:1.0611em;vertical-align:-0.247em;",[28,36031,36033,36036],{"className":36032},[274],[28,36034,32913],{"className":36035,"style":459},[274,275],[28,36037,36039],{"className":36038},[556],[28,36040,36042,36079],{"className":36041},[560,561],[28,36043,36045,36076],{"className":36044},[565],[28,36046,36048,36059],{"className":36047,"style":5477},[569],[28,36049,36050,36053],{"style":32995},[28,36051],{"className":36052,"style":578},[577],[28,36054,36056],{"className":36055},[582,583,584,585],[28,36057,20787],{"className":36058,"style":20803},[274,275,585],[28,36060,36061,36064],{"style":3094},[28,36062],{"className":36063,"style":578},[577],[28,36065,36067],{"className":36066},[582,583,584,585],[28,36068,36070,36073],{"className":36069},[274,585],[28,36071,5074],{"className":36072},[274,585],[28,36074,1244],{"className":36075},[274,585],[28,36077,593],{"className":36078},[592],[28,36080,36082],{"className":36081},[565],[28,36083,36085],{"className":36084,"style":33294},[569],[28,36086],{},[28,36088],{"className":36089,"style":280},[279],[28,36091,35992],{"className":36092},[284],[28,36094],{"className":36095,"style":280},[279],[28,36097,36099,36102],{"className":36098},[265],[28,36100],{"className":36101,"style":20926},[269],[28,36103,36105,36108],{"className":36104},[274],[28,36106,32913],{"className":36107,"style":459},[274,275],[28,36109,36111],{"className":36110},[556],[28,36112,36114,36134],{"className":36113},[560,561],[28,36115,36117,36131],{"className":36116},[565],[28,36118,36120],{"className":36119,"style":23070},[569],[28,36121,36122,36125],{"style":20948},[28,36123],{"className":36124,"style":578},[577],[28,36126,36128],{"className":36127},[582,583,584,585],[28,36129,20756],{"className":36130},[274,275,585],[28,36132,593],{"className":36133},[592],[28,36135,36137],{"className":36136},[565],[28,36138,36140],{"className":36139,"style":20967},[569],[28,36141],{}," 就是答案——按排名对齐。多维没有这种排序，所有困难都从这里开始。",[11,36144,36145],{},"高斯之间有一个著名的闭式解：",[28,36147,36149],{"className":36148,"translate":193},[192],[28,36150,36152,36323],{"className":36151,"translate":193},[197],[28,36153,36155],{"className":36154},[201],[203,36156,36157],{"xmlns":205,"display":206},[208,36158,36159,36320],{},[211,36160,36161,36169,36171,36173,36175,36181,36183,36190,36192,36194,36196,36198,36204,36206,36212,36214,36216,36218,36220,36226,36228,36234,36240,36242,36248,36250,36256,36258,36264,36266,36268,36270,36284,36290,36304,36316,36318],{},[32198,36162,36163,36165,36167],{},[214,36164,22957],{},[224,36166,1253],{},[224,36168,1253],{},[217,36170,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,36172,24766],{"mathvariant":20826},[217,36174,43],{"stretchy":241},[501,36176,36177,36179],{},[214,36178,12983],{},[224,36180,1244],{},[217,36182,1683],{"separator":261},[501,36184,36185,36188],{},[214,36186,36187],{"mathvariant":484},"Σ",[224,36189,1244],{},[217,36191,1522],{"stretchy":241},[217,36193,1683],{"separator":261},[214,36195,24766],{"mathvariant":20826},[217,36197,43],{"stretchy":241},[501,36199,36200,36202],{},[214,36201,12983],{},[224,36203,1253],{},[217,36205,1683],{"separator":261},[501,36207,36208,36210],{},[214,36209,36187],{"mathvariant":484},[224,36211,1253],{},[217,36213,1522],{"stretchy":241},[217,36215,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,36217,83],{},[214,36219,24404],{"mathvariant":484},[501,36221,36222,36224],{},[214,36223,12983],{},[224,36225,1244],{},[217,36227,5074],{},[501,36229,36230,36232],{},[214,36231,12983],{},[224,36233,1253],{},[3054,36235,36236,36238],{},[214,36237,24404],{"mathvariant":484},[224,36239,1253],{},[217,36241,124],{},[211,36243,36244,36246],{},[214,36245,1241],{"mathvariant":484},[214,36247,1867],{"mathvariant":484},[217,36249,43],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[501,36251,36252,36254],{},[214,36253,36187],{"mathvariant":484},[224,36255,1244],{},[217,36257,124],{},[501,36259,36260,36262],{},[214,36261,36187],{"mathvariant":484},[224,36263,1253],{},[217,36265,5074],{},[224,36267,1253],{},[217,36269,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[32198,36271,36272,36274,36276],{},[214,36273,36187],{"mathvariant":484},[224,36275,1244],{},[211,36277,36278,36280,36282],{},[224,36279,1244],{},[214,36281,12219],{"mathvariant":484},[224,36283,1253],{},[501,36285,36286,36288],{},[214,36287,36187],{"mathvariant":484},[224,36289,1253],{},[32198,36291,36292,36294,36296],{},[214,36293,36187],{"mathvariant":484},[224,36295,1244],{},[211,36297,36298,36300,36302],{},[224,36299,1244],{},[214,36301,12219],{"mathvariant":484},[224,36303,1253],{},[3054,36305,36306,36308],{},[217,36307,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[211,36309,36310,36312,36314],{},[224,36311,1244],{},[214,36313,12219],{"mathvariant":484},[224,36315,1253],{},[217,36317,1522],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[214,36319,494],{"mathvariant":484},[253,36321,36322],{"encoding":255},"W_2^2\\bigl(\\mathcal 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",[28,37082,37084,37142],{"className":37083,"translate":193},[197],[28,37085,37087],{"className":37086},[201],[203,37088,37089],{"xmlns":205},[208,37090,37091,37139],{},[211,37092,37093,37095,37101,37103,37109,37115,37117,37119,37125,37127,37133],{},[217,37094,43],{"stretchy":241},[501,37096,37097,37099],{},[214,37098,12983],{},[224,37100,1244],{},[217,37102,5074],{},[501,37104,37105,37107],{},[214,37106,12983],{},[224,37108,1253],{},[3054,37110,37111,37113],{},[217,37112,1522],{"stretchy":241},[224,37114,1253],{},[217,37116,124],{},[217,37118,43],{"stretchy":241},[501,37120,37121,37123],{},[214,37122,491],{},[224,37124,1244],{},[217,37126,5074],{},[501,37128,37129,37131],{},[214,37130,491],{},[224,37132,1253],{},[3054,37134,37135,37137],{},[217,37136,1522],{"stretchy":241},[224,37138,1253],{},[253,37140,37141],{"encoding":255},"(m_1-m_2)^2+(\\sigma_1-\\sigma_2)^2",[28,37143,37145,37203,37287,37345],{"className":37144,"ariaHidden":261},[260],[28,37146,37148,37151,37154,37194,37197,37200],{"className":37147},[265],[28,37149],{"className":37150,"style":341},[269],[28,37152,43],{"className":37153},[348],[28,37155,37157,37160],{"className":37156},[274],[28,37158,12983],{"className":37159},[274,275],[28,37161,37163],{"className":37162},[556],[28,37164,37166,37186],{"className":37165},[560,561],[28,37167,37169,37183],{"className":37168},[565],[28,37170,37172],{"className":37171,"style":1285},[569],[28,37173,37174,37177],{"style":573},[28,37175],{"className":37176,"style":578},[577],[28,37178,37180],{"className":37179},[582,583,584,585],[28,37181,1244],{"className":37182},[274,585],[28,37184,593],{"className":37185},[592],[28,37187,37189],{"className":37188},[565],[28,37190,37192],{"className":37191,"style":600},[569],[28,37193],{},[28,37195],{"className":37196,"style":280},[279],[28,37198,5074],{"className":37199},[284],[28,37201],{"className":37202,"style":280},[279],[28,37204,37206,37209,37249,37278,37281,37284],{"className":37205},[265],[28,37207],{"className":37208,"style":31837},[269],[28,37210,37212,37215],{"className":37211},[274],[28,37213,12983],{"className":37214},[274,275],[28,37216,37218],{"className":37217},[556],[28,37219,37221,37241],{"className":37220},[560,561],[28,37222,37224,37238],{"className":37223},[565],[28,37225,37227],{"className":37226,"style":1285},[569],[28,37228,37229,37232],{"style":573},[28,37230],{"className":37231,"style":578},[577],[28,37233,37235],{"className":37234},[582,583,584,585],[28,37236,1253],{"className":37237},[274,585],[28,37239,593],{"className":37240},[592],[28,37242,37244],{"className":37243},[565],[28,37245,37247],{"className":37246,"style":600},[569],[28,37248],{},[28,37250,37252,37255],{"className":37251},[355],[28,37253,1522],{"className":37254},[355],[28,37256,37258],{"className":37257},[556],[28,37259,37261],{"className":37260},[560],[28,37262,37264],{"className":37263},[565],[28,37265,37267],{"className":37266,"style":5477},[569],[28,37268,37269,37272],{"style":3094},[28,37270],{"className":37271,"style":578},[577],[28,37273,37275],{"className":37274},[582,583,584,585],[28,37276,1253],{"className":37277},[274,585],[28,37279],{"className":37280,"style":280},[279],[28,37282,124],{"className":37283},[284],[28,37285],{"className":37286,"style":280},[279],[28,37288,37290,37293,37296,37336,37339,37342],{"className":37289},[265],[28,37291],{"className":37292,"style":341},[269],[28,37294,43],{"className":37295},[348],[28,37297,37299,37302],{"className":37298},[274],[28,37300,491],{"className":37301,"style":298},[274,275],[28,37303,37305],{"className":37304},[556],[28,37306,37308,37328],{"className":37307},[560,561],[28,37309,37311,37325],{"className":37310},[565],[28,37312,37314],{"className":37313,"style":1285},[569],[28,37315,37316,37319],{"style":3660},[28,37317],{"className":37318,"style":578},[577],[28,37320,37322],{"className":37321},[582,583,584,585],[28,37323,1244],{"className":37324},[274,585],[28,37326,593],{"className":37327},[592],[28,37329,37331],{"className":37330},[565],[28,37332,37334],{"className":37333,"style":600},[569],[28,37335],{},[28,37337],{"className":37338,"style":280},[279],[28,37340,5074],{"className":37341},[284],[28,37343],{"className":37344,"style":280},[279],[28,37346,37348,37351,37391],{"className":37347},[265],[28,37349],{"className":37350,"style":31837},[269],[28,37352,37354,37357],{"className":37353},[274],[28,37355,491],{"className":37356,"style":298},[274,275],[28,37358,37360],{"className":37359},[556],[28,37361,37363,37383],{"className":37362},[560,561],[28,37364,37366,37380],{"className":37365},[565],[28,37367,37369],{"className":37368,"style":1285},[569],[28,37370,37371,37374],{"style":3660},[28,37372],{"className":37373,"style":578},[577],[28,37375,37377],{"className":37376},[582,583,584,585],[28,37378,1253],{"className":37379},[274,585],[28,37381,593],{"className":37382},[592],[28,37384,37386],{"className":37385},[565],[28,37387,37389],{"className":37388,"style":600},[569],[28,37390],{},[28,37392,37394,37397],{"className":37393},[355],[28,37395,1522],{"className":37396},[355],[28,37398,37400],{"className":37399},[556],[28,37401,37403],{"className":37402},[560],[28,37404,37406],{"className":37405},[565],[28,37407,37409],{"className":37408,"style":5477},[569],[28,37410,37411,37414],{"style":3094},[28,37412],{"className":37413,"style":578},[577],[28,37415,37417],{"className":37416},[582,583,584,585],[28,37418,1253],{"className":37419},[274,585],"，验算一下：",[28,37422,37424,37508],{"className":37423,"translate":193},[197],[28,37425,37427],{"className":37426},[201],[203,37428,37429],{"xmlns":205},[208,37430,37431,37505],{},[211,37432,37433,37439,37441,37449,37451,37459,37461,37463,37465,37471,37477,37479,37481,37483,37485,37491,37493,37499],{},[211,37434,37435,37437],{},[214,37436,1241],{"mathvariant":484},[214,37438,1867],{"mathvariant":484},[217,37440,43],{"stretchy":241},[32198,37442,37443,37445,37447],{},[214,37444,491],{},[224,37446,1244],{},[224,37448,1253],{},[217,37450,124],{},[32198,37452,37453,37455,37457],{},[214,37454,491],{},[224,37456,1253],{},[224,37458,1253],{},[217,37460,5074],{},[224,37462,1253],{},[214,37464,889],{"mathvariant":484},[501,37466,37467,37469],{},[214,37468,491],{},[224,37470,1244],{},[501,37472,37473,37475],{},[214,37474,491],{},[224,37476,1253],{},[214,37478,889],{"mathvariant":484},[217,37480,1522],{"stretchy":241},[217,37482,83],{},[217,37484,43],{"stretchy":241},[501,37486,37487,37489],{},[214,37488,491],{},[224,37490,1244],{},[217,37492,5074],{},[501,37494,37495,37497],{},[214,37496,491],{},[224,37498,1253],{},[3054,37500,37501,37503],{},[217,37502,1522],{"stretchy":241},[224,37504,1253],{},[253,37506,37507],{"encoding":255},"\\mathrm{tr}(\\sigma_1^2+\\sigma_2^2-2|\\sigma_1\\sigma_2|)=(\\sigma_1-\\sigma_2)^2",[28,37509,37511,37588,37655,37760,37818],{"className":37510,"ariaHidden":261},[260],[28,37512,37514,37517,37523,37526,37579,37582,37585],{"className":37513},[265],[28,37515],{"className":37516,"style":31837},[269],[28,37518,37520],{"className":37519},[274],[28,37521,36754],{"className":37522},[274,1712],[28,37524,43],{"className":37525},[348],[28,37527,37529,37532],{"className":37528},[274],[28,37530,491],{"className":37531,"style":298},[274,275],[28,37533,37535],{"className":37534},[556],[28,37536,37538,37570],{"className":37537},[560,561],[28,37539,37541,37567],{"className":37540},[565],[28,37542,37544,37556],{"className":37543,"style":5477},[569],[28,37545,37547,37550],{"style":37546},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[28,37548],{"className":37549,"style":578},[577],[28,37551,37553],{"className":37552},[582,583,584,585],[28,37554,1244],{"className":37555},[274,585],[28,37557,37558,37561],{"style":3094},[28,37559],{"className":37560,"style":578},[577],[28,37562,37564],{"className":37563},[582,583,584,585],[28,37565,1253],{"className":37566},[274,585],[28,37568,593],{"className":37569},[592],[28,37571,37573],{"className":37572},[565],[28,37574,37577],{"className":37575,"style":37576},[569],"height:0.2481em;",[28,37578],{},[28,37580],{"className":37581,"style":280},[279],[28,37583,124],{"className":37584},[284],[28,37586],{"className":37587,"style":280},[279],[28,37589,37591,37595,37646,37649,37652],{"className":37590},[265],[28,37592],{"className":37593,"style":37594},[269],"height:1.0622em;vertical-align:-0.2481em;",[28,37596,37598,37601],{"className":37597},[274],[28,37599,491],{"className":37600,"style":298},[274,275],[28,37602,37604],{"className":37603},[556],[28,37605,37607,37638],{"className":37606},[560,561],[28,37608,37610,37635],{"className":37609},[565],[28,37611,37613,37624],{"className":37612,"style":5477},[569],[28,37614,37615,37618],{"style":37546},[28,37616],{"className":37617,"style":578},[577],[28,37619,37621],{"className":37620},[582,583,584,585],[28,37622,1253],{"className":37623},[274,585],[28,37625,37626,37629],{"style":3094},[28,37627],{"className":37628,"style":578},[577],[28,37630,37632],{"className":37631},[582,583,584,585],[28,37633,1253],{"className":37634},[274,585],[28,37636,593],{"className":37637},[592],[28,37639,37641],{"className":37640},[565],[28,37642,37644],{"className":37643,"style":37576},[569],[28,37645],{},[28,37647],{"className":37648,"style":280},[279],[28,37650,5074],{"className":37651},[284],[28,37653],{"className":37654,"style":280},[279],[28,37656,37658,37661,37665,37705,37745,37748,37751,37754,37757],{"className":37657},[265],[28,37659],{"className":37660,"style":341},[269],[28,37662,37664],{"className":37663},[274],"2∣",[28,37666,37668,37671],{"className":37667},[274],[28,37669,491],{"className":37670,"style":298},[274,275],[28,37672,37674],{"className":37673},[556],[28,37675,37677,37697],{"className":37676},[560,561],[28,37678,37680,37694],{"className":37679},[565],[28,37681,37683],{"className":37682,"style":1285},[569],[28,37684,37685,37688],{"style":3660},[28,37686],{"className":37687,"style":578},[577],[28,37689,37691],{"className":37690},[582,583,584,585],[28,37692,1244],{"className":37693},[274,585],[28,37695,593],{"className":37696},[592],[28,37698,37700],{"className":37699},[565],[28,37701,37703],{"className":37702,"style":600},[569],[28,37704],{},[28,37706,37708,37711],{"className":37707},[274],[28,37709,491],{"className":37710,"style":298},[274,275],[28,37712,37714],{"className":37713},[556],[28,37715,37717,37737],{"className":37716},[560,561],[28,37718,37720,37734],{"className":37719},[565],[28,37721,37723],{"className":37722,"style":1285},[569],[28,37724,37725,37728],{"style":3660},[28,37726],{"className":37727,"style":578},[577],[28,37729,37731],{"className":37730},[582,583,584,585],[28,37732,1253],{"className":37733},[274,585],[28,37735,593],{"className":37736},[592],[28,37738,37740],{"className":37739},[565],[28,37741,37743],{"className":37742,"style":600},[569],[28,37744],{},[28,37746,889],{"className":37747},[274],[28,37749,1522],{"className":37750},[355],[28,37752],{"className":37753,"style":302},[279],[28,37755,83],{"className":37756},[306],[28,37758],{"className":37759,"style":302},[279],[28,37761,37763,37766,37769,37809,37812,37815],{"className":37762},[265],[28,37764],{"className":37765,"style":341},[269],[28,37767,43],{"className":37768},[348],[28,37770,37772,37775],{"className":37771},[274],[28,37773,491],{"className":37774,"style":298},[274,275],[28,37776,37778],{"className":37777},[556],[28,37779,37781,37801],{"className":37780},[560,561],[28,37782,37784,37798],{"className":37783},[565],[28,37785,37787],{"className":37786,"style":1285},[569],[28,37788,37789,37792],{"style":3660},[28,37790],{"className":37791,"style":578},[577],[28,37793,37795],{"className":37794},[582,583,584,585],[28,37796,1244],{"className":37797},[274,585],[28,37799,593],{"className":37800},[592],[28,37802,37804],{"className":37803},[565],[28,37805,37807],{"className":37806,"style":600},[569],[28,37808],{},[28,37810],{"className":37811,"style":280},[279],[28,37813,5074],{"className":37814},[284],[28,37816],{"className":37817,"style":280},[279],[28,37819,37821,37824,37864],{"className":37820},[265],[28,37822],{"className":37823,"style":31837},[269],[28,37825,37827,37830],{"className":37826},[274],[28,37828,491],{"className":37829,"style":298},[274,275],[28,37831,37833],{"className":37832},[556],[28,37834,37836,37856],{"className":37835},[560,561],[28,37837,37839,37853],{"className":37838},[565],[28,37840,37842],{"className":37841,"style":1285},[569],[28,37843,37844,37847],{"style":3660},[28,37845],{"className":37846,"style":578},[577],[28,37848,37850],{"className":37849},[582,583,584,585],[28,37851,1253],{"className":37852},[274,585],[28,37854,593],{"className":37855},[592],[28,37857,37859],{"className":37858},[565],[28,37860,37862],{"className":37861,"style":600},[569],[28,37863],{},[28,37865,37867,37870],{"className":37866},[355],[28,37868,1522],{"className":37869},[355],[28,37871,37873],{"className":37872},[556],[28,37874,37876],{"className":37875},[560],[28,37877,37879],{"className":37878},[565],[28,37880,37882],{"className":37881,"style":5477},[569],[28,37883,37884,37887],{"style":3094},[28,37885],{"className":37886,"style":578},[577],[28,37888,37890],{"className":37889},[582,583,584,585],[28,37891,1253],{"className":37892},[274,585],"，对得上。取 ",[28,37895,37897,37921],{"className":37896,"translate":193},[197],[28,37898,37900],{"className":37899},[201],[203,37901,37902],{"xmlns":205},[208,37903,37904,37918],{},[211,37905,37906,37908,37910,37912,37914,37916],{},[214,37907,24766],{"mathvariant":20826},[217,37909,43],{"stretchy":241},[224,37911,89],{},[217,37913,1683],{"separator":261},[224,37915,1244],{},[217,37917,1522],{"stretchy":241},[253,37919,37920],{"encoding":255},"\\mathcal N(0,1)",[28,37922,37924],{"className":37923,"ariaHidden":261},[260],[28,37925,37927,37930,37933,37936,37939,37942,37945,37948],{"className":37926},[265],[28,37928],{"className":37929,"style":341},[269],[28,37931,24766],{"className":37932,"style":24834},[274,20866],[28,37934,43],{"className":37935},[348],[28,37937,89],{"className":37938},[274],[28,37940,1683],{"className":37941},[1723],[28,37943],{"className":37944,"style":530},[279],[28,37946,1244],{"className":37947},[274],[28,37949,1522],{"className":37950},[355]," 与 ",[28,37953,37955,37979],{"className":37954,"translate":193},[197],[28,37956,37958],{"className":37957},[201],[203,37959,37960],{"xmlns":205},[208,37961,37962,37976],{},[211,37963,37964,37966,37968,37970,37972,37974],{},[214,37965,24766],{"mathvariant":20826},[217,37967,43],{"stretchy":241},[224,37969,89],{},[217,37971,1683],{"separator":261},[224,37973,9215],{},[217,37975,1522],{"stretchy":241},[253,37977,37978],{"encoding":255},"\\mathcal N(0,4)",[28,37980,37982],{"className":37981,"ariaHidden":261},[260],[28,37983,37985,37988,37991,37994,37997,38000,38003,38006],{"className":37984},[265],[28,37986],{"className":37987,"style":341},[269],[28,37989,24766],{"className":37990,"style":24834},[274,20866],[28,37992,43],{"className":37993},[348],[28,37995,89],{"className":37996},[274],[28,37998,1683],{"className":37999},[1723],[28,38001],{"className":38002,"style":530},[279],[28,38004,9215],{"className":38005},[274],[28,38007,1522],{"className":38008},[355],"，得 ",[28,38011,38013,38035],{"className":38012,"translate":193},[197],[28,38014,38016],{"className":38015},[201],[203,38017,38018],{"xmlns":205},[208,38019,38020,38032],{},[211,38021,38022,38028,38030],{},[501,38023,38024,38026],{},[214,38025,22957],{},[224,38027,1253],{},[217,38029,83],{},[224,38031,1244],{},[253,38033,38034],{"encoding":255},"W_2=1",[28,38036,38038,38093],{"className":38037,"ariaHidden":261},[260],[28,38039,38041,38044,38084,38087,38090],{"className":38040},[265],[28,38042],{"className":38043,"style":25059},[269],[28,38045,38047,38050],{"className":38046},[274],[28,38048,22957],{"className":38049,"style":459},[274,275],[28,38051,38053],{"className":38052},[556],[28,38054,38056,38076],{"className":38055},[560,561],[28,38057,38059,38073],{"className":38058},[565],[28,38060,38062],{"className":38061,"style":1285},[569],[28,38063,38064,38067],{"style":20948},[28,38065],{"className":38066,"style":578},[577],[28,38068,38070],{"className":38069},[582,583,584,585],[28,38071,1253],{"className":38072},[274,585],[28,38074,593],{"className":38075},[592],[28,38077,38079],{"className":38078},[565],[28,38080,38082],{"className":38081,"style":600},[569],[28,38083],{},[28,38085],{"className":38086,"style":302},[279],[28,38088,83],{"className":38089},[306],[28,38091],{"className":38092,"style":302},[279],[28,38094,38096,38099],{"className":38095},[265],[28,38097],{"className":38098,"style":316},[269],[28,38100,1244],{"className":38101},[274],"，正好是标准差之差；而最优映射是 ",[28,38104,38106,38127],{"className":38105,"translate":193},[197],[28,38107,38109],{"className":38108},[201],[203,38110,38111],{"xmlns":205},[208,38112,38113,38124],{},[211,38114,38115,38117,38120,38122],{},[214,38116,176],{},[217,38118,38119],{},"↦",[224,38121,1253],{},[214,38123,176],{},[253,38125,38126],{"encoding":255},"x\\mapsto2x",[28,38128,38130,38149],{"className":38129,"ariaHidden":261},[260],[28,38131,38133,38137,38140,38143,38146],{"className":38132},[265],[28,38134],{"className":38135,"style":38136},[269],"height:0.522em;vertical-align:-0.011em;",[28,38138,176],{"className":38139},[274,275],[28,38141],{"className":38142,"style":302},[279],[28,38144,38119],{"className":38145},[306],[28,38147],{"className":38148,"style":302},[279],[28,38150,38152,38155,38158],{"className":38151},[265],[28,38153],{"className":38154,"style":316},[269],[28,38156,1253],{"className":38157},[274],[28,38159,176],{"className":38160},[274,275],"，位移的均方根也正好是 ",[28,38163,38165,38178],{"className":38164,"translate":193},[197],[28,38166,38168],{"className":38167},[201],[203,38169,38170],{"xmlns":205},[208,38171,38172,38176],{},[211,38173,38174],{},[224,38175,1244],{},[253,38177,1244],{"encoding":255},[28,38179,38181],{"className":38180,"ariaHidden":261},[260],[28,38182,38184,38187],{"className":38183},[265],[28,38185],{"className":38186,"style":316},[269],[28,38188,1244],{"className":38189},[274],"。两个数在这里是同一个数。",[11,38192,38193,38194,38401],{},"最优映射还是仿射的：",[28,38195,38197,38243],{"className":38196,"translate":193},[197],[28,38198,38200],{"className":38199},[201],[203,38201,38202],{"xmlns":205},[208,38203,38204,38240],{},[211,38205,38206,38208,38210,38212,38214,38216,38222,38224,38226,38228,38230,38232,38238],{},[214,38207,443],{},[217,38209,43],{"stretchy":241},[214,38211,176],{},[217,38213,1522],{"stretchy":241},[217,38215,83],{},[501,38217,38218,38220],{},[214,38219,12983],{},[224,38221,1253],{},[217,38223,124],{},[214,38225,903],{},[217,38227,43],{"stretchy":241},[214,38229,176],{},[217,38231,5074],{},[501,38233,38234,38236],{},[214,38235,12983],{},[224,38237,1244],{},[217,38239,1522],{"stretchy":241},[253,38241,38242],{"encoding":255},"T(x)=m_2+A(x-m_1)",[28,38244,38246,38273,38328,38352],{"className":38245,"ariaHidden":261},[260],[28,38247,38249,38252,38255,38258,38261,38264,38267,38270],{"className":38248},[265],[28,38250],{"className":38251,"style":341},[269],[28,38253,443],{"className":38254,"style":459},[274,275],[28,38256,43],{"className":38257},[348],[28,38259,176],{"className":38260},[274,275],[28,38262,1522],{"className":38263},[355],[28,38265],{"className":38266,"style":302},[279],[28,38268,83],{"className":38269},[306],[28,38271],{"className":38272,"style":302},[279],[28,38274,38276,38279,38319,38322,38325],{"className":38275},[265],[28,38277],{"className":38278,"style":25252},[269],[28,38280,38282,38285],{"className":38281},[274],[28,38283,12983],{"className":38284},[274,275],[28,38286,38288],{"className":38287},[556],[28,38289,38291,38311],{"className":38290},[560,561],[28,38292,38294,38308],{"className":38293},[565],[28,38295,38297],{"className":38296,"style":1285},[569],[28,38298,38299,38302],{"style":573},[28,38300],{"className":38301,"style":578},[577],[28,38303,38305],{"className":38304},[582,583,584,585],[28,38306,1253],{"className":38307},[274,585],[28,38309,593],{"className":38310},[592],[28,38312,38314],{"className":38313},[565],[28,38315,38317],{"className":38316,"style":600},[569],[28,38318],{},[28,38320],{"className":38321,"style":280},[279],[28,38323,124],{"className":38324},[284],[28,38326],{"className":38327,"style":280},[279],[28,38329,38331,38334,38337,38340,38343,38346,38349],{"className":38330},[265],[28,38332],{"className":38333,"style":341},[269],[28,38335,903],{"className":38336},[274,275],[28,38338,43],{"className":38339},[348],[28,38341,176],{"className":38342},[274,275],[28,38344],{"className":38345,"style":280},[279],[28,38347,5074],{"className":38348},[284],[28,38350],{"className":38351,"style":280},[279],[28,38353,38355,38358,38398],{"className":38354},[265],[28,38356],{"className":38357,"style":341},[269],[28,38359,38361,38364],{"className":38360},[274],[28,38362,12983],{"className":38363},[274,275],[28,38365,38367],{"className":38366},[556],[28,38368,38370,38390],{"className":38369},[560,561],[28,38371,38373,38387],{"className":38372},[565],[28,38374,38376],{"className":38375,"style":1285},[569],[28,38377,38378,38381],{"style":573},[28,38379],{"className":38380,"style":578},[577],[28,38382,38384],{"className":38383},[582,583,584,585],[28,38385,1244],{"className":38386},[274,585],[28,38388,593],{"className":38389},[592],[28,38391,38393],{"className":38392},[565],[28,38394,38396],{"className":38395,"style":600},[569],[28,38397],{},[28,38399,1522],{"className":38400},[355],"，其中",[28,38403,38405],{"className":38404,"translate":193},[192],[28,38406,38408,38506],{"className":38407,"translate":193},[197],[28,38409,38411],{"className":38410},[201],[203,38412,38413],{"xmlns":205,"display":206},[208,38414,38415,38503],{},[211,38416,38417,38419,38421,38437,38439,38453,38459,38473,38485,38501],{},[214,38418,903],{},[217,38420,83],{},[32198,38422,38423,38425,38427],{},[214,38424,36187],{"mathvariant":484},[224,38426,1244],{},[211,38428,38429,38431,38433,38435],{},[217,38430,5074],{},[224,38432,1244],{},[214,38434,12219],{"mathvariant":484},[224,38436,1253],{},[217,38438,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[32198,38440,38441,38443,38445],{},[214,38442,36187],{"mathvariant":484},[224,38444,1244],{},[211,38446,38447,38449,38451],{},[224,38448,1244],{},[214,38450,12219],{"mathvariant":484},[224,38452,1253],{},[501,38454,38455,38457],{},[214,38456,36187],{"mathvariant":484},[224,38458,1253],{},[32198,38460,38461,38463,38465],{},[214,38462,36187],{"mathvariant":484},[224,38464,1244],{},[211,38466,38467,38469,38471],{},[224,38468,1244],{},[214,38470,12219],{"mathvariant":484},[224,38472,1253],{},[3054,38474,38475,38477],{},[217,38476,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[211,38478,38479,38481,38483],{},[224,38480,1244],{},[214,38482,12219],{"mathvariant":484},[224,38484,1253],{},[32198,38486,38487,38489,38491],{},[214,38488,36187],{"mathvariant":484},[224,38490,1244],{},[211,38492,38493,38495,38497,38499],{},[217,38494,5074],{},[224,38496,1244],{},[214,38498,12219],{"mathvariant":484},[224,38500,1253],{},[214,38502,494],{"mathvariant":484},[253,38504,38505],{"encoding":255},"A=\\Sigma_1^{-1\u002F2}\\bigl(\\Sigma_1^{1\u002F2}\\Sigma_2\\Sigma_1^{1\u002F2}\\bigr)^{1\u002F2}\\Sigma_1^{-1\u002F2}.",[28,38507,38509,38527],{"className":38508,"ariaHidden":261},[260],[28,38510,38512,38515,38518,38521,38524],{"className":38511},[265],[28,38513],{"className":38514,"style":455},[269],[28,38516,903],{"className":38517},[274,275],[28,38519],{"className":38520,"style":302},[279],[28,38522,83],{"className":38523},[306],[28,38525],{"className":38526,"style":302},[279],[28,38528,38530,38534,38591,38597,38651,38691,38745,38780,38837],{"className":38529},[265],[28,38531],{"className":38532,"style":38533},[269],"height:1.4779em;vertical-align:-0.35em;",[28,38535,38537,38540],{"className":38536},[274],[28,38538,36187],{"className":38539},[274],[28,38541,38543],{"className":38542},[556],[28,38544,38546,38583],{"className":38545},[560,561],[28,38547,38549,38580],{"className":38548},[565],[28,38550,38552,38563],{"className":38551,"style":36898},[569],[28,38553,38554,38557],{"style":36901},[28,38555],{"className":38556,"style":578},[577],[28,38558,38560],{"className":38559},[582,583,584,585],[28,38561,1244],{"className":38562},[274,585],[28,38564,38565,38568],{"style":36913},[28,38566],{"className":38567,"style":578},[577],[28,38569,38571],{"className":38570},[582,583,584,585],[28,38572,38574,38577],{"className":38573},[274,585],[28,38575,5074],{"className":38576},[274,585],[28,38578,36926],{"className":38579},[274,585],[28,38581,593],{"className":38582},[592],[28,38584,38586],{"className":38585},[565],[28,38587,38589],{"className":38588,"style":36936},[569],[28,38590],{},[28,38592,38594],{"className":38593},[348],[28,38595,43],{"className":38596},[2169,2170],[28,38598,38600,38603],{"className":38599},[274],[28,38601,36187],{"className":38602},[274],[28,38604,38606],{"className":38605},[556],[28,38607,38609,38643],{"className":38608},[560,561],[28,38610,38612,38640],{"className":38611},[565],[28,38613,38615,38626],{"className":38614,"style":36898},[569],[28,38616,38617,38620],{"style":36901},[28,38618],{"className":38619,"style":578},[577],[28,38621,38623],{"className":38622},[582,583,584,585],[28,38624,1244],{"className":38625},[274,585],[28,38627,38628,38631],{"style":36913},[28,38629],{"className":38630,"style":578},[577],[28,38632,38634],{"className":38633},[582,583,584,585],[28,38635,38637],{"className":38636},[274,585],[28,38638,36926],{"className":38639},[274,585],[28,38641,593],{"className":38642},[592],[28,38644,38646],{"className":38645},[565],[28,38647,38649],{"className":38648,"style":36936},[569],[28,38650],{},[28,38652,38654,38657],{"className":38653},[274],[28,38655,36187],{"className":38656},[274],[28,38658,38660],{"className":38659},[556],[28,38661,38663,38683],{"className":38662},[560,561],[28,38664,38666,38680],{"className":38665},[565],[28,38667,38669],{"className":38668,"style":1285},[569],[28,38670,38671,38674],{"style":573},[28,38672],{"className":38673,"style":578},[577],[28,38675,38677],{"className":38676},[582,583,584,585],[28,38678,1253],{"className":38679},[274,585],[28,38681,593],{"className":38682},[592],[28,38684,38686],{"className":38685},[565],[28,38687,38689],{"className":38688,"style":600},[569],[28,38690],{},[28,38692,38694,38697],{"className":38693},[274],[28,38695,36187],{"className":38696},[274],[28,38698,38700],{"className":38699},[556],[28,38701,38703,38737],{"className":38702},[560,561],[28,38704,38706,38734],{"className":38705},[565],[28,38707,38709,38720],{"className":38708,"style":36898},[569],[28,38710,38711,38714],{"style":36901},[28,38712],{"className":38713,"style":578},[577],[28,38715,38717],{"className":38716},[582,583,584,585],[28,38718,1244],{"className":38719},[274,585],[28,38721,38722,38725],{"style":36913},[28,38723],{"className":38724,"style":578},[577],[28,38726,38728],{"className":38727},[582,583,584,585],[28,38729,38731],{"className":38730},[274,585],[28,38732,36926],{"className":38733},[274,585],[28,38735,593],{"className":38736},[592],[28,38738,38740],{"className":38739},[565],[28,38741,38743],{"className":38742,"style":36936},[569],[28,38744],{},[28,38746,38748,38754],{"className":38747},[355],[28,38749,38751],{"className":38750},[355],[28,38752,1522],{"className":38753},[2169,2170],[28,38755,38757],{"className":38756},[556],[28,38758,38760],{"className":38759},[560],[28,38761,38763],{"className":38762},[565],[28,38764,38766],{"className":38765,"style":37054},[569],[28,38767,38768,38771],{"style":33361},[28,38769],{"className":38770,"style":578},[577],[28,38772,38774],{"className":38773},[582,583,584,585],[28,38775,38777],{"className":38776},[274,585],[28,38778,36926],{"className":38779},[274,585],[28,38781,38783,38786],{"className":38782},[274],[28,38784,36187],{"className":38785},[274],[28,38787,38789],{"className":38788},[556],[28,38790,38792,38829],{"className":38791},[560,561],[28,38793,38795,38826],{"className":38794},[565],[28,38796,38798,38809],{"className":38797,"style":36898},[569],[28,38799,38800,38803],{"style":36901},[28,38801],{"className":38802,"style":578},[577],[28,38804,38806],{"className":38805},[582,583,584,585],[28,38807,1244],{"className":38808},[274,585],[28,38810,38811,38814],{"style":36913},[28,38812],{"className":38813,"style":578},[577],[28,38815,38817],{"className":38816},[582,583,584,585],[28,38818,38820,38823],{"className":38819},[274,585],[28,38821,5074],{"className":38822},[274,585],[28,38824,36926],{"className":38825},[274,585],[28,38827,593],{"className":38828},[592],[28,38830,38832],{"className":38831},[565],[28,38833,38835],{"className":38834,"style":36936},[569],[28,38836],{},[28,38838,494],{"className":38839},[274],[11,38841,38842],{},"写成操作就是“先白化再上色”，风格迁移里的 AdaIN 做的正是这个（对角版本）。所以“高斯之间的最优传输”这件事，工程界早就在做了，只是没有这么叫。",[11,38844,38845,38846,38916,38917,39145,39146,39215],{},"高斯集合配上 ",[28,38847,38849,38867],{"className":38848,"translate":193},[197],[28,38850,38852],{"className":38851},[201],[203,38853,38854],{"xmlns":205},[208,38855,38856,38864],{},[211,38857,38858],{},[501,38859,38860,38862],{},[214,38861,22957],{},[224,38863,1253],{},[253,38865,38866],{"encoding":255},"W_2",[28,38868,38870],{"className":38869,"ariaHidden":261},[260],[28,38871,38873,38876],{"className":38872},[265],[28,38874],{"className":38875,"style":25059},[269],[28,38877,38879,38882],{"className":38878},[274],[28,38880,22957],{"className":38881,"style":459},[274,275],[28,38883,38885],{"className":38884},[556],[28,38886,38888,38908],{"className":38887},[560,561],[28,38889,38891,38905],{"className":38890},[565],[28,38892,38894],{"className":38893,"style":1285},[569],[28,38895,38896,38899],{"style":20948},[28,38897],{"className":38898,"style":578},[577],[28,38900,38902],{"className":38901},[582,583,584,585],[28,38903,1253],{"className":38904},[274,585],[28,38906,593],{"className":38907},[592],[28,38909,38911],{"className":38910},[565],[28,38912,38914],{"className":38913,"style":600},[569],[28,38915],{}," 构成一个黎曼流形（Bures–Wasserstein 空间），两点间的测地线就是位移插值 ",[28,38918,38920,38972],{"className":38919,"translate":193},[197],[28,38921,38923],{"className":38922},[201],[203,38924,38925],{"xmlns":205},[208,38926,38927,38969],{},[211,38928,38929,38935,38937,38939,38941,38943,38945,38947,38949,38955,38957,38959,38961,38967],{},[501,38930,38931,38933],{},[214,38932,20756],{},[214,38934,1241],{},[217,38936,83],{},[217,38938,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,38940,43],{"stretchy":241},[224,38942,1244],{},[217,38944,5074],{},[214,38946,1241],{},[217,38948,1522],{"stretchy":241},[211,38950,38951,38953],{},[214,38952,1686],{"mathvariant":484},[214,38954,12988],{"mathvariant":484},[217,38956,124],{},[214,38958,1241],{},[214,38960,443],{},[501,38962,38963,38965],{},[217,38964,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,38966,20907],{"mathvariant":484},[214,38968,20756],{},[253,38970,38971],{"encoding":255},"\\mu_t=\\bigl((1-t)\\mathrm{id}+tT\\bigr)_\\#\\mu",[28,38973,38975,39030,39057,39085],{"className":38974,"ariaHidden":261},[260],[28,38976,38978,38981,39021,39024,39027],{"className":38977},[265],[28,38979],{"className":38980,"style":612},[269],[28,38982,38984,38987],{"className":38983},[274],[28,38985,20756],{"className":38986},[274,275],[28,38988,38990],{"className":38989},[556],[28,38991,38993,39013],{"className":38992},[560,561],[28,38994,38996,39010],{"className":38995},[565],[28,38997,38999],{"className":38998,"style":3978},[569],[28,39000,39001,39004],{"style":573},[28,39002],{"className":39003,"style":578},[577],[28,39005,39007],{"className":39006},[582,583,584,585],[28,39008,1241],{"className":39009},[274,275,585],[28,39011,593],{"className":39012},[592],[28,39014,39016],{"className":39015},[565],[28,39017,39019],{"className":39018,"style":600},[569],[28,39020],{},[28,39022],{"className":39023,"style":302},[279],[28,39025,83],{"className":39026},[306],[28,39028],{"className":39029,"style":302},[279],[28,39031,39033,39036,39042,39045,39048,39051,39054],{"className":39032},[265],[28,39034],{"className":39035,"style":2469},[269],[28,39037,39039],{"className":39038},[348],[28,39040,43],{"className":39041},[2169,2170],[28,39043,43],{"className":39044},[348],[28,39046,1244],{"className":39047},[274],[28,39049],{"className":39050,"style":280},[279],[28,39052,5074],{"className":39053},[284],[28,39055],{"className":39056,"style":280},[279],[28,39058,39060,39063,39066,39069,39076,39079,39082],{"className":39059},[265],[28,39061],{"className":39062,"style":341},[269],[28,39064,1241],{"className":39065},[274,275],[28,39067,1522],{"className":39068},[355],[28,39070,39072],{"className":39071},[274],[28,39073,39075],{"className":39074},[274,1712],"id",[28,39077],{"className":39078,"style":280},[279],[28,39080,124],{"className":39081},[284],[28,39083],{"className":39084,"style":280},[279],[28,39086,39088,39092,39096,39142],{"className":39087},[265],[28,39089],{"className":39090,"style":39091},[269],"height:1.3858em;vertical-align:-0.5358em;",[28,39093,39095],{"className":39094,"style":459},[274,275],"tT",[28,39097,39099,39105],{"className":39098},[355],[28,39100,39102],{"className":39101},[355],[28,39103,1522],{"className":39104},[2169,2170],[28,39106,39108],{"className":39107},[556],[28,39109,39111,39133],{"className":39110},[560,561],[28,39112,39114,39130],{"className":39113},[565],[28,39115,39118],{"className":39116,"style":39117},[569],"height:0.0864em;",[28,39119,39121,39124],{"style":39120},"top:-2.3003em;margin-right:0.05em;",[28,39122],{"className":39123,"style":578},[577],[28,39125,39127],{"className":39126},[582,583,584,585],[28,39128,20907],{"className":39129},[274,585],[28,39131,593],{"className":39132},[592],[28,39134,39136],{"className":39135},[565],[28,39137,39140],{"className":39138,"style":39139},[569],"height:0.5358em;",[28,39141],{},[28,39143,20756],{"className":39144},[274,275],"。这意味着我可以在分布空间里插值和求平均。我在写 Fréchet 均值那篇时说过，在没有向量结构的空间里“平均”要重新定义；",[28,39147,39149,39166],{"className":39148,"translate":193},[197],[28,39150,39152],{"className":39151},[201],[203,39153,39154],{"xmlns":205},[208,39155,39156,39164],{},[211,39157,39158],{},[501,39159,39160,39162],{},[214,39161,22957],{},[224,39163,1253],{},[253,39165,38866],{"encoding":255},[28,39167,39169],{"className":39168,"ariaHidden":261},[260],[28,39170,39172,39175],{"className":39171},[265],[28,39173],{"className":39174,"style":25059},[269],[28,39176,39178,39181],{"className":39177},[274],[28,39179,22957],{"className":39180,"style":459},[274,275],[28,39182,39184],{"className":39183},[556],[28,39185,39187,39207],{"className":39186},[560,561],[28,39188,39190,39204],{"className":39189},[565],[28,39191,39193],{"className":39192,"style":1285},[569],[28,39194,39195,39198],{"style":20948},[28,39196],{"className":39197,"style":578},[577],[28,39199,39201],{"className":39200},[582,583,584,585],[28,39202,1253],{"className":39203},[274,585],[28,39205,593],{"className":39206},[592],[28,39208,39210],{"className":39209},[565],[28,39211,39213],{"className":39212,"style":600},[569],[28,39214],{}," 空间是最典型的例子——两个高斯的中点要沿测地线走一半，不是把参数取平均。",[414,39217,39218],{"id":39218},"计算的代价",[11,39220,39221,39249],{},[28,39222,39224,39237],{"className":39223,"translate":193},[197],[28,39225,39227],{"className":39226},[201],[203,39228,39229],{"xmlns":205},[208,39230,39231,39235],{},[211,39232,39233],{},[214,39234,22957],{},[253,39236,22957],{"encoding":255},[28,39238,39240],{"className":39239,"ariaHidden":261},[260],[28,39241,39243,39246],{"className":39242},[265],[28,39244],{"className":39245,"style":455},[269],[28,39247,22957],{"className":39248,"style":459},[274,275]," 定义了“是什么”，但没告诉我“怎么算”。",[11,39251,39252,39336,39337,39399,39400,39462,39463,39491],{},[28,39253,39255,39277],{"className":39254,"translate":193},[197],[28,39256,39258],{"className":39257},[201],[203,39259,39260],{"xmlns":205},[208,39261,39262,39274],{},[211,39263,39264,39266,39268],{},[214,39265,3060],{},[217,39267,83],{},[3054,39269,39270,39272],{},[224,39271,226],{},[224,39273,9215],{},[253,39275,39276],{"encoding":255},"n=10^4",[28,39278,39280,39298],{"className":39279,"ariaHidden":261},[260],[28,39281,39283,39286,39289,39292,39295],{"className":39282},[265],[28,39284],{"className":39285,"style":645},[269],[28,39287,3060],{"className":39288},[274,275],[28,39290],{"className":39291,"style":302},[279],[28,39293,83],{"className":39294},[306],[28,39296],{"className":39297,"style":302},[279],[28,39299,39301,39304,39307],{"className":39300},[265],[28,39302],{"className":39303,"style":5477},[269],[28,39305,1244],{"className":39306},[274],[28,39308,39310,39313],{"className":39309},[274],[28,39311,89],{"className":39312},[274],[28,39314,39316],{"className":39315},[556],[28,39317,39319],{"className":39318},[560],[28,39320,39322],{"className":39321},[565],[28,39323,39325],{"className":39324,"style":5477},[569],[28,39326,39327,39330],{"style":3094},[28,39328],{"className":39329,"style":578},[577],[28,39331,39333],{"className":39332},[582,583,584,585],[28,39334,9215],{"className":39335},[274,585]," 个样本对 ",[28,39338,39340,39358],{"className":39339,"translate":193},[197],[28,39341,39343],{"className":39342},[201],[203,39344,39345],{"xmlns":205},[208,39346,39347,39355],{},[211,39348,39349],{},[3054,39350,39351,39353],{},[224,39352,226],{},[224,39354,9215],{},[253,39356,39357],{"encoding":255},"10^4",[28,39359,39361],{"className":39360,"ariaHidden":261},[260],[28,39362,39364,39367,39370],{"className":39363},[265],[28,39365],{"className":39366,"style":5477},[269],[28,39368,1244],{"className":39369},[274],[28,39371,39373,39376],{"className":39372},[274],[28,39374,89],{"className":39375},[274],[28,39377,39379],{"className":39378},[556],[28,39380,39382],{"className":39381},[560],[28,39383,39385],{"className":39384},[565],[28,39386,39388],{"className":39387,"style":5477},[569],[28,39389,39390,39393],{"style":3094},[28,39391],{"className":39392,"style":578},[577],[28,39394,39396],{"className":39395},[582,583,584,585],[28,39397,9215],{"className":39398},[274,585]," 个样本，代价矩阵就是 ",[28,39401,39403,39421],{"className":39402,"translate":193},[197],[28,39404,39406],{"className":39405},[201],[203,39407,39408],{"xmlns":205},[208,39409,39410,39418],{},[211,39411,39412],{},[3054,39413,39414,39416],{},[224,39415,226],{},[224,39417,5236],{},[253,39419,39420],{"encoding":255},"10^8",[28,39422,39424],{"className":39423,"ariaHidden":261},[260],[28,39425,39427,39430,39433],{"className":39426},[265],[28,39428],{"className":39429,"style":5477},[269],[28,39431,1244],{"className":39432},[274],[28,39434,39436,39439],{"className":39435},[274],[28,39437,89],{"className":39438},[274],[28,39440,39442],{"className":39441},[556],[28,39443,39445],{"className":39444},[560],[28,39446,39448],{"className":39447},[565],[28,39449,39451],{"className":39450,"style":5477},[569],[28,39452,39453,39456],{"style":3094},[28,39454],{"className":39455,"style":578},[577],[28,39457,39459],{"className":39458},[582,583,584,585],[28,39460,5236],{"className":39461},[274,585]," 个数，float32 存下来 400 MB，还没开始迭代。我最早在自己机器上试 OT，把 ",[28,39464,39466,39479],{"className":39465,"translate":193},[197],[28,39467,39469],{"className":39468},[201],[203,39470,39471],{"xmlns":205},[208,39472,39473,39477],{},[211,39474,39475],{},[214,39476,3060],{},[253,39478,3060],{"encoding":255},[28,39480,39482],{"className":39481,"ariaHidden":261},[260],[28,39483,39485,39488],{"className":39484},[265],[28,39486],{"className":39487,"style":645},[269],[28,39489,3060],{"className":39490},[274,275]," 开到 5000 就被内存劝退了。",[11,39493,39494,39495,39570,39571,39599,39600,39628],{},"离散 OT 是线性规划，实践上是 ",[28,39496,39498,39523],{"className":39497,"translate":193},[197],[28,39499,39501],{"className":39500},[201],[203,39502,39503],{"xmlns":205},[208,39504,39505,39520],{},[211,39506,39507,39510,39512,39518],{},[214,39508,39509],{},"O",[217,39511,43],{"stretchy":241},[3054,39513,39514,39516],{},[214,39515,3060],{},[224,39517,3400],{},[217,39519,1522],{"stretchy":241},[253,39521,39522],{"encoding":255},"O(n^3)",[28,39524,39526],{"className":39525,"ariaHidden":261},[260],[28,39527,39529,39532,39535,39538,39567],{"className":39528},[265],[28,39530],{"className":39531,"style":31837},[269],[28,39533,39509],{"className":39534,"style":2279},[274,275],[28,39536,43],{"className":39537},[348],[28,39539,39541,39544],{"className":39540},[274],[28,39542,3060],{"className":39543},[274,275],[28,39545,39547],{"className":39546},[556],[28,39548,39550],{"className":39549},[560],[28,39551,39553],{"className":39552},[565],[28,39554,39556],{"className":39555,"style":5477},[569],[28,39557,39558,39561],{"style":3094},[28,39559],{"className":39560,"style":578},[577],[28,39562,39564],{"className":39563},[582,583,584,585],[28,39565,3400],{"className":39566},[274,585],[28,39568,1522],{"className":39569},[355]," 量级的算法。更麻烦的是解落在多面体的顶点上，",[28,39572,39574,39587],{"className":39573,"translate":193},[197],[28,39575,39577],{"className":39576},[201],[203,39578,39579],{"xmlns":205},[208,39580,39581,39585],{},[211,39582,39583],{},[214,39584,22365],{},[253,39586,23325],{"encoding":255},[28,39588,39590],{"className":39589,"ariaHidden":261},[260],[28,39591,39593,39596],{"className":39592},[265],[28,39594],{"className":39595,"style":645},[269],[28,39597,22365],{"className":39598,"style":298},[274,275]," 很稀疏，而 ",[28,39601,39603,39616],{"className":39602,"translate":193},[197],[28,39604,39606],{"className":39605},[201],[203,39607,39608],{"xmlns":205},[208,39609,39610,39614],{},[211,39611,39612],{},[214,39613,22957],{},[253,39615,22957],{"encoding":255},[28,39617,39619],{"className":39618,"ariaHidden":261},[260],[28,39620,39622,39625],{"className":39621},[265],[28,39623],{"className":39624,"style":455},[269],[28,39626,22957],{"className":39627,"style":459},[274,275]," 作为代价矩阵的函数是分段线性的，梯度分段常数、在切换点跳变。这种“硬”信号在优化里很难用。",[11,39630,39631],{},"Cuturi 在 2013 年的做法是加熵正则：",[28,39633,39635],{"className":39634,"translate":193},[192],[28,39636,39638,39732],{"className":39637,"translate":193},[197],[28,39639,39641],{"className":39640},[201],[203,39642,39643],{"xmlns":205,"display":206},[208,39644,39645,39729],{},[211,39646,39647,39674,39676,39679,39681,39683,39685,39688,39690,39692,39702,39712,39715,39717,39727],{},[1771,39648,39649,39656],{},[211,39650,39651,39654],{},[214,39652,39653],{},"min",[217,39655,21407],{},[211,39657,39658,39660,39662,39664,39666,39668,39670,39672],{},[214,39659,22365],{},[217,39661,12054],{},[214,39663,22347],{"mathvariant":484},[217,39665,43],{"stretchy":241},[214,39667,20756],{},[217,39669,1683],{"separator":261},[214,39671,20787],{},[217,39673,1522],{"stretchy":241},[228,39675,14860],{},[217,39677,39678],{"stretchy":241},"⟨",[214,39680,15828],{},[217,39682,1683],{"separator":261},[214,39684,22365],{},[217,39686,39687],{"stretchy":241},"⟩",[217,39689,124],{},[214,39691,24625],{},[1771,39693,39694,39696],{},[217,39695,25442],{},[211,39697,39698,39700],{},[214,39699,1686],{},[214,39701,7907],{},[501,39703,39704,39706],{},[214,39705,22365],{},[211,39707,39708,39710],{},[214,39709,1686],{},[214,39711,7907],{},[214,39713,39714],{},"log",[217,39716,21407],{},[501,39718,39719,39721],{},[214,39720,22365],{},[211,39722,39723,39725],{},[214,39724,1686],{},[214,39726,7907],{},[214,39728,494],{"mathvariant":484},[253,39730,39731],{"encoding":255},"\\min_{\\pi\\in\\Pi(\\mu,\\nu)}\\ \\langle C,\\pi\\rangle+\\varepsilon\\sum_{ij}\\pi_{ij}\\log\\pi_{ij}.",[28,39733,39735,39841],{"className":39734,"ariaHidden":261},[260],[28,39736,39738,39742,39811,39814,39817,39820,39823,39826,39829,39832,39835,39838],{"className":39737},[265],[28,39739],{"className":39740,"style":39741},[269],"height:1.716em;vertical-align:-0.966em;",[28,39743,39745],{"className":39744},[21329,21485],[28,39746,39748,39803],{"className":39747},[560,561],[28,39749,39751,39800],{"className":39750},[565],[28,39752,39755,39790],{"className":39753,"style":39754},[569],"height:0.6679em;",[28,39756,39757,39760],{"style":23141},[28,39758],{"className":39759,"style":1914},[577],[28,39761,39763],{"className":39762},[582,583,584,585],[28,39764,39766,39769,39772,39775,39778,39781,39784,39787],{"className":39765},[274,585],[28,39767,22365],{"className":39768,"style":298},[274,275,585],[28,39770,12054],{"className":39771},[306,585],[28,39773,22347],{"className":39774},[274,585],[28,39776,43],{"className":39777},[348,585],[28,39779,20756],{"className":39780},[274,275,585],[28,39782,1683],{"className":39783},[1723,585],[28,39785,20787],{"className":39786,"style":20803},[274,275,585],[28,39788,1522],{"className":39789},[355,585],[28,39791,39792,39795],{"style":1926},[28,39793],{"className":39794,"style":1914},[577],[28,39796,39797],{},[28,39798,39653],{"className":39799},[21329],[28,39801,593],{"className":39802},[592],[28,39804,39806],{"className":39805},[565],[28,39807,39809],{"className":39808,"style":23196},[569],[28,39810],{},[28,39812,14860],{"className":39813},[279],[28,39815,39678],{"className":39816},[348],[28,39818,15828],{"className":39819,"style":28331},[274,275],[28,39821,1683],{"className":39822},[1723],[28,39824],{"className":39825,"style":530},[279],[28,39827,22365],{"className":39828,"style":298},[274,275],[28,39830,39687],{"className":39831},[355],[28,39833],{"className":39834,"style":280},[279],[28,39836,124],{"className":39837},[284],[28,39839],{"className":39840,"style":280},[279],[28,39842,39844,39847,39850,39853,39900,39903,39946,39949,39956,39959,40002],{"className":39843},[265],[28,39845],{"className":39846,"style":30259},[269],[28,39848,24625],{"className":39849},[274,275],[28,39851],{"className":39852,"style":530},[279],[28,39854,39856],{"className":39855},[21329,21485],[28,39857,39859,39892],{"className":39858},[560,561],[28,39860,39862,39889],{"className":39861},[565],[28,39863,39865,39879],{"className":39864,"style":25713},[569],[28,39866,39867,39870],{"style":25716},[28,39868],{"className":39869,"style":25720},[577],[28,39871,39873],{"className":39872},[582,583,584,585],[28,39874,39876],{"className":39875},[274,585],[28,39877,25397],{"className":39878,"style":8057},[274,275,585],[28,39880,39881,39884],{"style":25732},[28,39882],{"className":39883,"style":25720},[577],[28,39885,39886],{},[28,39887,25442],{"className":39888},[21329,21330,21547],[28,39890,593],{"className":39891},[592],[28,39893,39895],{"className":39894},[565],[28,39896,39898],{"className":39897,"style":25750},[569],[28,39899],{},[28,39901],{"className":39902,"style":530},[279],[28,39904,39906,39909],{"className":39905},[274],[28,39907,22365],{"className":39908,"style":298},[274,275],[28,39910,39912],{"className":39911},[556],[28,39913,39915,39938],{"className":39914},[560,561],[28,39916,39918,39935],{"className":39917},[565],[28,39919,39921],{"className":39920,"style":3881},[569],[28,39922,39923,39926],{"style":3660},[28,39924],{"className":39925,"style":578},[577],[28,39927,39929],{"className":39928},[582,583,584,585],[28,39930,39932],{"className":39931},[274,585],[28,39933,25397],{"className":39934,"style":8057},[274,275,585],[28,39936,593],{"className":39937},[592],[28,39939,39941],{"className":39940},[565],[28,39942,39944],{"className":39943,"style":20967},[569],[28,39945],{},[28,39947],{"className":39948,"style":530},[279],[28,39950,39952,39953],{"className":39951},[21329],"lo",[28,39954,34347],{"style":39955},"margin-right:0.0139em;",[28,39957],{"className":39958,"style":530},[279],[28,39960,39962,39965],{"className":39961},[274],[28,39963,22365],{"className":39964,"style":298},[274,275],[28,39966,39968],{"className":39967},[556],[28,39969,39971,39994],{"className":39970},[560,561],[28,39972,39974,39991],{"className":39973},[565],[28,39975,39977],{"className":39976,"style":3881},[569],[28,39978,39979,39982],{"style":3660},[28,39980],{"className":39981,"style":578},[577],[28,39983,39985],{"className":39984},[582,583,584,585],[28,39986,39988],{"className":39987},[274,585],[28,39989,25397],{"className":39990,"style":8057},[274,275,585],[28,39992,593],{"className":39993},[592],[28,39995,39997],{"className":39996},[565],[28,39998,40000],{"className":39999,"style":20967},[569],[28,40001],{},[28,40003,494],{"className":40004},[274],[11,40006,40007],{},"熵项把解从顶点“推”到内部，变成严格正的光滑解。用拉格朗日乘子可以解出",[28,40009,40011],{"className":40010,"translate":193},[192],[28,40012,40014,40103],{"className":40013,"translate":193},[197],[28,40015,40017],{"className":40016},[201],[203,40018,40019],{"xmlns":205,"display":206},[208,40020,40021,40100],{},[211,40022,40023,40033,40035,40041,40051,40057,40059,40061,40071,40073,40076,40078,40080,40082,40092,40094,40096,40098],{},[501,40024,40025,40027],{},[214,40026,22365],{},[211,40028,40029,40031],{},[214,40030,1686],{},[214,40032,7907],{},[217,40034,83],{},[501,40036,40037,40039],{},[214,40038,3936],{},[214,40040,1686],{},[501,40042,40043,40045],{},[214,40044,24390],{},[211,40046,40047,40049],{},[214,40048,1686],{},[214,40050,7907],{},[501,40052,40053,40055],{},[214,40054,1693],{},[214,40056,7907],{},[217,40058,1683],{"separator":261},[279,40060],{"width":4053},[501,40062,40063,40065],{},[214,40064,24390],{},[211,40066,40067,40069],{},[214,40068,1686],{},[214,40070,7907],{},[217,40072,83],{},[214,40074,40075],{},"exp",[217,40077,21407],{},[217,40079,43],{"stretchy":241},[217,40081,5074],{},[501,40083,40084,40086],{},[214,40085,15828],{},[211,40087,40088,40090],{},[214,40089,1686],{},[214,40091,7907],{},[214,40093,12219],{"mathvariant":484},[214,40095,24625],{},[217,40097,1522],{"stretchy":241},[217,40099,1683],{"separator":261},[253,40101,40102],{"encoding":255},"\\pi_{ij}=u_iK_{ij}v_j,\\qquad K_{ij}=\\exp(-C_{ij}\u002F\\varepsilon),",[28,40104,40106,40164,40354],{"className":40105,"ariaHidden":261},[260],[28,40107,40109,40112,40155,40158,40161],{"className":40108},[265],[28,40110],{"className":40111,"style":25364},[269],[28,40113,40115,40118],{"className":40114},[274],[28,40116,22365],{"className":40117,"style":298},[274,275],[28,40119,40121],{"className":40120},[556],[28,40122,40124,40147],{"className":40123},[560,561],[28,40125,40127,40144],{"className":40126},[565],[28,40128,40130],{"className":40129,"style":3881},[569],[28,40131,40132,40135],{"style":3660},[28,40133],{"className":40134,"style":578},[577],[28,40136,40138],{"className":40137},[582,583,584,585],[28,40139,40141],{"className":40140},[274,585],[28,40142,25397],{"className":40143,"style":8057},[274,275,585],[28,40145,593],{"className":40146},[592],[28,40148,40150],{"className":40149},[565],[28,40151,40153],{"className":40152,"style":20967},[569],[28,40154],{},[28,40156],{"className":40157,"style":302},[279],[28,40159,83],{"className":40160},[306],[28,40162],{"className":40163,"style":302},[279],[28,40165,40167,40170,40210,40253,40293,40296,40299,40302,40345,40348,40351],{"className":40166},[265],[28,40168],{"className":40169,"style":20926},[269],[28,40171,40173,40176],{"className":40172},[274],[28,40174,3936],{"className":40175},[274,275],[28,40177,40179],{"className":40178},[556],[28,40180,40182,40202],{"className":40181},[560,561],[28,40183,40185,40199],{"className":40184},[565],[28,40186,40188],{"className":40187,"style":3881},[569],[28,40189,40190,40193],{"style":573},[28,40191],{"className":40192,"style":578},[577],[28,40194,40196],{"className":40195},[582,583,584,585],[28,40197,1686],{"className":40198},[274,275,585],[28,40200,593],{"className":40201},[592],[28,40203,40205],{"className":40204},[565],[28,40206,40208],{"className":40207,"style":600},[569],[28,40209],{},[28,40211,40213,40216],{"className":40212},[274],[28,40214,24390],{"className":40215,"style":28331},[274,275],[28,40217,40219],{"className":40218},[556],[28,40220,40222,40245],{"className":40221},[560,561],[28,40223,40225,40242],{"className":40224},[565],[28,40226,40228],{"className":40227,"style":3881},[569],[28,40229,40230,40233],{"style":28346},[28,40231],{"className":40232,"style":578},[577],[28,40234,40236],{"className":40235},[582,583,584,585],[28,40237,40239],{"className":40238},[274,585],[28,40240,25397],{"className":40241,"style":8057},[274,275,585],[28,40243,593],{"className":40244},[592],[28,40246,40248],{"className":40247},[565],[28,40249,40251],{"className":40250,"style":20967},[569],[28,40252],{},[28,40254,40256,40259],{"className":40255},[274],[28,40257,1693],{"className":40258,"style":298},[274,275],[28,40260,40262],{"className":40261},[556],[28,40263,40265,40285],{"className":40264},[560,561],[28,40266,40268,40282],{"className":40267},[565],[28,40269,40271],{"className":40270,"style":3881},[569],[28,40272,40273,40276],{"style":3660},[28,40274],{"className":40275,"style":578},[577],[28,40277,40279],{"className":40278},[582,583,584,585],[28,40280,7907],{"className":40281,"style":8057},[274,275,585],[28,40283,593],{"className":40284},[592],[28,40286,40288],{"className":40287},[565],[28,40289,40291],{"className":40290,"style":20967},[569],[28,40292],{},[28,40294,1683],{"className":40295},[1723],[28,40297],{"className":40298,"style":4222},[279],[28,40300],{"className":40301,"style":530},[279],[28,40303,40305,40308],{"className":40304},[274],[28,40306,24390],{"className":40307,"style":28331},[274,275],[28,40309,40311],{"className":40310},[556],[28,40312,40314,40337],{"className":40313},[560,561],[28,40315,40317,40334],{"className":40316},[565],[28,40318,40320],{"className":40319,"style":3881},[569],[28,40321,40322,40325],{"style":28346},[28,40323],{"className":40324,"style":578},[577],[28,40326,40328],{"className":40327},[582,583,584,585],[28,40329,40331],{"className":40330},[274,585],[28,40332,25397],{"className":40333,"style":8057},[274,275,585],[28,40335,593],{"className":40336},[592],[28,40338,40340],{"className":40339},[565],[28,40341,40343],{"className":40342,"style":20967},[569],[28,40344],{},[28,40346],{"className":40347,"style":302},[279],[28,40349,83],{"className":40350},[306],[28,40352],{"className":40353,"style":302},[279],[28,40355,40357,40360,40363,40366,40369,40412,40415,40418,40421],{"className":40356},[265],[28,40358],{"className":40359,"style":21101},[269],[28,40361,40075],{"className":40362},[21329],[28,40364,43],{"className":40365},[348],[28,40367,5074],{"className":40368},[274],[28,40370,40372,40375],{"className":40371},[274],[28,40373,15828],{"className":40374,"style":28331},[274,275],[28,40376,40378],{"className":40377},[556],[28,40379,40381,40404],{"className":40380},[560,561],[28,40382,40384,40401],{"className":40383},[565],[28,40385,40387],{"className":40386,"style":3881},[569],[28,40388,40389,40392],{"style":28346},[28,40390],{"className":40391,"style":578},[577],[28,40393,40395],{"className":40394},[582,583,584,585],[28,40396,40398],{"className":40397},[274,585],[28,40399,25397],{"className":40400,"style":8057},[274,275,585],[28,40402,593],{"className":40403},[592],[28,40405,40407],{"className":40406},[565],[28,40408,40410],{"className":40409,"style":20967},[569],[28,40411],{},[28,40413,12219],{"className":40414},[274],[28,40416,24625],{"className":40417},[274,275],[28,40419,1522],{"className":40420},[355],[28,40422,1683],{"className":40423},[1723],[11,40425,40426,40467,40468,177,40545,40653],{},[28,40427,40429,40446],{"className":40428,"translate":193},[197],[28,40430,40432],{"className":40431},[201],[203,40433,40434],{"xmlns":205},[208,40435,40436,40444],{},[211,40437,40438,40440,40442],{},[214,40439,3936],{},[217,40441,1683],{"separator":261},[214,40443,1693],{},[253,40445,26820],{"encoding":255},[28,40447,40449],{"className":40448,"ariaHidden":261},[260],[28,40450,40452,40455,40458,40461,40464],{"className":40451},[265],[28,40453],{"className":40454,"style":612},[269],[28,40456,3936],{"className":40457},[274,275],[28,40459,1683],{"className":40460},[1723],[28,40462],{"className":40463,"style":530},[279],[28,40465,1693],{"className":40466,"style":298},[274,275]," 由边缘约束定出，交替更新 ",[28,40469,40471,40500],{"className":40470,"translate":193},[197],[28,40472,40474],{"className":40473},[201],[203,40475,40476],{"xmlns":205},[208,40477,40478,40497],{},[211,40479,40480,40482,40485,40487,40489,40491,40493,40495],{},[214,40481,3936],{},[217,40483,40484],{},"←",[214,40486,238],{},[214,40488,12219],{"mathvariant":484},[217,40490,43],{"stretchy":241},[214,40492,24390],{},[214,40494,1693],{},[217,40496,1522],{"stretchy":241},[253,40498,40499],{"encoding":255},"u\\leftarrow a\u002F(Kv)",[28,40501,40503,40521],{"className":40502,"ariaHidden":261},[260],[28,40504,40506,40509,40512,40515,40518],{"className":40505},[265],[28,40507],{"className":40508,"style":645},[269],[28,40510,3936],{"className":40511},[274,275],[28,40513],{"className":40514,"style":302},[279],[28,40516,40484],{"className":40517},[306],[28,40519],{"className":40520,"style":302},[279],[28,40522,40524,40527,40530,40533,40536,40539,40542],{"className":40523},[265],[28,40525],{"className":40526,"style":341},[269],[28,40528,238],{"className":40529},[274,275],[28,40531,12219],{"className":40532},[274],[28,40534,43],{"className":40535},[348],[28,40537,24390],{"className":40538,"style":28331},[274,275],[28,40540,1693],{"className":40541,"style":298},[274,275],[28,40543,1522],{"className":40544},[355],[28,40546,40548,40580],{"className":40547,"translate":193},[197],[28,40549,40551],{"className":40550},[201],[203,40552,40553],{"xmlns":205},[208,40554,40555,40577],{},[211,40556,40557,40559,40561,40563,40565,40567,40573,40575],{},[214,40558,1693],{},[217,40560,40484],{},[214,40562,3820],{},[214,40564,12219],{"mathvariant":484},[217,40566,43],{"stretchy":241},[3054,40568,40569,40571],{},[214,40570,24390],{},[214,40572,884],{"mathvariant":484},[214,40574,3936],{},[217,40576,1522],{"stretchy":241},[253,40578,40579],{"encoding":255},"v\\leftarrow b\u002F(K^\\top u)",[28,40581,40583,40601],{"className":40582,"ariaHidden":261},[260],[28,40584,40586,40589,40592,40595,40598],{"className":40585},[265],[28,40587],{"className":40588,"style":645},[269],[28,40590,1693],{"className":40591,"style":298},[274,275],[28,40593],{"className":40594,"style":302},[279],[28,40596,40484],{"className":40597},[306],[28,40599],{"className":40600,"style":302},[279],[28,40602,40604,40608,40611,40614,40617,40647,40650],{"className":40603},[265],[28,40605],{"className":40606,"style":40607},[269],"height:1.0991em;vertical-align:-0.25em;",[28,40609,3820],{"className":40610},[274,275],[28,40612,12219],{"className":40613},[274],[28,40615,43],{"className":40616},[348],[28,40618,40620,40623],{"className":40619},[274],[28,40621,24390],{"className":40622,"style":28331},[274,275],[28,40624,40626],{"className":40625},[556],[28,40627,40629],{"className":40628},[560],[28,40630,40632],{"className":40631},[565],[28,40633,40636],{"className":40634,"style":40635},[569],"height:0.8491em;",[28,40637,40638,40641],{"style":3094},[28,40639],{"className":40640,"style":578},[577],[28,40642,40644],{"className":40643},[582,583,584,585],[28,40645,884],{"className":40646},[274,585],[28,40648,3936],{"className":40649},[274,275],[28,40651,1522],{"className":40652},[355],"，每次都是矩阵—向量乘法。这就是 Sinkhorn 迭代。它的意义不只是快，而是可微、可并到 GPU 上，于是能塞进神经网络当一层用。",[11,40655,40656,40684,40685,40713,40714,40742,40743,40771,40772,12514,40823,40851],{},[28,40657,40659,40672],{"className":40658,"translate":193},[197],[28,40660,40662],{"className":40661},[201],[203,40663,40664],{"xmlns":205},[208,40665,40666,40670],{},[211,40667,40668],{},[214,40669,24625],{},[253,40671,24628],{"encoding":255},[28,40673,40675],{"className":40674,"ariaHidden":261},[260],[28,40676,40678,40681],{"className":40677},[265],[28,40679],{"className":40680,"style":645},[269],[28,40682,24625],{"className":40683},[274,275]," 可以读作一个旋钮。