[{"data":1,"prerenderedAt":72554},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun":3,"i-ci:tag":-1,"i-mingcute:time-line":-1},{"post":4,"nextPost":52829,"prevPost":52841},{"id":5,"title":6,"body":7,"description":13,"draft":52826,"enableCodeHighlight":52826,"enableCodeHightlight":52826,"enableComment":52827,"enableD2":52826,"enableKatex":52827,"extension":52828,"image":52812,"important":52826,"location":52829,"meta":52830,"navigation":52827,"ogImage":52829,"path":52831,"seo":52832,"stem":52833,"summary":52834,"tags":52835,"time":52839,"__hash__":52840},"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":9,"toc":52811},"minimark",[10,14,17,21,24,27,408,752,835,902,1139,1392,1515,1864,2476,2479,2979,3098,3466,3499,3671,4046,4747,4910,5165,5197,5301,5304,5307,5370,5589,5592,6985,7250,8023,8026,9610,10133,10751,10754,10757,11112,11545,11872,11875,12248,12251,12254,12381,12606,12801,12992,13065,13588,13769,14355,14750,15158,15876,16360,16363,17297,18411,18623,19061,19064,19438,19441,19472,19716,19854,19857,20230,20233,20651,20882,21081,21563,22120,22175,22307,22595,22785,22892,22895,23113,23146,23372,23375,23378,23772,23775,23860,24600,24975,24978,25115,26061,26064,26067,26070,26273,26446,26616,27090,27530,27633,27998,28580,28583,29400,29403,29406,29467,29470,29702,30144,30525,31020,31023,32380,32503,32661,33137,33906,33909,33913,34066,34267,34472,34595,35162,35165,35226,35513,35684,36678,36681,37167,37243,37246,37709,37942,38343,40166,40242,40245,40473,40964,40967,41418,41922,41925,42064,42294,42384,42387,42992,43396,44128,44469,44811,45592,45595,45598,45601,45604,45669,45836,45986,46430,46433,46598,47053,47056,47429,47491,48118,49067,49170,49528,49729,49800,50597,50805,50808,50811,50814,50929,51417,51449,51452,52028,52031,52034,52188,52440,52443,52446,52449,52452,52484,52487,52805,52808],[11,12,13],"p",{},"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”。",[11,15,16],{},"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。",[18,19,20],"h2",{"id":20},"搬土",[11,22,23],{},"先算一道运输题。",[11,25,26],{},"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？",[11,28,29,30,77,78,137,138,190,191,330,331,407],{},"设甲运往 A 的吨数是 ",[31,32,36,58],"span",{"className":33,"translate":35},[34],"katex","no",[31,37,40],{"className":38},[39],"katex-mathml",[41,42,44],"math",{"xmlns":43},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[45,46,47,54],"semantics",{},[48,49,50],"mrow",{},[51,52,53],"mi",{},"x",[55,56,53],"annotation",{"encoding":57},"application\u002Fx-tex",[31,59,63],{"className":60,"ariaHidden":62},[61],"katex-html","true",[31,64,67,72],{"className":65},[66],"base",[31,68],{"className":69,"style":71},[70],"strut","height:0.4306em;",[31,73,53],{"className":74},[75,76],"mord","mathnormal","。甲的 3 吨要分完，所以甲到 B 是 ",[31,79,81,103],{"className":80,"translate":35},[34],[31,82,84],{"className":83},[39],[41,85,86],{"xmlns":43},[45,87,88,100],{},[48,89,90,94,98],{},[91,92,93],"mn",{},"3",[95,96,97],"mo",{},"−",[51,99,53],{},[55,101,102],{"encoding":57},"3-x",[31,104,106,128],{"className":105,"ariaHidden":62},[61],[31,107,109,113,116,121,125],{"className":108},[66],[31,110],{"className":111,"style":112},[70],"height:0.7278em;vertical-align:-0.0833em;",[31,114,93],{"className":115},[75],[31,117],{"className":118,"style":120},[119],"mspace","margin-right:0.2222em;",[31,122,97],{"className":123},[124],"mbin",[31,126],{"className":127,"style":120},[119],[31,129,131,134],{"className":130},[66],[31,132],{"className":133,"style":71},[70],[31,135,53],{"className":136},[75,76],"；A 要 2 吨，所以乙到 A 是 ",[31,139,141,160],{"className":140,"translate":35},[34],[31,142,144],{"className":143},[39],[41,145,146],{"xmlns":43},[45,147,148,157],{},[48,149,150,153,155],{},[91,151,152],{},"2",[95,154,97],{},[51,156,53],{},[55,158,159],{"encoding":57},"2-x",[31,161,163,181],{"className":162,"ariaHidden":62},[61],[31,164,166,169,172,175,178],{"className":165},[66],[31,167],{"className":168,"style":112},[70],[31,170,152],{"className":171},[75],[31,173],{"className":174,"style":120},[119],[31,176,97],{"className":177},[124],[31,179],{"className":180,"style":120},[119],[31,182,184,187],{"className":183},[66],[31,185],{"className":186,"style":71},[70],[31,188,53],{"className":189},[75,76],"；乙只有 1 吨，所以乙到 B 是 ",[31,192,194,233],{"className":193,"translate":35},[34],[31,195,197],{"className":196},[39],[41,198,199],{"xmlns":43},[45,200,201,230],{},[48,202,203,206,208,212,214,216,218,221,224,226,228],{},[91,204,205],{},"1",[95,207,97],{},[95,209,211],{"stretchy":210},"false","(",[91,213,152],{},[95,215,97],{},[51,217,53],{},[95,219,220],{"stretchy":210},")",[95,222,223],{},"=",[51,225,53],{},[95,227,97],{},[91,229,205],{},[55,231,232],{"encoding":57},"1-(2-x)=x-1",[31,234,236,254,277,301,320],{"className":235,"ariaHidden":62},[61],[31,237,239,242,245,248,251],{"className":238},[66],[31,240],{"className":241,"style":112},[70],[31,243,205],{"className":244},[75],[31,246],{"className":247,"style":120},[119],[31,249,97],{"className":250},[124],[31,252],{"className":253,"style":120},[119],[31,255,257,261,265,268,271,274],{"className":256},[66],[31,258],{"className":259,"style":260},[70],"height:1em;vertical-align:-0.25em;",[31,262,211],{"className":263},[264],"mopen",[31,266,152],{"className":267},[75],[31,269],{"className":270,"style":120},[119],[31,272,97],{"className":273},[124],[31,275],{"className":276,"style":120},[119],[31,278,280,283,286,290,294,298],{"className":279},[66],[31,281],{"className":282,"style":260},[70],[31,284,53],{"className":285},[75,76],[31,287,220],{"className":288},[289],"mclose",[31,291],{"className":292,"style":293},[119],"margin-right:0.2778em;",[31,295,223],{"className":296},[297],"mrel",[31,299],{"className":300,"style":293},[119],[31,302,304,308,311,314,317],{"className":303},[66],[31,305],{"className":306,"style":307},[70],"height:0.6667em;vertical-align:-0.0833em;",[31,309,53],{"className":310},[75,76],[31,312],{"className":313,"style":120},[119],[31,315,97],{"className":316},[124],[31,318],{"className":319,"style":120},[119],[31,321,323,327],{"className":322},[66],[31,324],{"className":325,"style":326},[70],"height:0.6444em;",[31,328,205],{"className":329},[75],"。四个数都得非负，于是 ",[31,332,334,357],{"className":333,"translate":35},[34],[31,335,337],{"className":336},[39],[41,338,339],{"xmlns":43},[45,340,341,354],{},[48,342,343,345,348,350,352],{},[91,344,205],{},[95,346,347],{},"≤",[51,349,53],{},[95,351,347],{},[91,353,152],{},[55,355,356],{"encoding":57},"1\\le x\\le 2",[31,358,360,379,398],{"className":359,"ariaHidden":62},[61],[31,361,363,367,370,373,376],{"className":362},[66],[31,364],{"className":365,"style":366},[70],"height:0.7804em;vertical-align:-0.136em;",[31,368,205],{"className":369},[75],[31,371],{"className":372,"style":293},[119],[31,374,347],{"className":375},[297],[31,377],{"className":378,"style":293},[119],[31,380,382,386,389,392,395],{"className":381},[66],[31,383],{"className":384,"style":385},[70],"height:0.7719em;vertical-align:-0.136em;",[31,387,53],{"className":388},[75,76],[31,390],{"className":391,"style":293},[119],[31,393,347],{"className":394},[297],[31,396],{"className":397,"style":293},[119],[31,399,401,404],{"className":400},[66],[31,402],{"className":403,"style":326},[70],[31,405,152],{"className":406},[75],"。总运费",[31,409,412],{"className":410,"translate":35},[411],"katex-display",[31,413,415,499],{"className":414,"translate":35},[34],[31,416,418],{"className":417},[39],[41,419,421],{"xmlns":43,"display":420},"block",[45,422,423,496],{},[48,424,425,427,430,432,435,437,439,441,443,445,447,449,451,453,455,457,459,461,463,465,467,469,471,473,475,477,479,481,483,486,488,490,492],{},[51,426,53],{},[95,428,429],{},"⋅",[91,431,205],{},[95,433,434],{},"+",[95,436,211],{"stretchy":210},[91,438,93],{},[95,440,97],{},[51,442,53],{},[95,444,220],{"stretchy":210},[95,446,429],{},[91,448,152],{},[95,450,434],{},[95,452,211],{"stretchy":210},[91,454,152],{},[95,456,97],{},[51,458,53],{},[95,460,220],{"stretchy":210},[95,462,429],{},[91,464,93],{},[95,466,434],{},[95,468,211],{"stretchy":210},[51,470,53],{},[95,472,97],{},[91,474,205],{},[95,476,220],{"stretchy":210},[95,478,429],{},[91,480,205],{},[95,482,223],{},[91,484,485],{},"11",[95,487,97],{},[91,489,93],{},[51,491,53],{},[51,493,495],{"mathvariant":494},"normal",".",[55,497,498],{"encoding":57},"x\\cdot1+(3-x)\\cdot2+(2-x)\\cdot3+(x-1)\\cdot1=11-3x .",[31,500,502,521,539,560,581,599,620,641,659,680,701,719,737],{"className":501,"ariaHidden":62},[61],[31,503,505,509,512,515,518],{"className":504},[66],[31,506],{"className":507,"style":508},[70],"height:0.4445em;",[31,510,53],{"className":511},[75,76],[31,513],{"className":514,"style":120},[119],[31,516,429],{"className":517},[124],[31,519],{"className":520,"style":120},[119],[31,522,524,527,530,533,536],{"className":523},[66],[31,525],{"className":526,"style":112},[70],[31,528,205],{"className":529},[75],[31,531],{"className":532,"style":120},[119],[31,534,434],{"className":535},[124],[31,537],{"className":538,"style":120},[119],[31,540,542,545,548,551,554,557],{"className":541},[66],[31,543],{"className":544,"style":260},[70],[31,546,211],{"className":547},[264],[31,549,93],{"className":550},[75],[31,552],{"className":553,"style":120},[119],[31,555,97],{"className":556},[124],[31,558],{"className":559,"style":120},[119],[31,561,563,566,569,572,575,578],{"className":562},[66],[31,564],{"className":565,"style":260},[70],[31,567,53],{"className":568},[75,76],[31,570,220],{"className":571},[289],[31,573],{"className":574,"style":120},[119],[31,576,429],{"className":577},[124],[31,579],{"className":580,"style":120},[119],[31,582,584,587,590,593,596],{"className":583},[66],[31,585],{"className":586,"style":112},[70],[31,588,152],{"className":589},[75],[31,591],{"className":592,"style":120},[119],[31,594,434],{"className":595},[124],[31,597],{"className":598,"style":120},[119],[31,600,602,605,608,611,614,617],{"className":601},[66],[31,603],{"className":604,"style":260},[70],[31,606,211],{"className":607},[264],[31,609,152],{"className":610},[75],[31,612],{"className":613,"style":120},[119],[31,615,97],{"className":616},[124],[31,618],{"className":619,"style":120},[119],[31,621,623,626,629,632,635,638],{"className":622},[66],[31,624],{"className":625,"style":260},[70],[31,627,53],{"className":628},[75,76],[31,630,220],{"className":631},[289],[31,633],{"className":634,"style":120},[119],[31,636,429],{"className":637},[124],[31,639],{"className":640,"style":120},[119],[31,642,644,647,650,653,656],{"className":643},[66],[31,645],{"className":646,"style":112},[70],[31,648,93],{"className":649},[75],[31,651],{"className":652,"style":120},[119],[31,654,434],{"className":655},[124],[31,657],{"className":658,"style":120},[119],[31,660,662,665,668,671,674,677],{"className":661},[66],[31,663],{"className":664,"style":260},[70],[31,666,211],{"className":667},[264],[31,669,53],{"className":670},[75,76],[31,672],{"className":673,"style":120},[119],[31,675,97],{"className":676},[124],[31,678],{"className":679,"style":120},[119],[31,681,683,686,689,692,695,698],{"className":682},[66],[31,684],{"className":685,"style":260},[70],[31,687,205],{"className":688},[75],[31,690,220],{"className":691},[289],[31,693],{"className":694,"style":120},[119],[31,696,429],{"className":697},[124],[31,699],{"className":700,"style":120},[119],[31,702,704,707,710,713,716],{"className":703},[66],[31,705],{"className":706,"style":326},[70],[31,708,205],{"className":709},[75],[31,711],{"className":712,"style":293},[119],[31,714,223],{"className":715},[297],[31,717],{"className":718,"style":293},[119],[31,720,722,725,728,731,734],{"className":721},[66],[31,723],{"className":724,"style":112},[70],[31,726,485],{"className":727},[75],[31,729],{"className":730,"style":120},[119],[31,732,97],{"className":733},[124],[31,735],{"className":736,"style":120},[119],[31,738,740,743,746,749],{"className":739},[66],[31,741],{"className":742,"style":326},[70],[31,744,93],{"className":745},[75],[31,747,53],{"className":748},[75,76],[31,750,495],{"className":751},[75],[11,753,754,782,783,834],{},[31,755,757,770],{"className":756,"translate":35},[34],[31,758,760],{"className":759},[39],[41,761,762],{"xmlns":43},[45,763,764,768],{},[48,765,766],{},[51,767,53],{},[55,769,53],{"encoding":57},[31,771,773],{"className":772,"ariaHidden":62},[61],[31,774,776,779],{"className":775},[66],[31,777],{"className":778,"style":71},[70],[31,780,53],{"className":781},[75,76]," 越大越省，取 ",[31,784,786,804],{"className":785,"translate":35},[34],[31,787,789],{"className":788},[39],[41,790,791],{"xmlns":43},[45,792,793,801],{},[48,794,795,797,799],{},[51,796,53],{},[95,798,223],{},[91,800,152],{},[55,802,803],{"encoding":57},"x=2",[31,805,807,825],{"className":806,"ariaHidden":62},[61],[31,808,810,813,816,819,822],{"className":809},[66],[31,811],{"className":812,"style":71},[70],[31,814,53],{"className":815},[75,76],[31,817],{"className":818,"style":293},[119],[31,820,223],{"className":821},[297],[31,823],{"className":824,"style":293},[119],[31,826,828,831],{"className":827},[66],[31,829],{"className":830,"style":326},[70],[31,832,152],{"className":833},[75],"，总运费 5 块：甲到 A 运 2 吨，甲到 B 运 1 吨，乙到 B 运 1 吨。",[11,836,837,838,869,870,901],{},"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 ",[31,839,841,856],{"className":840,"translate":35},[34],[31,842,844],{"className":843},[39],[41,845,846],{"xmlns":43},[45,847,848,853],{},[48,849,850],{},[51,851,852],{},"μ",[55,854,855],{"encoding":57},"\\mu",[31,857,859],{"className":858,"ariaHidden":62},[61],[31,860,862,866],{"className":861},[66],[31,863],{"className":864,"style":865},[70],"height:0.625em;vertical-align:-0.1944em;",[31,867,852],{"className":868},[75,76]," 和一个坑 ",[31,871,873,888],{"className":872,"translate":35},[34],[31,874,876],{"className":875},[39],[41,877,878],{"xmlns":43},[45,879,880,885],{},[48,881,882],{},[51,883,884],{},"ν",[55,886,887],{"encoding":57},"\\nu",[31,889,891],{"className":890,"ariaHidden":62},[61],[31,892,894,897],{"className":893},[66],[31,895],{"className":896,"style":71},[70],[31,898,884],{"className":899,"style":900},[75,76],"margin-right:0.0637em;","，怎么搬最省。",[11,903,904,905,990,991,1107,1108,1138],{},"他的写法是找一个映射 ",[31,906,908,936],{"className":907,"translate":35},[34],[31,909,911],{"className":910},[39],[41,912,913],{"xmlns":43},[45,914,915,933],{},[48,916,917,920,923,927,930],{},[51,918,919],{},"T",[95,921,922],{},":",[51,924,926],{"mathvariant":925},"script","X",[95,928,929],{},"→",[51,931,932],{"mathvariant":925},"Y",[55,934,935],{"encoding":57},"T:\\mathcal