取三个点搬到三个点：",[28,40686,40688,40701],{"className":40687,"translate":193},[197],[28,40689,40691],{"className":40690},[201],[203,40692,40693],{"xmlns":205},[208,40694,40695,40699],{},[211,40696,40697],{},[214,40698,24625],{},[253,40700,24628],{"encoding":255},[28,40702,40704],{"className":40703,"ariaHidden":261},[260],[28,40705,40707,40710],{"className":40706},[265],[28,40708],{"className":40709,"style":645},[269],[28,40711,24625],{"className":40712},[274,275]," 很大时，",[28,40715,40717,40730],{"className":40716,"translate":193},[197],[28,40718,40720],{"className":40719},[201],[203,40721,40722],{"xmlns":205},[208,40723,40724,40728],{},[211,40725,40726],{},[214,40727,24390],{},[253,40729,24390],{"encoding":255},[28,40731,40733],{"className":40732,"ariaHidden":261},[260],[28,40734,40736,40739],{"className":40735},[265],[28,40737],{"className":40738,"style":455},[269],[28,40740,24390],{"className":40741,"style":28331},[274,275]," 接近全 ",[28,40744,40746,40759],{"className":40745,"translate":193},[197],[28,40747,40749],{"className":40748},[201],[203,40750,40751],{"xmlns":205},[208,40752,40753,40757],{},[211,40754,40755],{},[224,40756,1244],{},[253,40758,1244],{"encoding":255},[28,40760,40762],{"className":40761,"ariaHidden":261},[260],[28,40763,40765,40768],{"className":40764},[265],[28,40766],{"className":40767,"style":316},[269],[28,40769,1244],{"className":40770},[274]," 矩阵，每个点都往每处送一点，解是稠密的、模糊的；",[28,40773,40775,40793],{"className":40774,"translate":193},[197],[28,40776,40778],{"className":40777},[201],[203,40779,40780],{"xmlns":205},[208,40781,40782,40790],{},[211,40783,40784,40786,40788],{},[214,40785,24625],{},[217,40787,950],{},[224,40789,89],{},[253,40791,40792],{"encoding":255},"\\varepsilon\\to0",[28,40794,40796,40814],{"className":40795,"ariaHidden":261},[260],[28,40797,40799,40802,40805,40808,40811],{"className":40798},[265],[28,40800],{"className":40801,"style":645},[269],[28,40803,24625],{"className":40804},[274,275],[28,40806],{"className":40807,"style":302},[279],[28,40809,950],{"className":40810},[306],[28,40812],{"className":40813,"style":302},[279],[28,40815,40817,40820],{"className":40816},[265],[28,40818],{"className":40819,"style":316},[269],[28,40821,89],{"className":40822},[274],[28,40824,40826,40839],{"className":40825,"translate":193},[197],[28,40827,40829],{"className":40828},[201],[203,40830,40831],{"xmlns":205},[208,40832,40833,40837],{},[211,40834,40835],{},[214,40836,24390],{},[253,40838,24390],{"encoding":255},[28,40840,40842],{"className":40841,"ariaHidden":261},[260],[28,40843,40845,40848],{"className":40844},[265],[28,40846],{"className":40847,"style":455},[269],[28,40849,24390],{"className":40850,"style":28331},[274,275]," 接近单位矩阵，解退化成“就近硬匹配”，也就是原始 OT。中间是连续插值。",[11,40853,21267,40854,40905,40906,40984,40985,41324],{},[28,40855,40857,40875],{"className":40856,"translate":193},[197],[28,40858,40860],{"className":40859},[201],[203,40861,40862],{"xmlns":205},[208,40863,40864,40872],{},[211,40865,40866,40868,40870],{},[214,40867,24625],{},[217,40869,130],{},[224,40871,89],{},[253,40873,40874],{"encoding":255},"\\varepsilon>0",[28,40876,40878,40896],{"className":40877,"ariaHidden":261},[260],[28,40879,40881,40884,40887,40890,40893],{"className":40880},[265],[28,40882],{"className":40883,"style":2616},[269],[28,40885,24625],{"className":40886},[274,275],[28,40888],{"className":40889,"style":302},[279],[28,40891,130],{"className":40892},[306],[28,40894],{"className":40895,"style":302},[279],[28,40897,40899,40902],{"className":40898},[265],[28,40900],{"className":40901,"style":316},[269],[28,40903,89],{"className":40904},[274]," 时它不再是度量。最直接的一点：把 ",[28,40907,40909,40941],{"className":40908,"translate":193},[197],[28,40910,40912],{"className":40911},[201],[203,40913,40914],{"xmlns":205},[208,40915,40916,40938],{},[211,40917,40918,40920,40922,40932,40934,40936],{},[214,40919,22365],{},[217,40921,83],{},[211,40923,40924,40926,40928,40930],{},[214,40925,12988],{"mathvariant":484},[214,40927,1686],{"mathvariant":484},[214,40929,238],{"mathvariant":484},[214,40931,34347],{"mathvariant":484},[217,40933,43],{"stretchy":241},[214,40935,238],{},[217,40937,1522],{"stretchy":241},[253,40939,40940],{"encoding":255},"\\pi=\\mathrm{diag}(a)",[28,40942,40944,40962],{"className":40943,"ariaHidden":261},[260],[28,40945,40947,40950,40953,40956,40959],{"className":40946},[265],[28,40948],{"className":40949,"style":645},[269],[28,40951,22365],{"className":40952,"style":298},[274,275],[28,40954],{"className":40955,"style":302},[279],[28,40957,83],{"className":40958},[306],[28,40960],{"className":40961,"style":302},[279],[28,40963,40965,40968,40975,40978,40981],{"className":40964},[265],[28,40966],{"className":40967,"style":341},[269],[28,40969,40971],{"className":40970},[274],[28,40972,40974],{"className":40973,"style":39955},[274,1712],"diag",[28,40976,43],{"className":40977},[348],[28,40979,238],{"className":40980},[274,275],[28,40982,1522],{"className":40983},[355]," 代入，",[28,40986,40988,41046],{"className":40987,"translate":193},[197],[28,40989,40991],{"className":40990},[201],[203,40992,40993],{"xmlns":205},[208,40994,40995,41043],{},[211,40996,40997,41003,41005,41007,41009,41011,41013,41015,41017,41023,41029,41031,41033,41039,41041],{},[501,40998,40999,41001],{},[214,41000,22957],{},[214,41002,24625],{},[217,41004,43],{"stretchy":241},[214,41006,20756],{},[217,41008,1683],{"separator":261},[214,41010,20756],{},[217,41012,1522],{"stretchy":241},[217,41014,83],{},[214,41016,24625],{},[501,41018,41019,41021],{},[217,41020,25442],{},[214,41022,1686],{},[501,41024,41025,41027],{},[214,41026,238],{},[214,41028,1686],{},[214,41030,39714],{},[217,41032,21407],{},[501,41034,41035,41037],{},[214,41036,238],{},[214,41038,1686],{},[217,41040,1854],{"mathvariant":484},[224,41042,89],{},[253,41044,41045],{"encoding":255},"W_\\varepsilon(\\mu,\\mu)=\\varepsilon\\sum_i a_i\\log a_i\\ne0",[28,41047,41049,41122,41315],{"className":41048,"ariaHidden":261},[260],[28,41050,41052,41055,41095,41098,41101,41104,41107,41110,41113,41116,41119],{"className":41051},[265],[28,41053],{"className":41054,"style":341},[269],[28,41056,41058,41061],{"className":41057},[274],[28,41059,22957],{"className":41060,"style":459},[274,275],[28,41062,41064],{"className":41063},[556],[28,41065,41067,41087],{"className":41066},[560,561],[28,41068,41070,41084],{"className":41069},[565],[28,41071,41073],{"className":41072,"style":23070},[569],[28,41074,41075,41078],{"style":20948},[28,41076],{"className":41077,"style":578},[577],[28,41079,41081],{"className":41080},[582,583,584,585],[28,41082,24625],{"className":41083},[274,275,585],[28,41085,593],{"className":41086},[592],[28,41088,41090],{"className":41089},[565],[28,41091,41093],{"className":41092,"style":600},[569],[28,41094],{},[28,41096,43],{"className":41097},[348],[28,41099,20756],{"className":41100},[274,275],[28,41102,1683],{"className":41103},[1723],[28,41105],{"className":41106,"style":530},[279],[28,41108,20756],{"className":41109},[274,275],[28,41111,1522],{"className":41112},[355],[28,41114],{"className":41115,"style":302},[279],[28,41117,83],{"className":41118},[306],[28,41120],{"className":41121,"style":302},[279],[28,41123,41125,41129,41132,41135,41179,41182,41222,41225,41230,41233,41273,41276,41312],{"className":41124},[265],[28,41126],{"className":41127,"style":41128},[269],"height:1.0497em;vertical-align:-0.2997em;",[28,41130,24625],{"className":41131},[274,275],[28,41133],{"className":41134,"style":530},[279],[28,41136,41138,41142],{"className":41137},[21329],[28,41139,25442],{"className":41140,"style":41141},[21329,21330,21331],"position:relative;top:0em;",[28,41143,41145],{"className":41144},[556],[28,41146,41148,41170],{"className":41147},[560,561],[28,41149,41151,41167],{"className":41150},[565],[28,41152,41155],{"className":41153,"style":41154},[569],"height:0.162em;",[28,41156,41158,41161],{"style":41157},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[28,41159],{"className":41160,"style":578},[577],[28,41162,41164],{"className":41163},[582,583,584,585],[28,41165,1686],{"className":41166},[274,275,585],[28,41168,593],{"className":41169},[592],[28,41171,41173],{"className":41172},[565],[28,41174,41177],{"className":41175,"style":41176},[569],"height:0.2997em;",[28,41178],{},[28,41180],{"className":41181,"style":530},[279],[28,41183,41185,41188],{"className":41184},[274],[28,41186,238],{"className":41187},[274,275],[28,41189,41191],{"className":41190},[556],[28,41192,41194,41214],{"className":41193},[560,561],[28,41195,41197,41211],{"className":41196},[565],[28,41198,41200],{"className":41199,"style":3881},[569],[28,41201,41202,41205],{"style":573},[28,41203],{"className":41204,"style":578},[577],[28,41206,41208],{"className":41207},[582,583,584,585],[28,41209,1686],{"className":41210},[274,275,585],[28,41212,593],{"className":41213},[592],[28,41215,41217],{"className":41216},[565],[28,41218,41220],{"className":41219,"style":600},[569],[28,41221],{},[28,41223],{"className":41224,"style":530},[279],[28,41226,39952,41228],{"className":41227},[21329],[28,41229,34347],{"style":39955},[28,41231],{"className":41232,"style":530},[279],[28,41234,41236,41239],{"className":41235},[274],[28,41237,238],{"className":41238},[274,275],[28,41240,41242],{"className":41241},[556],[28,41243,41245,41265],{"className":41244},[560,561],[28,41246,41248,41262],{"className":41247},[565],[28,41249,41251],{"className":41250,"style":3881},[569],[28,41252,41253,41256],{"style":573},[28,41254],{"className":41255,"style":578},[577],[28,41257,41259],{"className":41258},[582,583,584,585],[28,41260,1686],{"className":41261},[274,275,585],[28,41263,593],{"className":41264},[592],[28,41266,41268],{"className":41267},[565],[28,41269,41271],{"className":41270,"style":600},[569],[28,41272],{},[28,41274],{"className":41275,"style":302},[279],[28,41277,41279,41306,41309],{"className":41278},[306],[28,41280,41282],{"className":41281},[306],[28,41283,41285],{"className":41284},[274,2359],[28,41286,41288],{"className":41287},[2363],[28,41289,41291,41294,41303],{"className":41290},[2367],[28,41292],{"className":41293,"style":2371},[269],[28,41295,41297],{"className":41296},[2375],[28,41298,41300],{"className":41299},[274],[28,41301,2382],{"className":41302},[306],[28,41304],{"className":41305},[2386],[28,41307],{"className":41308},[279,2390],[28,41310,83],{"className":41311},[306],[28,41313],{"className":41314,"style":302},[279],[28,41316,41318,41321],{"className":41317},[265],[28,41319],{"className":41320,"style":316},[269],[28,41322,89],{"className":41323},[274],"。所以工程上常用的 Sinkhorn 散度要先去偏：",[28,41326,41328],{"className":41327,"translate":193},[192],[28,41329,41331,41443],{"className":41330,"translate":193},[197],[28,41332,41334],{"className":41333},[201],[203,41335,41336],{"xmlns":205,"display":206},[208,41337,41338,41440],{},[211,41339,41340,41346,41348,41350,41352,41354,41356,41358,41368,41370,41372,41374,41376,41378,41380,41388,41398,41400,41402,41404,41406,41408,41410,41418,41428,41430,41432,41434,41436,41438],{},[501,41341,41342,41344],{},[214,41343,6730],{},[214,41345,24625],{},[217,41347,43],{"stretchy":241},[214,41349,20756],{},[217,41351,1683],{"separator":261},[214,41353,20787],{},[217,41355,1522],{"stretchy":241},[217,41357,83],{},[501,41359,41360,41366],{},[211,41361,41362,41364],{},[214,41363,39509],{"mathvariant":484},[214,41365,443],{"mathvariant":484},[214,41367,24625],{},[217,41369,43],{"stretchy":241},[214,41371,20756],{},[217,41373,1683],{"separator":261},[214,41375,20787],{},[217,41377,1522],{"stretchy":241},[217,41379,5074],{},[5027,41381,41382],{"scriptlevel":89,"displaystyle":241},[21928,41383,41384,41386],{},[224,41385,1244],{},[224,41387,1253],{},[501,41389,41390,41396],{},[211,41391,41392,41394],{},[214,41393,39509],{"mathvariant":484},[214,41395,443],{"mathvariant":484},[214,41397,24625],{},[217,41399,43],{"stretchy":241},[214,41401,20756],{},[217,41403,1683],{"separator":261},[214,41405,20756],{},[217,41407,1522],{"stretchy":241},[217,41409,5074],{},[5027,41411,41412],{"scriptlevel":89,"displaystyle":241},[21928,41413,41414,41416],{},[224,41415,1244],{},[224,41417,1253],{},[501,41419,41420,41426],{},[211,41421,41422,41424],{},[214,41423,39509],{"mathvariant":484},[214,41425,443],{"mathvariant":484},[214,41427,24625],{},[217,41429,43],{"stretchy":241},[214,41431,20787],{},[217,41433,1683],{"separator":261},[214,41435,20787],{},[217,41437,1522],{"stretchy":241},[214,41439,494],{"mathvariant":484},[253,41441,41442],{"encoding":255},"S_\\varepsilon(\\mu,\\nu)=\\mathrm{OT}_\\varepsilon(\\mu,\\nu)-\\tfrac12\\mathrm{OT}_\\varepsilon(\\mu,\\mu)-\\tfrac12\\mathrm{OT}_\\varepsilon(\\nu,\\nu).",[28,41444,41446,41520,41598,41742],{"className":41445,"ariaHidden":261},[260],[28,41447,41449,41452,41493,41496,41499,41502,41505,41508,41511,41514,41517],{"className":41448},[265],[28,41450],{"className":41451,"style":341},[269],[28,41453,41455,41458],{"className":41454},[274],[28,41456,6730],{"className":41457,"style":6776},[274,275],[28,41459,41461],{"className":41460},[556],[28,41462,41464,41485],{"className":41463},[560,561],[28,41465,41467,41482],{"className":41466},[565],[28,41468,41470],{"className":41469,"style":23070},[569],[28,41471,41473,41476],{"style":41472},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[28,41474],{"className":41475,"style":578},[577],[28,41477,41479],{"className":41478},[582,583,584,585],[28,41480,24625],{"className":41481},[274,275,585],[28,41483,593],{"className":41484},[592],[28,41486,41488],{"className":41487},[565],[28,41489,41491],{"className":41490,"style":600},[569],[28,41492],{},[28,41494,43],{"className":41495},[348],[28,41497,20756],{"className":41498},[274,275],[28,41500,1683],{"className":41501},[1723],[28,41503],{"className":41504,"style":530},[279],[28,41506,20787],{"className":41507,"style":20803},[274,275],[28,41509,1522],{"className":41510},[355],[28,41512],{"className":41513,"style":302},[279],[28,41515,83],{"className":41516},[306],[28,41518],{"className":41519,"style":302},[279],[28,41521,41523,41526,41571,41574,41577,41580,41583,41586,41589,41592,41595],{"className":41522},[265],[28,41524],{"className":41525,"style":341},[269],[28,41527,41529,41536],{"className":41528},[274],[28,41530,41532],{"className":41531},[274],[28,41533,41535],{"className":41534},[274,1712],"OT",[28,41537,41539],{"className":41538},[556],[28,41540,41542,41563],{"className":41541},[560,561],[28,41543,41545,41560],{"className":41544},[565],[28,41546,41548],{"className":41547,"style":23070},[569],[28,41549,41551,41554],{"style":41550},"top:-2.55em;margin-right:0.05em;",[28,41552],{"className":41553,"style":578},[577],[28,41555,41557],{"className":41556},[582,583,584,585],[28,41558,24625],{"className":41559},[274,275,585],[28,41561,593],{"className":41562},[592],[28,41564,41566],{"className":41565},[565],[28,41567,41569],{"className":41568,"style":600},[569],[28,41570],{},[28,41572,43],{"className":41573},[348],[28,41575,20756],{"className":41576},[274,275],[28,41578,1683],{"className":41579},[1723],[28,41581],{"className":41582,"style":530},[279],[28,41584,20787],{"className":41585,"style":20803},[274,275],[28,41587,1522],{"className":41588},[355],[28,41590],{"className":41591,"style":280},[279],[28,41593,5074],{"className":41594},[284],[28,41596],{"className":41597,"style":280},[279],[28,41599,41601,41604,41672,41715,41718,41721,41724,41727,41730,41733,41736,41739],{"className":41600},[265],[28,41602],{"className":41603,"style":21986},[269],[28,41605,41607,41610,41669],{"className":41606},[274],[28,41608],{"className":41609},[348,8285],[28,41611,41613],{"className":41612},[21928],[28,41614,41616,41661],{"className":41615},[560,561],[28,41617,41619,41658],{"className":41618},[565],[28,41620,41622,41636,41644],{"className":41621,"style":22005},[569],[28,41623,41624,41627],{"style":22008},[28,41625],{"className":41626,"style":1914},[577],[28,41628,41630],{"className":41629},[582,583,584,585],[28,41631,41633],{"className":41632},[274,585],[28,41634,1253],{"className":41635},[274,585],[28,41637,41638,41641],{"style":22023},[28,41639],{"className":41640,"style":1914},[577],[28,41642],{"className":41643,"style":22031},[22030],[28,41645,41646,41649],{"style":22034},[28,41647],{"className":41648,"style":1914},[577],[28,41650,41652],{"className":41651},[582,583,584,585],[28,41653,41655],{"className":41654},[274,585],[28,41656,1244],{"className":41657},[274,585],[28,41659,593],{"className":41660},[592],[28,41662,41664],{"className":41663},[565],[28,41665,41667],{"className":41666,"style":22056},[569],[28,41668],{},[28,41670],{"className":41671},[355,8285],[28,41673,41675,41681],{"className":41674},[274],[28,41676,41678],{"className":41677},[274],[28,41679,41535],{"className":41680},[274,1712],[28,41682,41684],{"className":41683},[556],[28,41685,41687,41707],{"className":41686},[560,561],[28,41688,41690,41704],{"className":41689},[565],[28,41691,41693],{"className":41692,"style":23070},[569],[28,41694,41695,41698],{"style":41550},[28,41696],{"className":41697,"style":578},[577],[28,41699,41701],{"className":41700},[582,583,584,585],[28,41702,24625],{"className":41703},[274,275,585],[28,41705,593],{"className":41706},[592],[28,41708,41710],{"className":41709},[565],[28,41711,41713],{"className":41712,"style":600},[569],[28,41714],{},[28,41716,43],{"className":41717},[348],[28,41719,20756],{"className":41720},[274,275],[28,41722,1683],{"className":41723},[1723],[28,41725],{"className":41726,"style":530},[279],[28,41728,20756],{"className":41729},[274,275],[28,41731,1522],{"className":41732},[355],[28,41734],{"className":41735,"style":280},[279],[28,41737,5074],{"className":41738},[284],[28,41740],{"className":41741,"style":280},[279],[28,41743,41745,41748,41816,41859,41862,41865,41868,41871,41874,41877],{"className":41744},[265],[28,41746],{"className":41747,"style":21986},[269],[28,41749,41751,41754,41813],{"className":41750},[274],[28,41752],{"className":41753},[348,8285],[28,41755,41757],{"className":41756},[21928],[28,41758,41760,41805],{"className":41759},[560,561],[28,41761,41763,41802],{"className":41762},[565],[28,41764,41766,41780,41788],{"className":41765,"style":22005},[569],[28,41767,41768,41771],{"style":22008},[28,41769],{"className":41770,"style":1914},[577],[28,41772,41774],{"className":41773},[582,583,584,585],[28,41775,41777],{"className":41776},[274,585],[28,41778,1253],{"className":41779},[274,585],[28,41781,41782,41785],{"style":22023},[28,41783],{"className":41784,"style":1914},[577],[28,41786],{"className":41787,"style":22031},[22030],[28,41789,41790,41793],{"style":22034},[28,41791],{"className":41792,"style":1914},[577],[28,41794,41796],{"className":41795},[582,583,584,585],[28,41797,41799],{"className":41798},[274,585],[28,41800,1244],{"className":41801},[274,585],[28,41803,593],{"className":41804},[592],[28,41806,41808],{"className":41807},[565],[28,41809,41811],{"className":41810,"style":22056},[569],[28,41812],{},[28,41814],{"className":41815},[355,8285],[28,41817,41819,41825],{"className":41818},[274],[28,41820,41822],{"className":41821},[274],[28,41823,41535],{"className":41824},[274,1712],[28,41826,41828],{"className":41827},[556],[28,41829,41831,41851],{"className":41830},[560,561],[28,41832,41834,41848],{"className":41833},[565],[28,41835,41837],{"className":41836,"style":23070},[569],[28,41838,41839,41842],{"style":41550},[28,41840],{"className":41841,"style":578},[577],[28,41843,41845],{"className":41844},[582,583,584,585],[28,41846,24625],{"className":41847},[274,275,585],[28,41849,593],{"className":41850},[592],[28,41852,41854],{"className":41853},[565],[28,41855,41857],{"className":41856,"style":600},[569],[28,41858],{},[28,41860,43],{"className":41861},[348],[28,41863,20787],{"className":41864,"style":20803},[274,275],[28,41866,1683],{"className":41867},[1723],[28,41869],{"className":41870,"style":530},[279],[28,41872,20787],{"className":41873,"style":20803},[274,275],[28,41875,1522],{"className":41876},[355],[28,41878,494],{"className":41879},[274],[11,41881,41882,41883,41934],{},"去偏之后才非负、才在 ",[28,41884,41886,41904],{"className":41885,"translate":193},[197],[28,41887,41889],{"className":41888},[201],[203,41890,41891],{"xmlns":205},[208,41892,41893,41901],{},[211,41894,41895,41897,41899],{},[214,41896,20756],{},[217,41898,83],{},[214,41900,20787],{},[253,41902,41903],{"encoding":255},"\\mu=\\nu",[28,41905,41907,41925],{"className":41906,"ariaHidden":261},[260],[28,41908,41910,41913,41916,41919,41922],{"className":41909},[265],[28,41911],{"className":41912,"style":612},[269],[28,41914,20756],{"className":41915},[274,275],[28,41917],{"className":41918,"style":302},[279],[28,41920,83],{"className":41921},[306],[28,41923],{"className":41924,"style":302},[279],[28,41926,41928,41931],{"className":41927},[265],[28,41929],{"className":41930,"style":645},[269],[28,41932,20787],{"className":41933,"style":20803},[274,275]," 处为零。",[11,41936,41937,41938,42008,42009,42037,42038,42066],{},"还有一个更根本的代价，来自样本。经验分布 ",[28,41939,41941,41959],{"className":41940,"translate":193},[197],[28,41942,41944],{"className":41943},[201],[203,41945,41946],{"xmlns":205},[208,41947,41948,41956],{},[211,41949,41950],{},[501,41951,41952,41954],{},[214,41953,20756],{},[214,41955,3060],{},[253,41957,41958],{"encoding":255},"\\mu_n",[28,41960,41962],{"className":41961,"ariaHidden":261},[260],[28,41963,41965,41968],{"className":41964},[265],[28,41966],{"className":41967,"style":612},[269],[28,41969,41971,41974],{"className":41970},[274],[28,41972,20756],{"className":41973},[274,275],[28,41975,41977],{"className":41976},[556],[28,41978,41980,42000],{"className":41979},[560,561],[28,41981,41983,41997],{"className":41982},[565],[28,41984,41986],{"className":41985,"style":23070},[569],[28,41987,41988,41991],{"style":573},[28,41989],{"className":41990,"style":578},[577],[28,41992,41994],{"className":41993},[582,583,584,585],[28,41995,3060],{"className":41996},[274,275,585],[28,41998,593],{"className":41999},[592],[28,42001,42003],{"className":42002},[565],[28,42004,42006],{"className":42005,"style":600},[569],[28,42007],{}," 到真分布 ",[28,42010,42012,42025],{"className":42011,"translate":193},[197],[28,42013,42015],{"className":42014},[201],[203,42016,42017],{"xmlns":205},[208,42018,42019,42023],{},[211,42020,42021],{},[214,42022,20756],{},[253,42024,20759],{"encoding":255},[28,42026,42028],{"className":42027,"ariaHidden":261},[260],[28,42029,42031,42034],{"className":42030},[265],[28,42032],{"className":42033,"style":612},[269],[28,42035,20756],{"className":42036},[274,275]," 的收敛速度，在 ",[28,42039,42041,42054],{"className":42040,"translate":193},[197],[28,42042,42044],{"className":42043},[201],[203,42045,42046],{"xmlns":205},[208,42047,42048,42052],{},[211,42049,42050],{},[214,42051,12988],{},[253,42053,12988],{"encoding":255},[28,42055,42057],{"className":42056,"ariaHidden":261},[260],[28,42058,42060,42063],{"className":42059},[265],[28,42061],{"className":42062,"style":7448},[269],[28,42064,12988],{"className":42065},[274,275]," 维空间里大致是",[28,42068,42070],{"className":42069,"translate":193},[192],[28,42071,42073,42147],{"className":42072,"translate":193},[197],[28,42074,42076],{"className":42075},[201],[203,42077,42078],{"xmlns":205,"display":206},[208,42079,42080,42144],{},[211,42081,42082,42085,42087,42093,42095,42101,42103,42105,42107,42109,42112,42114,42128,42130,42132,42134,42136,42138,42140,42142],{},[214,42083,42084],{"mathvariant":12057},"E",[228,42086,488],{},[501,42088,42089,42091],{},[214,42090,22957],{},[214,42092,11],{},[217,42094,43],{"stretchy":241},[501,42096,42097,42099],{},[214,42098,20756],{},[214,42100,3060],{},[217,42102,1683],{"separator":261},[214,42104,20756],{},[217,42106,1522],{"stretchy":241},[228,42108,14860],{},[217,42110,42111],{},"≲",[228,42113,14860],{},[3054,42115,42116,42118],{},[214,42117,3060],{},[211,42119,42120,42122,42124,42126],{},[217,42121,5074],{},[224,42123,1244],{},[214,42125,12219],{"mathvariant":484},[214,42127,12988],{},[279,42129],{"width":4053},[217,42131,43],{"stretchy":241},[214,42133,12988],{},[217,42135,130],{},[224,42137,1253],{},[214,42139,11],{},[217,42141,1522],{"stretchy":241},[214,42143,494],{"mathvariant":484},[253,42145,42146],{"encoding":255},"\\mathbb E\\,W_p(\\mu_n,\\mu)\\ \\lesssim\\ n^{-1\u002Fd}\\qquad(d>2p).",[28,42148,42150,42273,42337],{"className":42149,"ariaHidden":261},[260],[28,42151,42153,42156,42159,42162,42202,42205,42245,42248,42251,42254,42257,42260,42263,42267,42270],{"className":42152},[265],[28,42154],{"className":42155,"style":21101},[269],[28,42157,42084],{"className":42158},[274,12195],[28,42160],{"className":42161,"style":530},[279],[28,42163,42165,42168],{"className":42164},[274],[28,42166,22957],{"className":42167,"style":459},[274,275],[28,42169,42171],{"className":42170},[556],[28,42172,42174,42194],{"className":42173},[560,561],[28,42175,42177,42191],{"className":42176},[565],[28,42178,42180],{"className":42179,"style":23070},[569],[28,42181,42182,42185],{"style":20948},[28,42183],{"className":42184,"style":578},[577],[28,42186,42188],{"className":42187},[582,583,584,585],[28,42189,11],{"className":42190},[274,275,585],[28,42192,593],{"className":42193},[592],[28,42195,42197],{"className":42196},[565],[28,42198,42200],{"className":42199,"style":20967},[569],[28,42201],{},[28,42203,43],{"className":42204},[348],[28,42206,42208,42211],{"className":42207},[274],[28,42209,20756],{"className":42210},[274,275],[28,42212,42214],{"className":42213},[556],[28,42215,42217,42237],{"className":42216},[560,561],[28,42218,42220,42234],{"className":42219},[565],[28,42221,42223],{"className":42222,"style":23070},[569],[28,42224,42225,42228],{"style":573},[28,42226],{"className":42227,"style":578},[577],[28,42229,42231],{"className":42230},[582,583,584,585],[28,42232,3060],{"className":42233},[274,275,585],[28,42235,593],{"className":42236},[592],[28,42238,42240],{"className":42239},[565],[28,42241,42243],{"className":42242,"style":600},[569],[28,42244],{},[28,42246,1683],{"className":42247},[1723],[28,42249],{"className":42250,"style":530},[279],[28,42252,20756],{"className":42253},[274,275],[28,42255,1522],{"className":42256},[355],[28,42258,14860],{"className":42259},[279],[28,42261],{"className":42262,"style":302},[279],[28,42264,42111],{"className":42265},[306,42266],"amsrm",[28,42268,14860],{"className":42269},[279],[28,42271],{"className":42272,"style":302},[279],[28,42274,42276,42280,42319,42322,42325,42328,42331,42334],{"className":42275},[265],[28,42277],{"className":42278,"style":42279},[269],"height:1.188em;vertical-align:-0.25em;",[28,42281,42283,42286],{"className":42282},[274],[28,42284,3060],{"className":42285},[274,275],[28,42287,42289],{"className":42288},[556],[28,42290,42292],{"className":42291},[560],[28,42293,42295],{"className":42294},[565],[28,42296,42299],{"className":42297,"style":42298},[569],"height:0.938em;",[28,42300,42301,42304],{"style":4880},[28,42302],{"className":42303,"style":578},[577],[28,42305,42307],{"className":42306},[582,583,584,585],[28,42308,42310,42313,42316],{"className":42309},[274,585],[28,42311,5074],{"className":42312},[274,585],[28,42314,23875],{"className":42315},[274,585],[28,42317,12988],{"className":42318},[274,275,585],[28,42320],{"className":42321,"style":4222},[279],[28,42323,43],{"className":42324},[348],[28,42326,12988],{"className":42327},[274,275],[28,42329],{"className":42330,"style":302},[279],[28,42332,130],{"className":42333},[306],[28,42335],{"className":42336,"style":302},[279],[28,42338,42340,42343,42346,42349,42352],{"className":42339},[265],[28,42341],{"className":42342,"style":341},[269],[28,42344,1253],{"className":42345},[274],[28,42347,11],{"className":42348},[274,275],[28,42350,1522],{"className":42351},[355],[28,42353,494],{"className":42354},[274],[11,42356,42357,42358,42409,42410,42480,42481,42543],{},"指数太糟糕了。",[28,42359,42361,42379],{"className":42360,"translate":193},[197],[28,42362,42364],{"className":42363},[201],[203,42365,42366],{"xmlns":205},[208,42367,42368,42376],{},[211,42369,42370,42372,42374],{},[214,42371,12988],{},[217,42373,83],{},[224,42375,77],{},[253,42377,42378],{"encoding":255},"d=100",[28,42380,42382,42400],{"className":42381,"ariaHidden":261},[260],[28,42383,42385,42388,42391,42394,42397],{"className":42384},[265],[28,42386],{"className":42387,"style":7448},[269],[28,42389,12988],{"className":42390},[274,275],[28,42392],{"className":42393,"style":302},[279],[28,42395,83],{"className":42396},[306],[28,42398],{"className":42399,"style":302},[279],[28,42401,42403,42406],{"className":42402},[265],[28,42404],{"className":42405,"style":316},[269],[28,42407,77],{"className":42408},[274]," 时误差按 ",[28,42411,42413,42436],{"className":42412,"translate":193},[197],[28,42414,42416],{"className":42415},[201],[203,42417,42418],{"xmlns":205},[208,42419,42420,42433],{},[211,42421,42422],{},[3054,42423,42424,42426],{},[214,42425,3060],{},[211,42427,42428,42430],{},[217,42429,5074],{},[224,42431,42432],{},"0.01",[253,42434,42435],{"encoding":255},"n^{-0.01}",[28,42437,42439],{"className":42438,"ariaHidden":261},[260],[28,42440,42442,42445],{"className":42441},[265],[28,42443],{"className":42444,"style":5477},[269],[28,42446,42448,42451],{"className":42447},[274],[28,42449,3060],{"className":42450},[274,275],[28,42452,42454],{"className":42453},[556],[28,42455,42457],{"className":42456},[560],[28,42458,42460],{"className":42459},[565],[28,42461,42463],{"className":42462,"style":5477},[569],[28,42464,42465,42468],{"style":3094},[28,42466],{"className":42467,"style":578},[577],[28,42469,42471],{"className":42470},[582,583,584,585],[28,42472,42474,42477],{"className":42473},[274,585],[28,42475,5074],{"className":42476},[274,585],[28,42478,42432],{"className":42479},[274,585]," 衰减，想把误差减半，样本量要乘 ",[28,42482,42484,42502],{"className":42483,"translate":193},[197],[28,42485,42487],{"className":42486},[201],[203,42488,42489],{"xmlns":205},[208,42490,42491,42499],{},[211,42492,42493],{},[3054,42494,42495,42497],{},[224,42496,1253],{},[224,42498,77],{},[253,42500,42501],{"encoding":255},"2^{100}",[28,42503,42505],{"className":42504,"ariaHidden":261},[260],[28,42506,42508,42511],{"className":42507},[265],[28,42509],{"className":42510,"style":5477},[269],[28,42512,42514,42517],{"className":42513},[274],[28,42515,1253],{"className":42516},[274],[28,42518,42520],{"className":42519},[556],[28,42521,42523],{"className":42522},[560],[28,42524,42526],{"className":42525},[565],[28,42527,42529],{"className":42528,"style":5477},[569],[28,42530,42531,42534],{"style":3094},[28,42532],{"className":42533,"style":578},[577],[28,42535,42537],{"className":42536},[582,583,584,585],[28,42538,42540],{"className":42539},[274,585],[28,42541,77],{"className":42542},[274,585],"。所以“直接在高维空间算 Wasserstein”在绝大多数真实问题上不可行。",[11,42545,42546,42547,42575,42576,42650],{},"三条出路我都用过：把数据先降到低维再算；用 sliced Wasserstein，把分布投影到随机一维方向上算 ",[28,42548,42550,42563],{"className":42549,"translate":193},[197],[28,42551,42553],{"className":42552},[201],[203,42554,42555],{"xmlns":205},[208,42556,42557,42561],{},[211,42558,42559],{},[214,42560,22957],{},[253,42562,22957],{"encoding":255},[28,42564,42566],{"className":42565,"ariaHidden":261},[260],[28,42567,42569,42572],{"className":42568},[265],[28,42570],{"className":42571,"style":455},[269],[28,42573,22957],{"className":42574,"style":459},[274,275],"，再对方向平均（一维的收敛是 ",[28,42577,42579,42605],{"className":42578,"translate":193},[197],[28,42580,42582],{"className":42581},[201],[203,42583,42584],{"xmlns":205},[208,42585,42586,42602],{},[211,42587,42588],{},[3054,42589,42590,42592],{},[214,42591,3060],{},[211,42593,42594,42596,42598,42600],{},[217,42595,5074],{},[224,42597,1244],{},[214,42599,12219],{"mathvariant":484},[224,42601,1253],{},[253,42603,42604],{"encoding":255},"n^{-1\u002F2}",[28,42606,42608],{"className":42607,"ariaHidden":261},[260],[28,42609,42611,42615],{"className":42610},[265],[28,42612],{"className":42613,"style":42614},[269],"height:0.888em;",[28,42616,42618,42621],{"className":42617},[274],[28,42619,3060],{"className":42620},[274,275],[28,42622,42624],{"className":42623},[556],[28,42625,42627],{"className":42626},[560],[28,42628,42630],{"className":42629},[565],[28,42631,42633],{"className":42632,"style":42614},[569],[28,42634,42635,42638],{"style":3094},[28,42636],{"className":42637,"style":578},[577],[28,42639,42641],{"className":42640},[582,583,584,585],[28,42642,42644,42647],{"className":42643},[274,585],[28,42645,5074],{"className":42646},[274,585],[28,42648,36926],{"className":42649},[274,585],"，一下就正常了）；或者用低秩分解把代价矩阵近似成低秩。共同点是不要在原始高维空间里做全秩的 OT。",[414,42652,42653],{"id":42653},"单纯形上的尺子",[11,42655,42656,42657,2693,42711,42764,42765,2693,42818,2923],{},"我拿两个模型对同一张图跑一遍，输出分别是 ",[28,42658,42660,42684],{"className":42659,"translate":193},[197],[28,42661,42663],{"className":42662},[201],[203,42664,42665],{"xmlns":205},[208,42666,42667,42681],{},[211,42668,42669,42671,42674,42676,42679],{},[217,42670,43],{"stretchy":241},[224,42672,42673],{},"0.90",[217,42675,1683],{"separator":261},[224,42677,42678],{},"0.10",[217,42680,1522],{"stretchy":241},[253,42682,42683],{"encoding":255},"(0.90,0.10)",[28,42685,42687],{"className":42686,"ariaHidden":261},[260],[28,42688,42690,42693,42696,42699,42702,42705,42708],{"className":42689},[265],[28,42691],{"className":42692,"style":341},[269],[28,42694,43],{"className":42695},[348],[28,42697,42673],{"className":42698},[274],[28,42700,1683],{"className":42701},[1723],[28,42703],{"className":42704,"style":530},[279],[28,42706,42678],{"className":42707},[274],[28,42709,1522],{"className":42710},[355],[28,42712,42714,42737],{"className":42713,"translate":193},[197],[28,42715,42717],{"className":42716},[201],[203,42718,42719],{"xmlns":205},[208,42720,42721,42734],{},[211,42722,42723,42725,42728,42730,42732],{},[217,42724,43],{"stretchy":241},[224,42726,42727],{},"0.99",[217,42729,1683],{"separator":261},[224,42731,42432],{},[217,42733,1522],{"stretchy":241},[253,42735,42736],{"encoding":255},"(0.99,0.01)",[28,42738,42740],{"className":42739,"ariaHidden":261},[260],[28,42741,42743,42746,42749,42752,42755,42758,42761],{"className":42742},[265],[28,42744],{"className":42745,"style":341},[269],[28,42747,43],{"className":42748},[348],[28,42750,42727],{"className":42751},[274],[28,42753,1683],{"className":42754},[1723],[28,42756],{"className":42757,"style":530},[279],[28,42759,42432],{"className":42760},[274],[28,42762,1522],{"className":42763},[355],"。另一张图上，两个模型的输出是 ",[28,42766,42768,42791],{"className":42767,"translate":193},[197],[28,42769,42771],{"className":42770},[201],[203,42772,42773],{"xmlns":205},[208,42774,42775,42788],{},[211,42776,42777,42779,42782,42784,42786],{},[217,42778,43],{"stretchy":241},[224,42780,42781],{},"0.50",[217,42783,1683],{"separator":261},[224,42785,42781],{},[217,42787,1522],{"stretchy":241},[253,42789,42790],{"encoding":255},"(0.50,0.50)",[28,42792,42794],{"className":42793,"ariaHidden":261},[260],[28,42795,42797,42800,42803,42806,42809,42812,42815],{"className":42796},[265],[28,42798],{"className":42799,"style":341},[269],[28,42801,43],{"className":42802},[348],[28,42804,42781],{"className":42805},[274],[28,42807,1683],{"className":42808},[1723],[28,42810],{"className":42811,"style":530},[279],[28,42813,42781],{"className":42814},[274],[28,42816,1522],{"className":42817},[355],[28,42819,42821,42845],{"className":42820,"translate":193},[197],[28,42822,42824],{"className":42823},[201],[203,42825,42826],{"xmlns":205},[208,42827,42828,42842],{},[211,42829,42830,42832,42835,42837,42840],{},[217,42831,43],{"stretchy":241},[224,42833,42834],{},"0.59",[217,42836,1683],{"separator":261},[224,42838,42839],{},"0.41",[217,42841,1522],{"stretchy":241},[253,42843,42844],{"encoding":255},"(0.59,0.41)",[28,42846,42848],{"className":42847,"ariaHidden":261},[260],[28,42849,42851,42854,42857,42860,42863,42866,42869],{"className":42850},[265],[28,42852],{"className":42853,"style":341},[269],[28,42855,43],{"className":42856},[348],[28,42858,42834],{"className":42859},[274],[28,42861,1683],{"className":42862},[1723],[28,42864],{"className":42865,"style":530},[279],[28,42867,42839],{"className":42868},[274],[28,42870,1522],{"className":42871},[355],[11,42873,42874,42875,42904],{},"两组变化的欧氏距离都是 ",[28,42876,42878,42892],{"className":42877,"translate":193},[197],[28,42879,42881],{"className":42880},[201],[203,42882,42883],{"xmlns":205},[208,42884,42885,42890],{},[211,42886,42887],{},[224,42888,42889],{},"0.09",[253,42891,42889],{"encoding":255},[28,42893,42895],{"className":42894,"ariaHidden":261},[260],[28,42896,42898,42901],{"className":42897},[265],[28,42899],{"className":42900,"style":316},[269],[28,42902,42889],{"className":42903},[274],"。但我的直觉说它们不是一回事：前者是从“犹豫”变成“笃定”，后者只是从不偏不倚稍微偏了一点。",[11,42906,42907,42908,24711,42961,6904,42990,24711,43043,43072,43073,12434,43101,43129],{},"这种直觉能不能写成数？能。把 Bernoulli 分布按 Fisher–Rao 距离量一下：",[28,42909,42911,42934],{"className":42910,"translate":193},[197],[28,42912,42914],{"className":42913},[201],[203,42915,42916],{"xmlns":205},[208,42917,42918,42931],{},[211,42919,42920,42922,42925,42927,42929],{},[217,42921,43],{"stretchy":241},[224,42923,42924],{},"0.5",[217,42926,1683],{"separator":261},[224,42928,42834],{},[217,42930,1522],{"stretchy":241},[253,42932,42933],{"encoding":255},"(0.5,0.59)",[28,42935,42937],{"className":42936,"ariaHidden":261},[260],[28,42938,42940,42943,42946,42949,42952,42955,42958],{"className":42939},[265],[28,42941],{"className":42942,"style":341},[269],[28,42944,43],{"className":42945},[348],[28,42947,42924],{"className":42948},[274],[28,42950,1683],{"className":42951},[1723],[28,42953],{"className":42954,"style":530},[279],[28,42956,42834],{"className":42957},[274],[28,42959,1522],{"className":42960},[355],[28,42962,42964,42978],{"className":42963,"translate":193},[197],[28,42965,42967],{"className":42966},[201],[203,42968,42969],{"xmlns":205},[208,42970,42971,42976],{},[211,42972,42973],{},[224,42974,42975],{},"0.18",[253,42977,42975],{"encoding":255},[28,42979,42981],{"className":42980,"ariaHidden":261},[260],[28,42982,42984,42987],{"className":42983},[265],[28,42985],{"className":42986,"style":316},[269],[28,42988,42975],{"className":42989},[274],[28,42991,42993,43016],{"className":42992,"translate":193},[197],[28,42994,42996],{"className":42995},[201],[203,42997,42998],{"xmlns":205},[208,42999,43000,43013],{},[211,43001,43002,43004,43007,43009,43011],{},[217,43003,43],{"stretchy":241},[224,43005,43006],{},"0.9",[217,43008,1683],{"separator":261},[224,43010,42727],{},[217,43012,1522],{"stretchy":241},[253,43014,43015],{"encoding":255},"(0.9,0.99)",[28,43017,43019],{"className":43018,"ariaHidden":261},[260],[28,43020,43022,43025,43028,43031,43034,43037,43040],{"className":43021},[265],[28,43023],{"className":43024,"style":341},[269],[28,43026,43],{"className":43027},[348],[28,43029,43006],{"className":43030},[274],[28,43032,1683],{"className":43033},[1723],[28,43035],{"className":43036,"style":530},[279],[28,43038,42727],{"className":43039},[274],[28,43041,1522],{"className":43042},[355],[28,43044,43046,43060],{"className":43045,"translate":193},[197],[28,43047,43049],{"className":43048},[201],[203,43050,43051],{"xmlns":205},[208,43052,43053,43058],{},[211,43054,43055],{},[224,43056,43057],{},"0.44",[253,43059,43057],{"encoding":255},[28,43061,43063],{"className":43062,"ariaHidden":261},[260],[28,43064,43066,43069],{"className":43065},[265],[28,43067],{"className":43068,"style":316},[269],[28,43070,43057],{"className":43071},[274],"，后者是前者的两倍半。同样的坐标变化，在概率接近 ",[28,43074,43076,43089],{"className":43075,"translate":193},[197],[28,43077,43079],{"className":43078},[201],[203,43080,43081],{"xmlns":205},[208,43082,43083,43087],{},[211,43084,43085],{},[224,43086,89],{},[253,43088,89],{"encoding":255},[28,43090,43092],{"className":43091,"ariaHidden":261},[260],[28,43093,43095,43098],{"className":43094},[265],[28,43096],{"className":43097,"style":316},[269],[28,43099,89],{"className":43100},[274],[28,43102,43104,43117],{"className":43103,"translate":193},[197],[28,43105,43107],{"className":43106},[201],[203,43108,43109],{"xmlns":205},[208,43110,43111,43115],{},[211,43112,43113],{},[224,43114,1244],{},[253,43116,1244],{"encoding":255},[28,43118,43120],{"className":43119,"ariaHidden":261},[260],[28,43121,43123,43126],{"className":43122},[265],[28,43124],{"className":43125,"style":316},[269],[28,43127,1244],{"className":43128},[274]," 的地方要走更远的路——那里是单纯形的“边缘”，流形在那边被拉长了。",[11,43131,43132],{},"那么这把尺子是怎么来的？",[11,43134,43135],{},"模型的输出住在单纯形",[28,43137,43139],{"className":43138,"translate":193},[192],[28,43140,43142,43217],{"className":43141,"translate":193},[197],[28,43143,43145],{"className":43144},[201],[203,43146,43147],{"xmlns":205,"display":206},[208,43148,43149,43214],{},[211,43150,43151,43164,43166,43168,43170,43172,43178,43180,43182,43188,43190,43192,43194,43196,43202,43208,43210,43212],{},[3054,43152,43153,43156],{},[214,43154,43155],{"mathvariant":484},"Δ",[211,43157,43158,43160,43162],{},[214,43159,24390],{},[217,43161,5074],{},[224,43163,1244],{},[217,43165,83],{},[217,43167,7916],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[214,43169,11],{},[217,43171,12054],{},[3054,43173,43174,43176],{},[214,43175,34579],{"mathvariant":12057},[214,43177,24390],{},[217,43179,877],{},[228,43181,14860],{},[501,43183,43184,43186],{},[214,43185,11],{},[214,43187,1686],{},[217,43189,9572],{},[224,43191,89],{},[217,43193,1683],{"separator":261},[228,43195,14860],{},[1771,43197,43198,43200],{},[217,43199,25442],{},[214,43201,1686],{},[501,43203,43204,43206],{},[214,43205,11],{},[214,43207,1686],{},[217,43209,83],{},[224,43211,1244],{},[217,43213,14883],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[253,43215,43216],{"encoding":255},"\\Delta^{K-1}=\\Bigl\\{p\\in\\mathbb R^K:\\ p_i\\ge0,\\ \\sum_i p_i=1\\Bigr\\}",[28,43218,43220,43274,43298,43345,43400,43514],{"className":43219,"ariaHidden":261},[260],[28,43221,43223,43227,43265,43268,43271],{"className":43222},[265],[28,43224],{"className":43225,"style":43226},[269],"height:0.8913em;",[28,43228,43230,43233],{"className":43229},[274],[28,43231,43155],{"className":43232},[274],[28,43234,43236],{"className":43235},[556],[28,43237,43239],{"className":43238},[560],[28,43240,43242],{"className":43241},[565],[28,43243,43245],{"className":43244,"style":43226},[569],[28,43246,43247,43250],{"style":4880},[28,43248],{"className":43249,"style":578},[577],[28,43251,43253],{"className":43252},[582,583,584,585],[28,43254,43256,43259,43262],{"className":43255},[274,585],[28,43257,24390],{"className":43258,"style":28331},[274,275,585],[28,43260,5074],{"className":43261},[284,585],[28,43263,1244],{"className":43264},[274,585],[28,43266],{"className":43267,"style":302},[279],[28,43269,83],{"className":43270},[306],[28,43272],{"className":43273,"style":302},[279],[28,43275,43277,43280,43286,43289,43292,43295],{"className":43276},[265],[28,43278],{"className":43279,"style":22499},[269],[28,43281,43283],{"className":43282},[348],[28,43284,7916],{"className":43285},[2169,22506],[28,43287,11],{"className":43288},[274,275],[28,43290],{"className":43291,"style":302},[279],[28,43293,12054],{"className":43294},[306],[28,43296],{"className":43297,"style":302},[279],[28,43299,43301,43304,43333,43336,43339,43342],{"className":43300},[265],[28,43302],{"className":43303,"style":43226},[269],[28,43305,43307,43310],{"className":43306},[274],[28,43308,34579],{"className":43309},[274,12195],[28,43311,43313],{"className":43312},[556],[28,43314,43316],{"className":43315},[560],[28,43317,43319],{"className":43318},[565],[28,43320,43322],{"className":43321,"style":43226},[569],[28,43323,43324,43327],{"style":4880},[28,43325],{"className":43326,"style":578},[577],[28,43328,43330],{"className":43329},[582,583,584,585],[28,43331,24390],{"className":43332,"style":28331},[274,275,585],[28,43334],{"className":43335,"style":302},[279],[28,43337,877],{"className":43338},[306],[28,43340,14860],{"className":43341},[279],[28,43343],{"className":43344,"style":302},[279],[28,43346,43348,43351,43391,43394,43397],{"className":43347},[265],[28,43349],{"className":43350,"style":2450},[269],[28,43352,43354,43357],{"className":43353},[274],[28,43355,11],{"className":43356},[274,275],[28,43358,43360],{"className":43359},[556],[28,43361,43363,43383],{"className":43362},[560,561],[28,43364,43366,43380],{"className":43365},[565],[28,43367,43369],{"className":43368,"style":3881},[569],[28,43370,43371,43374],{"style":573},[28,43372],{"className":43373,"style":578},[577],[28,43375,43377],{"className":43376},[582,583,584,585],[28,43378,1686],{"className":43379},[274,275,585],[28,43381,593],{"className":43382},[592],[28,43384,43386],{"className":43385},[565],[28,43387,43389],{"className":43388,"style":600},[569],[28,43390],{},[28,43392],{"className":43393,"style":302},[279],[28,43395,9572],{"className":43396},[306],[28,43398],{"className":43399,"style":302},[279],[28,43401,43403,43406,43409,43412,43415,43418,43462,43465,43505,43508,43511],{"className":43402},[265],[28,43404],{"className":43405,"style":30055},[269],[28,43407,89],{"className":43408},[274],[28,43410,1683],{"className":43411},[1723],[28,43413,14860],{"className":43414},[279],[28,43416],{"className":43417,"style":530},[279],[28,43419,43421],{"className":43420},[21329,21485],[28,43422,43424,43454],{"className":43423},[560,561],[28,43425,43427,43451],{"className":43426},[565],[28,43428,43430,43441],{"className":43429,"style":25713},[569],[28,43431,43432,43435],{"style":25716},[28,43433],{"className":43434,"style":25720},[577],[28,43436,43438],{"className":43437},[582,583,584,585],[28,43439,1686],{"className":43440},[274,275,585],[28,43442,43443,43446],{"style":25732},[28,43444],{"className":43445,"style":25720},[577],[28,43447,43448],{},[28,43449,25442],{"className":43450},[21329,21330,21547],[28,43452,593],{"className":43453},[592],[28,43455,43457],{"className":43456},[565],[28,43458,43460],{"className":43459,"style":26154},[569],[28,43461],{},[28,43463],{"className":43464,"style":530},[279],[28,43466,43468,43471],{"className":43467},[274],[28,43469,11],{"className":43470},[274,275],[28,43472,43474],{"className":43473},[556],[28,43475,43477,43497],{"className":43476},[560,561],[28,43478,43480,43494],{"className":43479},[565],[28,43481,43483],{"className":43482,"style":3881},[569],[28,43484,43485,43488],{"style":573},[28,43486],{"className":43487,"style":578},[577],[28,43489,43491],{"className":43490},[582,583,584,585],[28,43492,1686],{"className":43493},[274,275,585],[28,43495,593],{"className":43496},[592],[28,43498,43500],{"className":43499},[565],[28,43501,43503],{"className":43502,"style":600},[569],[28,43504],{},[28,43506],{"className":43507,"style":302},[279],[28,43509,83],{"className":43510},[306],[28,43512],{"className":43513,"style":302},[279],[28,43515,43517,43520,43523],{"className":43516},[265],[28,43518],{"className":43519,"style":22499},[269],[28,43521,1244],{"className":43522},[274],[28,43524,43526],{"className":43525},[355],[28,43527,14883],{"className":43528},[2169,22506],[11,43530,43531],{},"里。问题在于这个空间有边界：越靠近顶点，同样的坐标变化对应的“实质变化”越大。欧氏距离看不见这件事。",[11,43533,43534,43535,43616],{},"Rao 在 1945 年的想法是：把参数化的分布族 ",[28,43536,43538,43561],{"className":43537,"translate":193},[197],[28,43539,43541],{"className":43540},[201],[203,43542,43543],{"xmlns":205},[208,43544,43545,43558],{},[211,43546,43547,43549,43556],{},[217,43548,7916],{"stretchy":241},[501,43550,43551,43553],{},[214,43552,11],{},[214,43554,43555],{},"θ",[217,43557,14883],{"stretchy":241},[253,43559,43560],{"encoding":255},"\\{p_\\theta\\}",[28,43562,43564],{"className":43563,"ariaHidden":261},[260],[28,43565,43567,43570,43573,43613],{"className":43566},[265],[28,43568],{"className":43569,"style":341},[269],[28,43571,7916],{"className":43572},[348],[28,43574,43576,43579],{"className":43575},[274],[28,43577,11],{"className":43578},[274,275],[28,43580,43582],{"className":43581},[556],[28,43583,43585,43605],{"className":43584},[560,561],[28,43586,43588,43602],{"className":43587},[565],[28,43589,43591],{"className":43590,"style":20945},[569],[28,43592,43593,43596],{"style":573},[28,43594],{"className":43595,"style":578},[577],[28,43597,43599],{"className":43598},[582,583,584,585],[28,43600,43555],{"className":43601,"style":2279},[274,275,585],[28,43603,593],{"className":43604},[592],[28,43606,43608],{"className":43607},[565],[28,43609,43611],{"className":43610,"style":600},[569],[28,43612],{},[28,43614,14883],{"className":43615},[355]," 本身看成一个流形，在上面装一把由数据决定的尺子。这把尺子就是 Fisher 信息矩阵",[28,43618,43620],{"className":43619,"translate":193},[192],[28,43621,43623,43744],{"className":43622,"translate":193},[197],[28,43624,43626],{"className":43625},[201],[203,43627,43628],{"xmlns":205,"display":206},[208,43629,43630,43741],{},[211,43631,43632,43642,43644,43646,43648,43650,43660,43662,43669,43671,43673,43679,43681,43687,43689,43691,43697,43699,43701,43703,43713,43715,43721,43727,43729,43731,43737,43739],{},[501,43633,43634,43636],{},[214,43635,34347],{},[211,43637,43638,43640],{},[214,43639,1686],{},[214,43641,7907],{},[217,43643,43],{"stretchy":241},[214,43645,43555],{},[217,43647,1522],{"stretchy":241},[217,43649,83],{},[501,43651,43652,43654],{},[214,43653,42084],{"mathvariant":12057},[501,43655,43656,43658],{},[214,43657,11],{},[214,43659,43555],{},[217,43661,73],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[501,43663,43664,43667],{},[214,43665,43666],{"mathvariant":484},"∂",[214,43668,1686],{},[214,43670,39714],{},[217,43672,21407],{},[501,43674,43675,43677],{},[214,43676,11],{},[214,43678,43555],{},[217,43680,20336],{},[501,43682,43683,43685],{},[214,43684,43666],{"mathvariant":484},[214,43686,7907],{},[214,43688,39714],{},[217,43690,21407],{},[501,43692,43693,43695],{},[214,43694,11],{},[214,43696,43555],{},[217,43698,246],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,43700,83],{},[217,43702,5074],{},[501,43704,43705,43707],{},[214,43706,42084],{"mathvariant":12057},[501,43708,43709,43711],{},[214,43710,11],{},[214,43712,43555],{},[217,43714,73],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[501,43716,43717,43719],{},[214,43718,43666],{"mathvariant":484},[214,43720,1686],{},[501,43722,43723,43725],{},[214,43724,43666],{"mathvariant":484},[214,43726,7907],{},[214,43728,39714],{},[217,43730,21407],{},[501,43732,43733,43735],{},[214,43734,11],{},[214,43736,43555],{},[217,43738,246],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,43740,1683],{"separator":261},[253,43742,43743],{"encoding":255},"g_{ij}(\\theta)=\\mathbb E_{p_\\theta}\\bigl[\\partial_i\\log p_\\theta\\cdot\\partial_j\\log p_\\theta\\bigr]\n=-\\mathbb E_{p_\\theta}\\bigl[\\partial_i\\partial_j\\log p_\\theta\\bigr],",[28,43745,43747,43814,44010,44122],{"className":43746,"ariaHidden":261},[260],[28,43748,43750,43753,43796,43799,43802,43805,43808,43811],{"className":43749},[265],[28,43751],{"className":43752,"style":21101},[269],[28,43754,43756,43759],{"className":43755},[274],[28,43757,34347],{"className":43758,"style":298},[274,275],[28,43760,43762],{"className":43761},[556],[28,43763,43765,43788],{"className":43764},[560,561],[28,43766,43768,43785],{"className":43767},[565],[28,43769,43771],{"className":43770,"style":3881},[569],[28,43772,43773,43776],{"style":3660},[28,43774],{"className":43775,"style":578},[577],[28,43777,43779],{"className":43778},[582,583,584,585],[28,43780,43782],{"className":43781},[274,585],[28,43783,25397],{"className":43784,"style":8057},[274,275,585],[28,43786,593],{"className":43787},[592],[28,43789,43791],{"className":43790},[565],[28,43792,43794],{"className":43793,"style":20967},[569],[28,43795],{},[28,43797,43],{"className":43798},[348],[28,43800,43555],{"className":43801,"style":2279},[274,275],[28,43803,1522],{"className":43804},[355],[28,43806],{"className":43807,"style":302},[279],[28,43809,83],{"className":43810},[306],[28,43812],{"className":43813,"style":302},[279],[28,43815,43817,43820,43902,43908,43950,43953,43958,43961,44001,44004,44007],{"className":43816},[265],[28,43818],{"className":43819,"style":2469},[269],[28,43821,43823,43826],{"className":43822},[274],[28,43824,42084],{"className":43825},[274,12195],[28,43827,43829],{"className":43828},[556],[28,43830,43832,43894],{"className":43831},[560,561],[28,43833,43835,43891],{"className":43834},[565],[28,43836,43838],{"className":43837,"style":23070},[569],[28,43839,43840,43843],{"style":573},[28,43841],{"className":43842,"style":578},[577],[28,43844,43846],{"className":43845},[582,583,584,585],[28,43847,43849],{"className":43848},[274,585],[28,43850,43852,43855],{"className":43851},[274,585],[28,43853,11],{"className":43854},[274,275,585],[28,43856,43858],{"className":43857},[556],[28,43859,43861,43882],{"className":43860},[560,561],[28,43862,43864,43879],{"className":43863},[565],[28,43865,43867],{"className":43866,"style":31564},[569],[28,43868,43870,43873],{"style":43869},"top:-2.3488em;margin-left:0em;margin-right:0.0714em;",[28,43871],{"className":43872,"style":31571},[577],[28,43874,43876],{"className":43875},[582,31575,2170,585],[28,43877,43555],{"className":43878,"style":2279},[274,275,585],[28,43880,593],{"className":43881},[592],[28,43883,43885],{"className":43884},[565],[28,43886,43889],{"className":43887,"style":43888},[569],"height:0.1512em;",[28,43890],{},[28,43892,593],{"className":43893},[592],[28,43895,43897],{"className":43896},[565],[28,43898,43900],{"className":43899,"style":20967},[569],[28,43901],{},[28,43903,43905],{"className":43904},[348],[28,43906,73],{"className":43907},[2169,2170],[28,43909,43911,43915],{"className":43910},[274],[28,43912,43666],{"className":43913,"style":43914},[274],"margin-right:0.0556em;",[28,43916,43918],{"className":43917},[556],[28,43919,43921,43942],{"className":43920},[560,561],[28,43922,43924,43939],{"className":43923},[565],[28,43925,43927],{"className":43926,"style":3881},[569],[28,43928,43930,43933],{"style":43929},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[28,43931],{"className":43932,"style":578},[577],[28,43934,43936],{"className":43935},[582,583,584,585],[28,43937,1686],{"className":43938},[274,275,585],[28,43940,593],{"className":43941},[592],[28,43943,43945],{"className":43944},[565],[28,43946,43948],{"className":43947,"style":600},[569],[28,43949],{},[28,43951],{"className":43952,"style":530},[279],[28,43954,39952,43956],{"className":43955},[21329],[28,43957,34347],{"style":39955},[28,43959],{"className":43960,"style":530},[279],[28,43962,43964,43967],{"className":43963},[274],[28,43965,11],{"className":43966},[274,275],[28,43968,43970],{"className":43969},[556],[28,43971,43973,43993],{"className":43972},[560,561],[28,43974,43976,43990],{"className":43975},[565],[28,43977,43979],{"className":43978,"style":20945},[569],[28,43980,43981,43984],{"style":573},[28,43982],{"className":43983,"style":578},[577],[28,43985,43987],{"className":43986},[582,583,584,585],[28,43988,43555],{"className":43989,"style":2279},[274,275,585],[28,43991,593],{"className":43992},[592],[28,43994,43996],{"className":43995},[565],[28,43997,43999],{"className":43998,"style":600},[569],[28,44000],{},[28,44002],{"className":44003,"style":280},[279],[28,44005,20336],{"className":44006},[284],[28,44008],{"className":44009,"style":280},[279],[28,44011,44013,44016,44056,44059,44064,44067,44107,44113,44116,44119],{"className":44012},[265],[28,44014],{"className":44015,"style":2469},[269],[28,44017,44019,44022],{"className":44018},[274],[28,44020,43666],{"className":44021,"style":43914},[274],[28,44023,44025],{"className":44024},[556],[28,44026,44028,44048],{"className":44027},[560,561],[28,44029,44031,44045],{"className":44030},[565],[28,44032,44034],{"className":44033,"style":3881},[569],[28,44035,44036,44039],{"style":43929},[28,44037],{"className":44038,"style":578},[577],[28,44040,44042],{"className":44041},[582,583,584,585],[28,44043,7907],{"className":44044,"style":8057},[274,275,585],[28,44046,593],{"className":44047},[592],[28,44049,44051],{"className":44050},[565],[28,44052,44054],{"className":44053,"style":20967},[569],[28,44055],{},[28,44057],{"className":44058,"style":530},[279],[28,44060,39952,44062],{"className":44061},[21329],[28,44063,34347],{"style":39955},[28,44065],{"className":44066,"style":530},[279],[28,44068,44070,44073],{"className":44069},[274],[28,44071,11],{"className":44072},[274,275],[28,44074,44076],{"className":44075},[556],[28,44077,44079,44099],{"className":44078},[560,561],[28,44080,44082,44096],{"className":44081},[565],[28,44083,44085],{"className":44084,"style":20945},[569],[28,44086,44087,44090],{"style":573},[28,44088],{"className":44089,"style":578},[577],[28,44091,44093],{"className":44092},[582,583,584,585],[28,44094,43555],{"className":44095,"style":2279},[274,275,585],[28,44097,593],{"className":44098},[592],[28,44100,44102],{"className":44101},[565],[28,44103,44105],{"className":44104,"style":600},[569],[28,44106],{},[28,44108,44110],{"className":44109},[355],[28,44111,246],{"className":44112},[2169,2170],[28,44114],{"className":44115,"style":302},[279],[28,44117,83],{"className":44118},[306],[28,44120],{"className":44121,"style":302},[279],[28,44123,44125,44128,44131,44211,44217,44257,44297,44300,44305,44308,44348,44354],{"className":44124},[265],[28,44126],{"className":44127,"style":2469},[269],[28,44129,5074],{"className":44130},[274],[28,44132,44134,44137],{"className":44133},[274],[28,44135,42084],{"className":44136},[274,12195],[28,44138,44140],{"className":44139},[556],[28,44141,44143,44203],{"className":44142},[560,561],[28,44144,44146,44200],{"className":44145},[565],[28,44147,44149],{"className":44148,"style":23070},[569],[28,44150,44151,44154],{"style":573},[28,44152],{"className":44153,"style":578},[577],[28,44155,44157],{"className":44156},[582,583,584,585],[28,44158,44160],{"className":44159},[274,585],[28,44161,44163,44166],{"className":44162},[274,585],[28,44164,11],{"className":44165},[274,275,585],[28,44167,44169],{"className":44168},[556],[28,44170,44172,44192],{"className":44171},[560,561],[28,44173,44175,44189],{"className":44174},[565],[28,44176,44178],{"className":44177,"style":31564},[569],[28,44179,44180,44183],{"style":43869},[28,44181],{"className":44182,"style":31571},[577],[28,44184,44186],{"className":44185},[582,31575,2170,585],[28,44187,43555],{"className":44188,"style":2279},[274,275,585],[28,44190,593],{"className":44191},[592],[28,44193,44195],{"className":44194},[565],[28,44196,44198],{"className":44197,"style":43888},[569],[28,44199],{},[28,44201,593],{"className":44202},[592],[28,44204,44206],{"className":44205},[565],[28,44207,44209],{"className":44208,"style":20967},[569],[28,44210],{},[28,44212,44214],{"className":44213},[348],[28,44215,73],{"className":44216},[2169,2170],[28,44218,44220,44223],{"className":44219},[274],[28,44221,43666],{"className":44222,"style":43914},[274],[28,44224,44226],{"className":44225},[556],[28,44227,44229,44249],{"className":44228},[560,561],[28,44230,44232,44246],{"className":44231},[565],[28,44233,44235],{"className":44234,"style":3881},[569],[28,44236,44237,44240],{"style":43929},[28,44238],{"className":44239,"style":578},[577],[28,44241,44243],{"className":44242},[582,583,584,585],[28,44244,1686],{"className":44245},[274,275,585],[28,44247,593],{"className":44248},[592],[28,44250,44252],{"className":44251},[565],[28,44253,44255],{"className":44254,"style":600},[569],[28,44256],{},[28,44258,44260,44263],{"className":44259},[274],[28,44261,43666],{"className":44262,"style":43914},[274],[28,44264,44266],{"className":44265},[556],[28,44267,44269,44289],{"className":44268},[560,561],[28,44270,44272,44286],{"className":44271},[565],[28,44273,44275],{"className":44274,"style":3881},[569],[28,44276,44277,44280],{"style":43929},[28,44278],{"className":44279,"style":578},[577],[28,44281,44283],{"className":44282},[582,583,584,585],[28,44284,7907],{"className":44285,"style":8057},[274,275,585],[28,44287,593],{"className":44288},[592],[28,44290,44292],{"className":44291},[565],[28,44293,44295],{"className":44294,"style":20967},[569],[28,44296],{},[28,44298],{"className":44299,"style":530},[279],[28,44301,39952,44303],{"className":44302},[21329],[28,44304,34347],{"style":39955},[28,44306],{"className":44307,"style":530},[279],[28,44309,44311,44314],{"className":44310},[274],[28,44312,11],{"className":44313},[274,275],[28,44315,44317],{"className":44316},[556],[28,44318,44320,44340],{"className":44319},[560,561],[28,44321,44323,44337],{"className":44322},[565],[28,44324,44326],{"className":44325,"style":20945},[569],[28,44327,44328,44331],{"style":573},[28,44329],{"className":44330,"style":578},[577],[28,44332,44334],{"className":44333},[582,583,584,585],[28,44335,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",[28,44361,44363,44397],{"className":44362,"translate":193},[197],[28,44364,44366],{"className":44365},[201],[203,44367,44368],{"xmlns":205},[208,44369,44370,44394],{},[211,44371,44372,44374,44376,44382,44384,44386,44388,44390,44392],{},[214,44373,42084],{"mathvariant":12057},[217,44375,73],{"stretchy":241},[501,44377,44378,44380],{},[214,44379,43666],{"mathvariant":484},[214,44381,1686],{},[214,44383,39714],{},[217,44385,21407],{},[214,44387,11],{},[217,44389,246],{"stretchy":241},[217,44391,83],{},[224,44393,89],{},[253,44395,44396],{"encoding":255},"\\mathbb E[\\partial_i\\log p]=0",[28,44398,44400,44478],{"className":44399,"ariaHidden":261},[260],[28,44401,44403,44406,44409,44412,44452,44455,44460,44463,44466,44469,44472,44475],{"className":44402},[265],[28,44404],{"className":44405,"style":341},[269],[28,44407,42084],{"className":44408},[274,12195],[28,44410,73],{"className":44411},[348],[28,44413,44415,44418],{"className":44414},[274],[28,44416,43666],{"className":44417,"style":43914},[274],[28,44419,44421],{"className":44420},[556],[28,44422,44424,44444],{"className":44423},[560,561],[28,44425,44427,44441],{"className":44426},[565],[28,44428,44430],{"className":44429,"style":3881},[569],[28,44431,44432,44435],{"style":43929},[28,44433],{"className":44434,"style":578},[577],[28,44436,44438],{"className":44437},[582,583,584,585],[28,44439,1686],{"className":44440},[274,275,585],[28,44442,593],{"className":44443},[592],[28,44445,44447],{"className":44446},[565],[28,44448,44450],{"className":44449,"style":600},[569],[28,44451],{},[28,44453],{"className":44454,"style":530},[279],[28,44456,39952,44458],{"className":44457},[21329],[28,44459,34347],{"style":39955},[28,44461],{"className":44462,"style":530},[279],[28,44464,11],{"className":44465},[274,275],[28,44467,246],{"className":44468},[355],[28,44470],{"className":44471,"style":302},[279],[28,44473,83],{"className":44474},[306],[28,44476],{"className":44477,"style":302},[279],[28,44479,44481,44484],{"className":44480},[265],[28,44482],{"className":44483,"style":316},[269],[28,44485,89],{"className":44486},[274],"，因为对 ",[28,44489,44491,44505],{"className":44490,"translate":193},[197],[28,44492,44494],{"className":44493},[201],[203,44495,44496],{"xmlns":205},[208,44497,44498,44502],{},[211,44499,44500],{},[214,44501,43555],{},[253,44503,44504],{"encoding":255},"\\theta",[28,44506,44508],{"className":44507,"ariaHidden":261},[260],[28,44509,44511,44514],{"className":44510},[265],[28,44512],{"className":44513,"style":7448},[269],[28,44515,43555],{"className":44516,"style":2279},[274,275]," 求导再积分，",[28,44519,44521,44571],{"className":44520,"translate":193},[197],[28,44522,44524],{"className":44523},[201],[203,44525,44526],{"xmlns":205},[208,44527,44528,44568],{},[211,44529,44530,44532,44538,44540,44542,44544,44546,44548,44554,44556,44558,44560,44562,44564,44566],{},[217,44531,21282],{},[501,44533,44534,44536],{},[214,44535,43666],{"mathvariant":484},[214,44537,1686],{},[214,44539,11],{},[228,44541,488],{},[214,44543,12988],{},[214,44545,176],{},[217,44547,83],{},[501,44549,44550,44552],{},[214,44551,43666],{"mathvariant":484},[214,44553,1686],{},[217,44555,21282],{},[214,44557,11],{},[228,44559,488],{},[214,44561,12988],{},[214,44563,176],{},[217,44565,83],{},[224,44567,89],{},[253,44569,44570],{"encoding":255},"\\int\\partial_i p\\,dx=\\partial_i\\int p\\,dx=0",[28,44572,44574,44647,44723],{"className":44573,"ariaHidden":261},[260],[28,44575,44577,44580,44583,44586,44626,44629,44632,44635,44638,44641,44644],{"className":44576},[265],[28,44578],{"className":44579,"style":21325},[269],[28,44581,21282],{"className":44582,"style":21332},[21329,21330,21331],[28,44584],{"className":44585,"style":530},[279],[28,44587,44589,44592],{"className":44588},[274],[28,44590,43666],{"className":44591,"style":43914},[274],[28,44593,44595],{"className":44594},[556],[28,44596,44598,44618],{"className":44597},[560,561],[28,44599,44601,44615],{"className":44600},[565],[28,44602,44604],{"className":44603,"style":3881},[569],[28,44605,44606,44609],{"style":43929},[28,44607],{"className":44608,"style":578},[577],[28,44610,44612],{"className":44611},[582,583,584,585],[28,44613,1686],{"className":44614},[274,275,585],[28,44616,593],{"className":44617},[592],[28,44619,44621],{"className":44620},[565],[28,44622,44624],{"className":44623,"style":600},[569],[28,44625],{},[28,44627,11],{"className":44628},[274,275],[28,44630],{"className":44631,"style":530},[279],[28,44633,12988],{"className":44634},[274,275],[28,44636,176],{"className":44637},[274,275],[28,44639],{"className":44640,"style":302},[279],[28,44642,83],{"className":44643},[306],[28,44645],{"className":44646,"style":302},[279],[28,44648,44650,44653,44693,44696,44699,44702,44705,44708,44711,44714,44717,44720],{"className":44649},[265],[28,44651],{"className":44652,"style":21325},[269],[28,44654,44656,44659],{"className":44655},[274],[28,44657,43666],{"className":44658,"style":43914},[274],[28,44660,44662],{"className":44661},[556],[28,44663,44665,44685],{"className":44664},[560,561],[28,44666,44668,44682],{"className":44667},[565],[28,44669,44671],{"className":44670,"style":3881},[569],[28,44672,44673,44676],{"style":43929},[28,44674],{"className":44675,"style":578},[577],[28,44677,44679],{"className":44678},[582,583,584,585],[28,44680,1686],{"className":44681},[274,275,585],[28,44683,593],{"className":44684},[592],[28,44686,44688],{"className":44687},[565],[28,44689,44691],{"className":44690,"style":600},[569],[28,44692],{},[28,44694],{"className":44695,"style":530},[279],[28,44697,21282],{"className":44698,"style":21332},[21329,21330,21331],[28,44700],{"className":44701,"style":530},[279],[28,44703,11],{"className":44704},[274,275],[28,44706],{"className":44707,"style":530},[279],[28,44709,12988],{"className":44710},[274,275],[28,44712,176],{"className":44713},[274,275],[28,44715],{"className":44716,"style":302},[279],[28,44718,83],{"className":44719},[306],[28,44721],{"className":44722,"style":302},[279],[28,44724,44726,44729],{"className":44725},[265],[28,44727],{"className":44728,"style":316},[269],[28,44730,89],{"className":44731},[274],[11,44733,44734],{},"为什么是它？有三个我觉得足够有说服力的理由。",[11,44736,44737,44738,44842,44843,44871],{},"第一，它是 KL 的二阶展开。把 ",[28,44739,44741,44771],{"className":44740,"translate":193},[197],[28,44742,44744],{"className":44743},[201],[203,44745,44746],{"xmlns":205},[208,44747,44748,44768],{},[211,44749,44750,44752,44754],{},[214,44751,39714],{},[217,44753,21407],{},[501,44755,44756,44758],{},[214,44757,11],{},[211,44759,44760,44762,44764,44766],{},[214,44761,43555],{},[217,44763,124],{},[214,44765,12988],{},[214,44767,43555],{},[253,44769,44770],{"encoding":255},"\\log p_{\\theta+d\\theta}",[28,44772,44774],{"className":44773,"ariaHidden":261},[260],[28,44775,44777,44781,44786,44789],{"className":44776},[265],[28,44778],{"className":44779,"style":44780},[269],"height:0.9028em;vertical-align:-0.2083em;",[28,44782,39952,44784],{"className":44783},[21329],[28,44785,34347],{"style":39955},[28,44787],{"className":44788,"style":530},[279],[28,44790,44792,44795],{"className":44791},[274],[28,44793,11],{"className":44794},[274,275],[28,44796,44798],{"className":44797},[556],[28,44799,44801,44833],{"className":44800},[560,561],[28,44802,44804,44830],{"className":44803},[565],[28,44805,44807],{"className":44806,"style":20945},[569],[28,44808,44809,44812],{"style":573},[28,44810],{"className":44811,"style":578},[577],[28,44813,44815],{"className":44814},[582,583,584,585],[28,44816,44818,44821,44824,44827],{"className":44817},[274,585],[28,44819,43555],{"className":44820,"style":2279},[274,275,585],[28,44822,124],{"className":44823},[284,585],[28,44825,12988],{"className":44826},[274,275,585],[28,44828,43555],{"className":44829,"style":2279},[274,275,585],[28,44831,593],{"className":44832},[592],[28,44834,44836],{"className":44835},[565],[28,44837,44840],{"className":44838,"style":44839},[569],"height:0.2083em;",[28,44841],{}," 在 ",[28,44844,44846,44859],{"className":44845,"translate":193},[197],[28,44847,44849],{"className":44848},[201],[203,44850,44851],{"xmlns":205},[208,44852,44853,44857],{},[211,44854,44855],{},[214,44856,43555],{},[253,44858,44504],{"encoding":255},[28,44860,44862],{"className":44861,"ariaHidden":261},[260],[28,44863,44865,44868],{"className":44864},[265],[28,44866],{"className":44867,"style":7448},[269],[28,44869,43555],{"className":44870,"style":2279},[274,275]," 处展开到二阶，一阶项的期望为零，剩下",[28,44873,44875],{"className":44874,"translate":193},[192],[28,44876,44878,45068],{"className":44877,"translate":193},[197],[28,44879,44881],{"className":44880},[201],[203,44882,44883],{"xmlns":205,"display":206},[208,44884,44885,45065],{},[211,44886,44887,44893,44895,44901,44903,44917,44919,44921,44923,44931,44933,44935,44937,44943,44949,44951,44953,44959,44961,44963,44965,44971,44973,44975,44981,44983,44985,44987,44989,44991,44993,44999,45001,45003,45011,45013,45023,45025,45027,45033,45035,45037,45043,45045,45047,45049,45051,45053,45055,45061,45063],{},[211,44888,44889,44891],{},[214,44890,24390],{"mathvariant":484},[214,44892,24393],{"mathvariant":484},[217,44894,43],{"stretchy":241},[501,44896,44897,44899],{},[214,44898,11],{},[214,44900,43555],{},[214,44902,24404],{"mathvariant":484},[501,44904,44905,44907],{},[214,44906,11],{},[211,44908,44909,44911,44913,44915],{},[214,44910,43555],{},[217,44912,124],{},[214,44914,12988],{},[214,44916,43555],{},[217,44918,1522],{"stretchy":241},[217,44920,83],{},[217,44922,5074],{},[5027,44924,44925],{"scriptlevel":89,"displaystyle":241},[21928,44926,44927,44929],{},[224,44928,1244],{},[224,44930,1253],{},[228,44932,488],{},[214,44934,42084],{"mathvariant":12057},[217,44936,73],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[501,44938,44939,44941],{},[214,44940,43666],{"mathvariant":484},[214,44942,1686],{},[501,44944,44945,44947],{},[214,44946,43666],{"mathvariant":484},[214,44948,7907],{},[214,44950,39714],{},[217,44952,21407],{},[501,44954,44955,44957],{},[214,44956,11],{},[214,44958,43555],{},[217,44960,246],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[228,44962,488],{},[214,44964,12988],{},[3054,44966,44967,44969],{},[214,44968,43555],{},[214,44970,1686],{},[228,44972,488],{},[214,44974,12988],{},[3054,44976,44977,44979],{},[214,44978,43555],{},[214,44980,7907],{},[217,44982,124],{},[214,44984,39509],{},[217,44986,43],{"stretchy":241},[214,44988,24404],{"mathvariant":484},[214,44990,12988],{},[214,44992,43555],{},[3054,44994,44995,44997],{},[214,44996,24404],{"mathvariant":484},[224,44998,3400],{},[217,45000,1522],{"stretchy":241},[217,45002,83],{},[5027,45004,45005],{"scriptlevel":89,"displaystyle":241},[21928,45006,45007,45009],{},[224,45008,1244],{},[224,45010,1253],{},[228,45012,488],{},[501,45014,45015,45017],{},[214,45016,34347],{},[211,45018,45019,45021],{},[214,45020,1686],{},[214,45022,7907],{},[228,45024,488],{},[214,45026,12988],{},[3054,45028,45029,45031],{},[214,45030,43555],{},[214,45032,1686],{},[228,45034,488],{},[214,45036,12988],{},[3054,45038,45039,45041],{},[214,45040,43555],{},[214,45042,7907],{},[217,45044,124],{},[214,45046,39509],{},[217,45048,43],{"stretchy":241},[214,45050,24404],{"mathvariant":484},[214,45052,12988],{},[214,45054,43555],{},[3054,45056,45057,45059],{},[214,45058,24404],{"mathvariant":484},[224,45060,3400],{},[217,45062,1522],{"stretchy":241},[214,45064,494],{"mathvariant":484},[253,45066,45067],{"encoding":255},"\\mathrm{KL}(p_\\theta\\|p_{\\theta+d\\theta})\n=-\\tfrac12\\,\\mathbb E\\bigl[\\partial_i\\partial_j\\log p_\\theta\\bigr]\\,d\\theta^i\\,d\\theta^j+O(\\|d\\theta\\|^3)\n=\\tfrac12\\,g_{ij}\\,d\\theta^i\\,d\\theta^j+O(\\|d\\theta\\|^3).",[28,45069,45071,45193,45500,45562,45762],{"className":45070,"ariaHidden":261},[260],[28,45072,45074,45077,45083,45086,45126,45129,45181,45184,45187,45190],{"className":45073},[265],[28,45075],{"className":45076,"style":341},[269],[28,45078,45080],{"className":45079},[274],[28,45081,24438],{"className":45082},[274,1712],[28,45084,43],{"className":45085},[348],[28,45087,45089,45092],{"className":45088},[274],[28,45090,11],{"className":45091},[274,275],[28,45093,45095],{"className":45094},[556],[28,45096,45098,45118],{"className":45097},[560,561],[28,45099,45101,45115],{"className":45100},[565],[28,45102,45104],{"className":45103,"style":20945},[569],[28,45105,45106,45109],{"style":573},[28,45107],{"className":45108,"style":578},[577],[28,45110,45112],{"className":45111},[582,583,584,585],[28,45113,43555],{"className":45114,"style":2279},[274,275,585],[28,45116,593],{"className":45117},[592],[28,45119,45121],{"className":45120},[565],[28,45122,45124],{"className":45123,"style":600},[569],[28,45125],{},[28,45127,24404],{"className":45128},[274],[28,45130,45132,45135],{"className":45131},[274],[28,45133,11],{"className":45134},[274,275],[28,45136,45138],{"className":45137},[556],[28,45139,45141,45173],{"className":45140},[560,561],[28,45142,45144,45170],{"className":45143},[565],[28,45145,45147],{"className":45146,"style":20945},[569],[28,45148,45149,45152],{"style":573},[28,45150],{"className":45151,"style":578},[577],[28,45153,45155],{"className":45154},[582,583,584,585],[28,45156,45158,45161,45164,45167],{"className":45157},[274,585],[28,45159,43555],{"className":45160,"style":2279},[274,275,585],[28,45162,124],{"className":45163},[284,585],[28,45165,12988],{"className":45166},[274,275,585],[28,45168,43555],{"className":45169,"style":2279},[274,275,585],[28,45171,593],{"className":45172},[592],[28,45174,45176],{"className":45175},[565],[28,45177,45179],{"className":45178,"style":44839},[569],[28,45180],{},[28,45182,1522],{"className":45183},[355],[28,45185],{"className":45186,"style":302},[279],[28,45188,83],{"className":45189},[306],[28,45191],{"className":45192,"style":302},[279],[28,45194,45196,45200,45203,45271,45274,45277,45283,45323,45363,45366,45371,45374,45414,45420,45423,45426,45456,45459,45462,45491,45494,45497],{"className":45195},[265],[28,45197],{"className":45198,"style":45199},[269],"height:1.2247em;vertical-align:-0.35em;",[28,45201,5074],{"className":45202},[274],[28,45204,45206,45209,45268],{"className":45205},[274],[28,45207],{"className":45208},[348,8285],[28,45210,45212],{"className":45211},[21928],[28,45213,45215,45260],{"className":45214},[560,561],[28,45216,45218,45257],{"className":45217},[565],[28,45219,45221,45235,45243],{"className":45220,"style":22005},[569],[28,45222,45223,45226],{"style":22008},[28,45224],{"className":45225,"style":1914},[577],[28,45227,45229],{"className":45228},[582,583,584,585],[28,45230,45232],{"className":45231},[274,585],[28,45233,1253],{"className":45234},[274,585],[28,45236,45237,45240],{"style":22023},[28,45238],{"className":45239,"style":1914},[577],[28,45241],{"className":45242,"style":22031},[22030],[28,45244,45245,45248],{"style":22034},[28,45246],{"className":45247,"style":1914},[577],[28,45249,45251],{"className":45250},[582,583,584,585],[28,45252,45254],{"className":45253},[274,585],[28,45255,1244],{"className":45256},[274,585],[28,45258,593],{"className":45259},[592],[28,45261,45263],{"className":45262},[565],[28,45264,45266],{"className":45265,"style":22056},[569],[28,45267],{},[28,45269],{"className":45270},[355,8285],[28,45272],{"className":45273,"style":530},[279],[28,45275,42084],{"className":45276},[274,12195],[28,45278,45280],{"className":45279},[348],[28,45281,73],{"className":45282},[2169,2170],[28,45284,45286,45289],{"className":45285},[274],[28,45287,43666],{"className":45288,"style":43914},[274],[28,45290,45292],{"className":45291},[556],[28,45293,45295,45315],{"className":45294},[560,561],[28,45296,45298,45312],{"className":45297},[565],[28,45299,45301],{"className":45300,"style":3881},[569],[28,45302,45303,45306],{"style":43929},[28,45304],{"className":45305,"style":578},[577],[28,45307,45309],{"className":45308},[582,583,584,585],[28,45310,1686],{"className":45311},[274,275,585],[28,45313,593],{"className":45314},[592],[28,45316,45318],{"className":45317},[565],[28,45319,45321],{"className":45320,"style":600},[569],[28,45322],{},[28,45324,45326,45329],{"className":45325},[274],[28,45327,43666],{"className":45328,"style":43914},[274],[28,45330,45332],{"className":45331},[556],[28,45333,45335,45355],{"className":45334},[560,561],[28,45336,45338,45352],{"className":45337},[565],[28,45339,45341],{"className":45340,"style":3881},[569],[28,45342,45343,45346],{"style":43929},[28,45344],{"className":45345,"style":578},[577],[28,45347,45349],{"className":45348},[582,583,584,585],[28,45350,7907],{"className":45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KL 虽然整体上不对称，局部却退化成一个二次型，二次型的矩阵就是 Fisher。反过来说，任何“局部像 KL”的距离都会给出同一个度量。",[11,45822,45823],{},"第二，Chentsov 定理说：单纯形上的黎曼度量，若在充分统计量（更准确地说，Markov 态射）下保持不变，则差一个常数因子必为 Fisher–Rao。换句话说，这不是我随手挑的度量，而是在“只依赖分布、不依赖参数化”这个要求下唯一的选择。",[11,45825,45826],{},"第三，对指数族它恰好是势函数的 Hessian，下一节说。",[11,45828,45829,45830,6904],{},"把 Bernoulli 族算一遍。",[28,45831,45833,45885],{"className":45832,"translate":193},[197],[28,45834,45836],{"className":45835},[201],[203,45837,45838],{"xmlns":205},[208,45839,45840,45882],{},[211,45841,45842,45844,45846,45848,45850,45852,45854,45856,45862,45864,45866,45868,45870],{},[214,45843,11],{},[217,45845,43],{"stretchy":241},[214,45847,176],{},[217,45849,889],{},[214,45851,43555],{},[217,45853,1522],{"stretchy":241},[217,45855,83],{},[3054,45857,45858,45860],{},[214,45859,43555],{},[214,45861,176],{},[217,45863,43],{"stretchy":241},[224,45865,1244],{},[217,45867,5074],{},[214,45869,43555],{},[3054,45871,45872,45874],{},[217,45873,1522],{"stretchy":241},[211,45875,45876,45878,45880],{},[224,45877,1244],{},[217,45879,5074],{},[214,45881,176],{},[253,45883,45884],{"encoding":255},"p(x\\mid\\theta)=\\theta^x(1-\\theta)^{1-x}",[28,45886,45888,45912,45933,45983],{"className":45887,"ariaHidden":261},[260],[28,45889,45891,45894,45897,45900,45903,45906,45909],{"className":45890},[265],[28,45892],{"className":45893,"style":341},[269],[28,45895,11],{"className":45896},[274,275],[28,45898,43],{"className":45899},[348],[28,45901,176],{"className":45902},[274,275],[28,45904],{"className":45905,"style":302},[279],[28,45907,889],{"className":45908},[306],[28,45910],{"className":45911,"style":302},[279],[28,45913,45915,45918,45921,45924,45927,45930],{"className":45914},[265],[28,45916],{"className":45917,"style":341},[269],[28,45919,43555],{"className":45920,"style":2279},[274,275],[28,45922,1522],{"className":45923},[355],[28,45925],{"className":45926,"style":302},[279],[28,45928,83],{"className":45929},[306],[28,45931],{"className":45932,"style":302},[279],[28,45934,45936,45939,45968,45971,45974,45977,45980],{"className":45935},[265],[28,45937],{"className":45938,"style":341},[269],[28,45940,45942,45945],{"className":45941},[274],[28,45943,43555],{"className":45944,"style":2279},[274,275],[28,45946,45948],{"className":45947},[556],[28,45949,45951],{"className":45950},[560],[28,45952,45954],{"className":45953},[565],[28,45955,45957],{"className":45956,"style":3073},[569],[28,45958,45959,45962],{"style":3094},[28,45960],{"className":45961,"style":578},[577],[28,45963,45965],{"className":45964},[582,583,584,585],[28,45966,176],{"className":45967},[274,275,585],[28,45969,43],{"className":45970},[348],[28,45972,1244],{"className":45973},[274],[28,45975],{"className":45976,"style":280},[279],[28,45978,5074],{"className":45979},[284],[28,45981],{"className":45982,"style":280},[279],[28,45984,45986,45989,45992],{"className":45985},[265],[28,45987],{"className":45988,"style":31837},[269],[28,45990,43555],{"className":45991,"style":2279},[274,275],[28,45993,45995,45998],{"className":45994},[355],[28,45996,1522],{"className":45997},[355],[28,45999,46001],{"className":46000},[556],[28,46002,46004],{"className":46003},[560],[28,46005,46007],{"className":46006},[565],[28,46008,46010],{"className":46009,"style":5477},[569],[28,46011,46012,46015],{"style":3094},[28,46013],{"className":46014,"style":578},[577],[28,46016,46018],{"className":46017},[582,583,584,585],[28,46019,46021,46024,46027],{"className":46020},[274,585],[28,46022,1244],{"className":46023},[274,585],[28,46025,5074],{"className":46026},[284,585],[28,46028,176],{"className":46029},[274,275,585],[28,46031,46033],{"className":46032,"translate":193},[192],[28,46034,46036,46078],{"className":46035,"translate":193},[197],[28,46037,46039],{"className":46038},[201],[203,46040,46041],{"xmlns":205,"display":206},[208,46042,46043,46075],{},[211,46044,46045,46047,46049,46051,46053,46055,46073],{},[214,46046,34347],{},[217,46048,43],{"stretchy":241},[214,46050,43555],{},[217,46052,1522],{"stretchy":241},[217,46054,83],{},[21928,46056,46057,46059],{},[224,46058,1244],{},[211,46060,46061,46063,46065,46067,46069,46071],{},[214,46062,43555],{},[217,46064,43],{"stretchy":241},[224,46066,1244],{},[217,46068,5074],{},[214,46070,43555],{},[217,46072,1522],{"stretchy":241},[214,46074,494],{"mathvariant":484},[253,46076,46077],{"encoding":255},"g(\\theta)=\\frac{1}{\\theta(1-\\theta)} .",[28,46079,46081,46108],{"className":46080,"ariaHidden":261},[260],[28,46082,46084,46087,46090,46093,46096,46099,46102,46105],{"className":46083},[265],[28,46085],{"className":46086,"style":341},[269],[28,46088,34347],{"className":46089,"style":298},[274,275],[28,46091,43],{"className":46092},[348],[28,46094,43555],{"className":46095,"style":2279},[274,275],[28,46097,1522],{"className":46098},[355],[28,46100],{"className":46101,"style":302},[279],[28,46103,83],{"className":46104},[306],[28,46106],{"className":46107,"style":302},[279],[28,46109,46111,46115,46200],{"className":46110},[265],[28,46112],{"className":46113,"style":46114},[269],"height:2.2574em;vertical-align:-0.936em;",[28,46116,46118,46121,46197],{"className":46117},[274],[28,46119],{"className":46120},[348,8285],[28,46122,46124],{"className":46123},[21928],[28,46125,46127,46188],{"className":46126},[560,561],[28,46128,46130,46185],{"className":46129},[565],[28,46131,46134,46166,46174],{"className":46132,"style":46133},[569],"height:1.3214em;",[28,46135,46136,46139],{"style":24920},[28,46137],{"className":46138,"style":1914},[577],[28,46140,46142,46145,46148,46151,46154,46157,46160,46163],{"className":46141},[274],[28,46143,43555],{"className":46144,"style":2279},[274,275],[28,46146,43],{"className":46147},[348],[28,46149,1244],{"className":46150},[274],[28,46152],{"className":46153,"style":280},[279],[28,46155,5074],{"className":46156},[284],[28,46158],{"className":46159,"style":280},[279],[28,46161,43555],{"className":46162,"style":2279},[274,275],[28,46164,1522],{"className":46165},[355],[28,46167,46168,46171],{"style":22023},[28,46169],{"className":46170,"style":1914},[577],[28,46172],{"className":46173,"style":22031},[22030],[28,46175,46176,46179],{"style":24940},[28,46177],{"className":46178,"style":1914},[577],[28,46180,46182],{"className":46181},[274],[28,46183,1244],{"className":46184},[274],[28,46186,593],{"className":46187},[592],[28,46189,46191],{"className":46190},[565],[28,46192,46195],{"className":46193,"style":46194},[569],"height:0.936em;",[28,46196],{},[28,46198],{"className":46199},[355,8285],[28,46201,494],{"className":46202},[274],[11,46204,46205,46206,46369],{},"弧长 ",[28,46207,46209,46250],{"className":46208,"translate":193},[197],[28,46210,46212],{"className":46211},[201],[203,46213,46214],{"xmlns":205},[208,46215,46216,46247],{},[211,46217,46218,46220,46222,46224,46226,46228,46230],{},[214,46219,12988],{},[214,46221,1664],{},[217,46223,83],{},[214,46225,12988],{},[214,46227,43555],{},[214,46229,12219],{"mathvariant":484},[46231,46232,46233],"msqrt",{},[211,46234,46235,46237,46239,46241,46243,46245],{},[214,46236,43555],{},[217,46238,43],{"stretchy":241},[224,46240,1244],{},[217,46242,5074],{},[214,46244,43555],{},[217,46246,1522],{"stretchy":241},[253,46248,46249],{"encoding":255},"ds=d\\theta\u002F\\sqrt{\\theta(1-\\theta)}",[28,46251,46253,46274],{"className":46252,"ariaHidden":261},[260],[28,46254,46256,46259,46262,46265,46268,46271],{"className":46255},[265],[28,46257],{"className":46258,"style":7448},[269],[28,46260,12988],{"className":46261},[274,275],[28,46263,1664],{"className":46264},[274,275],[28,46266],{"className":46267,"style":302},[279],[28,46269,83],{"className":46270},[306],[28,46272],{"className":46273,"style":302},[279],[28,46275,46277,46281,46284,46287,46290],{"className":46276},[265],[28,46278],{"className":46279,"style":46280},[269],"height:1.24em;vertical-align:-0.305em;",[28,46282,12988],{"className":46283},[274,275],[28,46285,43555],{"className":46286,"style":2279},[274,275],[28,46288,12219],{"className":46289},[274],[28,46291,46294],{"className":46292},[274,46293],"sqrt",[28,46295,46297,46360],{"className":46296},[560,561],[28,46298,46300,46357],{"className":46299},[565],[28,46301,46304,46339],{"className":46302,"style":46303},[569],"height:0.935em;",[28,46305,46308,46311],{"className":46306,"style":46307},[1945],"top:-3.2em;",[28,46309],{"className":46310,"style":33134},[577],[28,46312,46315,46318,46321,46324,46327,46330,46333,46336],{"className":46313,"style":46314},[274],"padding-left:1em;",[28,46316,43555],{"className":46317,"style":2279},[274,275],[28,46319,43],{"className":46320},[348],[28,46322,1244],{"className":46323},[274],[28,46325],{"className":46326,"style":280},[279],[28,46328,5074],{"className":46329},[284],[28,46331],{"className":46332,"style":280},[279],[28,46334,43555],{"className":46335,"style":2279},[274,275],[28,46337,1522],{"className":46338},[355],[28,46340,46342,46345],{"style":46341},"top:-2.895em;",[28,46343],{"className":46344,"style":33134},[577],[28,46346,46350],{"className":46347,"style":46349},[46348],"hide-tail","min-width:1.02em;height:1.28em;",[1961,46351,46354],{"xmlns":1963,"width":1964,"height":46352,"viewBox":46353,"preserveAspectRatio":1967},"1.28em","0 0 400000 1296",[1969,46355],{"d":46356},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 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的距离是 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",[28,47043,47045,47063],{"className":47044,"translate":193},[197],[28,47046,47048],{"className":47047},[201],[203,47049,47050],{"xmlns":205},[208,47051,47052,47060],{},[211,47053,47054,47056,47058],{},[214,47055,43555],{},[217,47057,950],{},[224,47059,89],{},[253,47061,47062],{"encoding":255},"\\theta\\to0",[28,47064,47066,47084],{"className":47065,"ariaHidden":261},[260],[28,47067,47069,47072,47075,47078,47081],{"className":47068},[265],[28,47070],{"className":47071,"style":7448},[269],[28,47073,43555],{"className":47074,"style":2279},[274,275],[28,47076],{"className":47077,"style":302},[279],[28,47079,950],{"className":47080},[306],[28,47082],{"className":47083,"style":302},[279],[28,47085,47087,47090],{"className":47086},[265],[28,47088],{"className":47089,"style":316},[269],[28,47091,89],{"className":47092},[274],[28,47094,47096,47114],{"className":47095,"translate":193},[197],[28,47097,47099],{"className":47098},[201],[203,47100,47101],{"xmlns":205},[208,47102,47103,47111],{},[211,47104,47105,47107,47109],{},[214,47106,43555],{},[217,47108,950],{},[224,47110,1244],{},[253,47112,47113],{"encoding":255},"\\theta\\to1",[28,47115,47117,47135],{"className":47116,"ariaHidden":261},[260],[28,47118,47120,47123,47126,47129,47132],{"className":47119},[265],[28,47121],{"className":47122,"style":7448},[269],[28,47124,43555],{"className":47125,"style":2279},[274,275],[28,47127],{"className":47128,"style":302},[279],[28,47130,950],{"className":47131},[306],[28,47133],{"className":47134,"style":302},[279],[28,47136,47138,47141],{"className":47137},[265],[28,47139],{"className":47140,"style":316},[269],[28,47142,1244],{"className":47143},[274],[28,47145,47147,47160],{"className":47146,"translate":193},[197],[28,47148,47150],{"className":47149},[201],[203,47151,47152],{"xmlns":205},[208,47153,47154,47158],{},[211,47155,47156],{},[214,47157,22365],{},[253,47159,23325],{"encoding":255},[28,47161,47163],{"className":47162,"ariaHidden":261},[260],[28,47164,47166,47169],{"className":47165},[265],[28,47167],{"className":47168,"style":645},[269],[28,47170,22365],{"className":47171,"style":298},[274,275],"，有限。这一点我很喜欢：从“几乎确定是 0”走到“几乎确定是 1”，不管两端有多极端，路径长度都是 ",[28,47174,47176,47189],{"className":47175,"translate":193},[197],[28,47177,47179],{"className":47178},[201],[203,47180,47181],{"xmlns":205},[208,47182,47183,47187],{},[211,47184,47185],{},[214,47186,22365],{},[253,47188,23325],{"encoding":255},[28,47190,47192],{"className":47191,"ariaHidden":261},[260],[28,47193,47195,47198],{"className":47194},[265],[28,47196],{"className":47197,"style":645},[269],[28,47199,22365],{"className":47200,"style":298},[274,275],"；而 ",[28,47203,47205,47240],{"className":47204,"translate":193},[197],[28,47206,47208],{"className":47207},[201],[203,47209,47210],{"xmlns":205},[208,47211,47212,47237],{},[211,47213,47214,47220,47222,47225,47227,47230,47232,47234],{},[211,47215,47216,47218],{},[214,47217,24390],{"mathvariant":484},[214,47219,24393],{"mathvariant":484},[217,47221,43],{"stretchy":241},[224,47223,47224],{},"0.999",[214,47226,24404],{"mathvariant":484},[224,47228,47229],{},"0.001",[217,47231,1522],{"stretchy":241},[217,47233,46977],{},[224,47235,47236],{},"6.9",[253,47238,47239],{"encoding":255},"\\mathrm{KL}(0.999\\|0.001)\\approx6.9",[28,47241,47243,47274],{"className":47242,"ariaHidden":261},[260],[28,47244,47246,47249,47255,47258,47262,47265,47268,47271],{"className":47245},[265],[28,47247],{"className":47248,"style":341},[269],[28,47250,47252],{"className":47251},[274],[28,47253,24438],{"className":47254},[274,1712],[28,47256,43],{"className":47257},[348],[28,47259,47261],{"className":47260},[274],"0.999∥0.001",[28,47263,1522],{"className":47264},[355],[28,47266],{"className":47267,"style":302},[279],[28,47269,46977],{"className":47270},[306],[28,47272],{"className":47273,"style":302},[279],[28,47275,47277,47280],{"className":47276},[265],[28,47278],{"className":47279,"style":316},[269],[28,47281,47236],{"className":47282},[274]," 并且随极端程度一起发散。几何距离和信息量在这里分道扬镳。",[11,47285,47286,47287,47386],{},"那个 ",[28,47288,47290,47312],{"className":47289,"translate":193},[197],[28,47291,47293],{"className":47292},[201],[203,47294,47295],{"xmlns":205},[208,47296,47297,47309],{},[211,47298,47299,47301,47303,47305],{},[224,47300,1253],{},[214,47302,46413],{},[217,47304,21407],{},[46231,47306,47307],{},[214,47308,43555],{},[253,47310,47311],{"encoding":255},"2\\arcsin\\sqrt\\theta",[28,47313,47315],{"className":47314,"ariaHidden":261},[260],[28,47316,47318,47322,47325,47328,47331,47334],{"className":47317},[265],[28,47319],{"className":47320,"style":47321},[269],"height:1.04em;vertical-align:-0.1078em;",[28,47323,1253],{"className":47324},[274],[28,47326],{"className":47327,"style":530},[279],[28,47329,46413],{"className":47330},[21329],[28,47332],{"className":47333,"style":530},[279],[28,47335,47337],{"className":47336},[274,46293],[28,47338,47340,47377],{"className":47339},[560,561],[28,47341,47343,47374],{"className":47342},[565],[28,47344,47347,47357],{"className":47345,"style":47346},[569],"height:0.9322em;",[28,47348,47350,47353],{"className":47349,"style":1926},[1945],[28,47351],{"className":47352,"style":1914},[577],[28,47354,43555],{"className":47355,"style":47356},[274,275],"margin-right:0.0278em;padding-left:0.833em;",[28,47358,47360,47363],{"style":47359},"top:-2.8922em;",[28,47361],{"className":47362,"style":1914},[577],[28,47364,47367],{"className":47365,"style":47366},[46348],"min-width:0.853em;height:1.08em;",[1961,47368,47371],{"xmlns":1963,"width":1964,"height":47369,"viewBox":47370,"preserveAspectRatio":1967},"1.08em","0 0 400000 1080",[1969,47372],{"d":47373},"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",[28,47375,593],{"className":47376},[592],[28,47378,47380],{"className":47379},[565],[28,47381,47384],{"className":47382,"style":47383},[569],"height:0.1078em;",[28,47385],{}," 不是巧合。对一般的离散分布，",[28,47388,47390],{"className":47389,"translate":193},[192],[28,47391,47393,47462],{"className":47392,"translate":193},[197],[28,47394,47396],{"className":47395},[201],[203,47397,47398],{"xmlns":205,"display":206},[208,47399,47400,47459],{},[211,47401,47402,47412,47414,47416,47418,47420,47422,47424,47426,47429,47431,47433,47439,47455,47457],{},[501,47403,47404,47406],{},[214,47405,12988],{},[211,47407,47408,47410],{},[214,47409,32913],{"mathvariant":484},[214,47411,34579],{"mathvariant":484},[217,47413,43],{"stretchy":241},[214,47415,11],{},[217,47417,1683],{"separator":261},[214,47419,1829],{},[217,47421,1522],{"stretchy":241},[217,47423,83],{},[224,47425,1253],{},[214,47427,47428],{},"arccos",[217,47430,21407],{},[217,47432,43],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[1771,47434,47435,47437],{},[217,47436,25442],{},[214,47438,1686],{},[46231,47440,47441],{},[211,47442,47443,47449],{},[501,47444,47445,47447],{},[214,47446,11],{},[214,47448,1686],{},[501,47450,47451,47453],{},[214,47452,1829],{},[214,47454,1686],{},[217,47456,1522],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[217,47458,1683],{"separator":261},[253,47460,47461],{"encoding":255},"d_{\\mathrm{FR}}(p,q)=2\\arccos\\Bigl(\\sum_i\\sqrt{p_iq_i}\\Bigr),",[28,47463,47465,47545],{"className":47464,"ariaHidden":261},[260],[28,47466,47468,47471,47518,47521,47524,47527,47530,47533,47536,47539,47542],{"className":47467},[265],[28,47469],{"className":47470,"style":341},[269],[28,47472,47474,47477],{"className":47473},[274],[28,47475,12988],{"className":47476},[274,275],[28,47478,47480],{"className":47479},[556],[28,47481,47483,47510],{"className":47482},[560,561],[28,47484,47486,47507],{"className":47485},[565],[28,47487,47489],{"className":47488,"style":570},[569],[28,47490,47491,47494],{"style":573},[28,47492],{"className":47493,"style":578},[577],[28,47495,47497],{"className":47496},[582,583,584,585],[28,47498,47500],{"className":47499},[274,585],[28,47501,47503],{"className":47502},[274,585],[28,47504,47506],{"className":47505},[274,1712,585],"FR",[28,47508,593],{"className":47509},[592],[28,47511,47513],{"className":47512},[565],[28,47514,47516],{"className":47515,"style":600},[569],[28,47517],{},[28,47519,43],{"className":47520},[348],[28,47522,11],{"className":47523},[274,275],[28,47525,1683],{"className":47526},[1723],[28,47528],{"className":47529,"style":530},[279],[28,47531,1829],{"className":47532,"style":298},[274,275],[28,47534,1522],{"className":47535},[355],[28,47537],{"className":47538,"style":302},[279],[28,47540,83],{"className":47541},[306],[28,47543],{"className":47544,"style":302},[279],[28,47546,47548,47552,47555,47558,47561,47567,47611,47614,47741,47747],{"className":47547},[265],[28,47549],{"className":47550,"style":47551},[269],"height:2.4277em;vertical-align:-1.2777em;",[28,47553,1253],{"className":47554},[274],[28,47556],{"className":47557,"style":530},[279],[28,47559,47428],{"className":47560},[21329],[28,47562,47564],{"className":47563},[348],[28,47565,43],{"className":47566},[2169,22506],[28,47568,47570],{"className":47569},[21329,21485],[28,47571,47573,47603],{"className":47572},[560,561],[28,47574,47576,47600],{"className":47575},[565],[28,47577,47579,47590],{"className":47578,"style":25713},[569],[28,47580,47581,47584],{"style":25716},[28,47582],{"className":47583,"style":25720},[577],[28,47585,47587],{"className":47586},[582,583,584,585],[28,47588,1686],{"className":47589},[274,275,585],[28,47591,47592,47595],{"style":25732},[28,47593],{"className":47594,"style":25720},[577],[28,47596,47597],{},[28,47598,25442],{"className":47599},[21329,21330,21547],[28,47601,593],{"className":47602},[592],[28,47604,47606],{"className":47605},[565],[28,47607,47609],{"className":47608,"style":26154},[569],[28,47610],{},[28,47612],{"className":47613,"style":530},[279],[28,47615,47617],{"className":47616},[274,46293],[28,47618,47620,47732],{"className":47619},[560,561],[28,47621,47623,47729],{"className":47622},[565],[28,47624,47626,47716],{"className":47625,"style":6798},[569],[28,47627,47629,47632],{"className":47628,"style":1926},[1945],[28,47630],{"className":47631,"style":1914},[577],[28,47633,47636,47676],{"className":47634,"style":47635},[274],"padding-left:0.833em;",[28,47637,47639,47642],{"className":47638},[274],[28,47640,11],{"className":47641},[274,275],[28,47643,47645],{"className":47644},[556],[28,47646,47648,47668],{"className":47647},[560,561],[28,47649,47651,47665],{"className":47650},[565],[28,47652,47654],{"className":47653,"style":3881},[569],[28,47655,47656,47659],{"style":573},[28,47657],{"className":47658,"style":578},[577],[28,47660,47662],{"className":47661},[582,583,584,585],[28,47663,1686],{"className":47664},[274,275,585],[28,47666,593],{"className":47667},[592],[28,47669,47671],{"className":47670},[565],[28,47672,47674],{"className":47673,"style":600},[569],[28,47675],{},[28,47677,47679,47682],{"className":47678},[274],[28,47680,1829],{"className":47681,"style":298},[274,275],[28,47683,47685],{"className":47684},[556],[28,47686,47688,47708],{"className":47687},[560,561],[28,47689,47691,47705],{"className":47690},[565],[28,47692,47694],{"className":47693,"style":3881},[569],[28,47695,47696,47699],{"style":3660},[28,47697],{"className":47698,"style":578},[577],[28,47700,47702],{"className":47701},[582,583,584,585],[28,47703,1686],{"className":47704},[274,275,585],[28,47706,593],{"className":47707},[592],[28,47709,47711],{"className":47710},[565],[28,47712,47714],{"className":47713,"style":600},[569],[28,47715],{},[28,47717,47719,47722],{"style":47718},"top:-2.7119em;",[28,47720],{"className":47721,"style":1914},[577],[28,47723,47725],{"className":47724,"style":47366},[46348],[1961,47726,47727],{"xmlns":1963,"width":1964,"height":47369,"viewBox":47370,"preserveAspectRatio":1967},[1969,47728],{"d":47373},[28,47730,593],{"className":47731},[592],[28,47733,47735],{"className":47734},[565],[28,47736,47739],{"className":47737,"style":47738},[569],"height:0.2881em;",[28,47740],{},[28,47742,47744],{"className":47743},[355],[28,47745,1522],{"className":47746},[2169,22506],[28,47748,1683],{"className":47749},[1723],[11,47751,47752,47753,47879,47880,47940,47941,47969,47970,48116,48117,48331],{},"而映射 ",[28,47754,47756,47782],{"className":47755,"translate":193},[197],[28,47757,47759],{"className":47758},[201],[203,47760,47761],{"xmlns":205},[208,47762,47763,47779],{},[211,47764,47765,47767,47769,47771,47773,47775],{},[214,47766,11],{},[217,47768,38119],{},[214,47770,3936],{},[217,47772,83],{},[224,47774,1253],{},[46231,47776,47777],{},[214,47778,11],{},[253,47780,47781],{"encoding":255},"p\\mapsto u=2\\sqrt p",[28,47783,47785,47804,47822],{"className":47784,"ariaHidden":261},[260],[28,47786,47788,47792,47795,47798,47801],{"className":47787},[265],[28,47789],{"className":47790,"style":47791},[269],"height:0.7054em;vertical-align:-0.1944em;",[28,47793,11],{"className":47794},[274,275],[28,47796],{"className":47797,"style":302},[279],[28,47799,38119],{"className":47800},[306],[28,47802],{"className":47803,"style":302},[279],[28,47805,47807,47810,47813,47816,47819],{"className":47806},[265],[28,47808],{"className":47809,"style":645},[269],[28,47811,3936],{"className":47812},[274,275],[28,47814],{"className":47815,"style":302},[279],[28,47817,83],{"className":47818},[306],[28,47820],{"className":47821,"style":302},[279],[28,47823,47825,47829,47832],{"className":47824},[265],[28,47826],{"className":47827,"style":47828},[269],"height:1.04em;vertical-align:-0.3369em;",[28,47830,1253],{"className":47831},[274],[28,47833,47835],{"className":47834},[274,46293],[28,47836,47838,47870],{"className":47837},[560,561],[28,47839,47841,47867],{"className":47840},[565],[28,47842,47845,47854],{"className":47843,"style":47844},[569],"height:0.7031em;",[28,47846,47848,47851],{"className":47847,"style":1926},[1945],[28,47849],{"className":47850,"style":1914},[577],[28,47852,11],{"className":47853,"style":47635},[274,275],[28,47855,47857,47860],{"style":47856},"top:-2.6631em;",[28,47858],{"className":47859,"style":1914},[577],[28,47861,47863],{"className":47862,"style":47366},[46348],[1961,47864,47865],{"xmlns":1963,"width":1964,"height":47369,"viewBox":47370,"preserveAspectRatio":1967},[1969,47866],{"d":47373},[28,47868,593],{"className":47869},[592],[28,47871,47873],{"className":47872},[565],[28,47874,47877],{"className":47875,"style":47876},[569],"height:0.3369em;",[28,47878],{}," 把单纯形嵌入到 ",[28,47881,47883,47901],{"className":47882,"translate":193},[197],[28,47884,47886],{"className":47885},[201],[203,47887,47888],{"xmlns":205},[208,47889,47890,47898],{},[211,47891,47892],{},[3054,47893,47894,47896],{},[214,47895,34579],{"mathvariant":12057},[214,47897,24390],{},[253,47899,47900],{"encoding":255},"\\mathbb R^K",[28,47902,47904],{"className":47903,"ariaHidden":261},[260],[28,47905,47907,47911],{"className":47906},[265],[28,47908],{"className":47909,"style":47910},[269],"height:0.8413em;",[28,47912,47914,47917],{"className":47913},[274],[28,47915,34579],{"className":47916},[274,12195],[28,47918,47920],{"className":47919},[556],[28,47921,47923],{"className":47922},[560],[28,47924,47926],{"className":47925},[565],[28,47927,47929],{"className":47928,"style":47910},[569],[28,47930,47931,47934],{"style":3094},[28,47932],{"className":47933,"style":578},[577],[28,47935,47937],{"className":47936},[582,583,584,585],[28,47938,24390],{"className":47939,"style":28331},[274,275,585]," 中半径为 ",[28,47942,47944,47957],{"className":47943,"translate":193},[197],[28,47945,47947],{"className":47946},[201],[203,47948,47949],{"xmlns":205},[208,47950,47951,47955],{},[211,47952,47953],{},[224,47954,1253],{},[253,47956,1253],{"encoding":255},[28,47958,47960],{"className":47959,"ariaHidden":261},[260],[28,47961,47963,47966],{"className":47962},[265],[28,47964],{"className":47965,"style":316},[269],[28,47967,1253],{"className":47968},[274]," 的球面上（因为 ",[28,47971,47973,48003],{"className":47972,"translate":193},[197],[28,47974,47976],{"className":47975},[201],[203,47977,47978],{"xmlns":205},[208,47979,47980,48000],{},[211,47981,47982,47988,47990,47996,47998],{},[501,47983,47984,47986],{},[217,47985,25442],{},[214,47987,1686],{},[224,47989,9215],{},[501,47991,47992,47994],{},[214,47993,11],{},[214,47995,1686],{},[217,47997,83],{},[224,47999,9215],{},[253,48001,48002],{"encoding":255},"\\sum_i4p_i=4",[28,48004,48006,48107],{"className":48005,"ariaHidden":261},[260],[28,48007,48009,48012,48052,48055,48058,48098,48101,48104],{"className":48008},[265],[28,48010],{"className":48011,"style":41128},[269],[28,48013,48015,48018],{"className":48014},[21329],[28,48016,25442],{"className":48017,"style":41141},[21329,21330,21331],[28,48019,48021],{"className":48020},[556],[28,48022,48024,48044],{"className":48023},[560,561],[28,48025,48027,48041],{"className":48026},[565],[28,48028,48030],{"className":48029,"style":41154},[569],[28,48031,48032,48035],{"style":41157},[28,48033],{"className":48034,"style":578},[577],[28,48036,48038],{"className":48037},[582,583,584,585],[28,48039,1686],{"className":48040},[274,275,585],[28,48042,593],{"className":48043},[592],[28,48045,48047],{"className":48046},[565],[28,48048,48050],{"className":48049,"style":41176},[569],[28,48051],{},[28,48053],{"className":48054,"style":530},[279],[28,48056,9215],{"className":48057},[274],[28,48059,48061,48064],{"className":48060},[274],[28,48062,11],{"className":48063},[274,275],[28,48065,48067],{"className":48066},[556],[28,48068,48070,48090],{"className":48069},[560,561],[28,48071,48073,48087],{"className":48072},[565],[28,48074,48076],{"className":48075,"style":3881},[569],[28,48077,48078,48081],{"style":573},[28,48079],{"className":48080,"style":578},[577],[28,48082,48084],{"className":48083},[582,583,584,585],[28,48085,1686],{"className":48086},[274,275,585],[28,48088,593],{"className":48089},[592],[28,48091,48093],{"className":48092},[565],[28,48094,48096],{"className":48095,"style":600},[569],[28,48097],{},[28,48099],{"className":48100,"style":302},[279],[28,48102,83],{"className":48103},[306],[28,48105],{"className":48106,"style":302},[279],[28,48108,48110,48113],{"className":48109},[265],[28,48111],{"className":48112,"style":316},[269],[28,48114,9215],{"className":48115},[274],"），Fisher–Rao 距离正好是球面上的大圆距离。这个嵌入叫 Hellinger 嵌入，",[28,48118,48120,48154],{"className":48119,"translate":193},[197],[28,48121,48123],{"className":48122},[201],[203,48124,48125],{"xmlns":205},[208,48126,48127,48151],{},[211,48128,48129,48135],{},[501,48130,48131,48133],{},[217,48132,25442],{},[214,48134,1686],{},[46231,48136,48137],{},[211,48138,48139,48145],{},[501,48140,48141,48143],{},[214,48142,11],{},[214,48144,1686],{},[501,48146,48147,48149],{},[214,48148,1829],{},[214,48150,1686],{},[253,48152,48153],{"encoding":255},"\\sum_i\\sqrt{p_iq_i}",[28,48155,48157],{"className":48156,"ariaHidden":261},[260],[28,48158,48160,48164,48204,48207],{"className":48159},[265],[28,48161],{"className":48162,"style":48163},[269],"height:1.0869em;vertical-align:-0.3369em;",[28,48165,48167,48170],{"className":48166},[21329],[28,48168,25442],{"className":48169,"style":41141},[21329,21330,21331],[28,48171,48173],{"className":48172},[556],[28,48174,48176,48196],{"className":48175},[560,561],[28,48177,48179,48193],{"className":48178},[565],[28,48180,48182],{"className":48181,"style":41154},[569],[28,48183,48184,48187],{"style":41157},[28,48185],{"className":48186,"style":578},[577],[28,48188,48190],{"className":48189},[582,583,584,585],[28,48191,1686],{"className":48192},[274,275,585],[28,48194,593],{"className":48195},[592],[28,48197,48199],{"className":48198},[565],[28,48200,48202],{"className":48201,"style":41176},[569],[28,48203],{},[28,48205],{"className":48206,"style":530},[279],[28,48208,48210],{"className":48209},[274,46293],[28,48211,48213,48323],{"className":48212},[560,561],[28,48214,48216,48320],{"className":48215},[565],[28,48217,48219,48308],{"className":48218,"style":47844},[569],[28,48220,48222,48225],{"className":48221,"style":1926},[1945],[28,48223],{"className":48224,"style":1914},[577],[28,48226,48228,48268],{"className":48227,"style":47635},[274],[28,48229,48231,48234],{"className":48230},[274],[28,48232,11],{"className":48233},[274,275],[28,48235,48237],{"className":48236},[556],[28,48238,48240,48260],{"className":48239},[560,561],[28,48241,48243,48257],{"className":48242},[565],[28,48244,48246],{"className":48245,"style":3881},[569],[28,48247,48248,48251],{"style":573},[28,48249],{"className":48250,"style":578},[577],[28,48252,48254],{"className":48253},[582,583,584,585],[28,48255,1686],{"className":48256},[274,275,585],[28,48258,593],{"className":48259},[592],[28,48261,48263],{"className":48262},[565],[28,48264,48266],{"className":48265,"style":600},[569],[28,48267],{},[28,48269,48271,48274],{"className":48270},[274],[28,48272,1829],{"className":48273,"style":298},[274,275],[28,48275,48277],{"className":48276},[556],[28,48278,48280,48300],{"className":48279},[560,561],[28,48281,48283,48297],{"className":48282},[565],[28,48284,48286],{"className":48285,"style":3881},[569],[28,48287,48288,48291],{"style":3660},[28,48289],{"className":48290,"style":578},[577],[28,48292,48294],{"className":48293},[582,583,584,585],[28,48295,1686],{"className":48296},[274,275,585],[28,48298,593],{"className":48299},[592],[28,48301,48303],{"className":48302},[565],[28,48304,48306],{"className":48305,"style":600},[569],[28,48307],{},[28,48309,48310,48313],{"style":47856},[28,48311],{"className":48312,"style":1914},[577],[28,48314,48316],{"className":48315,"style":47366},[46348],[1961,48317,48318],{"xmlns":1963,"width":1964,"height":47369,"viewBox":47370,"preserveAspectRatio":1967},[1969,48319],{"d":47373},[28,48321,593],{"className":48322},[592],[28,48324,48326],{"className":48325},[565],[28,48327,48329],{"className":48328,"style":47876},[569],[28,48330],{}," 叫 Bhattacharyya 系数。",[11,48333,48334],{},"有了球面，测地线就清楚了。两点之间的大圆弧写回单纯形是",[28,48336,48338],{"className":48337,"translate":193},[192],[28,48339,48341,48459],{"className":48340,"translate":193},[197],[28,48342,48344],{"className":48343},[201],[203,48345,48346],{"xmlns":205,"display":206},[208,48347,48348,48456],{},[211,48349,48350,48356,48358,48398,48400,48402,48408,48410,48416,48418,48420,48422,48424,48426,48428,48436,48438,48440,48448,48454],{},[501,48351,48352,48354],{},[214,48353,11],{},[214,48355,1241],{},[217,48357,83],{},[21928,48359,48360,48392],{},[211,48361,48362,48364,48366,48368,48370,48372,48374,48378,48380,48382,48386],{},[217,48363,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,48365,43],{"stretchy":241},[224,48367,1244],{},[217,48369,5074],{},[214,48371,1241],{},[217,48373,1522],{"stretchy":241},[46231,48375,48376],{},[214,48377,11],{},[217,48379,124],{},[214,48381,1241],{},[46231,48383,48384],{},[214,48385,1829],{},[3054,48387,48388,48390],{},[217,48389,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[224,48391,1253],{},[501,48393,48394,48396],{},[214,48395,12058],{},[214,48397,1241],{},[217,48399,1683],{"separator":261},[279,48401],{"width":4053},[501,48403,48404,48406],{},[214,48405,12058],{},[214,48407,1241],{},[217,48409,83],{},[1771,48411,48412,48414],{},[217,48413,25442],{},[214,48415,1686],{},[217,48417,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,48419,43],{"stretchy":241},[224,48421,1244],{},[217,48423,5074],{},[214,48425,1241],{},[217,48427,1522],{"stretchy":241},[46231,48429,48430],{},[501,48431,48432,48434],{},[214,48433,11],{},[214,48435,1686],{},[217,48437,124],{},[214,48439,1241],{},[46231,48441,48442],{},[501,48443,48444,48446],{},[214,48445,1829],{},[214,48447,1686],{},[3054,48449,48450,48452],{},[217,48451,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[224,48453,1253],{},[214,48455,494],{"mathvariant":484},[253,48457,48458],{"encoding":255},"p_t=\\frac{\\bigl((1-t)\\sqrt p+t\\sqrt q\\bigr)^2}{Z_t},\\qquad\nZ_t=\\sum_i\\bigl((1-t)\\sqrt{p_i}+t\\sqrt{q_i}\\bigr)^2 .",[28,48460,48462,48517,48845,48916,49022],{"className":48461,"ariaHidden":261},[260],[28,48463,48465,48468,48508,48511,48514],{"className":48464},[265],[28,48466],{"className":48467,"style":612},[269],[28,48469,48471,48474],{"className":48470},[274],[28,48472,11],{"className":48473},[274,275],[28,48475,48477],{"className":48476},[556],[28,48478,48480,48500],{"className":48479},[560,561],[28,48481,48483,48497],{"className":48482},[565],[28,48484,48486],{"className":48485,"style":3978},[569],[28,48487,48488,48491],{"style":573},[28,48489],{"className":48490,"style":578},[577],[28,48492,48494],{"className":48493},[582,583,584,585],[28,48495,1241],{"className":48496},[274,275,585],[28,48498,593],{"className":48499},[592],[28,48501,48503],{"className":48502},[565],[28,48504,48506],{"className":48505,"style":600},[569],[28,48507],{},[28,48509],{"className":48510,"style":302},[279],[28,48512,83],{"className":48513},[306],[28,48515],{"className":48516,"style":302},[279],[28,48518,48520,48524,48787,48790,48793,48796,48836,48839,48842],{"className":48519},[265],[28,48521],{"className":48522,"style":48523},[269],"height:2.63em;vertical-align:-0.836em;",[28,48525,48527,48530,48784],{"className":48526},[274],[28,48528],{"className":48529},[348,8285],[28,48531,48533],{"className":48532},[21928],[28,48534,48536,48775],{"className":48535},[560,561],[28,48537,48539,48772],{"className":48538},[565],[28,48540,48543,48593,48602],{"className":48541,"style":48542},[569],"height:1.794em;",[28,48544,48546,48550],{"style":48545},"top:-2.368em;",[28,48547],{"className":48548,"style":48549},[577],"height:3.054em;",[28,48551,48553],{"className":48552},[274],[28,48554,48556,48559],{"className":48555},[274],[28,48557,12058],{"className":48558,"style":28331},[274,275],[28,48560,48562],{"className":48561},[556],[28,48563,48565,48585],{"className":48564},[560,561],[28,48566,48568,48582],{"className":48567},[565],[28,48569,48571],{"className":48570,"style":3978},[569],[28,48572,48573,48576],{"style":28346},[28,48574],{"className":48575,"style":578},[577],[28,48577,48579],{"className":48578},[582,583,584,585],[28,48580,1241],{"className":48581},[274,275,585],[28,48583,593],{"className":48584},[592],[28,48586,48588],{"className":48587},[565],[28,48589,48591],{"className":48590,"style":600},[569],[28,48592],{},[28,48594,48596,48599],{"style":48595},"top:-3.284em;",[28,48597],{"className":48598,"style":48549},[577],[28,48600],{"className":48601,"style":22031},[22030],[28,48603,48605,48608],{"style":48604},"top:-3.794em;",[28,48606],{"className":48607,"style":48549},[577],[28,48609,48611,48617,48620,48623,48626,48629,48632,48635,48638,48682,48685,48688,48691,48694,48739],{"className":48610},[274],[28,48612,48614],{"className":48613},[348],[28,48615,43],{"className":48616},[2169,2170],[28,48618,43],{"className":48619},[348],[28,48621,1244],{"className":48622},[274],[28,48624],{"className":48625,"style":280},[279],[28,48627,5074],{"className":48628},[284],[28,48630],{"className":48631,"style":280},[279],[28,48633,1241],{"className":48634},[274,275],[28,48636,1522],{"className":48637},[355],[28,48639,48641],{"className":48640},[274,46293],[28,48642,48644,48674],{"className":48643},[560,561],[28,48645,48647,48671],{"className":48646},[565],[28,48648,48650,48659],{"className":48649,"style":47844},[569],[28,48651,48653,48656],{"className":48652,"style":1926},[1945],[28,48654],{"className":48655,"style":1914},[577],[28,48657,11],{"className":48658,"style":47635},[274,275],[28,48660,48661,48664],{"style":47856},[28,48662],{"className":48663,"style":1914},[577],[28,48665,48667],{"className":48666,"style":47366},[46348],[1961,48668,48669],{"xmlns":1963,"width":1964,"height":47369,"viewBox":47370,"preserveAspectRatio":1967},[1969,48670],{"d":47373},[28,48672,593],{"className":48673},[592],[28,48675,48677],{"className":48676},[565],[28,48678,48680],{"className":48679,"style":47876},[569],[28,48681],{},[28,48683],{"className":48684,"style":280},[279],[28,48686,124],{"className":48687},[284],[28,48689],{"className":48690,"style":280},[279],[28,48692,1241],{"className":48693},[274,275],[28,48695,48697],{"className":48696},[274,46293],[28,48698,48700,48731],{"className":48699},[560,561],[28,48701,48703,48728],{"className":48702},[565],[28,48704,48706,48716],{"className":48705,"style":47844},[569],[28,48707,48709,48712],{"className":48708,"style":1926},[1945],[28,48710],{"className":48711,"style":1914},[577],[28,48713,1829],{"className":48714,"style":48715},[274,275],"margin-right:0.0359em;padding-left:0.833em;",[28,48717,48718,48721],{"style":47856},[28,48719],{"className":48720,"style":1914},[577],[28,48722,48724],{"className":48723,"style":47366},[46348],[1961,48725,48726],{"xmlns":1963,"width":1964,"height":47369,"viewBox":47370,"preserveAspectRatio":1967},[1969,48727],{"d":47373},[28,48729,593],{"className":48730},[592],[28,48732,48734],{"className":48733},[565],[28,48735,48737],{"className":48736,"style":47876},[569],[28,48738],{},[28,48740,48742,48748],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],[28,49039,49041,49105],{"className":49040},[565],[28,49042,49044,49093],{"className":49043,"style":6798},[569],[28,49045,49047,49050],{"className":49046,"style":1926},[1945],[28,49048],{"className":49049,"style":1914},[577],[28,49051,49053],{"className":49052,"style":47635},[274],[28,49054,49056,49059],{"className":49055},[274],[28,49057,1829],{"className":49058,"style":298},[274,275],[28,49060,49062],{"className":49061},[556],[28,49063,49065,49085],{"className":49064},[560,561],[28,49066,49068,49082],{"className":49067},[565],[28,49069,49071],{"className":49070,"style":3881},[569],[28,49072,49073,49076],{"style":3660},[28,49074],{"className":49075,"style":578},[577],[28,49077,49079],{"className":49078},[582,583,584,585],[28,49080,1686],{"className":49081},[274,275,585],[28,49083,593],{"className":49084},[592],[28,49086,49088],{"className":49087},[565],[28,49089,49091],{"className":49090,"style":600},[569],[28,49092],{},[28,49094,49095,49098],{"style":47718},[28,49096],{"className":49097,"style":1914},[577],[28,49099,49101],{"className":49100,"style":47366},[46348],[1961,49102,49103],{"xmlns":1963,"width":1964,"height":47369,"viewBox":47370,"preserveAspectRatio":1967},[1969,49104],{"d":47373},[28,49106,593],{"className":49107},[592],[28,49109,49111],{"className":49110},[565],[28,49112,49114],{"className":49113,"style":47738},[569],[28,49115],{},[28,49117,49119,49125],{"className":49118},[355],[28,49120,49122],{"className":49121},[355],[28,49123,1522],{"className":49124},[2169,2170],[28,49126,49128],{"className":49127},[556],[28,49129,49131],{"className":49130},[560],[28,49132,49134],{"className":49133},[565],[28,49135,49137],{"className":49136,"style":48760},[569],[28,49138,49139,49142],{"style":33361},[28,49140],{"className":49141,"style":578},[577],[28,49143,49145],{"className":49144},[582,583,584,585],[28,49146,1253],{"className":49147},[274,585],[28,49149,494],{"className":49150},[274],[11,49152,49153],{},"这不是线性插值：概率质量走的是一条弯的路。如果我在单纯形上按大圆距离做最近原型分类，两个原型之间的“垂直平分线”是一条大圆，投影回单纯形是弯的；而按线性插值它会是直的。这就是我理解的“非欧氏边界”最干净的来源。",[414,49155,49156],{"id":49156},"指数族",[11,49158,49159,49160,49188,49189,49217],{},"我抛一枚硬币 100 次，60 次正面。这句话里真正有用的信息只有两个数：100 和 60。给定正面概率 ",[28,49161,49163,49176],{"className":49162,"translate":193},[197],[28,49164,49166],{"className":49165},[201],[203,49167,49168],{"xmlns":205},[208,49169,49170,49174],{},[211,49171,49172],{},[214,49173,43555],{},[253,49175,44504],{"encoding":255},[28,49177,49179],{"className":49178,"ariaHidden":261},[260],[28,49180,49182,49185],{"className":49181},[265],[28,49183],{"className":49184,"style":7448},[269],[28,49186,43555],{"className":49187,"style":2279},[274,275],"，正面次数服从二项分布，而样本对 ",[28,49190,49192,49205],{"className":49191,"translate":193},[197],[28,49193,49195],{"className":49194},[201],[203,49196,49197],{"xmlns":205},[208,49198,49199,49203],{},[211,49200,49201],{},[214,49202,43555],{},[253,49204,44504],{"encoding":255},[28,49206,49208],{"className":49207,"ariaHidden":261},[260],[28,49209,49211,49214],{"className":49210},[265],[28,49212],{"className":49213,"style":7448},[269],[28,49215,43555],{"className":49216,"style":2279},[274,275]," 的全部影响都通过“正面次数”这个线性统计量进去——这就是充分统计量。",[11,49219,49220],{},"指数族把这个结构写成一个模板：",[28,49222,49224],{"className":49223,"translate":193},[192],[28,49225,49227,49296],{"className":49226,"translate":193},[197],[28,49228,49230],{"className":49229},[201],[203,49231,49232],{"xmlns":205,"display":206},[208,49233,49234,49293],{},[211,49235,49236,49238,49240,49242,49244,49246,49248,49250,49253,49255,49257,49259,49261,49263,49265,49271,49273,49275,49277,49279,49281,49283,49285,49287,49289,49291],{},[214,49237,11],{},[217,49239,43],{"stretchy":241},[214,49241,176],{},[217,49243,889],{},[214,49245,43555],{},[217,49247,1522],{"stretchy":241},[217,49249,83],{},[214,49251,49252],{},"h",[217,49254,43],{"stretchy":241},[214,49256,176],{},[217,49258,1522],{"stretchy":241},[214,49260,40075],{},[217,49262,21407],{},[217,49264,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[3054,49266,49267,49269],{},[214,49268,43555],{},[214,49270,884],{"mathvariant":484},[214,49272,443],{},[217,49274,43],{"stretchy":241},[214,49276,176],{},[217,49278,1522],{"stretchy":241},[217,49280,5074],{},[214,49282,27871],{},[217,49284,43],{"stretchy":241},[214,49286,43555],{},[217,49288,1522],{"stretchy":241},[217,49290,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,49292,494],{"mathvariant":484},[253,49294,49295],{"encoding":255},"p(x\\mid\\theta)=h(x)\\exp\\bigl(\\theta^\\top T(x)-\\psi(\\theta)\\bigr).",[28,49297,49299,49323,49344,49426],{"className":49298,"ariaHidden":261},[260],[28,49300,49302,49305,49308,49311,49314,49317,49320],{"className":49301},[265],[28,49303],{"className":49304,"style":341},[269],[28,49306,11],{"className":49307},[274,275],[28,49309,43],{"className":49310},[348],[28,49312,176],{"className":49313},[274,275],[28,49315],{"className":49316,"style":302},[279],[28,49318,889],{"className":49319},[306],[28,49321],{"className":49322,"style":302},[279],[28,49324,49326,49329,49332,49335,49338,49341],{"className":49325},[265],[28,49327],{"className":49328,"style":341},[269],[28,49330,43555],{"className":49331,"style":2279},[274,275],[28,49333,1522],{"className":49334},[355],[28,49336],{"className":49337,"style":302},[279],[28,49339,83],{"className":49340},[306],[28,49342],{"className":49343,"style":302},[279],[28,49345,49347,49351,49354,49357,49360,49363,49366,49369,49375,49405,49408,49411,49414,49417,49420,49423],{"className":49346},[265],[28,49348],{"className":49349,"style":49350},[269],"height:1.2491em;vertical-align:-0.35em;",[28,49352,49252],{"className":49353},[274,275],[28,49355,43],{"className":49356},[348],[28,49358,176],{"className":49359},[274,275],[28,49361,1522],{"className":49362},[355],[28,49364],{"className":49365,"style":530},[279],[28,49367,40075],{"className":49368},[21329],[28,49370,49372],{"className":49371},[348],[28,49373,43],{"className":49374},[2169,2170],[28,49376,49378,49381],{"className":49377},[274],[28,49379,43555],{"className":49380,"style":2279},[274,275],[28,49382,49384],{"className":49383},[556],[28,49385,49387],{"className":49386},[560],[28,49388,49390],{"className":49389},[565],[28,49391,49394],{"className":49392,"style":49393},[569],"height:0.8991em;",[28,49395,49396,49399],{"style":4880},[28,49397],{"className":49398,"style":578},[577],[28,49400,49402],{"className":49401},[582,583,584,585],[28,49403,884],{"className":49404},[274,585],[28,49406,443],{"className":49407,"style":459},[274,275],[28,49409,43],{"className":49410},[348],[28,49412,176],{"className":49413},[274,275],[28,49415,1522],{"className":49416},[355],[28,49418],{"className":49419,"style":280},[279],[28,49421,5074],{"className":49422},[284],[28,49424],{"className":49425,"style":280},[279],[28,49427,49429,49432,49435,49438,49441,49444,49450],{"className":49428},[265],[28,49430],{"className":49431,"style":2469},[269],[28,49433,27871],{"className":49434,"style":298},[274,275],[28,49436,43],{"className":49437},[348],[28,49439,43555],{"className":49440,"style":2279},[274,275],[28,49442,1522],{"className":49443},[355],[28,49445,49447],{"className":49446},[355],[28,49448,1522],{"className":49449},[2169,2170],[28,49451,494],{"className":49452},[274],[11,49454,49455,49483,49484,49512,49513,49542,49543,177,49609,177,49757,2923],{},[28,49456,49458,49471],{"className":49457,"translate":193},[197],[28,49459,49461],{"className":49460},[201],[203,49462,49463],{"xmlns":205},[208,49464,49465,49469],{},[211,49466,49467],{},[214,49468,443],{},[253,49470,443],{"encoding":255},[28,49472,49474],{"className":49473,"ariaHidden":261},[260],[28,49475,49477,49480],{"className":49476},[265],[28,49478],{"className":49479,"style":455},[269],[28,49481,443],{"className":49482,"style":459},[274,275]," 是充分统计量，",[28,49485,49487,49500],{"className":49486,"translate":193},[197],[28,49488,49490],{"className":49489},[201],[203,49491,49492],{"xmlns":205},[208,49493,49494,49498],{},[211,49495,49496],{},[214,49497,43555],{},[253,49499,44504],{"encoding":255},[28,49501,49503],{"className":49502,"ariaHidden":261},[260],[28,49504,49506,49509],{"className":49505},[265],[28,49507],{"className":49508,"style":7448},[269],[28,49510,43555],{"className":49511,"style":2279},[274,275]," 是自然参数，",[28,49514,49516,49530],{"className":49515,"translate":193},[197],[28,49517,49519],{"className":49518},[201],[203,49520,49521],{"xmlns":205},[208,49522,49523,49527],{},[211,49524,49525],{},[214,49526,27871],{},[253,49528,49529],{"encoding":255},"\\psi",[28,49531,49533],{"className":49532,"ariaHidden":261},[260],[28,49534,49536,49539],{"className":49535},[265],[28,49537],{"className":49538,"style":2371},[269],[28,49540,27871],{"className":49541,"style":298},[274,275]," 是对数配分函数，负责把总质量归一化成 1。