X\\to\\mathcal Y",[31,937,939,959,979],{"className":938,"ariaHidden":62},[61],[31,940,942,946,950,953,956],{"className":941},[66],[31,943],{"className":944,"style":945},[70],"height:0.6833em;",[31,947,919],{"className":948,"style":949},[75,76],"margin-right:0.1389em;",[31,951],{"className":952,"style":293},[119],[31,954,922],{"className":955},[297],[31,957],{"className":958,"style":293},[119],[31,960,962,965,970,973,976],{"className":961},[66],[31,963],{"className":964,"style":945},[70],[31,966,926],{"className":967,"style":969},[75,968],"mathcal","margin-right:0.1464em;",[31,971],{"className":972,"style":293},[119],[31,974,929],{"className":975},[297],[31,977],{"className":978,"style":293},[119],[31,980,982,986],{"className":981},[66],[31,983],{"className":984,"style":985},[70],"height:0.7805em;vertical-align:-0.0972em;",[31,987,932],{"className":988,"style":989},[75,968],"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 ",[31,992,994,1020],{"className":993,"translate":35},[34],[31,995,997],{"className":996},[39],[41,998,999],{"xmlns":43},[45,1000,1001,1017],{},[48,1002,1003,1011,1013,1015],{},[1004,1005,1006,1008],"msub",{},[51,1007,919],{},[51,1009,1010],{"mathvariant":494},"#",[51,1012,852],{},[95,1014,223],{},[51,1016,884],{},[55,1018,1019],{"encoding":57},"T_\\#\\mu=\\nu",[31,1021,1023,1098],{"className":1022,"ariaHidden":62},[61],[31,1024,1026,1030,1086,1089,1092,1095],{"className":1025},[66],[31,1027],{"className":1028,"style":1029},[70],"height:0.9694em;vertical-align:-0.2861em;",[31,1031,1033,1036],{"className":1032},[75],[31,1034,919],{"className":1035,"style":949},[75,76],[31,1037,1040],{"className":1038},[1039],"msupsub",[31,1041,1045,1077],{"className":1042},[1043,1044],"vlist-t","vlist-t2",[31,1046,1049,1072],{"className":1047},[1048],"vlist-r",[31,1050,1054],{"className":1051,"style":1053},[1052],"vlist","height:0.3361em;",[31,1055,1057,1062],{"style":1056},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[31,1058],{"className":1059,"style":1061},[1060],"pstrut","height:2.7em;",[31,1063,1069],{"className":1064},[1065,1066,1067,1068],"sizing","reset-size6","size3","mtight",[31,1070,1010],{"className":1071},[75,1068],[31,1073,1076],{"className":1074},[1075],"vlist-s","​",[31,1078,1080],{"className":1079},[1048],[31,1081,1084],{"className":1082,"style":1083},[1052],"height:0.2861em;",[31,1085],{},[31,1087,852],{"className":1088},[75,76],[31,1090],{"className":1091,"style":293},[119],[31,1093,223],{"className":1094},[297],[31,1096],{"className":1097,"style":293},[119],[31,1099,1101,1104],{"className":1100},[66],[31,1102],{"className":1103,"style":71},[70],[31,1105,884],{"className":1106,"style":900},[75,76],"，即对任意可测集 ",[31,1109,1111,1125],{"className":1110,"translate":35},[34],[31,1112,1114],{"className":1113},[39],[41,1115,1116],{"xmlns":43},[45,1117,1118,1123],{},[48,1119,1120],{},[51,1121,1122],{},"B",[55,1124,1122],{"encoding":57},[31,1126,1128],{"className":1127,"ariaHidden":62},[61],[31,1129,1131,1134],{"className":1130},[66],[31,1132],{"className":1133,"style":945},[70],[31,1135,1122],{"className":1136,"style":1137},[75,76],"margin-right:0.0502em;","，",[31,1140,1142],{"className":1141,"translate":35},[411],[31,1143,1145,1214],{"className":1144,"translate":35},[34],[31,1146,1148],{"className":1147},[39],[41,1149,1150],{"xmlns":43,"display":420},[45,1151,1152,1211],{},[48,1153,1154,1156,1162,1164,1166,1168,1170,1172,1174,1176,1179,1190,1192,1194,1196,1198,1200,1202,1204,1206,1208],{},[95,1155,211],{"stretchy":210},[1004,1157,1158,1160],{},[51,1159,919],{},[51,1161,1010],{"mathvariant":494},[51,1163,852],{},[95,1165,220],{"stretchy":210},[95,1167,211],{"stretchy":210},[51,1169,1122],{},[95,1171,220],{"stretchy":210},[95,1173,223],{},[51,1175,852],{},[95,1177,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},"1.2em",[1180,1181,1182,1184],"msup",{},[51,1183,919],{},[48,1185,1186,1188],{},[95,1187,97],{},[91,1189,205],{},[95,1191,211],{"stretchy":210},[51,1193,1122],{},[95,1195,220],{"stretchy":210},[95,1197,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,1199,223],{},[51,1201,884],{},[95,1203,211],{"stretchy":210},[51,1205,1122],{},[95,1207,220],{"stretchy":210},[95,1209,1210],{"separator":62},",",[55,1212,1213],{"encoding":57},"(T_\\#\\mu)(B)=\\mu\\bigl(T^{-1}(B)\\bigr)=\\nu(B),",[31,1215,1217,1291,1370],{"className":1216,"ariaHidden":62},[61],[31,1218,1220,1224,1227,1267,1270,1273,1276,1279,1282,1285,1288],{"className":1219},[66],[31,1221],{"className":1222,"style":1223},[70],"height:1.0361em;vertical-align:-0.2861em;",[31,1225,211],{"className":1226},[264],[31,1228,1230,1233],{"className":1229},[75],[31,1231,919],{"className":1232,"style":949},[75,76],[31,1234,1236],{"className":1235},[1039],[31,1237,1239,1259],{"className":1238},[1043,1044],[31,1240,1242,1256],{"className":1241},[1048],[31,1243,1245],{"className":1244,"style":1053},[1052],[31,1246,1247,1250],{"style":1056},[31,1248],{"className":1249,"style":1061},[1060],[31,1251,1253],{"className":1252},[1065,1066,1067,1068],[31,1254,1010],{"className":1255},[75,1068],[31,1257,1076],{"className":1258},[1075],[31,1260,1262],{"className":1261},[1048],[31,1263,1265],{"className":1264,"style":1083},[1052],[31,1266],{},[31,1268,852],{"className":1269},[75,76],[31,1271,220],{"className":1272},[289],[31,1274,211],{"className":1275},[264],[31,1277,1122],{"className":1278,"style":1137},[75,76],[31,1280,220],{"className":1281},[289],[31,1283],{"className":1284,"style":293},[119],[31,1286,223],{"className":1287},[297],[31,1289],{"className":1290,"style":293},[119],[31,1292,1294,1298,1301,1309,1346,1349,1352,1355,1361,1364,1367],{"className":1293},[66],[31,1295],{"className":1296,"style":1297},[70],"height:1.2141em;vertical-align:-0.35em;",[31,1299,852],{"className":1300},[75,76],[31,1302,1304],{"className":1303},[264],[31,1305,211],{"className":1306},[1307,1308],"delimsizing","size1",[31,1310,1312,1315],{"className":1311},[75],[31,1313,919],{"className":1314,"style":949},[75,76],[31,1316,1318],{"className":1317},[1039],[31,1319,1321],{"className":1320},[1043],[31,1322,1324],{"className":1323},[1048],[31,1325,1328],{"className":1326,"style":1327},[1052],"height:0.8641em;",[31,1329,1331,1334],{"style":1330},"top:-3.113em;margin-right:0.05em;",[31,1332],{"className":1333,"style":1061},[1060],[31,1335,1337],{"className":1336},[1065,1066,1067,1068],[31,1338,1340,1343],{"className":1339},[75,1068],[31,1341,97],{"className":1342},[75,1068],[31,1344,205],{"className":1345},[75,1068],[31,1347,211],{"className":1348},[264],[31,1350,1122],{"className":1351,"style":1137},[75,76],[31,1353,220],{"className":1354},[289],[31,1356,1358],{"className":1357},[289],[31,1359,220],{"className":1360},[1307,1308],[31,1362],{"className":1363,"style":293},[119],[31,1365,223],{"className":1366},[297],[31,1368],{"className":1369,"style":293},[119],[31,1371,1373,1376,1379,1382,1385,1388],{"className":1372},[66],[31,1374],{"className":1375,"style":260},[70],[31,1377,884],{"className":1378,"style":900},[75,76],[31,1380,211],{"className":1381},[264],[31,1383,1122],{"className":1384,"style":1137},[75,76],[31,1386,220],{"className":1387},[289],[31,1389,1210],{"className":1390},[1391],"mpunct",[11,1393,1394,1395,1514],{},"代价是 ",[31,1396,1398,1447],{"className":1397,"translate":35},[34],[31,1399,1401],{"className":1400},[39],[41,1402,1403],{"xmlns":43},[45,1404,1405,1444],{},[48,1406,1407,1410,1413,1415,1417,1419,1421,1423,1425,1427,1429,1433,1436,1438,1440,1442],{},[95,1408,1409],{},"∫",[51,1411,1412],{},"c",[95,1414,211],{"stretchy":210},[51,1416,53],{},[95,1418,1210],{"separator":62},[51,1420,919],{},[95,1422,211],{"stretchy":210},[51,1424,53],{},[95,1426,220],{"stretchy":210},[95,1428,220],{"stretchy":210},[1430,1431,1432],"mtext",{}," ",[51,1434,1435],{},"d",[51,1437,852],{},[95,1439,211],{"stretchy":210},[51,1441,53],{},[95,1443,220],{"stretchy":210},[55,1445,1446],{"encoding":57},"\\int c(x,T(x))\\,d\\mu(x)",[31,1448,1450],{"className":1449,"ariaHidden":62},[61],[31,1451,1453,1457,1464,1468,1471,1474,1477,1480,1483,1486,1489,1492,1496,1499,1502,1505,1508,1511],{"className":1452},[66],[31,1454],{"className":1455,"style":1456},[70],"height:1.1111em;vertical-align:-0.3061em;",[31,1458,1409],{"className":1459,"style":1463},[1460,1461,1462],"mop","op-symbol","small-op","margin-right:0.1945em;position:relative;top:-0.0006em;",[31,1465],{"className":1466,"style":1467},[119],"margin-right:0.1667em;",[31,1469,1412],{"className":1470},[75,76],[31,1472,211],{"className":1473},[264],[31,1475,53],{"className":1476},[75,76],[31,1478,1210],{"className":1479},[1391],[31,1481],{"className":1482,"style":1467},[119],[31,1484,919],{"className":1485,"style":949},[75,76],[31,1487,211],{"className":1488},[264],[31,1490,53],{"className":1491},[75,76],[31,1493,1495],{"className":1494},[289],"))",[31,1497],{"className":1498,"style":1467},[119],[31,1500,1435],{"className":1501},[75,76],[31,1503,852],{"className":1504},[75,76],[31,1506,211],{"className":1507},[264],[31,1509,53],{"className":1510},[75,76],[31,1512,220],{"className":1513},[289],"。于是 Monge 问题就是",[31,1516,1518],{"className":1517,"translate":35},[411],[31,1519,1521,1607],{"className":1520,"translate":35},[34],[31,1522,1524],{"className":1523},[39],[41,1525,1526],{"xmlns":43,"display":420},[45,1527,1528,1604],{},[48,1529,1530,1543,1546,1552,1554,1556,1558,1560,1562,1564,1566,1568,1570,1572,1574,1576,1578,1580,1582,1585,1588,1590,1596,1598,1600,1602],{},[1531,1532,1533,1541],"munder",{},[48,1534,1535,1538],{},[51,1536,1537],{},"inf",[95,1539,1540],{},"⁡",[51,1542,919],{},[1430,1544,1545],{}," ",[1004,1547,1548,1550],{},[95,1549,1409],{},[51,1551,926],{"mathvariant":925},[51,1553,1412],{},[95,1555,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,1557,53],{},[95,1559,1210],{"separator":62},[51,1561,919],{},[95,1563,211],{"stretchy":210},[51,1565,53],{},[95,1567,220],{"stretchy":210},[95,1569,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1430,1571,1432],{},[51,1573,1435],{},[51,1575,852],{},[95,1577,211],{"stretchy":210},[51,1579,53],{},[95,1581,220],{"stretchy":210},[119,1583],{"width":1584},"1em",[1430,1586,1587],{},"s.t.",[119,1589],{"width":1584},[1004,1591,1592,1594],{},[51,1593,919],{},[51,1595,1010],{"mathvariant":494},[51,1597,852],{},[95,1599,223],{},[51,1601,884],{},[51,1603,495],{"mathvariant":494},[55,1605,1606],{"encoding":57},"\\inf_{T}\\ \\int_{\\mathcal X} c\\bigl(x,T(x)\\bigr)\\,d\\mu(x)\n\\quad\\text{s.t.}\\quad T_\\#\\mu=\\nu .",[31,1608,1610,1852],{"className":1609,"ariaHidden":62},[61],[31,1611,1613,1617,1675,1678,1681,1729,1732,1735,1741,1744,1747,1750,1753,1756,1759,1762,1768,1771,1774,1777,1780,1783,1786,1790,1797,1800,1840,1843,1846,1849],{"className":1612},[66],[31,1614],{"className":1615,"style":1616},[70],"height:2.2719em;vertical-align:-0.9119em;",[31,1618,1621],{"className":1619},[1460,1620],"op-limits",[31,1622,1624,1666],{"className":1623},[1043,1044],[31,1625,1627,1663],{"className":1626},[1048],[31,1628,1631,1647],{"className":1629,"style":1630},[1052],"height:0.6944em;",[31,1632,1634,1638],{"style":1633},"top:-2.3557em;margin-left:0em;",[31,1635],{"className":1636,"style":1637},[1060],"height:3em;",[31,1639,1641],{"className":1640},[1065,1066,1067,1068],[31,1642,1644],{"className":1643},[75,1068],[31,1645,919],{"className":1646,"style":949},[75,76,1068],[31,1648,1650,1653],{"style":1649},"top:-3em;",[31,1651],{"className":1652,"style":1637},[1060],[31,1654,1655],{},[31,1656,1658,1659],{"className":1657},[1460],"in",[31,1660,1662],{"style":1661},"margin-right:0.0778em;","f",[31,1664,1076],{"className":1665},[1075],[31,1667,1669],{"className":1668},[1048],[31,1670,1673],{"className":1671,"style":1672},[1052],"height:0.7443em;",[31,1674],{},[31,1676,1545],{"className":1677},[119],[31,1679],{"className":1680,"style":1467},[119],[31,1682,1684,1689],{"className":1683},[1460],[31,1685,1409],{"className":1686,"style":1688},[1460,1461,1687],"large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[31,1690,1692],{"className":1691},[1039],[31,1693,1695,1720],{"className":1694},[1043,1044],[31,1696,1698,1717],{"className":1697},[1048],[31,1699,1702],{"className":1700,"style":1701},[1052],"height:-0.4336em;",[31,1703,1705,1708],{"style":1704},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[31,1706],{"className":1707,"style":1061},[1060],[31,1709,1711],{"className":1710},[1065,1066,1067,1068],[31,1712,1714],{"className":1713},[75,1068],[31,1715,926],{"className":1716,"style":969},[75,968,1068],[31,1718,1076],{"className":1719},[1075],[31,1721,1723],{"className":1722},[1048],[31,1724,1727],{"className":1725,"style":1726},[1052],"height:0.9119em;",[31,1728],{},[31,1730],{"className":1731,"style":1467},[119],[31,1733,1412],{"className":1734},[75,76],[31,1736,1738],{"className":1737},[264],[31,1739,211],{"className":1740},[1307,1308],[31,1742,53],{"className":1743},[75,76],[31,1745,1210],{"className":1746},[1391],[31,1748],{"className":1749,"style":1467},[119],[31,1751,919],{"className":1752,"style":949},[75,76],[31,1754,211],{"className":1755},[264],[31,1757,53],{"className":1758},[75,76],[31,1760,220],{"className":1761},[289],[31,1763,1765],{"className":1764},[289],[31,1766,220],{"className":1767},[1307,1308],[31,1769],{"className":1770,"style":1467},[119],[31,1772,1435],{"className":1773},[75,76],[31,1775,852],{"className":1776},[75,76],[31,1778,211],{"className":1779},[264],[31,1781,53],{"className":1782},[75,76],[31,1784,220],{"className":1785},[289],[31,1787],{"className":1788,"style":1789},[119],"margin-right:1em;",[31,1791,1794],{"className":1792},[75,1793],"text",[31,1795,1587],{"className":1796},[75],[31,1798],{"className":1799,"style":1789},[119],[31,1801,1803,1806],{"className":1802},[75],[31,1804,919],{"className":1805,"style":949},[75,76],[31,1807,1809],{"className":1808},[1039],[31,1810,1812,1832],{"className":1811},[1043,1044],[31,1813,1815,1829],{"className":1814},[1048],[31,1816,1818],{"className":1817,"style":1053},[1052],[31,1819,1820,1823],{"style":1056},[31,1821],{"className":1822,"style":1061},[1060],[31,1824,1826],{"className":1825},[1065,1066,1067,1068],[31,1827,1010],{"className":1828},[75,1068],[31,1830,1076],{"className":1831},[1075],[31,1833,1835],{"className":1834},[1048],[31,1836,1838],{"className":1837,"style":1083},[1052],[31,1839],{},[31,1841,852],{"className":1842},[75,76],[31,1844],{"className":1845,"style":293},[119],[31,1847,223],{"className":1848},[297],[31,1850],{"className":1851,"style":293},[119],[31,1853,1855,1858,1861],{"className":1854},[66],[31,1856],{"className":1857,"style":71},[70],[31,1859,884],{"className":1860,"style":900},[75,76],[31,1862,495],{"className":1863},[75],[11,1865,1866,1867,1895,1896,1924,1925,1953,1954,2053,2054,2372,2373,2417,2418,2446,2447,2475],{},"这个提法有一个硬伤：",[31,1868,1870,1883],{"className":1869,"translate":35},[34],[31,1871,1873],{"className":1872},[39],[41,1874,1875],{"xmlns":43},[45,1876,1877,1881],{},[48,1878,1879],{},[51,1880,919],{},[55,1882,919],{"encoding":57},[31,1884,1886],{"className":1885,"ariaHidden":62},[61],[31,1887,1889,1892],{"className":1888},[66],[31,1890],{"className":1891,"style":945},[70],[31,1893,919],{"className":1894,"style":949},[75,76]," 必须是确定性的，一个产地只能有一条固定去向。