抛硬币就是 ",[28,49544,49546,49570],{"className":49545,"translate":193},[197],[28,49547,49549],{"className":49548},[201],[203,49550,49551],{"xmlns":205},[208,49552,49553,49567],{},[211,49554,49555,49557,49559,49561,49563,49565],{},[214,49556,443],{},[217,49558,43],{"stretchy":241},[214,49560,176],{},[217,49562,1522],{"stretchy":241},[217,49564,83],{},[214,49566,176],{},[253,49568,49569],{"encoding":255},"T(x)=x",[28,49571,49573,49600],{"className":49572,"ariaHidden":261},[260],[28,49574,49576,49579,49582,49585,49588,49591,49594,49597],{"className":49575},[265],[28,49577],{"className":49578,"style":341},[269],[28,49580,443],{"className":49581,"style":459},[274,275],[28,49583,43],{"className":49584},[348],[28,49586,176],{"className":49587},[274,275],[28,49589,1522],{"className":49590},[355],[28,49592],{"className":49593,"style":302},[279],[28,49595,83],{"className":49596},[306],[28,49598],{"className":49599,"style":302},[279],[28,49601,49603,49606],{"className":49602},[265],[28,49604],{"className":49605,"style":645},[269],[28,49607,176],{"className":49608},[274,275],[28,49610,49612,49644],{"className":49611,"translate":193},[197],[28,49613,49615],{"className":49614},[201],[203,49616,49617],{"xmlns":205},[208,49618,49619,49641],{},[211,49620,49621,49623,49625,49627,49629],{},[214,49622,43555],{},[217,49624,83],{},[214,49626,39714],{},[217,49628,21407],{},[21928,49630,49631,49633],{},[214,49632,11],{},[211,49634,49635,49637,49639],{},[224,49636,1244],{},[217,49638,5074],{},[214,49640,11],{},[253,49642,49643],{"encoding":255},"\\theta=\\log\\frac{p}{1-p}",[28,49645,49647,49665],{"className":49646,"ariaHidden":261},[260],[28,49648,49650,49653,49656,49659,49662],{"className":49649},[265],[28,49651],{"className":49652,"style":7448},[269],[28,49654,43555],{"className":49655,"style":2279},[274,275],[28,49657],{"className":49658,"style":302},[279],[28,49660,83],{"className":49661},[306],[28,49663],{"className":49664,"style":302},[279],[28,49666,49668,49672,49677,49680],{"className":49667},[265],[28,49669],{"className":49670,"style":49671},[269],"height:1.2286em;vertical-align:-0.4811em;",[28,49673,39952,49675],{"className":49674},[21329],[28,49676,34347],{"style":39955},[28,49678],{"className":49679,"style":530},[279],[28,49681,49683,49686,49754],{"className":49682},[274],[28,49684],{"className":49685},[348,8285],[28,49687,49689],{"className":49688},[21928],[28,49690,49692,49745],{"className":49691},[560,561],[28,49693,49695,49742],{"className":49694},[565],[28,49696,49699,49719,49727],{"className":49697,"style":49698},[569],"height:0.7475em;",[28,49700,49701,49704],{"style":22008},[28,49702],{"className":49703,"style":1914},[577],[28,49705,49707],{"className":49706},[582,583,584,585],[28,49708,49710,49713,49716],{"className":49709},[274,585],[28,49711,1244],{"className":49712},[274,585],[28,49714,5074],{"className":49715},[284,585],[28,49717,11],{"className":49718},[274,275,585],[28,49720,49721,49724],{"style":22023},[28,49722],{"className":49723,"style":1914},[577],[28,49725],{"className":49726,"style":22031},[22030],[28,49728,49730,49733],{"style":49729},"top:-3.4461em;",[28,49731],{"className":49732,"style":1914},[577],[28,49734,49736],{"className":49735},[582,583,584,585],[28,49737,49739],{"className":49738},[274,585],[28,49740,11],{"className":49741},[274,275,585],[28,49743,593],{"className":49744},[592],[28,49746,49748],{"className":49747},[565],[28,49749,49752],{"className":49750,"style":49751},[569],"height:0.4811em;",[28,49753],{},[28,49755],{"className":49756},[355,8285],[28,49758,49760,49800],{"className":49759,"translate":193},[197],[28,49761,49763],{"className":49762},[201],[203,49764,49765],{"xmlns":205},[208,49766,49767,49797],{},[211,49768,49769,49771,49773,49775,49777,49779,49781,49783,49785,49787,49789,49795],{},[214,49770,27871],{},[217,49772,43],{"stretchy":241},[214,49774,43555],{},[217,49776,1522],{"stretchy":241},[217,49778,83],{},[214,49780,39714],{},[217,49782,21407],{},[217,49784,43],{"stretchy":241},[224,49786,1244],{},[217,49788,124],{},[3054,49790,49791,49793],{},[214,49792,1667],{},[214,49794,43555],{},[217,49796,1522],{"stretchy":241},[253,49798,49799],{"encoding":255},"\\psi(\\theta)=\\log(1+e^\\theta)",[28,49801,49803,49830,49856],{"className":49802,"ariaHidden":261},[260],[28,49804,49806,49809,49812,49815,49818,49821,49824,49827],{"className":49805},[265],[28,49807],{"className":49808,"style":341},[269],[28,49810,27871],{"className":49811,"style":298},[274,275],[28,49813,43],{"className":49814},[348],[28,49816,43555],{"className":49817,"style":2279},[274,275],[28,49819,1522],{"className":49820},[355],[28,49822],{"className":49823,"style":302},[279],[28,49825,83],{"className":49826},[306],[28,49828],{"className":49829,"style":302},[279],[28,49831,49833,49836,49841,49844,49847,49850,49853],{"className":49832},[265],[28,49834],{"className":49835,"style":341},[269],[28,49837,39952,49839],{"className":49838},[21329],[28,49840,34347],{"style":39955},[28,49842,43],{"className":49843},[348],[28,49845,1244],{"className":49846},[274],[28,49848],{"className":49849,"style":280},[279],[28,49851,124],{"className":49852},[284],[28,49854],{"className":49855,"style":280},[279],[28,49857,49859,49862,49891],{"className":49858},[265],[28,49860],{"className":49861,"style":40607},[269],[28,49863,49865,49868],{"className":49864},[274],[28,49866,1667],{"className":49867},[274,275],[28,49869,49871],{"className":49870},[556],[28,49872,49874],{"className":49873},[560],[28,49875,49877],{"className":49876},[565],[28,49878,49880],{"className":49879,"style":40635},[569],[28,49881,49882,49885],{"style":3094},[28,49883],{"className":49884,"style":578},[577],[28,49886,49888],{"className":49887},[582,583,584,585],[28,49889,43555],{"className":49890,"style":2279},[274,275,585],[28,49892,1522],{"className":49893},[355],[11,49895,49896,49924,49925,6904,50066,50245,50246,50274],{},[28,49897,49899,49912],{"className":49898,"translate":193},[197],[28,49900,49902],{"className":49901},[201],[203,49903,49904],{"xmlns":205},[208,49905,49906,49910],{},[211,49907,49908],{},[214,49909,27871],{},[253,49911,49529],{"encoding":255},[28,49913,49915],{"className":49914,"ariaHidden":261},[260],[28,49916,49918,49921],{"className":49917},[265],[28,49919],{"className":49920,"style":2371},[269],[28,49922,27871],{"className":49923,"style":298},[274,275]," 的性质很好：",[28,49926,49928,49970],{"className":49927,"translate":193},[197],[28,49929,49931],{"className":49930},[201],[203,49932,49933],{"xmlns":205},[208,49934,49935,49967],{},[211,49936,49937,49939,49941,49943,49945,49947,49949,49955,49957,49959,49961,49963,49965],{},[214,49938,31947],{"mathvariant":484},[214,49940,27871],{},[217,49942,43],{"stretchy":241},[214,49944,43555],{},[217,49946,1522],{"stretchy":241},[217,49948,83],{},[501,49950,49951,49953],{},[214,49952,42084],{"mathvariant":12057},[214,49954,43555],{},[217,49956,73],{"stretchy":241},[214,49958,443],{},[217,49960,43],{"stretchy":241},[214,49962,20827],{},[217,49964,1522],{"stretchy":241},[217,49966,246],{"stretchy":241},[253,49968,49969],{"encoding":255},"\\nabla\\psi(\\theta)=\\mathbb E_\\theta[T(X)]",[28,49971,49973,50003],{"className":49972,"ariaHidden":261},[260],[28,49974,49976,49979,49982,49985,49988,49991,49994,49997,50000],{"className":49975},[265],[28,49977],{"className":49978,"style":341},[269],[28,49980,31947],{"className":49981},[274],[28,49983,27871],{"className":49984,"style":298},[274,275],[28,49986,43],{"className":49987},[348],[28,49989,43555],{"className":49990,"style":2279},[274,275],[28,49992,1522],{"className":49993},[355],[28,49995],{"className":49996,"style":302},[279],[28,49998,83],{"className":49999},[306],[28,50001],{"className":50002,"style":302},[279],[28,50004,50006,50009,50049,50052,50055,50058,50062],{"className":50005},[265],[28,50007],{"className":50008,"style":341},[269],[28,50010,50012,50015],{"className":50011},[274],[28,50013,42084],{"className":50014},[274,12195],[28,50016,50018],{"className":50017},[556],[28,50019,50021,50041],{"className":50020},[560,561],[28,50022,50024,50038],{"className":50023},[565],[28,50025,50027],{"className":50026,"style":20945},[569],[28,50028,50029,50032],{"style":573},[28,50030],{"className":50031,"style":578},[577],[28,50033,50035],{"className":50034},[582,583,584,585],[28,50036,43555],{"className":50037,"style":2279},[274,275,585],[28,50039,593],{"className":50040},[592],[28,50042,50044],{"className":50043},[565],[28,50045,50047],{"className":50046,"style":600},[569],[28,50048],{},[28,50050,73],{"className":50051},[348],[28,50053,443],{"className":50054,"style":459},[274,275],[28,50056,43],{"className":50057},[348],[28,50059,20827],{"className":50060,"style":50061},[274,275],"margin-right:0.0785em;",[28,50063,50065],{"className":50064},[355],")]",[28,50067,50069,50121],{"className":50068,"translate":193},[197],[28,50070,50072],{"className":50071},[201],[203,50073,50074],{"xmlns":205},[208,50075,50076,50118],{},[211,50077,50078,50084,50086,50088,50090,50092,50094,50106,50108,50110,50112,50114,50116],{},[3054,50079,50080,50082],{},[214,50081,31947],{"mathvariant":484},[224,50083,1253],{},[214,50085,27871],{},[217,50087,43],{"stretchy":241},[214,50089,43555],{},[217,50091,1522],{"stretchy":241},[217,50093,83],{},[501,50095,50096,50104],{},[211,50097,50098,50100,50102],{},[214,50099,15828],{"mathvariant":484},[214,50101,3933],{"mathvariant":484},[214,50103,1693],{"mathvariant":484},[214,50105,43555],{},[217,50107,73],{"stretchy":241},[214,50109,443],{},[217,50111,43],{"stretchy":241},[214,50113,20827],{},[217,50115,1522],{"stretchy":241},[217,50117,246],{"stretchy":241},[253,50119,50120],{"encoding":255},"\\nabla^2\\psi(\\theta)=\\mathrm{Cov}_\\theta[T(X)]",[28,50122,50124,50180],{"className":50123,"ariaHidden":261},[260],[28,50125,50127,50130,50159,50162,50165,50168,50171,50174,50177],{"className":50126},[265],[28,50128],{"className":50129,"style":31837},[269],[28,50131,50133,50136],{"className":50132},[274],[28,50134,31947],{"className":50135},[274],[28,50137,50139],{"className":50138},[556],[28,50140,50142],{"className":50141},[560],[28,50143,50145],{"className":50144},[565],[28,50146,50148],{"className":50147,"style":5477},[569],[28,50149,50150,50153],{"style":3094},[28,50151],{"className":50152,"style":578},[577],[28,50154,50156],{"className":50155},[582,583,584,585],[28,50157,1253],{"className":50158},[274,585],[28,50160,27871],{"className":50161,"style":298},[274,275],[28,50163,43],{"className":50164},[348],[28,50166,43555],{"className":50167,"style":2279},[274,275],[28,50169,1522],{"className":50170},[355],[28,50172],{"className":50173,"style":302},[279],[28,50175,83],{"className":50176},[306],[28,50178],{"className":50179,"style":302},[279],[28,50181,50183,50186,50230,50233,50236,50239,50242],{"className":50182},[265],[28,50184],{"className":50185,"style":341},[269],[28,50187,50189,50196],{"className":50188},[274],[28,50190,50192],{"className":50191},[274],[28,50193,50195],{"className":50194,"style":39955},[274,1712],"Cov",[28,50197,50199],{"className":50198},[556],[28,50200,50202,50222],{"className":50201},[560,561],[28,50203,50205,50219],{"className":50204},[565],[28,50206,50208],{"className":50207,"style":20945},[569],[28,50209,50210,50213],{"style":41550},[28,50211],{"className":50212,"style":578},[577],[28,50214,50216],{"className":50215},[582,583,584,585],[28,50217,43555],{"className":50218,"style":2279},[274,275,585],[28,50220,593],{"className":50221},[592],[28,50223,50225],{"className":50224},[565],[28,50226,50228],{"className":50227,"style":600},[569],[28,50229],{},[28,50231,73],{"className":50232},[348],[28,50234,443],{"className":50235,"style":459},[274,275],[28,50237,43],{"className":50238},[348],[28,50240,20827],{"className":50241,"style":50061},[274,275],[28,50243,50065],{"className":50244},[355],"。第二个等式说明 Fisher 度量就是 ",[28,50247,50249,50262],{"className":50248,"translate":193},[197],[28,50250,50252],{"className":50251},[201],[203,50253,50254],{"xmlns":205},[208,50255,50256,50260],{},[211,50257,50258],{},[214,50259,27871],{},[253,50261,49529],{"encoding":255},[28,50263,50265],{"className":50264,"ariaHidden":261},[260],[28,50266,50268,50271],{"className":50267},[265],[28,50269],{"className":50270,"style":2371},[269],[28,50272,27871],{"className":50273,"style":298},[274,275]," 的 Hessian，上一节里“度量由凸函数给出”的说法在这里变成了具体的东西。",[11,50276,50277,50278,50350,50351,50379,50380,50608,50609,50710,50711,50739,50740,50769],{},"令 ",[28,50279,50281,50308],{"className":50280,"translate":193},[197],[28,50282,50284],{"className":50283},[201],[203,50285,50286],{"xmlns":205},[208,50287,50288,50305],{},[211,50289,50290,50293,50295,50297,50299,50301,50303],{},[214,50291,50292],{},"η",[217,50294,83],{},[214,50296,31947],{"mathvariant":484},[214,50298,27871],{},[217,50300,43],{"stretchy":241},[214,50302,43555],{},[217,50304,1522],{"stretchy":241},[253,50306,50307],{"encoding":255},"\\eta=\\nabla\\psi(\\theta)",[28,50309,50311,50329],{"className":50310,"ariaHidden":261},[260],[28,50312,50314,50317,50320,50323,50326],{"className":50313},[265],[28,50315],{"className":50316,"style":612},[269],[28,50318,50292],{"className":50319,"style":298},[274,275],[28,50321],{"className":50322,"style":302},[279],[28,50324,83],{"className":50325},[306],[28,50327],{"className":50328,"style":302},[279],[28,50330,50332,50335,50338,50341,50344,50347],{"className":50331},[265],[28,50333],{"className":50334,"style":341},[269],[28,50336,31947],{"className":50337},[274],[28,50339,27871],{"className":50340,"style":298},[274,275],[28,50342,43],{"className":50343},[348],[28,50345,43555],{"className":50346,"style":2279},[274,275],[28,50348,1522],{"className":50349},[355]," 为期望参数，它与 ",[28,50352,50354,50367],{"className":50353,"translate":193},[197],[28,50355,50357],{"className":50356},[201],[203,50358,50359],{"xmlns":205},[208,50360,50361,50365],{},[211,50362,50363],{},[214,50364,43555],{},[253,50366,44504],{"encoding":255},[28,50368,50370],{"className":50369,"ariaHidden":261},[260],[28,50371,50373,50376],{"className":50372},[265],[28,50374],{"className":50375,"style":7448},[269],[28,50377,43555],{"className":50378,"style":2279},[274,275]," 之间是 Legendre 变换的关系：",[28,50381,50383,50442],{"className":50382,"translate":193},[197],[28,50384,50386],{"className":50385},[201],[203,50387,50388],{"xmlns":205},[208,50389,50390,50439],{},[211,50391,50392,50399,50401,50403,50405,50407,50417,50419,50425,50427,50429,50431,50433,50435,50437],{},[3054,50393,50394,50396],{},[214,50395,27871],{},[217,50397,50398],{},"∗",[217,50400,43],{"stretchy":241},[214,50402,50292],{},[217,50404,1522],{"stretchy":241},[217,50406,83],{},[501,50408,50409,50415],{},[211,50410,50411,50413],{},[214,50412,30596],{},[217,50414,21407],{},[214,50416,43555],{},[217,50418,43],{"stretchy":241},[3054,50420,50421,50423],{},[214,50422,43555],{},[214,50424,884],{"mathvariant":484},[214,50426,50292],{},[217,50428,5074],{},[214,50430,27871],{},[217,50432,43],{"stretchy":241},[214,50434,43555],{},[217,50436,1522],{"stretchy":241},[217,50438,1522],{"stretchy":241},[253,50440,50441],{"encoding":255},"\\psi^*(\\eta)=\\sup_\\theta(\\theta^\\top\\eta-\\psi(\\theta))",[28,50443,50445,50499,50590],{"className":50444,"ariaHidden":261},[260],[28,50446,50448,50451,50481,50484,50487,50490,50493,50496],{"className":50447},[265],[28,50449],{"className":50450,"style":341},[269],[28,50452,50454,50457],{"className":50453},[274],[28,50455,27871],{"className":50456,"style":298},[274,275],[28,50458,50460],{"className":50459},[556],[28,50461,50463],{"className":50462},[560],[28,50464,50466],{"className":50465},[565],[28,50467,50470],{"className":50468,"style":50469},[569],"height:0.6887em;",[28,50471,50472,50475],{"style":3094},[28,50473],{"className":50474,"style":578},[577],[28,50476,50478],{"className":50477},[582,583,584,585],[28,50479,50398],{"className":50480},[284,585],[28,50482,43],{"className":50483},[348],[28,50485,50292],{"className":50486,"style":298},[274,275],[28,50488,1522],{"className":50489},[355],[28,50491],{"className":50492,"style":302},[279],[28,50494,83],{"className":50495},[306],[28,50497],{"className":50498,"style":302},[279],[28,50500,50502,50505,50546,50549,50578,50581,50584,50587],{"className":50501},[265],[28,50503],{"className":50504,"style":40607},[269],[28,50506,50508,50511],{"className":50507},[21329],[28,50509,30596],{"className":50510},[21329],[28,50512,50514],{"className":50513},[556],[28,50515,50517,50538],{"className":50516},[560,561],[28,50518,50520,50535],{"className":50519},[565],[28,50521,50524],{"className":50522,"style":50523},[569],"height:0.242em;",[28,50525,50526,50529],{"style":22604},[28,50527],{"className":50528,"style":578},[577],[28,50530,50532],{"className":50531},[582,583,584,585],[28,50533,43555],{"className":50534,"style":2279},[274,275,585],[28,50536,593],{"className":50537},[592],[28,50539,50541],{"className":50540},[565],[28,50542,50544],{"className":50543,"style":22626},[569],[28,50545],{},[28,50547,43],{"className":50548},[348],[28,50550,50552,50555],{"className":50551},[274],[28,50553,43555],{"className":50554,"style":2279},[274,275],[28,50556,50558],{"className":50557},[556],[28,50559,50561],{"className":50560},[560],[28,50562,50564],{"className":50563},[565],[28,50565,50567],{"className":50566,"style":40635},[569],[28,50568,50569,50572],{"style":3094},[28,50570],{"className":50571,"style":578},[577],[28,50573,50575],{"className":50574},[582,583,584,585],[28,50576,884],{"className":50577},[274,585],[28,50579,50292],{"className":50580,"style":298},[274,275],[28,50582],{"className":50583,"style":280},[279],[28,50585,5074],{"className":50586},[284],[28,50588],{"className":50589,"style":280},[279],[28,50591,50593,50596,50599,50602,50605],{"className":50592},[265],[28,50594],{"className":50595,"style":341},[269],[28,50597,27871],{"className":50598,"style":298},[274,275],[28,50600,43],{"className":50601},[348],[28,50603,43555],{"className":50604,"style":2279},[274,275],[28,50606,21363],{"className":50607},[355],"，且 ",[28,50610,50612,50642],{"className":50611,"translate":193},[197],[28,50613,50615],{"className":50614},[201],[203,50616,50617],{"xmlns":205},[208,50618,50619,50639],{},[211,50620,50621,50623,50629,50631,50633,50635,50637],{},[214,50622,31947],{"mathvariant":484},[3054,50624,50625,50627],{},[214,50626,27871],{},[217,50628,50398],{},[217,50630,43],{"stretchy":241},[214,50632,50292],{},[217,50634,1522],{"stretchy":241},[217,50636,83],{},[214,50638,43555],{},[253,50640,50641],{"encoding":255},"\\nabla\\psi^*(\\eta)=\\theta",[28,50643,50645,50701],{"className":50644,"ariaHidden":261},[260],[28,50646,50648,50651,50654,50683,50686,50689,50692,50695,50698],{"className":50647},[265],[28,50649],{"className":50650,"style":341},[269],[28,50652,31947],{"className":50653},[274],[28,50655,50657,50660],{"className":50656},[274],[28,50658,27871],{"className":50659,"style":298},[274,275],[28,50661,50663],{"className":50662},[556],[28,50664,50666],{"className":50665},[560],[28,50667,50669],{"className":50668},[565],[28,50670,50672],{"className":50671,"style":50469},[569],[28,50673,50674,50677],{"style":3094},[28,50675],{"className":50676,"style":578},[577],[28,50678,50680],{"className":50679},[582,583,584,585],[28,50681,50398],{"className":50682},[284,585],[28,50684,43],{"className":50685},[348],[28,50687,50292],{"className":50688,"style":298},[274,275],[28,50690,1522],{"className":50691},[355],[28,50693],{"className":50694,"style":302},[279],[28,50696,83],{"className":50697},[306],[28,50699],{"className":50700,"style":302},[279],[28,50702,50704,50707],{"className":50703},[265],[28,50705],{"className":50706,"style":7448},[269],[28,50708,43555],{"className":50709,"style":2279},[274,275],"。所以同一个分布有两套坐标，自然参数 ",[28,50712,50714,50727],{"className":50713,"translate":193},[197],[28,50715,50717],{"className":50716},[201],[203,50718,50719],{"xmlns":205},[208,50720,50721,50725],{},[211,50722,50723],{},[214,50724,43555],{},[253,50726,44504],{"encoding":255},[28,50728,50730],{"className":50729,"ariaHidden":261},[260],[28,50731,50733,50736],{"className":50732},[265],[28,50734],{"className":50735,"style":7448},[269],[28,50737,43555],{"className":50738,"style":2279},[274,275]," 和期望参数 ",[28,50741,50743,50757],{"className":50742,"translate":193},[197],[28,50744,50746],{"className":50745},[201],[203,50747,50748],{"xmlns":205},[208,50749,50750,50754],{},[211,50751,50752],{},[214,50753,50292],{},[253,50755,50756],{"encoding":255},"\\eta",[28,50758,50760],{"className":50759,"ariaHidden":261},[260],[28,50761,50763,50766],{"className":50762},[265],[28,50764],{"className":50765,"style":612},[269],[28,50767,50292],{"className":50768,"style":298},[274,275],"，互为对偶。",[11,50771,50772],{},"现在算两个成员之间的 KL：",[28,50774,50776],{"className":50775,"translate":193},[192],[28,50777,50779,51091],{"className":50778,"translate":193},[197],[28,50780,50782],{"className":50781},[201],[203,50783,50784],{"xmlns":205,"display":206},[208,50785,50786,51088],{},[5015,50787,50788,50870,50946,51016],{"rowspacing":5017,"columnalign":5018,"columnspacing":5019},[5021,50789,50790,50824],{},[5024,50791,50792],{},[5027,50793,50794],{"scriptlevel":89,"displaystyle":261},[211,50795,50796,50802,50804,50810,50812,50822],{},[211,50797,50798,50800],{},[214,50799,24390],{"mathvariant":484},[214,50801,24393],{"mathvariant":484},[217,50803,43],{"stretchy":241},[501,50805,50806,50808],{},[214,50807,11],{},[214,50809,43555],{},[214,50811,24404],{"mathvariant":484},[501,50813,50814,50816],{},[214,50815,11],{},[3054,50817,50818,50820],{},[214,50819,43555],{},[217,50821,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[217,50823,1522],{"stretchy":241},[5024,50825,50826],{},[5027,50827,50828],{"scriptlevel":89,"displaystyle":261},[211,50829,50830,50832,50834,50840,50842,50844,50846,50852,50854,50856,50858,50868],{},[211,50831],{},[217,50833,83],{},[501,50835,50836,50838],{},[214,50837,42084],{"mathvariant":12057},[214,50839,43555],{},[217,50841,73],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,50843,39714],{},[217,50845,21407],{},[501,50847,50848,50850],{},[214,50849,11],{},[214,50851,43555],{},[217,50853,5074],{},[214,50855,39714],{},[217,50857,21407],{},[501,50859,50860,50862],{},[214,50861,11],{},[3054,50863,50864,50866],{},[214,50865,43555],{},[217,50867,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[217,50869,246],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[5021,50871,50872,50878],{},[5024,50873,50874],{},[5027,50875,50876],{"scriptlevel":89,"displaystyle":261},[211,50877],{},[5024,50879,50880],{},[5027,50881,50882],{"scriptlevel":89,"displaystyle":261},[211,50883,50884,50886,50888,50894,50896,50898,50900,50902,50908,50914,50916,50918,50920,50922,50924,50926,50928,50930,50932,50934,50936,50942,50944],{},[211,50885],{},[217,50887,83],{},[501,50889,50890,50892],{},[214,50891,42084],{"mathvariant":12057},[214,50893,43555],{},[217,50895,73],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,50897,43],{"stretchy":241},[214,50899,43555],{},[217,50901,5074],{},[3054,50903,50904,50906],{},[214,50905,43555],{},[217,50907,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[3054,50909,50910,50912],{},[217,50911,1522],{"stretchy":241},[214,50913,884],{"mathvariant":484},[214,50915,443],{},[217,50917,246],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,50919,5074],{},[217,50921,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,50923,27871],{},[217,50925,43],{"stretchy":241},[214,50927,43555],{},[217,50929,1522],{"stretchy":241},[217,50931,5074],{},[214,50933,27871],{},[217,50935,43],{"stretchy":241},[3054,50937,50938,50940],{},[214,50939,43555],{},[217,50941,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[217,50943,1522],{"stretchy":241},[217,50945,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[5021,50947,50948,50954],{},[5024,50949,50950],{},[5027,50951,50952],{"scriptlevel":89,"displaystyle":261},[211,50953],{},[5024,50955,50956],{},[5027,50957,50958],{"scriptlevel":89,"displaystyle":261},[211,50959,50960,50962,50964,50966,50968,50970,50976,50982,50984,50986,50988,50990,50992,50994,50996,50998,51000,51002,51004,51006,51008,51014],{},[211,50961],{},[217,50963,83],{},[217,50965,43],{"stretchy":241},[214,50967,43555],{},[217,50969,5074],{},[3054,50971,50972,50974],{},[214,50973,43555],{},[217,50975,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[3054,50977,50978,50980],{},[217,50979,1522],{"stretchy":241},[214,50981,884],{"mathvariant":484},[214,50983,31947],{"mathvariant":484},[214,50985,27871],{},[217,50987,43],{"stretchy":241},[214,50989,43555],{},[217,50991,1522],{"stretchy":241},[217,50993,5074],{},[214,50995,27871],{},[217,50997,43],{"stretchy":241},[214,50999,43555],{},[217,51001,1522],{"stretchy":241},[217,51003,124],{},[214,51005,27871],{},[217,51007,43],{"stretchy":241},[3054,51009,51010,51012],{},[214,51011,43555],{},[217,51013,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[217,51015,1522],{"stretchy":241},[5021,51017,51018,51024],{},[5024,51019,51020],{},[5027,51021,51022],{"scriptlevel":89,"displaystyle":261},[211,51023],{},[5024,51025,51026],{},[5027,51027,51028],{"scriptlevel":89,"displaystyle":261},[211,51029,51030,51032,51034,51036,51038,51044,51046,51048,51050,51052,51054,51056,51058,51060,51062,51064,51066,51072,51074,51080,51082,51084,51086],{},[211,51031],{},[217,51033,83],{},[214,51035,27871],{},[217,51037,43],{"stretchy":241},[3054,51039,51040,51042],{},[214,51041,43555],{},[217,51043,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[217,51045,1522],{"stretchy":241},[217,51047,5074],{},[214,51049,27871],{},[217,51051,43],{"stretchy":241},[214,51053,43555],{},[217,51055,1522],{"stretchy":241},[217,51057,5074],{},[214,51059,31947],{"mathvariant":484},[214,51061,27871],{},[217,51063,43],{"stretchy":241},[214,51065,43555],{},[3054,51067,51068,51070],{},[217,51069,1522],{"stretchy":241},[214,51071,884],{"mathvariant":484},[217,51073,43],{"stretchy":241},[3054,51075,51076,51078],{},[214,51077,43555],{},[217,51079,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[217,51081,5074],{},[214,51083,43555],{},[217,51085,1522],{"stretchy":241},[214,51087,494],{"mathvariant":484},[253,51089,51090],{"encoding":255},"\\begin{aligned}\n\\mathrm{KL}(p_\\theta\\|p_{\\theta'})\n&=\\mathbb E_\\theta\\bigl[\\log p_\\theta-\\log p_{\\theta'}\\bigr]\\\\\n&=\\mathbb E_\\theta\\bigl[(\\theta-\\theta')^\\top T\\bigr]-\\bigl(\\psi(\\theta)-\\psi(\\theta')\\bigr)\\\\\n&=(\\theta-\\theta')^\\top\\nabla\\psi(\\theta)-\\psi(\\theta)+\\psi(\\theta')\\\\\n&=\\psi(\\theta')-\\psi(\\theta)-\\nabla\\psi(\\theta)^\\top(\\theta'-\\theta).\n\\end{aligned}",[28,51092,51094],{"className":51093,"ariaHidden":261},[260],[28,51095,51097,51101],{"className":51096},[265],[28,51098],{"className":51099,"style":51100},[269],"height:5.8873em;vertical-align:-2.6937em;",[28,51102,51104],{"className":51103},[274],[28,51105,51107,51297],{"className":51106},[5015],[28,51108,51110],{"className":51109},[5137],[28,51111,51113,51288],{"className":51112},[560,561],[28,51114,51116,51285],{"className":51115},[565],[28,51117,51120,51258,51267,51276],{"className":51118,"style":51119},[569],"height:3.1937em;",[28,51121,51123,51126],{"style":51122},"top:-5.3437em;",[28,51124],{"className":51125,"style":1914},[577],[28,51127,51129,51135,51138,51178,51181,51255],{"className":51128},[274],[28,51130,51132],{"className":51131},[274],[28,51133,24438],{"className":51134},[274,1712],[28,51136,43],{"className":51137},[348],[28,51139,51141,51144],{"className":51140},[274],[28,51142,11],{"className":51143},[274,275],[28,51145,51147],{"className":51146},[556],[28,51148,51150,51170],{"className":51149},[560,561],[28,51151,51153,51167],{"className":51152},[565],[28,51154,51156],{"className":51155,"style":20945},[569],[28,51157,51158,51161],{"style":573},[28,51159],{"className":51160,"style":578},[577],[28,51162,51164],{"className":51163},[582,583,584,585],[28,51165,43555],{"className":51166,"style":2279},[274,275,585],[28,51168,593],{"className":51169},[592],[28,51171,51173],{"className":51172},[565],[28,51174,51176],{"className":51175,"style":600},[569],[28,51177],{},[28,51179,24404],{"className":51180},[274],[28,51182,51184,51187],{"className":51183},[274],[28,51185,11],{"className":51186},[274,275],[28,51188,51190],{"className":51189},[556],[28,51191,51193,51247],{"className":51192},[560,561],[28,51194,51196,51244],{"className":51195},[565],[28,51197,51199],{"className":51198,"style":20945},[569],[28,51200,51201,51204],{"style":573},[28,51202],{"className":51203,"style":578},[577],[28,51205,51207],{"className":51206},[582,583,584,585],[28,51208,51210],{"className":51209},[274,585],[28,51211,51213,51216],{"className":51212},[274,585],[28,51214,43555],{"className":51215,"style":2279},[274,275,585],[28,51217,51219],{"className":51218},[556],[28,51220,51222],{"className":51221},[560],[28,51223,51225],{"className":51224},[565],[28,51226,51229],{"className":51227,"style":51228},[569],"height:0.6828em;",[28,51230,51232,51235],{"style":51231},"top:-2.786em;margin-right:0.0714em;",[28,51233],{"className":51234,"style":31571},[577],[28,51236,51238],{"className":51237},[582,31575,2170,585],[28,51239,51241],{"className":51240},[274,585],[28,51242,6739],{"className":51243},[274,585],[28,51245,593],{"className":51246},[592],[28,51248,51250],{"className":51249},[565],[28,51251,51253],{"className":51252,"style":600},[569],[28,51254],{},[28,51256,1522],{"className":51257},[355],[28,51259,51261,51264],{"style":51260},"top:-3.7846em;",[28,51262],{"className":51263,"style":1914},[577],[28,51265],{"className":51266},[274],[28,51268,51270,51273],{"style":51269},"top:-2.2254em;",[28,51271],{"className":51272,"style":1914},[577],[28,51274],{"className":51275},[274],[28,51277,51279,51282],{"style":51278},"top:-0.6663em;",[28,51280],{"className":51281,"style":1914},[577],[28,51283],{"className":51284},[274],[28,51286,593],{"className":51287},[592],[28,51289,51291],{"className":51290},[565],[28,51292,51295],{"className":51293,"style":51294},[569],"height:2.6937em;",[28,51296],{},[28,51298,51300],{"className":51299},[5205],[28,51301,51303,52122],{"className":51302},[560,561],[28,51304,51306,52119],{"className":51305},[565],[28,51307,51309,51518,51752,51934],{"className":51308,"style":51119},[569],[28,51310,51311,51314],{"style":51122},[28,51312],{"className":51313,"style":1914},[577],[28,51315,51317,51320,51323,51326,51329,51369,51375,51380,51383,51423,51426,51429,51432,51437,51440,51512],{"className":51316},[274],[28,51318],{"className":51319},[274],[28,51321],{"className":51322,"style":302},[279],[28,51324,83],{"className":51325},[306],[28,51327],{"className":51328,"style":302},[279],[28,51330,51332,51335],{"className":51331},[274],[28,51333,42084],{"className":51334},[274,12195],[28,51336,51338],{"className":51337},[556],[28,51339,51341,51361],{"className":51340},[560,561],[28,51342,51344,51358],{"className":51343},[565],[28,51345,51347],{"className":51346,"style":20945},[569],[28,51348,51349,51352],{"style":573},[28,51350],{"className":51351,"style":578},[577],[28,51353,51355],{"className":51354},[582,583,584,585],[28,51356,43555],{"className":51357,"style":2279},[274,275,585],[28,51359,593],{"className":51360},[592],[28,51362,51364],{"className":51363},[565],[28,51365,51367],{"className":51366,"style":600},[569],[28,51368],{},[28,51370,51372],{"className":51371},[348],[28,51373,73],{"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",[28,52134,52136,52149],{"className":52135,"translate":193},[197],[28,52137,52139],{"className":52138},[201],[203,52140,52141],{"xmlns":205},[208,52142,52143,52147],{},[211,52144,52145],{},[214,52146,27871],{},[253,52148,49529],{"encoding":255},[28,52150,52152],{"className":52151,"ariaHidden":261},[260],[28,52153,52155,52158],{"className":52154},[265],[28,52156],{"className":52157,"style":2371},[269],[28,52159,27871],{"className":52160,"style":298},[274,275],[28,52162,52164,52177],{"className":52163,"translate":193},[197],[28,52165,52167],{"className":52166},[201],[203,52168,52169],{"xmlns":205},[208,52170,52171,52175],{},[211,52172,52173],{},[214,52174,43555],{},[253,52176,44504],{"encoding":255},[28,52178,52180],{"className":52179,"ariaHidden":261},[260],[28,52181,52183,52186],{"className":52182},[265],[28,52184],{"className":52185,"style":7448},[269],[28,52187,43555],{"className":52188,"style":2279},[274,275]," 处的切平面在 ",[28,52191,52193,52211],{"className":52192,"translate":193},[197],[28,52194,52196],{"className":52195},[201],[203,52197,52198],{"xmlns":205},[208,52199,52200,52208],{},[211,52201,52202],{},[3054,52203,52204,52206],{},[214,52205,43555],{},[217,52207,6739],{"mathvariant":484,"lspace":5019,"rspace":5019},[253,52209,52210],{"encoding":255},"\\theta'",[28,52212,52214],{"className":52213,"ariaHidden":261},[260],[28,52215,52217,52220],{"className":52216},[265],[28,52218],{"className":52219,"style":6798},[269],[28,52221,52223,52226],{"className":52222},[274],[28,52224,43555],{"className":52225,"style":2279},[274,275],[28,52227,52229],{"className":52228},[556],[28,52230,52232],{"className":52231},[560],[28,52233,52235],{"className":52234},[565],[28,52236,52238],{"className":52237,"style":6798},[569],[28,52239,52240,52243],{"style":3094},[28,52241],{"className":52242,"style":578},[577],[28,52244,52246],{"className":52245},[582,583,584,585],[28,52247,52249],{"className":52248},[274,585],[28,52250,6739],{"className":52251},[274,585]," 处高出的那部分。它有名字：Bregman 散度。",[11,52254,52255,52256,6904,52307,52410],{},"拿方差固定为 1 的高斯族验一下。此时 ",[28,52257,52259,52277],{"className":52258,"translate":193},[197],[28,52260,52262],{"className":52261},[201],[203,52263,52264],{"xmlns":205},[208,52265,52266,52274],{},[211,52267,52268,52270,52272],{},[214,52269,43555],{},[217,52271,83],{},[214,52273,20756],{},[253,52275,52276],{"encoding":255},"\\theta=\\mu",[28,52278,52280,52298],{"className":52279,"ariaHidden":261},[260],[28,52281,52283,52286,52289,52292,52295],{"className":52282},[265],[28,52284],{"className":52285,"style":7448},[269],[28,52287,43555],{"className":52288,"style":2279},[274,275],[28,52290],{"className":52291,"style":302},[279],[28,52293,83],{"className":52294},[306],[28,52296],{"className":52297,"style":302},[279],[28,52299,52301,52304],{"className":52300},[265],[28,52302],{"className":52303,"style":612},[269],[28,52305,20756],{"className":52306},[274,275],[28,52308,52310,52342],{"className":52309,"translate":193},[197],[28,52311,52313],{"className":52312},[201],[203,52314,52315],{"xmlns":205},[208,52316,52317,52339],{},[211,52318,52319,52321,52323,52325,52327,52329,52335,52337],{},[214,52320,27871],{},[217,52322,43],{"stretchy":241},[214,52324,43555],{},[217,52326,1522],{"stretchy":241},[217,52328,83],{},[3054,52330,52331,52333],{},[214,52332,43555],{},[224,52334,1253],{},[214,52336,12219],{"mathvariant":484},[224,52338,1253],{},[253,52340,52341],{"encoding":255},"\\psi(\\theta)=\\theta^2\u002F2",[28,52343,52345,52372],{"className":52344,"ariaHidden":261},[260],[28,52346,52348,52351,52354,52357,52360,52363,52366,52369],{"className":52347},[265],[28,52349],{"className":52350,"style":341},[269],[28,52352,27871],{"className":52353,"style":298},[274,275],[28,52355,43],{"className":52356},[348],[28,52358,43555],{"className":52359,"style":2279},[274,275],[28,52361,1522],{"className":52362},[355],[28,52364],{"className":52365,"style":302},[279],[28,52367,83],{"className":52368},[306],[28,52370],{"className":52371,"style":302},[279],[28,52373,52375,52378,52407],{"className":52374},[265],[28,52376],{"className":52377,"style":31837},[269],[28,52379,52381,52384],{"className":52380},[274],[28,52382,43555],{"className":52383,"style":2279},[274,275],[28,52385,52387],{"className":52386},[556],[28,52388,52390],{"className":52389},[560],[28,52391,52393],{"className":52392},[565],[28,52394,52396],{"className":52395,"style":5477},[569],[28,52397,52398,52401],{"style":3094},[28,52399],{"className":52400,"style":578},[577],[28,52402,52404],{"className":52403},[582,583,584,585],[28,52405,1253],{"className":52406},[274,585],[28,52408,35142],{"className":52409},[274],"，手算",[28,52412,52414],{"className":52413,"translate":193},[192],[28,52415,52417,52505],{"className":52416,"translate":193},[197],[28,52418,52420],{"className":52419},[201],[203,52421,52422],{"xmlns":205,"display":206},[208,52423,52424,52502],{},[211,52425,52426,52432,52434,52436,52438,52444,52446,52448,52450,52452,52454,52456,52462,52464,52466,52468,52470,52472,52500],{},[211,52427,52428,52430],{},[214,52429,24390],{"mathvariant":484},[214,52431,24393],{"mathvariant":484},[217,52433,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,52435,24766],{"mathvariant":20826},[217,52437,43],{"stretchy":241},[501,52439,52440,52442],{},[214,52441,43555],{},[224,52443,1244],{},[217,52445,1683],{"separator":261},[224,52447,1244],{},[217,52449,1522],{"stretchy":241},[217,52451,24404],{"fence":241,"stretchy":261,"minsize":1810,"maxsize":1810},[214,52453,24766],{"mathvariant":20826},[217,52455,43],{"stretchy":241},[501,52457,52458,52460],{},[214,52459,43555],{},[224,52461,1253],{},[217,52463,1683],{"separator":261},[224,52465,1244],{},[217,52467,1522],{"stretchy":241},[217,52469,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,52471,83],{},[21928,52473,52474,52498],{},[211,52475,52476,52478,52484,52486,52492],{},[217,52477,43],{"stretchy":241},[501,52479,52480,52482],{},[214,52481,43555],{},[224,52483,1244],{},[217,52485,5074],{},[501,52487,52488,52490],{},[214,52489,43555],{},[224,52491,1253],{},[3054,52493,52494,52496],{},[217,52495,1522],{"stretchy":241},[224,52497,1253],{},[224,52499,1253],{},[217,52501,1683],{"separator":261},[253,52503,52504],{"encoding":255},"\\mathrm{KL}\\bigl(\\mathcal N(\\theta_1,1)\\big\\|\\mathcal N(\\theta_2,1)\\bigr)=\\frac{(\\theta_1-\\theta_2)^2}{2},",[28,52506,52508,52698],{"className":52507,"ariaHidden":261},[260],[28,52509,52511,52514,52520,52526,52529,52532,52572,52575,52578,52581,52584,52625,52628,52631,52671,52674,52677,52680,52683,52689,52692,52695],{"className":52510},[265],[28,52512],{"className":52513,"style":2469},[269],[28,52515,52517],{"className":52516},[274],[28,52518,24438],{"className":52519},[274,1712],[28,52521,52523],{"className":52522},[348],[28,52524,43],{"className":52525},[2169,2170],[28,52527,24766],{"className":52528,"style":24834},[274,20866],[28,52530,43],{"className":52531},[348],[28,52533,52535,52538],{"className":52534},[274],[28,52536,43555],{"className":52537,"style":2279},[274,275],[28,52539,52541],{"className":52540},[556],[28,52542,52544,52564],{"className":52543},[560,561],[28,52545,52547,52561],{"className":52546},[565],[28,52548,52550],{"className":52549,"style":1285},[569],[28,52551,52552,52555],{"style":46480},[28,52553],{"className":52554,"style":578},[577],[28,52556,52558],{"className":52557},[582,583,584,585],[28,52559,1244],{"className":52560},[274,585],[28,52562,593],{"className":52563},[592],[28,52565,52567],{"className":52566},[565],[28,52568,52570],{"className":52569,"style":600},[569],[28,52571],{},[28,52573,1683],{"className":52574},[1723],[28,52576],{"className":52577,"style":530},[279],[28,52579,1244],{"className":52580},[274],[28,52582,1522],{"className":52583},[355],[28,52585,52587],{"className":52586},[274],[28,52588,52590],{"className":52589},[2169,33118],[28,52591,52593,52617],{"className":52592},[560,561],[28,52594,52596,52614],{"className":52595},[565],[28,52597,52599],{"className":52598,"style":2091},[569],[28,52600,52601,52604],{"style":33130},[28,52602],{"className":52603,"style":33134},[577],[28,52605,52607],{"style":52606},"width:0.556em;height:1.2em;",[1961,52608,52611],{"xmlns":1963,"width":52609,"height":1810,"viewBox":52610},"0.556em","0 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h43z",[28,52615,593],{"className":52616},[592],[28,52618,52620],{"className":52619},[565],[28,52621,52623],{"className":52622,"style":33154},[569],[28,52624],{},[28,52626,24766],{"className":52627,"style":24834},[274,20866],[28,52629,43],{"className":52630},[348],[28,52632,52634,52637],{"className":52633},[274],[28,52635,43555],{"className":52636,"style":2279},[274,275],[28,52638,52640],{"className":52639},[556],[28,52641,52643,52663],{"className":52642},[560,561],[28,52644,52646,52660],{"className":52645},[565],[28,52647,52649],{"className":52648,"style":1285},[569],[28,52650,52651,52654],{"style":46480},[28,52652],{"className":52653,"style":578},[577],[28,52655,52657],{"className":52656},[582,583,584,585],[28,52658,1253],{"className":52659},[274,585],[28,52661,593],{"className":52662},[592],[28,52664,52666],{"className":52665},[565],[28,52667,52669],{"className":52668,"style":600},[569],[28,52670],{},[28,52672,1683],{"className":52673},[1723],[28,52675],{"className":52676,"style":530},[279],[28,52678,1244],{"className":52679},[274],[28,52681,1522],{"className":52682},[355],[28,52684,52686],{"className":52685},[355],[28,52687,1522],{"className":52688},[2169,2170],[28,52690],{"className":52691,"style":302},[279],[28,52693,83],{"className":52694},[306],[28,52696],{"className":52697,"style":302},[279],[28,52699,52701,52704,52884],{"className":52700},[265],[28,52702],{"className":52703,"style":24898},[269],[28,52705,52707,52710,52881],{"className":52706},[274],[28,52708],{"className":52709},[348,8285],[28,52711,52713],{"className":52712},[21928],[28,52714,52716,52873],{"className":52715},[560,561],[28,52717,52719,52870],{"className":52718},[565],[28,52720,52722,52733,52741],{"className":52721,"style":24917},[569],[28,52723,52724,52727],{"style":24920},[28,52725],{"className":52726,"style":1914},[577],[28,52728,52730],{"className":52729},[274],[28,52731,1253],{"className":52732},[274],[28,52734,52735,52738],{"style":22023},[28,52736],{"className":52737,"style":1914},[577],[28,52739],{"className":52740,"style":22031},[22030],[28,52742,52743,52746],{"style":24940},[28,52744],{"className":52745,"style":1914},[577],[28,52747,52749,52752,52792,52795,52798,52801,52841],{"className":52748},[274],[28,52750,43],{"className":52751},[348],[28,52753,52755,52758],{"className":52754},[274],[28,52756,43555],{"className":52757,"style":2279},[274,275],[28,52759,52761],{"className":52760},[556],[28,52762,52764,52784],{"className":52763},[560,561],[28,52765,52767,52781],{"className":52766},[565],[28,52768,52770],{"className":52769,"style":1285},[569],[28,52771,52772,52775],{"style":46480},[28,52773],{"className":52774,"style":578},[577],[28,52776,52778],{"className":52777},[582,583,584,585],[28,52779,1244],{"className":52780},[274,585],[28,52782,593],{"className":52783},[592],[28,52785,52787],{"className":52786},[565],[28,52788,52790],{"className":52789,"style":600},[569],[28,52791],{},[28,52793],{"className":52794,"style":280},[279],[28,52796,5074],{"className":52797},[284],[28,52799],{"className":52800,"style":280},[279],[28,52802,52804,52807],{"className":52803},[274],[28,52805,43555],{"className":52806,"style":2279},[274,275],[28,52808,52810],{"className":52809},[556],[28,52811,52813,52833],{"className":52812},[560,561],[28,52814,52816,52830],{"className":52815},[565],[28,52817,52819],{"className":52818,"style":1285},[569],[28,52820,52821,52824],{"style":46480},[28,52822],{"className":52823,"style":578},[577],[28,52825,52827],{"className":52826},[582,583,584,585],[28,52828,1253],{"className":52829},[274,585],[28,52831,593],{"className":52832},[592],[28,52834,52836],{"className":52835},[565],[28,52837,52839],{"className":52838,"style":600},[569],[28,52840],{},[28,52842,52844,52847],{"className":52843},[355],[28,52845,1522],{"className":52846},[355],[28,52848,52850],{"className":52849},[556],[28,52851,52853],{"className":52852},[560],[28,52854,52856],{"className":52855},[565],[28,52857,52859],{"className":52858,"style":5477},[569],[28,52860,52861,52864],{"style":3094},[28,52862],{"className":52863,"style":578},[577],[28,52865,52867],{"className":52866},[582,583,584,585],[28,52868,1253],{"className":52869},[274,585],[28,52871,593],{"className":52872},[592],[28,52874,52876],{"className":52875},[565],[28,52877,52879],{"className":52878,"style":24985},[569],[28,52880],{},[28,52882],{"className":52883},[355,8285],[28,52885,1683],{"className":52886},[1723],[11,52888,52889,52890,52918,52919,53655],{},"而 ",[28,52891,52893,52906],{"className":52892,"translate":193},[197],[28,52894,52896],{"className":52895},[201],[203,52897,52898],{"xmlns":205},[208,52899,52900,52904],{},[211,52901,52902],{},[214,52903,27871],{},[253,52905,49529],{"encoding":255},[28,52907,52909],{"className":52908,"ariaHidden":261},[260],[28,52910,52912,52915],{"className":52911},[265],[28,52913],{"className":52914,"style":2371},[269],[28,52916,27871],{"className":52917,"style":298},[274,275]," 的 Bregman 散度是 ",[28,52920,52922,53026],{"className":52921,"translate":193},[197],[28,52923,52925],{"className":52924},[201],[203,52926,52927],{"xmlns":205},[208,52928,52929,53023],{},[211,52930,52931,52939,52947,52949,52957,52965,52967,52973,52975,52981,52983,52989,52991,52993,53001,53003,53009,53011,53017],{},[5027,52932,52933],{"scriptlevel":89,"displaystyle":241},[21928,52934,52935,52937],{},[224,52936,1244],{},[224,52938,1253],{},[32198,52940,52941,52943,52945],{},[214,52942,43555],{},[224,52944,1253],{},[224,52946,1253],{},[217,52948,5074],{},[5027,52950,52951],{"scriptlevel":89,"displaystyle":241},[21928,52952,52953,52955],{},[224,52954,1244],{},[224,52956,1253],{},[32198,52958,52959,52961,52963],{},[214,52960,43555],{},[224,52962,1244],{},[224,52964,1253],{},[217,52966,5074],{},[501,52968,52969,52971],{},[214,52970,43555],{},[224,52972,1244],{},[217,52974,43],{"stretchy":241},[501,52976,52977,52979],{},[214,52978,43555],{},[224,52980,1253],{},[217,52982,5074],{},[501,52984,52985,52987],{},[214,52986,43555],{},[224,52988,1244],{},[217,52990,1522],{"stretchy":241},[217,52992,83],{},[5027,52994,52995],{"scriptlevel":89,"displaystyle":241},[21928,52996,52997,52999],{},[224,52998,1244],{},[224,53000,1253],{},[217,53002,43],{"stretchy":241},[501,53004,53005,53007],{},[214,53006,43555],{},[224,53008,1253],{},[217,53010,5074],{},[501,53012,53013,53015],{},[214,53014,43555],{},[224,53016,1244],{},[3054,53018,53019,53021],{},[217,53020,1522],{"stretchy":241},[224,53022,1253],{},[253,53024,53025],{"encoding":255},"\\tfrac12\\theta_2^2-\\tfrac12\\theta_1^2-\\theta_1(\\theta_2-\\theta_1)=\\tfrac12(\\theta_2-\\theta_1)^2",[28,53027,53029,53164,53298,53396,53454,53580],{"className":53028,"ariaHidden":261},[260],[28,53030,53032,53035,53103,53155,53158,53161],{"className":53031},[265],[28,53033],{"className":53034,"style":21986},[269],[28,53036,53038,53041,53100],{"className":53037},[274],[28,53039],{"className":53040},[348,8285],[28,53042,53044],{"className":53043},[21928],[28,53045,53047,53092],{"className":53046},[560,561],[28,53048,53050,53089],{"className":53049},[565],[28,53051,53053,53067,53075],{"className":53052,"style":22005},[569],[28,53054,53055,53058],{"style":22008},[28,53056],{"className":53057,"style":1914},[577],[28,53059,53061],{"className":53060},[582,583,584,585],[28,53062,53064],{"className":53063},[274,585],[28,53065,1253],{"className":53066},[274,585],[28,53068,53069,53072],{"style":22023},[28,53070],{"className":53071,"style":1914},[577],[28,53073],{"className":53074,"style":22031},[22030],[28,53076,53077,53080],{"style":22034},[28,53078],{"className":53079,"style":1914},[577],[28,53081,53083],{"className":53082},[582,583,584,585],[28,53084,53086],{"className":53085},[274,585],[28,53087,1244],{"className":53088},[274,585],[28,53090,593],{"className":53091},[592],[28,53093,53095],{"className":53094},[565],[28,53096,53098],{"className":53097,"style":22056},[569],[28,53099],{},[28,53101],{"className":53102},[355,8285],[28,53104,53106,53109],{"className":53105},[274],[28,53107,43555],{"className":53108,"style":2279},[274,275],[28,53110,53112],{"className":53111},[556],[28,53113,53115,53147],{"className":53114},[560,561],[28,53116,53118,53144],{"className":53117},[565],[28,53119,53121,53133],{"className":53120,"style":5477},[569],[28,53122,53124,53127],{"style":53123},"top:-2.4519em;margin-left:-0.0278em;margin-right:0.05em;",[28,53125],{"className":53126,"style":578},[577],[28,53128,53130],{"className":53129},[582,583,584,585],[28,53131,1253],{"className":53132},[274,585],[28,53134,53135,53138],{"style":3094},[28,53136],{"className":53137,"style":578},[577],[28,53139,53141],{"className":53140},[582,583,584,585],[28,53142,1253],{"className":53143},[274,585],[28,53145,593],{"className":53146},[592],[28,53148,53150],{"className":53149},[565],[28,53151,53153],{"className":53152,"style":37576},[569],[28,53154],{},[28,53156],{"className":53157,"style":280},[279],[28,53159,5074],{"className":53160},[284],[28,53162],{"className":53163,"style":280},[279],[28,53165,53167,53170,53238,53289,53292,53295],{"className":53166},[265],[28,53168],{"className":53169,"style":21986},[269],[28,53171,53173,53176,53235],{"className":53172},[274],[28,53174],{"className":53175},[348,8285],[28,53177,53179],{"className":53178},[21928],[28,53180,53182,53227],{"className":53181},[560,561],[28,53183,53185,53224],{"className":53184},[565],[28,53186,53188,53202,53210],{"className":53187,"style":22005},[569],[28,53189,53190,53193],{"style":22008},[28,53191],{"className":53192,"style":1914},[577],[28,53194,53196],{"className":53195},[582,583,584,585],[28,53197,53199],{"className":53198},[274,585],[28,53200,1253],{"className":53201},[274,585],[28,53203,53204,53207],{"style":22023},[28,53205],{"className":53206,"style":1914},[577],[28,53208],{"className":53209,"style":22031},[22030],[28,53211,53212,53215],{"style":22034},[28,53213],{"className":53214,"style":1914},[577],[28,53216,53218],{"className":53217},[582,583,584,585],[28,53219,53221],{"className":53220},[274,585],[28,53222,1244],{"className":53223},[274,585],[28,53225,593],{"className":53226},[592],[28,53228,53230],{"className":53229},[565],[28,53231,53233],{"className":53232,"style":22056},[569],[28,53234],{},[28,53236],{"className":53237},[355,8285],[28,53239,53241,53244],{"className":53240},[274],[28,53242,43555],{"className":53243,"style":2279},[274,275],[28,53245,53247],{"className":53246},[556],[28,53248,53250,53281],{"className":53249},[560,561],[28,53251,53253,53278],{"className":53252},[565],[28,53254,53256,53267],{"className":53255,"style":5477},[569],[28,53257,53258,53261],{"style":53123},[28,53259],{"className":53260,"style":578},[577],[28,53262,53264],{"className":53263},[582,583,584,585],[28,53265,1244],{"className":53266},[274,585],[28,53268,53269,53272],{"style":3094},[28,53270],{"className":53271,"style":578},[577],[28,53273,53275],{"className":53274},[582,583,584,585],[28,53276,1253],{"className":53277},[274,585],[28,53279,593],{"className":53280},[592],[28,53282,53284],{"className":53283},[565],[28,53285,53287],{"className":53286,"style":37576},[569],[28,53288],{},[28,53290],{"className":53291,"style":280},[279],[28,53293,5074],{"className":53294},[284],[28,53296],{"className":53297,"style":280},[279],[28,53299,53301,53304,53344,53347,53387,53390,53393],{"className":53300},[265],[28,53302],{"className":53303,"style":341},[269],[28,53305,53307,53310],{"className":53306},[274],[28,53308,43555],{"className":53309,"style":2279},[274,275],[28,53311,53313],{"className":53312},[556],[28,53314,53316,53336],{"className":53315},[560,561],[28,53317,53319,53333],{"className":53318},[565],[28,53320,53322],{"className":53321,"style":1285},[569],[28,53323,53324,53327],{"style":46480},[28,53325],{"className":53326,"style":578},[577],[28,53328,53330],{"className":53329},[582,583,584,585],[28,53331,1244],{"className":53332},[274,585],[28,53334,593],{"className":53335},[592],[28,53337,53339],{"className":53338},[565],[28,53340,53342],{"className":53341,"style":600},[569],[28,53343],{},[28,53345,43],{"className":53346},[348],[28,53348,53350,53353],{"className":53349},[274],[28,53351,43555],{"className":53352,"style":2279},[274,275],[28,53354,53356],{"className":53355},[556],[28,53357,53359,53379],{"className":53358},[560,561],[28,53360,53362,53376],{"className":53361},[565],[28,53363,53365],{"className":53364,"style":1285},[569],[28,53366,53367,53370],{"style":46480},[28,53368],{"className":53369,"style":578},[577],[28,53371,53373],{"className":53372},[582,583,584,585],[28,53374,1253],{"className":53375},[274,585],[28,53377,593],{"className":53378},[592],[28,53380,53382],{"className":53381},[565],[28,53383,53385],{"className":53384,"style":600},[569],[28,53386],{},[28,53388],{"className":53389,"style":280},[279],[28,53391,5074],{"className":53392},[284],[28,53394],{"className":53395,"style":280},[279],[28,53397,53399,53402,53442,53445,53448,53451],{"className":53398},[265],[28,53400],{"className":53401,"style":341},[269],[28,53403,53405,53408],{"className":53404},[274],[28,53406,43555],{"className":53407,"style":2279},[274,275],[28,53409,53411],{"className":53410},[556],[28,53412,53414,53434],{"className":53413},[560,561],[28,53415,53417,53431],{"className":53416},[565],[28,53418,53420],{"className":53419,"style":1285},[569],[28,53421,53422,53425],{"style":46480},[28,53423],{"className":53424,"style":578},[577],[28,53426,53428],{"className":53427},[582,583,584,585],[28,53429,1244],{"className":53430},[274,585],[28,53432,593],{"className":53433},[592],[28,53435,53437],{"className":53436},[565],[28,53438,53440],{"className":53439,"style":600},[569],[28,53441],{},[28,53443,1522],{"className":53444},[355],[28,53446],{"className":53447,"style":302},[279],[28,53449,83],{"className":53450},[306],[28,53452],{"className":53453,"style":302},[279],[28,53455,53457,53460,53528,53531,53571,53574,53577],{"className":53456},[265],[28,53458],{"className":53459,"style":21986},[269],[28,53461,53463,53466,53525],{"className":53462},[274],[28,53464],{"className":53465},[348,8285],[28,53467,53469],{"className":53468},[21928],[28,53470,53472,53517],{"className":53471},[560,561],[28,53473,53475,53514],{"className":53474},[565],[28,53476,53478,53492,53500],{"className":53477,"style":22005},[569],[28,53479,53480,53483],{"style":22008},[28,53481],{"className":53482,"style":1914},[577],[28,53484,53486],{"className":53485},[582,583,584,585],[28,53487,53489],{"className":53488},[274,585],[28,53490,1253],{"className":53491},[274,585],[28,53493,53494,53497],{"style":22023},[28,53495],{"className":53496,"style":1914},[577],[28,53498],{"className":53499,"style":22031},[22030],[28,53501,53502,53505],{"style":22034},[28,53503],{"className":53504,"style":1914},[577],[28,53506,53508],{"className":53507},[582,583,584,585],[28,53509,53511],{"className":53510},[274,585],[28,53512,1244],{"className":53513},[274,585],[28,53515,593],{"className":53516},[592],[28,53518,53520],{"className":53519},[565],[28,53521,53523],{"className":53522,"style":22056},[569],[28,53524],{},[28,53526],{"className":53527},[355,8285],[28,53529,43],{"className":53530},[348],[28,53532,53534,53537],{"className":53533},[274],[28,53535,43555],{"className":53536,"style":2279},[274,275],[28,53538,53540],{"className":53539},[556],[28,53541,53543,53563],{"className":53542},[560,561],[28,53544,53546,53560],{"className":53545},[565],[28,53547,53549],{"className":53548,"style":1285},[569],[28,53550,53551,53554],{"style":46480},[28,53552],{"className":53553,"style":578},[577],[28,53555,53557],{"className":53556},[582,583,584,585],[28,53558,1253],{"className":53559},[274,585],[28,53561,593],{"className":53562},[592],[28,53564,53566],{"className":53565},[565],[28,53567,53569],{"className":53568,"style":600},[569],[28,53570],{},[28,53572],{"className":53573,"style":280},[279],[28,53575,5074],{"className":53576},[284],[28,53578],{"className":53579,"style":280},[279],[28,53581,53583,53586,53626],{"className":53582},[265],[28,53584],{"className":53585,"style":31837},[269],[28,53587,53589,53592],{"className":53588},[274],[28,53590,43555],{"className":53591,"style":2279},[274,275],[28,53593,53595],{"className":53594},[556],[28,53596,53598,53618],{"className":53597},[560,561],[28,53599,53601,53615],{"className":53600},[565],[28,53602,53604],{"className":53603,"style":1285},[569],[28,53605,53606,53609],{"style":46480},[28,53607],{"className":53608,"style":578},[577],[28,53610,53612],{"className":53611},[582,583,584,585],[28,53613,1244],{"className":53614},[274,585],[28,53616,593],{"className":53617},[592],[28,53619,53621],{"className":53620},[565],[28,53622,53624],{"className":53623,"style":600},[569],[28,53625],{},[28,53627,53629,53632],{"className":53628},[355],[28,53630,1522],{"className":53631},[355],[28,53633,53635],{"className":53634},[556],[28,53636,53638],{"className":53637},[560],[28,53639,53641],{"className":53640},[565],[28,53642,53644],{"className":53643,"style":5477},[569],[28,53645,53646,53649],{"style":3094},[28,53647],{"className":53648,"style":578},[577],[28,53650,53652],{"className":53651},[582,583,584,585],[28,53653,1253],{"className":53654},[274,585],"。同一个数。",[11,53657,53658],{},"所以在这类模型里，KL 不是某种抽象的“信息量”，它就是 Bregman 散度；而它的 Hessian 就是 Fisher 度量。散度和度量是同一个凸函数的两个侧面。这是我写这篇博客最想讲清楚的一件事。",[414,53660,53662],{"id":53661},"bregman","Bregman",[11,53664,53665,53666,53740,53741,53815],{},"先看一个会让人不舒服的例子。",[28,53667,53669,53695],{"className":53668,"translate":193},[197],[28,53670,53672],{"className":53671},[201],[203,53673,53674],{"xmlns":205},[208,53675,53676,53692],{},[211,53677,53678,53680,53682,53684,53686,53688,53690],{},[214,53679,11],{},[217,53681,83],{},[217,53683,43],{"stretchy":241},[224,53685,1244],{},[217,53687,1683],{"separator":261},[224,53689,89],{},[217,53691,1522],{"stretchy":241},[253,53693,53694],{"encoding":255},"p=(1,0)",[28,53696,53698,53716],{"className":53697,"ariaHidden":261},[260],[28,53699,53701,53704,53707,53710,53713],{"className":53700},[265],[28,53702],{"className":53703,"style":612},[269],[28,53705,11],{"className":53706},[274,275],[28,53708],{"className":53709,"style":302},[279],[28,53711,83],{"className":53712},[306],[28,53714],{"className":53715,"style":302},[279],[28,53717,53719,53722,53725,53728,53731,53734,53737],{"className":53718},[265],[28,53720],{"className":53721,"style":341},[269],[28,53723,43],{"className":53724},[348],[28,53726,1244],{"className":53727},[274],[28,53729,1683],{"className":53730},[1723],[28,53732],{"className":53733,"style":530},[279],[28,53735,89],{"className":53736},[274],[28,53738,1522],{"className":53739},[355]," 是“真相”，",[28,53742,53744,53770],{"className":53743,"translate":193},[197],[28,53745,53747],{"className":53746},[201],[203,53748,53749],{"xmlns":205},[208,53750,53751,53767],{},[211,53752,53753,53755,53757,53759,53761,53763,53765],{},[214,53754,1829],{},[217,53756,83],{},[217,53758,43],{"stretchy":241},[224,53760,42924],{},[217,53762,1683],{"separator":261},[224,53764,42924],{},[217,53766,1522],{"stretchy":241},[253,53768,53769],{"encoding":255},"q=(0.5,0.5)",[28,53771,53773,53791],{"className":53772,"ariaHidden":261},[260],[28,53774,53776,53779,53782,53785,53788],{"className":53775},[265],[28,53777],{"className":53778,"style":612},[269],[28,53780,1829],{"className":53781,"style":298},[274,275],[28,53783],{"className":53784,"style":302},[279],[28,53786,83],{"className":53787},[306],[28,53789],{"className":53790,"style":302},[279],[28,53792,53794,53797,53800,53803,53806,53809,53812],{"className":53793},[265],[28,53795],{"className":53796,"style":341},[269],[28,53798,43],{"className":53799},[348],[28,53801,42924],{"className":53802},[274],[28,53804,1683],{"className":53805},[1723],[28,53807],{"className":53808,"style":530},[279],[28,53810,42924],{"className":53811},[274],[28,53813,1522],{"className":53814},[355]," 是“模型的预测”。",[28,53817,53819],{"className":53818,"translate":193},[192],[28,53820,53822,53891],{"className":53821,"translate":193},[197],[28,53823,53825],{"className":53824},[201],[203,53826,53827],{"xmlns":205,"display":206},[208,53828,53829,53888],{},[211,53830,53831,53837,53839,53841,53843,53845,53847,53849,53851,53853,53855,53857,53860,53862,53864,53870,53872,53874,53876,53878,53880,53882,53884,53886],{},[211,53832,53833,53835],{},[214,53834,24390],{"mathvariant":484},[214,53836,24393],{"mathvariant":484},[217,53838,43],{"stretchy":241},[214,53840,11],{},[214,53842,24404],{"mathvariant":484},[214,53844,1829],{},[217,53846,1522],{"stretchy":241},[217,53848,83],{},[214,53850,39714],{},[217,53852,21407],{},[224,53854,1253],{},[217,53856,46977],{},[224,53858,53859],{},"0.693",[217,53861,1683],{"separator":261},[279,53863],{"width":4053},[211,53865,53866,53868],{},[214,53867,24390],{"mathvariant":484},[214,53869,24393],{"mathvariant":484},[217,53871,43],{"stretchy":241},[214,53873,1829],{},[214,53875,24404],{"mathvariant":484},[214,53877,11],{},[217,53879,1522],{"stretchy":241},[217,53881,83],{},[217,53883,124],{},[214,53885,24419],{"mathvariant":484},[214,53887,494],{"mathvariant":484},[253,53889,53890],{"encoding":255},"\\mathrm{KL}(p\\|q)=\\log2\\approx0.693,\n\\qquad\n\\mathrm{KL}(q\\|p)=+\\infty .",[28,53892,53894,53930,53956,54004],{"className":53893,"ariaHidden":261},[260],[28,53895,53897,53900,53906,53909,53912,53915,53918,53921,53924,53927],{"className":53896},[265],[28,53898],{"className":53899,"style":341},[269],[28,53901,53903],{"className":53902},[274],[28,53904,24438],{"className":53905},[274,1712],[28,53907,43],{"className":53908},[348],[28,53910,11],{"className":53911},[274,275],[28,53913,24404],{"className":53914},[274],[28,53916,1829],{"className":53917,"style":298},[274,275],[28,53919,1522],{"className":53920},[355],[28,53922],{"className":53923,"style":302},[279],[28,53925,83],{"className":53926},[306],[28,53928],{"className":53929,"style":302},[279],[28,53931,53933,53936,53941,53944,53947,53950,53953],{"className":53932},[265],[28,53934],{"className":53935,"style":2371},[269],[28,53937,39952,53939],{"className":53938},[21329],[28,53940,34347],{"style":39955},[28,53942],{"className":53943,"style":530},[279],[28,53945,1253],{"className":53946},[274],[28,53948],{"className":53949,"style":302},[279],[28,53951,46977],{"className":53952},[306],[28,53954],{"className":53955,"style":302},[279],[28,53957,53959,53962,53965,53968,53971,53974,53980,53983,53986,53989,53992,53995,53998,54001],{"className":53958},[265],[28,53960],{"className":53961,"style":341},[269],[28,53963,53859],{"className":53964},[274],[28,53966,1683],{"className":53967},[1723],[28,53969],{"className":53970,"style":4222},[279],[28,53972],{"className":53973,"style":530},[279],[28,53975,53977],{"className":53976},[274],[28,53978,24438],{"className":53979},[274,1712],[28,53981,43],{"className":53982},[348],[28,53984,1829],{"className":53985,"style":298},[274,275],[28,53987,24404],{"className":53988},[274],[28,53990,11],{"className":53991},[274,275],[28,53993,1522],{"className":53994},[355],[28,53996],{"className":53997,"style":302},[279],[28,53999,83],{"className":54000},[306],[28,54002],{"className":54003,"style":302},[279],[28,54005,54007,54010,54013],{"className":54006},[265],[28,54008],{"className":54009,"style":270},[269],[28,54011,124],{"className":54012},[274],[28,54014,54016],{"className":54015},[274],"∞.",[11,54018,54019,54020,54048,54049,54077,54078,54106,54107,54135,54136,54106,54164,54192,54193,54221],{},"同一个“差”，从左往右量是 0.69，从右往左量是无穷。原因不难想：",[28,54021,54023,54036],{"className":54022,"translate":193},[197],[28,54024,54026],{"className":54025},[201],[203,54027,54028],{"xmlns":205},[208,54029,54030,54034],{},[211,54031,54032],{},[214,54033,11],{},[253,54035,11],{"encoding":255},[28,54037,54039],{"className":54038,"ariaHidden":261},[260],[28,54040,54042,54045],{"className":54041},[265],[28,54043],{"className":54044,"style":612},[269],[28,54046,11],{"className":54047},[274,275]," 在第二位上放了 0，而 ",[28,54050,54052,54065],{"className":54051,"translate":193},[197],[28,54053,54055],{"className":54054},[201],[203,54056,54057],{"xmlns":205},[208,54058,54059,54063],{},[211,54060,54061],{},[214,54062,1829],{},[253,54064,1829],{"encoding":255},[28,54066,54068],{"className":54067,"ariaHidden":261},[260],[28,54069,54071,54074],{"className":54070},[265],[28,54072],{"className":54073,"style":612},[269],[28,54075,1829],{"className":54076,"style":298},[274,275]," 在那里放了 0.5，用 ",[28,54079,54081,54094],{"className":54080,"translate":193},[197],[28,54082,54084],{"className":54083},[201],[203,54085,54086],{"xmlns":205},[208,54087,54088,54092],{},[211,54089,54090],{},[214,54091,11],{},[253,54093,11],{"encoding":255},[28,54095,54097],{"className":54096,"ariaHidden":261},[260],[28,54098,54100,54103],{"className":54099},[265],[28,54101],{"className":54102,"style":612},[269],[28,54104,11],{"className":54105},[274,275]," 去看 ",[28,54108,54110,54123],{"className":54109,"translate":193},[197],[28,54111,54113],{"className":54112},[201],[203,54114,54115],{"xmlns":205},[208,54116,54117,54121],{},[211,54118,54119],{},[214,54120,1829],{},[253,54122,1829],{"encoding":255},[28,54124,54126],{"className":54125,"ariaHidden":261},[260],[28,54127,54129,54132],{"className":54128},[265],[28,54130],{"className":54131,"style":612},[269],[28,54133,1829],{"className":54134,"style":298},[274,275],"，只是多花了 0.69 nat；用 ",[28,54137,54139,54152],{"className":54138,"translate":193},[197],[28,54140,54142],{"className":54141},[201],[203,54143,54144],{"xmlns":205},[208,54145,54146,54150],{},[211,54147,54148],{},[214,54149,1829],{},[253,54151,1829],{"encoding":255},[28,54153,54155],{"className":54154,"ariaHidden":261},[260],[28,54156,54158,54161],{"className":54157},[265],[28,54159],{"className":54160,"style":612},[269],[28,54162,1829],{"className":54163,"style":298},[274,275],[28,54165,54167,54180],{"className":54166,"translate":193},[197],[28,54168,54170],{"className":54169},[201],[203,54171,54172],{"xmlns":205},[208,54173,54174,54178],{},[211,54175,54176],{},[214,54177,11],{},[253,54179,11],{"encoding":255},[28,54181,54183],{"className":54182,"ariaHidden":261},[260],[28,54184,54186,54189],{"className":54185},[265],[28,54187],{"className":54188,"style":612},[269],[28,54190,11],{"className":54191},[274,275],"，等于说“我给了一个 ",[28,54194,54196,54209],{"className":54195,"translate":193},[197],[28,54197,54199],{"className":54198},[201],[203,54200,54201],{"xmlns":205},[208,54202,54203,54207],{},[211,54204,54205],{},[214,54206,11],{},[253,54208,11],{"encoding":255},[28,54210,54212],{"className":54211,"ariaHidden":261},[260],[28,54213,54215,54218],{"className":54214},[265],[28,54216],{"className":54217,"style":612},[269],[28,54219,11],{"className":54220},[274,275]," 认为绝不可能的东西分配了概率”，这种错误在 KL 里罚到无穷。",[11,54223,54224,54225,54286,54287,54315,54316,54344],{},"这正是交叉熵写成 ",[28,54226,54228,54256],{"className":54227,"translate":193},[197],[28,54229,54231],{"className":54230},[201],[203,54232,54233],{"xmlns":205},[208,54234,54235,54253],{},[211,54236,54237,54243,54245,54247,54249,54251],{},[211,54238,54239,54241],{},[214,54240,24390],{"mathvariant":484},[214,54242,24393],{"mathvariant":484},[217,54244,43],{"stretchy":241},[214,54246,180],{},[214,54248,24404],{"mathvariant":484},[214,54250,1829],{},[217,54252,1522],{"stretchy":241},[253,54254,54255],{"encoding":255},"\\mathrm{KL}(y\\|q)",[28,54257,54259],{"className":54258,"ariaHidden":261},[260],[28,54260,54262,54265,54271,54274,54277,54280,54283],{"className":54261},[265],[28,54263],{"className":54264,"style":341},[269],[28,54266,54268],{"className":54267},[274],[28,54269,24438],{"className":54270},[274,1712],[28,54272,43],{"className":54273},[348],[28,54275,180],{"className":54276,"style":298},[274,275],[28,54278,24404],{"className":54279},[274],[28,54281,1829],{"className":54282,"style":298},[274,275],[28,54284,1522],{"className":54285},[355]," 而不是反过来的原因：",[28,54288,54290,54303],{"className":54289,"translate":193},[197],[28,54291,54293],{"className":54292},[201],[203,54294,54295],{"xmlns":205},[208,54296,54297,54301],{},[211,54298,54299],{},[214,54300,180],{},[253,54302,180],{"encoding":255},[28,54304,54306],{"className":54305,"ariaHidden":261},[260],[28,54307,54309,54312],{"className":54308},[265],[28,54310],{"className":54311,"style":612},[269],[28,54313,180],{"className":54314,"style":298},[274,275]," 是真相，模型 ",[28,54317,54319,54332],{"className":54318,"translate":193},[197],[28,54320,54322],{"className":54321},[201],[203,54323,54324],{"xmlns":205},[208,54325,54326,54330],{},[211,54327,54328],{},[214,54329,1829],{},[253,54331,1829],{"encoding":255},[28,54333,54335],{"className":54334,"ariaHidden":261},[260],[28,54336,54338,54341],{"className":54337},[265],[28,54339],{"className":54340,"style":612},[269],[28,54342,1829],{"className":54343,"style":298},[274,275]," 不该在真答案上给 0 概率。",[11,54346,54347,54348,54561,54562,54743,54744,54824,54825,54853,54854,2693,54883,54911],{},"再看一个尺度上的例子。