如果 ",[31,1897,1899,1912],{"className":1898,"translate":35},[34],[31,1900,1902],{"className":1901},[39],[41,1903,1904],{"xmlns":43},[45,1905,1906,1910],{},[48,1907,1908],{},[51,1909,852],{},[55,1911,855],{"encoding":57},[31,1913,1915],{"className":1914,"ariaHidden":62},[61],[31,1916,1918,1921],{"className":1917},[66],[31,1919],{"className":1920,"style":865},[70],[31,1922,852],{"className":1923},[75,76]," 在某个点上有原子，而 ",[31,1926,1928,1941],{"className":1927,"translate":35},[34],[31,1929,1931],{"className":1930},[39],[41,1932,1933],{"xmlns":43},[45,1934,1935,1939],{},[48,1936,1937],{},[51,1938,884],{},[55,1940,887],{"encoding":57},[31,1942,1944],{"className":1943,"ariaHidden":62},[61],[31,1945,1947,1950],{"className":1946},[66],[31,1948],{"className":1949,"style":71},[70],[31,1951,884],{"className":1952,"style":900},[75,76]," 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，",[31,1955,1957,1981],{"className":1956,"translate":35},[34],[31,1958,1960],{"className":1959},[39],[41,1961,1962],{"xmlns":43},[45,1963,1964,1978],{},[48,1965,1966,1968,1970],{},[51,1967,852],{},[95,1969,223],{},[1004,1971,1972,1975],{},[51,1973,1974],{},"δ",[91,1976,1977],{},"0",[55,1979,1980],{"encoding":57},"\\mu=\\delta_0",[31,1982,1984,2002],{"className":1983,"ariaHidden":62},[61],[31,1985,1987,1990,1993,1996,1999],{"className":1986},[66],[31,1988],{"className":1989,"style":865},[70],[31,1991,852],{"className":1992},[75,76],[31,1994],{"className":1995,"style":293},[119],[31,1997,223],{"className":1998},[297],[31,2000],{"className":2001,"style":293},[119],[31,2003,2005,2009],{"className":2004},[66],[31,2006],{"className":2007,"style":2008},[70],"height:0.8444em;vertical-align:-0.15em;",[31,2010,2012,2016],{"className":2011},[75],[31,2013,1974],{"className":2014,"style":2015},[75,76],"margin-right:0.0379em;",[31,2017,2019],{"className":2018},[1039],[31,2020,2022,2044],{"className":2021},[1043,1044],[31,2023,2025,2041],{"className":2024},[1048],[31,2026,2029],{"className":2027,"style":2028},[1052],"height:0.3011em;",[31,2030,2032,2035],{"style":2031},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[31,2033],{"className":2034,"style":1061},[1060],[31,2036,2038],{"className":2037},[1065,1066,1067,1068],[31,2039,1977],{"className":2040},[75,1068],[31,2042,1076],{"className":2043},[1075],[31,2045,2047],{"className":2046},[1048],[31,2048,2051],{"className":2049,"style":2050},[1052],"height:0.15em;",[31,2052],{},"；而两个厂各要一半，",[31,2055,2057,2105],{"className":2056,"translate":35},[34],[31,2058,2060],{"className":2059},[39],[41,2061,2062],{"xmlns":43},[45,2063,2064,2102],{},[48,2065,2066,2068,2070,2080,2086,2088,2096],{},[51,2067,884],{},[95,2069,223],{},[2071,2072,2073],"mstyle",{"scriptlevel":1977,"displaystyle":210},[2074,2075,2076,2078],"mfrac",{},[91,2077,205],{},[91,2079,152],{},[1004,2081,2082,2084],{},[51,2083,1974],{},[91,2085,205],{},[95,2087,434],{},[2071,2089,2090],{"scriptlevel":1977,"displaystyle":210},[2074,2091,2092,2094],{},[91,2093,205],{},[91,2095,152],{},[1004,2097,2098,2100],{},[51,2099,1974],{},[91,2101,152],{},[55,2103,2104],{"encoding":57},"\\nu=\\tfrac12\\delta_1+\\tfrac12\\delta_2",[31,2106,2108,2126,2258],{"className":2107,"ariaHidden":62},[61],[31,2109,2111,2114,2117,2120,2123],{"className":2110},[66],[31,2112],{"className":2113,"style":71},[70],[31,2115,884],{"className":2116,"style":900},[75,76],[31,2118],{"className":2119,"style":293},[119],[31,2121,223],{"className":2122},[297],[31,2124],{"className":2125,"style":293},[119],[31,2127,2129,2133,2209,2249,2252,2255],{"className":2128},[66],[31,2130],{"className":2131,"style":2132},[70],"height:1.1901em;vertical-align:-0.345em;",[31,2134,2136,2140,2206],{"className":2135},[75],[31,2137],{"className":2138},[264,2139],"nulldelimiter",[31,2141,2143],{"className":2142},[2074],[31,2144,2146,2197],{"className":2145},[1043,1044],[31,2147,2149,2194],{"className":2148},[1048],[31,2150,2153,2168,2179],{"className":2151,"style":2152},[1052],"height:0.8451em;",[31,2154,2156,2159],{"style":2155},"top:-2.655em;",[31,2157],{"className":2158,"style":1637},[1060],[31,2160,2162],{"className":2161},[1065,1066,1067,1068],[31,2163,2165],{"className":2164},[75,1068],[31,2166,152],{"className":2167},[75,1068],[31,2169,2171,2174],{"style":2170},"top:-3.23em;",[31,2172],{"className":2173,"style":1637},[1060],[31,2175],{"className":2176,"style":2178},[2177],"frac-line","border-bottom-width:0.04em;",[31,2180,2182,2185],{"style":2181},"top:-3.394em;",[31,2183],{"className":2184,"style":1637},[1060],[31,2186,2188],{"className":2187},[1065,1066,1067,1068],[31,2189,2191],{"className":2190},[75,1068],[31,2192,205],{"className":2193},[75,1068],[31,2195,1076],{"className":2196},[1075],[31,2198,2200],{"className":2199},[1048],[31,2201,2204],{"className":2202,"style":2203},[1052],"height:0.345em;",[31,2205],{},[31,2207],{"className":2208},[289,2139],[31,2210,2212,2215],{"className":2211},[75],[31,2213,1974],{"className":2214,"style":2015},[75,76],[31,2216,2218],{"className":2217},[1039],[31,2219,2221,2241],{"className":2220},[1043,1044],[31,2222,2224,2238],{"className":2223},[1048],[31,2225,2227],{"className":2226,"style":2028},[1052],[31,2228,2229,2232],{"style":2031},[31,2230],{"className":2231,"style":1061},[1060],[31,2233,2235],{"className":2234},[1065,1066,1067,1068],[31,2236,205],{"className":2237},[75,1068],[31,2239,1076],{"className":2240},[1075],[31,2242,2244],{"className":2243},[1048],[31,2245,2247],{"className":2246,"style":2050},[1052],[31,2248],{},[31,2250],{"className":2251,"style":120},[119],[31,2253,434],{"className":2254},[124],[31,2256],{"className":2257,"style":120},[119],[31,2259,2261,2264,2332],{"className":2260},[66],[31,2262],{"className":2263,"style":2132},[70],[31,2265,2267,2270,2329],{"className":2266},[75],[31,2268],{"className":2269},[264,2139],[31,2271,2273],{"className":2272},[2074],[31,2274,2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必须同时等于 ",[31,2419,2421,2434],{"className":2420,"translate":35},[34],[31,2422,2424],{"className":2423},[39],[41,2425,2426],{"xmlns":43},[45,2427,2428,2432],{},[48,2429,2430],{},[91,2431,205],{},[55,2433,205],{"encoding":57},[31,2435,2437],{"className":2436,"ariaHidden":62},[61],[31,2438,2440,2443],{"className":2439},[66],[31,2441],{"className":2442,"style":326},[70],[31,2444,205],{"className":2445},[75]," 和 ",[31,2448,2450,2463],{"className":2449,"translate":35},[34],[31,2451,2453],{"className":2452},[39],[41,2454,2455],{"xmlns":43},[45,2456,2457,2461],{},[48,2458,2459],{},[91,2460,152],{},[55,2462,152],{"encoding":57},[31,2464,2466],{"className":2465,"ariaHidden":62},[61],[31,2467,2469,2472],{"className":2468},[66],[31,2470],{"className":2471,"style":326},[70],[31,2473,152],{"className":2474},[75],"，函数做不到。",[11,2477,2478],{},"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",[31,2480,2482],{"className":2481,"translate":35},[411],[31,2483,2485,2610],{"className":2484,"translate":35},[34],[31,2486,2488],{"className":2487},[39],[41,2489,2490],{"xmlns":43,"display":420},[45,2491,2492,2607],{},[48,2493,2494,2497,2499,2501,2503,2505,2507,2509,2513,2516,2519,2522,2524,2526,2529,2531,2533,2535,2537,2539,2541,2558,2564,2566,2568,2570,2572,2574,2576,2590,2596,2598,2600,2602,2605],{},[51,2495,2496],{"mathvariant":494},"Π",[95,2498,211],{"stretchy":210},[51,2500,852],{},[95,2502,1210],{"separator":62},[51,2504,884],{},[95,2506,220],{"stretchy":210},[95,2508,223],{},[95,2510,2512],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},"1.8em","{",[51,2514,2515],{},"π",[95,2517,2518],{},"∈",[51,2520,2521],{"mathvariant":925},"P",[95,2523,211],{"stretchy":210},[51,2525,926],{"mathvariant":925},[95,2527,2528],{},"×",[51,2530,932],{"mathvariant":925},[95,2532,220],{"stretchy":210},[1430,2534,1545],{},[95,2536,922],{},[1430,2538,1545],{},[95,2540,211],{"stretchy":210},[1004,2542,2543,2556],{},[48,2544,2545,2547,2550,2553],{},[51,2546,11],{"mathvariant":494},[51,2548,2549],{"mathvariant":494},"r",[51,2551,2552],{"mathvariant":494},"o",[51,2554,2555],{"mathvariant":494},"j",[51,2557,926],{"mathvariant":925},[1004,2559,2560,2562],{},[95,2561,220],{"stretchy":210},[51,2563,1010],{"mathvariant":494},[51,2565,2515],{},[95,2567,223],{},[51,2569,852],{},[95,2571,1210],{"separator":62},[1430,2573,1545],{},[95,2575,211],{"stretchy":210},[1004,2577,2578,2588],{},[48,2579,2580,2582,2584,2586],{},[51,2581,11],{"mathvariant":494},[51,2583,2549],{"mathvariant":494},[51,2585,2552],{"mathvariant":494},[51,2587,2555],{"mathvariant":494},[51,2589,932],{"mathvariant":925},[1004,2591,2592,2594],{},[95,2593,220],{"stretchy":210},[51,2595,1010],{"mathvariant":494},[51,2597,2515],{},[95,2599,223],{},[51,2601,884],{},[95,2603,2604],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},"}",[95,2606,1210],{"separator":62},[55,2608,2609],{"encoding":57},"\\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\\},",[31,2611,2613,2649,2676,2700,2727,2840,2961],{"className":2612,"ariaHidden":62},[61],[31,2614,2616,2619,2622,2625,2628,2631,2634,2637,2640,2643,2646],{"className":2615},[66],[31,2617],{"className":2618,"style":260},[70],[31,2620,2496],{"className":2621},[75],[31,2623,211],{"className":2624},[264],[31,2626,852],{"className":2627},[75,76],[31,2629,1210],{"className":2630},[1391],[31,2632],{"className":2633,"style":1467},[119],[31,2635,884],{"className":2636,"style":900},[75,76],[31,2638,220],{"className":2639},[289],[31,2641],{"className":2642,"style":293},[119],[31,2644,223],{"className":2645},[297],[31,2647],{"className":2648,"style":293},[119],[31,2650,2652,2656,2663,2667,2670,2673],{"className":2651},[66],[31,2653],{"className":2654,"style":2655},[70],"height:1.8em;vertical-align:-0.65em;",[31,2657,2659],{"className":2658},[264],[31,2660,2512],{"className":2661},[1307,2662],"size2",[31,2664,2515],{"className":2665,"style":2666},[75,76],"margin-right:0.0359em;",[31,2668],{"className":2669,"style":293},[119],[31,2671,2518],{"className":2672},[297],[31,2674],{"className":2675,"style":293},[119],[31,2677,2679,2682,2685,2688,2691,2694,2697],{"className":2678},[66],[31,2680],{"className":2681,"style":260},[70],[31,2683,2521],{"className":2684,"style":989},[75,968],[31,2686,211],{"className":2687},[264],[31,2689,926],{"className":2690,"style":969},[75,968],[31,2692],{"className":2693,"style":120},[119],[31,2695,2528],{"className":2696},[124],[31,2698],{"className":2699,"style":120},[119],[31,2701,2703,2706,2709,2712,2715,2718,2721,2724],{"className":2702},[66],[31,2704],{"className":2705,"style":260},[70],[31,2707,932],{"className":2708,"style":989},[75,968],[31,2710,220],{"className":2711},[289],[31,2713,1545],{"className":2714},[119],[31,2716],{"className":2717,"style":293},[119],[31,2719,922],{"className":2720},[297],[31,2722,1545],{"className":2723},[119],[31,2725],{"className":2726,"style":293},[119],[31,2728,2730,2733,2736,2787,2828,2831,2834,2837],{"className":2729},[66],[31,2731],{"className":2732,"style":1223},[70],[31,2734,211],{"className":2735},[264],[31,2737,2739,2747],{"className":2738},[75],[31,2740,2742],{"className":2741},[75],[31,2743,2746],{"className":2744},[75,2745],"mathrm","proj",[31,2748,2750],{"className":2749},[1039],[31,2751,2753,2778],{"className":2752},[1043,1044],[31,2754,2756,2775],{"className":2755},[1048],[31,2757,2760],{"className":2758,"style":2759},[1052],"height:0.2342em;",[31,2761,2763,2766],{"style":2762},"top:-2.4559em;margin-right:0.05em;",[31,2764],{"className":2765,"style":1061},[1060],[31,2767,2769],{"className":2768},[1065,1066,1067,1068],[31,2770,2772],{"className":2771},[75,1068],[31,2773,926],{"className":2774,"style":969},[75,968,1068],[31,2776,1076],{"className":2777},[1075],[31,2779,2781],{"className":2780},[1048],[31,2782,2785],{"className":2783,"style":2784},[1052],"height:0.2441em;",[31,2786],{},[31,2788,2790,2793],{"className":2789},[289],[31,2791,220],{"className":2792},[289],[31,2794,2796],{"className":2795},[1039],[31,2797,2799,2820],{"className":2798},[1043,1044],[31,2800,2802,2817],{"className":2801},[1048],[31,2803,2805],{"className":2804,"style":1053},[1052],[31,2806,2808,2811],{"style":2807},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[31,2809],{"className":2810,"style":1061},[1060],[31,2812,2814],{"className":2813},[1065,1066,1067,1068],[31,2815,1010],{"className":2816},[75,1068],[31,2818,1076],{"className":2819},[1075],[31,2821,2823],{"className":2822},[1048],[31,2824,2826],{"className":2825,"style":1083},[1052],[31,2827],{},[31,2829,2515],{"className":2830,"style":2666},[75,76],[31,2832],{"className":2833,"style":293},[119],[31,2835,223],{"className":2836},[297],[31,2838],{"className":2839,"style":293},[119],[31,2841,2843,2847,2850,2853,2856,2859,2862,2909,2949,2952,2955,2958],{"className":2842},[66],[31,2844],{"className":2845,"style":2846},[70],"height:1.0622em;vertical-align:-0.3122em;",[31,2848,852],{"className":2849},[75,76],[31,2851,1210],{"className":2852},[1391],[31,2854,1545],{"className":2855},[119],[31,2857],{"className":2858,"style":1467},[119],[31,2860,211],{"className":2861},[264],[31,2863,2865,2871],{"className":2864},[75],[31,2866,2868],{"className":2867},[75],[31,2869,2746],{"className":2870},[75,2745],[31,2872,2874],{"className":2873},[1039],[31,2875,2877,2900],{"className":2876},[1043,1044],[31,2878,2880,2897],{"className":2879},[1048],[31,2881,2883],{"className":2882,"style":2759},[1052],[31,2884,2885,2888],{"style":2762},[31,2886],{"className":2887,"style":1061},[1060],[31,2889,2891],{"className":2890},[1065,1066,1067,1068],[31,2892,2894],{"className":2893},[75,1068],[31,2895,932],{"className":2896,"style":989},[75,968,1068],[31,2898,1076],{"className":2899},[1075],[31,2901,2903],{"className":2902},[1048],[31,2904,2907],{"className":2905,"style":2906},[1052],"height:0.3122em;",[31,2908],{},[31,2910,2912,2915],{"className":2911},[289],[31,2913,220],{"className":2914},[289],[31,2916,2918],{"cla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读作“从 ",[31,3041,3043,3056],{"className":3042,"translate":35},[34],[31,3044,3046],{"className":3045},[39],[41,3047,3048],{"xmlns":43},[45,3049,3050,3054],{},[48,3051,3052],{},[51,3053,53],{},[55,3055,53],{"encoding":57},[31,3057,3059],{"className":3058,"ariaHidden":62},[61],[31,3060,3062,3065],{"className":3061},[66],[31,3063],{"className":3064,"style":71},[70],[31,3066,53],{"className":3067},[75,76]," 搬到 ",[31,3070,3072,3085],{"className":3071,"translate":35},[34],[31,3073,3075],{"className":3074},[39],[41,3076,3077],{"xmlns":43},[45,3078,3079,3083],{},[48,3080,3081],{},[51,3082,3003],{},[55,3084,3003],{"encoding":57},[31,3086,3088],{"className":3087,"ariaHidden":62},[61],[31,3089,3091,3094],{"className":3090},[66],[31,3092],{"className":3093,"style":865},[70],[31,3095,3003],{"className":3096,"style":2666},[75,76]," 的那部分质量”，于是",[31,3099,3101],{"className":3100,"translate":35},[411],[31,3102,3104,3203],{"className":3103,"translate":35},[34],[31,3105,3107],{"className":3106},[39],[41,3108,3109],{"xmlns":43,"display":420},[45,3110,3111,3200],{},[48,3112,3113,3120,3122,3124,3126,3128,3130,3132,3158,3170,3172,3174,3176,3178,3180,3182,3184,3186,3188,3190,3192,3194,3196,3198],{},[1004,3114,3115,3118],{},[51,3116,3117],{},"W",[51,3119,1412],{},[95,3121,211],{"stretchy":210},[51,3123,852],{},[95,3125,1210],{"separator":62},[51,3127,884],{},[95,3129,220],{"stretchy":210},[95,3131,223],{},[1531,3133,3134,3140],{},[48,3135,3136,3138],{},[51,3137,1537],{},[95,3139,1540],{},[48,3141,3142,3144,3146,3148,3150,3152,3154,3156],{},[51,3143,2515],{},[95,3145,2518],{},[51,3147,2496],{"mathvariant":494},[95,3149,211],{"stretchy":210},[51,3151,852],{},[95,3153,1210],{"separator":62},[51,3155,884],{},[95,3157,220],{"stretchy":210},[1004,3159,3160,3162],{},[95,3161,1409],{},[48,3163,3164,3166,3168],{},[51,3165,926],{"mathvariant":925},[95,3167,2528],{},[51,3169,932],{"mathvariant":925},[51,3171,1412],{},[95,3173,211],{"stretchy":210},[51,3175,53],{},[95,3177,1210],{"separator":62},[51,3179,3003],{},[95,3181,220],{"stretchy":210},[1430,3183,1432],{},[51,3185,1435],{},[51,3187,2515],{},[95,3189,211],{"stretchy":210},[51,3191,53],{},[95,3193,1210],{"separator":62},[51,3195,3003],{},[95,3197,220],{"stretchy":210},[51,3199,495],{"mathvariant":494},[55,3201,3202],{"encoding":57},"W_c(\\mu,\\nu)=\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int_{\\mathcal X\\times\\mathcal 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就是那张运输表。",[11,3500,3501,3502,3641,3642,3670],{},"取 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是度量），则",[31,3672,3674],{"className":3673,"translate":35},[411],[31,3675,3677,3774],{"className":3676,"translate":35},[34],[31,3678,3680],{"className":3679},[39],[41,3681,3682],{"xmlns":43,"display":420},[45,3683,3684,3771],{},[48,3685,3686,3692,3694,3696,3698,3700,3702,3704],{},[1004,3687,3688,3690],{},[51,3689,3117],{},[51,3691,11],{},[95,3693,211],{"stretchy":210},[51,3695,852],{},[95,3697,1210],{"separator":62},[51,3699,884],{},[95,3701,220],{"stretchy":210},[95,3703,223],{},[1180,3705,3706,3762],{},[48,3707,3708,3710,3736,3738,3740,3742,3744,3746,3748,3754,3756,3758,3760],{},[95,3709,211],{"fence":62},[1531,3711,3712,3718],{},[48,3713,3714,3716],{},[51,3715,1537],{},[95,3717,1540],{},[48,3719,3720,3722,3724,3726,3728,3730,3732,3734],{},[51,3721,2515],{},[95,3723,2518],{},[51,3725,2496],{"mathvariant":494},[95,3727,211],{"stretchy":210},[51,3729,852],{},[95,3731,1210],{"separator":62},[51,3733,884],{},[95,3735,220],{"stretchy":210},[95,3737,1409],{},[51,3739,1435],{},[95,3741,211],{"stretchy":210},[51,3743,53],{},[95,3745,1210],{"separator":62},[51,3747,3003],{},[1180,3749,3750,3752],{},[95,3751,220],{"stretchy":210},[51,3753,11],{},[1430,3755,1432],{},[51,3757,1435],{},[51,3759,2515],{},[95,3761,220],{"fence":62},[48,3763,3764,3766,3769],{},[91,3765,205],{},[51,3767,3768],{"mathvariant":494},"\u002F",[51,3770,11],{},[55,3772,3773],{"encoding":57},"W_p(\\mu,\\nu)=\\left(\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int 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上的一个真度量。这一点比看起来重要，因为 KL 不是。取 ",[31,4137,4139,4175],{"className":4138,"translate":35},[34],[31,4140,4142],{"className":4141},[39],[41,4143,4144],{"xmlns":43},[45,4145,4146,4172],{},[48,4147,4148,4150,4152,4158,4160,4162,4164,4166],{},[51,4149,852],{},[95,4151,223],{},[1004,4153,4154,4156],{},[51,4155,1974],{},[51,4157,53],{},[95,4159,1210],{"separator":62},[1430,4161,1545],{},[51,4163,884],{},[95,4165,223],{},[1004,4167,4168,4170],{},[51,4169,1974],{},[51,4171,3003],{},[55,4173,4174],{"encoding":57},"\\mu=\\delta_x,\\ 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是个有限的数，而 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要么发散要么给出零梯度，",[31,4719,4721,4734],{"className":4720,"translate":35},[34],[31,4722,4724],{"className":4723},[39],[41,4725,4726],{"xmlns":43},[45,4727,4728,4732],{},[48,4729,4730],{},[51,4731,3117],{},[55,4733,3117],{"encoding":57},[31,4735,4737],{"className":4736,"ariaHidden":62},[61],[31,4738,4740,4743],{"className":4739},[66],[31,4741],{"className":4742,"style":945},[70],[31,4744,3117],{"className":4745,"style":949},[75,76]," 却仍然告诉你“隔了多远”。",[11,4748,4749,4750,4778,4779,4809,4810,4880,4881,4909],{},"平移的例子更直白。把 ",[31,4751,4753,4766],{"className":4752,"translate":35},[34],[31,4754,4756],{"className":4755},[39],[41,4757,4758],{"xmlns":43},[45,4759,4760,4764],{},[48,4761,4762],{},[51,4763,852],{},[55,4765,855],{"encoding":57},[31,4767,4769],{"className":4768,"ariaHidden":62},[61],[31,4770,4772,4775],{"className":4771},[66],[31,4773],{"className":4774,"style":865},[70],[31,4776,852],{"className":4777},[75,76]," 整体挪 ",[31,4780,4782,4797],{"className":4781,"translate":35},[34],[31,4783,4785],{"className":4784},[39],[41,4786,4787],{"xmlns":43},[45,4788,4789,4794],{},[48,4790,4791],{},[51,4792,4793],{},"ε",[55,4795,4796],{"encoding":57},"\\varepsilon",[31,4798,4800],{"className":4799,"ariaHidden":62},[61],[31,4801,4803,4806],{"className":4802},[66],[31,4804],{"className":4805,"style":71},[70],[31,4807,4793],{"className":4808},[75,76],"：",[31,4811,4813,4831],{"className":4812,"translate":35},[34],[31,4814,4816],{"className":4815},[39],[41,4817,4818],{"xmlns":43},[45,4819,4820,4828],{},[48,4821,4822],{},[1004,4823,4824,4826],{},[51,4825,3117],{},[51,4827,11],{},[55,4829,4830],{"encoding":57},"W_p",[31,4832,4834],{"className":4833,"ariaHidden":62},[61],[31,4835,4837,4840],{"className":4836},[66],[31,4838],{"className":4839,"style":1029},[70],[31,4841,4843,4846],{"className":4842},[75],[31,4844,3117],{"className":4845,"style":949},[75,76],[31,4847,4849],{"className":4848},[1039],[31,4850,4852,4872],{"className":4851},[1043,1044],[31,4853,4855,4869],{"className":4854},[1048],[31,4856,4858],{"className":4857,"style":3230},[1052],[31,4859,4860,4863],{"style":1056},[31,4861],{"className":4862,"style":1061},[1060],[31,4864,4866],{"className":4865},[1065,1066,1067,1068],[31,4867,11],{"className":4868},[75,76,1068],[31,4870,1076],{"className":4871},[1075],[31,4873,4875],{"className":4874},[1048],[31,4876,4878],{"className":4877,"style":1083},[1052],[31,4879],{}," 是 ",[31,4882,4884,4897],{"className":4883,"translate":35},[34],[31,4885,4887],{"className":4886},[39],[41,4888,4889],{"xmlns":43},[45,4890,4891,4895],{},[48,4892,4893],{},[51,4894,4793],{},[55,4896,4796],{"encoding":57},[31,4898,4900],{"className":4899,"ariaHidden":62},[61],[31,4901,4903,4906],{"className":4902},[66],[31,4904],{"className":4905,"style":71},[70],[31,4907,4793],{"className":4908},[75,76]," 量级，而",[31,4911,4913],{"className":4912,"translate":35},[411],[31,4914,4916,4979],{"className":4915,"translate":35},[34],[31,4917,4919],{"className":4918},[39],[41,4920,4921],{"xmlns":43,"display":420},[45,4922,4923,4976],{},[48,4924,4925,4931,4933,4936,4938,4940,4942,4944,4946,4948,4950,4952,4954,4956,4958,4960,4962,4964,4974],{},[48,4926,4927,4929],{},[51,4928,4558],{"mathvariant":494},[51,4930,4561],{"mathvariant":494},[95,4932,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,4934,4935],{"mathvariant":925},"N",[95,4937,211],{"stretchy":210},[91,4939,1977],{},[95,4941,1210],{"separator":62},[91,4943,205],{},[95,4945,220],{"stretchy":210},[51,4947,4572],{"mathvariant":494},[51,4949,4935],{"mathvariant":925},[95,4951,211],{"stretchy":210},[51,4953,4793],{},[95,4955,1210],{"separator":62},[91,4957,205],{},[95,4959,220],{"stretchy":210},[95,4961,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,4963,223],{},[2074,4965,4966,4972],{},[1180,4967,4968,4970],{},[51,4969,4793],{},[91,4971,152],{},[91,4973,152],{},[95,4975,1210],{"separator":62},[55,4977,4978],{"encoding":57},"\\mathrm{KL}\\bigl(\\mathcal N(0,1)\\|\\mathcal N(\\varepsilon,1)\\bigr)=\\frac{\\varepsilon^2}{2},",[31,4980,4982,5062],{"className":4981,"ariaHidden":62},[61],[31,4983,4985,4989,4995,5001,5005,5008,5011,5014,5017,5020,5023,5026,5029,5032,5035,5038,5041,5044,5047,5053,5056,5059],{"className":4984},[66],[31,4986],{"className":4987,"style":4988},[70],"height:1.2em;vertical-align:-0.35em;",[31,4990,4992],{"className":4991},[75],[31,4993,4606],{"className":4994},[75,2745],[31,4996,4998],{"className":4997},[264],[31,4999,211],{"className":5000},[1307,1308],[31,5002,4935],{"className":5003,"style":5004},[75,968],"margin-right:0.1474em;",[31,5006,211],{"className":5007},[264],[31,5009,1977],{"className":5010},[75],[31,5012,1210],{"className":5013},[1391],[31,5015],{"className":5016,"style":1467},[119],[31,5018,205],{"className":5019},[75],[31,5021,220],{"className":5022},[289],[31,5024,4572],{"className":5025},[75],[31,5027,4935],{"className":5028,"style":5004},[75,968],[31,5030,211],{"className":5031},[264],[31,5033,4793],{"className":5034},[75,76],[31,5036,1210],{"className":5037},[1391],[31,5039],{"className":5040,"style":1467},[119],[31,5042,205],{"className":5043},[75],[31,5045,220],{"className":5046},[289],[31,5048,5050],{"className":5049},[289],[31,5051,220],{"className":5052},[1307,1308],[31,5054],{"className":5055,"style":293},[119],[31,5057,223],{"className":5058},[297],[31,5060],{"className":5061,"style":293},[119],[31,5063,5065,5069,5162],{"className":5064},[66],[31,5066],{"className":5067,"style":5068},[70],"height:2.1771em;vertical-align:-0.686em;",[31,5070,5072,5075,5159],{"className":5071},[75],[31,5073],{"className":5074},[264,2139],[31,5076,5078],{"className":5077},[2074],[31,5079,5081,5150],{"className":5080},[1043,1044],[31,5082,5084,5147],{"className":5083},[1048],[31,5085,5088,5100,5108],{"className":5086,"style":5087},[1052],"height:1.4911em;",[31,5089,5091,5094],{"style":5090},"top:-2.314em;",[31,5092],{"className":5093,"style":1637},[1060],[31,5095,5097],{"className":5096},[75],[31,5098,152],{"className":5099},[75],[31,5101,5102,5105],{"style":2170},[31,5103],{"className":5104,"style":1637},[1060],[31,5106],{"className":5107,"style":2178},[2177],[31,5109,5111,5114],{"style":5110},"top:-3.677em;",[31,5112],{"className":5113,"style":1637},[1060],[31,5115,5117],{"className":5116},[75],[31,5118,5120,5123],{"className":5119},[75],[31,5121,4793],{"className":5122},[75,76],[31,5124,5126],{"className":5125},[1039],[31,5127,5129],{"className":5128},[1043],[31,5130,5132],{"className":5131},[1048],[31,5133,5136],{"className":5134,"style":5135},[1052],"height:0.8141em;",[31,5137,5138,5141],{"style":3631},[31,5139],{"className":5140,"style":1061},[1060],[31,5142,5144],{"className":5143},[1065,1066,1067,1068],[31,5145,152],{"className":5146},[75,1068],[31,5148,1076],{"className":5149},[1075],[31,5151,5153],{"className":5152},[1048],[31,5154,5157],{"className":5155,"style":5156},[1052],"height:0.686em;",[31,5158],{},[31,5160],{"className":5161},[289,2139],[31,5163,1210],{"className":5164},[1391],[11,5166,5167,5168,5196],{},"是二阶的。所以小位移时 KL 的梯度趋近于零，",[31,5169,5171,5184],{"className":5170,"translate":35},[34],[31,5172,5174],{"className":5173},[39],[41,5175,5176],{"xmlns":43},[45,5177,5178,5182],{},[48,5179,5180],{},[51,5181,3117],{},[55,5183,3117],{"encoding":57},[31,5185,5187],{"className":5186,"ariaHidden":62},[61],[31,5188,5190,5193],{"className":5189},[66],[31,5191],{"className":5192,"style":945},[70],[31,5194,3117],{"className":5195,"style":949},[75,76]," 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。",[11,5198,5199,5200,5271,5272,5300],{},"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 ",[31,5201,5203,5221],{"className":5202,"translate":35},[34],[31,5204,5206],{"className":5205},[39],[41,5207,5208],{"xmlns":43},[45,5209,5210,5218],{},[48,5211,5212],{},[1004,5213,5214,5216],{},[51,5215,3117],{},[91,5217,205],{},[55,5219,5220],{"encoding":57},"W_1",[31,5222,5224],{"className":5223,"ariaHidden":62},[61],[31,5225,5227,5231],{"className":5226},[66],[31,5228],{"className":5229,"style":5230},[70],"height:0.8333em;vertical-align:-0.15em;",[31,5232,5234,5237],{"className":5233},[75],[31,5235,3117],{"className":5236,"style":949},[75,76],[31,5238,5240],{"className":5239},[1039],[31,5241,5243,5263],{"className":5242},[1043,1044],[31,5244,5246,5260],{"className":5245},[1048],[31,5247,5249],{"className":5248,"style":2028},[1052],[31,5250,5251,5254],{"style":1056},[31,5252],{"className":5253,"style":1061},[1060],[31,5255,5257],{"className":5256},[1065,1066,1067,1068],[31,5258,205],{"className":5259},[75,1068],[31,5261,1076],{"className":5262},[1075],[31,5264,5266],{"className":5265},[1048],[31,5267,5269],{"className":5268,"style":2050},[1052],[31,5270],{},"，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，",[31,5273,5275,5288],{"className":5274,"translate":35},[34],[31,5276,5278],{"className":5277},[39],[41,5279,5280],{"xmlns":43},[45,5281,5282,5286],{},[48,5283,5284],{},[51,5285,3117],{},[55,5287,3117],{"encoding":57},[31,5289,5291],{"className":5290,"ariaHidden":62},[61],[31,5292,5294,5297],{"className":5293},[66],[31,5295],{"className":5296,"style":945},[70],[31,5298,3117],{"className":5299,"style":949},[75,76]," 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。",[18,5302,5303],{"id":5303},"对偶",[11,5305,5306],{},"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？",[11,5308,5309,5310,5339,5340,5369],{},"不去枚举所有运输表。