Itakura–Saito 散度 ",[28,54349,54351,54413],{"className":54350,"translate":193},[197],[28,54352,54354],{"className":54353},[201],[203,54355,54356],{"xmlns":205},[208,54357,54358,54410],{},[211,54359,54360,54372,54374,54376,54378,54380,54382,54384,54386,54388,54390,54392,54394,54396,54398,54400,54402,54404,54406,54408],{},[501,54361,54362,54365],{},[214,54363,54364],{},"D",[211,54366,54367,54370],{},[214,54368,54369],{"mathvariant":484},"I",[214,54371,6730],{"mathvariant":484},[217,54373,43],{"stretchy":241},[214,54375,176],{},[214,54377,24404],{"mathvariant":484},[214,54379,180],{},[217,54381,1522],{"stretchy":241},[217,54383,83],{},[214,54385,176],{},[214,54387,12219],{"mathvariant":484},[214,54389,180],{},[217,54391,5074],{},[214,54393,39714],{},[217,54395,21407],{},[217,54397,43],{"stretchy":241},[214,54399,176],{},[214,54401,12219],{"mathvariant":484},[214,54403,180],{},[217,54405,1522],{"stretchy":241},[217,54407,5074],{},[224,54409,1244],{},[253,54411,54412],{"encoding":255},"D_{\\mathrm{IS}}(x\\|y)=x\u002Fy-\\log(x\u002Fy)-1",[28,54414,54416,54493,54517,54552],{"className":54415,"ariaHidden":261},[260],[28,54417,54419,54422,54469,54472,54475,54478,54481,54484,54487,54490],{"className":54418},[265],[28,54420],{"className":54421,"style":341},[269],[28,54423,54425,54428],{"className":54424},[274],[28,54426,54364],{"className":54427,"style":2279},[274,275],[28,54429,54431],{"className":54430},[556],[28,54432,54434,54461],{"className":54433},[560,561],[28,54435,54437,54458],{"className":54436},[565],[28,54438,54440],{"className":54439,"style":570},[569],[28,54441,54442,54445],{"style":46480},[28,54443],{"className":54444,"style":578},[577],[28,54446,54448],{"className":54447},[582,583,584,585],[28,54449,54451],{"className":54450},[274,585],[28,54452,54454],{"className":54453},[274,585],[28,54455,54457],{"className":54456},[274,1712,585],"IS",[28,54459,593],{"className":54460},[592],[28,54462,54464],{"className":54463},[565],[28,54465,54467],{"className":54466,"style":600},[569],[28,54468],{},[28,54470,43],{"className":54471},[348],[28,54473,176],{"className":54474},[274,275],[28,54476,24404],{"className":54477},[274],[28,54479,180],{"className":54480,"style":298},[274,275],[28,54482,1522],{"className":54483},[355],[28,54485],{"className":54486,"style":302},[279],[28,54488,83],{"className":54489},[306],[28,54491],{"className":54492,"style":302},[279],[28,54494,54496,54499,54502,54505,54508,54511,54514],{"className":54495},[265],[28,54497],{"className":54498,"style":341},[269],[28,54500,176],{"className":54501},[274,275],[28,54503,12219],{"className":54504},[274],[28,54506,180],{"className":54507,"style":298},[274,275],[28,54509],{"className":54510,"style":280},[279],[28,54512,5074],{"className":54513},[284],[28,54515],{"className":54516,"style":280},[279],[28,54518,54520,54523,54528,54531,54534,54537,54540,54543,54546,54549],{"className":54519},[265],[28,54521],{"className":54522,"style":341},[269],[28,54524,39952,54526],{"className":54525},[21329],[28,54527,34347],{"style":39955},[28,54529,43],{"className":54530},[348],[28,54532,176],{"className":54533},[274,275],[28,54535,12219],{"className":54536},[274],[28,54538,180],{"className":54539,"style":298},[274,275],[28,54541,1522],{"className":54542},[355],[28,54544],{"className":54545,"style":280},[279],[28,54547,5074],{"className":54548},[284],[28,54550],{"className":54551,"style":280},[279],[28,54553,54555,54558],{"className":54554},[265],[28,54556],{"className":54557,"style":316},[269],[28,54559,1244],{"className":54560},[274],"：把音量从 1 减半到 0.5，",[28,54563,54565,54608],{"className":54564,"translate":193},[197],[28,54566,54568],{"className":54567},[201],[203,54569,54570],{"xmlns":205},[208,54571,54572,54605],{},[211,54573,54574,54584,54586,54588,54590,54592,54594,54596,54598,54600,54602],{},[501,54575,54576,54578],{},[214,54577,54364],{},[211,54579,54580,54582],{},[214,54581,54369],{"mathvariant":484},[214,54583,6730],{"mathvariant":484},[217,54585,83],{},[224,54587,1253],{},[217,54589,5074],{},[214,54591,39714],{},[217,54593,21407],{},[224,54595,1253],{},[217,54597,5074],{},[224,54599,1244],{},[217,54601,83],{},[224,54603,54604],{},"0.307",[253,54606,54607],{"encoding":255},"D_{\\mathrm{IS}}=2-\\log2-1=0.307",[28,54609,54611,54672,54690,54716,54734],{"className":54610,"ariaHidden":261},[260],[28,54612,54614,54617,54663,54666,54669],{"className":54613},[265],[28,54615],{"className":54616,"style":25059},[269],[28,54618,54620,54623],{"className":54619},[274],[28,54621,54364],{"className":54622,"style":2279},[274,275],[28,54624,54626],{"className":54625},[556],[28,54627,54629,54655],{"className":54628},[560,561],[28,54630,54632,54652],{"className":54631},[565],[28,54633,54635],{"className":54634,"style":570},[569],[28,54636,54637,54640],{"style":46480},[28,54638],{"className":54639,"style":578},[577],[28,54641,54643],{"className":54642},[582,583,584,585],[28,54644,54646],{"className":54645},[274,585],[28,54647,54649],{"className":54648},[274,585],[28,54650,54457],{"className":54651},[274,1712,585],[28,54653,593],{"className":54654},[592],[28,54656,54658],{"className":54657},[565],[28,54659,54661],{"className":54660,"style":600},[569],[28,54662],{},[28,54664],{"className":54665,"style":302},[279],[28,54667,83],{"className":54668},[306],[28,54670],{"className":54671,"style":302},[279],[28,54673,54675,54678,54681,54684,54687],{"className":54674},[265],[28,54676],{"className":54677,"style":6853},[269],[28,54679,1253],{"className":54680},[274],[28,54682],{"className":54683,"style":280},[279],[28,54685,5074],{"className":54686},[284],[28,54688],{"className":54689,"style":280},[279],[28,54691,54693,54696,54701,54704,54707,54710,54713],{"className":54692},[265],[28,54694],{"className":54695,"style":2371},[269],[28,54697,39952,54699],{"className":54698},[21329],[28,54700,34347],{"style":39955},[28,54702],{"className":54703,"style":530},[279],[28,54705,1253],{"className":54706},[274],[28,54708],{"className":54709,"style":280},[279],[28,54711,5074],{"className":54712},[284],[28,54714],{"className":54715,"style":280},[279],[28,54717,54719,54722,54725,54728,54731],{"className":54718},[265],[28,54720],{"className":54721,"style":316},[269],[28,54723,1244],{"className":54724},[274],[28,54726],{"className":54727,"style":302},[279],[28,54729,83],{"className":54730},[306],[28,54732],{"className":54733,"style":302},[279],[28,54735,54737,54740],{"className":54736},[265],[28,54738],{"className":54739,"style":316},[269],[28,54741,54604],{"className":54742},[274],"；从 2 减半到 1，",[28,54745,54747,54769],{"className":54746,"translate":193},[197],[28,54748,54750],{"className":54749},[201],[203,54751,54752],{"xmlns":205},[208,54753,54754,54766],{},[211,54755,54756],{},[501,54757,54758,54760],{},[214,54759,54364],{},[211,54761,54762,54764],{},[214,54763,54369],{"mathvariant":484},[214,54765,6730],{"mathvariant":484},[253,54767,54768],{"encoding":255},"D_{\\mathrm{IS}}",[28,54770,54772],{"className":54771,"ariaHidden":261},[260],[28,54773,54775,54778],{"className":54774},[265],[28,54776],{"className":54777,"style":25059},[269],[28,54779,54781,54784],{"className":54780},[274],[28,54782,54364],{"className":54783,"style":2279},[274,275],[28,54785,54787],{"className":54786},[556],[28,54788,54790,54816],{"className":54789},[560,561],[28,54791,54793,54813],{"className":54792},[565],[28,54794,54796],{"className":54795,"style":570},[569],[28,54797,54798,54801],{"style":46480},[28,54799],{"className":54800,"style":578},[577],[28,54802,54804],{"className":54803},[582,583,584,585],[28,54805,54807],{"className":54806},[274,585],[28,54808,54810],{"className":54809},[274,585],[28,54811,54457],{"className":54812},[274,1712,585],[28,54814,593],{"className":54815},[592],[28,54817,54819],{"className":54818},[565],[28,54820,54822],{"className":54821,"style":600},[569],[28,54823],{}," 还是 ",[28,54826,54828,54841],{"className":54827,"translate":193},[197],[28,54829,54831],{"className":54830},[201],[203,54832,54833],{"xmlns":205},[208,54834,54835,54839],{},[211,54836,54837],{},[224,54838,54604],{},[253,54840,54604],{"encoding":255},[28,54842,54844],{"className":54843,"ariaHidden":261},[260],[28,54845,54847,54850],{"className":54846},[265],[28,54848],{"className":54849,"style":316},[269],[28,54851,54604],{"className":54852},[274],"。同样的比例变化，距离一样。而欧氏距离给的是 ",[28,54855,54857,54871],{"className":54856,"translate":193},[197],[28,54858,54860],{"className":54859},[201],[203,54861,54862],{"xmlns":205},[208,54863,54864,54869],{},[211,54865,54866],{},[224,54867,54868],{},"0.25",[253,54870,54868],{"encoding":255},[28,54872,54874],{"className":54873,"ariaHidden":261},[260],[28,54875,54877,54880],{"className":54876},[265],[28,54878],{"className":54879,"style":316},[269],[28,54881,54868],{"className":54882},[274],[28,54884,54886,54899],{"className":54885,"translate":193},[197],[28,54887,54889],{"className":54888},[201],[203,54890,54891],{"xmlns":205},[208,54892,54893,54897],{},[211,54894,54895],{},[224,54896,1244],{},[253,54898,1244],{"encoding":255},[28,54900,54902],{"className":54901,"ariaHidden":261},[260],[28,54903,54905,54908],{"className":54904},[265],[28,54906],{"className":54907,"style":316},[269],[28,54909,1244],{"className":54910},[274],"，差了四倍——因为欧氏觉得“从 2 到 1 变化更大”，可听感上这明明是同一件事。",[11,54913,54914],{},"这两个例子指向同一件事：距离可以不对称，也可以不按绝对差来量。",[11,54916,54917,54918,54946,54947,6904],{},"一般地，给定凸函数 ",[28,54919,54921,54934],{"className":54920,"translate":193},[197],[28,54922,54924],{"className":54923},[201],[203,54925,54926],{"xmlns":205},[208,54927,54928,54932],{},[211,54929,54930],{},[214,54931,1511],{},[253,54933,1511],{"encoding":255},[28,54935,54937],{"className":54936,"ariaHidden":261},[260],[28,54938,54940,54943],{"className":54939},[265],[28,54941],{"className":54942,"style":2371},[269],[28,54944,1511],{"className":54945,"style":1546},[274,275]," 和可微点 ",[28,54948,54950,54963],{"className":54949,"translate":193},[197],[28,54951,54953],{"className":54952},[201],[203,54954,54955],{"xmlns":205},[208,54956,54957,54961],{},[211,54958,54959],{},[214,54960,180],{},[253,54962,180],{"encoding":255},[28,54964,54966],{"className":54965,"ariaHidden":261},[260],[28,54967,54969,54972],{"className":54968},[265],[28,54970],{"className":54971,"style":612},[269],[28,54973,180],{"className":54974,"style":298},[274,275],[28,54976,54978],{"className":54977,"translate":193},[192],[28,54979,54981,55057],{"className":54980,"translate":193},[197],[28,54982,54984],{"className":54983},[201],[203,54985,54986],{"xmlns":205,"display":206},[208,54987,54988,55054],{},[211,54989,54990,54996,54998,55000,55002,55004,55006,55008,55010,55012,55014,55016,55018,55020,55022,55024,55026,55028,55030,55032,55034,55036,55042,55044,55046,55048,55050,55052],{},[501,54991,54992,54994],{},[214,54993,54364],{},[214,54995,1511],{},[217,54997,43],{"stretchy":241},[214,54999,176],{},[214,55001,24404],{"mathvariant":484},[214,55003,180],{},[217,55005,1522],{"stretchy":241},[217,55007,83],{},[214,55009,1511],{},[217,55011,43],{"stretchy":241},[214,55013,176],{},[217,55015,1522],{"stretchy":241},[217,55017,5074],{},[214,55019,1511],{},[217,55021,43],{"stretchy":241},[214,55023,180],{},[217,55025,1522],{"stretchy":241},[217,55027,5074],{},[214,55029,31947],{"mathvariant":484},[214,55031,1511],{},[217,55033,43],{"stretchy":241},[214,55035,180],{},[3054,55037,55038,55040],{},[217,55039,1522],{"stretchy":241},[214,55041,884],{"mathvariant":484},[217,55043,43],{"stretchy":241},[214,55045,176],{},[217,55047,5074],{},[214,55049,180],{},[217,55051,1522],{"stretchy":241},[214,55053,494],{"mathvariant":484},[253,55055,55056],{"encoding":255},"D_f(x\\|y)=f(x)-f(y)-\\nabla f(y)^\\top(x-y).",[28,55058,55060,55130,55157,55184,55247],{"className":55059,"ariaHidden":261},[260],[28,55061,55063,55066,55106,55109,55112,55115,55118,55121,55124,55127],{"className":55062},[265],[28,55064],{"className":55065,"style":21101},[269],[28,55067,55069,55072],{"className":55068},[274],[28,55070,54364],{"className":55071,"style":2279},[274,275],[28,55073,55075],{"className":55074},[556],[28,55076,55078,55098],{"className":55077},[560,561],[28,55079,55081,55095],{"className":55080},[565],[28,55082,55084],{"className":55083,"style":20945},[569],[28,55085,55086,55089],{"style":46480},[28,55087],{"className":55088,"style":578},[577],[28,55090,55092],{"className":55091},[582,583,584,585],[28,55093,1511],{"className":55094,"style":1546},[274,275,585],[28,55096,593],{"className":55097},[592],[28,55099,55101],{"className":55100},[565],[28,55102,55104],{"className":55103,"style":20967},[569],[28,55105],{},[28,55107,43],{"className":55108},[348],[28,55110,176],{"className":55111},[274,275],[28,55113,24404],{"className":55114},[274],[28,55116,180],{"className":55117,"style":298},[274,275],[28,55119,1522],{"className":55120},[355],[28,55122],{"className":55123,"style":302},[279],[28,55125,83],{"className":55126},[306],[28,55128],{"className":55129,"style":302},[279],[28,55131,55133,55136,55139,55142,55145,55148,55151,55154],{"className":55132},[265],[28,55134],{"className":55135,"style":341},[269],[28,55137,1511],{"className":55138,"style":1546},[274,275],[28,55140,43],{"className":55141},[348],[28,55143,176],{"className":55144},[274,275],[28,55146,1522],{"className":55147},[355],[28,55149],{"className":55150,"style":280},[279],[28,55152,5074],{"className":55153},[284],[28,55155],{"className":55156,"style":280},[279],[28,55158,55160,55163,55166,55169,55172,55175,55178,55181],{"className":55159},[265],[28,55161],{"className":55162,"style":341},[269],[28,55164,1511],{"className":55165,"style":1546},[274,275],[28,55167,43],{"className":55168},[348],[28,55170,180],{"className":55171,"style":298},[274,275],[28,55173,1522],{"className":55174},[355],[28,55176],{"className":55177,"style":280},[279],[28,55179,5074],{"className":55180},[284],[28,55182],{"className":55183,"style":280},[279],[28,55185,55187,55191,55194,55197,55200,55203,55232,55235,55238,55241,55244],{"className":55186},[265],[28,55188],{"className":55189,"style":55190},[269],"height:1.1491em;vertical-align:-0.25em;",[28,55192,31947],{"className":55193},[274],[28,55195,1511],{"className":55196,"style":1546},[274,275],[28,55198,43],{"className":55199},[348],[28,55201,180],{"className":55202,"style":298},[274,275],[28,55204,55206,55209],{"className":55205},[355],[28,55207,1522],{"className":55208},[355],[28,55210,55212],{"className":55211},[556],[28,55213,55215],{"className":55214},[560],[28,55216,55218],{"className":55217},[565],[28,55219,55221],{"className":55220,"style":49393},[569],[28,55222,55223,55226],{"style":4880},[28,55224],{"className":55225,"style":578},[577],[28,55227,55229],{"className":55228},[582,583,584,585],[28,55230,884],{"className":55231},[274,585],[28,55233,43],{"className":55234},[348],[28,55236,176],{"className":55237},[274,275],[28,55239],{"className":55240,"style":280},[279],[28,55242,5074],{"className":55243},[284],[28,55245],{"className":55246,"style":280},[279],[28,55248,55250,55253,55256,55259],{"className":55249},[265],[28,55251],{"className":55252,"style":341},[269],[28,55254,180],{"className":55255,"style":298},[274,275],[28,55257,1522],{"className":55258},[355],[28,55260,494],{"className":55261},[274],[11,55263,55264,55265,55293,55294,55322,55323,55351,55352,55380,55381,55432],{},"几何上，这是 ",[28,55266,55268,55281],{"className":55267,"translate":193},[197],[28,55269,55271],{"className":55270},[201],[203,55272,55273],{"xmlns":205},[208,55274,55275,55279],{},[211,55276,55277],{},[214,55278,1511],{},[253,55280,1511],{"encoding":255},[28,55282,55284],{"className":55283,"ariaHidden":261},[260],[28,55285,55287,55290],{"className":55286},[265],[28,55288],{"className":55289,"style":2371},[269],[28,55291,1511],{"className":55292,"style":1546},[274,275]," 的图像在 ",[28,55295,55297,55310],{"className":55296,"translate":193},[197],[28,55298,55300],{"className":55299},[201],[203,55301,55302],{"xmlns":205},[208,55303,55304,55308],{},[211,55305,55306],{},[214,55307,176],{},[253,55309,176],{"encoding":255},[28,55311,55313],{"className":55312,"ariaHidden":261},[260],[28,55314,55316,55319],{"className":55315},[265],[28,55317],{"className":55318,"style":645},[269],[28,55320,176],{"className":55321},[274,275]," 处比 ",[28,55324,55326,55339],{"className":55325,"translate":193},[197],[28,55327,55329],{"className":55328},[201],[203,55330,55331],{"xmlns":205},[208,55332,55333,55337],{},[211,55334,55335],{},[214,55336,180],{},[253,55338,180],{"encoding":255},[28,55340,55342],{"className":55341,"ariaHidden":261},[260],[28,55343,55345,55348],{"className":55344},[265],[28,55346],{"className":55347,"style":612},[269],[28,55349,180],{"className":55350,"style":298},[274,275]," 处的切平面高出多少。",[28,55353,55355,55368],{"className":55354,"translate":193},[197],[28,55356,55358],{"className":55357},[201],[203,55359,55360],{"xmlns":205},[208,55361,55362,55366],{},[211,55363,55364],{},[214,55365,1511],{},[253,55367,1511],{"encoding":255},[28,55369,55371],{"className":55370,"ariaHidden":261},[260],[28,55372,55374,55377],{"className":55373},[265],[28,55375],{"className":55376,"style":2371},[269],[28,55378,1511],{"className":55379,"style":1546},[274,275]," 凸所以非负，严格凸所以只在 ",[28,55382,55384,55402],{"className":55383,"translate":193},[197],[28,55385,55387],{"className":55386},[201],[203,55388,55389],{"xmlns":205},[208,55390,55391,55399],{},[211,55392,55393,55395,55397],{},[214,55394,176],{},[217,55396,83],{},[214,55398,180],{},[253,55400,55401],{"encoding":255},"x=y",[28,55403,55405,55423],{"className":55404,"ariaHidden":261},[260],[28,55406,55408,55411,55414,55417,55420],{"className":55407},[265],[28,55409],{"className":55410,"style":645},[269],[28,55412,176],{"className":55413},[274,275],[28,55415],{"className":55416,"style":302},[279],[28,55418,83],{"className":55419},[306],[28,55421],{"className":55422,"style":302},[279],[28,55424,55426,55429],{"className":55425},[265],[28,55427],{"className":55428,"style":612},[269],[28,55430,180],{"className":55431,"style":298},[274,275]," 时为零。它不对称，也不满足三角不等式，所以我不叫它距离，叫散度。",[11,55434,55435,55436,55618,55619,5901,55786,56003,56004,5901,56065,56241,56242,56425],{},"几个我常用的：",[28,55437,55439,55479],{"className":55438,"translate":193},[197],[28,55440,55442],{"className":55441},[201],[203,55443,55444],{"xmlns":205},[208,55445,55446,55476],{},[211,55447,55448,55450,55452,55454,55456,55458,55466,55468,55470],{},[214,55449,1511],{},[217,55451,43],{"stretchy":241},[214,55453,176],{},[217,55455,1522],{"stretchy":241},[217,55457,83],{},[5027,55459,55460],{"scriptlevel":89,"displaystyle":241},[21928,55461,55462,55464],{},[224,55463,1244],{},[224,55465,1253],{},[214,55467,24404],{"mathvariant":484},[214,55469,176],{},[3054,55471,55472,55474],{},[214,55473,24404],{"mathvariant":484},[224,55475,1253],{},[253,55477,55478],{"encoding":255},"f(x)=\\tfrac12\\|x\\|^2",[28,55480,55482,55509],{"className":55481,"ariaHidden":261},[260],[28,55483,55485,55488,55491,55494,55497,55500,55503,55506],{"className":55484},[265],[28,55486],{"className":55487,"style":341},[269],[28,55489,1511],{"className":55490,"style":1546},[274,275],[28,55492,43],{"className":55493},[348],[28,55495,176],{"className":55496},[274,275],[28,55498,1522],{"className":55499},[355],[28,55501],{"className":55502,"style":302},[279],[28,55504,83],{"className":55505},[306],[28,55507],{"className":55508,"style":302},[279],[28,55510,55512,55515,55583,55586,55589],{"className":55511},[265],[28,55513],{"className":55514,"style":21986},[269],[28,55516,55518,55521,55580],{"className":55517},[274],[28,55519],{"className":55520},[348,8285],[28,55522,55524],{"className":55523},[21928],[28,55525,55527,55572],{"className":55526},[560,561],[28,55528,55530,55569],{"className":55529},[565],[28,55531,55533,55547,55555],{"className":55532,"style":22005},[569],[28,55534,55535,55538],{"style":22008},[28,55536],{"className":55537,"style":1914},[577],[28,55539,55541],{"className":55540},[582,583,584,585],[28,55542,55544],{"className":55543},[274,585],[28,55545,1253],{"className":55546},[274,585],[28,55548,55549,55552],{"style":22023},[28,55550],{"className":55551,"style":1914},[577],[28,55553],{"className":55554,"style":22031},[22030],[28,55556,55557,55560],{"style":22034},[28,55558],{"className":55559,"style":1914},[577],[28,55561,55563],{"className":55562},[582,583,584,585],[28,55564,55566],{"className":55565},[274,585],[28,55567,1244],{"className":55568},[274,585],[28,55570,593],{"className":55571},[592],[28,55573,55575],{"className":55574},[565],[28,55576,55578],{"className":55577,"style":22056},[569],[28,55579],{},[28,55581],{"className":55582},[355,8285],[28,55584,24404],{"className":55585},[274],[28,55587,176],{"className":55588},[274,275],[28,55590,55592,55595],{"className":55591},[274],[28,55593,24404],{"className":55594},[274],[28,55596,55598],{"className":55597},[556],[28,55599,55601],{"className":55600},[560],[28,55602,55604],{"className":55603},[565],[28,55605,55607],{"className":55606,"style":5477},[569],[28,55608,55609,55612],{"style":3094},[28,55610],{"className":55611,"style":578},[577],[28,55613,55615],{"className":55614},[582,583,584,585],[28,55616,1253],{"className":55617},[274,585]," 给 ",[28,55620,55622,55656],{"className":55621,"translate":193},[197],[28,55623,55625],{"className":55624},[201],[203,55626,55627],{"xmlns":205},[208,55628,55629,55653],{},[211,55630,55631,55639,55641,55643,55645,55647],{},[5027,55632,55633],{"scriptlevel":89,"displaystyle":241},[21928,55634,55635,55637],{},[224,55636,1244],{},[224,55638,1253],{},[214,55640,24404],{"mathvariant":484},[214,55642,176],{},[217,55644,5074],{},[214,55646,180],{},[3054,55648,55649,55651],{},[214,55650,24404],{"mathvariant":484},[224,55652,1253],{},[253,55654,55655],{"encoding":255},"\\tfrac12\\|x-y\\|^2",[28,55657,55659,55748],{"className":55658,"ariaHidden":261},[260],[28,55660,55662,55665,55733,55736,55739,55742,55745],{"className":55661},[265],[28,55663],{"className":55664,"style":21986},[269],[28,55666,55668,55671,55730],{"className":55667},[274],[28,55669],{"className":55670},[348,8285],[28,55672,55674],{"className":55673},[21928],[28,55675,55677,55722],{"className":55676},[560,561],[28,55678,55680,55719],{"className":55679},[565],[28,55681,55683,55697,55705],{"className":55682,"style":22005},[569],[28,55684,55685,55688],{"style":22008},[28,55686],{"className":55687,"style":1914},[577],[28,55689,55691],{"className":55690},[582,583,584,585],[28,55692,55694],{"className":55693},[274,585],[28,55695,1253],{"className":55696},[274,585],[28,55698,55699,55702],{"style":22023},[28,55700],{"className":55701,"style":1914},[577],[28,55703],{"className":55704,"style":22031},[22030],[28,55706,55707,55710],{"style":22034},[28,55708],{"className":55709,"style":1914},[577],[28,55711,55713],{"className":55712},[582,583,584,585],[28,55714,55716],{"className":55715},[274,585],[28,55717,1244],{"className":55718},[274,585],[28,55720,593],{"className":55721},[592],[28,55723,55725],{"className":55724},[565],[28,55726,55728],{"className":55727,"style":22056},[569],[28,55729],{},[28,55731],{"className":55732},[355,8285],[28,55734,24404],{"className":55735},[274],[28,55737,176],{"className":55738},[274,275],[28,55740],{"className":55741,"style":280},[279],[28,55743,5074],{"className":55744},[284],[28,55746],{"className":55747,"style":280},[279],[28,55749,55751,55754,55757],{"className":55750},[265],[28,55752],{"className":55753,"style":31837},[269],[28,55755,180],{"className":55756,"style":298},[274,275],[28,55758,55760,55763],{"className":55759},[274],[28,55761,24404],{"className":55762},[274],[28,55764,55766],{"className":55765},[556],[28,55767,55769],{"className":55768},[560],[28,55770,55772],{"className":55771},[565],[28,55773,55775],{"className":55774,"style":5477},[569],[28,55776,55777,55780],{"style":3094},[28,55778],{"className":55779,"style":578},[577],[28,55781,55783],{"className":55782},[582,583,584,585],[28,55784,1253],{"className":55785},[274,585],[28,55787,55789,55833],{"className":55788,"translate":193},[197],[28,55790,55792],{"className":55791},[201],[203,55793,55794],{"xmlns":205},[208,55795,55796,55830],{},[211,55797,55798,55800,55802,55804,55806,55808,55814,55820,55822,55824],{},[214,55799,1511],{},[217,55801,43],{"stretchy":241},[214,55803,11],{},[217,55805,1522],{"stretchy":241},[217,55807,83],{},[501,55809,55810,55812],{},[217,55811,25442],{},[214,55813,1686],{},[501,55815,55816,55818],{},[214,55817,11],{},[214,55819,1686],{},[214,55821,39714],{},[217,55823,21407],{},[501,55825,55826,55828],{},[214,55827,11],{},[214,55829,1686],{},[253,55831,55832],{"encoding":255},"f(p)=\\sum_ip_i\\log p_i",[28,55834,55836,55863],{"className":55835,"ariaHidden":261},[260],[28,55837,55839,55842,55845,55848,55851,55854,55857,55860],{"className":55838},[265],[28,55840],{"className":55841,"style":341},[269],[28,55843,1511],{"className":55844,"style":1546},[274,275],[28,55846,43],{"className":55847},[348],[28,55849,11],{"className":55850},[274,275],[28,55852,1522],{"className":55853},[355],[28,55855],{"className":55856,"style":302},[279],[28,55858,83],{"className":55859},[306],[28,55861],{"className":55862,"style":302},[279],[28,55864,55866,55869,55909,55912,55952,55955,55960,55963],{"className":55865},[265],[28,55867],{"className":55868,"style":41128},[269],[28,55870,55872,55875],{"className":55871},[21329],[28,55873,25442],{"className":55874,"style":41141},[21329,21330,21331],[28,55876,55878],{"className":55877},[556],[28,55879,55881,55901],{"className":55880},[560,561],[28,55882,55884,55898],{"className":55883},[565],[28,55885,55887],{"className":55886,"style":41154},[569],[28,55888,55889,55892],{"style":41157},[28,55890],{"className":55891,"style":578},[577],[28,55893,55895],{"className":55894},[582,583,584,585],[28,55896,1686],{"className":55897},[274,275,585],[28,55899,593],{"className":55900},[592],[28,55902,55904],{"className":55903},[565],[28,55905,55907],{"className":55906,"style":41176},[569],[28,55908],{},[28,55910],{"className":55911,"style":530},[279],[28,55913,55915,55918],{"className":55914},[274],[28,55916,11],{"className":55917},[274,275],[28,55919,55921],{"className":55920},[556],[28,55922,55924,55944],{"className":55923},[560,561],[28,55925,55927,55941],{"className":55926},[565],[28,55928,55930],{"className":55929,"style":3881},[569],[28,55931,55932,55935],{"style":573},[28,55933],{"className":55934,"style":578},[577],[28,55936,55938],{"className":55937},[582,583,584,585],[28,55939,1686],{"className":55940},[274,275,585],[28,55942,593],{"className":55943},[592],[28,55945,55947],{"className":55946},[565],[28,55948,55950],{"className":55949,"style":600},[569],[28,55951],{},[28,55953],{"className":55954,"style":530},[279],[28,55956,39952,55958],{"className":55957},[21329],[28,55959,34347],{"style":39955},[28,55961],{"className":55962,"style":530},[279],[28,55964,55966,55969],{"className":55965},[274],[28,55967,11],{"className":55968},[274,275],[28,55970,55972],{"className":55971},[556],[28,55973,55975,55995],{"className":55974},[560,561],[28,55976,55978,55992],{"className":55977},[565],[28,55979,55981],{"className":55980,"style":3881},[569],[28,55982,55983,55986],{"style":573},[28,55984],{"className":55985,"style":578},[577],[28,55987,55989],{"className":55988},[582,583,584,585],[28,55990,1686],{"className":55991},[274,275,585],[28,55993,593],{"className":55994},[592],[28,55996,55998],{"className":55997},[565],[28,55999,56001],{"className":56000,"style":600},[569],[28,56002],{},"（负熵）给 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给 Itakura–Saito；",[28,56243,56245,56285],{"className":56244,"translate":193},[197],[28,56246,56248],{"className":56247},[201],[203,56249,56250],{"xmlns":205},[208,56251,56252,56282],{},[211,56253,56254,56256,56258,56260,56262,56264,56272,56278,56280],{},[214,56255,1511],{},[217,56257,43],{"stretchy":241},[214,56259,176],{},[217,56261,1522],{"stretchy":241},[217,56263,83],{},[5027,56265,56266],{"scriptlevel":89,"displaystyle":241},[21928,56267,56268,56270],{},[224,56269,1244],{},[224,56271,1253],{},[3054,56273,56274,56276],{},[214,56275,176],{},[214,56277,884],{"mathvariant":484},[214,56279,903],{},[214,56281,176],{},[253,56283,56284],{"encoding":255},"f(x)=\\tfrac12x^\\top 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给 Mahalanobis 距离。",[11,56427,56428],{},"它在机器学习里出现的频率比多数人以为的高。交叉熵",[28,56430,56432],{"className":56431,"translate":193},[192],[28,56433,56435,56552],{"className":56434,"translate":193},[197],[28,56436,56438],{"className":56437},[201],[203,56439,56440],{"xmlns":205,"display":206},[208,56441,56442,56549],{},[211,56443,56444,56450,56452,56454,56456,56458,56460,56462,56464,56470,56476,56478,56480,56486,56488,56494,56496,56498,56500,56502,56504,56506,56509,56511,56513,56515,56517,56527,56529,56531,56533,56535,56537,56539,56541,56543,56545,56547],{},[211,56445,56446,56448],{},[214,56447,15828],{"mathvariant":484},[214,56449,42084],{"mathvariant":484},[217,56451,43],{"stretchy":241},[214,56453,180],{},[217,56455,1683],{"separator":261},[214,56457,1829],{},[217,56459,1522],{"stretchy":241},[217,56461,83],{},[217,56463,5074],{},[1771,56465,56466,56468],{},[217,56467,25442],{},[214,56469,1686],{},[501,56471,56472,56474],{},[214,56473,180],{},[214,56475,1686],{},[214,56477,39714],{},[217,56479,21407],{},[501,56481,56482,56484],{},[214,56483,1829],{},[214,56485,1686],{},[217,56487,83],{},[211,56489,56490,56492],{},[214,56491,24390],{"mathvariant":484},[214,56493,24393],{"mathvariant":484},[217,56495,43],{"stretchy":241},[214,56497,180],{},[214,56499,24404],{"mathvariant":484},[214,56501,1829],{},[217,56503,1522],{"stretchy":241},[217,56505,124],{},[214,56507,56508],{},"H",[217,56510,43],{"stretchy":241},[214,56512,180],{},[217,56514,1522],{"stretchy":241},[217,56516,83],{},[501,56518,56519,56521],{},[214,56520,54364],{},[211,56522,56523,56525],{},[217,56524,5074],{},[214,56526,56508],{},[217,56528,43],{"stretchy":241},[214,56530,180],{},[214,56532,24404],{"mathvariant":484},[214,56534,1829],{},[217,56536,1522],{"stretchy":241},[217,56538,124],{},[214,56540,56508],{},[217,56542,43],{"stretchy":241},[214,56544,180],{},[217,56546,1522],{"stretchy":241},[217,56548,1683],{"separator":261},[253,56550,56551],{"encoding":255},"\\mathrm{CE}(y,q)=-\\sum_iy_i\\log q_i=\\mathrm{KL}(y\\|q)+H(y)=D_{-H}(y\\|q)+H(y),",[28,56553,56555,56595,56754,56790,56818,56894],{"className":56554,"ariaHidden":261},[260],[28,56556,56558,56561,56568,56571,56574,56577,56580,56583,56586,56589,56592],{"className":56557},[265],[28,56559],{"className":56560,"style":341},[269],[28,56562,56564],{"className":56563},[274],[28,56565,56567],{"className":56566},[274,1712],"CE",[28,56569,43],{"className":56570},[348],[28,56572,180],{"className":56573,"style":298},[274,275],[28,56575,1683],{"className":56576},[1723],[28,56578],{"className":56579,"style":530},[279],[28,56581,1829],{"className":56582,"style":298},[274,275],[28,56584,1522],{"className":56585},[355],[28,56587],{"className":56588,"style":302},[279],[28,56590,83],{"className":56591},[306],[28,56593],{"className":56594,"style":302},[279],[28,56596,56598,56601,56604,56607,56651,56654,56694,56697,56702,56705,56745,56748,56751],{"className":56597},[265],[28,56599],{"className":56600,"style":30055},[269],[28,56602,5074],{"className":56603},[274],[28,56605],{"className":56606,"style":530},[279],[28,56608,56610],{"className":56609},[21329,21485],[28,56611,56613,56643],{"className":56612},[560,561],[28,56614,56616,56640],{"className":56615},[565],[28,56617,56619,56630],{"className":56618,"style":25713},[569],[28,56620,56621,56624],{"style":25716},[28,56622],{"className":56623,"style":25720},[577],[28,56625,56627],{"className":56626},[582,583,584,585],[28,56628,1686],{"className":56629},[274,275,585],[28,56631,56632,56635],{"style":25732},[28,56633],{"className":56634,"style":25720},[577],[28,56636,56637],{},[28,56638,25442],{"className":56639},[21329,21330,21547],[28,56641,593],{"className":56642},[592],[28,56644,56646],{"className":56645},[565],[28,56647,56649],{"className":56648,"style":26154},[569],[28,56650],{},[28,56652],{"className":56653,"style":530},[279],[28,56655,56657,56660],{"className":56656},[274],[28,56658,180],{"className":56659,"style":298},[274,275],[28,56661,56663],{"className":56662},[556],[28,56664,56666,56686],{"className":56665},[560,561],[28,56667,56669,56683],{"className":56668},[565],[28,56670,56672],{"className":56671,"style":3881},[569],[28,56673,56674,56677],{"style":3660},[28,56675],{"className":56676,"style":578},[577],[28,56678,56680],{"className":56679},[582,583,584,585],[28,56681,1686],{"className":56682},[274,275,585],[28,56684,593],{"className":56685},[592],[28,56687,56689],{"className":56688},[565],[28,56690,56692],{"className":56691,"style":600},[569],[28,56693],{},[28,56695],{"className":56696,"style":530},[279],[28,56698,39952,56700],{"className":56699},[21329],[28,56701,34347],{"style":39955},[28,56703],{"className":56704,"style":530},[279],[28,56706,56708,56711],{"className":56707},[274],[28,56709,1829],{"className":56710,"style":298},[274,275],[28,56712,56714],{"className":56713},[556],[28,56715,56717,56737],{"className":56716},[560,561],[28,56718,56720,56734],{"className":56719},[565],[28,56721,56723],{"className":56722,"style":3881},[569],[28,56724,56725,56728],{"style":3660},[28,56726],{"className":56727,"style":578},[577],[28,56729,56731],{"className":56730},[582,583,584,585],[28,56732,1686],{"className":56733},[274,275,585],[28,56735,593],{"className":56736},[592],[28,56738,56740],{"className":56739},[565],[28,56741,56743],{"className":56742,"style":600},[569],[28,56744],{},[28,56746],{"className":56747,"style":302},[279],[28,56749,83],{"className":56750},[306],[28,56752],{"className":56753,"style":302},[279],[28,56755,56757,56760,56766,56769,56772,56775,56778,56781,56784,56787],{"className":56756},[265],[28,56758],{"className":56759,"style":341},[269],[28,56761,56763],{"className":56762},[274],[28,56764,24438],{"className":56765},[274,1712],[28,56767,43],{"className":56768},[348],[28,56770,180],{"className":56771,"style":298},[274,275],[28,56773,24404],{"className":56774},[274],[28,56776,1829],{"className":56777,"style":298},[274,275],[28,56779,1522],{"className":56780},[355],[28,56782],{"className":56783,"style":280},[279],[28,56785,124],{"className":56786},[284],[28,56788],{"className":56789,"style":280},[279],[28,56791,56793,56796,56800,56803,56806,56809,56812,56815],{"className":56792},[265],[28,56794],{"className":56795,"style":341},[269],[28,56797,56508],{"className":56798,"style":56799},[274,275],"margin-right:0.0813em;",[28,56801,43],{"className":56802},[348],[28,56804,180],{"className":56805,"style":298},[274,275],[28,56807,1522],{"className":56808},[355],[28,56810],{"className":56811,"style":302},[279],[28,56813,83],{"className":56814},[306],[28,56816],{"className":56817,"style":302},[279],[28,56819,56821,56824,56870,56873,56876,56879,56882,56885,56888,56891],{"className":56820},[265],[28,56822],{"className":56823,"style":341},[269],[28,56825,56827,56830],{"className":56826},[274],[28,56828,54364],{"className":56829,"style":2279},[274,275],[28,56831,56833],{"className":56832},[556],[28,56834,56836,56862],{"className":56835},[560,561],[28,56837,56839,56859],{"className":56838},[565],[28,56840,56842],{"className":56841,"style":570},[569],[28,56843,56844,56847],{"style":46480},[28,56845],{"className":56846,"style":578},[577],[28,56848,56850],{"className":56849},[582,583,584,585],[28,56851,56853,56856],{"className":56852},[274,585],[28,56854,5074],{"className":56855},[274,585],[28,56857,56508],{"className":56858,"style":56799},[274,275,585],[28,56860,593],{"className":56861},[592],[28,56863,56865],{"className":56864},[565],[28,56866,56868],{"className":56867,"style":44839},[569],[28,56869],{},[28,56871,43],{"className":56872},[348],[28,56874,180],{"className":56875,"style":298},[274,275],[28,56877,24404],{"className":56878},[274],[28,56880,1829],{"className":56881,"style":298},[274,275],[28,56883,1522],{"className":56884},[355],[28,56886],{"className":56887,"style":280},[279],[28,56889,124],{"className":56890},[284],[28,56892],{"className":56893,"style":280},[279],[28,56895,56897,56900,56903,56906,56909,56912],{"className":56896},[265],[28,56898],{"className":56899,"style":341},[269],[28,56901,56508],{"className":56902,"style":56799},[274,275],[28,56904,43],{"className":56905},[348],[28,56907,180],{"className":56908,"style":298},[274,275],[28,56910,1522],{"className":56911},[355],[28,56913,1683],{"className":56914},[1723],[11,56916,56917,56918,56962,56963,2923],{},"所以训练一个分类器、一个语言模型，最小化交叉熵就是在最小化一个 Bregman 散度；标签固定时 ",[28,56919,56921,56941],{"className":56920,"translate":193},[197],[28,56922,56924],{"className":56923},[201],[203,56925,56926],{"xmlns":205},[208,56927,56928,56938],{},[211,56929,56930,56932,56934,56936],{},[214,56931,56508],{},[217,56933,43],{"stretchy":241},[214,56935,180],{},[217,56937,1522],{"stretchy":241},[253,56939,56940],{"encoding":255},"H(y)",[28,56942,56944],{"className":56943,"ariaHidden":261},[260],[28,56945,56947,56950,56953,56956,56959],{"className":56946},[265],[28,56948],{"className":56949,"style":341},[269],[28,56951,56508],{"className":56952,"style":56799},[274,275],[28,56954,43],{"className":56955},[348],[28,56957,180],{"className":56958,"style":298},[274,275],[28,56960,1522],{"className":56961},[355]," 是常数，梯度完全来自 ",[28,56964,56966,56979],{"className":56965,"translate":193},[197],[28,56967,56969],{"className":56968},[201],[203,56970,56971],{"xmlns":205},[208,56972,56973,56977],{},[211,56974,56975],{},[214,56976,54364],{},[253,56978,54364],{"encoding":255},[28,56980,56982],{"className":56981,"ariaHidden":261},[260],[28,56983,56985,56988],{"className":56984},[265],[28,56986],{"className":56987,"style":455},[269],[28,56989,54364],{"className":56990,"style":2279},[274,275],[11,56992,56993],{},"Bregman 散度还有一个我很喜欢的性质：它的质心是算术平均。对任意 Bregman 散度，",[28,56995,56997],{"className":56996,"translate":193},[192],[28,56998,57000,57077],{"className":56999,"translate":193},[197],[28,57001,57003],{"className":57002},[201],[203,57004,57005],{"xmlns":205,"display":206},[208,57006,57007,57074],{},[211,57008,57009,57012,57014,57024,57026,57032,57038,57040,57046,57048,57050,57052,57054,57060,57066,57072],{},[214,57010,57011],{},"arg",[217,57013,21407],{},[1771,57015,57016,57022],{},[211,57017,57018,57020],{},[214,57019,39653],{},[217,57021,21407],{},[214,57023,22],{},[228,57025,14860],{},[1771,57027,57028,57030],{},[217,57029,25442],{},[214,57031,1686],{},[501,57033,57034,57036],{},[214,57035,54364],{},[214,57037,1511],{},[217,57039,43],{"stretchy":241},[501,57041,57042,57044],{},[214,57043,176],{},[214,57045,1686],{},[214,57047,24404],{"mathvariant":484},[214,57049,22],{},[217,57051,1522],{"stretchy":241},[217,57053,83],{},[21928,57055,57056,57058],{},[224,57057,1244],{},[214,57059,3060],{},[1771,57061,57062,57064],{},[217,57063,25442],{},[214,57065,1686],{},[501,57067,57068,57070],{},[214,57069,176],{},[214,57071,1686],{},[214,57073,494],{"mathvariant":484},[253,57075,57076],{"encoding":255},"\\arg\\min_c\\ \\sum_iD_f(x_i\\|c)=\\frac1n\\sum_ix_i .",[28,57078,57080,57295],{"className":57079,"ariaHidden":261},[260],[28,57081,57083,57086,57092,57095,57141,57144,57147,57191,57194,57234,57237,57277,57280,57283,57286,57289,57292],{"className":57082},[265],[28,57084],{"className":57085,"style":30055},[269],[28,57087,57089,57090],{"className":57088},[21329],"ar",[28,57091,34347],{"style":39955},[28,57093],{"className":57094,"style":530},[279],[28,57096,57098],{"className":57097},[21329,21485],[28,57099,57101,57132],{"className":57100},[560,561],[28,57102,57104,57129],{"className":57103},[565],[28,57105,57107,57119],{"className":57106,"style":39754},[569],[28,57108,57110,57113],{"style":57109},"top:-2.4em;margin-left:0em;",[28,57111],{"className":57112,"style":1914},[577],[28,57114,57116],{"className":57115},[582,583,584,585],[28,57117,22],{"className":57118},[274,275,585],[28,57120,57121,57124],{"style":1926},[28,57122],{"className":57123,"style":1914},[577],[28,57125,57126],{},[28,57127,39653],{"className":57128},[21329],[28,57130,593],{"className":57131},[592],[28,57133,57135],{"className":57134},[565],[28,57136,57139],{"className":57137,"style":57138},[569],"height:0.7em;",[28,57140],{},[28,57142,14860],{"className":57143},[279],[28,57145],{"className":57146,"style":530},[279],[28,57148,57150],{"className":57149},[21329,21485],[28,57151,57153,57183],{"className":57152},[560,561],[28,57154,57156,57180],{"className":57155},[565],[28,57157,57159,57170],{"className":57158,"style":25713},[569],[28,57160,57161,57164],{"style":25716},[28,57162],{"className":57163,"style":25720},[577],[28,57165,57167],{"className":57166},[582,583,584,585],[28,57168,1686],{"className":57169},[274,275,585],[28,57171,57172,57175],{"style":25732},[28,57173],{"className":57174,"style":25720},[577],[28,57176,57177],{},[28,57178,25442],{"className":57179},[21329,21330,21547],[28,57181,593],{"className":57182},[592],[28,57184,57186],{"className":57185},[565],[28,57187,57189],{"className":57188,"style":26154},[569],[28,57190],{},[28,57192],{"className":57193,"style":530},[279],[28,57195,57197,57200],{"className":57196},[274],[28,57198,54364],{"className":57199,"style":2279},[274,275],[28,57201,57203],{"className":57202},[556],[28,57204,57206,57226],{"className":57205},[560,561],[28,57207,57209,57223],{"className":57208},[565],[28,57210,57212],{"className":57211,"style":20945},[569],[28,57213,57214,57217],{"style":46480},[28,57215],{"className":57216,"style":578},[577],[28,57218,57220],{"className":57219},[582,583,584,585],[28,57221,1511],{"className":57222,"style":1546},[274,275,585],[28,57224,593],{"className":57225},[592],[28,57227,57229],{"className":57228},[565],[28,57230,57232],{"className":57231,"style":20967},[569],[28,57233],{},[28,57235,43],{"className":57236},[348],[28,57238,57240,57243],{"className":57239},[274],[28,57241,176],{"className":57242},[274,275],[28,57244,57246],{"className":57245},[556],[28,57247,57249,57269],{"className":57248},[560,561],[28,57250,57252,57266],{"className":57251},[565],[28,57253,57255],{"className":57254,"style":3881},[569],[28,57256,57257,57260],{"style":573},[28,57258],{"className":57259,"style":578},[577],[28,57261,57263],{"className":57262},[582,583,584,585],[28,57264,1686],{"className":57265},[274,275,585],[28,57267,593],{"className":57268},[592],[28,57270,57272],{"className":57271},[565],[28,57273,57275],{"className":57274,"style":600},[569],[28,57276],{},[28,57278,24404],{"className":57279},[274],[28,57281,22],{"className":57282},[274,275],[28,57284,1522],{"className":57285},[355],[28,57287],{"className":57288,"style":302},[279],[28,57290,83],{"className":57291},[306],[28,57293],{"className":57294,"style":302},[279],[28,57296,57298,57302,57364,57367,57411,57414,57454],{"className":57297},[265],[28,57299],{"className":57300,"style":57301},[269],"height:2.5991em;vertical-align:-1.2777em;",[28,57303,57305,57308,57361],{"className":57304},[274],[28,57306],{"className":57307},[348,8285],[28,57309,57311],{"className":57310},[21928],[28,57312,57314,57353],{"className":57313},[560,561],[28,57315,57317,57350],{"className":57316},[565],[28,57318,57320,57331,57339],{"className":57319,"style":46133},[569],[28,57321,57322,57325],{"style":24920},[28,57323],{"className":57324,"style":1914},[577],[28,57326,57328],{"className":57327},[274],[28,57329,3060],{"className":57330},[274,275],[28,57332,57333,57336],{"style":22023},[28,57334],{"className":57335,"style":1914},[577],[28,57337],{"className":57338,"style":22031},[22030],[28,57340,57341,57344],{"style":24940},[28,57342],{"className":57343,"style":1914},[577],[28,57345,57347],{"className":57346},[274],[28,57348,1244],{"className":57349},[274],[28,57351,593],{"className":57352},[592],[28,57354,57356],{"className":57355},[565],[28,57357,57359],{"className":57358,"style":24985},[569],[28,57360],{},[28,57362],{"className":57363},[355,8285],[28,57365],{"className":57366,"style":530},[279],[28,57368,57370],{"className":57369},[21329,21485],[28,57371,57373,57403],{"className":57372},[560,561],[28,57374,57376,57400],{"className":57375},[565],[28,57377,57379,57390],{"className":57378,"style":25713},[569],[28,57380,57381,57384],{"style":25716},[28,57382],{"className":57383,"style":25720},[577],[28,57385,57387],{"className":57386},[582,583,584,585],[28,57388,1686],{"className":57389},[274,275,585],[28,57391,57392,57395],{"style":25732},[28,57393],{"className":57394,"style":25720},[577],[28,57396,57397],{},[28,57398,25442],{"className":57399},[21329,21330,21547],[28,57401,593],{"className":57402},[592],[28,57404,57406],{"className":57405},[565],[28,57407,57409],{"className":57408,"style":26154},[569],[28,57410],{},[28,57412],{"className":57413,"style":530},[279],[28,57415,57417,57420],{"className":57416},[274],[28,57418,176],{"className":57419},[274,275],[28,57421,57423],{"className":57422},[556],[28,57424,57426,57446],{"className":57425},[560,561],[28,57427,57429,57443],{"className":57428},[565],[28,57430,57432],{"className":57431,"style":3881},[569],[28,57433,57434,57437],{"style":573},[28,57435],{"className":57436,"style":578},[577],[28,57438,57440],{"className":57439},[582,583,584,585],[28,57441,1686],{"className":57442},[274,275,585],[28,57444,593],{"className":57445},[592],[28,57447,57449],{"className":57448},[565],[28,57450,57452],{"className":57451,"style":600},[569],[28,57453],{},[28,57455,494],{"className":57456},[274],[11,57458,57459,57460,57689],{},"求导得到 ",[28,57461,57463,57513],{"className":57462,"translate":193},[197],[28,57464,57466],{"className":57465},[201],[203,57467,57468],{"xmlns":205},[208,57469,57470,57510],{},[211,57471,57472,57478,57484,57486,57488,57490,57492,57494,57500,57502,57504,57506,57508],{},[501,57473,57474,57476],{},[217,57475,25442],{},[214,57477,1686],{},[3054,57479,57480,57482],{},[214,57481,31947],{"mathvariant":484},[224,57483,1253],{},[214,57485,1511],{},[217,57487,43],{"stretchy":241},[214,57489,22],{},[217,57491,1522],{"stretchy":241},[217,57493,43],{"stretchy":241},[501,57495,57496,57498],{},[214,57497,176],{},[214,57499,1686],{},[217,57501,5074],{},[214,57503,22],{},[217,57505,1522],{"stretchy":241},[217,57507,83],{},[224,57509,89],{},[253,57511,57512],{"encoding":255},"\\sum_i\\nabla^2f(c)(x_i-c)=0",[28,57514,57516,57659,57680],{"className":57515,"ariaHidden":261},[260],[28,57517,57519,57523,57563,57566,57595,57598,57601,57604,57607,57610,57650,57653,57656],{"className":57518},[265],[28,57520],{"className":57521,"style":57522},[269],"height:1.1138em;vertical-align:-0.2997em;",[28,57524,57526,57529],{"className":57525},[21329],[28,57527,25442],{"className":57528,"style":41141},[21329,21330,21331],[28,57530,57532],{"className":57531},[556],[28,57533,57535,57555],{"className":57534},[560,561],[28,57536,57538,57552],{"className":57537},[565],[28,57539,57541],{"className":57540,"style":41154},[569],[28,57542,57543,57546],{"style":41157},[28,57544],{"className":57545,"style":578},[577],[28,57547,57549],{"className":57548},[582,583,584,585],[28,57550,1686],{"className":57551},[274,275,585],[28,57553,593],{"className":57554},[592],[28,57556,57558],{"className":57557},[565],[28,57559,57561],{"className":57560,"style":41176},[569],[28,57562],{},[28,57564],{"className":57565,"style":530},[279],[28,57567,57569,57572],{"className":57568},[274],[28,57570,31947],{"className":57571},[274],[28,57573,57575],{"className":57574},[556],[28,57576,57578],{"className":57577},[560],[28,57579,57581],{"className":57580},[565],[28,57582,57584],{"className":57583,"style":5477},[569],[28,57585,57586,57589],{"style":3094},[28,57587],{"className":57588,"style":578},[577],[28,57590,57592],{"className":57591},[582,583,584,585],[28,57593,1253],{"className":57594},[274,585],[28,57596,1511],{"className":57597,"style":1546},[274,275],[28,57599,43],{"className":57600},[348],[28,57602,22],{"className":57603},[274,275],[28,57605,1522],{"className":57606},[355],[28,57608,43],{"className":57609},[348],[28,57611,57613,57616],{"className":57612},[274],[28,57614,176],{"className":57615},[274,275],[28,57617,57619],{"className":57618},[556],[28,57620,57622,57642],{"className":57621},[560,561],[28,57623,57625,57639],{"className":57624},[565],[28,57626,57628],{"className":57627,"style":3881},[569],[28,57629,57630,57633],{"style":573},[28,57631],{"className":57632,"style":578},[577],[28,57634,57636],{"className":57635},[582,583,584,585],[28,57637,1686],{"className":57638},[274,275,585],[28,57640,593],{"className":57641},[592],[28,57643,57645],{"className":57644},[565],[28,57646,57648],{"className":57647,"style":600},[569],[28,57649],{},[28,57651],{"className":57652,"style":280},[279],[28,57654,5074],{"className":57655},[284],[28,57657],{"className":57658,"style":280},[279],[28,57660,57662,57665,57668,57671,57674,57677],{"className":57661},[265],[28,57663],{"className":57664,"style":341},[269],[28,57666,22],{"className":57667},[274,275],[28,57669,1522],{"className":57670},[355],[28,57672],{"className":57673,"style":302},[279],[28,57675,83],{"className":57676},[306],[28,57678],{"className":57679,"style":302},[279],[28,57681,57683,57686],{"className":57682},[265],[28,57684],{"className":57685,"style":316},[269],[28,57687,89],{"className":57688},[274],"，结论就是这一行。所以换散度不会改变“平均”这件事，但会改变“谁离谁近”这件事。",[11,57691,57692,57693,57817,57818,37951,57954,58090],{},"这一点值得展开，因为它关系到分类边界。设两个类原型 ",[28,57694,57696,57722],{"className":57695,"translate":193},[197],[28,57697,57699],{"className":57698},[201],[203,57700,57701],{"xmlns":205},[208,57702,57703,57719],{},[211,57704,57705,57711,57713],{},[501,57706,57707,57709],{},[214,57708,22],{},[224,57710,1244],{},[217,57712,1683],{"separator":261},[501,57714,57715,57717],{},[214,57716,22],{},[224,57718,1253],{},[253,57720,57721],{"encoding":255},"c_1,c_2",[28,57723,57725],{"className":57724,"ariaHidden":261},[260],[28,57726,57728,57731,57771,57774,57777],{"className":57727},[265],[28,57729],{"className":57730,"style":612},[269],[28,57732,57734,57737],{"className":57733},[274],[28,57735,22],{"className":57736},[274,275],[28,57738,57740],{"className":57739},[556],[28,57741,57743,57763],{"className":57742},[560,561],[28,57744,57746,57760],{"className":57745},[565],[28,57747,57749],{"className":57748,"style":1285},[569],[28,57750,57751,57754],{"style":573},[28,57752],{"className":57753,"style":578},[577],[28,57755,57757],{"className":57756},[582,583,584,585],[28,57758,1244],{"className":57759},[274,585],[28,57761,593],{"className":57762},[592],[28,57764,57766],{"className":57765},[565],[28,57767,57769],{"className":57768,"style":600},[569],[28,57770],{},[28,57772,1683],{"className":57773},[1723],[28,57775],{"className":57776,"style":530},[279],[28,57778,57780,57783],{"className":57779},[274],[28,57781,22],{"className":57782},[274,275],[28,57784,57786],{"className":57785},[556],[28,57787,57789,57809],{"className":57788},[560,561],[28,57790,57792,57806],{"className":57791},[565],[28,57793,57795],{"className":57794,"style":1285},[569],[28,57796,57797,57800],{"style":573},[28,57798],{"className":57799,"style":578},[577],[28,57801,57803],{"className":57802},[582,583,584,585],[28,57804,1253],{"className":57805},[274,585],[28,57807,593],{"className":57808},[592],[28,57810,57812],{"className":57811},[565],[28,57813,57815],{"className":57814,"style":600},[569],[28,57816],{},"，最近原型判类就是比较 ",[28,57819,57821,57853],{"className":57820,"translate":193},[197],[28,57822,57824],{"className":57823},[201],[203,57825,57826],{"xmlns":205},[208,57827,57828,57850],{},[211,57829,57830,57836,57838,57840,57842,57848],{},[501,57831,57832,57834],{},[214,57833,54364],{},[214,57835,1511],{},[217,57837,43],{"stretchy":241},[214,57839,176],{},[214,57841,24404],{"mathvariant":484},[501,57843,57844,57846],{},[214,57845,22],{},[224,57847,1244],{},[217,57849,1522],{"stretchy":241},[253,57851,57852],{"encoding":255},"D_f(x\\|c_1)",[28,57854,57856],{"className":57855,"ariaHidden":261},[260],[28,57857,57859,57862,57902,57905,57908,57911,57951],{"className":57858},[265],[28,57860],{"className":57861,"style":21101},[269],[28,57863,57865,57868],{"className":57864},[274],[28,57866,54364],{"className":57867,"style":2279},[274,275],[28,57869,57871],{"className":57870},[556],[28,57872,57874,57894],{"className":57873},[560,561],[28,57875,57877,57891],{"className":57876},[565],[28,57878,57880],{"className":57879,"style":20945},[569],[28,57881,57882,57885],{"style":46480},[28,57883],{"className":57884,"style":578},[577],[28,57886,57888],{"className":57887},[582,583,584,585],[28,57889,1511],{"className":57890,"style":1546},[274,275,585],[28,57892,593],{"className":57893},[592],[28,57895,57897],{"className":57896},[565],[28,57898,57900],{"className":57899,"style":20967},[569],[28,57901],{},[28,57903,43],{"className":57904},[348],[28,57906,176],{"className":57907},[274,275],[28,57909,24404],{"className":57910},[274],[28,57912,57914,57917],{"className":57913},[274],[28,57915,22],{"className":57916},[274,275],[28,57918,57920],{"className":57919},[556],[28,57921,57923,57943],{"className":57922},[560,561],[28,57924,57926,57940],{"className":57925},[565],[28,57927,57929],{"className":57928,"style":1285},[569],[28,57930,57931,57934],{"style":573},[28,57932],{"className":57933,"style":578},[577],[28,57935,57937],{"className":57936},[582,583,584,585],[28,57938,1244],{"className":57939},[274,585],[28,57941,593],{"className":57942},[592],[28,57944,57946],{"className":57945},[565],[28,57947,57949],{"className":57948,"style":600},[569],[28,57950],{},[28,57952,1522],{"className":57953},[355],[28,57955,57957,57989],{"className":57956,"translate":193},[197],[28,57958,57960],{"className":57959},[201],[203,57961,57962],{"xmlns":205},[208,57963,57964,57986],{},[211,57965,57966,57972,57974,57976,57978,57984],{},[501,57967,57968,57970],{},[214,57969,54364],{},[214,57971,1511],{},[217,57973,43],{"stretchy":241},[214,57975,176],{},[214,57977,24404],{"mathvariant":484},[501,57979,57980,57982],{},[214,57981,22],{},[224,57983,1253],{},[217,57985,1522],{"stretchy":241},[253,57987,57988],{"encoding":255},"D_f(x\\|c_2)",[28,57990,57992],{"className":57991,"ariaHidden":261},[260],[28,57993,57995,57998,58038,58041,58044,58047,58087],{"className":57994},[265],[28,57996],{"className":57997,"style":21101},[269],[28,57999,58001,58004],{"className":58000},[274],[28,58002,54364],{"className":58003,"style":2279},[274,275],[28,58005,58007],{"className":58006},[556],[28,58008,58010,58030],{"className":58009},[560,561],[28,58011,58013,58027],{"className":58012},[565],[28,58014,58016],{"className":58015,"style":20945},[569],[28,58017,58018,58021],{"style":46480},[28,58019],{"className":58020,"style":578},[577],[28,58022,58024],{"className":58023},[582,583,584,585],[28,58025,1511],{"className":58026,"style":1546},[274,275,585],[28,58028,593],{"className":58029},[592],[28,58031,58033],{"className":58032},[565],[28,58034,58036],{"className":58035,"style":20967},[569],[28,58037],{},[28,58039,43],{"className":58040},[348],[28,58042,176],{"className":58043},[274,275],[28,58045,24404],{"className":58046},[274],[28,58048,58050,58053],{"className":58049},[274],[28,58051,22],{"className":58052},[274,275],[28,58054,58056],{"className":58055},[556],[28,58057,58059,58079],{"className":58058},[560,561],[28,58060,58062,58076],{"className":58061},[565],[28,58063,58065],{"className":58064,"style":1285},[569],[28,58066,58067,58070],{"style":573},[28,58068],{"className":58069,"style":578},[577],[28,58071,58073],{"className":58072},[582,583,584,585],[28,58074,1253],{"className":58075},[274,585],[28,58077,593],{"className":58078},[592],[28,58080,58082],{"className":58081},[565],[28,58083,58085],{"className":58084,"style":600},[569],[28,58086],{},[28,58088,1522],{"className":58089},[355],"。把定义代进去，",[28,58092,58094],{"className":58093,"translate":193},[192],[28,58095,58097,58460],{"className":58096,"translate":193},[197],[28,58098,58100],{"className":58099},[201],[203,58101,58102],{"xmlns":205,"display":206},[208,58103,58104,58457],{},[5015,58105,58106,58164,58304],{"rowspacing":5017,"columnalign":5018,"columnspacing":5019},[5021,58107,58108,58114],{},[5024,58109,58110],{},[5027,58111,58112],{"scriptlevel":89,"displaystyle":261},[211,58113],{},[5024,58115,58116],{},[5027,58117,58118],{"scriptlevel":89,"displaystyle":261},[211,58119,58120,58122,58128,58130,58132,58134,58140,58142,58144,58150,58152,58154,58156,58162],{},[211,58121],{},[501,58123,58124,58126],{},[214,58125,54364],{},[214,58127,1511],{},[217,58129,43],{"stretchy":241},[214,58131,176],{},[214,58133,24404],{"mathvariant":484},[501,58135,58136,58138],{},[214,58137,22],{},[224,58139,1244],{},[217,58141,1522],{"stretchy":241},[217,58143,5074],{},[501,58145,58146,58148],{},[214,58147,54364],{},[214,58149,1511],{},[217,58151,43],{"stretchy":241},[214,58153,176],{},[214,58155,24404],{"mathvariant":484},[501,58157,58158,58160],{},[214,58159,22],{},[224,58161,1253],{},[217,58163,1522],{"stretchy":241},[5021,58165,58166,58172],{},[5024,58167,58168],{},[5027,58169,58170],{"scriptlevel":89,"displaystyle":261},[211,58171],{},[5024,58173,58174],{},[5027,58175,58176],{"scriptlevel":89,"displaystyle":261},[211,58177,58178,58180,58182,58184,58186,58188,58190,58192,58194,58196,58198,58204,58206,58208,58210,58212,58214,58220,58226,58228,58230,58232,58238,58240,58242,58244,58246,58248,58250,58252,58254,58256,58258,58260,58266,58268,58270,58272,58274,58276,58282,58288,58290,58292,58294,58300,58302],{},[211,58179],{},[217,58181,83],{},[217,58183,73],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,58185,1511],{},[217,58187,43],{"stretchy":241},[214,58189,176],{},[217,58191,1522],{"stretchy":241},[217,58193,5074],{},[214,58195,1511],{},[217,58197,43],{"stretchy":241},[501,58199,58200,58202],{},[214,58201,22],{},[224,58203,1244],{},[217,58205,1522],{"stretchy":241},[217,58207,5074],{},[214,58209,31947],{"mathvariant":484},[214,58211,1511],{},[217,58213,43],{"stretchy":241},[501,58215,58216,58218],{},[214,58217,22],{},[224,58219,1244],{},[3054,58221,58222,58224],{},[217,58223,1522],{"stretchy":241},[214,58225,884],{"mathvariant":484},[217,58227,43],{"stretchy":241},[214,58229,176],{},[217,58231,5074],{},[501,58233,58234,58236],{},[214,58235,22],{},[224,58237,1244],{},[217,58239,1522],{"stretchy":241},[217,58241,246],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[217,58243,5074],{},[217,58245,73],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,58247,1511],{},[217,58249,43],{"stretchy":241},[214,58251,176],{},[217,58253,1522],{"stretchy":241},[217,58255,5074],{},[214,58257,1511],{},[217,58259,43],{"stretchy":241},[501,58261,58262,58264],{},[214,58263,22],{},[224,58265,1253],{},[217,58267,1522],{"stretchy":241},[217,58269,5074],{},[214,58271,31947],{"mathvariant":484},[214,58273,1511],{},[217,58275,43],{"stretchy":241},[501,58277,58278,58280],{},[214,58279,22],{},[224,58281,1253],{},[3054,58283,58284,58286],{},[217,58285,1522],{"stretchy":241},[214,58287,884],{"mathvariant":484},[217,58289,43],{"stretchy":241},[214,58291,176],{},[217,58293,5074],{},[501,58295,58296,58298],{},[214,58297,22],{},[224,58299,1253],{},[217,58301,1522],{"stretchy":241},[217,58303,246],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[5021,58305,58306,58312],{},[5024,58307,58308],{},[5027,58309,58310],{"scriptlevel":89,"displaystyle":261},[211,58311],{},[5024,58313,58314],{},[5027,58315,58316],{"scriptlevel":89,"displaystyle":261},[211,58317,58318,58320,58322,58413,58415,58417,58419,58421,58423,58429,58431,58433,58435,58437,58439,58445,58447,58453,58455],{},[211,58319],{},[217,58321,83],{},[1771,58323,58324,58410],{},[1771,58325,58326,58408],{},[211,58327,58328,58330,58332,58334,58340,58342,58344,58346,58348,58354,58356,58358,58360,58362,58364,58370,58376,58382,58384,58386,58388,58390,58396,58402],{},[217,58329,5074],{},[214,58331,1511],{},[217,58333,43],{"stretchy":241},[501,58335,58336,58338],{},[214,58337,22],{},[224,58339,1244],{},[217,58341,1522],{"stretchy":241},[217,58343,124],{},[214,58345,1511],{},[217,58347,43],{"stretchy":241},[501,58349,58350,58352],{},[214,58351,22],{},[224,58353,1253],{},[217,58355,1522],{"stretchy":241},[217,58357,124],{},[214,58359,31947],{"mathvariant":484},[214,58361,1511],{},[217,58363,43],{"stretchy":241},[501,58365,58366,58368],{},[214,58367,22],{},[224,58369,1244],{},[3054,58371,58372,58374],{},[217,58373,1522],{"stretchy":241},[214,58375,884],{"mathvariant":484},[501,58377,58378,58380],{},[214,58379,22],{},[224,58381,1244],{},[217,58383,5074],{},[214,58385,31947],{"mathvariant":484},[214,58387,1511],{},[217,58389,43],{"stretchy":241},[501,58391,58392,58394],{},[214,58393,22],{},[224,58395,1253],{},[3054,58397,58398,58400],{},[217,58399,1522],{"stretchy":241},[214,58401,884],{"mathvariant":484},[501,58403,58404,58406],{},[214,58405,22],{},[224,58407,1253],{},[217,58409,1793],{"stretchy":261},[228,58411,58412],{},"常数",[217,58414,124],{},[217,58416,43],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,58418,31947],{"mathvariant":484},[214,58420,1511],{},[217,58422,43],{"stretchy":241},[501,58424,58425,58427],{},[214,58426,22],{},[224,58428,1253],{},[217,58430,1522],{"stretchy":241},[217,58432,5074],{},[214,58434,31947],{"mathvariant":484},[214,58436,1511],{},[217,58438,43],{"stretchy":241},[501,58440,58441,58443],{},[214,58442,22],{},[224,58444,1244],{},[217,58446,1522],{"stretchy":241},[3054,58448,58449,58451],{},[217,58450,1522],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[214,58452,884],{"mathvariant":484},[214,58454,176],{},[214,58456,494],{"mathvariant":484},[253,58458,58459],{"encoding":255},"\\begin{aligned}\n&D_f(x\\|c_1)-D_f(x\\|c_2)\\\\\n&=\\bigl[f(x)-f(c_1)-\\nabla f(c_1)^\\top(x-c_1)\\bigr]\n -\\bigl[f(x)-f(c_2)-\\nabla f(c_2)^\\top(x-c_2)\\bigr]\\\\\n&=\\underbrace{-f(c_1)+f(c_2)+\\nabla f(c_1)^\\top c_1-\\nabla f(c_2)^\\top c_2}_{\\text{常数}}\n +\\bigl(\\nabla f(c_2)-\\nabla f(c_1)\\bigr)^\\top x .\n\\end{aligned}",[28,58461,58463],{"className":58462,"ariaHidden":261},[260],[28,58464,58466,58470],{"className":58465},[265],[28,58467],{"className":58468,"style":58469},[269],"height:5.7244em;vertical-align:-2.6122em;",[28,58471,58473],{"className":58472},[274],[28,58474,58476,58529],{"className":58475},[5015],[28,58477,58479],{"className":58478},[5137],[28,58480,58482,58520],{"className":58481},[560,561],[28,58483,58485,58517],{"className":58484},[565],[28,58486,58489,58499,58508],{"className":58487,"style":58488},[569],"height:3.1122em;",[28,58490,58492,58496],{"style":58491},"top:-5.3612em;",[28,58493],{"className":58494,"style":58495},[577],"height:3.089em;",[28,58497],{"className":58498},[274],[28,58500,58502,58505],{"style":58501},"top:-3.8021em;",[28,58503],{"className":58504,"style":58495},[577],[28,58506],{"className":58507},[274],[28,58509,58511,58514],{"style":58510},"top:-2.0531em;",[28,58512],{"className":58513,"style":58495},[577],[28,58515],{"className":58516},[274],[28,58518,593],{"className":58519},[592],[28,58521,58523],{"className":58522},[565],[28,58524,58527],{"className":58525,"style":58526},[569],"height:2.6122em;",[28,58528],{},[28,58530,58532],{"className":58531},[5205],[28,58533,58535,59888],{"className":58534},[560,561],[28,58536,58538,59885],{"className":58537},[565],[28,58539,58541,58745,59228],{"className":58540,"style":58488},[569],[28,58542,58543,58546],{"style":58491},[28,58544],{"className":58545,"style":58495},[577],[28,58547,58549,58552,58592,58595,58598,58601,58641,58644,58647,58650,58653,58693,58696,58699,58702,58742],{"className":58548},[274],[28,58550],{"className":58551},[274],[28,58553,58555,58558],{"className":58554},[274],[28,58556,54364],{"className":58557,"style":2279},[274,275],[28,58559,58561],{"className":58560},[556],[28,58562,58564,58584],{"className":58563},[560,561],[28,58565,58567,58581],{"className":58566},[565],[28,58568,58570],{"className":58569,"style":20945},[569],[28,58571,58572,58575],{"style":46480},[28,58573],{"className":58574,"style":578},[577],[28,58576,58578],{"className":58577},[582,583,584,585],[28,58579,1511],{"className":58580,"style":1546},[274,275,585],[28,58582,593],{"className":58583},[592],[28,58585,58587],{"className":58586},[565],[28,58588,58590],{"className":58589,"style":20967},[569],[28,58591],{},[28,58593,43],{"className":58594},[348],[28,58596,176],{"className":58597},[274,275],[28,58599,24404],{"className":58600},[274],[28,58602,58604,58607],{"className":58603},[274],[28,58605,22],{"className":58606},[274,275],[28,58608,58610],{"className":58609},[556],[28,58611,58613,58633],{"className":58612},[560,561],[28,58614,58616,58630],{"className":58615},[565],[28,58617,58619],{"className":58618,"style":1285},[569],[28,58620,58621,58624],{"style":573},[28,58622],{"className":58623,"style":578},[577],[28,58625,58627],{"className":58626},[582,583,584,585],[28,58628,1244],{"className":58629},[274,585],[28,58631,593],{"className":58632},[592],[28,58634,58636],{"className":58635},[565],[28,58637,58639],{"className":58638,"style":600},[569],[28,58640],{},[28,58642,1522],{"className":58643},[355],[28,58645],{"className":58646,"style":280},[279],[28,58648,5074],{"className":58649},[284],[28,58651],{"className":58652,"style":280},[279],[28,58654,58656,58659],{"className":58655},[274],[28,58657,54364],{"className":58658,"style":2279},[274,275],[28,58660,58662],{"className":58661},[556],[28,58663,58665,58685],{"className":58664},[560,561],[28,58666,58668,58682],{"className":58667},[565],[28,58669,58671],{"className":58670,"style":20945},[569],[28,58672,58673,58676],{"style":46480},[28,58674],{"className":58675,"style":578},[577],[28,58677,58679],{"className":58678},[582,583,584,585],[28,58680,1511],{"className":58681,"style":1546},[274,275,585],[28,58683,593],{"className":58684},[592],[28,58686,58688],{"className":58687},[565],[28,58689,58691],{"className":58690,"style":20967},[569],[28,58692],{},[28,58694,43],{"className":58695},[348],[28,58697,176],{"className":58698},[274,275],[28,58700,24404],{"className":58701},[274],[28,58703,58705,58708],{"className":58704},[274],[28,58706,22],{"className":58707},[274,275],[28,58709,58711],{"className":58710},[556],[28,58712,58714,58734],{"className":58713},[560,561],[28,58715,58717,58731],{"className":58716},[565],[28,58718,58720],{"className":58719,"style":1285},[569],[28,58721,58722,58725],{"style":573},[28,58723],{"className":58724,"style":578},[577],[28,58726,58728],{"className":58727},[582,583,584,585],[28,58729,1253],{"className":58730},[274,585],[28,58732,593],{"className":58733},[592],[28,58735,58737],{"className":58736},[565],[28,58738,58740],{"className":58739,"style":600},[569],[28,58741],{},[28,58743,1522],{"className":58744},[355],[28,58746,58747,58750],{"style":58501},[28,58748],{"className":58749,"style":58495},[577],[28,58751,58753,58756,58759,58762,58765,58771,58774,58777,58780,58783,58786,58789,58792,58795,58798,58838,58841,58844,58847,58850,58853,58856,58859,58899,58928,58931,589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那一项消掉了，剩下的是 ",[28,59943,59945,59958],{"className":59944,"translate":193},[197],[28,59946,59948],{"className":59947},[201],[203,59949,59950],{"xmlns":205},[208,59951,59952,59956],{},[211,59953,59954],{},[214,59955,176],{},[253,59957,176],{"encoding":255},[28,59959,59961],{"className":59960,"ariaHidden":261},[260],[28,59962,59964,59967],{"className":59963},[265],[28,59965],{"className":59966,"style":645},[269],[28,59968,176],{"className":59969},[274,275]," 的仿射函数。所以边界仍然是一个超平面。",[11,59972,59973],{},"这个结论有点扫兴，但我觉得它比“Bregman 给出弯曲边界”更有用：Bregman 改变的不是假设空间，而是“垂直于谁”——这个超平面垂直于 Hessian 度量下的法向，而不是欧氏法向。换句话说，它换了一把尺子，没有换刀。",[11,59975,59976,59977,60028,60029,2693,60058,60087,60088,60141,60142,60171,60172,60201],{},"具体到二维单纯形（其实就是线段 ",[28,59978,59980,60001],{"className":59979,"translate":193},[197],[28,59981,59983],{"className":59982},[201],[203,59984,59985],{"xmlns":205},[208,59986,59987,59999],{},[211,59988,59989,59991,59993,59995,59997],{},[217,59990,73],{"stretchy":241},[224,59992,89],{},[217,59994,1683],{"separator":261},[224,59996,1244],{},[217,59998,246],{"stretchy":241},[253,60000,34973],{"encoding":255},[28,60002,60004],{"className":60003,"ariaHidden":261},[260],[28,60005,60007,60010,60013,60016,60019,60022,60025],{"className":60006},[265],[28,60008],{"className":60009,"style":341},[269],[28,60011,73],{"className":60012},[348],[28,60014,89],{"className":60015},[274],[28,60017,1683],{"className":60018},[1723],[28,60020],{"className":60021,"style":530},[279],[28,60023,1244],{"className":60024},[274],[28,60026,246],{"className":60027},[355],"）上，两个类原型 ",[28,60030,60032,60046],{"className":60031,"translate":193},[197],[28,60033,60035],{"className":60034},[201],[203,60036,60037],{"xmlns":205},[208,60038,60039,60044],{},[211,60040,60041],{},[224,60042,60043],{},"0.2",[253,60045,60043],{"encoding":255},[28,60047,60049],{"className":60048,"ariaHidden":261},[260],[28,60050,60052,60055],{"className":60051},[265],[28,60053],{"className":60054,"style":316},[269],[28,60056,60043],{"className":60057},[274],[28,60059,60061,60075],{"className":60060,"translate":193},[197],[28,60062,60064],{"className":60063},[201],[203,60065,60066],{"xmlns":205},[208,60067,60068,60073],{},[211,60069,60070],{},[224,60071,60072],{},"0.6",[253,60074,60072],{"encoding":255},[28,60076,60078],{"className":60077,"ariaHidden":261},[260],[28,60079,60081,60084],{"className":60080},[265],[28,60082],{"className":60083,"style":316},[269],[28,60085,60072],{"className":60086},[274],"，用 KL 判类的分界点在 ",[28,60089,60091,60110],{"className":60090,"translate":193},[197],[28,60092,60094],{"className":60093},[201],[203,60095,60096],{"xmlns":205},[208,60097,60098,60107],{},[211,60099,60100,60102,60104],{},[214,60101,11],{},[217,60103,46977],{},[224,60105,60106],{},"0.387",[253,60108,60109],{"encoding":255},"p\\approx0.387",[28,60111,60113,60132],{"className":60112,"ariaHidden":261},[260],[28,60114,60116,60120,60123,60126,60129],{"className":60115},[265],[28,60117],{"className":60118,"style":60119},[269],"height:0.6776em;vertical-align:-0.1944em;",[28,60121,11],{"className":60122},[274,275],[28,60124],{"className":60125,"style":302},[279],[28,60127,46977],{"className":60128},[306],[28,60130],{"className":60131,"style":302},[279],[28,60133,60135,60138],{"className":60134},[265],[28,60136],{"className":60137,"style":316},[269],[28,60139,60106],{"className":60140},[274],"；欧氏中点是 ",[28,60143,60145,60159],{"className":60144,"translate":193},[197],[28,60146,60148],{"className":60147},[201],[203,60149,60150],{"xmlns":205},[208,60151,60152,60157],{},[211,60153,60154],{},[224,60155,60156],{},"0.4",[253,60158,60156],{"encoding":255},[28,60160,60162],{"className":60161,"ariaHidden":261},[260],[28,60163,60165,60168],{"className":60164},[265],[28,60166],{"className":60167,"style":316},[269],[28,60169,60156],{"className":60170},[274],"；而在 logit 坐标里取中点又得到 ",[28,60173,60175,60189],{"className":60174,"translate":193},[197],[28,60176,60178],{"className":60177},[201],[203,60179,60180],{"xmlns":205},[208,60181,60182,60187],{},[211,60183,60184],{},[224,60185,60186],{},"0.380",[253,60188,60186],{"encoding":255},[28,60190,60192],{"className":60191,"ariaHidden":261},[260],[28,60193,60195,60198],{"className":60194},[265],[28,60196],{"className":60197,"style":316},[269],[28,60199,60186],{"className":60200},[274],"。三个数，三把尺子，没有哪把是“错”的，取决于你认为什么该被同等对待。",[11,60203,60204,60205,11318,60309,44842,60634,60662,60663,60691],{},"真正弯的边界来自两个地方。一是把比较放在约束集上：单纯形加 Fisher–Rao，就是上一节的大圆。二是把它放在非线性映射之后：如果 ",[28,60206,60208,60254],{"className":60207,"translate":193},[197],[28,60209,60211],{"className":60210},[201],[203,60212,60213],{"xmlns":205},[208,60214,60215,60251],{},[211,60216,60217,60219,60221,60223,60225,60227,60243,60245,60247,60249],{},[214,60218,1829],{},[217,60220,43],{"stretchy":241},[214,60222,176],{},[217,60224,1522],{"stretchy":241},[217,60226,83],{},[211,60228,60229,60231,60233,60235,60237,60239,60241],{},[214,60230,1664],{"mathvariant":484},[214,60232,3933],{"mathvariant":484},[214,60234,1511],{"mathvariant":484},[214,60236,1241],{"mathvariant":484},[214,60238,12983],{"mathvariant":484},[214,60240,238],{"mathvariant":484},[214,60242,176],{"mathvariant":484},[217,60244,43],{"stretchy":241},[214,60246,22957],{},[214,60248,176],{},[217,60250,1522],{"stretchy":241},[253,60252,60253],{"encoding":255},"q(x)=\\mathrm{softmax}(Wx)",[28,60255,60257,60284],{"className":60256,"ariaHidden":261},[260],[28,60258,60260,60263,60266,60269,60272,60275,60278,60281],{"className":60259},[265],[28,60261],{"className":60262,"style":341},[269],[28,60264,1829],{"className":60265,"style":298},[274,275],[28,60267,43],{"className":60268},[348],[28,60270,176],{"className":60271},[274,275],[28,60273,1522],{"className":60274},[355],[28,60276],{"className":60277,"style":302},[279],[28,60279,83],{"className":60280},[306],[28,60282],{"className":60283,"style":302},[279],[28,60285,60287,60290,60297,60300,60303,60306],{"className":60286},[265],[28,60288],{"className":60289,"style":341},[269],[28,60291,60293],{"className":60292},[274],[28,60294,60296],{"className":60295},[274,1712],"softmax",[28,60298,43],{"className":60299},[348],[28,60301,22957],{"className":60302,"style":459},[274,275],[28,60304,176],{"className":60305},[274,275],[28,60307,1522],{"className":60308},[355],[28,60310,60312,60386],{"className":60311,"translate":193},[197],[28,60313,60315],{"className":60314},[201],[203,60316,60317],{"xmlns":205},[208,60318,60319,60383],{},[211,60320,60321,60323,60325,60327,60333,60335,60337,60339,60341,60343,60345,60351,60353,60355,60361,60363,60365,60367,60369,60371,60373,60379,60381],{},[217,60322,7916],{"stretchy":241},[214,60324,176],{},[217,60326,877],{},[501,60328,60329,60331],{},[214,60330,54364],{},[214,60332,1511],{},[217,60334,43],{"stretchy":241},[214,60336,1829],{},[217,60338,43],{"stretchy":241},[214,60340,176],{},[217,60342,1522],{"stretchy":241},[214,60344,24404],{"mathvariant":484},[501,60346,60347,60349],{},[214,60348,22],{},[224,60350,1244],{},[217,60352,1522],{"stretchy":241},[217,60354,83],{},[501,60356,60357,60359],{},[214,60358,54364],{},[214,60360,1511],{},[217,60362,43],{"stretchy":241},[214,60364,1829],{},[217,60366,43],{"stretchy":241},[214,60368,176],{},[217,60370,1522],{"stretchy":241},[214,60372,24404],{"mathvariant":484},[501,60374,60375,60377],{},[214,60376,22],{},[224,60378,1253],{},[217,60380,1522],{"stretchy":241},[217,60382,14883],{"stretchy":241},[253,60384,60385],{"encoding":255},"\\{x:D_f(q(x)\\|c_1)=D_f(q(x)\\|c_2)\\}",[28,60387,60389,60410,60526],{"className":60388,"ariaHidden":261},[260],[28,60390,60392,60395,60398,60401,60404,60407],{"className":60391},[265],[28,60393],{"className":60394,"style":341},[269],[28,60396,7916],{"className":60397},[348],[28,60399,176],{"className":60400},[274,275],[28,60402],{"className":60403,"style":302},[279],[28,60405,877],{"className":60406},[306],[28,60408],{"className":60409,"style":302},[279],[28,60411,60413,60416,60456,60459,60462,60465,60468,60471,60474,60514,60517,60520,60523],{"className":60412},[265],[28,60414],{"className":60415,"style":21101},[269],[28,60417,60419,60422],{"className":60418},[274],[28,60420,54364],{"className":60421,"style":2279},[274,275],[28,60423,60425],{"className":60424},[556],[28,60426,60428,60448],{"className":60427},[560,561],[28,60429,60431,60445],{"className":60430},[565],[28,60432,60434],{"className":60433,"style":20945},[569],[28,60435,60436,60439],{"style":46480},[28,60437],{"className":60438,"style":578},[577],[28,60440,60442],{"className":60441},[582,583,584,585],[28,60443,1511],{"className":60444,"style":1546},[274,275,585],[28,60446,593],{"className":60447},[592],[28,60449,60451],{"className":60450},[565],[28,60452,60454],{"className":60453,"style":20967},[569],[28,60455],{},[28,60457,43],{"className":60458},[348],[28,60460,1829],{"className":60461,"style":298},[274,275],[28,60463,43],{"className":60464},[348],[28,60466,176],{"className":60467},[274,275],[28,60469,1522],{"className":60470},[355],[28,60472,24404],{"className":60473},[274],[28,60475,60477,60480],{"className":60476},[274],[28,60478,22],{"className":60479},[274,275],[28,60481,60483],{"className":60482},[556],[28,60484,60486,60506],{"className":60485},[560,561],[28,60487,60489,60503],{"className":60488},[565],[28,60490,60492],{"className":60491,"style":1285},[569],[28,60493,60494,60497],{"style":573},[28,60495],{"className":60496,"style":578},[577],[28,60498,60500],{"className":60499},[582,583,584,585],[28,60501,1244],{"className":60502},[274,585],[28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空间是超平面，在 ",[28,60664,60666,60679],{"className":60665,"translate":193},[197],[28,60667,60669],{"className":60668},[201],[203,60670,60671],{"xmlns":205},[208,60672,60673,60677],{},[211,60674,60675],{},[214,60676,176],{},[253,60678,176],{"encoding":255},[28,60680,60682],{"className":60681,"ariaHidden":261},[260],[28,60683,60685,60688],{"className":60684},[265],[28,60686],{"className":60687,"style":645},[269],[28,60689,176],{"className":60690},[274,275]," 空间是它的原像，一般是一张弯的曲面。我见到的“非欧氏边界”基本都是后者。",[11,60693,60694],{},"Bregman 还给出两个优化工具。一个是镜像下降：",[28,60696,60698],{"className":60697,"translate":193},[192],[28,60699,60701,60787],{"className":60700,"translate":193},[197],[28,60702,60704],{"className":60703},[201],[203,60705,60706],{"xmlns":205,"display":206},[208,60707,60708,60784],{},[211,60709,60710,60722,60724,60726,60728,60738,60740,60742,60748,60750,60752,60754,60756,60762,60768,60770,60772,60774,60780,60782],{},[501,60711,60712,60714],{},[214,60713,176],{},[211,60715,60716,60718,60720],{},[214,60717,1241],{},[217,60719,124],{},[224,60721,1244],{},[217,60723,83],{},[214,60725,57011],{},[217,60727,21407],{},[1771,60729,60730,60736],{},[211,60731,60732,60734],{},[214,60733,39653],{},[217,60735,21407],{},[214,60737,176],{},[228,60739,14860],{},[217,60741,39678],{"stretchy":241},[501,60743,60744,60746],{},[214,60745,34347],{},[214,60747,1241],{},[217,60749,1683],{"separator":261},[214,60751,176],{},[217,60753,39687],{"stretchy":241},[217,60755,124],{},[21928,60757,60758,60760],{},[224,60759,1244],{},[214,60761,50292],{},[501,60763,60764,60766],{},[214,60765,54364],{},[214,60767,1511],{},[217,60769,43],{"stretchy":241},[214,60771,176],{},[214,60773,24404],{"mathvariant":484},[501,60775,60776,60778],{},[214,60777,176],{},[214,60779,1241],{},[217,60781,1522],{"stretchy":241},[217,60783,1683],{"separator":261},[253,60785,60786],{"encoding":255},"x_{t+1}=\\arg\\min_x\\ \\langle g_t,x\\rangle+\\frac1\\eta D_f(x\\|x_t),",[28,60788,60790,60855,60981],{"className":60789,"ariaHidden":261},[260],[28,60791,60793,60797,60846,60849,60852],{"className":60792},[265],[28,60794],{"className":60795,"style":60796},[269],"height:0.6389em;vertical-align:-0.2083em;",[28,60798,60800,60803],{"className":60799},[274],[28,60801,176],{"className":60802},[274,275],[28,60804,60806],{"className":60805},[556],[28,60807,60809,60838],{"className":60808},[560,561],[28,60810,60812,60835],{"className":60811},[565],[28,60813,60815],{"className":60814,"style":1285},[569],[28,60816,60817,60820],{"style":573},[28,60818],{"className":60819,"style":578},[577],[28,60821,60823],{"className":60822},[582,583,584,585],[28,60824,60826,60829,60832],{"className":60825},[274,585],[28,60827,1241],{"className":60828},[274,275,585],[28,60830,124],{"className":60831},[284,585],[28,60833,1244],{"className":60834},[274,585],[28,60836,593],{"className":60837},[592],[28,60839,60841],{"className":60840},[565],[28,60842,60844],{"className":60843,"style":44839},[569],[28,60845],{},[28,60847],{"className":60848,"style":302},[279],[28,60850,83],{"className":60851},[306],[28,60853],{"className":60854,"style":302},[279],[28,60856,60858,60862,60867,60870,60914,60917,60920,60960,60963,60966,60969,60972,60975,60978],{"className":60857},[265],[28,60859],{"className":60860,"style":60861},[269],"height:1.45em;vertical-align:-0.7em;",[28,60863,57089,60865],{"className":60864},[21329],[28,60866,34347],{"style":39955},[28,60868],{"className":60869,"style":530},[279],[28,60871,60873],{"className":60872},[21329,21485],[28,60874,60876,60906],{"className":60875},[560,561],[28,60877,60879,60903],{"className":60878},[565],[28,60880,60882,60893],{"className":60881,"style":39754},[569],[28,60883,60884,60887],{"style":57109},[28,60885],{"className":60886,"style":1914},[577],[28,60888,60890],{"className":60889},[582,583,584,585],[28,60891,176],{"className":60892},[274,275,585],[28,60894,60895,60898],{"style":1926},[28,60896],{"className":60897,"style":1914},[577],[28,60899,60900],{},[28,60901,39653],{"className":60902},[21329],[28,60904,593],{"className":60905},[592],[28,60907,60909],{"className":60908},[565],[28,60910,60912],{"className":60911,"style":57138},[569],[28,60913],{},[28,60915,14860],{"className":60916},[279],[28,60918,39678],{"className":60919},[348],[28,60921,60923,60926],{"className":60922},[274],[28,60924,34347],{"className":60925,"style":298},[274,275],[28,60927,60929],{"className":60928},[556],[28,60930,60932,60952],{"className":60931},[560,561],[28,60933,60935,60949],{"className":60934},[565],[28,60936,60938],{"className":60937,"style":3978},[569],[28,60939,60940,60943],{"style":3660},[28,60941],{"className":60942,"style":578},[577],[28,60944,60946],{"className":60945},[582,583,584,585],[28,60947,1241],{"className":60948},[274,275,585],[28,60950,593],{"className":60951},[592],[28,60953,60955],{"className":60954},[565],[28,60956,60958],{"className":60957,"style":600},[569],[28,60959],{},[28,60961,1683],{"className":60962},[1723],[28,60964],{"className":60965,"style":530},[279],[28,60967,176],{"className":60968},[274,275],[28,60970,39687],{"className":60971},[355],[28,60973],{"className":60974,"style":280},[279],[28,60976,124],{"className":60977},[284],[28,60979],{"className":60980,"style":280},[279],[28,60982,60984,60988,61051,61091,61094,61097,61100,61140,61143],{"className":60983},[265],[28,60985],{"className":60986,"style":60987},[269],"height:2.2019em;vertical-align:-0.8804em;",[28,60989,60991,60994,61048],{"className":60990},[274],[28,60992],{"className":60993},[348,8285],[28,60995,60997],{"className":60996},[21928],[28,60998,61000,61039],{"className":60999},[560,561],[28,61001,61003,61036],{"className":61002},[565],[28,61004,61006,61017,61025],{"className":61005,"style":46133},[569],[28,61007,61008,61011],{"style":24920},[28,61009],{"className":61010,"style":1914},[577],[28,61012,61014],{"className":61013},[274],[28,61015,50292],{"className":61016,"style":298},[274,275],[28,61018,61019,61022],{"style":22023},[28,61020],{"className":61021,"style":1914},[577],[28,61023],{"className":61024,"style":22031},[22030],[28,61026,61027,61030],{"style":24940},[28,61028],{"className":61029,"style":1914},[577],[28,61031,61033],{"className":61032},[274],[28,61034,1244],{"className":61035},[274],[28,61037,593],{"className":61038},[592],[28,61040,61042],{"className":61041},[565],[28,61043,61046],{"className":61044,"style":61045},[569],"height:0.8804em;",[28,61047],{},[28,61049],{"className":61050},[355,8285],[28,61052,61054,61057],{"className":61053},[274],[28,61055,54364],{"className":61056,"style":2279},[274,275],[28,61058,61060],{"className":61059},[556],[28,61061,61063,61083],{"className":61062},[560,561],[28,61064,61066,61080],{"className":61065},[565],[28,61067,61069],{"className":61068,"style":20945},[569],[28,61070,61071,61074],{"style":46480},[28,61072],{"className":61073,"style":578},[577],[28,61075,61077],{"className":61076},[582,583,584,585],[28,61078,1511],{"className":61079,"style":1546},[274,275,585],[28,61081,593],{"className":61082},[592],[28,61084,61086],{"className":61085},[565],[28,61087,61089],{"className":61088,"style":20967},[569],[28,61090],{},[28,61092,43],{"className":61093},[348],[28,61095,176],{"className":61096},[274,275],[28,61098,24404],{"className":61099},[274],[28,61101,61103,61106],{"className":61102},[274],[28,61104,176],{"className":61105},[274,275],[28,61107,61109],{"className":61108},[556],[28,61110,61112,61132],{"className":61111},[560,561],[28,61113,61115,61129],{"className":61114},[565],[28,61116,61118],{"className":61117,"style":3978},[569],[28,61119,61120,61123],{"style":573},[28,61121],{"className":61122,"style":578},[577],[28,61124,61126],{"className":61125},[582,583,584,585],[28,61127,1241],{"className":61128},[274,275,585],[28,61130,593],{"className":61131},[592],[28,61133,61135],{"className":61134},[565],[28,61136,61138],{"className":61137,"style":600},[569],[28,61139],{},[28,61141,1522],{"className":61142},[355],[28,61144,1683],{"className":61145},[1723],[11,61147,61148,61149,61177,61178,61345,61346,61374,61375,61620,61621,61649],{},"直觉是“走一步，但别离原来的位置太远”，而“太远”用 ",[28,61150,61152,61165],{"className":61151,"translate":193},[197],[28,61153,61155],{"className":61154},[201],[203,61156,61157],{"xmlns":205},[208,61158,61159,61163],{},[211,61160,61161],{},[214,61162,1511],{},[253,61164,1511],{"encoding":255},[28,61166,61168],{"className":61167,"ariaHidden":261},[260],[28,61169,61171,61174],{"className":61170},[265],[28,61172],{"className":61173,"style":2371},[269],[28,61175,1511],{"className":61176,"style":1546},[274,275]," 来量。取 ",[28,61179,61181,61215],{"className":61180,"translate":193},[197],[28,61182,61184],{"className":61183},[201],[203,61185,61186],{"xmlns":205},[208,61187,61188,61212],{},[211,61189,61190,61192,61194,61202,61204,61206],{},[214,61191,1511],{},[217,61193,83],{},[5027,61195,61196],{"scriptlevel":89,"displaystyle":241},[21928,61197,61198,61200],{},[224,61199,1244],{},[224,61201,1253],{},[214,61203,24404],{"mathvariant":484},[214,61205,176],{},[3054,61207,61208,61210],{},[214,61209,24404],{"mathvariant":484},[224,61211,1253],{},[253,61213,61214],{"encoding":255},"f=\\tfrac12\\|x\\|^2",[28,61216,61218,61236],{"className":61217,"ariaHidden":261},[260],[28,61219,61221,61224,61227,61230,61233],{"className":61220},[265],[28,61222],{"className":61223,"style":2371},[269],[28,61225,1511],{"className":61226,"style":1546},[274,275],[28,61228],{"className":61229,"style":302},[279],[28,61231,83],{"className":61232},[306],[28,61234],{"className":61235,"style":302},[279],[28,61237,61239,61242,61310,61313,61316],{"className":61238},[265],[28,61240],{"className":61241,"style":21986},[269],[28,61243,61245,61248,61307],{"className":61244},[274],[28,61246],{"className":61247},[348,8285],[28,61249,61251],{"className":61250},[21928],[28,61252,61254,61299],{"className":61253},[560,561],[28,61255,61257,61296],{"className":61256},[565],[28,61258,61260,61274,61282],{"className":61259,"style":22005},[569],[28,61261,61262,61265],{"style":22008},[28,61263],{"className":61264,"style":1914},[577],[28,61266,61268],{"className":61267},[582,583,584,585],[28,61269,61271],{"className":61270},[274,585],[28,61272,1253],{"className":61273},[274,585],[28,61275,61276,61279],{"style":22023},[28,61277],{"className":61278,"style":1914},[577],[28,61280],{"className":61281,"style":22031},[22030],[28,61283,61284,61287],{"style":22034},[28,61285],{"className":61286,"style":1914},[577],[28,61288,61290],{"className":61289},[582,583,584,585],[28,61291,61293],{"className":61292},[274,585],[28,61294,1244],{"className":61295},[274,585],[28,61297,593],{"className":61298},[592],[28,61300,61302],{"className":61301},[565],[28,61303,61305],{"className":61304,"style":22056},[569],[28,61306],{},[28,61308],{"className":61309},[355,8285],[28,61311,24404],{"className":61312},[274],[28,61314,176],{"className":61315},[274,275],[28,61317,61319,61322],{"className":61318},[274],[28,61320,24404],{"className":61321},[274],[28,61323,61325],{"className":61324},[556],[28,61326,61328],{"className":61327},[560],[28,61329,61331],{"className":61330},[565],[28,61332,61334],{"className":61333,"style":5477},[569],[28,61335,61336,61339],{"style":3094},[28,61337],{"className":61338,"style":578},[577],[28,61340,61342],{"className":61341},[582,583,584,585],[28,61343,1253],{"className":61344},[274,585]," 就是梯度下降；取 ",[28,61347,61349,61362],{"className":61348,"translate":193},[197],[28,61350,61352],{"className":61351},[201],[203,61353,61354],{"xmlns":205},[208,61355,61356,61360],{},[211,61357,61358],{},[214,61359,1511],{},[253,61361,1511],{"encoding":255},[28,61363,61365],{"className":61364,"ariaHidden":261},[260],[28,61366,61368,61371],{"className":61367},[265],[28,61369],{"className":61370,"style":2371},[269],[28,61372,1511],{"className":61373,"style":1546},[274,275]," 为负熵，在单纯形上得到指数梯度／乘法权重更新，更新式是乘性的，天然保持非负。另一个是自然梯度：",[28,61376,61378,61436],{"className":61377,"translate":193},[197],[28,61379,61381],{"className":61380},[201],[203,61382,61383],{"xmlns":205},[208,61384,61385,61433],{},[211,61386,61387,61399,61401,61407,61409,61411,61413,61415,61417,61419,61429,61431],{},[501,61388,61389,61391],{},[214,61390,43555],{},[211,61392,61393,61395,61397],{},[214,61394,1241],{},[217,61396,124],{},[224,61398,1244],{},[217,61400,83],{},[501,61402,61403,61405],{},[214,61404,43555],{},[214,61406,1241],{},[217,61408,5074],{},[214,61410,50292],{},[228,61412,488],{},[214,61414,35082],{},[217,61416,43],{"stretchy":241},[214,61418,43555],{},[3054,61420,61421,61423],{},[217,61422,1522],{"stretchy":241},[211,61424,61425,61427],{},[217,61426,5074],{},[224,61428,1244],{},[214,61430,31947],{"mathvariant":484},[214,61432,24393],{},[253,61434,61435],{"encoding":255},"\\theta_{t+1}=\\theta_t-\\eta\\,G(\\theta)^{-1}\\nabla L",[28,61437,61439,61503,61558],{"className":61438,"ariaHidden":261},[260],[28,61440,61442,61445,61494,61497,61500],{"className":61441},[265],[28,61443],{"className":61444,"style":44780},[269],[28,61446,61448,61451],{"className":61447},[274],[28,61449,43555],{"className":61450,"style":2279},[274,275],[28,61452,61454],{"className":61453},[556],[28,61455,61457,61486],{"className":61456},[560,561],[28,61458,61460,61483],{"className":61459},[565],[28,61461,61463],{"className":61462,"style":1285},[569],[28,61464,61465,61468],{"style":46480},[28,61466],{"className":61467,"style":578},[577],[28,61469,61471],{"className":61470},[582,583,584,585],[28,61472,61474,61477,61480],{"className":61473},[274,585],[28,61475,1241],{"className":61476},[274,275,585],[28,61478,124],{"className":61479},[284,585],[28,61481,1244],{"className":61482},[274,585],[28,61484,593],{"className":61485},[592],[28,61487,61489],{"className":61488},[565],[28,61490,61492],{"className":61491,"style":44839},[569],[28,61493],{},[28,61495],{"className":61496,"style":302},[279],[28,61498,83],{"className":61499},[306],[28,61501],{"className":61502,"style":302},[279],[28,61504,61506,61509,61549,61552,61555],{"className":61505},[265],[28,61507],{"className":61508,"style":21865},[269],[28,61510,61512,61515],{"className":61511},[274],[28,61513,43555],{"className":61514,"style":2279},[274,275],[28,61516,61518],{"className":61517},[556],[28,61519,61521,61541],{"className":61520},[560,561],[28,61522,61524,61538],{"className":61523},[565],[28,61525,61527],{"className":61526,"style":3978},[569],[28,61528,61529,61532],{"style":46480},[28,61530],{"className":61531,"style":578},[577],[28,61533,61535],{"className":61534},[582,583,584,585],[28,61536,1241],{"className":61537},[274,275,585],[28,61539,593],{"className":61540},[592],[28,61542,61544],{"className":61543},[565],[28,61545,61547],{"className":61546,"style":600},[569],[28,61548],{},[28,61550],{"className":61551,"style":280},[279],[28,61553,5074],{"className":61554},[284],[28,61556],{"className":61557,"style":280},[279],[28,61559,61561,61564,61567,61570,61573,61576,61579,61614,61617],{"className":61560},[265],[28,61562],{"className":61563,"style":31837},[269],[28,61565,50292],{"className":61566,"style":298},[274,275],[28,61568],{"className":61569,"style":530},[279],[28,61571,35082],{"className":61572},[274,275],[28,61574,43],{"className":61575},[348],[28,61577,43555],{"className":61578,"style":2279},[274,275],[28,61580,61582,61585],{"className":61581},[355],[28,61583,1522],{"className":61584},[355],[28,61586,61588],{"className":61587},[556],[28,61589,61591],{"className":61590},[560],[28,61592,61594],{"className":61593},[565],[28,61595,61597],{"className":61596,"style":5477},[569],[28,61598,61599,61602],{"style":3094},[28,61600],{"className":61601,"style":578},[577],[28,61603,61605],{"className":61604},[582,583,584,585],[28,61606,61608,61611],{"className":61607},[274,585],[28,61609,5074],{"className":61610},[274,585],[28,61612,1244],{"className":61613},[274,585],[28,61615,31947],{"className":61616},[274],[28,61618,24393],{"className":61619},[274,275],"，其中 ",[28,61622,61624,61637],{"className":61623,"translate":193},[197],[28,61625,61627],{"className":61626},[201],[203,61628,61629],{"xmlns":205},[208,61630,61631,61635],{},[211,61632,61633],{},[214,61634,35082],{},[253,61636,35082],{"encoding":255},[28,61638,61640],{"className":61639,"ariaHidden":261},[260],[28,61641,61643,61646],{"className":61642},[265],[28,61644],{"className":61645,"style":455},[269],[28,61647,35082],{"className":61648},[274,275]," 是 Fisher。它是在 Fisher–Rao 度量下做最速下降，而不是在欧氏度量下。对 softmax 回归，自然梯度会消掉 logits 尺度带来的平台期——因为在 Fisher 度量下，那个尺度根本不算“距离”。",[414,61651,61652],{"id":61652},"点态密度",[11,61654,61655,61656,61761,61762,61790],{},"手上有五个一维样本：",[28,61657,61659,61701],{"className":61658,"translate":193},[197],[28,61660,61662],{"className":61661},[201],[203,61663,61664],{"xmlns":205},[208,61665,61666,61698],{},[211,61667,61668,61670,61672,61674,61677,61679,61681,61684,61686,61688,61691,61693,61695],{},[224,61669,43006],{},[217,61671,1683],{"separator":261},[228,61673,14860],{},[224,61675,61676],{},"1.0",[217,61678,1683],{"separator":261},[228,61680,14860],{},[224,61682,61683],{},"1.1",[217,61685,1683],{"separator":261},[228,61687,14860],{},[224,61689,61690],{},"5.0",[217,61692,1683],{"separator":261},[228,61694,14860],{},[224,61696,61697],{},"5.1",[253,61699,61700],{"encoding":255},"0.9,\\ 1.0,\\ 1.1,\\ 5.0,\\ 5.1",[28,61702,61704],{"className":61703,"ariaHidden":261},[260],[28,61705,61707,61710,61713,61716,61719,61722,61725,61728,61731,61734,61737,61740,61743,61746,61749,61752,61755,61758],{"className":61706},[265],[28,61708],{"className":61709,"style":8494},[269],[28,61711,43006],{"className":61712},[274],[28,61714,1683],{"className":61715},[1723],[28,61717,14860],{"className":61718},[279],[28,61720],{"className":61721,"style":530},[279],[28,61723,61676],{"className":61724},[274],[28,61726,1683],{"className":61727},[1723],[28,61729,14860],{"className":61730},[279],[28,61732],{"className":61733,"style":530},[279],[28,61735,61683],{"className":61736},[274],[28,61738,1683],{"className":61739},[1723],[28,61741,14860],{"className":61742},[279],[28,61744],{"className":61745,"style":530},[279],[28,61747,61690],{"className":61748},[274],[28,61750,1683],{"className":61751},[1723],[28,61753,14860],{"className":61754},[279],[28,61756],{"className":61757,"style":530},[279],[28,61759,61697],{"className":61760},[274],"。肉眼看出两簇，中间在 ",[28,61763,61765,61778],{"className":61764,"translate":193},[197],[28,61766,61768],{"className":61767},[201],[203,61769,61770],{"xmlns":205},[208,61771,61772,61776],{},[211,61773,61774],{},[224,61775,3400],{},[253,61777,3400],{"encoding":255},[28,61779,61781],{"className":61780,"ariaHidden":261},[260],[28,61782,61784,61787],{"className":61783},[265],[28,61785],{"className":61786,"style":316},[269],[28,61788,3400],{"className":61789},[274]," 附近是空的。",[11,61792,61793,61794,61823,61824,61852,61853,61904,61905,61934,61935,61986,61987,62016,62017,62020],{},"现在用核密度估计去拟合，只用高斯核、只改带宽。带宽取 ",[28,61795,61797,61811],{"className":61796,"translate":193},[197],[28,61798,61800],{"className":61799},[201],[203,61801,61802],{"xmlns":205},[208,61803,61804,61809],{},[211,61805,61806],{},[224,61807,61808],{},"0.1",[253,61810,61808],{"encoding":255},[28,61812,61814],{"className":61813,"ariaHidden":261},[260],[28,61815,61817,61820],{"className":61816},[265],[28,61818],{"className":61819,"style":316},[269],[28,61821,61808],{"className":61822},[274],"，每个点都是一个尖峰，曲线在两个簇内部剧烈起伏；带宽取 ",[28,61825,61827,61840],{"className":61826,"translate":193},[197],[28,61828,61830],{"className":61829},[201],[203,61831,61832],{"xmlns":205},[208,61833,61834,61838],{},[211,61835,61836],{},[224,61837,251],{},[253,61839,251],{"encoding":255},[28,61841,61843],{"className":61842,"ariaHidden":261},[260],[28,61844,61846,61849],{"className":61845},[265],[28,61847],{"className":61848,"style":316},[269],[28,61850,251],{"className":61851},[274],"，曲线被抹平。抹平到什么程度？算一下会发现，",[28,61854,61856,61874],{"className":61855,"translate":193},[197],[28,61857,61859],{"className":61858},[201],[203,61860,61861],{"xmlns":205},[208,61862,61863,61871],{},[211,61864,61865,61867,61869],{},[214,61866,176],{},[217,61868,83],{},[224,61870,3400],{},[253,61872,61873],{"encoding":255},"x=3",[28,61875,61877,61895],{"className":61876,"ariaHidden":261},[260],[28,61878,61880,61883,61886,61889,61892],{"className":61879},[265],[28,61881],{"className":61882,"style":645},[269],[28,61884,176],{"className":61885},[274,275],[28,61887],{"className":61888,"style":302},[279],[28,61890,83],{"className":61891},[306],[28,61893],{"className":61894,"style":302},[279],[28,61896,61898,61901],{"className":61897},[265],[28,61899],{"className":61900,"style":316},[269],[28,61902,3400],{"className":61903},[274]," 那个空洞处的密度大约是 ",[28,61906,61908,61922],{"className":61907,"translate":193},[197],[28,61909,61911],{"className":61910},[201],[203,61912,61913],{"xmlns":205},[208,61914,61915,61920],{},[211,61916,61917],{},[224,61918,61919],{},"0.073",[253,61921,61919],{"encoding":255},[28,61923,61925],{"className":61924,"ariaHidden":261},[260],[28,61926,61928,61931],{"className":61927},[265],[28,61929],{"className":61930,"style":316},[269],[28,61932,61919],{"className":61933},[274],"，而簇心 ",[28,61936,61938,61956],{"className":61937,"translate":193},[197],[28,61939,61941],{"className":61940},[201],[203,61942,61943],{"xmlns":205},[208,61944,61945,61953],{},[211,61946,61947,61949,61951],{},[214,61948,176],{},[217,61950,83],{},[224,61952,1244],{},[253,61954,61955],{"encoding":255},"x=1",[28,61957,61959,61977],{"className":61958,"ariaHidden":261},[260],[28,61960,61962,61965,61968,61971,61974],{"className":61961},[265],[28,61963],{"className":61964,"style":645},[269],[28,61966,176],{"className":61967},[274,275],[28,61969],{"className":61970,"style":302},[279],[28,61972,83],{"className":61973},[306],[28,61975],{"className":61976,"style":302},[279],[28,61978,61980,61983],{"className":61979},[265],[28,61981],{"className":61982,"style":316},[269],[28,61984,1244],{"className":61985},[274]," 处是 ",[28,61988,61990,62004],{"className":61989,"translate":193},[197],[28,61991,61993],{"className":61992},[201],[203,61994,61995],{"xmlns":205},[208,61996,61997,62002],{},[211,61998,61999],{},[224,62000,62001],{},"0.071",[253,62003,62001],{"encoding":255},[28,62005,62007],{"className":62006,"ariaHidden":261},[260],[28,62008,62010,62013],{"className":62009},[265],[28,62011],{"className":62012,"style":316},[269],[28,62014,62001],{"className":62015},[274],"——",[183,62018,62019],{},"空洞反而比簇心更“密”","。因为核太宽，五个点全都够到了中间，而簇心只够到自己人和远处那两个点的尾巴。",[11,62022,62023,62024,62052,62053,62081,62082,62110],{},"这个例子里，带宽就是分辨率。",[28,62025,62027,62040],{"className":62026,"translate":193},[197],[28,62028,62030],{"className":62029},[201],[203,62031,62032],{"xmlns":205},[208,62033,62034,62038],{},[211,62035,62036],{},[214,62037,49252],{},[253,62039,49252],{"encoding":255},[28,62041,62043],{"className":62042,"ariaHidden":261},[260],[28,62044,62046,62049],{"className":62045},[265],[28,62047],{"className":62048,"style":7448},[269],[28,62050,49252],{"className":62051},[274,275]," 太小，方差爆炸；",[28,62054,62056,62069],{"className":62055,"translate":193},[197],[28,62057,62059],{"className":62058},[201],[203,62060,62061],{"xmlns":205},[208,62062,62063,62067],{},[211,62064,62065],{},[214,62066,49252],{},[253,62068,49252],{"encoding":255},[28,62070,62072],{"className":62071,"ariaHidden":261},[260],[28,62073,62075,62078],{"className":62074},[265],[28,62076],{"className":62077,"style":7448},[269],[28,62079,49252],{"className":62080},[274,275]," 太大，两个类之间的空隙被填平，一个明显偏离的样本反而显得很正常。我用 KDE 做异常检测时最常踩的坑就是这个：调 ",[28,62083,62085,62098],{"className":62084,"translate":193},[197],[28,62086,62088],{"className":62087},[201],[203,62089,62090],{"xmlns":205},[208,62091,62092,62096],{},[211,62093,62094],{},[214,62095,49252],{},[253,62097,49252],{"encoding":255},[28,62099,62101],{"className":62100,"ariaHidden":261},[260],[28,62102,62104,62107],{"className":62103},[265],[28,62105],{"className":62106,"style":7448},[269],[28,62108,49252],{"className":62109},[274,275]," 其实是在调“我允许多大的类间间隙”。",[11,62112,62113],{},"形式上是",[28,62115,62117],{"className":62116,"translate":193},[192],[28,62118,62120,62235],{"className":62119,"translate":193},[197],[28,62121,62123],{"className":62122},[201],[203,62124,62125],{"xmlns":205,"display":206},[208,62126,62127,62232],{},[211,62128,62129,62141,62143,62145,62147,62149,62155,62170,62176,62178,62180,62182,62188,62190,62192,62194,62200,62202,62204,62206,62208,62218,62220,62222,62224,62226,62228,62230],{},[501,62130,62131,62139],{},[62132,62133,62134,62136],"mover",{"accent":261},[214,62135,11],{},[217,62137,62138],{},"^",[214,62140,49252],{},[217,62142,43],{"stretchy":241},[214,62144,176],{},[217,62146,1522],{"stretchy":241},[217,62148,83],{},[21928,62150,62151,62153],{},[224,62152,1244],{},[214,62154,3060],{},[62156,62157,62158,62160,62168],"munderover",{},[217,62159,25442],{},[211,62161,62162,62164,62166],{},[214,62163,1686],{},[217,62165,83],{},[224,62167,1244],{},[214,62169,3060],{},[501,62171,62172,62174],{},[214,62173,24390],{},[214,62175,49252],{},[217,62177,43],{"stretchy":241},[214,62179,176],{},[217,62181,5074],{},[501,62183,62184,62186],{},[214,62185,176],{},[214,62187,1686],{},[217,62189,1522],{"stretchy":241},[217,62191,1683],{"separator":261},[279,62193],{"width":4053},[501,62195,62196,62198],{},[214,62197,24390],{},[214,62199,49252],{},[217,62201,43],{"stretchy":241},[214,62203,3936],{},[217,62205,1522],{"stretchy":241},[217,62207,83],{},[3054,62209,62210,62212],{},[214,62211,49252],{},[211,62213,62214,62216],{},[217,62215,5074],{},[214,62217,12988],{},[214,62219,24390],{},[217,62221,43],{"stretchy":241},[214,62223,3936],{},[214,62225,12219],{"mathvariant":484},[214,62227,49252],{},[217,62229,1522],{"stretchy":241},[214,62231,494],{"mathvariant":484},[253,62233,62234],{"encoding":255},"\\hat p_h(x)=\\frac1n\\sum_{i=1}^nK_h(x-x_i),\\qquad K_h(u)=h^{-d}K(u\u002Fh).",[28,62236,62238,62345,62541,62657],{"className":62237,"ariaHidden":261},[260],[28,62239,62241,62244,62327,62330,62333,62336,62339,62342],{"className":62240},[265],[28,62242],{"className":62243,"style":341},[269],[28,62245,62247,62293],{"className":62246},[274],[28,62248,62251],{"className":62249},[274,62250],"accent",[28,62252,62254,62284],{"className":62253},[560,561],[28,62255,62257,62281],{"className":62256},[565],[28,62258,62260,62268],{"className":62259,"style":7448},[569],[28,62261,62262,62265],{"style":1926},[28,62263],{"className":62264,"style":1914},[577],[28,62266,11],{"className":62267},[274,275],[28,62269,62270,62273],{"style":1926},[28,62271],{"className":62272,"style":1914},[577],[28,62274,62278],{"className":62275,"style":62277},[62276],"accent-body","left:-0.1667em;",[28,62279,62138],{"className":62280},[274],[28,62282,593],{"className":62283},[592],[28,62285,62287],{"className":62286},[565],[28,62288,62291],{"className":62289,"style":62290},[569],"height:0.1944em;",[28,62292],{},[28,62294,62296],{"className":62295},[556],[28,62297,62299,62319],{"className":62298},[560,561],[28,62300,62302,62316],{"className":62301},[565],[28,62303,62305],{"className":62304,"style":20945},[569],[28,62306,62307,62310],{"style":573},[28,62308],{"className":62309,"style":578},[577],[28,62311,62313],{"className":62312},[582,583,584,585],[28,62314,49252],{"className":62315},[274,275,585],[28,62317,593],{"className":62318},[592],[28,62320,62322],{"className":62321},[565],[28,62323,62325],{"className":62324,"style":600},[569],[28,62326],{},[28,62328,43],{"className":62329},[348],[28,62331,176],{"className":62332},[274,275],[28,62334,1522],{"className":62335},[355],[28,62337],{"className":62338,"style":302},[279],[28,62340,83],{"className":62341},[306],[28,62343],{"className":62344,"style":302},[279],[28,62346,62348,62352,62414,62417,62483,62486,62526,62529,62532,62535,62538],{"className":62347},[265],[28,62349],{"className":62350,"style":62351},[269],"height:2.9291em;vertical-align:-1.2777em;",[28,62353,62355,62358,62411],{"className":62354},[274],[28,62356],{"className":62357},[348,8285],[28,62359,62361],{"className":62360},[21928],[28,62362,62364,62403],{"className":62363},[560,561],[28,62365,62367,62400],{"className":62366},[565],[28,62368,62370,62381,62389],{"className":62369,"style":46133},[569],[28,62371,62372,62375],{"style":24920},[28,62373],{"className":62374,"style":1914},[577],[28,62376,62378],{"className":62377},[274],[28,62379,3060],{"className":62380},[274,275],[28,62382,62383,62386],{"style":22023},[28,62384],{"className":62385,"style":1914},[577],[28,62387],{"className":62388,"style":22031},[22030],[28,62390,62391,62394],{"style":24940},[28,62392],{"className":62393,"style":1914},[577],[28,62395,62397],{"className":62396},[274],[28,62398,1244],{"className":62399},[274],[28,62401,593],{"className":62402},[592],[28,62404,62406],{"className":62405},[565],[28,62407,62409],{"className":62408,"style":24985},[569],[28,62410],{},[28,62412],{"className":62413},[355,8285],[28,62415],{"className":62416,"style":530},[279],[28,62418,62420],{"className":62419},[21329,21485],[28,62421,62423,62475],{"className":62422},[560,561],[28,62424,62426,62472],{"className":62425},[565],[28,62427,62430,62450,62460],{"className":62428,"style":62429},[569],"height:1.6514em;",[28,62431,62432,62435],{"style":25716},[28,62433],{"className":62434,"style":25720},[577],[28,62436,62438],{"className":62437},[582,583,584,585],[28,62439,62441,62444,62447],{"className":62440},[274,585],[28,62442,1686],{"className":62443},[274,275,585],[28,62445,83],{"className":62446},[306,585],[28,62448,1244],{"className":62449},[274,585],[28,62451,62452,62455],{"style":25732},[28,62453],{"className":62454,"style":25720},[577],[28,62456,62457],{},[28,62458,25442],{"className":62459},[21329,21330,21547],[28,62461,62463,62466],{"style":62462},"top:-4.3em;margin-left:0em;",[28,62464],{"className":62465,"style":25720},[577],[28,62467,62469],{"className":62468},[582,583,584,585],[28,62470,3060],{"className":62471},[274,275,585],[28,62473,593],{"className":62474},[592],[28,62476,62478],{"className":62477},[565],[28,62479,62481],{"className":62480,"style":26154},[569],[28,62482],{},[28,62484],{"className":62485,"style":530},[279],[28,62487,62489,62492],{"className":62488},[274],[28,62490,24390],{"className":62491,"style":28331},[274,275],[28,62493,62495],{"className":62494},[556],[28,62496,62498,62518],{"className":62497},[560,561],[28,62499,62501,62515],{"className":62500},[565],[28,62502,62504],{"className":62503,"style":20945},[569],[28,62505,62506,62509],{"style":28346},[28,62507],{"className":62508,"style":578},[577],[28,62510,62512],{"className":62511},[582,583,584,585],[28,62513,49252],{"className":62514},[274,275,585],[28,62516,593],{"className":62517},[592],[28,62519,62521],{"className":62520},[565],[28,62522,62524],{"className":62523,"style":600},[569],[28,62525],{},[28,62527,43],{"className":62528},[348],[28,62530,176],{"className":62531},[274,275],[28,62533],{"className":62534,"style":280},[279],[28,62536,5074],{"className":62537},[284],[28,62539],{"className":62540,"style":280},[279],[28,62542,62544,62547,62587,62590,62593,62596,62599,62639,62642,62645,62648,62651,62654],{"className":62543},[265],[28,62545],{"className":62546,"style":341},[269],[28,62548,62550,62553],{"className":62549},[274],[28,62551,176],{"className":62552},[274,275],[28,62554,62556],{"className":62555},[556],[28,62557,62559,62579],{"className":62558},[560,561],[28,62560,62562,62576],{"className":62561},[565],[28,62563,62565],{"className":62564,"style":3881},[569],[28,62566,62567,62570],{"style":573},[28,62568],{"className":62569,"style":578},[577],[28,62571,62573],{"className":62572},[582,583,584,585],[28,62574,1686],{"className":62575},[274,275,585],[28,62577,593],{"className":62578},[592],[28,62580,62582],{"className":62581},[565],[28,62583,62585],{"className":62584,"style":600},[569],[28,62586],{},[28,62588,1522],{"className":62589},[355],[28,62591,1683],{"className":62592},[1723],[28,62594],{"className":62595,"style":4222},[279],[28,62597],{"className":62598,"style":530},[279],[28,62600,62602,62605],{"className":62601},[274],[28,62603,24390],{"className":62604,"style":28331},[274,275],[28,62606,62608],{"className":62607},[556],[28,62609,62611,62631],{"className":62610},[560,561],[28,62612,62614,62628],{"className":62613},[565],[28,62615,62617],{"className":62616,"style":20945},[569],[28,62618,62619,62622],{"style":28346},[28,62620],{"className":62621,"style":578},[577],[28,62623,62625],{"className":62624},[582,583,584,585],[28,62626,49252],{"className":62627},[274,275,585],[28,62629,593],{"className":62630},[592],[28,62632,62634],{"className":62633},[565],[28,62635,62637],{"className":62636,"style":600},[569],[28,62638],{},[28,62640,43],{"className":62641},[348],[28,62643,3936],{"className":62644},[274,275],[28,62646,1522],{"className":62647},[355],[28,62649],{"className":62650,"style":302},[279],[28,62652,83],{"className":62653},[306],[28,62655],{"className":62656,"style":302},[279],[28,62658,62660,62663,62698,62701,62704,62707,62710,62713,62716],{"className":62659},[265],[28,62661],{"className":62662,"style":55190},[269],[28,62664,62666,62669],{"className":62665},[274],[28,62667,49252],{"className":62668},[274,275],[28,62670,62672],{"className":62671},[556],[28,62673,62675],{"className":62674},[560],[28,62676,62678],{"className":62677},[565],[28,62679,62681],{"className":62680,"style":49393},[569],[28,62682,62683,62686],{"style":4880},[28,62684],{"className":62685,"style":578},[577],[28,62687,62689],{"className":62688},[582,583,584,585],[28,62690,62692,62695],{"className":62691},[274,585],[28,62693,5074],{"className":62694},[274,585],[28,62696,12988],{"className":62697},[274,275,585],[28,62699,24390],{"className":62700,"style":28331},[274,275],[28,62702,43],{"className":62703},[348],[28,62705,3936],{"className":62706},[274,275],[28,62708,12219],{"className":62709},[274],[28,62711,49252],{"className":62712},[274,275],[28,62714,1522],{"className":62715},[355],[28,62717,494],{"className":62718},[274],[11,62720,62721,62722,6904,62781,6904,62968,63122],{},"它的偏差和方差可以算出来（记 ",[28,62723,62725,62745],{"className":62724,"translate":193},[197],[28,62726,62728],{"className":62727},[201],[203,62729,62730],{"xmlns":205},[208,62731,62732,62742],{},[211,62733,62734,62736,62738,62740],{},[217,62735,21282],{},[214,62737,24390],{},[217,62739,83],{},[224,62741,1244],{},[253,62743,62744],{"encoding":255},"\\int K=1",[28,62746,62748,62772],{"className":62747,"ariaHidden":261},[260],[28,62749,62751,62754,62757,62760,62763,62766,62769],{"className":62750},[265],[28,62752],{"className":62753,"style":21325},[269],[28,62755,21282],{"className":62756,"style":21332},[21329,21330,21331],[28,62758],{"className":62759,"style":530},[279],[28,62761,24390],{"className":62762,"style":28331},[274,275],[28,62764],{"className":62765,"style":302},[279],[28,62767,83],{"className":62768},[306],[28,62770],{"className":62771,"style":302},[279],[28,62773,62775,62778],{"className":62774},[265],[28,62776],{"className":62777,"style":316},[269],[28,62779,1244],{"className":62780},[274],[28,62782,62784,62830],{"className":62783,"translate":193},[197],[28,62785,62787],{"className":62786},[201],[203,62788,62789],{"xmlns":205},[208,62790,62791,62827],{},[211,62792,62793,62795,62797,62799,62805,62807,62809,62811,62813,62815,62817,62819],{},[217,62794,21282],{},[214,62796,24404],{"mathvariant":484},[214,62798,3936],{},[3054,62800,62801,62803],{},[214,62802,24404],{"mathvariant":484},[224,62804,1253],{},[214,62806,24390],{},[217,62808,43],{"stretchy":241},[214,62810,3936],{},[217,62812,1522],{"stretchy":241},[214,62814,12988],{},[214,62816,3936],{},[217,62818,83],{},[32198,62820,62821,62823,62825],{},[214,62822,491],{},[214,62824,24390],{},[224,62826,1253],{},[253,62828,62829],{"encoding":255},"\\int\\|u\\|^2K(u)du=\\sigma_K^2",[28,62831,62833,62908],{"className":62832,"ariaHidden":261},[260],[28,62834,62836,62840,62843,62846,62849,62852,62881,62884,62887,62890,62893,62896,62899,62902,62905],{"className":62835},[265],[28,62837],{"className":62838,"style":62839},[269],"height:1.1202em;vertical-align:-0.3061em;",[28,62841,21282],{"className":62842,"style":21332},[21329,21330,21331],[28,62844],{"className":62845,"style":530},[279],[28,62847,24404],{"className":62848},[274],[28,62850,3936],{"className":62851},[274,275],[28,62853,62855,62858],{"className":62854},[274],[28,62856,24404],{"className":62857},[274],[28,62859,62861],{"className":62860},[556],[28,62862,62864],{"className":62863},[560],[28,62865,62867],{"className":62866},[565],[28,62868,62870],{"className":62869,"style":5477},[569],[28,62871,62872,62875],{"style":3094},[28,62873],{"className":62874,"style":578},[577],[28,62876,62878],{"className":62877},[582,583,584,585],[28,62879,1253],{"className":62880},[274,585],[28,62882,24390],{"className":62883,"style":28331},[274,275],[28,62885,43],{"className":62886},[348],[28,62888,3936],{"className":62889},[274,275],[28,62891,1522],{"className":62892},[355],[28,62894,12988],{"className":62895},[274,275],[28,62897,3936],{"className":62898},[274,275],[28,62900],{"className":62901,"style":302},[279],[28,62903,83],{"className":62904},[306],[28,62906],{"className":62907,"style":302},[279],[28,62909,62911,62915],{"className":62910},[265],[28,62912],{"className":62913,"style":62914},[269],"height:1.0894em;vertical-align:-0.2753em;",[28,62916,62918,62921],{"className":62917},[274],[28,62919,491],{"className":62920,"style":298},[274,275],[28,62922,62924],{"className":62923},[556],[28,62925,62927,62959],{"className":62926},[560,561],[28,62928,62930,62956],{"className":62929},[565],[28,62931,62933,62945],{"className":62932,"style":5477},[569],[28,62934,62936,62939],{"style":62935},"top:-2.4247em;margin-left:-0.0359em;margin-right:0.05em;",[28,62937],{"className":62938,"style":578},[577],[28,62940,62942],{"className":62941},[582,583,584,585],[28,62943,24390],{"className":62944,"style":28331},[274,275,585],[28,62946,62947,62950],{"style":3094},[28,62948],{"className":62949,"style":578},[577],[28,62951,62953],{"className":62952},[582,583,584,585],[28,62954,1253],{"className":62955},[274,585],[28,62957,593],{"className":62958},[592],[28,62960,62962],{"className":62961},[565],[28,62963,62966],{"className":62964,"style":62965},[569],"height:0.2753em;",[28,62967],{},[28,62969,62971,63005],{"className":62970,"translate":193},[197],[28,62972,62974],{"className":62973},[201],[203,62975,62976],{"xmlns":205},[208,62977,62978,63002],{},[211,62979,62980,62982,62984,62992,62994,62996],{},[214,62981,24404],{"mathvariant":484},[214,62983,24390],{},[32198,62985,62986,62988,62990],{},[214,62987,24404],{"mathvariant":484},[224,62989,1253],{},[224,62991,1253],{},[217,62993,83],{},[217,62995,21282],{},[3054,62997,62998,63000],{},[214,62999,24390],{},[224,63001,1253],{},[253,63003,63004],{"encoding":255},"\\|K\\|_2^2=\\int K^2",[28,63006,63008,63081],{"className":63007,"ariaHidden":261},[260],[28,63009,63011,63014,63017,63020,63072,63075,63078],{"className":63010},[265],[28,63012],{"className":63013,"style":31837},[269],[28,63015,24404],{"className":63016},[274],[28,63018,24390],{"className":63019,"style":28331},[274,275],[28,63021,63023,63026],{"className":63022},[274],[28,63024,24404],{"className":63025},[274],[28,63027,63029],{"className":63028},[556],[28,63030,63032,63064],{"className":63031},[560,561],[28,63033,63035,63061],{"className":63034},[565],[28,63036,63038,63050],{"className":63037,"style":5477},[569],[28,63039,63041,63044],{"style":63040},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[28,63042],{"className":63043,"style":578},[577],[28,63045,63047],{"className":63046},[582,583,584,585],[28,63048,1253],{"className":63049},[274,585],[28,63051,63052,63055],{"style":3094},[28,63053],{"className":63054,"style":578},[577],[28,63056,63058],{"className":63057},[582,583,584,585],[28,63059,1253],{"className":63060},[274,585],[28,63062,593],{"className":63063},[592],[28,63065,63067],{"className":63066},[565],[28,63068,63070],{"className":63069,"style":37576},[569],[28,63071],{},[28,63073],{"className":63074,"style":302},[279],[28,63076,83],{"className":63077},[306],[28,63079],{"className":63080,"style":302},[279],[28,63082,63084,63087,63090,63093],{"className":63083},[265],[28,63085],{"className":63086,"style":62839},[269],[28,63088,21282],{"className":63089,"style":21332},[21329,21330,21331],[28,63091],{"className":63092,"style":530},[279],[28,63094,63096,63099],{"className":63095},[274],[28,63097,24390],{"className":63098,"style":28331},[274,275],[28,63100,63102],{"className":63101},[556],[28,63103,63105],{"className":63104},[560],[28,63106,63108],{"className":63107},[565],[28,63109,63111],{"className":63110,"style":5477},[569],[28,63112,63113,63116],{"style":3094},[28,63114],{"className":63115,"style":578},[577],[28,63117,63119],{"className":63118},[582,583,584,585],[28,63120,1253],{"className":63121},[274,585],"）：",[28,63124,63126],{"className":63125,"translate":193},[192],[28,63127,63129,63271],{"className":63128,"translate":193},[197],[28,63130,63132],{"className":63131},[201],[203,63133,63134],{"xmlns":205,"display":206},[208,63135,63136,63268],{},[211,63137,63138,63140,63142,63152,63154,63156,63158,63160,63162,63164,63166,63168,63170,63190,63192,63194,63196,63198,63200,63202,63204,63212,63214,63224,63226,63228,63230,63232,63266],{},[214,63139,42084],{"mathvariant":12057},[228,63141,488],{},[501,63143,63144,63150],{},[62132,63145,63146,63148],{"accent":261},[214,63147,11],{},[217,63149,62138],{},[214,63151,49252],{},[217,63153,43],{"stretchy":241},[214,63155,176],{},[217,63157,1522],{"stretchy":241},[217,63159,5074],{},[214,63161,11],{},[217,63163,43],{"stretchy":241},[214,63165,176],{},[217,63167,1522],{"stretchy":241},[217,63169,46977],{},[21928,63171,63172,63188],{},[211,63173,63174,63180],{},[3054,63175,63176,63178],{},[214,63177,49252],{},[224,63179,1253],{},[32198,63181,63182,63184,63186],{},[214,63183,491],{},[214,63185,24390],{},[224,63187,1253],{},[224,63189,1253],{},[214,63191,43155],{"mathvariant":484},[214,63193,11],{},[217,63195,43],{"stretchy":241},[214,63197,176],{},[217,63199,1522],{"stretchy":241},[217,63201,1683],{"separator":261},[279,63203],{"width":4053},[211,63205,63206,63208,63210],{},[214,63207,15866],{"mathvariant":484},[214,63209,238],{"mathvariant":484},[214,63211,1867],{"mathvariant":484},[228,63213,488],{},[501,63215,63216,63222],{},[62132,63217,63218,63220],{"accent":261},[214,63219,11],{},[217,63221,62138],{},[214,63223,49252],{},[217,63225,43],{"stretchy":241},[214,63227,176],{},[217,63229,1522],{"stretchy":241},[217,63231,46977],{},[21928,63233,63234,63256],{},[211,63235,63236,63238,63240,63242,63244,63246,63248],{},[214,63237,11],{},[217,63239,43],{"stretchy":241},[214,63241,176],{},[217,63243,1522],{"stretchy":241},[214,63245,24404],{"mathvariant":484},[214,63247,24390],{},[32198,63249,63250,63252,63254],{},[214,63251,24404],{"mathvariant":484},[224,63253,1253],{},[224,63255,1253],{},[211,63257,63258,63260],{},[214,63259,3060],{},[3054,63261,63262,63264],{},[214,63263,49252],{},[214,63265,12988],{},[214,63267,494],{"mathvariant":484},[253,63269,63270],{"encoding":255},"\\mathbb E\\,\\hat p_h(x)-p(x)\\approx\\frac{h^2\\sigma_K^2}{2}\\Delta p(x),\n\\qquad\n\\mathrm{Var}\\,\\hat p_h(x)\\approx\\frac{p(x)\\|K\\|_2^2}{nh^d}.",[28,63272,63274,63383,63410,63686],{"className":63273,"ariaHidden":261},[260],[28,63275,63277,63280,63283,63286,63365,63368,63371,63374,63377,63380],{"className":63276},[265],[28,63278],{"className":63279,"style":341},[269],[28,63281,42084],{"className":63282},[274,12195],[28,63284],{"className":63285,"style":530},[279],[28,63287,63289,63331],{"className":63288},[274],[28,63290,63292],{"className":63291},[274,62250],[28,63293,63295,63323],{"className":63294},[560,561],[28,63296,63298,63320],{"className":63297},[565],[28,63299,63301,63309],{"className":63300,"style":7448},[569],[28,63302,63303,63306],{"style":1926},[28,63304],{"className":63305,"style":1914},[577],[28,63307,11],{"className":63308},[274,275],[28,63310,63311,63314],{"style":1926},[28,63312],{"className":63313,"style":1914},[577],[28,63315,63317],{"className":63316,"style":62277},[62276],[28,63318,62138],{"className":63319},[274],[28,63321,593],{"className":63322},[592],[28,63324,63326],{"className":63325},[565],[28,63327,63329],{"className":63328,"style":62290},[569],[28,63330],{},[28,63332,63334],{"className":63333},[556],[28,63335,63337,63357],{"className":63336},[560,561],[28,63338,63340,63354],{"className":63339},[565],[28,63341,63343],{"className":63342,"style":20945},[569],[28,63344,63345,63348],{"style":573},[28,63346],{"className":63347,"style":578},[577],[28,63349,63351],{"className":63350},[582,583,584,585],[28,63352,49252],{"className":63353},[274,275,585],[28,63355,593],{"className":63356},[592],[28,63358,63360],{"className":63359},[565],[28,63361,63363],{"className":63362,"style":600},[569],[28,63364],{},[28,63366,43],{"className":63367},[348],[28,63369,176],{"className":63370},[274,275],[28,63372,1522],{"className":63373},[355],[28,63375],{"className":63376,"style":280},[279],[28,63378,5074],{"className":63379},[284],[28,63381],{"className":63382,"style":280},[279],[28,63384,63386,63389,63392,63395,63398,63401,63404,63407],{"className":63385},[265],[28,63387],{"className":63388,"style":341},[269],[28,63390,11],{"className":63391},[274,275],[28,63393,43],{"className":63394},[348],[28,63396,176],{"className":63397},[274,275],[28,63399,1522],{"className":63400},[355],[28,63402],{"className":63403,"style":302},[279],[28,63405,46977],{"className":63406},[306],[28,63408],{"className":63409,"style":302},[279],[28,63411,63413,63416,63555,63558,63561,63564,63567,63570,63573,63576,63579,63586,63589,63668,63671,63674,63677,63680,63683],{"className":63412},[265],[28,63414],{"className":63415,"style":24898},[269],[28,63417,63419,63422,63552],{"className":63418},[274],[28,63420],{"className":63421},[348,8285],[28,63423,63425],{"className":63424},[21928],[28,63426,63428,63544],{"className":63427},[560,561],[28,63429,63431,63541],{"className":63430},[565],[28,63432,63434,63445,63453],{"className":63433,"style":24917},[569],[28,63435,63436,63439],{"style":24920},[28,63437],{"className":63438,"style":1914},[577],[28,63440,63442],{"className":63441},[274],[28,63443,1253],{"className":63444},[274],[28,63446,63447,63450],{"style":22023},[28,63448],{"className":63449,"style":1914},[577],[28,63451],{"className":63452,"style":22031},[22030],[28,63454,63455,63458],{"style":24940},[28,63456],{"className":63457,"style":1914},[577],[28,63459,63461,63490],{"className":63460},[274],[28,63462,63464,63467],{"className":63463},[274],[28,63465,49252],{"className":63466},[274,275],[28,63468,63470],{"className":63469},[556],[28,63471,63473],{"className":63472},[560],[28,63474,63476],{"className":63475},[565],[28,63477,63479],{"className":63478,"style":5477},[569],[28,63480,63481,63484],{"style":3094},[28,63482],{"className":63483,"style":578},[577],[28,63485,63487],{"className":63486},[582,583,584,585],[28,63488,1253],{"className":63489},[274,585],[28,63491,63493,63496],{"className":63492},[274],[28,63494,491],{"className":63495,"style":298},[274,275],[28,63497,63499],{"className":63498},[556],[28,63500,63502,63533],{"className":63501},[560,561],[28,63503,63505,63530],{"className":63504},[565],[28,63506,63508,63519],{"className":63507,"style":5477},[569],[28,63509,63510,63513],{"style":62935},[28,63511],{"className":63512,"style":578},[577],[28,63514,63516],{"className":63515},[582,583,584,585],[28,63517,24390],{"className":63518,"style":28331},[274,275,585],[28,63520,63521,63524],{"style":3094},[28,63522],{"className":63523,"style":578},[577],[28,63525,63527],{"className":63526},[582,583,584,585],[28,63528,1253],{"className":63529},[274,585],[28,63531,593],{"className":63532},[592],[28,63534,63536],{"className":63535},[565],[28,63537,63539],{"className":63538,"style":62965},[569],[28,63540],{},[28,63542,593],{"className":63543},[592],[28,63545,63547],{"className":63546},[565],[28,63548,63550],{"className":63549,"style":24985},[569],[28,63551],{},[28,63553],{"className":63554},[355,8285],[28,63556,43155],{"className":63557},[274],[28,63559,11],{"className":63560},[274,275],[28,63562,43],{"className":63563},[348],[28,635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",[28,63858,63860,63873],{"className":63859,"translate":193},[197],[28,63861,63863],{"className":63862},[201],[203,63864,63865],{"xmlns":205},[208,63866,63867,63871],{},[211,63868,63869],{},[214,63870,49252],{},[253,63872,49252],{"encoding":255},[28,63874,63876],{"className":63875,"ariaHidden":261},[260],[28,63877,63879,63882],{"className":63878},[265],[28,63880],{"className":63881,"style":7448},[269],[28,63883,49252],{"className":63884},[274,275]," 增、一个随 ",[28,63887,63889,63902],{"className":63888,"translate":193},[197],[28,63890,63892],{"className":63891},[201],[203,63893,63894],{"xmlns":205},[208,63895,63896,63900],{},[211,63897,63898],{},[214,63899,49252],{},[253,63901,49252],{"encoding":255},[28,63903,63905],{"className":63904,"ariaHidden":261},[260],[28,63906,63908,63911],{"className":63907},[265],[28,63909],{"className":63910,"style":7448},[269],[28,63912,49252],{"className":63913},[274,275]," 减，最优带宽 ",[28,63916,63918,63961],{"className":63917,"translate":193},[197],[28,63919,63921],{"className":63920},[201],[203,63922,63923],{"xmlns":205},[208,63924,63925,63958],{},[211,63926,63927,63933,63936],{},[3054,63928,63929,63931],{},[214,63930,49252],{},[217,63932,50398],{},[217,63934,63935],{},"∝",[3054,63937,63938,63940],{},[214,63939,3060],{},[211,63941,63942,63944,63946,63948,63950,63952,63954,63956],{},[217,63943,5074],{},[224,63945,1244],{},[214,63947,12219],{"mathvariant":484},[217,63949,43],{"stretchy":241},[214,63951,12988],{},[217,63953,124],{},[224,63955,9215],{},[217,63957,1522],{"stretchy":241},[253,63959,63960],{"encoding":255},"h^*\\propto n^{-1\u002F(d+4)}",[28,63962,63964,64008],{"className":63963,"ariaHidden":261},[260],[28,63965,63967,63970,63999,64002,64005],{"className":63966},[265],[28,63968],{"className":63969,"style":7448},[269],[28,63971,63973,63976],{"className":63972},[274],[28,63974,49252],{"className":63975},[274,275],[28,63977,63979],{"className":63978},[556],[28,63980,63982],{"className":63981},[560],[28,63983,63985],{"className":63984},[565],[28,63986,63988],{"className":63987,"style":50469},[569],[28,63989,63990,63993],{"style":3094},[28,63991],{"className":63992,"style":578},[577],[28,63994,63996],{"className":63995},[582,583,584,585],[28,63997,50398],{"className":63998},[284,585],[28,64000],{"className":64001,"style":302},[279],[28,64003,63935],{"className":64004},[306],[28,64006],{"className":64007,"style":302},[279],[28,64009,64011,64014],{"className":64010},[265],[28,64012],{"className":64013,"style":42614},[269],[28,64015,64017,64020],{"className":64016},[274],[28,64018,3060],{"className":64019},[274,275],[28,64021,64023],{"className":64022},[556],[28,64024,64026],{"className":64025},[560],[28,64027,64029],{"className":64028},[565],[28,64030,64032],{"className":64031,"style":42614},[569],[28,64033,64034,64037],{"style":3094},[28,64035],{"className":64036,"style":578},[577],[28,64038,64040],{"className":64039},[582,583,584,585],[28,64041,64043,64046,64049,64052,64055,64058,64061],{"className":64042},[274,585],[28,64044,5074],{"className":64045},[274,585],[28,64047,23875],{"className":64048},[274,585],[28,64050,43],{"className":64051},[348,585],[28,64053,12988],{"className":64054},[274,275,585],[28,64056,124],{"className":64057},[284,585],[28,64059,9215],{"className":64060},[274,585],[28,64062,1522],{"className":64063},[355,585],"，此时 ",[28,64066,64068,64113],{"className":64067,"translate":193},[197],[28,64069,64071],{"className":64070},[201],[203,64072,64073],{"xmlns":205},[208,64074,64075,64110],{},[211,64076,64077,64086,64088],{},[211,64078,64079,64082,64084],{},[214,64080,64081],{"mathvariant":484},"M",[214,64083,6730],{"mathvariant":484},[214,64085,42084],{"mathvariant":484},[217,64087,63935],{},[3054,64089,64090,64092],{},[214,64091,3060],{},[211,64093,64094,64096,64098,64100,64102,64104,64106,64108],{},[217,64095,5074],{},[224,64097,9215],{},[214,64099,12219],{"mathvariant":484},[217,64101,43],{"stretchy":241},[214,64103,12988],{},[217,64105,124],{},[224,64107,9215],{},[217,64109,1522],{"stretchy":241},[253,64111,64112],{"encoding":255},"\\mathrm{MSE}\\propto n^{-4\u002F(d+4)}",[28,64114,64116,64138],{"className":64115,"ariaHidden":261},[260],[28,64117,64119,64122,64129,64132,64135],{"className":64118},[265],[28,64120],{"className":64121,"style":455},[269],[28,64123,64125],{"className":64124},[274],[28,64126,64128],{"className":64127},[274,1712],"MSE",[28,64130],{"className":64131,"style":302},[279],[28,64133,63935],{"className":64134},[306],[28,64136],{"className":64137,"style":302},[279],[28,64139,64141,64144],{"className":64140},[265],[28,64142],{"className":64143,"style":42614},[269],[28,64145,64147,64150],{"className":64146},[274],[28,64148,3060],{"className":64149},[274,275],[28,64151,64153],{"className":64152},[556],[28,64154,64156],{"className":64155},[560],[28,64157,64159],{"className":64158},[565],[28,64160,64162],{"className":64161,"style":42614},[569],[28,64163,64164,64167],{"style":3094},[28,64165],{"className":64166,"style":578},[577],[28,64168,64170],{"className":64169},[582,583,584,585],[28,64171,64173,64176,64180,64183,64186,64189,64192],{"className":64172},[274,585],[28,64174,5074],{"className":64175},[274,585],[28,64177,64179],{"className":64178},[274,585],"4\u002F",[28,64181,43],{"className":64182},[348,585],[28,64184,12988],{"className":64185},[274,275,585],[28,64187,124],{"className":64188},[284,585],[28,64190,9215],{"className":64191},[274,585],[28,64193,1522],{"className":64194},[355,585],[11,64196,64197,64248,64249,64323,64324,64248,64375,64449,64450,64535],{},[28,64198,64200,64218],{"className":64199,"translate":193},[197],[28,64201,64203],{"className":64202},[201],[203,64204,64205],{"xmlns":205},[208,64206,64207,64215],{},[211,64208,64209,64211,64213],{},[214,64210,12988],{},[217,64212,83],{},[224,64214,1244],{},[253,64216,64217],{"encoding":255},"d=1",[28,64219,64221,64239],{"className":64220,"ariaHidden":261},[260],[28,64222,64224,64227,64230,64233,64236],{"className":64223},[265],[28,64225],{"className":64226,"style":7448},[269],[28,64228,12988],{"className":64229},[274,275],[28,64231],{"className":64232,"style":302},[279],[28,64234,83],{"className":64235},[306],[28,64237],{"className":64238,"style":302},[279],[28,64240,64242,64245],{"className":64241},[265],[28,64243],{"className":64244,"style":316},[269],[28,64246,1244],{"className":64247},[274]," 时是 ",[28,64250,64252,64278],{"className":64251,"translate":193},[197],[28,64253,64255],{"className":64254},[201],[203,64256,64257],{"xmlns":205},[208,64258,64259,64275],{},[211,64260,64261],{},[3054,64262,64263,64265],{},[214,64264,3060],{},[211,64266,64267,64269,64271,64273],{},[217,64268,5074],{},[224,64270,9215],{},[214,64272,12219],{"mathvariant":484},[224,64274,251],{},[253,64276,64277],{"encoding":255},"n^{-4\u002F5}",[28,64279,64281],{"className":64280,"ariaHidden":261},[260],[28,64282,64284,64287],{"className":64283},[265],[28,64285],{"className":64286,"style":42614},[269],[28,64288,64290,64293],{"className":64289},[274],[28,64291,3060],{"className":64292},[274,275],[28,64294,64296],{"className":64295},[556],[28,64297,64299],{"className":64298},[560],[28,64300,64302],{"className":64301},[565],[28,64303,64305],{"className":64304,"style":42614},[569],[28,64306,64307,64310],{"style":3094},[28,64308],{"className":64309,"style":578},[577],[28,64311,64313],{"className":64312},[582,583,584,585],[28,64314,64316,64319],{"className":64315},[274,585],[28,64317,5074],{"className":64318},[274,585],[28,64320,64322],{"className":64321},[274,585],"4\u002F5","，很不错。",[28,64325,64327,64345],{"className":64326,"translate":193},[197],[28,64328,64330],{"className":64329},[201],[203,64331,64332],{"xmlns":205},[208,64333,64334,64342],{},[211,64335,64336,64338,64340],{},[214,64337,12988],{},[217,64339,83],{},[224,64341,226],{},[253,64343,64344],{"encoding":255},"d=10",[28,64346,64348,64366],{"className":64347,"ariaHidden":261},[260],[28,64349,64351,64354,64357,64360,64363],{"className":64350},[265],[28,64352],{"className":64353,"style":7448},[269],[28,64355,12988],{"className":64356},[274,275],[28,64358],{"className":64359,"style":302},[279],[28,64361,83],{"className":64362},[306],[28,64364],{"className":64365,"style":302},[279],[28,64367,64369,64372],{"className":64368},[265],[28,64370],{"className":64371,"style":316},[269],[28,64373,226],{"className":64374},[274],[28,64376,64378,64404],{"className":64377,"translate":193},[197],[28,64379,64381],{"className":64380},[201],[203,64382,64383],{"xmlns":205},[208,64384,64385,64401],{},[211,64386,64387],{},[3054,64388,64389,64391],{},[214,64390,3060],{},[211,64392,64393,64395,64397,64399],{},[217,64394,5074],{},[224,64396,1253],{},[214,64398,12219],{"mathvariant":484},[224,64400,9376],{},[253,64402,64403],{"encoding":255},"n^{-2\u002F7}",[28,64405,64407],{"className":64406,"ariaHidden":261},[260],[28,64408,64410,64413],{"className":64409},[265],[28,64411],{"className":64412,"style":42614},[269],[28,64414,64416,64419],{"className":64415},[274],[28,64417,3060],{"className":64418},[274,275],[28,64420,64422],{"className":64421},[556],[28,64423,64425],{"className":64424},[560],[28,64426,64428],{"className":64427},[565],[28,64429,64431],{"className":64430,"style":42614},[569],[28,64432,64433,64436],{"style":3094},[28,64434],{"className":64435,"style":578},[577],[28,64437,64439],{"className":64438},[582,583,584,585],[28,64440,64442,64445],{"className":64441},[274,585],[28,64443,5074],{"className":64444},[274,585],[28,64446,64448],{"className":64447},[274,585],"2\u002F7","，想把误差减半，样本量要乘 ",[28,64451,64453,64476],{"className":64452,"translate":193},[197],[28,64454,64456],{"className":64455},[201],[203,64457,64458],{"xmlns":205},[208,64459,64460,64473],{},[211,64461,64462,64469,64471],{},[3054,64463,64464,64466],{},[224,64465,1253],{},[224,64467,64468],{},"3.5",[217,64470,46977],{},[224,64472,20391],{},[253,64474,64475],{"encoding":255},"2^{3.5}\\approx11",[28,64477,64479,64526],{"className":64478,"ariaHidden":261},[260],[28,64480,64482,64485,64517,64520,64523],{"className":64481},[265],[28,64483],{"className":64484,"style":5477},[269],[28,64486,64488,64491],{"className":64487},[274],[28,64489,1253],{"className":64490},[274],[28,64492,64494],{"className":64493},[556],[28,64495,64497],{"className":64496},[560],[28,64498,64500],{"className":64499},[565],[28,64501,64503],{"className":64502,"style":5477},[569],[28,64504,64505,64508],{"style":3094},[28,64506],{"className":64507,"style":578},[577],[28,64509,64511],{"className":64510},[582,583,584,585],[28,64512,64514],{"className":64513},[274,585],[28,64515,64468],{"className":64516},[274,585],[28,64518],{"className":64519,"style":302},[279],[28,64521,46977],{"className":64522},[306],[28,64524],{"className":64525,"style":302},[279],[28,64527,64529,64532],{"className":64528},[265],[28,64530],{"className":64531,"style":316},[269],[28,64533,20391],{"className":64534},[274],"。