我给每个矿定一个“提货加价” ",[31,5311,5313,5327],{"className":5312,"translate":35},[34],[31,5314,5316],{"className":5315},[39],[41,5317,5318],{"xmlns":43},[45,5319,5320,5325],{},[48,5321,5322],{},[51,5323,5324],{},"u",[55,5326,5324],{"encoding":57},[31,5328,5330],{"className":5329,"ariaHidden":62},[61],[31,5331,5333,5336],{"className":5332},[66],[31,5334],{"className":5335,"style":71},[70],[31,5337,5324],{"className":5338},[75,76],"，每个厂定一个“到货减价” ",[31,5341,5343,5357],{"className":5342,"translate":35},[34],[31,5344,5346],{"className":5345},[39],[41,5347,5348],{"xmlns":43},[45,5349,5350,5355],{},[48,5351,5352],{},[51,5353,5354],{},"v",[55,5356,5354],{"encoding":57},[31,5358,5360],{"className":5359,"ariaHidden":62},[61],[31,5361,5363,5366],{"className":5362},[66],[31,5364],{"className":5365,"style":71},[70],[31,5367,5354],{"className":5368,"style":2666},[75,76],"，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，",[31,5371,5373],{"className":5372,"translate":35},[411],[31,5374,5376,5417],{"className":5375,"translate":35},[34],[31,5377,5379],{"className":5378},[39],[41,5380,5381],{"xmlns":43,"display":420},[45,5382,5383,5414],{},[48,5384,5385,5392,5394,5400,5402,5412],{},[1004,5386,5387,5389],{},[51,5388,5324],{},[51,5390,5391],{},"i",[95,5393,434],{},[1004,5395,5396,5398],{},[51,5397,5354],{},[51,5399,2555],{},[95,5401,347],{},[1004,5403,5404,5406],{},[51,5405,1412],{},[48,5407,5408,5410],{},[51,5409,5391],{},[51,5411,2555],{},[51,5413,495],{"mathvariant":494},[55,5415,5416],{"encoding":57},"u_i+v_j\\le c_{ij}.",[31,5418,5420,5477,5535],{"className":5419,"ariaHidden":62},[61],[31,5421,5423,5427,5468,5471,5474],{"className":5422},[66],[31,5424],{"className":5425,"style":5426},[70],"height:0.7333em;vertical-align:-0.15em;",[31,5428,5430,5433],{"className":5429},[75],[31,5431,5324],{"className":5432},[75,76],[31,5434,5436],{"className":5435},[1039],[31,5437,5439,5460],{"className":5438},[1043,1044],[31,5440,5442,5457],{"className":5441},[1048],[31,5443,5446],{"className":5444,"style":5445},[1052],"height:0.3117em;",[31,5447,5448,5451],{"style":2807},[31,5449],{"className":5450,"style":1061},[1060],[31,5452,5454],{"className":5453},[1065,1066,1067,1068],[31,5455,5391],{"className":5456},[75,76,1068],[31,5458,1076],{"className":5459},[1075],[31,5461,5463],{"className":5462},[1048],[31,5464,5466],{"className":5465,"style":2050},[1052],[31,5467],{},[31,5469],{"className":5470,"style":120},[119],[31,5472,434],{"className":5473},[124],[31,5475],{"className":5476,"style":120},[119],[31,5478,5480,5484,5526,5529,5532],{"className":5479},[66],[31,5481],{"className":5482,"style":5483},[70],"height:0.9221em;vertical-align:-0.2861em;",[31,5485,5487,5490],{"className":5486},[75],[31,5488,5354],{"className":5489,"style":2666},[75,76],[31,5491,5493],{"className":5492},[1039],[31,5494,5496,5518],{"className":5495},[1043,1044],[31,5497,5499,5515],{"className":5498},[1048],[31,5500,5502],{"className":5501,"style":5445},[1052],[31,5503,5505,5508],{"style":5504},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[31,5506],{"className":5507,"style":1061},[1060],[31,5509,5511],{"className":5510},[1065,1066,1067,1068],[31,5512,2555],{"className":5513,"style":5514},[75,76,1068],"margin-right:0.0572em;",[31,5516,1076],{"className":5517},[1075],[31,5519,5521],{"className":5520},[1048],[31,5522,5524],{"className":5523,"style":1083},[1052],[31,5525],{},[31,5527],{"className":5528,"style":293},[119],[31,5530,347],{"className":5531},[297],[31,5533],{"className":5534,"style":293},[119],[31,5536,5538,5542,5586],{"className":5537},[66],[31,5539],{"className":5540,"style":5541},[70],"height:0.7167em;vertical-align:-0.2861em;",[31,5543,5545,5548],{"className":5544},[75],[31,5546,1412],{"className":5547},[75,76],[31,5549,5551],{"className":5550},[1039],[31,5552,5554,5578],{"className":5553},[1043,1044],[31,5555,5557,5575],{"className":5556},[1048],[31,5558,5560],{"className":5559,"style":5445},[1052],[31,5561,5562,5565],{"style":2807},[31,5563],{"className":5564,"style":1061},[1060],[31,5566,5568],{"className":5567},[1065,1066,1067,1068],[31,5569,5571],{"className":5570},[75,1068],[31,5572,5574],{"className":5573,"style":5514},[75,76,1068],"ij",[31,5576,1076],{"className":5577},[1075],[31,5579,5581],{"className":5580},[1048],[31,5582,5584],{"className":5583,"style":1083},[1052],[31,5585],{},[31,5587,495],{"className":5588},[75],[11,5590,5591],{},"那么任何一张运单的总价都满足",[31,5593,5595],{"className":5594,"translate":35},[411],[31,5596,5598,5848],{"className":5597,"translate":35},[34],[31,5599,5601],{"className":5600},[39],[41,5602,5603],{"xmlns":43,"display":420},[45,5604,5605,5845],{},[5606,5607,5611,5701,5785],"mtable",{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},"0.25em","right 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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 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",[31,6989,6991,7009],{"className":6990,"translate":35},[34],[31,6992,6994],{"className":6993},[39],[41,6995,6996],{"xmlns":43},[45,6997,6998,7006],{},[48,6999,7000,7002,7004],{},[51,7001,5324],{},[95,7003,1210],{"separator":62},[51,7005,5354],{},[55,7007,7008],{"encoding":57},"u,v",[31,7010,7012],{"className":7011,"ariaHidden":62},[61],[31,7013,7015,7018,7021,7024,7027],{"className":7014},[66],[31,7016],{"className":7017,"style":865},[70],[31,7019,5324],{"className":7020},[75,76],[31,7022,1210],{"className":7023},[1391],[31,7025],{"className":7026,"style":1467},[119],[31,7028,5354],{"className":7029,"style":2666},[75,76]," 有关，跟运单无关；而中间那步把 ",[31,7032,7034,7047],{"className":7033,"translate":35},[34],[31,7035,7037],{"className":7036},[39],[41,7038,7039],{"xmlns":43},[45,7040,7041,7045],{},[48,7042,7043],{},[51,7044,2515],{},[55,7046,3485],{"encoding":57},[31,7048,7050],{"className":7049,"ariaHidden":62},[61],[31,7051,7053,7056],{"className":7052},[66],[31,7054],{"className":7055,"style":71},[70],[31,7057,2515],{"className":7058,"style":2666},[75,76]," 的边缘和换成了产量 ",[31,7061,7063,7081],{"className":7062,"translate":35},[34],[31,7064,7066],{"className":7065},[39],[41,7067,7068],{"xmlns":43},[45,7069,7070,7078],{},[48,7071,7072],{},[1004,7073,7074,7076],{},[51,7075,5813],{},[51,7077,5391],{},[55,7079,7080],{"encoding":57},"a_i",[31,7082,7084],{"className":7083,"ariaHidden":62},[61],[31,7085,7087,7091],{"className":7086},[66],[31,7088],{"className":7089,"style":7090},[70],"height:0.5806em;vertical-align:-0.15em;",[31,7092,7094,7097],{"className":7093},[75],[31,7095,5813],{"className":7096},[75,76],[31,7098,7100],{"className":7099},[1039],[31,7101,7103,7123],{"className":7102},[1043,1044],[31,7104,7106,7120],{"className":7105},[1048],[31,7107,7109],{"className":7108,"style":5445},[1052],[31,7110,7111,7114],{"style":2807},[31,7112],{"className":7113,"style":1061},[1060],[31,7115,7117],{"className":7116},[1065,1066,1067,1068],[31,7118,5391],{"className":7119},[75,76,1068],[31,7121,1076],{"className":7122},[1075],[31,7124,7126],{"className":7125},[1048],[31,7127,7129],{"className":7128,"style":2050},[1052],[31,7130],{},"、需求 ",[31,7133,7135,7153],{"className":7134,"translate":35},[34],[31,7136,7138],{"className":7137},[39],[41,7139,7140],{"xmlns":43},[45,7141,7142,7150],{},[48,7143,7144],{},[1004,7145,7146,7148],{},[51,7147,5834],{},[51,7149,2555],{},[55,7151,7152],{"encoding":57},"b_j",[31,7154,7156],{"className":7155,"ariaHidden":62},[61],[31,7157,7159,7162],{"className":7158},[66],[31,7160],{"className":7161,"style":4270},[70],[31,7163,7165,7168],{"className":7164},[75],[31,7166,5834],{"className":7167},[75,76],[31,7169,7171],{"className":7170},[1039],[31,7172,7174,7194],{"className":7173},[1043,1044],[31,7175,7177,7191],{"className":7176},[1048],[31,7178,7180],{"className":7179,"style":5445},[1052],[31,7181,7182,7185],{"style":2807},[31,7183],{"className":7184,"style":1061},[1060],[31,7186,7188],{"className":7187},[1065,1066,1067,1068],[31,7189,2555],{"className":7190,"style":5514},[75,76,1068],[31,7192,1076],{"className":7193},[1075],[31,7195,7197],{"className":7196},[1048],[31,7198,7200],{"className":7199,"style":1083},[1052],[31,7201],{},"。所以只要我找到一组 ",[31,7204,7206,7223],{"className":7205,"translate":35},[34],[31,7207,7209],{"className":7208},[39],[41,7210,7211],{"xmlns":43},[45,7212,7213,7221],{},[48,7214,7215,7217,7219],{},[51,7216,5324],{},[95,7218,1210],{"separator":62},[51,7220,5354],{},[55,7222,7008],{"encoding":57},[31,7224,7226],{"className":7225,"ariaHidden":62},[61],[31,7227,7229,7232,7235,7238,7241],{"className":7228},[66],[31,7230],{"className":7231,"style":865},[70],[31,7233,5324],{"className":7234},[75,76],[31,7236,1210],{"className":7237},[1391],[31,7239],{"className":7240,"style":1467},[119],[31,7242,5354],{"className":7243,"style":2666},[75,76],"，使这个和等于 5，我就",[7246,7247,7248],"strong",{},"证明","了运费不可能低于 5 块。",[11,7251,7252,7253,1138,7332,7406,7407,7502,7503,7598,7599,7699,7700,7800,7801,8022],{},"试一试：取 ",[31,7254,7256,7284],{"className":7255,"translate":35},[34],[31,7257,7259],{"className":7258},[39],[41,7260,7261],{"xmlns":43},[45,7262,7263,7281],{},[48,7264,7265,7267,7269,7271,7273,7275,7277,7279],{},[51,7266,5324],{},[95,7268,223],{},[95,7270,211],{"stretchy":210},[91,7272,1977],{},[95,7274,1210],{"separator":62},[95,7276,97],{},[91,7278,205],{},[95,7280,220],{"stretchy":210},[55,7282,7283],{"encoding":57},"u=(0,-1)",[31,7285,7287,7305],{"className":7286,"ariaHidden":62},[61],[31,7288,7290,7293,7296,7299,7302],{"className":7289},[66],[31,7291],{"className":7292,"style":71},[70],[31,7294,5324],{"className":7295},[75,76],[31,7297],{"className":7298,"style":293},[119],[31,7300,223],{"className":7301},[297],[31,7303],{"className":7304,"style":293},[119],[31,7306,7308,7311,7314,7317,7320,7323,7326,7329],{"className":7307},[66],[31,7309],{"className":7310,"style":260},[70],[31,7312,211],{"className":7313},[264],[31,7315,1977],{"className":7316},[75],[31,7318,1210],{"className":7319},[1391],[31,7321],{"className":7322,"style":1467},[119],[31,7324,97],{"className":7325},[75],[31,7327,205],{"className":7328},[75],[31,7330,220],{"className":7331},[289],[31,7333,7335,7361],{"className":7334,"translate":35},[34],[31,7336,7338],{"className":7337},[39],[41,7339,7340],{"xmlns":43},[45,7341,7342,7358],{},[48,7343,7344,7346,7348,7350,7352,7354,7356],{},[51,7345,5354],{},[95,7347,223],{},[95,7349,211],{"stretchy":210},[91,7351,205],{},[95,7353,1210],{"separator":62},[91,7355,152],{},[95,7357,220],{"stretchy":210},[55,7359,7360],{"encoding":57},"v=(1,2)",[31,7362,7364,7382],{"className":7363,"ariaHidden":62},[61],[31,7365,7367,7370,7373,7376,7379],{"className":7366},[66],[31,7368],{"className":7369,"style":71},[70],[31,7371,5354],{"className":7372,"style":2666},[75,76],[31,7374],{"className":7375,"style":293},[119],[31,7377,223],{"className":7378},[297],[31,7380],{"className":7381,"style":293},[119],[31,7383,7385,7388,7391,7394,7397,7400,7403],{"className":7384},[66],[31,7386],{"className":7387,"style":260},[70],[31,7389,211],{"className":7390},[264],[31,7392,205],{"className":7393},[75],[31,7395,1210],{"className":7396},[1391],[31,7398],{"className":7399,"style":1467},[119],[31,7401,152],{"className":7402},[75],[31,7404,220],{"className":7405},[289],"。四条路线逐一检查：甲到 A 是 ",[31,7408,7410,7436],{"className":7409,"translate":35},[34],[31,7411,7413],{"className":7412},[39],[41,7414,7415],{"xmlns":43},[45,7416,7417,7433],{},[48,7418,7419,7421,7423,7425,7427,7429,7431],{},[91,7420,1977],{},[95,7422,434],{},[91,7424,205],{},[95,7426,223],{},[91,7428,205],{},[95,7430,347],{},[91,7432,205],{},[55,7434,7435],{"encoding":57},"0+1=1\\le 1",[31,7437,7439,7457,7475,7493],{"className":7438,"ariaHidden":62},[61],[31,7440,7442,7445,7448,7451,7454],{"className":7441},[66],[31,7443],{"className":7444,"style":112},[70],[31,7446,1977],{"className":7447},[75],[31,7449],{"className":7450,"style":120},[119],[31,7452,434],{"className":7453},[124],[31,7455],{"className":7456,"style":120},[119],[31,7458,7460,7463,7466,7469,7472],{"className":7459},[66],[31,7461],{"className":7462,"style":326},[70],[31,7464,205],{"className":7465},[75],[31,7467],{"className":7468,"style":293},[119],[31,7470,223],{"className":7471},[297],[31,7473],{"className":7474,"style":293},[119],[31,7476,7478,7481,7484,7487,7490],{"className":7477},[66],[31,7479],{"className":7480,"style":366},[70],[31,7482,205],{"className":7483},[75],[31,7485],{"className":7486,"style":293},[119],[31,7488,347],{"className":7489},[297],[31,7491],{"className":7492,"style":293},[119],[31,7494,7496,7499],{"className":7495},[66],[31,7497],{"className":7498,"style":326},[70],[31,7500,205],{"className":7501},[75],"，甲到 B 是 ",[31,7504,7506,7532],{"className":7505,"translate":35},[34],[31,7507,7509],{"className":7508},[39],[41,7510,7511],{"xmlns":43},[45,7512,7513,7529],{},[48,7514,7515,7517,7519,7521,7523,7525,7527],{},[91,7516,1977],{},[95,7518,434],{},[91,7520,152],{},[95,7522,223],{},[91,7524,152],{},[95,7526,347],{},[91,7528,152],{},[55,7530,7531],{"encoding":57},"0+2=2\\le 2",[31,7533,7535,7553,7571,7589],{"className":7534,"ariaHidden":62},[61],[31,7536,7538,7541,7544,7547,7550],{"className":7537},[66],[31,7539],{"className":7540,"style":112},[70],[31,7542,1977],{"className":7543},[75],[31,7545],{"className":7546,"style":120},[119],[31,7548,434],{"className":7549},[124],[31,7551],{"className":7552,"style":120},[119],[31,7554,7556,7559,7562,7565,7568],{"className":7555},[66],[31,7557],{"className":7558,"style":326},[70],[31,7560,152],{"className":7561},[75],[31,7563],{"className":7564,"style":293},[119],[31,7566,223],{"className":7567},[297],[31,7569],{"className":7570,"style":293},[119],[31,7572,7574,7577,7580,7583,7586],{"className":7573},[66],[31,7575],{"className":7576,"style":366},[70],[31,7578,152],{"className":7579},[75],[31,7581],{"className":7582,"style":293},[119],[31,7584,347],{"className":7585},[297],[31,7587],{"className":7588,"style":293},[119],[31,7590,7592,7595],{"className":7591},[66],[31,7593],{"className":7594,"style":326},[70],[31,7596,152],{"className":7597},[75],"，乙到 A 是 ",[31,7600,7602,7630],{"className":7601,"translate":35},[34],[31,7603,7605],{"className":7604},[39],[41,7606,7607],{"xmlns":43},[45,7608,7609,7627],{},[48,7610,7611,7613,7615,7617,7619,7621,7623,7625],{},[95,7612,97],{},[91,7614,205],{},[95,7616,434],{},[91,7618,205],{},[95,7620,223],{},[91,7622,1977],{},[95,7624,347],{},[91,7626,93],{},[55,7628,7629],{"encoding":57},"-1+1=0\\le 3",[31,7631,7633,7654,7672,7690],{"className":7632,"ariaHidden":62},[61],[31,7634,7636,7639,7642,7645,7648,7651],{"className":7635},[66],[31,7637],{"className":7638,"style":112},[70],[31,7640,97],{"className":7641},[75],[31,7643,205],{"className":7644},[75],[31,7646],{"className":7647,"style":120},[119],[31,7649,434],{"className":7650},[124],[31,7652],{"className":7653,"style":120},[119],[31,7655,7657,7660,7663,7666,7669],{"className":7656},[66],[31,7658],{"className":7659,"style":326},[70],[31,7661,205],{"className":7662},[75],[31,7664],{"className":7665,"style":293},[119],[31,7667,223],{"className":7668},[297],[31,7670],{"className":7671,"style":293},[119],[31,7673,7675,7678,7681,7684,7687],{"className":7674},[66],[31,7676],{"className":7677,"style":366},[70],[31,7679,1977],{"className":7680},[75],[31,7682],{"className":7683,"style":293},[119],[31,7685,347],{"className":7686},[297],[31,7688],{"className":7689,"style":293},[119],[31,7691,7693,7696],{"className":7692},[66],[31,7694],{"className":7695,"style":326},[70],[31,7697,93],{"className":7698},[75],"，乙到 B 是 ",[31,7701,7703,7731],{"className":7702,"translate":35},[34],[31,7704,7706],{"className":7705},[39],[41,7707,7708],{"xmlns":43},[45,7709,7710,7728],{},[48,7711,7712,7714,7716,7718,7720,7722,7724,7726],{},[95,7713,97],{},[91,7715,205],{},[95,7717,434],{},[91,7719,152],{},[95,7721,223],{},[91,7723,205],{},[95,7725,347],{},[91,7727,205],{},[55,7729,7730],{"encoding":57},"-1+2=1\\le 1",[31,7732,7734,7755,7773,7791],{"className":7733,"ariaHidden":62},[61],[31,7735,7737,7740,7743,7746,7749,7752],{"className":7736},[66],[31,7738],{"className":7739,"style":112},[70],[31,7741,97],{"className":7742},[75],[31,7744,205],{"className":7745},[75],[31,7747],{"className":7748,"style":120},[119],[31,7750,434],{"className":7751},[124],[31,7753],{"className":7754,"style":120},[119],[31,7756,7758,7761,7764,7767,7770],{"className":7757},[66],[31,7759],{"className":7760,"style":326},[70],[31,7762,152],{"className":7763},[75],[31,7765],{"className":7766,"style":293},[119],[31,7768,223],{"className":7769},[297],[31,7771],{"className":7772,"style":293},[119],[31,7774,7776,7779,7782,7785,7788],{"className":7775},[66],[31,7777],{"className":7778,"style":366},[70],[31,7780,205],{"className":7781},[75],[31,7783],{"className":7784,"style":293},[119],[31,7786,347],{"className":7787},[297],[31,7789],{"className":7790,"style":293},[119],[31,7792,7794,7797],{"className":7793},[66],[31,7795],{"className":7796,"style":326},[70],[31,7798,205],{"className":7799},[75],"，全部成立。而 ",[31,7802,7804,7857],{"className":7803,"translate":35},[34],[31,7805,7807],{"className":7806},[39],[41,7808,7809],{"xmlns":43},[45,7810,7811,7854],{},[48,7812,7813,7815,7817,7819,7821,7823,7825,7827,7829,7831,7833,7835,7837,7839,7841,7843,7845,7847,7849,7851],{},[91,7814,93],{},[95,7816,429],{},[91,7818,1977],{},[95,7820,434],{},[91,7822,205],{},[95,7824,429],{},[95,7826,211],{"stretchy":210},[95,7828,97],{},[91,7830,205],{},[95,7832,220],{"stretchy":210},[95,7834,434],{},[91,7836,152],{},[95,7838,429],{},[91,7840,205],{},[95,7842,434],{},[91,7844,152],{},[95,7846,429],{},[91,7848,152],{},[95,7850,223],{},[91,7852,7853],{},"5",[55,7855,7856],{"encoding":57},"3\\cdot0+1\\cdot(-1)+2\\cdot1+2\\cdot2=5",[31,7858,7860,7878,7896,7914,7941,7959,7977,7995,8013],{"className":7859,"ariaHidden":62},[61],[31,7861,7863,7866,7869,7872,7875],{"className":7862},[66],[31,7864],{"className":7865,"style":326},[70],[31,7867,93],{"className":7868},[75],[31,7870],{"className":7871,"style":120},[119],[31,7873,429],{"className":7874},[124],[31,7876],{"className":7877,"style":120},[119],[31,7879,7881,7884,7887,7890,7893],{"className":7880},[66],[31,7882],{"className":7883,"style":112},[70],[31,7885,1977],{"className":7886},[75],[31,7888],{"className":7889,"style":120},[119],[31,7891,434],{"className":7892},[124],[31,7894],{"className":7895,"style":120},[119],[31,7897,7899,7902,7905,7908,7911],{"className":7898},[66],[31,7900],{"className":7901,"style":326},[70],[31,7903,205],{"className":7904},[75],[31,7906],{"className":7907,"style":120},[119],[31,7909,429],{"className":7910},[124],[31,7912],{"className":7913,"style":120},[119],[31,7915,7917,7920,7923,7926,7929,7932,7935,7938],{"className":7916},[66],[31,7918],{"className":7919,"style":260},[70],[31,7921,211],{"className":7922},[264],[31,7924,97],{"className":7925},[75],[31,7927,205],{"className":7928},[75],[31,7930,220],{"className":7931},[289],[31,7933],{"className":7934,"style":120},[119],[31,7936,434],{"className":7937},[124],[31,7939],{"className":7940,"style":120},[119],[31,7942,7944,7947,7950,7953,7956],{"className":7943},[66],[31,7945],{"className":7946,"style":326},[70],[31,7948,152],{"className":7949},[75],[31,7951],{"className":7952,"style":120},[119],[31,7954,429],{"className":7955},[124],[31,7957],{"className":7958,"style":120},[119],[31,7960,7962,7965,7968,7971,7974],{"className":7961},[66],[31,7963],{"className":7964,"style":112},[70],[31,7966,205],{"className":7967},[75],[31,7969],{"className":7970,"style":120},[119],[31,7972,434],{"className":7973},[124],[31,7975],{"className":7976,"style":120},[119],[31,7978,7980,7983,7986,7989,7992],{"className":7979},[66],[31,7981],{"className":7982,"style":326},[70],[31,7984,152],{"className":7985},[75],[31,7987],{"className":7988,"style":120},[119],[31,7990,429],{"className":7991},[124],[31,7993],{"className":7994,"style":120},[119],[31,7996,7998,8001,8004,8007,8010],{"className":7997},[66],[31,7999],{"className":8000,"style":326},[70],[31,8002,152],{"className":8003},[75],[31,8005],{"className":8006,"style":293},[119],[31,8008,223],{"className":8009},[297],[31,8011],{"className":8012,"style":293},[119],[31,8014,8016,8019],{"className":8015},[66],[31,8017],{"className":8018,"style":326},[70],[31,8020,7853],{"className":8021},[75],"。证书找到了。",[11,8024,8025],{},"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：",[31,8027,8029],{"className":8028,"translate":35},[411],[31,8030,8032,8305],{"className":8031,"translate":35},[34],[31,8033,8035],{"className":8034},[39],[41,8036,8037],{"xmlns":43,"display":420},[45,8038,8039,8302],{},[5606,8040,8041,8192],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,8042,8043,8067],{},[5615,8044,8045],{},[2071,8046,8047],{"scriptlevel":1977,"displaystyle":62},[48,8048,8049,8051,8053,8055,8057,8060,8062,8065],{},[51,8050,4561],{},[95,8052,211],{"stretchy":210},[51,8054,2515],{},[95,8056,1210],{"separator":62},[51,8058,8059],{},"φ",[95,8061,1210],{"separator":62},[51,8063,8064],{},"ψ",[95,8066,220],{"stretchy":210},[5615,8068,8069],{},[2071,8070,8071],{"scriptlevel":1977,"displaystyle":62},[48,8072,8073,8075,8077,8087,8097,8108,8110,8116,8122,8124,8130,8132,8138,8148,8150,8152,8158,8164,8166,8172,8174,8180,8190],{},[48,8074],{},[95,8076,223],{},[1531,8078,8079,8081],{},[95,8080,5625],{},[48,8082,8083,8085],{},[51,8084,5391],{},[51,8086,2555],{},[1004,8088,8089,8091],{},[51,8090,2515],{},[48,8092,8093,8095],{},[51,8094,5391],{},[51,8096,2555],{},[1004,8098,8099,8102],{},[51,8100,8101],{},"C",[48,8103,8104,8106],{},[51,8105,5391],{},[51,8107,2555],{},[95,8109,434],{},[1531,8111,8112,8114],{},[95,8113,5625],{},[51,8115,5391],{},[1004,8117,8118,8120],{},[51,8119,8059],{},[51,8121,5391],{},[95,8123,211],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[1004,8125,8126,8128],{},[51,8127,5813],{},[51,8129,5391],{},[95,8131,97],{},[1531,8133,8134,8136],{},[95,8135,5625],{},[51,8137,2555],{},[1004,8139,8140,8142],{},[51,8141,2515],{},[48,8143,8144,8146],{},[51,8145,5391],{},[51,8147,2555],{},[95,8149,220],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[95,8151,434],{},[1531,8153,8154,8156],{},[95,8155,5625],{},[51,8157,2555],{},[1004,8159,8160,8162],{},[51,8161,8064],{},[51,8163,2555],{},[95,8165,211],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[1004,8167,8168,8170],{},[51,8169,5834],{},[51,8171,2555],{},[95,8173,97],{},[1531,8175,8176,8178],{},[95,8177,5625],{},[51,8179,5391],{},[1004,8181,8182,8184],{},[51,8183,2515],{},[48,8185,8186,8188],{},[51,8187,5391],{},[51,8189,2555],{},[95,8191,220],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[5612,8193,8194,8200],{},[5615,8195,8196],{},[2071,8197,8198],{"scriptlevel":1977,"displaystyle":62},[48,8199],{},[5615,8201,8202],{},[2071,8203,8204],{"scriptlevel":1977,"displaystyle":62},[48,8205,8206,8208,8210,8216,8222,8228,8230,8236,8242,8248,8250,8260,8270,8272,8282,8284,8290,8292,8298,8300],{},[48,8207],{},[95,8209,223],{},[1531,8211,8212,8214],{},[95,8213,5625],{},[51,8215,5391],{},[1004,8217,8218,8220],{},[51,8219,5813],{},[51,8221,5391],{},[1004,8223,8224,8226],{},[51,8225,8059],{},[51,8227,5391],{},[95,8229,434],{},[1531,8231,8232,8234],{},[95,8233,5625],{},[51,8235,2555],{},[1004,8237,8238,8240],{},[51,8239,5834],{},[51,8241,2555],{},[1004,8243,8244,8246],{},[51,8245,8064],{},[51,8247,2555],{},[95,8249,434],{},[1531,8251,8252,8254],{},[95,8253,5625],{},[48,8255,8256,8258],{},[51,8257,5391],{},[51,8259,2555],{},[1004,8261,8262,8264],{},[51,8263,2515],{},[48,8265,8266,8268],{},[51,8267,5391],{},[51,8269,2555],{},[95,8271,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,8273,8274,8276],{},[51,8275,8101],{},[48,8277,8278,8280],{},[51,8279,5391],{},[51,8281,2555],{},[95,8283,97],{},[1004,8285,8286,8288],{},[51,8287,8059],{},[51,8289,5391],{},[95,8291,97],{},[1004,8293,8294,8296],{},[51,8295,8064],{},[51,8297,2555],{},[95,8299,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,8301,495],{"mathvariant":494},[55,8303,8304],{"encoding":57},"\\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}",[31,8306,8308],{"className":8307,"ariaHidden":62},[61],[31,8309,8311,8315],{"className":8310},[66],[31,8312],{"className":8313,"style":8314},[70],"height:5.3276em;vertical-align:-2.4138em;",[31,8316,8318],{"className":8317},[75],[31,8319,8321,8394],{"className":8320},[5606],[31,8322,8324],{"className":8323},[5867],[31,8325,8327,8385],{"className":8326},[1043,1044],[31,8328,8330,8382],{"className":8329},[1048],[31,8331,8334,8373],{"className":8332,"style":8333},[1052],"height:2.9138em;",[31,8335,8337,8340],{"style":8336},"top:-4.9138em;",[31,8338],{"className":8339,"style":5884},[1060],[31,8341,8343,8346,8349,8352,8355,8358,8361,8364,8367,8370],{"className":8342},[75],[31,8344,4561],{"className":8345},[75,76],[31,8347,211],{"className":8348},[264],[31,8350,2515],{"className":8351,"style":2666},[75,76],[31,8353,1210],{"className":8354},[1391],[31,8356],{"className":8357,"style":1467},[119],[31,8359,8059],{"className":8360},[75,76],[31,8362,1210],{"className":8363},[1391],[31,8365],{"className":8366,"style":1467},[119],[31,8368,8064],{"className":8369,"style":2666},[75,76],[31,8371,220],{"className":8372},[289],[31,8374,8376,8379],{"style":8375},"top:-2.15em;",[31,8377],{"className":8378,"style":5884},[1060],[31,8380],{"className":8381},[75],[31,8383,1076],{"className":8384},[1075],[31,8386,8388],{"className":8387},[1048],[31,8389,8392],{"className":8390,"style":8391},[1052],"height:2.4138em;",[31,8393],{},[31,8395,8397],{"className":8396},[6062],[31,8398,8400,9602],{"className":8399},[1043,1044],[31,8401,8403,9599],{"className":8402},[1048],[31,8404,8406,9058],{"className":8405,"style":8333},[1052],[31,8407,8408,8411],{"style":8336},[31,8409],{"className":8410,"style":5884},[1060],[31,8412,8414,8417,8420,8423,8426,8473,8476,8519,8564,8567,8570,8573,8617,8620,8660,8666,8706,8709,8712,8715,8759,8762,8805,8811,8814,8817,8820,8864,8867,8907,8913,8953,8956,8959,8962,9006,9009,9052],{"className":8413},[75],[31,8415],{"className":8416},[75],[31,8418],{"className":8419,"style":293},[119],[31,8421,223],{"className":8422},[297],[31,8424],{"className":8425,"style":293},[119],[31,8427,8429],{"className":8428},[1460,1620],[31,8430,8432,8465],{"className":8431},[1043,1044],[31,8433,8435,8462],{"className":8434},[1048],[31,8436,8438,8452],{"className":8437,"style":5900},[1052],[31,8439,8440,8443],{"style":5903},[31,8441],{"className":8442,"style":5907},[1060],[31,8444,8446],{"className":8445},[1065,1066,1067,1068],[31,8447,8449],{"className":8448},[75,1068],[31,8450,5574],{"className":8451,"style":5514},[75,76,1068],[31,8453,8454,8457],{"style":5919},[31,8455],{"className":8456,"style":5907},[1060],[31,8458,8459],{},[31,8460,5625],{"className":8461},[1460,1461,1687],[31,8463,1076],{"className":8464},[1075],[31,8466,8468],{"className":8467},[1048],[31,8469,8471],{"className":8470,"style":5937},[1052],[31,8472],{},[31,8474],{"className":8475,"style":1467},[119],[31,8477,8479,8482],{"className":8478},[75],[31,8480,2515],{"className":8481,"style":2666},[75,76],[31,8483,8485],{"className":8484},[1039],[31,8486,8488,8511],{"className":8487},[1043,1044],[31,8489,8491,8508],{"className":8490},[1048],[31,8492,8494],{"className":8493,"style":5445},[1052],[31,8495,8496,8499],{"style":5504},[31,8497],{"className":8498,"style":1061},[1060],[31,8500,8502],{"className":8501},[1065,1066,1067,1068],[31,8503,8505],{"className":8504},[75,1068],[31,8506,5574],{"className":8507,"style":5514},[75,76,1068],[31,8509,1076],{"className":8510},[1075],[31,8512,8514],{"className":8513},[1048],[31,8515,8517],{"className":8516,"style":1083},[1052],[31,8518],{},[31,8520,8522,8526],{"className":8521},[75],[31,8523,8101],{"className":8524,"style":8525},[75,76],"margin-right:0.0715em;",[31,8527,8529],{"className":8528},[1039],[31,8530,8532,8556],{"className":8531},[1043,1044],[31,8533,8535,8553],{"className":8534},[1048],[31,8536,8538],{"className":8537,"style":5445},[1052],[31,8539,8541,8544],{"style":8540},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[31,8542],{"className":8543,"style":1061},[1060],[31,8545,8547],{"className":8546},[1065,1066,1067,1068],[31,8548,8550],{"className":8549},[75,1068],[31,8551,5574],{"className":8552,"style":5514},[75,76,1068],[31,8554,1076],{"className":8555},[1075],[31,8557,8559],{"className":8558},[1048],[31,8560,8562],{"className":8561,"style":1083},[1052],[31,8563],{},[31,8565],{"className":8566,"style":120},[119],[31,8568,434],{"className":8569},[124],[31,8571],{"className":8572,"style":120},[119],[31,8574,8576],{"className":8575},[1460,1620],[31,8577,8579,8609],{"className":8578},[1043,1044],[31,8580,8582,8606],{"className":8581},[1048],[31,8583,8585,8596],{"className":8584,"style":5900},[1052],[31,8586,8587,8590],{"style":5903},[31,8588],{"className":8589,"style":5907},[1060],[31,8591,8593],{"className":8592},[1065,1066,1067,1068],[31,8594,5391],{"className":8595},[75,76,1068],[31,8597,8598,8601],{"style":5919},[31,8599