这个数字我建议背下来：每加几个维度，KDE 就从“能用”滑向“不能用”。",[11,64537,64538,64539,64638,64639,64731,64732,64804,64805,64906,64907,65050,65051,65122,65123,64248,65173,2923],{},"但 KDE 在高维还有一层跟带宽无关的麻烦。考虑标准正态 ",[28,64540,64542,64570],{"className":64541,"translate":193},[197],[28,64543,64545],{"className":64544},[201],[203,64546,64547],{"xmlns":205},[208,64548,64549,64567],{},[211,64550,64551,64553,64555,64557,64559,64565],{},[214,64552,24766],{"mathvariant":20826},[217,64554,43],{"stretchy":241},[224,64556,89],{},[217,64558,1683],{"separator":261},[501,64560,64561,64563],{},[214,64562,54369],{},[214,64564,12988],{},[217,64566,1522],{"stretchy":241},[253,64568,64569],{"encoding":255},"\\mathcal N(0,I_d)",[28,64571,64573],{"className":64572,"ariaHidden":261},[260],[28,64574,64576,64579,64582,64585,64588,64591,64594,64635],{"className":64575},[265],[28,64577],{"className":64578,"style":341},[269],[28,64580,24766],{"className":64581,"style":24834},[274,20866],[28,64583,43],{"className":64584},[348],[28,64586,89],{"className":64587},[274],[28,64589,1683],{"className":64590},[1723],[28,64592],{"className":64593,"style":530},[279],[28,64595,64597,64600],{"className":64596},[274],[28,64598,54369],{"className":64599,"style":50061},[274,275],[28,64601,64603],{"className":64602},[556],[28,64604,64606,64627],{"className":64605},[560,561],[28,64607,64609,64624],{"className":64608},[565],[28,64610,64612],{"className":64611,"style":20945},[569],[28,64613,64615,64618],{"style":64614},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[28,64616],{"className":64617,"style":578},[577],[28,64619,64621],{"className":64620},[582,583,584,585],[28,64622,12988],{"className":64623},[274,275,585],[28,64625,593],{"className":64626},[592],[28,64628,64630],{"className":64629},[565],[28,64631,64633],{"className":64632,"style":600},[569],[28,64634],{},[28,64636,1522],{"className":64637},[355],"。众数在原点的密度是 ",[28,64640,64642,64674],{"className":64641,"translate":193},[197],[28,64643,64645],{"className":64644},[201],[203,64646,64647],{"xmlns":205},[208,64648,64649,64671],{},[211,64650,64651,64653,64655,64657],{},[217,64652,43],{"stretchy":241},[224,64654,1253],{},[214,64656,22365],{},[3054,64658,64659,64661],{},[217,64660,1522],{"stretchy":241},[211,64662,64663,64665,64667,64669],{},[217,64664,5074],{},[214,64666,12988],{},[214,64668,12219],{"mathvariant":484},[224,64670,1253],{},[253,64672,64673],{"encoding":255},"(2\\pi)^{-d\u002F2}",[28,64675,64677],{"className":64676,"ariaHidden":261},[260],[28,64678,64680,64684,64687,64690,64693],{"className":64679},[265],[28,64681],{"className":64682,"style":64683},[269],"height:1.138em;vertical-align:-0.25em;",[28,64685,43],{"className":64686},[348],[28,64688,1253],{"className":64689},[274],[28,64691,22365],{"className":64692,"style":298},[274,275],[28,64694,64696,64699],{"className":64695},[355],[28,64697,1522],{"className":64698},[355],[28,64700,64702],{"className":64701},[556],[28,64703,64705],{"className":64704},[560],[28,64706,64708],{"className":64707},[565],[28,64709,64711],{"className":64710,"style":42614},[569],[28,64712,64713,64716],{"style":3094},[28,64714],{"className":64715,"style":578},[577],[28,64717,64719],{"className":64718},[582,583,584,585],[28,64720,64722,64725,64728],{"className":64721},[274,585],[28,64723,5074],{"className":64724},[274,585],[28,64726,12988],{"className":64727},[274,275,585],[28,64729,35142],{"className":64730},[274,585],"；而一个“典型”样本落在半径 ",[28,64733,64735,64751],{"className":64734,"translate":193},[197],[28,64736,64738],{"className":64737},[201],[203,64739,64740],{"xmlns":205},[208,64741,64742,64748],{},[211,64743,64744],{},[46231,64745,64746],{},[214,64747,12988],{},[253,64749,64750],{"encoding":255},"\\sqrt d",[28,64752,64754],{"className":64753,"ariaHidden":261},[260],[28,64755,64757,64760],{"className":64756},[265],[28,64758],{"className":64759,"style":47321},[269],[28,64761,64763],{"className":64762},[274,46293],[28,64764,64766,64796],{"className":64765},[560,561],[28,64767,64769,64793],{"className":64768},[565],[28,64770,64772,64781],{"className":64771,"style":47346},[569],[28,64773,64775,64778],{"className":64774,"style":1926},[1945],[28,64776],{"className":64777,"style":1914},[577],[28,64779,12988],{"className":64780,"style":47635},[274,275],[28,64782,64783,64786],{"style":47359},[28,64784],{"className":64785,"style":1914},[577],[28,64787,64789],{"className":64788,"style":47366},[46348],[1961,64790,64791],{"xmlns":1963,"width":1964,"height":47369,"viewBox":47370,"preserveAspectRatio":1967},[1969,64792],{"d":47373},[28,64794,593],{"className":64795},[592],[28,64797,64799],{"className":64798},[565],[28,64800,64802],{"className":64801,"style":47383},[569],[28,64803],{}," 的球壳上（因为 ",[28,64806,64808,64838],{"className":64807,"translate":193},[197],[28,64809,64811],{"className":64810},[201],[203,64812,64813],{"xmlns":205},[208,64814,64815,64835],{},[211,64816,64817,64819,64821,64827,64829,64831,64833],{},[214,64818,24404],{"mathvariant":484},[214,64820,20827],{},[3054,64822,64823,64825],{},[214,64824,24404],{"mathvariant":484},[224,64826,1253],{},[214,64828,12219],{"mathvariant":484},[214,64830,12988],{},[217,64832,950],{},[224,64834,1244],{},[253,64836,64837],{"encoding":255},"\\|X\\|^2\u002Fd\\to1",[28,64839,64841,64897],{"className":64840,"ariaHidden":261},[260],[28,64842,64844,64847,64850,64853,64882,64885,64888,64891,64894],{"className":64843},[265],[28,64845],{"className":64846,"style":31837},[269],[28,64848,24404],{"className":64849},[274],[28,64851,20827],{"className":64852,"style":50061},[274,275],[28,64854,64856,64859],{"className":64855},[274],[28,64857,24404],{"className":64858},[274],[28,64860,64862],{"className":64861},[556],[28,64863,64865],{"className":64864},[560],[28,64866,64868],{"className":64867},[565],[28,64869,64871],{"className":64870,"style":5477},[569],[28,64872,64873,64876],{"style":3094},[28,64874],{"className":64875,"style":578},[577],[28,64877,64879],{"className":64878},[582,583,584,585],[28,64880,1253],{"className":64881},[274,585],[28,64883,12219],{"className":64884},[274],[28,64886,12988],{"className":64887},[274,275],[28,64889],{"className":64890,"style":302},[279],[28,64892,950],{"className":64893},[306],[28,64895],{"className":64896,"style":302},[279],[28,64898,64900,64903],{"className":64899},[265],[28,64901],{"className":64902,"style":316},[269],[28,64904,1244],{"className":64905},[274],"），那里的密度是 ",[28,64908,64910,64956],{"className":64909,"translate":193},[197],[28,64911,64913],{"className":64912},[201],[203,64914,64915],{"xmlns":205},[208,64916,64917,64953],{},[211,64918,64919,64921,64923,64925,64939],{},[217,64920,43],{"stretchy":241},[224,64922,1253],{},[214,64924,22365],{},[3054,64926,64927,64929],{},[217,64928,1522],{"stretchy":241},[211,64930,64931,64933,64935,64937],{},[217,64932,5074],{},[214,64934,12988],{},[214,64936,12219],{"mathvariant":484},[224,64938,1253],{},[3054,64940,64941,64943],{},[214,64942,1667],{},[211,64944,64945,64947,64949,64951],{},[217,64946,5074],{},[214,64948,12988],{},[214,64950,12219],{"mathvariant":484},[224,64952,1253],{},[253,64954,64955],{"encoding":255},"(2\\pi)^{-d\u002F2}e^{-d\u002F2}",[28,64957,64959],{"className":64958,"ariaHidden":261},[260],[28,64960,64962,64965,64968,64971,64974,65012],{"className":64961},[265],[28,64963],{"className":64964,"style":64683},[269],[28,64966,43],{"className":64967},[348],[28,64969,1253],{"className":64970},[274],[28,64972,22365],{"className":64973,"style":298},[274,275],[28,64975,64977,64980],{"className":64976},[355],[28,64978,1522],{"className":64979},[355],[28,64981,64983],{"className":64982},[556],[28,64984,64986],{"className":64985},[560],[28,64987,64989],{"className":64988},[565],[28,64990,64992],{"className":64991,"style":42614},[569],[28,64993,64994,64997],{"style":3094},[28,64995],{"className":64996,"style":578},[577],[28,64998,65000],{"className":64999},[582,583,584,585],[28,65001,65003,65006,65009],{"className":65002},[274,585],[28,65004,5074],{"className":65005},[274,585],[28,65007,12988],{"className":65008},[274,275,585],[28,65010,35142],{"className":65011},[274,585],[28,65013,65015,65018],{"className":65014},[274],[28,65016,1667],{"className":65017},[274,275],[28,65019,65021],{"className":65020},[556],[28,65022,65024],{"className":65023},[560],[28,65025,65027],{"className":65026},[565],[28,65028,65030],{"className":65029,"style":42614},[569],[28,65031,65032,65035],{"style":3094},[28,65033],{"className":65034,"style":578},[577],[28,65036,65038],{"className":65037},[582,583,584,585],[28,65039,65041,65044,65047],{"className":65040},[274,585],[28,65042,5074],{"className":65043},[274,585],[28,65045,12988],{"className":65046},[274,275,585],[28,65048,35142],{"className":65049},[274,585],"。两者差 ",[28,65052,65054,65078],{"className":65053,"translate":193},[197],[28,65055,65057],{"className":65056},[201],[203,65058,65059],{"xmlns":205},[208,65060,65061,65075],{},[211,65062,65063],{},[3054,65064,65065,65067],{},[214,65066,1667],{},[211,65068,65069,65071,65073],{},[214,65070,12988],{},[214,65072,12219],{"mathvariant":484},[224,65074,1253],{},[253,65076,65077],{"encoding":255},"e^{d\u002F2}",[28,65079,65081],{"className":65080,"ariaHidden":261},[260],[28,65082,65084,65087],{"className":65083},[265],[28,65085],{"className":65086,"style":42614},[269],[28,65088,65090,65093],{"className":65089},[274],[28,65091,1667],{"className":65092},[274,275],[28,65094,65096],{"className":65095},[556],[28,65097,65099],{"className":65098},[560],[28,65100,65102],{"className":65101},[565],[28,65103,65105],{"className":65104,"style":42614},[569],[28,65106,65107,65110],{"style":3094},[28,65108],{"className":65109,"style":578},[577],[28,65111,65113],{"className":65112},[582,583,584,585],[28,65114,65116,65119],{"className":65115},[274,585],[28,65117,12988],{"className":65118},[274,275,585],[28,65120,35142],{"className":65121},[274,585]," 倍，",[28,65124,65126,65143],{"className":65125,"translate":193},[197],[28,65127,65129],{"className":65128},[201],[203,65130,65131],{"xmlns":205},[208,65132,65133,65141],{},[211,65134,65135,65137,65139],{},[214,65136,12988],{},[217,65138,83],{},[224,65140,77],{},[253,65142,42378],{"encoding":255},[28,65144,65146,65164],{"className":65145,"ariaHidden":261},[260],[28,65147,65149,65152,65155,65158,65161],{"className":65148},[265],[28,65150],{"className":65151,"style":7448},[269],[28,65153,12988],{"className":65154},[274,275],[28,65156],{"className":65157,"style":302},[279],[28,65159,83],{"className":65160},[306],[28,65162],{"className":65163,"style":302},[279],[28,65165,65167,65170],{"className":65166},[265],[28,65168],{"className":65169,"style":316},[269],[28,65171,77],{"className":65172},[274],[28,65174,65176,65208],{"className":65175,"translate":193},[197],[28,65177,65179],{"className":65178},[201],[203,65180,65181],{"xmlns":205},[208,65182,65183,65205],{},[211,65184,65185,65192,65194,65196,65198],{},[3054,65186,65187,65189],{},[214,65188,1667],{},[224,65190,65191],{},"50",[217,65193,46977],{},[224,65195,251],{},[217,65197,22376],{},[3054,65199,65200,65202],{},[224,65201,226],{},[224,65203,65204],{},"21",[253,65206,65207],{"encoding":255},"e^{50}\\approx5\\times10^{21}",[28,65209,65211,65258,65276],{"className":65210,"ariaHidden":261},[260],[28,65212,65214,65217,65249,65252,65255],{"className":65213},[265],[28,65215],{"className":65216,"style":5477},[269],[28,65218,65220,65223],{"className":65219},[274],[28,65221,1667],{"className":65222},[274,275],[28,65224,65226],{"className":65225},[556],[28,65227,65229],{"className":65228},[560],[28,65230,65232],{"className":65231},[565],[28,65233,65235],{"className":65234,"style":5477},[569],[28,65236,65237,65240],{"style":3094},[28,65238],{"className":65239,"style":578},[577],[28,65241,65243],{"className":65242},[582,583,584,585],[28,65244,65246],{"className":65245},[274,585],[28,65247,65191],{"className":65248},[274,585],[28,65250],{"className":65251,"style":302},[279],[28,65253,46977],{"className":65254},[306],[28,65256],{"className":65257,"style":302},[279],[28,65259,65261,65264,65267,65270,65273],{"className":65260},[265],[28,65262],{"className":65263,"style":6853},[269],[28,65265,251],{"className":65266},[274],[28,65268],{"className":65269,"style":280},[279],[28,65271,22376],{"className":65272},[284],[28,65274],{"className":65275,"style":280},[279],[28,65277,65279,65282,65285],{"className":65278},[265],[28,65280],{"className":65281,"style":5477},[269],[28,65283,1244],{"className":65284},[274],[28,65286,65288,65291],{"className":65287},[274],[28,65289,89],{"className":65290},[274],[28,65292,65294],{"className":65293},[556],[28,65295,65297],{"className":65296},[560],[28,65298,65300],{"className":65299},[565],[28,65301,65303],{"className":65302,"style":5477},[569],[28,65304,65305,65308],{"style":3094},[28,65306],{"className":65307,"style":578},[577],[28,65309,65311],{"className":65310},[582,583,584,585],[28,65312,65314],{"className":65313},[274,585],[28,65315,65204],{"className":65316},[274,585],[11,65318,65319],{},"有个很好记的画面：一个 100 维的橙子，果皮比果肉厚得多。球的体积几乎全在表面附近那层薄壳里，维度越高这层壳占的比例越大。所以“最典型的位置”不是圆心，而是那层壳。密度的最大值在圆心，可那里几乎没有质量。",[11,65321,65322],{},"这意味着什么？如果我用“密度低于阈值”判 OOD，而阈值按众数附近校准，那几乎会把所有正常样本都判成异常。这就是典型集悖论。",[11,65324,65325],{},"所以“低密度等于分布外”这个直觉在高维是错的。密度是点态的，回答的是“这个点附近有多挤”；我真正想问的是“这个点离已知类别的概率质量有多远”。后者是分布态的问题，得用前面那些工具。",[414,65327,65328],{"id":65328},"可逆变换",[11,65330,65331,65332,65335,65336,65392],{},"写采样代码时我总会用到同一招：",[25,65333,65334],{},"random()"," 只能给 ",[28,65337,65339,65362],{"className":65338,"translate":193},[197],[28,65340,65342],{"className":65341},[201],[203,65343,65344],{"xmlns":205},[208,65345,65346,65360],{},[211,65347,65348,65350,65352,65354,65356,65358],{},[214,65349,32617],{},[217,65351,43],{"stretchy":241},[224,65353,89],{},[217,65355,1683],{"separator":261},[224,65357,1244],{},[217,65359,1522],{"stretchy":241},[253,65361,32689],{"encoding":255},[28,65363,65365],{"className":65364,"ariaHidden":261},[260],[28,65366,65368,65371,65374,65377,65380,65383,65386,65389],{"className":65367},[265],[28,65369],{"className":65370,"style":341},[269],[28,65372,32617],{"className":65373,"style":32643},[274,275],[28,65375,43],{"className":65376},[348],[28,65378,89],{"className":65379},[274],[28,65381,1683],{"className":65382},[1723],[28,65384],{"className":65385,"style":530},[279],[28,65387,1244],{"className":65388},[274],[28,65390,1522],{"className":65391},[355],"，可我想要指数分布，于是取",[28,65394,65396],{"className":65395,"translate":193},[192],[28,65397,65399,65440],{"className":65398,"translate":193},[197],[28,65400,65402],{"className":65401},[201],[203,65403,65404],{"xmlns":205,"display":206},[208,65405,65406,65437],{},[211,65407,65408,65410,65412,65414,65435],{},[214,65409,176],{},[217,65411,83],{},[217,65413,5074],{},[21928,65415,65416,65432],{},[211,65417,65418,65420,65422,65424,65426,65428,65430],{},[214,65419,39714],{},[217,65421,21407],{},[217,65423,43],{"stretchy":241},[224,65425,1244],{},[217,65427,5074],{},[214,65429,3936],{},[217,65431,1522],{"stretchy":241},[214,65433,65434],{},"λ",[214,65436,494],{"mathvariant":484},[253,65438,65439],{"encoding":255},"x=-\\frac{\\log(1-u)}{\\lambda}.",[28,65441,65443,65461],{"className":65442,"ariaHidden":261},[260],[28,65444,65446,65449,65452,65455,65458],{"className":65445},[265],[28,65447],{"className":65448,"style":645},[269],[28,65450,176],{"className":65451},[274,275],[28,65453],{"className":65454,"style":302},[279],[28,65456,83],{"className":65457},[306],[28,65459],{"className":65460,"style":302},[279],[28,65462,65464,65468,65471,65557],{"className":65463},[265],[28,65465],{"className":65466,"style":65467},[269],"height:2.113em;vertical-align:-0.686em;",[28,65469,5074],{"className":65470},[274],[28,65472,65474,65477,65554],{"className":65473},[274],[28,65475],{"className":65476},[348,8285],[28,65478,65480],{"className":65479},[21928],[28,65481,65483,65546],{"className":65482},[560,561],[28,65484,65486,65543],{"className":65485},[565],[28,65487,65490,65501,65509],{"className":65488,"style":65489},[569],"height:1.427em;",[28,65491,65492,65495],{"style":24920},[28,65493],{"className":65494,"style":1914},[577],[28,65496,65498],{"className":65497},[274],[28,65499,65434],{"className":65500},[274,275],[28,65502,65503,65506],{"style":22023},[28,65504],{"className":65505,"style":1914},[577],[28,65507],{"className":65508,"style":22031},[22030],[28,65510,65511,65514],{"style":24940},[28,65512],{"className":65513,"style":1914},[577],[28,65515,65517,65522,65525,65528,65531,65534,65537,65540],{"className":65516},[274],[28,65518,39952,65520],{"className":65519},[21329],[28,65521,34347],{"style":39955},[28,65523,43],{"className":65524},[348],[28,65526,1244],{"className":65527},[274],[28,65529],{"className":65530,"style":280},[279],[28,65532,5074],{"className":65533},[284],[28,65535],{"className":65536,"style":280},[279],[28,65538,3936],{"className":65539},[274,275],[28,65541,1522],{"className":65542},[355],[28,65544,593],{"className":65545},[592],[28,65547,65549],{"className":65548},[565],[28,65550,65552],{"className":65551,"style":24985},[569],[28,65553],{},[28,65555],{"className":65556},[355,8285],[28,65558,494],{"className":65559},[274],[11,65561,65562,65563,24711,65681,65709],{},"这一个式子就是一个标准化流。反过来看，",[28,65564,65566,65598],{"className":65565,"translate":193},[197],[28,65567,65569],{"className":65568},[201],[203,65570,65571],{"xmlns":205},[208,65572,65573,65595],{},[211,65574,65575,65577,65579,65581,65583],{},[214,65576,3936],{},[217,65578,83],{},[224,65580,1244],{},[217,65582,5074],{},[3054,65584,65585,65587],{},[214,65586,1667],{},[211,65588,65589,65591,65593],{},[217,65590,5074],{},[214,65592,65434],{},[214,65594,176],{},[253,65596,65597],{"encoding":255},"u=1-e^{-\\lambda x}",[28,65599,65601,65619,65637],{"className":65600,"ariaHidden":261},[260],[28,65602,65604,65607,65610,65613,65616],{"className":65603},[265],[28,65605],{"className":65606,"style":645},[269],[28,65608,3936],{"className":65609},[274,275],[28,65611],{"className":65612,"style":302},[279],[28,65614,83],{"className":65615},[306],[28,65617],{"className":65618,"style":302},[279],[28,65620,65622,65625,65628,65631,65634],{"className":65621},[265],[28,65623],{"className":65624,"style":6853},[269],[28,65626,1244],{"className":65627},[274],[28,65629],{"className":65630,"style":280},[279],[28,65632,5074],{"className":65633},[284],[28,65635],{"className":65636,"style":280},[279],[28,65638,65640,65643],{"className":65639},[265],[28,65641],{"className":65642,"style":40635},[269],[28,65644,65646,65649],{"className":65645},[274],[28,65647,1667],{"className":65648},[274,275],[28,65650,65652],{"className":65651},[556],[28,65653,65655],{"className":65654},[560],[28,65656,65658],{"className":65657},[565],[28,65659,65661],{"className":65660,"style":40635},[569],[28,65662,65663,65666],{"style":3094},[28,65664],{"className":65665,"style":578},[577],[28,65667,65669],{"className":65668},[582,583,584,585],[28,65670,65672,65675,65678],{"className":65671},[274,585],[28,65673,5074],{"className":65674},[274,585],[28,65676,65434],{"className":65677},[274,275,585],[28,65679,176],{"className":65680},[274,275,585],[28,65682,65684,65697],{"className":65683,"translate":193},[197],[28,65685,65687],{"className":65686},[201],[203,65688,65689],{"xmlns":205},[208,65690,65691,65695],{},[211,65692,65693],{},[214,65694,176],{},[253,65696,176],{"encoding":255},[28,65698,65700],{"className":65699,"ariaHidden":261},[260],[28,65701,65703,65706],{"className":65702},[265],[28,65704],{"className":65705,"style":645},[269],[28,65707,176],{"className":65708},[274,275]," 的 CDF，而密度按变量替换是",[28,65711,65713],{"className":65712,"translate":193},[192],[28,65714,65716,65796],{"className":65715,"translate":193},[197],[28,65717,65719],{"className":65718},[201],[203,65720,65721],{"xmlns":205,"display":206},[208,65722,65723,65793],{},[211,65724,65725,65731,65733,65735,65737,65739,65745,65747,65749,65751,65771,65773,65775,65777,65779,65791],{},[501,65726,65727,65729],{},[214,65728,11],{},[214,65730,176],{},[217,65732,43],{"stretchy":241},[214,65734,176],{},[217,65736,1522],{"stretchy":241},[217,65738,83],{},[501,65740,65741,65743],{},[214,65742,11],{},[214,65744,3936],{},[217,65746,43],{"stretchy":241},[214,65748,3936],{},[217,65750,1522],{"stretchy":241},[211,65752,65753,65755,65769],{},[217,65754,889],{"fence":261},[21928,65756,65757,65763],{},[211,65758,65759,65761],{},[214,65760,12988],{},[214,65762,3936],{},[211,65764,65765,65767],{},[214,65766,12988],{},[214,65768,176],{},[217,65770,889],{"fence":261},[217,65772,83],{},[224,65774,1244],{},[217,65776,20336],{},[214,65778,65434],{},[3054,65780,65781,65783],{},[214,65782,1667],{},[211,65784,65785,65787,65789],{},[217,65786,5074],{},[214,65788,65434],{},[214,65790,176],{},[217,65792,1683],{"separator":261},[253,65794,65795],{"encoding":255},"p_x(x)=p_u(u)\\left|\\frac{du}{dx}\\right|=1\\cdot\\lambda e^{-\\lambda x},",[28,65797,65799,65863,66085,66103],{"className":65798,"ariaHidden":261},[260],[28,65800,65802,65805,65845,65848,65851,65854,65857,65860],{"className":65801},[265],[28,65803],{"className":65804,"style":341},[269],[28,65806,65808,65811],{"className":65807},[274],[28,65809,11],{"className":65810},[274,275],[28,65812,65814],{"className":65813},[556],[28,65815,65817,65837],{"className":65816},[560,561],[28,65818,65820,65834],{"className":65819},[565],[28,65821,65823],{"className":65822,"style":23070},[569],[28,65824,65825,65828],{"style":573},[28,65826],{"className":65827,"style":578},[577],[28,65829,65831],{"className":65830},[582,583,584,585],[28,65832,176],{"className":65833},[274,275,585],[28,65835,593],{"className":65836},[592],[28,65838,65840],{"className":65839},[565],[28,65841,65843],{"className":65842,"style":600},[569],[28,65844],{},[28,65846,43],{"className":65847},[348],[28,65849,176],{"className":65850},[274,275],[28,65852,1522],{"className":65853},[355],[28,65855],{"className":65856,"style":302},[279],[28,65858,83],{"className":65859},[306],[28,65861],{"className":65862,"style":302},[279],[28,65864,65866,65870,65910,65913,65916,65919,65922,66076,66079,66082],{"className":65865},[265],[28,65867],{"className":65868,"style":65869},[269],"height:2.4em;vertical-align:-0.95em;",[28,65871,65873,65876],{"className":65872},[274],[28,65874,11],{"className":65875},[274,275],[28,65877,65879],{"className":65878},[556],[28,65880,65882,65902],{"className":65881},[560,561],[28,65883,65885,65899],{"className":65884},[565],[28,65886,65888],{"className":65887,"style":23070},[569],[28,65889,65890,65893],{"style":573},[28,65891],{"className":65892,"style":578},[577],[28,65894,65896],{"className":65895},[582,583,584,585],[28,65897,3936],{"className":65898},[274,275,585],[28,65900,593],{"className":65901},[592],[28,65903,65905],{"className":65904},[565],[28,65906,65908],{"className":65907,"style":600},[569],[28,65909],{},[28,65911,43],{"className":65912},[348],[28,65914,3936],{"className":65915},[274,275],[28,65917,1522],{"className":65918},[355],[28,65920],{"className":65921,"style":530},[279],[28,65923,65925,65970,66039],{"className":65924},[1897],[28,65926,65928],{"className":65927},[348],[28,65929,65931],{"className":65930},[2169,33118],[28,65932,65934,65961],{"className":65933},[560,561],[28,65935,65937,65958],{"className":65936},[565],[28,65938,65941],{"className":65939,"style":65940},[569],"height:1.45em;",[28,65942,65944,65948],{"style":65943},"top:-3.45em;",[28,65945],{"className":65946,"style":65947},[577],"height:4.4em;",[28,65949,65951],{"style":65950},"width:0.333em;height:2.4em;",[1961,65952,65955],{"xmlns":1963,"width":33140,"height":65953,"viewBox":65954},"2.4em","0 0 333 2400",[1969,65956],{"d":65957},"M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 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Box–Muller 那篇时用的也是这套逻辑：先有均匀分布，再靠一个可逆变换把形状掰成想要的样子。",[11,66158,66159,66160,6904,66226,66319],{},"一般地，设 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p_x(x)=\\log p_z(z)-\\log\\bigl|\\det J_f(z)\\bigr|.",[28,66860,66862,66934,67006],{"className":66861,"ariaHidden":261},[260],[28,66863,66865,66868,66873,66876,66916,66919,66922,66925,66928,66931],{"className":66864},[265],[28,66866],{"className":66867,"style":341},[269],[28,66869,39952,66871],{"className":66870},[21329],[28,66872,34347],{"style":39955},[28,66874],{"className":66875,"style":530},[279],[28,66877,66879,66882],{"className":66878},[274],[28,66880,11],{"className":66881},[274,275],[28,66883,66885],{"className":66884},[556],[28,66886,66888,66908],{"className":66887},[560,561],[28,66889,66891,66905],{"className":66890},[565],[28,66892,66894],{"className":66893,"style":23070},[569],[28,66895,66896,66899],{"style":573},[28,66897],{"className":66898,"style":578},[577],[28,66900,66902],{"className":66901},[582,583,584,585],[28,66903,176],{"className":66904},[274,275,585],[28,66906,593],{"className":66907},[592],[28,66909,66911],{"className":66910},[565],[28,66912,66914],{"className":66913,"style":600},[569],[28,66915],{},[28,66917,43],{"className":66918},[348],[28,66920,176],{"className":66921},[274,275],[28,66923,1522],{"className":66924},[355],[28,66926],{"className":66927,"style":302},[279],[28,66929,83],{"className":66930},[306],[28,66932],{"className":66933,"style":302},[279],[28,66935,66937,66940,66945,66948,66988,66991,66994,66997,67000,67003],{"className":66936},[265],[28,66938],{"className":66939,"style":341},[269],[28,66941,39952,66943],{"className":66942},[21329],[28,66944,34347],{"style":39955},[28,66946],{"className":66947,"style":530},[279],[28,66949,66951,66954],{"className":66950},[274],[28,66952,11],{"className":66953},[274,275],[28,66955,66957],{"className":66956},[556],[28,66958,66960,66980],{"className":66959},[560,561],[28,66961,66963,66977],{"className":66962},[565],[28,66964,66966],{"className":66965,"style":23070},[569],[28,66967,66968,66971],{"style":573},[28,66969],{"className":66970,"style":578},[577],[28,66972,66974],{"className":66973},[582,583,584,585],[28,66975,4838],{"className":66976,"style":4956},[274,275,585],[28,66978,593],{"className":66979},[592],[28,66981,66983],{"className":66982},[565],[28,66984,66986],{"className":66985,"style":600},[569],[28,66987],{},[28,66989,43],{"className":66990},[348],[28,66992,4838],{"className":66993,"style":4956},[274,275],[28,66995,1522],{"className":66996},[355],[28,66998],{"className":66999,"style":280},[279],[28,67001,5074],{"className":67002},[284],[28,67004],{"className":67005,"style":280},[279],[28,67007,67009,67012,67017,67054,67057,67060,67102,67105,67108,67111,67148],{"className":67008},[265],[28,67010],{"className":67011,"style":2469},[269],[28,67013,39952,67015],{"className":67014},[21329],[28,67016,34347],{"style":39955},[28,67018,67020],{"className":67019},[348],[28,67021,67023],{"className":67022},[2169,33118],[28,67024,67026,67046],{"className":67025},[560,561],[28,67027,67029,67043],{"className":67028},[565],[28,67030,67032],{"className":67031,"style":2091},[569],[28,67033,67034,67037],{"style":33130},[28,67035],{"className":67036,"style":33134},[577],[28,67038,67039],{"style":33137},[1961,67040,67041],{"xmlns":1963,"width":33140,"height":1810,"viewBox":33141},[1969,67042],{"d":33144},[28,67044,593],{"className":67045},[592],[28,67047,67049],{"className":67048},[565],[28,67050,67052],{"className":67051,"style":33154},[569],[28,67053],{},[28,67055,66381],{"className":67056},[21329],[28,67058],{"className":67059,"style":530},[279],[28,67061,67063,67067],{"className":67062},[274],[28,67064,66843],{"className":67065,"style":67066},[274,275],"margin-right:0.0962em;",[28,67068,67070],{"className":67069},[556],[28,67071,67073,67094],{"className":67072},[560,561],[28,67074,67076,67091],{"className":67075},[565],[28,67077,67079],{"className":67078,"style":20945},[569],[28,67080,67082,67085],{"style":67081},"top:-2.55em;margin-left:-0.0962em;margin-right:0.05em;",[28,67083],{"className":67084,"style":578},[577],[28,67086,67088],{"className":67087},[582,583,584,585],[28,67089,1511],{"className":67090,"style":1546},[274,275,585],[28,67092,593],{"className":67093},[592],[28,67095,67097],{"className":67096},[565],[28,67098,67100],{"className":67099,"style":20967},[569],[28,67101],{},[28,67103,43],{"className":67104},[348],[28,67106,4838],{"className":67107,"style":4956},[274,275],[28,67109,1522],{"className":67110},[355],[28,67112,67114],{"className":67113},[355],[28,67115,67117],{"className":67116},[2169,33118],[28,67118,67120,67140],{"className":67119},[560,561],[28,67121,67123,67137],{"className":67122},[565],[28,67124,67126],{"className":67125,"style":2091},[569],[28,67127,67128,67131],{"style":33130},[28,67129],{"className":67130,"style":33134},[577],[28,67132,67133],{"style":33137},[1961,67134,67135],{"xmlns":1963,"width":33140,"height":1810,"viewBox":33141},[1969,67136],{"d":33144},[28,67138,593],{"className":67139},[592],[28,67141,67143],{"className":67142},[565],[28,67144,67146],{"className":67145,"style":33154},[569],[28,67147],{},[28,67149,494],{"className":67150},[274],[11,67152,47286,67153,67212],{},[28,67154,67156,67180],{"className":67155,"translate":193},[197],[28,67157,67159],{"className":67158},[201],[203,67160,67161],{"xmlns":205},[208,67162,67163,67177],{},[211,67164,67165,67167,67169,67171,67173,67175],{},[214,67166,39714],{},[217,67168,21407],{},[214,67170,889],{"mathvariant":484},[214,67172,66381],{},[217,67174,21407],{},[214,67176,889],{"mathvariant":484},[253,67178,67179],{"encoding":255},"\\log|\\det|",[28,67181,67183],{"className":67182,"ariaHidden":261},[260],[28,67184,67186,67189,67194,67197,67200,67203,67206,67209],{"className":67185},[265],[28,67187],{"className":67188,"style":341},[269],[28,67190,39952,67192],{"className":67191},[21329],[28,67193,34347],{"style":39955},[28,67195],{"className":67196,"style":530},[279],[28,67198,889],{"className":67199},[274],[28,67201],{"className":67202,"style":530},[279],[28,67204,66381],{"className":67205},[21329],[28,67207],{"className":67208,"style":530},[279],[28,67210,889],{"className":67211},[274]," 项就是“局部的伸缩率”：变换把一小块体积放大了多少，密度就要相应缩小多少。多层复合时 log-det 相加，所以加深网络不增加计算上的困难，只要每层的 Jacobian 行列式好算（比如三角矩阵）。",[11,67214,67215,67216,177,67295,6904,67386,67451,67452,67480,67481,6904,67560,67674,67675,2923],{},"一维的情形可以推到极致：任意一维分布都能写成逆 CDF。取 ",[28,67217,67219,67247],{"className":67218,"translate":193},[197],[28,67220,67222],{"className":67221},[201],[203,67223,67224],{"xmlns":205},[208,67225,67226,67244],{},[211,67227,67228,67230,67232,67234,67236,67238,67240,67242],{},[214,67229,4838],{},[217,67231,66242],{},[214,67233,32617],{},[217,67235,43],{"stretchy":241},[224,67237,89],{},[217,67239,1683],{"separator":261},[224,67241,1244],{},[217,67243,1522],{"stretchy":241},[253,67245,67246],{"encoding":255},"z\\sim U(0,1)",[28,67248,67250,67268],{"className":67249,"ariaHidden":261},[260],[28,67251,67253,67256,67259,67262,67265],{"className":67252},[265],[28,67254],{"className":67255,"style":645},[269],[28,67257,4838],{"className":67258,"style":4956},[274,275],[28,67260],{"className":67261,"style":302},[279],[28,67263,66242],{"className":67264},[306],[28,67266],{"className":67267,"style":302},[279],[28,67269,67271,67274,67277,67280,67283,67286,67289,67292],{"className":67270},[265],[28,67272],{"className":67273,"style":341},[269],[28,67275,32617],{"className":67276,"style":32643},[274,275],[28,67278,43],{"className":67279},[348],[28,67281,89],{"className":67282},[274],[28,67284,1683],{"className":67285},[1723],[28,67287],{"className":67288,"style":530},[279],[28,67290,1244],{"className":67291},[274],[28,67293,1522],{"className":67294},[355],[28,67296,67298,67324],{"className":67297,"translate":193},[197],[28,67299,67301],{"className":67300},[201],[203,67302,67303],{"xmlns":205},[208,67304,67305,67321],{},[211,67306,67307,67309,67311],{},[214,67308,1511],{},[217,67310,83],{},[3054,67312,67313,67315],{},[214,67314,32913],{},[211,67316,67317,67319],{},[217,67318,5074],{},[224,67320,1244],{},[253,67322,67323],{"encoding":255},"f=F^{-1}",[28,67325,67327,67345],{"className":67326,"ariaHidden":261},[260],[28,67328,67330,67333,67336,67339,67342],{"className":67329},[265],[28,67331],{"className":67332,"style":2371},[269],[28,67334,1511],{"className":67335,"style":1546},[274,275],[28,67337],{"className":67338,"style":302},[279],[28,67340,83],{"className":67341},[306],[28,67343],{"className":67344,"style":302},[279],[28,67346,67348,67351],{"className":67347},[265],[28,67349],{"className":67350,"style":5477},[269],[28,67352,67354,67357],{"className":67353},[274],[28,67355,32913],{"className":67356,"style":459},[274,275],[28,67358,67360],{"className":67359},[556],[28,67361,67363],{"className":67362},[560],[28,67364,67366],{"className":67365},[565],[28,67367,67369],{"className":67368,"style":5477},[569],[28,67370,67371,67374],{"style":3094},[28,67372],{"className":67373,"style":578},[577],[28,67375,67377],{"className":67376},[582,583,584,585],[28,67378,67380,67383],{"className":67379},[274,585],[28,67381,5074],{"className":67382},[274,585],[28,67384,1244],{"className":67385},[274,585],[28,67387,67389,67412],{"className":67388,"translate":193},[197],[28,67390,67392],{"className":67391},[201],[203,67393,67394],{"xmlns":205},[208,67395,67396,67410],{},[211,67397,67398,67400,67402,67404,67406,67408],{},[214,67399,176],{},[217,67401,83],{},[214,67403,1511],{},[217,67405,43],{"stretchy":241},[214,67407,4838],{},[217,67409,1522],{"stretchy":241},[253,67411,66186],{"encoding":255},[28,67413,67415,67433],{"className":67414,"ariaHidden":261},[260],[28,67416,67418,67421,67424,67427,67430],{"className":67417},[265],[28,67419],{"className":67420,"style":645},[269],[28,67422,176],{"className":67423},[274,275],[28,67425],{"className":67426,"style":302},[279],[28,67428,83],{"className":67429},[306],[28,67431],{"className":67432,"style":302},[279],[28,67434,67436,67439,67442,67445,67448],{"className":67435},[265],[28,67437],{"className":67438,"style":341},[269],[28,67440,1511],{"className":67441,"style":1546},[274,275],[28,67443,43],{"className":67444},[348],[28,67446,4838],{"className":67447,"style":4956},[274,275],[28,67449,1522],{"className":67450},[355]," 的分布就是 ",[28,67453,67455,67468],{"className":67454,"translate":193},[197],[28,67456,67458],{"className":67457},[201],[203,67459,67460],{"xmlns":205},[208,67461,67462,67466],{},[211,67463,67464],{},[214,67465,32913],{},[253,67467,32913],{"encoding":255},[28,67469,67471],{"className":67470,"ariaHidden":261},[260],[28,67472,67474,67477],{"className":67473},[265],[28,67475],{"className":67476,"style":455},[269],[28,67478,32913],{"className":67479,"style":459},[274,275],"，这是精确的，不是近似。取 ",[28,67482,67484,67512],{"className":67483,"translate":193},[197],[28,67485,67487],{"className":67486},[201],[203,67488,67489],{"xmlns":205},[208,67490,67491,67509],{},[211,67492,67493,67495,67497,67499,67501,67503,67505,67507],{},[214,67494,4838],{},[217,67496,66242],{},[214,67498,24766],{"mathvariant":20826},[217,67500,43],{"stretchy":241},[224,67502,89],{},[217,67504,1683],{"separator":261},[224,67506,1244],{},[217,67508,1522],{"stretchy":241},[253,67510,67511],{"encoding":255},"z\\sim\\mathcal N(0,1)",[28,67513,67515,67533],{"className":67514,"ariaHidden":261},[260],[28,67516,67518,67521,67524,67527,67530],{"className":67517},[265],[28,67519],{"className":67520,"style":645},[269],[28,67522,4838],{"className":67523,"style":4956},[274,275],[28,67525],{"className":67526,"style":302},[279],[28,67528,66242],{"className":67529},[306],[28,67531],{"className":67532,"style":302},[279],[28,67534,67536,67539,67542,67545,67548,67551,67554,67557],{"className":67535},[265],[28,67537],{"className":67538,"style":341},[269],[28,67540,24766],{"className":67541,"style":24834},[274,20866],[28,67543,43],{"className":67544},[348],[28,67546,89],{"className":67547},[274],[28,67549,1683],{"className":67550},[1723],[28,67552],{"className":67553,"style":530},[279],[28,67555,1244],{"className":67556},[274],[28,67558,1522],{"className":67559},[355],[28,67561,67563,67594],{"className":67562,"translate":193},[197],[28,67564,67566],{"className":67565},[201],[203,67567,67568],{"xmlns":205},[208,67569,67570,67591],{},[211,67571,67572,67574,67576,67586,67588],{},[214,67573,1511],{},[217,67575,83],{},[3054,67577,67578,67580],{},[214,67579,32913],{},[211,67581,67582,67584],{},[217,67583,5074],{},[224,67585,1244],{},[217,67587,35992],{},[214,67589,67590],{"mathvariant":484},"Φ",[253,67592,67593],{"encoding":255},"f=F^{-1}\\circ\\Phi",[28,67595,67597,67615,67665],{"className":67596,"ariaHidden":261},[260],[28,67598,67600,67603,67606,67609,67612],{"className":67599},[265],[28,67601],{"className":67602,"style":2371},[269],[28,67604,1511],{"className":67605,"style":1546},[274,275],[28,67607],{"className":67608,"style":302},[279],[28,67610,83],{"className":67611},[306],[28,67613],{"className":67614,"style":302},[279],[28,67616,67618,67621,67656,67659,67662],{"className":67617},[265],[28,67619],{"className":67620,"style":5477},[269],[28,67622,67624,67627],{"className":67623},[274],[28,67625,32913],{"className":67626,"style":459},[274,275],[28,67628,67630],{"className":67629},[556],[28,67631,67633],{"className":67632},[560],[28,67634,67636],{"className":67635},[565],[28,67637,67639],{"className":67638,"style":5477},[569],[28,67640,67641,67644],{"style":3094},[28,67642],{"className":67643,"style":578},[577],[28,67645,67647],{"className":67646},[582,583,584,585],[28,67648,67650,67653],{"className":67649},[274,585],[28,67651,5074],{"className":67652},[274,585],[28,67654,1244],{"className":67655},[274,585],[28,67657],{"className":67658,"style":280},[279],[28,67660,35992],{"className":67661},[284],[28,67663],{"className":67664,"style":280},[279],[28,67666,67668,67671],{"className":67667},[265],[28,67669],{"className":67670,"style":455},[269],[28,67672,67590],{"className":67673},[274]," 就得到任意一维分布。上面那个指数分布的例子就是 ",[28,67676,67678,67728],{"className":67677,"translate":193},[197],[28,67679,67681],{"className":67680},[201],[203,67682,67683],{"xmlns":205},[208,67684,67685,67725],{},[211,67686,67687,67697,67699,67701,67703,67705,67707,67709,67711,67713,67715,67717,67719,67721,67723],{},[3054,67688,67689,67691],{},[214,67690,32913],{},[211,67692,67693,67695],{},[217,67694,5074],{},[224,67696,1244],{},[217,67698,43],{"stretchy":241},[214,67700,3936],{},[217,67702,1522],{"stretchy":241},[217,67704,83],{},[217,67706,5074],{},[214,67708,39714],{},[217,67710,21407],{},[217,67712,43],{"stretchy":241},[224,67714,1244],{},[217,67716,5074],{},[214,67718,3936],{},[217,67720,1522],{"stretchy":241},[214,67722,12219],{"mathvariant":484},[214,67724,65434],{},[253,67726,67727],{"encoding":255},"F^{-1}(u)=-\\log(1-u)\u002F\\lambda",[28,67729,67731,67790,67822],{"className":67730,"ariaHidden":261},[260],[28,67732,67734,67737,67772,67775,67778,67781,67784,67787],{"className":67733},[265],[28,67735],{"className":67736,"style":31837},[269],[28,67738,67740,67743],{"className":67739},[274],[28,67741,32913],{"className":67742,"style":459},[274,275],[28,67744,67746],{"className":67745},[556],[28,67747,67749],{"className":67748},[560],[28,67750,67752],{"className":67751},[565],[28,67753,67755],{"className":67754,"style":5477},[569],[28,67756,67757,67760],{"style":3094},[28,67758],{"className":67759,"style":578},[577],[28,67761,67763],{"className":67762},[582,583,584,585],[28,67764,67766,67769],{"className":67765},[274,585],[28,67767,5074],{"className":67768},[274,585],[28,67770,1244],{"className":67771},[274,585],[28,67773,43],{"className":67774},[348],[28,67776,3936],{"className":67777},[274,275],[28,67779,1522],{"className":67780},[355],[28,67782],{"className":67783,"style":302},[279],[28,67785,83],{"className":67786},[306],[28,67788],{"className":67789,"style":302},[279],[28,67791,67793,67796,67799,67802,67807,67810,67813,67816,67819],{"className":67792},[265],[28,67794],{"className":67795,"style":341},[269],[28,67797,5074],{"className":67798},[274],[28,67800],{"className":67801,"style":530},[279],[28,67803,39952,67805],{"className":67804},[21329],[28,67806,34347],{"style":39955},[28,67808,43],{"className":67809},[348],[28,67811,1244],{"className":67812},[274],[28,67814],{"className":67815,"style":280},[279],[28,67817,5074],{"className":67818},[284],[28,67820],{"className":67821,"style":280},[279],[28,67823,67825,67828,67831,67834,67837],{"className":67824},[265],[28,67826],{"className":67827,"style":341},[269],[28,67829,3936],{"className":67830},[274,275],[28,67832,1522],{"className":67833},[355],[28,67835,12219],{"className":67836},[274],[28,67838,65434],{"className":67839},[274,275],[11,67841,67842,67843,68135,68136,68164,68165,68457,68458,68570,68571,68623,68624,68691,68692,68720,68721,68788],{},"举个能算的例子。",[28,67844,67846,67906],{"className":67845,"translate":193},[197],[28,67847,67849],{"className":67848},[201],[203,67850,67851],{"xmlns":205},[208,67852,67853,67903],{},[211,67854,67855,67857,67859,67861,67863,67865,67873,67875,67877,67879,67881,67883,67891,67893,67895,67897,67899,67901],{},[214,67856,11],{},[217,67858,43],{"stretchy":241},[214,67860,176],{},[217,67862,1522],{"stretchy":241},[217,67864,83],{},[5027,67866,67867],{"scriptlevel":89,"displaystyle":241},[21928,67868,67869,67871],{},[224,67870,1244],{},[224,67872,1253],{},[214,67874,510],{},[217,67876,43],{"stretchy":241},[214,67878,176],{},[217,67880,1522],{"stretchy":241},[217,67882,124],{},[5027,67884,67885],{"scriptlevel":89,"displaystyle":241},[21928,67886,67887,67889],{},[224,67888,1244],{},[224,67890,1253],{},[214,67892,510],{},[217,67894,43],{"stretchy":241},[214,67896,176],{},[217,67898,5074],{},[224,67900,3400],{},[217,67902,1522],{"stretchy":241},[253,67904,67905],{"encoding":255},"p(x)=\\tfrac12\\varphi(x)+\\tfrac12\\varphi(x-3)",[28,67907,67909,67936,68031,68123],{"className":67908,"ariaHidden":261},[260],[28,67910,67912,67915,67918,67921,67924,67927,67930,67933],{"className":67911},[265],[28,67913],{"className":67914,"style":341},[269],[28,67916,11],{"className":67917},[274,275],[28,67919,43],{"className":67920},[348],[28,67922,176],{"className":67923},[274,275],[28,67925,1522],{"className":67926},[355],[28,67928],{"className":67929,"style":302},[279],[28,67931,83],{"className":67932},[306],[28,67934],{"className":67935,"style":302},[279],[28,67937,67939,67942,68010,68013,68016,68019,68022,68025,68028],{"className":67938},[265],[28,67940],{"className":67941,"style":21986},[269],[28,67943,67945,67948,68007],{"className":67944},[274],[28,67946],{"className":67947},[348,8285],[28,67949,67951],{"className":67950},[21928],[28,67952,67954,67999],{"className":67953},[560,561],[28,67955,67957,67996],{"className":67956},[565],[28,67958,67960,67974,67982],{"className":67959,"style":22005},[569],[28,67961,67962,67965],{"style":22008},[28,67963],{"className":67964,"style":1914},[577],[28,67966,67968],{"className":67967},[582,583,584,585],[28,67969,67971],{"className":67970},[274,585],[28,67972,1253],{"className":67973},[274,585],[28,67975,67976,67979],{"style":22023},[28,67977],{"className":67978,"style":1914},[577],[28,67980],{"className":67981,"style":22031},[22030],[28,67983,67984,67987],{"style":22034},[28,67985],{"className":67986,"style":1914},[577],[28,67988,67990],{"className":67989},[582,583,584,585],[28,67991,67993],{"className":67992},[274,585],[28,67994,1244],{"className":67995},[274,585],[28,67997,593],{"className":67998},[592],[28,68000,68002],{"className":68001},[565],[28,68003,68005],{"className":68004,"style":22056},[569],[28,68006],{},[28,68008],{"className":68009},[355,8285],[28,68011,510],{"className":68012},[274,275],[28,68014,43],{"className":68015},[348],[28,68017,176],{"className":68018},[274,275],[28,68020,1522],{"className":68021},[355],[28,68023],{"className":68024,"style":280},[279],[28,68026,124],{"className":68027},[284],[28,68029],{"className":68030,"style":280},[279],[28,68032,68034,68037,68105,68108,68111,68114,68117,68120],{"className":68033},[265],[28,68035],{"className":68036,"style":21986},[269],[28,68038,68040,68043,68102],{"className":68039},[274],[28,68041],{"className":68042},[348,8285],[28,68044,68046],{"className":68045},[21928],[28,68047,68049,68094],{"className":68048},[560,561],[28,68050,68052,68091],{"className":68051},[565],[28,68053,68055,68069,68077],{"className":68054,"style":22005},[569],[28,68056,68057,68060],{"style":22008},[28,68058],{"className":68059,"style":1914},[577],[28,68061,68063],{"className":68062},[582,583,584,585],[28,68064,68066],{"className":68065},[274,585],[28,68067,1253],{"className":68068},[274,585],[28,68070,68071,68074],{"style":22023},[28,68072],{"className":68073,"style":1914},[577],[28,68075],{"className":68076,"style":22031},[22030],[28,68078,68079,68082],{"style":22034},[28,68080],{"className":68081,"style":1914},[577],[28,68083,68085],{"className":68084},[582,583,584,585],[28,68086,68088],{"className":68087},[274,585],[28,68089,1244],{"className":68090},[274,585],[28,68092,593],{"className":68093},[592],[28,68095,68097],{"className":68096},[565],[28,68098,68100],{"className":68099,"style":22056},[569],[28,68101],{},[28,68103],{"className":68104},[355,8285],[28,68106,510],{"className":68107},[274,275],[28,68109,43],{"className":68110},[348],[28,68112,176],{"className":68113},[274,275],[28,68115],{"className":68116,"style":280},[279],[28,68118,5074],{"className":68119},[284],[28,68121],{"className":68122,"style":280},[279],[28,68124,68126,68129,68132],{"className":68125},[265],[28,68127],{"className":68128,"style":341},[269],[28,68130,3400],{"className":68131},[274],[28,68133,1522],{"className":68134},[355],"，两个单位方差高斯，中心相距 ",[28,68137,68139,68152],{"className":68138,"translate":193},[197],[28,68140,68142],{"className":68141},[201],[203,68143,68144],{"xmlns":205},[208,68145,68146,68150],{},[211,68147,68148],{},[224,68149,3400],{},[253,68151,3400],{"encoding":255},[28,68153,68155],{"className":68154,"ariaHidden":261},[260],[28,68156,68158,68161],{"className":68157},[265],[28,68159],{"className":68160,"style":316},[269],[28,68162,3400],{"className":68163},[274],"。它的 CDF 是 ",[28,68166,68168,68228],{"className":68167,"translate":193},[197],[28,68169,68171],{"className":68170},[201],[203,68172,68173],{"xmlns":205},[208,68174,68175,68225],{},[211,68176,68177,68179,68181,68183,68185,68187,68195,68197,68199,68201,68203,68205,68213,68215,68217,68219,68221,68223],{},[214,68178,32913],{},[217,68180,43],{"stretchy":241},[214,68182,176],{},[217,68184,1522],{"stretchy":241},[217,68186,83],{},[5027,68188,68189],{"scriptlevel":89,"displaystyle":241},[21928,68190,68191,68193],{},[224,68192,1244],{},[224,68194,1253],{},[214,68196,67590],{"mathvariant":484},[217,68198,43],{"stretchy":241},[214,68200,176],{},[217,68202,1522],{"stretchy":241},[217,68204,124],{},[5027,68206,68207],{"scriptlevel":89,"displaystyle":241},[21928,68208,68209,68211],{},[224,68210,1244],{},[224,68212,1253],{},[214,68214,67590],{"mathvariant":484},[217,68216,43],{"stretchy":241},[214,68218,176],{},[217,68220,5074],{},[224,68222,3400],{},[217,68224,1522],{"stretchy":241},[253,68226,68227],{"encoding":255},"F(x)=\\tfrac12\\Phi(x)+\\tfrac12\\Phi(x-3)",[28,68229,68231,68258,68353,68445],{"className":68230,"ariaHidden":261},[260],[28,68232,68234,68237,68240,68243,68246,68249,68252,68255],{"className":68233},[265],[28,68235],{"className":68236,"style":341},[269],[28,68238,32913],{"className":68239,"style":459},[274,275],[28,68241,43],{"className":68242},[348],[28,68244,176],{"className":68245},[274,275],[28,68247,1522],{"className":68248},[355],[28,68250],{"className":68251,"style":302},[279],[28,68253,83],{"className":68254},[306],[28,68256],{"className":68257,"style":302},[279],[28,68259,68261,68264,68332,68335,68338,68341,68344,68347,68350],{"className":68260},[265],[28,68262],{"className":68263,"style":21986},[269],[28,68265,68267,68270,68329],{"className":68266},[274],[28,68268],{"className":68269},[348,8285],[28,68271,68273],{"className":68272},[21928],[28,68274,68276,68321],{"className":68275},[560,561],[28,68277,68279,68318],{"className":68278},[565],[28,68280,68282,68296,68304],{"className":68281,"style":22005},[569],[28,68283,68284,68287],{"style":22008},[28,68285],{"className":68286,"style":1914},[577],[28,68288,68290],{"className":68289},[582,583,584,585],[28,68291,68293],{"className":68292},[274,585],[28,68294,1253],{"className":68295},[274,585],[28,68297,68298,68301],{"style":22023},[28,68299],{"className":68300,"style":1914},[577],[28,68302],{"className":68303,"style":22031},[22030],[28,68305,68306,68309],{"style":22034},[28,68307],{"className":68308,"style":1914},[577],[28,68310,68312],{"className":68311},[582,583,584,585],[28,68313,68315],{"className":68314},[274,585],[28,68316,1244],{"className":68317},[274,585],[28,68319,593],{"className":68320},[592],[28,68322,68324],{"className":68323},[565],[28,68325,68327],{"className":68326,"style":22056},[569],[28,68328],{},[28,68330],{"className":68331},[355,8285],[28,68333,67590],{"className":68334},[274],[28,68336,43],{"className":68337},[348],[28,68339,176],{"className":68340},[274,275],[28,68342,1522],{"className":68343},[355],[28,68345],{"className":68346,"style":280},[279],[28,68348,124],{"className":68349},[284],[28,68351],{"className":68352,"style":280},[279],[28,68354,68356,68359,68427,68430,68433,68436,68439,68442],{"className":68355},[265],[28,68357],{"className":68358,"style":21986},[269],[28,68360,68362,68365,68424],{"className":68361},[274],[28,68363],{"className":68364},[348,8285],[28,68366,68368],{"className":68367},[21928],[28,68369,68371,68416],{"className":68370},[560,561],[28,68372,68374,68413],{"className":68373},[565],[28,68375,68377,68391,68399],{"className":68376,"style":22005},[569],[28,68378,68379,68382],{"style":22008},[28,68380],{"className":68381,"style":1914},[577],[28,68383,68385],{"className":68384},[582,583,584,585],[28,68386,68388],{"className":68387},[274,585],[28,68389,1253],{"className":68390},[274,585],[28,68392,68393,68396],{"style":22023},[28,68394],{"className":68395,"style":1914},[577],[28,68397],{"className":68398,"style":22031},[22030],[28,68400,68401,68404],{"style":22034},[28,68402],{"className":68403,"style":1914},[577],[28,68405,68407],{"className":68406},[582,583,584,585],[28,68408,68410],{"className":68409},[274,585],[28,68411,1244],{"className":68412},[274,585],[28,68414,593],{"className":68415},[592],[28,68417,68419],{"className":68418},[565],[28,68420,68422],{"className":68421,"style":22056},[569],[28,68423],{},[28,68425],{"className":68426},[355,8285],[28,68428,67590],{"className":68429},[274],[28,68431,43],{"className":68432},[348],[28,68434,176],{"className":68435},[274,275],[28,68437],{"className":68438,"style":280},[279],[28,68440,5074],{"className":68441},[284],[28,68443],{"className":68444,"style":280},[279],[28,68446,68448,68451,68454],{"className":68447},[265],[28,68449],{"className":68450,"style":341},[269],[28,68452,3400],{"className":68453},[274],[28,68455,1522],{"className":68456},[355],"，流就是 ",[28,68459,68461,68490],{"className":68460,"translate":193},[197],[28,68462,68464],{"className":68463},[201],[203,68465,68466],{"xmlns":205},[208,68467,68468,68488],{},[211,68469,68470,68472,68474,68484,68486],{},[214,68471,1511],{},[217,68473,83],{},[3054,68475,68476,68478],{},[214,68477,32913],{},[211,68479,68480,68482],{},[217,68481,5074],{},[224,68483,1244],{},[217,68485,35992],{},[214,68487,67590],{"mathvariant":484},[253,68489,67593],{"encoding":255},[28,68491,68493,68511,68561],{"className":68492,"ariaHidden":261},[260],[28,68494,68496,68499,68502,68505,68508],{"className":68495},[265],[28,68497],{"className":68498,"style":2371},[269],[28,68500,1511],{"className":68501,"style":1546},[274,275],[28,68503],{"className":68504,"style":302},[279],[28,68506,83],{"className":68507},[306],[28,68509],{"className":68510,"style":302},[279],[28,68512,68514,68517,68552,68555,68558],{"className":68513},[265],[28,68515],{"className":68516,"style":5477},[269],[28,68518,68520,68523],{"className":68519},[274],[28,68521,32913],{"className":68522,"style":459},[274,275],[28,68524,68526],{"className":68525},[556],[28,68527,68529],{"className":68528},[560],[28,68530,68532],{"className":68531},[565],[28,68533,68535],{"className":68534,"style":5477},[569],[28,68536,68537,68540],{"style":3094},[28,68538],{"className":68539,"style":578},[577],[28,68541,68543],{"className":68542},[582,583,584,585],[28,68544,68546,68549],{"className":68545},[274,585],[28,68547,5074],{"className":68548},[274,585],[28,68550,1244],{"className":68551},[274,585],[28,68553],{"className":68554,"style":280},[279],[28,68556,35992],{"className":68557},[284],[28,68559],{"className":68560,"style":280},[279],[28,68562,68564,68567],{"className":68563},[265],[28,68565],{"className":68566,"style":455},[269],[28,68568,67590],{"className":68569},[274],"。两峰之间 ",[28,68572,68574,68593],{"className":68573,"translate":193},[197],[28,68575,68577],{"className":68576},[201],[203,68578,68579],{"xmlns":205},[208,68580,68581,68590],{},[211,68582,68583,68585,68587],{},[214,68584,176],{},[217,68586,83],{},[224,68588,68589],{},"1.5",[253,68591,68592],{"encoding":255},"x=1.5",[28,68594,68596,68614],{"className":68595,"ariaHidden":261},[260],[28,68597,68599,68602,68605,68608,68611],{"className":68598},[265],[28,68600],{"className":68601,"style":645},[269],[28,68603,176],{"className":68604},[274,275],[28,68606],{"className":68607,"style":302},[279],[28,68609,83],{"className":68610},[306],[28,68612],{"className":68613,"style":302},[279],[28,68615,68617,68620],{"className":68616},[265],[28,68618],{"className":68619,"style":316},[269],[28,68621,68589],{"className":68622},[274]," 处的真实密度是 ",[28,68625,68627,68652],{"className":68626,"translate":193},[197],[28,68628,68630],{"className":68629},[201],[203,68631,68632],{"xmlns":205},[208,68633,68634,68649],{},[211,68635,68636,68638,68640,68642,68644,68646],{},[214,68637,510],{},[217,68639,43],{"stretchy":241},[224,68641,68589],{},[217,68643,1522],{"stretchy":241},[217,68645,46977],{},[224,68647,68648],{},"0.1295",[253,68650,68651],{"encoding":255},"\\varphi(1.5)\\approx0.1295",[28,68653,68655,68682],{"className":68654,"ariaHidden":261},[260],[28,68656,68658,68661,68664,68667,68670,68673,68676,68679],{"className":68657},[265],[28,68659],{"className":68660,"style":341},[269],[28,68662,510],{"className":68663},[274,275],[28,68665,43],{"className":68666},[348],[28,68668,68589],{"className":68669},[274],[28,68671,1522],{"className":68672},[355],[28,68674],{"className":68675,"style":302},[279],[28,68677,46977],{"className":68678},[306],[28,68680],{"className":68681,"style":302},[279],[28,68683,68685,68688],{"className":68684},[265],[28,68686],{"className":68687,"style":316},[269],[28,68689,68648],{"className":68690},[274],"；如果错当成单个高斯（中心在 ",[28,68693,68695,68708],{"className":68694,"translate":193},[197],[28,68696,68698],{"className":68697},[201],[203,68699,68700],{"xmlns":205},[208,68701,68702,68706],{},[211,68703,68704],{},[224,68705,68589],{},[253,68707,68589],{"encoding":255},[28,68709,68711],{"className":68710,"ariaHidden":261},[260],[28,68712,68714,68717],{"className":68713},[265],[28,68715],{"className":68716,"style":316},[269],[28,68718,68589],{"className":68719},[274],"、单峰），会给出 ",[28,68722,68724,68749],{"className":68723,"translate":193},[197],[28,68725,68727],{"className":68726},[201],[203,68728,68729],{"xmlns":205},[208,68730,68731,68746],{},[211,68732,68733,68735,68737,68739,68741,68743],{},[214,68734,510],{},[217,68736,43],{"stretchy":241},[224,68738,89],{},[217,68740,1522],{"stretchy":241},[217,68742,46977],{},[224,68744,68745],{},"0.3989",[253,68747,68748],{"encoding":255},"\\varphi(0)\\approx0.3989",[28,68750,68752,68779],{"className":68751,"ariaHidden":261},[260],[28,68753,68755,68758,68761,68764,68767,68770,68773,68776],{"className":68754},[265],[28,68756],{"className":68757,"style":341},[269],[28,68759,510],{"className":68760},[274,275],[28,68762,43],{"className":68763},[348],[28,68765,89],{"className":68766},[274],[28,68768,1522],{"className":68769},[355],[28,68771],{"className":68772,"style":302},[279],[28,68774,46977],{"className":68775},[306],[28,68777],{"className":68778,"style":302},[279],[28,68780,68782,68785],{"className":68781},[265],[28,68783],{"className":68784,"style":316},[269],[28,68786,68745],{"className":68787},[274],"，高估三倍。流模型能把这个“鞍部”学出来，靠的正是分段可逆映射，而不是更大的方差。",[11,68790,68791,68792,68891],{},"连续情形就是 Neural ODE：",[28,68793,68795,68831],{"className":68794,"translate":193},[197],[28,68796,68798],{"className":68797},[201],[203,68799,68800],{"xmlns":205},[208,68801,68802,68828],{},[211,68803,68804,68806,68808,68810,68812,68814,68816,68818,68820,68822,68824,68826],{},[214,68805,12988],{},[214,68807,4838],{},[214,68809,12219],{"mathvariant":484},[214,68811,12988],{},[214,68813,1241],{},[217,68815,83],{},[214,68817,1693],{},[217,68819,43],{"stretchy":241},[214,68821,4838],{},[217,68823,1683],{"separator":261},[214,68825,1241],{},[217,68827,1522],{"stretchy":241},[253,68829,68830],{"encoding":255},"dz\u002Fdt=v(z,t)",[28,68832,68834,68864],{"className":68833,"ariaHidden":261},[260],[28,68835,68837,68840,68843,68846,68849,68852,68855,68858,68861],{"className":68836},[265],[28,68838],{"className":68839,"style":341},[269],[28,68841,12988],{"className":68842},[274,275],[28,68844,4838],{"className":68845,"style":4956},[274,275],[28,68847,12219],{"className":68848},[274],[28,68850,12988],{"className":68851},[274,275],[28,68853,1241],{"className":68854},[274,275],[28,68856],{"className":68857,"style":302},[279],[28,68859,83],{"className":68860},[306],[28,68862],{"className":68863,"style":302},[279],[28,68865,68867,68870,68873,68876,68879,68882,68885,68888],{"className":68866},[265],[28,68868],{"className":68869,"style":341},[269],[28,68871,1693],{"className":68872,"style":298},[274,275],[28,68874,43],{"className":68875},[348],[28,68877,4838],{"className":68878,"style":4956},[274,275],[28,68880,1683],{"className":68881},[1723],[28,68883],{"className":68884,"style":530},[279],[28,68886,1241],{"className":68887},[274,275],[28,68889,1522],{"className":68890},[355],"，密度沿轨迹演化，",[28,68893,68895],{"className":68894,"translate":193},[192],[28,68896,68898,68970],{"className":68897,"translate":193},[197],[28,68899,68901],{"className":68900},[201],[203,68902,68903],{"xmlns":205,"display":206},[208,68904,68905,68967],{},[211,68906,68907,68917,68919,68921,68927,68929,68935,68937,68939,68941,68947,68949,68963,68965],{},[21928,68908,68909,68911],{},[214,68910,12988],{},[211,68912,68913,68915],{},[214,68914,12988],{},[214,68916,1241],{},[214,68918,39714],{},[217,68920,21407],{},[501,68922,68923,68925],{},[214,68924,11],{},[214,68926,1241],{},[217,68928,43],{"stretchy":241},[501,68930,68931,68933],{},[214,68932,4838],{},[214,68934,1241],{},[217,68936,1522],{"stretchy":241},[217,68938,83],{},[217,68940,5074],{},[211,68942,68943,68945],{},[214,68944,1241],{"mathvariant":484},[214,68946,1867],{"mathvariant":484},[217,68948,43],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[21928,68950,68951,68957],{},[211,68952,68953,68955],{},[214,68954,43666],{"mathvariant":484},[214,68956,1693],{},[211,68958,68959,68961],{},[214,68960,43666],{"mathvariant":484},[214,68962,4838],{},[217,68964,1522],{"fence":261,"stretchy":261,"minsize":22362,"maxsize":22362},[217,68966,1683],{"separator":261},[253,68968,68969],{"encoding":255},"\\frac{d}{dt}\\log p_t(z_t)=-\\mathrm{tr}\\Bigl(\\frac{\\partial v}{\\partial z}\\Bigr),",[28,68971,68973,69152],{"className":68972,"ariaHidden":261},[260],[28,68974,68976,68980,69045,69048,69053,69056,69096,69099,69140,69143,69146,69149],{"className":68975},[265],[28,68977],{"className":68978,"style":68979},[269],"height:2.0574em;vertical-align:-0.686em;",[28,68981,68983,68986,69042],{"className":68982},[274],[28,68984],{"className":68985},[348,8285],[28,68987,68989],{"className":68988},[21928],[28,68990,68992,69034],{"className":68991},[560,561],[28,68993,68995,69031],{"className":68994},[565],[28,68996,68998,69012,69020],{"className":68997,"style":65988},[569],[28,68999,69000,69003],{"style":24920},[28,69001],{"className":69002,"style":1914},[577],[28,69004,69006,69009],{"className":69005},[274],[28,69007,12988],{"className":69008},[274,275],[28,69010,1241],{"className":69011},[274,275],[28,69013,69014,69017],{"style":22023},[28,69015],{"className":69016,"style":1914},[577],[28,69018],{"className":69019,"style":22031},[22030],[28,69021,69022,69025],{"style":24940},[28,69023],{"className":69024,"style":1914},[577],[28,69026,69028],{"className":69027},[274],[28,69029,12988],{"className":69030},[274,275],[28,69032,593],{"className":69033},[592],[28,69035,69037],{"className":69036},[565],[28,69038,69040],{"className":69039,"style":24985},[569],[28,69041],{},[28,69043],{"className":69044},[355,8285],[28,69046],{"className":69047,"style":530},[279],[28,69049,39952,69051],{"className":69050},[21329],[28,69052,34347],{"style":39955},[28,69054],{"className":69055,"style":530},[279],[28,69057,69059,69062],{"className":69058},[274],[28,69060,11],{"className":69061},[274,275],[28,69063,69065],{"className":69064},[556],[28,69066,69068,69088],{"className":69067},[560,561],[28,69069,69071,69085],{"className":69070},[565],[28,69072,69074],{"className":69073,"style":3978},[569],[28,69075,69076,69079],{"style":573},[28,69077],{"className":69078,"style":578},[577],[28,69080,69082],{"className":69081},[582,583,584,585],[28,69083,1241],{"className":69084},[274,275,585],[28,69086,593],{"className":69087},[592],[28,69089,69091],{"className":69090},[565],[28,69092,69094],{"className":69093,"style":600},[569],[28,69095],{},[28,69097,43],{"className":69098},[348],[28,69100,69102,69105],{"className":69101},[274],[28,69103,4838],{"className":69104,"style":4956},[274,275],[28,69106,69108],{"className":69107},[556],[28,69109,69111,69132],{"className":69110},[560,561],[28,69112,69114,69129],{"className":69113},[565],[28,69115,69117],{"className":69116,"style":3978},[569],[28,69118,69120,69123],{"style":69119},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[28,69121],{"className":69122,"style":578},[577],[28,69124,69126],{"className":69125},[582,583,584,585],[28,69127,1241],{"className":69128},[274,275,585],[28,69130,593],{"className":69131},[592],[28,69133,69135],{"className":69134},[565],[28,69136,69138],{"className":69137,"style":600},[569],[28,69139],{},[28,69141,1522],{"className":69142},[355],[28,69144],{"className":69145,"style":302},[279],[28,69147,83],{"className":69148},[306],[28,69150],{"className":69151,"style":302},[279],[28,69153,69155,69158,69161,69167,69173,69241,69247],{"className":69154},[265],[28,69156],{"className":69157,"style":68979},[269],[28,69159,5074],{"className":69160},[274],[28,69162,69164],{"className":69163},[274],[28,69165,36754],{"className":69166},[274,1712],[28,69168,69170],{"className":69169},[348],[28,69171,43],{"className":69172},[2169,22506],[28,69174,69176,69179,69238],{"className":69175},[274],[28,69177],{"className":69178},[348,8285],[28,69180,69182],{"className":69181},[21928],[28,69183,69185,69230],{"className":69184},[560,561],[28,69186,69188,69227],{"className":69187},[565],[28,69189,69191,69205,69213],{"className":69190,"style":65988},[569],[28,69192,69193,69196],{"style":24920},[28,69194],{"className":69195,"style":1914},[577],[28,69197,69199,69202],{"className":69198},[274],[28,69200,43666],{"className":69201,"style":43914},[274],[28,69203,4838],{"className":69204,"style":4956},[274,275],[28,69206,69207,69210],{"style":22023},[28,69208],{"className":69209,"style":1914},[577],[28,69211],{"className":69212,"style":22031},[22030],[28,69214,69215,69218],{"style":24940},[28,69216],{"className":69217,"style":1914},[577],[28,69219,69221,69224],{"className":69220},[274],[28,69222,43666],{"className":69223,"style":43914},[274],[28,69225,1693],{"className":69226,"style":298},[274,275],[28,69228,593],{"className":69229},[592],[28,69231,69233],{"className":69232},[565],[28,69234,69236],{"className":69235,"style":24985},[569],[28,69237],{},[28,69239],{"className":69240},[355,8285],[28,69242,69244],{"className":69243},[355],[28,69245,1522],{"className":69246},[2169,22506],[28,69248,1683],{"className":69249},[1723],[11,69251,69252,69253,6904,69422,69450],{},"这其实是变量替换公式的无穷小版本，行列式取对数后变成迹。迹在高维不好算，实践中用 Hutchinson 估计 ",[28,69254,69256,69302],{"className":69255,"translate":193},[197],[28,69257,69259],{"className":69258},[201],[203,69260,69261],{"xmlns":205},[208,69262,69263,69299],{},[211,69264,69265,69271,69273,69275,69277,69279,69285,69287,69293,69295,69297],{},[211,69266,69267,69269],{},[214,69268,1241],{"mathvariant":484},[214,69270,1867],{"mathvariant":484},[217,69272,43],{"stretchy":241},[214,69274,903],{},[217,69276,1522],{"stretchy":241},[217,69278,46977],{},[501,69280,69281,69283],{},[214,69282,42084],{"mathvariant":12057},[214,69284,24625],{},[217,69286,73],{"stretchy":241},[3054,69288,69289,69291],{},[214,69290,24625],{},[214,69292,884],{"mathvariant":484},[214,69294,903],{},[214,69296,24625],{},[217,69298,246],{"stretchy":241},[253,69300,69301],{"encoding":255},"\\mathrm{tr}(A)\\approx\\mathbb E_\\varepsilon[\\varepsilon^\\top A\\varepsilon]",[28,69303,69305,69335],{"className":69304,"ariaHidden":261},[260],[28,69306,69308,69311,69317,69320,69323,69326,69329,69332],{"className":69307},[265],[28,69309],{"className":69310,"style":341},[269],[28,69312,69314],{"className":69313},[274],[28,69315,36754],{"className":69316},[274,1712],[28,69318,43],{"className":69319},[348],[28,69321,903],{"className":69322},[274,275],[28,69324,1522],{"className":69325},[355],[28,69327],{"className":69328,"style":302},[279],[28,69330,46977],{"className":69331},[306],[28,69333],{"className":69334,"style":302},[279],[28,69336,69338,69341,69381,69384,69413,69416,69419],{"className":69337},[265],[28,69339],{"className":69340,"style":40607},[269],[28,69342,69344,69347],{"className":69343},[274],[28,69345,42084],{"className":69346},[274,12195],[28,69348,69350],{"className":69349},[556],[28,69351,69353,69373],{"className":69352},[560,561],[28,69354,69356,69370],{"className":69355},[565],[28,69357,69359],{"className":69358,"style":23070},[569],[28,69360,69361,69364],{"style":573},[28,69362],{"className":69363,"style":578},[577],[28,69365,69367],{"className":69366},[582,583,584,585],[28,69368,24625],{"className":69369},[274,275,585],[28,69371,593],{"className":69372},[592],[28,69374,69376],{"className":69375},[565],[28,69377,69379],{"className":69378,"style":600},[569],[28,69380],{},[28,69382,73],{"className":69383},[348],[28,69385,69387,69390],{"className":69386},[274],[28,69388,24625],{"className":69389},[274,275],[28,69391,69393],{"className":69392},[556],[28,69394,69396],{"className":69395},[560],[28,69397,69399],{"className":69398},[565],[28,69400,69402],{"className":69401,"style":40635},[569],[28,69403,69404,69407],{"style":3094},[28,69405],{"className":69406,"style":578},[577],[28,69408,69410],{"className":69409},[582,583,584,585],[28,69411,884],{"className":69412},[274,585],[28,69414,903],{"className":69415},[274,275],[28,69417,24625],{"className":69418},[274,275],[28,69420,246],{"className":69421},[355],[28,69423,69425,69438],{"className":69424,"translate":193},[197],[28,69426,69428],{"className":69427},[201],[203,69429,69430],{"xmlns":205},[208,69431,69432,69436],{},[211,69433,69434],{},[214,69435,24625],{},[253,69437,24628],{"encoding":255},[28,69439,69441],{"className":69440,"ariaHidden":261},[260],[28,69442,69444,69447],{"className":69443},[265],[28,69445],{"className":69446,"style":645},[269],[28,69448,24625],{"className":69449},[274,275]," 取 Rademacher 向量，代价是方差。",[11,69452,69453,69454,69487,69488,69521],{},"流与最优传输有一个漂亮的接口。Brenier 说最优映射是 ",[28,69455,69457,69472],{"className":69456,"translate":193},[197],[28,69458,69460],{"className":69459},[201],[203,69461,69462],{"xmlns":205},[208,69463,69464,69470],{},[211,69465,69466,69468],{},[214,69467,31947],{"mathvariant":484},[214,69469,3936],{},[253,69471,32034],{"encoding":255},[28,69473,69475],{"className":69474,"ariaHidden":261},[260],[28,69476,69478,69481,69484],{"className":69477},[265],[28,69479],{"className":69480,"style":455},[269],[28,69482,31947],{"className":69483},[274],[28,69485,3936],{"className":69486},[274,275],"，流说数据由可逆映射得到；当映射取 ",[28,69489,69491,69506],{"className":69490,"translate":193},[197],[28,69492,69494],{"className":69493},[201],[203,69495,69496],{"xmlns":205},[208,69497,69498,69504],{},[211,69499,69500,69502],{},[214,69501,31947],{"mathvariant":484},[214,69503,3936],{},[253,69505,32034],{"encoding":255},[28,69507,69509],{"className":69508,"ariaHidden":261},[260],[28,69510,69512,69515,69518],{"className":69511},[265],[28,69513],{"className":69514,"style":455},[269],[28,69516,31947],{"className":69517},[274],[28,69519,3936],{"className":69520},[274,275]," 时两者重合，于是可以问：能不能直接学那个“最优”的流？Flow matching 就是这么做的。它的条件目标是",[28,69523,69525],{"className":69524,"translate":193},[192],[28,69526,69528,69656],{"className":69527,"translate":193},[197],[28,69529,69531],{"className":69530},[201],[203,69532,69533],{"xmlns":205,"display":206},[208,69534,69535,69653],{},[211,69536,69537,69539,69541,69565,69567,69573,69575,69581,69583,69585,69587,69589,69591,69597,69599,69605,69607,69613,69615,69617,69623,69625,69627,69629,69631,69633,69635,69641,69643,69645,69651],{},[214,69538,24393],{"mathvariant":20826},[217,69540,83],{},[501,69542,69543,69545],{},[214,69544,42084],{"mathvariant":12057},[211,69546,69547,69549,69551,69557,69559],{},[214,69548,1241],{},[217,69550,1683],{"separator":261},[501,69552,69553,69555],{},[214,69554,176],{},[224,69556,89],{},[217,69558,1683],{"separator":261},[501,69560,69561,69563],{},[214,69562,176],{},[224,69564,1244],{},[217,69566,24404],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[501,69568,69569,69571],{},[214,69570,1693],{},[214,69572,43555],{},[217,69574,43],{"stretchy":241},[501,69576,69577,69579],{},[214,69578,176],{},[214,69580,1241],{},[217,69582,1683],{"separator":261},[214,69584,1241],{},[217,69586,1522],{"stretchy":241},[217,69588,5074],{},[217,69590,43],{"stretchy":241},[501,69592,69593,69595],{},[214,69594,176],{},[224,69596,1244],{},[217,69598,5074],{},[501,69600,69601,69603],{},[214,69602,176],{},[224,69604,89],{},[217,69606,1522],{"stretchy":241},[3054,69608,69609,69611],{},[217,69610,24404],{"fence":261,"stretchy":261,"minsize":1810,"maxsize":1810},[224,69612,1253],{},[217,69614,1683],{"separator":261},[279,69616],{"width":4053},[501,69618,69619,69621],{},[214,69620,176],{},[214,69622,1241],{},[217,69624,83],{},[217,69626,43],{"stretchy":241},[224,69628,1244],{},[217,69630,5074],{},[214,69632,1241],{},[217,69634,1522],{"stretchy":241},[501,69636,69637,69639],{},[214,69638,176],{},[224,69640,89],{},[217,69642,124],{},[214,69644,1241],{},[501,69646,69647,69649],{},[214,69648,176],{},[224,69650,1244],{},[217,69652,1683],{"separator":261},[253,69654,69655],{"encoding":255},"\\mathcal L=\\mathbb E_{t,x_0,x_1}\\bigl\\|v_\\theta(x_t,t)-(x_1-x_0)\\bigr\\|^2,\n\\qquad x_t=(1-t)x_0+tx_1,",[28,69657,69659,69677,69956,70014,70184,70205,70266],{"className":69658,"ariaHidden":261},[260],[28,69660,69662,69665,69668,69671,69674],{"className":69661},[265],[28,69663],{"className":69664,"style":455},[269],[28,69666,24393],{"className":69667},[274,20866],[28,69669],{"className":69670,"style":302},[279],[28,69672,83],{"className":69673},[306],[28,69675],{"className":69676,"style":302},[279],[28,69678,69680,69683,69815,69852,69892,69895,69935,69938,69941,69944,69947,69950,69953],{"className":69679},[265],[28,69681],{"className":69682,"style":2469},[269],[28,69684,69686,69689],{"className":69685},[274],[28,69687,42084],{"className":69688},[274,12195],[28,69690,69692],{"className":69691},[556],[28,69693,69695,69807],{"className":69694},[560,561],[28,69696,69698,69804],{"className":69697},[565],[28,69699,69701],{"className":69700,"style":3978},[569],[28,69702,69703,69706],{"style":573},[28,69704],{"className":69705,"style":578},[577],[28,69707,69709],{"className":69708},[582,583,584,585],[28,69710,69712,69715,69718,69761,69764],{"className":69711},[274,585],[28,69713,1241],{"className":69714},[274,275,585],[28,69716,1683],{"className":69717},[1723,585],[28,69719,69721,69724],{"className":69720},[274,585],[28,69722,176],{"className":69723},[274,275,585],[28,69725,69727],{"className":69726},[556],[28,69728,69730,69752],{"className":69729},[560,561],[28,69731,69733,69749],{"className":69732},[565],[28,69734,69737],{"className":69735,"style":69736},[569],"height:0.3173em;",[28,69738,69740,69743],{"style":69739},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[28,69741],{"className":69742,"style":31571},[577],[28,69744,69746],{"className":69745},[582,31575,2170,585],[28,69747,89],{"className":69748},[274,585],[28,69750,593],{"className":69751},[592],[28,69753,69755],{"className":69754},[565],[28,69756,69759],{"className":69757,"style":69758},[569],"height:0.143em;",[28,69760],{},[28,69762,1683],{"className":69763},[1723,585],[28,69765,69767,69770],{"className":69766},[274,585],[28,69768,176],{"className":69769},[274,275,585],[28,69771,69773],{"className":69772},[556],[28,69774,69776,69796],{"className":69775},[560,561],[28,69777,69779,69793],{"className":69778},[565],[28,69780,69782],{"className":69781,"style":69736},[569],[28,69783,69784,69787],{"style":69739},[28,69785],{"className":69786,"style":31571},[577],[28,69788,69790],{"className":69789},[582,31575,2170,585],[28,69791,1244],{"className":69792},[274,585],[28,69794,593],{"className":69795},[592],[28,69797,69799],{"className":69798},[565],[28,69800,69802],{"className":69801,"style":69758},[569],[28,69803],{},[28,69805,593],{"className":69806},[592],[28,69808,69810],{"className":69809},[565],[28,69811,69813],{"className":69812,"style":20967},[569],[28,69814],{},[28,69816,69818],{"className":69817},[348],[28,69819,69821],{"className":69820},[2169,33118],[28,69822,69824,69844],{"className":69823},[560,561],[28,69825,69827,69841],{"className":69826},[565],[28,69828,69830],{"className":69829,"style":2091},[569],[28,69831,69832,69835],{"style":33130},[28,69833],{"className":69834,"style":33134},[577],[28,69836,69837],{"style":52606},[1961,69838,69839],{"xmlns":1963,"width":52609,"height":1810,"viewBox":52610},[1969,69840],{"d":52613},[28,69842,593],{"className":69843},[592],[28,69845,69847],{"className":69846},[565],[28,69848,69850],{"className":69849,"style":33154},[569],[28,69851],{},[28,69853,69855,69858],{"className":69854},[274],[28,69856,1693],{"className":69857,"style":298},[274,275],[28,69859,69861],{"className":69860},[556],[28,69862,69864,69884],{"className":69863},[560,561],[28,69865,69867,69881],{"className":69866},[565],[28,69868,69870],{"className":69869,"style":20945},[569],[28,69871,69872,69875],{"style":3660},[28,69873],{"className":69874,"style":578},[577],[28,69876,69878],{"className":69877},[582,583,584,585],[28,69879,43555],{"className":69880,"style":2279},[274,275,585],[28,69882,593],{"className":69883},[592],[28,69885,69887],{"className":69886},[565],[28,69888,69890],{"className":69889,"style":600},[569],[28,69891],{},[28,69893,43],{"className":69894},[348],[28,69896,69898,69901],{"className":69897},[274],[28,69899,176],{"className":69900},[274,275],[28,69902,69904],{"className":69903},[556],[28,69905,69907,69927],{"className":69906},[560,561],[28,69908,69910,69924],{"className":69909},[565],[28,69911,69913],{"className":69912,"style":3978},[569],[28,69914,69915,69918],{"style":573},[28,69916],{"className":69917,"style":578},[577],[28,69919,69921],{"className":69920},[582,583,584,585],[28,69922,1241],{"className":69923},[274,275,585],[28,69925,593],{"className":69926},[592],[28,69928,69930],{"className":69929},[565],[28,69931,69933],{"className":69932,"style":600},[569],[28,69934],{},[28,69936,1683],{"className":69937},[1723],[28,69939],{"className":69940,"style":530},[279],[28,69942,1241],{"className":69943},[274,275],[28,69945,1522],{"className":69946},[355],[28,69948],{"className":69949,"style":280},[279],[28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是噪声与数据的配对。如果配对来自 OT 计划，这条直线就是 ",[28,70457,70459,70476],{"className":70458,"translate":193},[197],[28,70460,70462],{"className":70461},[201],[203,70463,70464],{"xmlns":205},[208,70465,70466,70474],{},[211,70467,70468],{},[501,70469,70470,70472],{},[214,70471,22957],{},[224,70473,1253],{},[253,70475,38866],{"encoding":255},[28,70477,70479],{"className":70478,"ariaHidden":261},[260],[28,70480,70482,70485],{"className":70481},[265],[28,70483],{"className":70484,"style":25059},[269],[28,70486,70488,70491],{"className":70487},[274],[28,70489,22957],{"className":70490,"style":459},[274,275],[28,70492,70494],{"className":70493},[556],[28,70495,70497,70517],{"className":70496},[560,561],[28,70498,70500,70514],{"className":70499},[565],[28,70501,70503],{"className":70502,"style":1285},[569],[28,70504,70505,70508],{"style":20948},[28,70506],{"className":70507,"style":578},[577],[28,70509,70511],{"className":70510},[582,583,584,585],[28,70512,1253],{"className":70513},[274,585],[28,70515,593],{"className":70516},[592],[28,70518,70520],{"className":70519},[565],[28,70521,70523],{"className":70522,"style":600},[569],[28,70524],{}," 的测地线，轨迹最直，采样需要的步数最少。这是我看到的“最优传输”与“生成模型”结合得最干净的一处：OT 在这里不是损失函数，而是给流指定了走哪条路。",[11,70527,70528],{},"不过我要泼一盆冷水。流模型能给出很好的密度，但似然不等于“属于这个类”。Nalisnick 等人在 2019 年观察到，在图像上训练得很好的流模型给 OOD 样本的似然反而更高；原因还是典型集——模型把质量摊在一个高维薄壳上，OOD 样本落在壳外，却可能恰好落在密度不低的地方。所以流更适合做生成、做密度比、做精确的变量替换，而不是拿来当 OOD 分数直接用。",[414,70530,70531],{"id":70531},"落地",[11,70533,70534],{},"现在把这些工具放回机器学习里，按我实际见过、用过的顺序说。",[11,70536,70537,70538,70579,70580,70649],{},"生成模型是最直接的一站。GAN 的原始目标用 JS 散度，而 JS 在两个分布支撑不交时是个常数 ",[28,70539,70541,70559],{"className":70540,"translate":193},[197],[28,70542,70544],{"className":70543},[201],[203,70545,70546],{"xmlns":205},[208,70547,70548,70556],{},[211,70549,70550,70552,70554],{},[214,70551,39714],{},[217,70553,21407],{},[224,70555,1253],{},[253,70557,70558],{"encoding":255},"\\log2",[28,70560,70562],{"className":70561,"ariaHidden":261},[260],[28,70563,70565,70568,70573,70576],{"className":70564},[265],[28,70566],{"className":70567,"style":2371},[269],[28,70569,39952,70571],{"className":70570},[21329],[28,70572,34347],{"style":39955},[28,70574],{"className":70575,"style":530},[279],[28,70577,1253],{"className":70578},[274],"，梯度为零——这是早期 GAN 训练不稳定的理论解释之一。WGAN 换成 ",[28,70581,70583,70600],{"className":70582,"translate":193},[197],[28,70584,70586],{"className":70585},[201],[203,70587,70588],{"xmlns":205},[208,70589,70590,70598],{},[211,70591,70592],{},[501,70593,70594,70596],{},[214,70595,22957],{},[224,70597,1244],{},[253,70599,25049],{"encoding":255},[28,70601,70603],{"className":70602,"ariaHidden":261},[260],[28,70604,70606,70609],{"className":70605},[265],[28,70607],{"className":70608,"style":25059},[269],[28,70610,70612,70615],{"className":70611},[274],[28,70613,22957],{"className":70614,"style":459},[274,275],[28,70616,70618],{"className":70617},[556],[28,70619,70621,70641],{"className":70620},[560,561],[28,70622,70624,70638],{"className":70623},[565],[28,70625,70627],{"className":70626,"style":1285},[569],[28,70628,70629,70632],{"style":20948},[28,70630],{"className":70631,"style":578},[577],[28,70633,70635],{"className":70634},[582,583,584,585],[28,70636,1244],{"className":70637},[274,585],[28,70639,593],{"className":70640},[592],[28,70642,70644],{"className":70643},[565],[28,70645,70647],{"className":70646,"style":600},[569],[28,70648],{},"，用对偶式",[28,70651,70653],{"className":70652,"translate":193},[192],[28,70654,70656,70758],{"className":70655,"translate":193},[197],[28,70657,70659],{"className":70658},[201],[203,70660,70661],{"xmlns":205,"display":206},[208,70662,70663,70755],{},[211,70664,70665,70675,70705,70707,70717,70719,70721,70723,70725,70727,70729,70731,70741,70743,70745,70747,70749,70751,70753],{},[1771,70666,70667,70673],{},[211,70668,70669,70671],{},[214,70670,39653],{},[217,70672,21407],{},[214,70674,35082],{},[1771,70676,70677,70683],{},[211,70678,70679,70681],{},[214,70680,29958],{},[217,70682,21407],{},[211,70684,70685,70687,70689,70701,70703],{},[214,70686,24404],{"mathvariant":484},[214,70688,1511],{},[501,70690,70691,70693],{},[214,70692,24404],{"mathvariant":484},[211,70694,70695,70697,70699],{},[214,70696,24393],{"mathvariant":484},[214,70698,1686],{"mathvariant":484},[214,70700,11],{"mathvariant":484},[217,70702,1876],{},[224,70704,1244],{},[228,70706,14860],{},[501,70708,70709,70711],{},[214,70710,42084],{"mathvariant":12057},[501,70712,70713,70715],{},[214,70714,15825],{},[214,70716,1867],{},[217,70718,73],{"stretchy":241},[214,70720,1511],{},[217,70722,43],{"stretchy":241},[214,70724,176],{},[217,70726,1522],{"stretchy":241},[217,70728,246],{"stretchy":241},[217,70730,5074],{},[501,70732,70733,70735],{},[214,70734,42084],{"mathvariant":12057},[501,70736,70737,70739],{},[214,70738,15825],{},[214,70740,34347],{},[217,70742,73],{"stretchy":241},[214,70744,1511],{},[217,70746,43],{"stretchy":241},[214,70748,176],{},[217,70750,1522],{"stretchy":241},[217,70752,246],{"stretchy":241},[217,70754,1683],{"separator":261},[253,70756,70757],{"encoding":255},"\\min_G\\max_{\\|f\\|_{\\mathrm{Lip}}\\le1}\\ \\mathbb E_{P_r}[f(x)]-\\mathbb E_{P_g}[f(x)],",[28,70759,70761,71031],{"className":70760,"ariaHidden":261},[260],[28,70762,70764,70768,70812,70815,70918,70921,70924,71007,71010,71013,71016,71019,71022,71025,71028],{"className":70763},[265],[28,70765],{"className":70766,"style":70767},[269],"height:1.7385em;vertical-align:-0.9885em;",[28,70769,70771],{"className":70770},[21329,21485],[28,70772,70774,70804],{"className":70773},[560,561],[28,70775,70777,70801],{"className":70776},[565],[28,70778,70780,70791],{"className":70779,"style":39754},[569],[28,70781,70782,70785],{"style":21497},[28,70783],{"className":70784,"style":1914},[577],[28,70786,70788],{"className":70787},[582,583,584,585],[28,70789,35082],{"className":70790},[274,275,585],[28,70792,70793,70796],{"style":1926},[28,70794],{"className":70795,"style":1914},[577],[28,70797,70798],{},[28,70799,39653],{"className":70800},[21329],[28,70802,593],{"className":70803},[592],[28,70805,70807],{"className":70806},[565],[28,70808,70810],{"className":70809,"style":21532},[569],[28,70811],{},[28,70813],{"className":70814,"style":530},[279],[28,70816,70818],{"className":70817},[21329,21485],[28,70819,70821,70909],{"className":70820},[560,561],[28,70822,70824,70906],{"className":70823},[565],[28,70825,70827,70896],{"className":70826,"style":645},[569],[28,70828,70829,70832],{"style":23141},[28,70830],{"className":70831,"style":1914},[577],[28,70833,70835],{"className":70834},[582,583,584,585],[28,70836,70838,70841,70844,70890,70893],{"className":70837},[274,585],[28,70839,24404],{"className":70840},[274,585],[28,70842,1511],{"className":70843,"style":1546},[274,275,585],[28,70845,70847,70850],{"className":70846},[274,585],[28,70848,24404],{"className":70849},[274,585],[28,70851,70853],{"className":70852},[556],[28,70854,70856,70882],{"className":70855},[560,561],[28,70857,70859,70879],{"className":70858},[565],[28,70860,70862],{"className":70861,"style":31564},[569],[28,70863,70864,70867],{"style":31567},[28,70865],{"className":70866,"style":31571},[577],[28,70868,70870],{"className":70869},[582,31575,2170,585],[28,70871,70873],{"className":70872},[274,585],[28,70874,70876],{"className":70875},[274,585],[28,70877,31585],{"className":70878},[274,1712,585],[28,70880,593],{"className":70881},[592],[28,70883,70885],{"className":70884},[565],[28,70886,70888],{"className":70887,"style":31595},[569],[28,70889],{},[28,70891,1876],{"className":70892},[306,585],[28,70894,1244],{"className":70895},[274,585],[28,70897,70898,70901],{"style":1926},[28,70899],{"className":70900,"style":1914},[577],[28,70902,70903],{},[28,70904,29958],{"className":70905},[21329],[28,70907,593],{"className":70908},[592],[28,70910,70912],{"className":70911},[565],[28,70913,70916],{"className":70914,"style":70915},[569],"height:0.9885em;",[28,70917],{},[28,70919,14860],{"className":70920},[279],[28,70922],{"className":70923,"style":530},[279],[28,70925,70927,70930],{"className":70926},[274],[28,70928,42084],{"className":70929},[274,12195],[28,70931,70933],{"className":70932},[556],[28,70934,70936,70998],{"className":70935},[560,561],[28,70937,70939,70995],{"className":70938},[565],[28,70940,70942],{"className":70941,"style":570},[569],[28,70943,70944,70947],{"style":573},[28,70945],{"className":70946,"style":578},[577],[28,70948,70950],{"className":70949},[582,583,584,585],[28,70951,70953],{"className":70952},[274,585],[28,70954,70956,70959],{"className":70955},[274,585],[28,70957,15825],{"className":70958,"style":459},[274,275,585],[28,70960,70962],{"className":70961},[556],[28,70963,70965,70987],{"className":70964},[560,561],[28,70966,70968,70984],{"className":70967},[565],[28,70969,70972],{"className":70970,"style":70971},[569],"height:0.1645em;",[28,70973,70975,70978],{"style":70974},"top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;",[28,70976],{"className":70977,"style":31571},[577],[28,70979,70981],{"className":70980},[582,31575,2170,585],[28,70982,1867],{"className":70983,"style":2279},[274,275,585],[28,70985,593],{"className":70986},[592],[28,70988,70990],{"className":70989},[565],[28,70991,70993],{"className":70992,"style":69758},[569],[28,70994],{},[28,70996,593],{"className":70997},[592],[28,70999,71001],{"className":71000},[565],[28,71002,71005],{"className":71003,"style":71004},[569],"height:0.2501em;",[28,71006],{},[28,71008,73],{"className":71009},[348],[28,71011,1511],{"className":71012,"style":1546},[274,275],[28,71014,43],{"className":71015},[348],[28,71017,176],{"className":71018},[274,275],[28,71020,50065],{"className":71021},[355],[28,71023],{"className":71024,"style":280},[279],[28,71026,5074],{"className":71027},[284],[28,71029],{"className":71030,"style":280},[279],[28,71032,71034,71038,71120,71123,71126,71129,71132,71135],{"className":71033},[265],[28,71035],{"className":71036,"style":71037},[269],"height:1.0973em;vertical-align:-0.3473em;",[28,71039,71041,71044],{"className":71040},[274],[28,71042,42084],{"className":71043},[274,12195],[28,71045,71047],{"className":71046},[556],[28,71048,71050,71111],{"className":71049},[560,561],[28,71051,71053,71108],{"className":71052},[565],[28,71054,71056],{"className":71055,"style":570},[569],[28,71057,71058,71061],{"style":573},[28,71059],{"className":71060,"style":578},[577],[28,71062,71064],{"className":71063},[582,583,584,585],[28,71065,71067],{"className":71066},[274,585],[28,71068,71070,71073],{"className":71069},[274,585],[28,71071,15825],{"className":71072,"style":459},[274,275,585],[28,71074,71076],{"className":71075},[556],[28,71077,71079,71099],{"className":71078},[560,561],[28,71080,71082,71096],{"className":71081},[565],[28,71083,71085],{"className":71084,"style":70971},[569],[28,71086,71087,71090],{"style":70974},[28,71088],{"className":71089,"style":31571},[577],[28,71091,71093],{"className":71092},[582,31575,2170,585],[28,71094,34347],{"className":71095,"style":298},[274,275,585],[28,71097,593],{"className":71098},[592],[28,71100,71102],{"className":71101},[565],[28,71103,71106],{"className":71104,"style":71105},[569],"height:0.2819em;",[28,71107],{},[28,71109,593],{"className":71110},[592],[28,71112,71114],{"className":71113},[565],[28,71115,71118],{"className":71116,"style":71117},[569],"height:0.3473em;",[28,71119],{},[28,71121,73],{"className":71122},[348],[28,71124,1511],{"className":71125,"style":1546},[274,275],[28,71127,43],{"className":71128},[348],[28,71130,176],{"className":71131},[274,275],[28,71133,50065],{"className":71134},[355],[28,71136,1683],{"className":71137},[1723],[11,71139,71140,71141,71169],{},"critic ",[28,71142,71144,71157],{"className":71143,"translate":193},[197],[28,71145,71147],{"className":71146},[201],[203,71148,71149],{"xmlns":205},[208,71150,71151,71155],{},[211,71152,71153],{},[214,71154,1511],{},[253,71156,1511],{"encoding":255},[28,71158,71160],{"className":71159,"ariaHidden":261},[260],[28,71161,71163,71166],{"className":71162},[265],[28,71164],{"className":71165,"style":2371},[269],[28,71167,1511],{"className":71168,"style":1546},[274,275]," 就是 Kantorovich 势函数，Lipschitz 约束用 weight clipping 或 gradient penalty 近似。这个改动的实质是把一个饱和的散度换成一个不饱和的度量。",[11,71171,71172],{},"扩散模型和流匹配是同一件事的两种写法，都是把噪声分布运输到数据分布。区别在路径：扩散用固定的前向过程，流匹配可以自己选，选了 OT 路径就得到更直的轨迹和更少的采样步数。",[11,71174,71175,71176,71204,71205,71329,71330,71469,71470,71498,71499,71678,71679,71748],{},"域适应与分布偏移是 ",[28,71177,71179,71192],{"className":71178,"translate":193},[197],[28,71180,71182],{"className":71181},[201],[203,71183,71184],{"xmlns":205},[208,71185,71186,71190],{},[211,71187,71188],{},[214,71189,22957],{},[253,71191,22957],{"encoding":255},[28,71193,71195],{"className":71194,"ariaHidden":261},[260],[28,71196,71198,71201],{"className":71197},[265],[28,71199],{"className":71200,"style":455},[269],[28,71202,22957],{"className":71203,"style":459},[274,275]," 最能发挥的地方。给定源域和目标域的特征分布 ",[28,71206,71208,71234],{"className":71207,"translate":193},[197],[28,71209,71211],{"className":71210},[201],[203,71212,71213],{"xmlns":205},[208,71214,71215,71231],{},[211,71216,71217,71223,71225],{},[501,71218,71219,71221],{},[214,71220,20756],{},[214,71222,6730],{},[217,71224,1683],{"separator":261},[501,71226,71227,71229],{},[214,71228,20756],{},[214,71230,443],{},[253,71232,71233],{"encoding":255},"\\mu_S,\\mu_T",[28,71235,71237],{"className":71236,"ariaHidden":261},[260],[28,71238,71240,71243,71283,71286,71289],{"className":71239},[265],[28,71241],{"className":71242,"style":612},[269],[28,71244,71246,71249],{"className":71245},[274],[28,71247,20756],{"className":71248},[274,275],[28,71250,71252],{"className":71251},[556],[28,71253,71255,71275],{"className":71254},[560,561],[28,71256,71258,71272],{"className":71257},[565],[28,71259,71261],{"className":71260,"style":570},[569],[28,71262,71263,71266],{"style":573},[28,71264],{"className":71265,"style":578},[577],[28,71267,71269],{"className":71268},[582,583,584,585],[28,71270,6730],{"className":71271,"style":6776},[274,275,585],[28,71273,593],{"className":71274},[592],[28,71276,71278],{"className":71277},[565],[28,71279,71281],{"className":71280,"style":600},[569],[28,71282],{},[28,71284,1683],{"className":71285},[1723],[28,71287],{"className":71288,"style":530},[279],[28,71290,71292,71295],{"className":71291},[274],[28,71293,20756],{"className":71294},[274,275],[28,71296,71298],{"className":71297},[556],[28,71299,71301,71321],{"className":71300},[560,561],[28,71302,71304,71318],{"className":71303},[565],[28,71305,71307],{"className":71306,"style":570},[569],[28,71308,71309,71312],{"style":573},[28,71310],{"className":71311,"style":578},[577],[28,71313,71315],{"className":71314},[582,583,584,585],[28,71316,443],{"className":71317,"style":459},[274,275,585],[28,71319,593],{"className":71320},[592],[28,71322,71324],{"className":71323},[565],[28,71325,71327],{"className":71326,"style":600},[569],[28,71328],{},"，直接最小化 ",[28,71331,71333,71365],{"className":71332,"translate":193},[197],[28,71334,71336],{"className":71335},[201],[203,71337,71338],{"xmlns":205},[208,71339,71340,71362],{},[211,71341,71342,71344,71346,71352,71354,71360],{},[214,71343,22957],{},[217,71345,43],{"stretchy":241},[501,71347,71348,71350],{},[214,71349,20756],{},[214,71351,6730],{},[217,71353,1683],{"separator":261},[501,71355,71356,71358],{},[214,71357,20756],{},[214,71359,443],{},[217,71361,1522],{"stretchy":241},[253,71363,71364],{"encoding":255},"W(\\mu_S,\\mu_T)",[28,71366,71368],{"className":71367,"ariaHidden":261},[260],[28,71369,71371,71374,71377,71380,71420,71423,71426,71466],{"className":71370},[265],[28,71372],{"className":71373,"style":341},[269],[28,71375,22957],{"className":71376,"style":459},[274,275],[28,71378,43],{"className":71379},[348],[28,71381,71383,71386],{"className":71382},[274],[28,71384,20756],{"className":71385},[274,275],[28,71387,71389],{"className":71388},[556],[28,71390,71392,71412],{"className":71391},[560,561],[28,71393,71395,71409],{"className":71394},[565],[28,71396,71398],{"className":71397,"style":570},[569],[28,71399,71400,71403],{"style":573},[28,71401],{"className":71402,"style":578},[577],[28,71404,71406],{"className":71405},[582,583,584,585],[28,71407,6730],{"className":71408,"style":6776},[274,275,585],[28,71410,593],{"className":71411},[592],[28,71413,71415],{"className":71414},[565],[28,71416,71418],{"className":71417,"style":600},[569],[28,71419],{},[28,71421,1683],{"className":71422},[1723],[28,71424],{"className":71425,"style":530},[279],[28,71427,71429,71432],{"className":71428},[274],[28,71430,20756],{"className":71431},[274,275],[28,71433,71435],{"className":71434},[556],[28,71436,71438,71458],{"className":71437},[560,561],[28,71439,71441,71455],{"className":71440},[565],[28,71442,71444],{"className":71443,"style":570},[569],[28,71445,71446,71449],{"style":573},[28,71447],{"className":71448,"style":578},[577],[28,71450,71452],{"className":71451},[582,583,584,585],[28,71453,443],{"className":71454,"style":459},[274,275,585],[28,71456,593],{"className":71457},[592],[28,71459,71461],{"className":71460},[565],[28,71462,71464],{"className":71463,"style":600},[569],[28,71465],{},[28,71467,1522],{"className":71468},[355]," 就能对齐两个域，且不需要标签。对偶形式又让它变成对抗训练：一个网络学 ",[28,71471,71473,71486],{"className":71472,"translate":193},[197],[28,71474,71476],{"className":71475},[201],[203,71477,71478],{"xmlns":205},[208,71479,71480,71484],{},[211,71481,71482],{},[214,71483,510],{},[253,71485,666],{"encoding":255},[28,71487,71489],{"className":71488,"ariaHidden":261},[260],[28,71490,71492,71495],{"className":71491},[265],[28,71493],{"className":71494,"style":612},[269],[28,71496,510],{"className":71497},[274,275]," 去最大化 ",[28,71500,71502,71544],{"className":71501,"translate":193},[197],[28,71503,71505],{"className":71504},[201],[203,71506,71507],{"xmlns":205},[208,71508,71509,71541],{},[211,71510,71511,71513,71515,71517,71519,71525,71527,71529,71531,71533,71535],{},[217,71512,21282],{},[214,71514,510],{},[228,71516,488],{},[214,71518,12988],{},[501,71520,71521,71523],{},[214,71522,20756],{},[214,71524,6730],{},[217,71526,5074],{},[217,71528,21282],{},[214,71530,510],{},[228,71532,488],{},[214,71534,12988],{},[501,71536,71537,71539],{},[214,71538,20756],{},[214,71540,443],{},[253,71542,71543],{"encoding":255},"\\int\\varphi\\,d\\mu_S-\\int\\varphi\\,d\\mu_T",[28,71545,71547,71617],{"className":71546,"ariaHidden":261},[260],[28,71548,71550,71553,71556,71559,71562,71565,71568,71608,71611,71614],{"className":71549},[265],[28,71551],{"className":71552,"style":21325},[269],[28,71554,21282],{"className":71555,"style":21332},[21329,21330,21331],[28,71557],{"className":71558,"style":530},[279],[28,71560,510],{"className":71561},[274,275],[28,71563],{"className":71564,"style":530},[279],[28,71566,12988],{"className":71567},[274,275],[28,71569,71571,71574],{"className":71570},[274],[28,71572,20756],{"className":71573},[274,275],[28,71575,71577],{"className":71576},[556],[28,71578,71580,71600],{"className":71579},[560,561],[28,71581,71583,71597],{"className":71582},[565],[28,71584,71586],{"className":71585,"style":570},[569],[28,71587,71588,71591],{"style":573},[28,71589],{"className":71590,"style":578},[577],[28,71592,71594],{"className":71593},[582,583,584,585],[28,71595,6730],{"className":71596,"style":6776},[274,275,585],[28,71598,593],{"className":71599},[592],[28,71601,71603],{"className":71602},[565],[28,71604,71606],{"className":71605,"style":600},[569],[28,71607],{},[28,71609],{"className":71610,"style":280},[279],[28,71612,5074],{"className":71613},[284],[28,71615],{"className":71616,"style":280},[279],[28,71618,71620,71623,71626,71629,71632,71635,71638],{"className":71619},[265],[28,71621],{"className":71622,"style":21325},[269],[28,71624,21282],{"className":71625,"style":21332},[21329,21330,21331],[28,71627],{"className":71628,"style":530},[279],[28,71630,510],{"className":71631},[274,275],[28,71633],{"className":71634,"style":530},[279],[28,71636,12988],{"className":71637},[274,275],[28,71639,71641,71644],{"className":71640},[274],[28,71642,20756],{"className":71643},[274,275],[28,71645,71647],{"className":71646},[556],[28,71648,71650,71670],{"className":71649},[560,561],[28,71651,71653,71667],{"className":71652},[565],[28,71654,71656],{"className":71655,"style":570},[569],[28,71657,71658,71661],{"style":573},[28,71659],{"className":71660,"style":578},[577],[28,71662,71664],{"className":71663},[582,583,584,585],[28,71665,443],{"className":71666,"style":459},[274,275,585],[28,71668,593],{"className":71669},[592],[28,71671,71673],{"className":71672},[565],[28,71674,71676],{"className":71675,"style":600},[569],[28,71677],{},"，另一个网络调整特征表示去最小化这个差。多个源域时用 Wasserstein 重心，在 ",[28,71680,71682,71699],{"className":71681,"translate":193},[197],[28,71683,71685],{"className":71684},[201],[203,71686,71687],{"xmlns":205},[208,71688,71689,71697],{},[211,71690,71691],{},[501,71692,71693,71695],{},[214,71694,22957],{},[224,71696,1253],{},[253,71698,38866],{"encoding":255},[28,71700,71702],{"className":71701,"ariaHidden":261},[260],[28,71703,71705,71708],{"className":71704},[265],[28,71706],{"className":71707,"style":25059},[269],[28,71709,71711,71714],{"className":71710},[274],[28,71712,22957],{"className":71713,"style":459},[274,275],[28,71715,71717],{"className":71716},[556],[28,71718,71720,71740],{"className":71719},[560,561],[28,71721,71723,71737],{"className":71722},[565],[28,71724,71726],{"className":71725,"style":1285},[569],[28,71727,71728,71731],{"style":20948},[28,71729],{"className":71730,"style":578},[577],[28,71732,71734],{"className":71733},[582,583,584,585],[28,71735,1253],{"className":71736},[274,585],[28,71738,593],{"className":71739},[592],[28,71741,71743],{"className":71742},[565],[28,71744,71746],{"className":71745,"style":600},[569],[28,71747],{}," 空间里求“平均分布”，这正是 Fréchet 均值在分布空间里的实例。",[11,71750,71751],{},"OOD 检测，也就是我开头那个 0.99 的问题，我会分三层来做。",[11,71753,71754],{},"设想一条质检流水线，训练时只有“正常”“划痕”“凹陷”三类。现在来了一个从没见过的新缺陷。",[11,71756,71757,71758,71906],{},"点态一层比较样本密度：对每个类拟合 KDE 或流，取 ",[28,71759,71761,71795],{"className":71760,"translate":193},[197],[28,71762,71764],{"className":71763},[201],[203,71765,71766],{"xmlns":205},[208,71767,71768,71792],{},[211,71769,71770,71772,71774,71776,71786,71788,71790],{},[217,71771,5074],{},[214,71773,39714],{},[217,71775,21407],{},[501,71777,71778,71784],{},[62132,71779,71780,71782],{"accent":261},[214,71781,11],{},[217,71783,62138],{},[214,71785,7436],{},[217,71787,43],{"stretchy":241},[214,71789,176],{},[217,71791,1522],{"stretchy":241},[253,71793,71794],{"encoding":255},"-\\log\\hat p_k(x)",[28,71796,71798],{"className":71797,"ariaHidden":261},[260],[28,71799,71801,71804,71807,71810,71815,71818,71897,71900,71903],{"className":71800},[265],[28,71802],{"className":71803,"style":341},[269],[28,71805,5074],{"className":71806},[274],[28,71808],{"className":71809,"style":530},[279],[28,71811,39952,71813],{"className":71812},[21329],[28,71814,34347],{"style":39955},[28,71816],{"className":71817,"style":530},[279],[28,71819,71821,71863],{"className":71820},[274],[28,71822,71824],{"className":71823},[274,62250],[28,71825,71827,71855],{"className":71826},[560,561],[28,71828,71830,71852],{"className":71829},[565],[28,71831,71833,71841],{"className":71832,"style":7448},[569],[28,71834,71835,71838],{"style":1926},[28,71836],{"className":71837,"style":1914},[577],[28,71839,11],{"className":71840},[274,275],[28,71842,71843,71846],{"style":1926},[28,71844],{"className":71845,"style":1914},[577],[28,71847,71849],{"className":71848,"style":62277},[62276],[28,71850,62138],{"className":71851},[274],[28,71853,593],{"className":71854},[592],[28,71856,71858],{"className":71857},[565],[28,71859,71861],{"className":71860,"style":62290},[569],[28,71862],{},[28,71864,71866],{"className":71865},[556],[28,71867,71869,71889],{"className":71868},[560,561],[28,71870,71872,71886],{"className":71871},[565],[28,71873,71875],{"className":71874,"style":20945},[569],[28,71876,71877,71880],{"style":573},[28,71878],{"className":71879,"style":578},[577],[28,71881,71883],{"className":71882},[582,583,584,585],[28,71884,7436],{"className":71885,"style":7452},[274,275,585],[28,71887,593],{"className":71888},[592],[28,71890,71892],{"className":71891},[565],[28,71893,71895],{"className":71894,"style":600},[569],[28,71896],{},[28,71898,43],{"className":71899},[348],[28,71901,176],{"className":71902},[274,275],[28,71904,1522],{"className":71905},[355]," 作为分数。它便宜、可解释，但受典型集悖论和带宽影响，高维不可靠——上一节那个“空洞比簇心更密”的例子就是反例。",[11,71908,71909,71910,71979,71980,72056,72057,72085,72086,6904,72114,72158],{},"分布一层算测试批（或测试样本的邻域）到每个类条件分布的 ",[28,71911,71913,71930],{"className":71912,"translate":193},[197],[28,71914,71916],{"className":71915},[201],[203,71917,71918],{"xmlns":205},[208,71919,71920,71928],{},[211,71921,71922],{},[501,71923,71924,71926],{},[214,71925,22957],{},[224,71927,1253],{},[253,71929,38866],{"encoding":255},[28,71931,71933],{"className":71932,"ariaHidden":261},[260],[28,71934,71936,71939],{"className":71935},[265],[28,71937],{"className":71938,"style":25059},[269],[28,71940,71942,71945],{"className":71941},[274],[28,71943,22957],{"className":71944,"style":459},[274,275],[28,71946,71948],{"className":71947},[556],[28,71949,71951,71971],{"className":71950},[560,561],[28,71952,71954,71968],{"className":71953},[565],[28,71955,71957],{"className":71956,"style":1285},[569],[28,71958,71959,71962],{"style":20948},[28,71960],{"className":71961,"style":578},[577],[28,71963,71965],{"className":71964},[582,583,584,585],[28,71966,1253],{"className":71967},[274,585],[28,71969,593],{"className":71970},[592],[28,71972,71974],{"className":71973},[565],[28,71975,71977],{"className":71976,"style":600},[569],[28,71978],{},"，取最小者作为“离最近已知类的距离”。优点是它看整片质量而不是单个点；缺点是它需要一批样本，还有 ",[28,71981,71983,72009],{"className":71982,"translate":193},[197],[28,71984,71986],{"className":71985},[201],[203,71987,71988],{"xmlns":205},[208,71989,71990,72006],{},[211,71991,71992],{},[3054,71993,71994,71996],{},[214,71995,3060],{},[211,71997,71998,72000,72002,72004],{},[217,71999,5074],{},[224,72001,1244],{},[214,72003,12219],{"mathvariant":484},[214,72005,12988],{},[253,72007,72008],{"encoding":255},"n^{-1\u002Fd}",[28,72010,72012],{"className":72011,"ariaHidden":261},[260],[28,72013,72015,72018],{"className":72014},[265],[28,72016],{"className":72017,"style":42614},[269],[28,72019,72021,72024],{"className":72020},[274],[28,72022,3060],{"className":72023},[274,275],[28,72025,72027],{"className":72026},[556],[28,72028,72030],{"className":72029},[560],[28,72031,72033],{"className":72032},[565],[28,72034,72036],{"className":72035,"style":42614},[569],[28,72037,72038,72041],{"style":3094},[28,72039],{"className":72040,"style":578},[577],[28,72042,72044],{"className":72043},[582,583,584,585],[28,72045,72047,72050,72053],{"className":72046},[274,585],[28,72048,5074],{"className":72049},[274,585],[28,72051,23875],{"className":72052},[274,585],[28,72054,12988],{"className":72055},[274,275,585]," 的样本复杂度。实践中的折中是：在低维隐空间里算，或者用对偶势 ",[28,72058,72060,72073],{"className":72059,"translate":193},[197],[28,72061,72063],{"className":72062},[201],[203,72064,72065],{"xmlns":205},[208,72066,72067,72071],{},[211,72068,72069],{},[214,72070,510],{},[253,72072,666],{"encoding":255},[28,72074,72076],{"className":72075,"ariaHidden":261},[260],[28,72077,72079,72082],{"className":72078},[265],[28,72080],{"className":72081,"style":612},[269],[28,72083,510],{"className":72084},[274,275],"——一旦在一批数据上解出 ",[28,72087,72089,72102],{"className":72088,"translate":193},[197],[28,72090,72092],{"className":72091},[201],[203,72093,72094],{"xmlns":205},[208,72095,72096,72100],{},[211,72097,72098],{},[214,72099,510],{},[253,72101,666],{"encoding":255},[28,72103,72105],{"className":72104,"ariaHidden":261},[260],[28,72106,72108,72111],{"className":72107},[265],[28,72109],{"className":72110,"style":612},[269],[28,72112,510],{"className":72113},[274,275],[28,72115,72117,72137],{"className":72116,"translate":193},[197],[28,72118,72120],{"className":72119},[201],[203,72121,72122],{"xmlns":205},[208,72123,72124,72134],{},[211,72125,72126,72128,72130,72132],{},[214,72127,510],{},[217,72129,43],{"stretchy":241},[214,72131,176],{},[217,72133,1522],{"stretchy":241},[253,72135,72136],{"encoding":255},"\\varphi(x)",[28,72138,72140],{"className":72139,"ariaHidden":261},[260],[28,72141,72143,72146,72149,72152,72155],{"className":72142},[265],[28,72144],{"className":72145,"style":341},[269],[28,72147,510],{"className":72148},[274,275],[28,72150,43],{"className":72151},[348],[28,72153,176],{"className":72154},[274,275],[28,72156,1522],{"className":72157},[355]," 本身就是逐样本分数，可以拿去做单点判定。我很喜欢这个技巧，它把“批级别的距离”翻译成了“点级别的分数”，两个需求一次满足。",[11,72160,72161],{},"输出一层比较类别概率分布的差异：训练时记录每个类在验证集上的平均预测分布（单纯形上的一个点），测试时算测试批的平均预测分布与它的 Fisher–Rao 距离。距离大，说明模型在这批数据上的“说话方式”变了，即使每个样本的置信度都不低。这一层不需要标签也不需要密度估计，我觉得是性价比最高的一个。",[11,72163,72164],{},"分类与损失那一侧前面已经说了：交叉熵是 Bregman，自然梯度是 Fisher 预条件，镜像下降是 Bregman 投影。这三个东西实现上都很朴素，但把它们在理论上认成同一族，能省掉很多“为什么这个 trick 有效”的困惑。",[414,72166,72167],{"id":72167},"回到开头",[11,72169,72170],{},"回到那个 0.99。",[11,72172,72173,72174,72202],{},"置信度回答的是“在这几个类里最像哪个”，它是一次相对排序，分母被 softmax 强行归一化成 ",[28,72175,72177,72190],{"className":72176,"translate":193},[197],[28,72178,72180],{"className":72179},[201],[203,72181,72182],{"xmlns":205},[208,72183,72184,72188],{},[211,72185,72186],{},[224,72187,1244],{},[253,72189,1244],{"encoding":255},[28,72191,72193],{"className":72192,"ariaHidden":261},[260],[28,72194,72196,72199],{"className":72195},[265],[28,72197],{"className":72198,"style":316},[269],[28,72200,1244],{"className":72201},[274],"。分布几何回答的是“这个点离已知类别的概率质量有多远”，它是一个绝对量，不依赖“一定属于某个类”这个假设。这两件事不该互相替代，而应该互补：分类器给候选，分布几何判越界。",[11,72204,72205],{},"如果只留一个式子，我会留 Kantorovich 对偶：",[28,72207,72209],{"className":72208,"translate":193},[192],[28,72210,72212,72292],{"className":72211,"translate":193},[197],[28,72213,72215],{"className":72214},[201],[203,72216,72217],{"xmlns":205,"display":206},[208,72218,72219,72289],{},[211,72220,72221,72227,72229,72231,72233,72235,72237,72239,72269,72271,72273,72275,72277,72279,72281,72283,72285,72287],{},[501,72222,72223,72225],{},[214,72224,22957],{},[224,72226,1244],{},[217,72228,43],{"stretchy":241},[214,72230,20756],{},[217,72232,1683],{"separator":261},[214,72234,20787],{},[217,72236,1522],{"stretchy":241},[217,72238,83],{},[1771,72240,72241,72247],{},[211,72242,72243,72245],{},[214,72244,30596],{},[217,72246,21407],{},[211,72248,72249,72251,72253,72265,72267],{},[214,72250,24404],{"mathvariant":484},[214,72252,510],{},[501,72254,72255,72257],{},[214,72256,24404],{"mathvariant":484},[211,72258,72259,72261,72263],{},[214,72260,24393],{"mathvariant":484},[214,72262,1686],{"mathvariant":484},[214,72264,11],{"mathvariant":484},[217,72266,1876],{},[224,72268,1244],{},[217,72270,21282],{},[214,72272,510],{},[228,72274,488],{},[214,72276,12988],{},[217,72278,43],{"stretchy":241},[214,72280,20756],{},[217,72282,5074],{},[214,72284,20787],{},[217,72286,1522],{"stretchy":241},[214,72288,494],{"mathvariant":484},[253,72290,72291],{"encoding":255},"W_1(\\mu,\\nu)=\\sup_{\\|\\varphi\\|_{\\mathrm{Lip}}\\le1}\\int\\varphi\\,d(\\mu-\\nu).",[28,72293,72295,72368,72509],{"className":72294,"ariaHidden":261},[260],[28,72296,72298,72301,72341,72344,72347,72350,72353,72356,72359,72362,72365],{"className":72297},[265],[28,72299],{"className":72300,"style":341},[269],[28,72302,72304,72307],{"className":72303},[274],[28,72305,22957],{"className":72306,"style":459},[274,275],[28,72308,72310],{"className":72309},[556],[28,72311,72313,72333],{"className":72312},[560,561],[28,72314,72316,72330],{"className":72315},[565],[28,72317,72319],{"className":72318,"style":1285},[569],[28,72320,72321,72324],{"style":20948},[28,72322],{"className":72323,"style":578},[577],[28,72325,72327],{"className":72326},[582,583,584,585],[28,72328,1244],{"className":72329},[274,585],[28,72331,593],{"className":72332},[592],[28,72334,72336],{"className":72335},[565],[28,72337,72339],{"className":72338,"style":600},[569],[28,72340],{},[28,72342,43],{"className":72343},[348],[28,72345,20756],{"className":72346},[274,275],[28,72348,1683],{"className":72349},[1723],[28,72351],{"className":72352,"style":530},[279],[28,72354,20787],{"className":72355,"style":20803},[274,275],[28,72357,1522],{"className":72358},[355],[28,72360],{"className":72361,"style":302},[279],[28,72363,83],{"className":72364},[306],[28,72366],{"className":72367,"style":302},[279],[28,72369,72371,72374,72476,72479,72482,72485,72488,72491,72494,72497,72500,72503,72506],{"className":72370},[265],[28,72372],{"className":72373,"style":31515},[269],[28,72375,72377],{"className":72376},[21329,21485],[28,72378,72380,72468],{"className":72379},[560,561],[28,72381,72383,72465],{"className":72382},[565],[28,72384,72386,72455],{"className":72385,"style":645},[569],[28,72387,72388,72391],{"style":31530},[28,72389],{"className":72390,"style":1914},[577],[28,72392,72394],{"className":72393},[582,583,584,585],[28,72395,72397,72400,72403,72449,72452],{"className":72396},[274,585],[28,72398,24404],{"className":72399},[274,585],[28,72401,510],{"className":72402},[274,275,585],[28,72404,72406,72409],{"className":72405},[274,585],[28,72407,24404],{"className":72408},[274,585],[28,72410,72412],{"className":72411},[556],[28,72413,72415,72441],{"className":72414},[560,561],[28,72416,72418,72438],{"className":72417},[565],[28,72419,72421],{"className":72420,"style":31564},[569],[28,72422,72423,72426],{"style":31567},[28,72424],{"className":72425,"style":31571},[577],[28,72427,72429],{"className":72428},[582,31575,2170,585],[28,72430,72432],{"className":72431},[274,585],[28,72433,72435],{"className":72434},[274,585],[28,72436,31585],{"className":72437},[274,1712,585],[28,72439,593],{"className":72440},[592],[28,72442,72444],{"className":72443},[565],[28,72445,72447],{"className":72446,"style":31595},[569],[28,72448],{},[28,72450,1876],{"className":72451},[306,585],[28,72453,1244],{"className":72454},[274,585],[28,72456,72457,72460],{"style":1926},[28,72458],{"className":72459,"style":1914},[577],[28,72461,72462],{},[28,72463,30596],{"className":72464},[21329],[28,72466,593],{"className":72467},[592],[28,72469,72471],{"className":72470},[565],[28,72472,72474],{"className":72473,"style":31623},[569],[28,72475],{},[28,72477],{"className":72478,"style":530},[279],[28,72480,21282],{"className":72481,"style":21548},[21329,21330,21547],[28,72483],{"className":72484,"style":530},[279],[28,72486,510],{"className":72487},[274,275],[28,72489],{"className":72490,"style":530},[279],[28,72492,12988],{"className":72493},[274,275],[28,72495,43],{"className":72496},[348],[28,72498,20756],{"className":72499},[274,275],[28,72501],{"className":72502,"style":280},[279],[28,72504,5074],{"className":72505},[284],[28,72507],{"className":72508,"style":280},[279],[28,72510,72512,72515,72518,72521],{"className":72511},[265],[28,72513],{"className":72514,"style":341},[269],[28,72516,20787],{"className":72517,"style":20803},[274,275],[28,72519,1522],{"className":72520},[355],[28,72522,494],{"className":72523},[274],[11,72525,72526],{},"它把一个关于分布的优化问题，变成了一个关于函数的优化问题。分布是无穷维的、难参数化的；函数是可以拿神经网络拟合的。这一步转换，才是最优传输能从 1781 年的搬土问题走进今天每一篇 WGAN、每一次域适应、每一个流匹配模型的真正原因。",[11,72528,72529],{},"至于 Fisher–Rao 和 Bregman，它们给我的东西不太一样：不是算法，而是一种校准过的直觉——当我说“两个分布很近”的时候，我最好知道自己是拿哪把尺子在量，以及那把尺子在局部是不是真的像一个度量。",{"title":23,"searchDepth":64,"depth":64,"links":72531},[72532,72533,72534,72535,72536,72537,72538,72539,72540,72541,72542],{"id":19972,"depth":64,"text":19972},{"id":25132,"depth":64,"text":25132},{"id":32052,"depth":64,"text":32052},{"id":39218,"depth":64,"text":39218},{"id":42653,"depth":64,"text":42653},{"id":49156,"depth":64,"text":49156},{"id":53661,"depth":64,"text":53662},{"id":61652,"depth":64,"text":61652},{"id":65328,"depth":64,"text":65328},{"id":70531,"depth":64,"text":70531},{"id":72167,"depth":64,"text":72167},{},"\u002Fblog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun",{"title":19961,"description":19966},"blog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun","从 Kantorovich 对偶到 Fisher–Rao 度量、Bregman 散度、核密度估计与标准化流，用同一套分布几何的语言解释为什么分布层面的判据比分类器置信度更稳健。",[72549,72550,72551],"mathematics","machine-learning","statistics","2026-10-31T20:17:23+08:00","IAT5rYeOtscnEe8Mhh1eCeIF4PJR1Ewyc86Z_G_0ncI",{"id":72555,"title":72556,"body":72557,"description":72733,"draft":19946,"enableCodeHighlight":19946,"enableCodeHightlight":19946,"enableComment":19947,"enableD2":19946,"enableKatex":19946,"extension":19948,"image":23,"important":19946,"location":19949,"meta":72734,"navigation":19947,"ogImage":19949,"path":72735,"seo":72736,"stem":72737,"summary":72738,"tags":72739,"time":72741,"__hash__":72742},"blog\u002Fblog\u002F2026\u002F0012-shu-xue-yu-hua-jia.md","数学与画家",{"type":8,"value":72558,"toc":72731},[72559,72570,72600,72603,72606,72609,72612,72615,72618,72621,72624,72627,72630,72633,72636,72639,72642,72645,72648,72651,72654,72657,72660,72663,72666,72669,72672,72675,72678,72681,72684,72687,72690,72693,72696,72699,72702,72705,72708,72711,72714,72717,72720,72723,72726],[11,72560,72561,72562,72569],{},"前段时间，OpenAI 声称，他们用内部尚未正式发布的模型产出了七百多篇论文，「AI 接替数学家」的叙事于是再一次被推到台前。市场是狂热还是炒作，我已经疲于争论。那七百篇论文是否名副其实，也先放在一边。真正让我想写点东西的，",[238,72563,72568],{"href":72564,"rel":72565,"target":72567},"https:\u002F\u002Fx.com\u002FNoe1017_Monad\u002Fstatus\u002F2112008540592410951",[72566],"nofollow","_blank","是 Twitter 上一条获得不少认同的帖子","。它把数学结构说成人类为了迁就自己那点脑容量而被迫发明的拐杖，把 AI 说成不需要几何直观、可以直接站在高维展开式里往下看的怪物。最后一句尤其漂亮：我们终于拿到了上帝账本里的答案，却发现上帝算账根本不用微积分，他老人家是直接拿计算器硬按出来的。",[15772,72571,72572],{},[11,72573,72574,72575,72578,72579,72581,72582,72584,72585,72587,72588,72590,72591,72593,72594,72596,72597,72599],{},"所谓优雅的数学结构，本质上是人类为了迁就自己那点可怜的脑容量，被迫发明出来的拐杖。  ",[72576,72577],"br",{},"\n一个顶尖人类数学家的工作内存，撑死也就同时追踪五六个变量的交互。  ",[72576,72580],{},"\n当你在算一个偏微分方程或者组合不等式时，一旦展开到三阶以上，几十个交叉项涌出来，人脑就直接宕机了。  ",[72576,72583],{},"\n所以人类数学家极其厌恶展开高阶项，一旦看到高阶项发散，人类的本能反应是此路不通，赶紧退回来找对称性、找不变量、搞同调代数或者几何直观，试图不展开式子就把问题绕过去。  ",[72576,72586],{},"\n但 AI 没有这个生理缺陷。  ",[72576,72589],{},"\n对于上万个并发的 Agent 来说，同时追踪一万个五阶交叉项的系数变化，跟我们算 1+1 一样轻松。  ",[72576,72592],{},"\n在它的视野里，根本不需要什么几何直观，它直接站在高维的代数展开式里往下看，人类觉得像乱麻一样的百项高阶余项，在它眼里就是一张明明白白的线性方程组。  ",[72576,72595],{},"\n既然把式子硬展开到第七阶，再凑几个待定系数就能让所有讨厌的发散项精准归零，它为什么要费劲去帮你发明一个人类能看懂的新概念？  ",[72576,72598],{},"\n这 700 个结果背后最无奈的黑色幽默：我们终于拿到了上帝账本里的答案，却发现上帝算账根本不用微积分，他老人家是直接拿计算器硬按出来的。",[11,72601,72602],{},"这套论证值得认真对待，恰恰因为它错得很有代表性：它把人类的工作记忆、数学的审美偏好、AI 的暴力计算串成了一条看似严密的因果链。但这条链中间有一个关键的偷换——它把「人类因为认知限制而偏爱结构」偷换成了「结构只是认知限制的产物」。前者是发生学上的事实，后者是一个关于数学本体的判断。两者之间隔着一整个哲学鸿沟。",[11,72604,72605],{},"画家这个比喻，也许比「拐杖」更接近真相。画家画不出世界上的每一个光子，所以他发明了透视、明暗和构图。但这些都长在真实的结构上：透视是射影几何，明暗是光的传播，构图是视觉心理。一个能逐像素渲染世界的渲染器，仍然跑在同一套射影几何上。暴力计算并没有绕开结构。它把结构执行到了人眼跟不上的规模。数学里的「硬展开」也一样。",[11,72607,72608],{},"它对人类工作记忆的描述大体成立，对数学的判断则基本错了。结构就是数学本身，拐杖之说站不住。AI 同样依赖结构，只不过那种结构人读不懂。问题在于「能不能翻译」，美丑反而次要。而翻译，恰恰是当前 AI 数学产出链条上最缺失的一环。",[11,72610,72611],{},"先承认其中合理的部分。人类的工作记忆确实窄得可怜。Miller 在 1956 年给出著名的 7±2，后来 Cowan 又把它修正到 4±1 左右。数字有争议，但「窄」这个结论没有被推翻。数学家偏爱对称性、不变量、同调，厌恶把式子硬展开，很大程度上确实是认知经济学的产物：找不变量，本质上是把高维信息压缩成一两个人类握得住的符号。AI 在大规模符号展开和并行系数追踪上的优势也是真实的。AlphaTensor 找矩阵乘法算法、各种组合搜索里那些看似暴力、实则依赖结构化搜索空间的路线，确实绕开了人类偏好的那条美学路径。",[11,72613,72614],{},"问题出在下一步。从「人类因为认知限制而偏爱结构」推到「结构只是认知限制的产物」，这是一个发生学谬误。一个东西因为什么被发明出来，并不等于它是什么。望远镜是因为人眼太弱才被造出来的，但这不意味着望远镜里的木星卫星是人类的幻觉。十进制是因为人有十根手指才流行的，但这也不意味着素数分布依赖于手指的数量。数学结构也一样：它可能是因为人脑太窄才被优先发现的，但一旦被发现，它说的就是对象本身的事，与人类脑容量无关。",[11,72616,72617],{},"把 Galois 群说成「人类算不动五次方程，所以编个借口说它不可解」，是一种误解。它本身是一个定理：一个方程能用根式解，当且仅当它的 Galois 群可解。五次一般方程对应的对称群 S5 不是可解群，所以没有根式解。这个结论与人类的工作记忆无关。就算人类有无限的工作记忆，S5 依然不是可解群，这个定理依然成立。同样，Noether 定理把连续对称性和守恒律绑在一起。把它说成「人类看不懂，所以画个对称性的图安慰自己」，也是一种误解。它说的是物理世界的一条结构事实：如果世界没有这种对称性，相应的守恒律就不存在。同调群也常被误解成「空间太复杂，人类算不动，所以发明一个符号糊弄过去」。它探测的是空间的洞，而它作为不变量能被计算，本身依赖链复形这一整套结构。",[11,72619,72620],{},"此外，它的说服力有很大一部分来自修辞。「可怜的脑容量」「拐杖」「硬按」——这些词先把你放在一个需要被怜悯的位置，再让你接受结论。把修辞剥掉，论证的核心其实只有一句：人脑窄，AI 不窄，所以结构是拐杖。而这句话恰恰是跳步的。",[11,72622,72623],{},"退一步说，就算 AI 真的把式子硬展开到第七阶，把所有发散项精准归零，它拿到的只是一张针对这个特例的系数表，换一个情形就未必管用。它能告诉你「这里可以归零」，但不能告诉你「为什么这里必然可以归零」。前者是答案，后者才是数学。打个比方：给你七百个点，你总可以找到一个不超过 699 次的多项式，让它精确穿过每一个点。但这只是插值，够不上理解。它在七百个点之外没有任何预测力。硬展开加待定系数，本质上就是高维版本的插值——你用一个足够大的展开式，把眼前的项全部按住，然后宣布问题解决了。可数学要的是「为什么会这样，以及换一个情形会怎样」。「按住」远远不够。",[11,72625,72626],{},"照这个逻辑，AI 不需要几何直观，可以直接站在高维的代数展开式里往下看。这个画面假设存在一个「没有直观」的视角。但任何计算都发生在某种表示里，而表示本身就是一种结构。Transformer 的注意力是内容相关的路由，等变网络把对称性硬编码进架构，神经算子学的是函数空间之间的映射——这些做法都把结构写进了模型，只是藏得比较深。没有免费午餐定理早就说过，不存在对所有问题都最优的通用学习器。每一个能学的模型，都带着自己的归纳偏置。所谓「不需要直观的 AI」，只是把直观藏进了架构里，让人看不见而已。",[11,72628,72629],{},"AlphaTensor 常被拿来当例子，但它恰恰站在这个论证的反面。DeepMind 并没有枚举所有可能的算法。它在张量分解的结构化空间里做强化学习搜索。它能找到 Strassen 式的算法，正是因为搜索空间被结构约束过。把 AlphaTensor 说成「暴力硬算」的证据，是对它的误读。",[11,72631,72632],{},"更有意思的是，AI 自己的前沿正在往结构走，纯暴力那条路反而越走越窄。等变网络、神经算子、状态空间模型、注意力机制，都是在把某种先验写进模型。为什么？因为纯暴力在高维会指数爆炸。这个领域最聪明的一批人，正在用工程手段重新发明数学家一百年前就熟悉的那些东西：对称性、局部性、可分解性。AI 的研究者自己正在沿着那条路径追赶，「绕开审美」的预言并没有发生。",[11,72634,72635],{},"技术细节上也站不住。「高阶项发散」常被当作「项太多、人脑装不下」的同义词。但在很多场合，发散属于表示问题，跟项数多少关系不大：这个展开本身是一个渐近级数，并不收敛。渐近级数在前几项可以给出极好的近似，但继续展开不会逼近真值。你需要的是重求和、解析延拓，或者换一个表示。数学家看到发散就后退，是因为他们知道「继续展开」不会自动收敛。这时候该做的是换一种表示，把项数堆得更高没有用。AI 就算把发散项凑没了，也可能只找到一种形式上的消去，并未得到收敛的表示。",[11,72637,72638],{},"即便并行 Agent 真的能同时追踪一万个五阶交叉项，代价也不是零。算力、内存、通信都会随着项数组合爆炸。「并行」被当成了免费的午餐。更关键的是，这个天花板背后还有复杂性理论：在普遍接受的复杂性假设下，有些问题是内在难解的——问题本身的复杂度摆在那里，跟聪明与否关系不大。「人脑太小」和「问题太难」被混为一谈了。就算给一个无限工作记忆的理想智能体，它也会撞上指数爆炸。人类偏爱结构，既因为脑子小，也因为问题本身硬。",[11,72640,72641],{},"所以，AI 可以使用一种人读不懂的结构。硬展开同样有结构，只不过这种结构被执行到了人跟不上的规模。一个逐像素渲染的渲染器，算得比任何画家都细，却始终跑在射影几何上，把透视算到了人眼看不见的精度。AI 的硬展开也一样：它把代数结构展开到了人脑装不下的长度，却始终在代数结构之内。",[11,72643,72644],{},"「上帝拿计算器硬按」假设答案先于结构存在。它背后是一个很朴素的柏拉图式画面：答案早就躺在那里，结构只是人类够不着时用的梯子。但数学的实践不是这样。很多答案是在结构被发明出来之后才成为「答案」的。",[11,72646,72647],{},"一个猜想在 n=5 成立、在 n=6 崩掉，硬算给你两个数。结构告诉你边界在哪里，以及为什么边界在那里。两个数可以和无穷多个故事相容，结构帮你挑出正确的那个。更一般地，硬算只能处理有限多个情形。即使你在计算机上验证费马大定理到 n 等于十亿，也不构成对所有的 n 的证明。Wiles 的证明之所以成立，靠的是模形式这座桥，把问题接到了另一个世界。计算量的大小并非关键。",[11,72649,72650],{},"知道一个命题为真，和知道它为什么为真，不是一回事。前者是证明，后者是理解。硬算可以给你前者，未必给你后者。而且，硬算能证明的通常只是有限多个特例。一旦命题是全称的，你就需要结构、归纳或者某种不变量，才能跨过「有限」到「所有」的鸿沟。七百个结果里可能有一些本身就是关于有限对象的定理，本身就有价值。但如果它们大多是硬展开出来的特例，那么它们更像是一批精确的数值事实，未必是可迁移的定理。另外，「七百篇论文」和「七百个结果」不是一回事。前者可能包含大量增量工作，后者才是数学意义上的成果。这个区分本身就值得追问，但先放在一边。",[11,72652,72653],{},"数学史上早有「先有黑箱，后有理解」的先例。四色定理 1976 年由机器辅助证明，当时吵得很厉害。后来人类花了很多年去简化、去压缩，形式化验证也完成了，但「机器证明算不算理解」这个问题并没有完全消失。Kepler 猜想被 Hales 用大规模计算机辅助证明，后来又有 Flyspeck 项目做形式化验证。模式很清楚：先拿到一个可验证但不可读的答案，再花一代人把它翻译成人类握得住的结构。翻译可能失败，也可能要几代人，但「现在翻译不出来」从来不足以让人停下来。",[11,72655,72656],{},"而且，人类数学自己也早就遇到过不可读的问题。有限单群分类定理的证明上万页，散落在几百篇文章里，据说至今没有人从头到尾读完。这是人类自己造出来的黑箱。区别只在于规模：人类数学在二十世纪就已经开始生产超出个体阅读能力的结果，AI 只是把这个趋势推得更远。",[11,72658,72659],{},"黑箱当然有价值。它可以被引用、被使用、被当作下一步的踏脚石。但它的价值有一个天花板：验证成本无法摊销。一个你理解了的定理，可以在一段话里教给别人，可以在新问题里直接调用。一个你不理解的黑箱，每用一次都要重新跑一遍、重新信一遍。理解之所以重要，是因为它把验证成本压缩成了可复用的结构。没有理解，知识就不能以低成本累积。",[11,72661,72662],{},"把这种思路推到极端：也许黑箱就是数学的新常态。也许未来的数学单位会从定理变成证书。数学家追求的东西，也会从「看懂」转向「可验证、可组合、可规模化」。这是一个严肃的可能性，我不打算轻易否定它。但它比「AI 算得更快」是一个大得多的主张：它改变的是数学知识的定义，速度倒在其次。而且，就算走到那一步，信任基础设施本身仍然是结构：证书格式、证明助理、可复现性标准、社区共识。黑箱的未来并未取消结构，只不过把结构挪到了另一个地方。",[11,72664,72665],{},"至于审美，AI 正在用一个丑但有效的答案挑战「美是数学的先验判据」这个前提。这个前提站不住。",[11,72667,72668],{},"数学史上「丑的真理」屡见不鲜。三次方程的求根公式在 16 世纪就撞上了 casus irreducibilis：三个实根，却必须经过复数才能算出来。当时的数学家觉得这近乎荒谬。非欧几何刚出来的时候，高斯不敢公开发表。Cantor 的无穷集合遭到 Kronecker 的激烈反对。Weierstrass 造出处处连续、处处不可导的函数，被当时的许多人视为病态。Banach-Tarski 分球悖论更是像在说「数学不该长这样」。后来的处理方式，是扩写美的定义，把这些东西留下来。复数变成了复平面上的几何，非欧几何变成了曲率，病态函数催生了测度论，也逼着拓扑学重新审视连续与收敛。分球悖论则逼着人们去追问选择公理到底意味着什么。丑的东西待得够久，人类就会给它找一套它配得上的语言。语言找到了，它就不丑了。",[11,72670,72671],{},"所以，美更像事后的评价。数学家说一个证明美，通常是在说：它把原本分散的东西统一了，没有多余的假设，并且揭示了机制。这些都是理解完成之后的判词，很难充当发现之前的门槛。历史上真正被审美直觉挡在门外的结果——比如 Galois 生前没能发表的手稿——事后看，那是审美的失败，数学本身并没有失败。",[11,72673,72674],{},"但也不要走到另一个极端，把审美说成纯粹的非理性。审美其实是一个关于「可压缩性」的启发式：一个漂亮的证明，往往更容易推广、更容易迁移、更容易在别的问题里复用。它当然会犯错，但命中率并不低。数学家对丑的警惕，除了对陌生的恐惧，还包含一种合理怀疑：这东西也许无法一般化。把审美当成纯粹的偏见，这太便宜了。",[11,72676,72677],{},"所以，「能不能翻译」比「美不美」更值得问。丑是可以忍受的，只要它能在可接受的成本内被压缩成一套可迁移的结构。丑而无法批量翻译，才是真正的问题。对 AI 那七百个结果，关键在翻译成本：把它们变成人类可读的结构要花多大代价，这个代价能不能被规模化地摊薄。如果每个结果都需要一位数学家花十年去反推，那七百个结果的边际价值就取决于翻译的规模化程度——而这恰恰是 AI 可能帮上忙的地方：让 AI 去搜「能压缩这七百个结果的统一框架」。丑不可怕，可怕的是丑得无法批量处理。",[11,72679,72680],{},"「地球灭亡之前，人类没法理解的事情肯定存在。」这句话我同意。Gödel 的不完备定理保证了真而不可证的东西存在，复杂性理论则告诉我们，一大批问题在可接受的资源内算不动。但这句话不能推出「所以现在别理解了」。",[11,72682,72683],{},"从「存在无法理解的东西」推不出「眼前这七百个就是无法理解的」。这是一个很基本的逻辑错误：存在量词不能这样实例化。而且，可理解域的边界是未知的，历史上每一次都被推得比预期远得多。Hilbert 纲领失败了，但失败本身催生了证明论和可计算性理论，也深刻改变了模型论的问题意识。数学的边界更像一片雾。你往前走，雾就退一点。",[11,72685,72686],{},"理解有许多层次。我们理解热力学，不需要追踪每一个分子。同样，我们理解一个函数，不需要它的封闭形式。七百个结果可能在细粒度上不可读，但在粗粒度上仍然可以被理解：它们可能共享某种对称性、某种生成机制、某种还没被命名的结构。放弃理解需要「已经证明这里推不动」。「现在看着像一团乱麻」还不够。前者是数学，后者是情绪。",[11,72688,72689],{},"另外，AI 的方法太有针对性，这是数学界最实在的批评之一。以前一个理论出来，不仅解决目标问题，还会带来一整套工具箱，影响其他领域。现在的 AI 方法往往只解决目标问题，然后就结束了。",[11,72691,72692],{},"这条批评有它的道理。人类数学的溢出效应是结构性的。Galois 理论本来是为了解五次方程，产出的群论和域扩张后来成了整个代数的主干。Riemann 为了理解素数分布把复分析引进数论，他的几何工作又养活了微分几何和拓扑。同调代数从拓扑里长出来，最后变成了交换代数、代数几何、表示论的通用语法。目标问题是钩子，工具箱才是本体。数学界评价一个工作，隐含的标准是「它留下了什么可复用的东西」。「它证明了什么」只是起点。",[11,72694,72695],{},"AI 目前的产出恰好踩在这个标准的反面。AlphaTensor 找矩阵乘法算法、组合搜索、符号回归凑公式，共同特征是：针对特定实例、依赖大规模搜索、结果的可迁移性有限。它优化的是「这个问题有没有解、解是什么」。「这类问题的结构长什么样」则被放在一边。一个只针对单点的工具，边际价值确实低，尤其当它消耗的算力还能拿去做别的事。",[11,72697,72698],{},"但针对性太强，是 AI 的局限，还是当前阶段的局限？历史上「针对性太强」的批评也落在过人类自己头上。数值分析早期被纯数学家看不起，觉得是工程凑数，后来它长出了稳定性、收敛阶、条件数这一整套理论。概率论被当成赌博工具几百年，Kolmogorov 公理化之后才翻身。针对性强的工具，如果被反复使用，往往会倒逼出一般理论。差别在于：人类工具的一般化是被使用者的需求驱动的，慢但稳。AI 工具的一般化目前没有人系统性地做，因为发论文的激励在「又刷了一个纪录」，不在「把这套方法抽象成可复用的框架」。",[11,72700,72701],{},"所以，真正缺的，是有人负责把针对性的方法一般化。「AI 方法针对性太强」只是表象。人类数学家有同行评议和长期声誉在逼他们想「这东西能不能推广」。AI 数学的产出链条里，这个环节目前是断的。如果哪天真有团队认真做「把七百个硬展开结果反推成一个统一框架」，而且做成了，那这条批评就自动消解了。但这件事的难度不比发现那七百个结果低，甚至更高：归纳一个框架，靠的是猜对抽象层级，算力反而次要。而这恰恰是目前最说不清楚 AI 能不能做好的部分。",[11,72703,72704],{},"翻译是一项制度功能，很难靠某位数学家的英雄行为完成。人类数学之所以能把计算变成结构，靠的是一整套同行评议、教科书、研讨会和长期声誉的机器，在逼着每一代人去问「这东西能不能推广」。个别天才只是其中一环。AI 数学现在缺的，正是这台机器，算力和聪明的模型都还在其次。真正的瓶颈在分工，认知带宽不是主要问题。",[11,72706,72707],{},"这套论证的力量来自三个漂亮的偷换，把动机当成定义，把事后的判词当成先验的门槛，把答案当成理解。",[11,72709,72710],{},"但挑战是真的，只是位置放错了：我们把计算翻译成结构的制度，没有跟上计算的速度。「AI 比人算得快」只是表面。",[11,72712,72713],{},"画家画不出世界上的每一个光子，所以他发明透视。但透视是射影几何，跟「眼睛不行」没有关系。",[11,72715,72716],{},"今天最强的渲染器可以逐像素地硬算，但它仍然跑在透视的数学上。硬算并没有取消结构。它把结构算到了人眼跟不上的地方。",[11,72718,72719],{},"上帝或许真的在用计算器。但账本不是给计算器看的。人类接下来要做的，是翻译账本，让计算器的结果变得可读。",[11,72721,72722],{},"所以说，AI 在未来会彻底取代数学研究的劳动吗？",[11,72724,72725],{},"我认为不会，而且在未来很长一段时间里，都不会，至少在 AGI 诞生之前永远不会。",[15772,72727,72728],{},[11,72729,72730],{},"砸掉计算器解决不了问题。数学的下一份工作，也许要从「发现」转向「翻译」。",{"title":23,"searchDepth":64,"depth":64,"links":72732},[],"前段时间，OpenAI 声称，他们用内部尚未正式发布的模型产出了七百多篇论文，「AI 接替数学家」的叙事于是再一次被推到台前。市场是狂热还是炒作，我已经疲于争论。那七百篇论文是否名副其实，也先放在一边。真正让我想写点东西的，是 Twitter 上一条获得不少认同的帖子。它把数学结构说成人类为了迁就自己那点脑容量而被迫发明的拐杖，把 AI 说成不需要几何直观、可以直接站在高维展开式里往下看的怪物。最后一句尤其漂亮：我们终于拿到了上帝账本里的答案，却发现上帝算账根本不用微积分，他老人家是直接拿计算器硬按出来的。",{},"\u002Fblog\u002F2026\u002F0012-shu-xue-yu-hua-jia",{"title":72556,"description":72733},"blog\u002F2026\u002F0012-shu-xue-yu-hua-jia","结构本身就是定理，AI 也有结构，只是那种结构人读不懂。关键在于可翻译性，以及把计算翻译成结构的制度是否跟得上。硬算常给出有限情形的答案而非「为什么」，而把针对性的结果一般化、翻译成可复用结构的制度，在 AI 数学的产出链条里是断的。",[72740],"thoughts","2026-10-25T16:52:45+08:00","MTXFn_wz9eNH537cFyhWDAYMtvfSTEdfulMeMndlEyU",1794483690791]