],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",[31,9719,9721,9764],{"className":9720,"translate":35},[34],[31,9722,9724],{"className":9723},[39],[41,9725,9726],{"xmlns":43},[45,9727,9728,9761],{},[48,9729,9730,9740,9742,9748,9750,9756,9759],{},[1004,9731,9732,9734],{},[51,9733,8101],{},[48,9735,9736,9738],{},[51,9737,5391],{},[51,9739,2555],{},[95,9741,97],{},[1004,9743,9744,9746],{},[51,9745,8059],{},[51,9747,5391],{},[95,9749,97],{},[1004,9751,9752,9754],{},[51,9753,8064],{},[51,9755,2555],{},[95,9757,9758],{},"\u003C",[91,9760,1977],{},[55,9762,9763],{"encoding":57},"C_{ij}-\\varphi_i-\\psi_j\u003C0",[31,9765,9767,9825,9881,9936],{"className":9766,"ariaHidden":62},[61],[31,9768,9770,9773,9816,9819,9822],{"className":9769},[66],[31,9771],{"className":9772,"style":1029},[70],[31,9774,9776,9779],{"className":9775},[75],[31,9777,8101],{"className":9778,"style":8525},[75,76],[31,9780,9782],{"className":9781},[1039],[31,9783,9785,9808],{"className":9784},[1043,1044],[31,9786,9788,9805],{"className":9787},[1048],[31,9789,9791],{"className":9790,"style":5445},[1052],[31,9792,9793,9796],{"style":8540},[31,9794],{"className":9795,"style":1061},[1060],[31,9797,9799],{"className":9798},[1065,1066,1067,1068],[31,9800,9802],{"className":9801},[75,1068],[31,9803,5574],{"className":9804,"style":5514},[75,76,1068],[31,9806,1076],{"className":9807},[1075],[31,9809,9811],{"className":9810},[1048],[31,9812,9814],{"className":9813,"style":1083},[1052],[31,9815],{},[31,9817],{"className":9818,"style":120},[119],[31,9820,97],{"className":9821},[124],[31,9823],{"className":9824,"style":120},[119],[31,9826,9828,9832,9872,9875,9878],{"className":9827},[66],[31,9829],{"className":9830,"style":9831},[70],"height:0.7778em;vertical-align:-0.1944em;",[31,9833,9835,9838],{"className":9834},[75],[31,9836,8059],{"className":9837},[75,76],[31,9839,9841],{"className":9840},[1039],[31,9842,9844,9864],{"className":9843},[1043,1044],[31,9845,9847,9861],{"className":9846},[1048],[31,9848,9850],{"className":9849,"style":5445},[1052],[31,9851,9852,9855],{"style":2807},[31,9853],{"className":9854,"style":1061},[1060],[31,9856,9858],{"className":9857},[1065,1066,1067,1068],[31,9859,5391],{"className":9860},[75,76,1068],[31,9862,1076],{"className":9863},[1075],[31,9865,9867],{"className":9866},[1048],[31,9868,9870],{"className":9869,"style":2050},[1052],[31,9871],{},[31,9873],{"className":9874,"style":120},[119],[31,9876,97],{"className":9877},[124],[31,9879],{"className":9880,"style":120},[119],[31,9882,9884,9887,9927,9930,9933],{"className":9883},[66],[31,9885],{"className":9886,"style":4270},[70],[31,9888,9890,9893],{"className":9889},[75],[31,9891,8064],{"className":9892,"style":2666},[75,76],[31,9894,9896],{"className":9895},[1039],[31,9897,9899,9919],{"className":9898},[1043,1044],[31,9900,9902,9916],{"className":9901},[1048],[31,9903,9905],{"className":9904,"style":5445},[1052],[31,9906,9907,9910],{"style":5504},[31,9908],{"className":9909,"style":1061},[1060],[31,9911,9913],{"className":9912},[1065,1066,1067,1068],[31,9914,2555],{"className":9915,"style":5514},[75,76,1068],[31,9917,1076],{"className":9918},[1075],[31,9920,9922],{"className":9921},[1048],[31,9923,9925],{"className":9924,"style":1083},[1052],[31,9926],{},[31,9928],{"className":9929,"style":293},[119],[31,9931,9758],{"className":9932},[297],[31,9934],{"className":9935,"style":293},[119],[31,9937,9939,9942],{"className":9938},[66],[31,9940],{"className":9941,"style":326},[70],[31,9943,1977],{"className":9944},[75],"，令 ",[31,9947,9949,9975],{"className":9948,"translate":35},[34],[31,9950,9952],{"className":9951},[39],[41,9953,9954],{"xmlns":43},[45,9955,9956,9972],{},[48,9957,9958,9968,9970],{},[1004,9959,9960,9962],{},[51,9961,2515],{},[48,9963,9964,9966],{},[51,9965,5391],{},[51,9967,2555],{},[95,9969,929],{},[51,9971,4587],{"mathvariant":494},[55,9973,9974],{"encoding":57},"\\pi_{ij}\\to\\infty",[31,9976,9978,10036],{"className":9977,"ariaHidden":62},[61],[31,9979,9981,9984,10027,10030,10033],{"className":9980},[66],[31,9982],{"className":9983,"style":5541},[70],[31,9985,9987,9990],{"className":9986},[75],[31,9988,2515],{"className":9989,"style":2666},[75,76],[31,9991,9993],{"className":9992},[1039],[31,9994,9996,10019],{"className":9995},[1043,1044],[31,9997,9999,10016],{"className":9998},[1048],[31,10000,10002],{"className":10001,"style":5445},[1052],[31,10003,10004,10007],{"style":5504},[31,10005],{"className":10006,"style":1061},[1060],[31,10008,10010],{"className":10009},[1065,1066,1067,1068],[31,10011,10013],{"className":10012},[75,1068],[31,10014,5574],{"className":10015,"style":5514},[75,76,1068],[31,10017,1076],{"className":10018},[1075],[31,10020,10022],{"className":10021},[1048],[31,10023,10025],{"className":10024,"style":1083},[1052],[31,10026],{},[31,10028],{"className":10029,"style":293},[119],[31,10031,929],{"className":10032},[297],[31,10034],{"className":10035,"style":293},[119],[31,10037,10039,10042],{"className":10038},[66],[31,10040],{"className":10041,"style":71},[70],[31,10043,4587],{"className":10044},[75]," 就得到 ",[31,10047,10049,10065],{"className":10048,"translate":35},[34],[31,10050,10052],{"className":10051},[39],[41,10053,10054],{"xmlns":43},[45,10055,10056,10062],{},[48,10057,10058,10060],{},[95,10059,97],{},[51,10061,4587],{"mathvariant":494},[55,10063,10064],{"encoding":57},"-\\infty",[31,10066,10068],{"className":10067,"ariaHidden":62},[61],[31,10069,10071,10074,10077],{"className":10070},[66],[31,10072],{"className":10073,"style":307},[70],[31,10075,97],{"className":10076},[75],[31,10078,4587],{"className":10079},[75],"；否则下确界在 ",[31,10082,10084,10102],{"className":10083,"translate":35},[34],[31,10085,10087],{"className":10086},[39],[41,10088,10089],{"xmlns":43},[45,10090,10091,10099],{},[48,10092,10093,10095,10097],{},[51,10094,2515],{},[95,10096,223],{},[91,10098,1977],{},[55,10100,10101],{"encoding":57},"\\pi=0",[31,10103,10105,10123],{"className":10104,"ariaHidden":62},[61],[31,10106,10108,10111,10114,10117,10120],{"className":10107},[66],[31,10109],{"className":10110,"style":71},[70],[31,10112,2515],{"className":10113,"style":2666},[75,76],[31,10115],{"className":10116,"style":293},[119],[31,10118,223],{"className":10119},[297],[31,10121],{"className":10122,"style":293},[119],[31,10124,10126,10129],{"className":10125},[66],[31,10127],{"className":10128,"style":326},[70],[31,10130,1977],{"className":10131},[75]," 处取到。于是对偶问题就是",[31,10134,10136],{"className":10135,"translate":35},[411],[31,10137,10139,10242],{"className":10138,"translate":35},[34],[31,10140,10142],{"className":10141},[39],[41,10143,10144],{"xmlns":43,"display":420},[45,10145,10146,10239],{},[48,10147,10148,10165,10167,10173,10179,10185,10187,10193,10199,10205,10207,10209,10211,10217,10219,10225,10227,10237],{},[1531,10149,10150,10157],{},[48,10151,10152,10155],{},[51,10153,10154],{},"max",[95,10156,1540],{},[48,10158,10159,10161,10163],{},[51,10160,8059],{},[95,10162,1210],{"separator":62},[51,10164,8064],{},[1430,10166,1545],{},[1531,10168,10169,10171],{},[95,10170,5625],{},[51,10172,5391],{},[1004,10174,10175,10177],{},[51,10176,5813],{},[51,10178,5391],{},[1004,10180,10181,10183],{},[51,10182,8059],{},[51,10184,5391],{},[95,10186,434],{},[1531,10188,10189,10191],{},[95,10190,5625],{},[51,10192,2555],{},[1004,10194,10195,10197],{},[51,10196,5834],{},[51,10198,2555],{},[1004,10200,10201,10203],{},[51,10202,8064],{},[51,10204,2555],{},[119,10206],{"width":1584},[1430,10208,1587],{},[119,10210],{"width":1584},[1004,10212,10213,10215],{},[51,10214,8059],{},[51,10216,5391],{},[95,10218,434],{},[1004,10220,10221,10223],{},[51,10222,8064],{},[51,10224,2555],{},[95,10226,347],{},[1004,10228,10229,10231],{},[51,10230,8101],{},[48,10232,10233,10235],{},[51,10234,5391],{},[51,10236,2555],{},[95,10238,1210],{"separator":62},[55,10240,10241],{"encoding":57},"\\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},",[31,10243,10245,10449,10644,10699],{"className":10244,"ariaHidden":62},[61],[31,10246,10248,10252,10307,10310,10313,10357,10360,10400,10440,10443,10446],{"className":10247},[66],[31,10249],{"className":10250,"style":10251},[70],"height:2.3277em;vertical-align:-1.2777em;",[31,10253,10255],{"className":10254},[1460,1620],[31,10256,10258,10298],{"className":10257},[1043,1044],[31,10259,10261,10295],{"className":10260},[1048],[31,10262,10264,10285],{"className":10263,"style":71},[1052],[31,10265,10267,10270],{"style":10266},"top:-2.3479em;margin-left:0em;",[31,10268],{"className":10269,"style":1637},[1060],[31,10271,10273],{"className":10272},[1065,1066,1067,1068],[31,10274,10276,10279,10282],{"className":10275},[75,1068],[31,10277,8059],{"className":10278},[75,76,1068],[31,10280,1210],{"className":10281},[1391,1068],[31,10283,8064],{"className":10284,"style":2666},[75,76,1068],[31,10286,10287,10290],{"style":1649},[31,10288],{"className":10289,"style":1637},[1060],[31,10291,10292],{},[31,10293,10154],{"className":10294},[1460],[31,10296,1076],{"className":10297},[1075],[31,10299,10301],{"className":10300},[1048],[31,10302,10305],{"className":10303,"style":10304},[1052],"height:0.8882em;",[31,10306],{},[31,10308,1545],{"className":10309},[119],[31,10311],{"className":10312,"style":1467},[119],[31,10314,10316],{"className":10315},[1460,1620],[31,10317,10319,10349],{"className":10318},[1043,1044],[31,10320,10322,10346],{"className":10321},[1048],[31,10323,10325,10336],{"className":10324,"style":5900},[1052],[31,10326,10327,10330],{"style":5903},[31,10328],{"className":10329,"style":5907},[1060],[31,10331,10333],{"className":10332},[1065,1066,1067,1068],[31,10334,5391],{"className":10335},[75,76,1068],[31,10337,10338,10341],{"style":5919},[31,10339],{"className":10340,"style":5907},[1060],[31,10342,10343],{},[31,10344,5625],{"className":10345},[1460,1461,1687],[31,10347,1076],{"className":10348},[1075],[31,10350,10352],{"className":10351},[1048],[31,10353,10355],{"className":10354,"style":6342},[1052],[31,10356],{},[31,10358],{"className":10359,"style":1467},[119],[31,10361,10363,10366],{"className":10362},[75],[31,10364,5813],{"className":10365},[75,76],[31,10367,10369],{"className":10368},[1039],[31,10370,10372,10392],{"className":10371},[1043,1044],[31,10373,10375,10389],{"className":10374},[1048],[31,10376,10378],{"className":10377,"style":5445},[1052],[31,10379,10380,10383],{"style":2807},[31,10381],{"className":10382,"style":1061},[1060],[31,10384,10386],{"className":10385},[1065,1066,1067,1068],[31,10387,5391],{"className":10388},[75,76,1068],[31,10390,1076],{"className":10391},[1075],[31,10393,10395],{"className":10394},[1048],[31,10396,10398],{"className":10397,"style":2050},[1052],[31,10399],{},[31,10401,10403,10406],{"className":10402},[75],[31,10404,8059],{"className":10405},[75,76],[31,10407,10409],{"className":10408},[1039],[31,10410,10412,10432],{"className":10411},[1043,1044],[31,10413,10415,10429],{"className":10414},[1048],[31,10416,10418],{"className":10417,"style":5445},[1052],[31,10419,10420,10423],{"style":2807},[31,10421],{"className":10422,"style":1061},[1060],[31,10424,10426],{"className":10425},[1065,1066,1067,1068],[31,10427,5391],{"className":10428},[75,76,1068],[31,10430,1076],{"className":10431},[1075],[31,10433,10435],{"className":10434},[1048],[31,10436,10438],{"className":10437,"style":2050},[1052],[31,10439],{},[31,10441],{"className":10442,"style":120},[119],[31,10444,434],{"className":10445},[124],[31,10447],{"className":10448,"style":120},[119],[31,10450,10452,10456,10500,10503,10543,10583,10586,10592,10595,10635,10638,10641],{"className":10451},[66],[31,10453],{"className":10454,"style":10455},[70],"height:2.4638em;vertical-align:-1.4138em;",[31,10457,10459],{"className":10458},[1460,1620],[31,10460,10462,10492],{"className":10461},[1043,1044],[31,10463,10465,10489],{"className":10464},[1048],[31,10466,10468,10479],{"className":10467,"style":5900},[1052],[31,10469,10470,10473],{"style":5903},[31,10471],{"className":10472,"style":5907},[1060],[31,10474,10476],{"className":10475},[1065,1066,1067,1068],[31,10477,2555],{"className":10478,"style":5514},[75,76,1068],[31,10480,10481,10484],{"style":5919},[31,10482],{"className":10483,"style":5907},[1060],[31,10485,10486],{},[31,10487,5625],{"className":10488},[1460,1461,1687],[31,10490,1076],{"className":10491},[1075],[31,10493,10495],{"className":10494},[1048],[31,10496,10498],{"className":10497,"style":5937},[1052],[31,10499],{},[31,10501],{"className":10502,"style":1467},[119],[31,10504,10506,10509],{"className":10505},[75],[31,10507,5834],{"className":10508},[75,76],[31,10510,10512],{"className":10511},[1039],[31,10513,10515,10535],{"className":10514},[1043,1044],[31,10516,10518,10532],{"className":10517},[1048],[31,10519,10521],{"className":10520,"style":5445},[1052],[31,10522,10523,10526],{"style":2807},[31,10524],{"className":10525,"style":1061},[1060],[31,10527,10529],{"className":10528},[1065,1066,1067,1068],[31,105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:\\ \\varphi(x)+\\psi(y)\\le c(x,y)\\Bigr\\}.",[31,10865,10867,10940,10983,11022,11049,11076],{"className":10866,"ariaHidden":62},[61],[31,10868,10870,10873,10913,10916,10919,10922,10925,10928,10931,10934,10937],{"className":10869},[66],[31,10871],{"className":10872,"style":260},[70],[31,10874,10876,10879],{"className":10875},[75],[31,10877,3117],{"className":10878,"style":949},[75,76],[31,10880,10882],{"className":10881},[1039],[31,10883,10885,10905],{"className":10884},[1043,1044],[31,10886,10888,10902],{"className":10887},[1048],[31,10889,10891],{"className":10890,"style":3230},[1052],[31,10892,10893,10896],{"style":1056},[31,10894],{"className":10895,"style":1061},[1060],[31,10897,10899],{"className":10898},[1065,1066,1067,1068],[31,10900,1412],{"className":10901},[75,76,1068],[31,10903,1076],{"className":10904},[1075],[31,10906,10908],{"className":10907},[1048],[31,10909,10911],{"className":10910,"style":2050},[1052],[31,10912],{},[31,10914,211],{"className":10915},[264],[31,10917,852],{"className":10918},[75,76],[31,10920,1210],{"className":10921},[1391],[31,10923],{"className":10924,"style":1467},[119],[31,10926,884],{"className":10927,"style":900},[75,76],[31,10929,220],{"className":10930},[289],[31,10932],{"className":10933,"style":293},[119],[31,10935,223],{"className":10936},[297],[31,10938],{"className":10939,"style":293},[119],[31,10941,10943,10947,10950,10956,10959,10962,10965,10968,10971,10974,10977,10980],{"className":10942},[66],[31,10944],{"className":10945,"style":10946},[70],"height:2.2222em;vertical-align:-0.8622em;",[31,10948,10792],{"className":10949},[1460],[31,10951,10953],{"className":10952},[264],[31,10954,2512],{"className":10955},[1307,2662],[31,10957,1409],{"className":10958,"style":1688},[1460,1461,1687],[31,10960],{"className":10961,"style":1467},[119],[31,10963,8059],{"className":10964},[75,76],[31,10966],{"className":10967,"style":1467},[119],[31,10969,1435],{"className":10970},[75,76],[31,10972,852],{"className":10973},[75,76],[31,10975],{"className":10976,"style":120},[119],[31,10978,434],{"className":10979},[124],[31,10981],{"className":10982,"style":120},[119],[31,10984,10986,10989,10992,10995,10998,11001,11004,11007,11010,11013,11016,11019],{"className":10985},[66],[31,10987],{"className":10988,"style":10946},[70],[31,10990,1409],{"className":10991,"style":1688},[1460,1461,1687],[31,10993],{"className":10994,"style":1467},[119],[31,10996,8064],{"className":10997,"style":2666},[75,76],[31,10999],{"className":11000,"style":1467},[119],[31,11002,1435],{"className":11003},[75,76],[31,11005,884],{"className":11006,"style":900},[75,76],[31,11008,1545],{"className":11009},[119],[31,11011],{"className":11012,"style":293},[119],[31,11014,922],{"className":11015},[297],[31,11017,1545],{"className":11018},[119],[31,11020],{"className":11021,"style":293},[119],[31,11023,11025,11028,11031,11034,11037,11040,11043,11046],{"className":11024},[66],[31,11026],{"className":11027,"style":260},[70],[31,11029,8059],{"className":11030},[75,76],[31,11032,211],{"className":11033},[264],[31,11035,53],{"className":11036},[75,76],[31,11038,220],{"className":11039},[289],[31,11041],{"className":11042,"style":120},[119],[31,11044,434],{"className":11045},[124],[31,11047],{"className":11048,"style":120},[119],[31,11050,11052,11055,11058,11061,11064,11067,11070,11073],{"className":11051},[66],[31,11053],{"className":11054,"style":260},[70],[31,11056,8064],{"className":11057,"style":2666},[75,76],[31,11059,211],{"className":11060},[264],[31,11062,3003],{"className":11063,"style":2666},[75,76],[31,11065,220],{"className":11066},[289],[31,11068],{"className":11069,"style":293},[119],[31,11071,347],{"className":11072},[297],[31,11074],{"className":11075,"style":293},[119],[31,11077,11079,11082,11085,11088,11091,11094,11097,11100,11103,11109],{"className":11078},[66],[31,11080],{"className":11081,"style":2655},[70],[31,11083,1412],{"className":11084},[75,76],[31,11086,211],{"className":11087},[264],[31,11089,53],{"className":11090},[75,76],[31,11092,1210],{"className":11093},[1391],[31,11095],{"className":11096,"style":1467},[119],[31,11098,3003],{"className":11099,"style":2666},[75,76],[31,11101,220],{"className":11102},[289],[31,11104,11106],{"className":11105},[289],[31,11107,2604],{"className":11108},[1307,2662],[31,11110,495],{"className":11111},[75],[11,11113,3501,11114,11165,11166,11297,11298,11354,11355,11486,11487,4880,11516,11544],{},[31,11115,11117,11135],{"className":11116,"translate":35},[34],[31,11118,11120],{"className":11119},[39],[41,11121,11122],{"xmlns":43},[45,11123,11124,11132],{},[48,11125,11126,11128,11130],{},[51,11127,1412],{},[95,11129,223],{},[51,11131,1435],{},[55,11133,11134],{"encoding":57},"c=d",[31,11136,11138,11156],{"className":11137,"ariaHidden":62},[61],[31,11139,11141,11144,11147,11150,11153],{"className":11140},[66],[31,11142],{"className":11143,"style":71},[70],[31,11145,1412],{"className":11146},[75,76],[31,11148],{"className":11149,"style":293},[119],[31,11151,223],{"className":11152},[297],[3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",[31,11167,11169,11213],{"className":11168,"translate":35},[34],[31,11170,11172],{"className":11171},[39],[41,11173,11174],{"xmlns":43},[45,11175,11176,11210],{},[48,11177,11178,11180,11182,11184,11186,11188,11190,11192,11194,11196,11198,11200,11202,11204,11206,11208],{},[51,11179,8059],{},[95,11181,211],{"stretchy":210},[51,11183,53],{},[95,11185,220],{"stretchy":210},[95,11187,434],{},[51,11189,8064],{},[95,11191,211],{"stretchy":210},[51,11193,3003],{},[95,11195,220],{"stretchy":210},[95,11197,347],{},[51,11199,1435],{},[95,11201,211],{"stretchy":210},[51,11203,53],{},[95,11205,1210],{"separator":62},[51,11207,3003],{},[95,11209,220],{"stretchy":210},[55,11211,11212],{"encoding":57},"\\varphi(x)+\\psi(y)\\le d(x,y)",[31,11214,11216,11243,11270],{"className":11215,"ariaHidden":62},[61],[31,11217,11219,11222,11225,11228,11231,11234,11237,11240],{"className":11218},[66],[31,11220],{"className":11221,"style":260},[70],[31,11223,8059],{"className":11224},[75,76],[31,11226,211],{"className":11227},[264],[31,11229,53],{"className":11230},[75,76],[31,11232,220],{"className":11233},[289],[31,11235],{"className":11236,"style":120},[119],[31,11238,434],{"className":11239},[124],[31,11241],{"className":11242,"style":120},[119],[31,11244,11246,11249,11252,11255,11258,11261,11264,11267],{"className":11245},[66],[31,11247],{"className":11248,"style":260},[70],[31,11250,8064],{"className":11251,"style":2666},[75,76],[31,11253,211],{"className":11254},[264],[31,11256,3003],{"className":11257,"style":2666},[75,76],[31,11259,220],{"className":11260},[289],[31,11262],{"className":11263,"style":293},[119],[31,11265,347],{"className":11266},[297],[31,11268],{"className":11269,"style":293},[119],[31,11271,11273,11276,11279,11282,11285,11288,11291,11294],{"className":11272},[66],[31,11274],{"className":11275,"style":260},[70],[31,11277,1435],{"className":11278},[75,76],[31,11280,211],{"className":11281},[264],[31,11283,53],{"className":11284},[75,76],[31,11286,1210],{"className":11287},[1391],[31,11289],{"className":11290,"style":1467},[119],[31,11292,3003],{"className":11293,"style":2666},[75,76],[31,11295,220],{"className":11296},[289]," 允许我取 ",[31,11299,11301,11321],{"className":11300,"translate":35},[34],[31,11302,11304],{"className":11303},[39],[41,11305,11306],{"xmlns":43},[45,11307,11308,11318],{},[48,11309,11310,11312,11314,11316],{},[51,11311,8064],{},[95,11313,223],{},[95,11315,97],{},[51,11317,8059],{},[55,11319,11320],{"encoding":57},"\\psi=-\\varphi",[31,11322,11324,11342],{"className":11323,"ariaHidden":62},[61],[31,11325,11327,11330,11333,11336,11339],{"className":11326},[66],[31,11328],{"className":11329,"style":4202},[70],[31,11331,8064],{"className":11332,"style":2666},[75,76],[31,11334],{"className":11335,"style":293},[119],[31,11337,223],{"className":11338},[297],[31,11340],{"className":11341,"style":293},[119],[31,11343,11345,11348,11351],{"className":11344},[66],[31,11346],{"className":11347,"style":9831},[70],[31,11349,97],{"className":11350},[75],[31,11352,8059],{"className":11353},[75,76],"，此时它变成 ",[31,11356,11358,11402],{"className":11357,"translate":35},[34],[31,11359,11361],{"className":11360},[39],[41,11362,11363],{"xmlns":43},[45,11364,11365,11399],{},[48,11366,11367,11369,11371,11373,11375,11377,11379,11381,11383,11385,11387,11389,11391,11393,11395,11397],{},[51,11368,8059],{},[95,11370,211],{"stretchy":210},[51,11372,53],{},[95,11374,220],{"stretchy":210},[95,11376,97],{},[51,11378,8059],{},[95,11380,211],{"stretchy":210},[51,11382,3003],{},[95,11384,220],{"stretchy":210},[95,11386,347],{},[51,11388,1435],{},[95,11390,211],{"stretchy":210},[51,11392,53],{},[95,11394,1210],{"separator":62},[51,11396,3003],{},[95,11398,220],{"stretchy":210},[55,11400,11401],{"encoding":57},"\\varphi(x)-\\varphi(y)\\le d(x,y)",[31,11403,11405,11432,11459],{"className":11404,"ariaHidden":62},[61],[31,11406,11408,11411,11414,11417,11420,11423,11426,11429],{"className":11407},[66],[31,11409],{"className":11410,"style":260},[70],[31,11412,8059],{"className":11413},[75,76],[31,11415,211],{"className":11416},[264],[31,11418,53],{"className":11419},[75,76],[31,11421,220],{"className":11422},[289],[31,11424],{"className":11425,"style":120},[119],[31,11427,97],{"className":11428},[124],[31,11430],{"className":11431,"style":120},[119],[31,11433,11435,11438,11441,11444,11447,11450,11453,11456],{"className":11434},[66],[31,11436],{"className":11437,"style":260},[70],[31,11439,8059],{"className":11440},[75,76],[31,11442,211],{"className":11443},[264],[31,11445,3003],{"className":11446,"style":2666},[75,76],[31,11448,220],{"className":11449},[289],[31,11451],{"className":11452,"style":293},[119],[31,11454,347],{"className":11455},[297],[31,11457],{"className":11458,"style":293},[119],[31,11460,11462,11465,11468,11471,11474,11477,11480,11483],{"className":11461},[66],[31,11463],{"className":11464,"style":260},[70],[31,11466,1435],{"className":11467},[75,76],[31,11469,211],{"className":11470},[264],[31,11472,53],{"className":11473},[75,76],[31,11475,1210],{"className":11476},[1391],[31,11478],{"className":11479,"style":1467},[119],[31,11481,3003],{"className":11482,"style":2666},[75,76],[31,11484,220],{"className":11485},[289],"，也就是 ",[31,11488,11490,11504],{"className":11489,"translate":35},[34],[31,11491,11493],{"className":11492},[39],[41,11494,11495],{"xmlns":43},[45,11496,11497,11501],{},[48,11498,11499],{},[51,11500,8059],{},[55,11502,11503],{"encoding":57},"\\varphi",[31,11505,11507],{"className":11506,"ariaHidden":62},[61],[31,11508,11510,11513],{"className":11509},[66],[31,11511],{"className":11512,"style":865},[70],[31,11514,8059],{"className":11515},[75,76],[31,11517,11519,11532],{"className":11518,"translate":35},[34],[31,11520,11522],{"className":11521},[39],[41,11523,11524],{"xmlns":43},[45,11525,11526,11530],{},[48,11527,11528],{},[91,11529,205],{},[55,11531,205],{"encoding":57},[31,11533,11535],{"className":11534,"ariaHidden":62},[61],[31,11536,11538,11541],{"className":11537},[66],[31,11539],{"className":11540,"style":326},[70],[31,11542,205],{"className":11543},[75],"-Lipschitz 的。所以",[31,11546,11548],{"className":11547,"translate":35},[411],[31,11549,11551,11631],{"className":11550,"translate":35},[34],[31,11552,11554],{"className":11553},[39],[41,11555,11556],{"xmlns":43,"display":420},[45,11557,11558,11628],{},[48,11559,11560,11566,11568,11570,11572,11574,11576,11578,11608,11610,11612,11614,11616,11618,11620,11622,11624,11626],{},[1004,11561,11562,11564],{},[51,11563,3117],{},[91,11565,205],{},[95,11567,211],{"stretchy":210},[51,11569,852],{},[95,11571,1210],{"separator":62},[51,11573,884],{},[95,11575,220],{"stretchy":210},[95,11577,223],{},[1531,11579,11580,11586],{},[48,11581,11582,11584],{},[51,11583,10792],{},[95,11585,1540],{},[48,11587,11588,11590,11592,11604,11606],{},[51,11589,4572],{"mathvariant":494},[51,11591,8059],{},[1004,11593,11594,11596],{},[51,11595,4572],{"mathvariant":494},[48,11597,11598,11600,11602],{},[51,11599,4561],{"mathvariant":494},[51,11601,5391],{"mathvariant":494},[51,11603,11],{"mathvariant":494},[95,11605,347],{},[91,11607,205],{},[95,11609,1409],{},[51,11611,8059],{},[1430,11613,1432],{},[51,11615,1435],{},[95,11617,211],{"stretchy":210},[51,11619,852],{},[95,11621,97],{},[51,11623,884],{},[95,11625,220],{"stretchy":210},[51,11627,495],{"mathvariant":494},[55,11629,11630],{"encoding":57},"W_1(\\mu,\\nu)=\\sup_{\\|\\varphi\\|_{\\mathrm{Lip}}\\le 1}\\int\\varphi\\,d(\\mu-\\nu).",[31,11632,11634,11707,11857],{"className":11633,"ariaHidden":62},[61],[31,11635,11637,11640,11680,11683,11686,11689,11692,11695,11698,11701,11704],{"className":11636},[66],[31,11638],{"className":11639,"style":260},[70],[31,11641,11643,11646],{"className":11642},[75],[31,11644,3117],{"className":11645,"style":949},[75,76],[31,11647,11649],{"className":11648},[1039],[31,11650,11652,11672],{"className":11651},[1043,1044],[31,11653,11655,11669],{"className":11654},[1048],[31,11656,11658],{"className":11657,"style":2028},[1052],[31,11659,11660,11663],{"style":1056},[31,11661],{"className":11662,"style":1061},[1060],[31,11664,11666],{"className":11665},[1065,1066,1067,1068],[31,11667,205],{"className":11668},[75,1068],[31,11670,1076],{"className":11671},[1075],[31,11673,11675],{"className":11674},[1048],[31,11676,11678],{"className":11677,"style":2050},[1052],[31,11679],{},[31,11681,211],{"className":11682},[264],[31,11684,852],{"className":11685},[75,76],[31,11687,1210],{"className":11688},[1391],[31,11690],{"className":11691,"style":1467},[119],[31,11693,884],{"className":11694,"style":900},[75,76],[31,11696,220],{"className":11697},[289],[31,11699],{"className":11700,"style":293},[119],[31,11702,223],{"className":11703},[297],[31,11705],{"className":11706,"style":293},[119],[31,11708,11710,11714,11824,11827,11830,11833,11836,11839,11842,11845,11848,11851,11854],{"className":11709},[66],[31,11711],{"className":11712,"style":11713},[70],"height:2.543em;vertical-align:-1.183em;",[31,11715,11717],{"className":11716},[1460,1620],[31,11718,11720,11815],{"className":11719},[1043,1044],[31,11721,11723,11812],{"className":11722},[1048],[31,11724,11726,11802],{"className":11725,"style":71},[1052],[31,11727,11729,11732],{"style":11728},"top:-2.1146em;margin-left:0em;",[31,11730],{"className":11731,"style":1637},[1060],[31,11733,11735],{"className":11734},[1065,1066,1067,1068],[31,11736,11738,11741,11744,11796,11799],{"className":11737},[75,1068],[31,11739,4572],{"className":11740},[75,1068],[31,11742,8059],{"className":11743},[75,76,1068],[31,11745,11747,11750],{"className":11746},[75,1068],[31,11748,4572],{"className":11749},[75,1068],[31,11751,11753],{"className":11752},[1039],[31,11754,11756,11787],{"className":11755},[1043,1044],[31,11757,11759,11784],{"className":11758},[1048],[31,11760,11763],{"className":11761,"style":11762},[1052],"height:0.3448em;",[31,11764,11766,11770],{"style":11765},"top:-2.3567em;margin-left:0em;margin-right:0.0714em;",[31,11767],{"className":11768,"style":11769},[1060],"height:2.5em;",[31,11771,11774],{"className":11772},[1065,11773,1308,1068],"reset-size3",[31,11775,11777],{"className":11776},[75,1068],[31,11778,11780],{"className":11779},[75,1068],[31,11781,11783],{"className":11782},[75,2745,1068],"Lip",[31,11785,1076],{"className":11786},[1075],[31,11788,11790],{"className":11789},[1048],[31,11791,11794],{"className":11792,"style":11793},[1052],"height:0.2822em;",[31,11795],{},[31,11797,347],{"className":11798},[297,1068],[31,11800,205],{"className":11801},[75,1068],[31,11803,11804,11807],{"style":1649},[31,11805],{"className":11806,"style":1637},[1060],[31,11808,11809],{},[31,11810,10792],{"className":11811},[1460],[31,11813,1076],{"className":11814},[1075],[31,11816,11818],{"className":11817},[1048],[31,11819,11822],{"className":11820,"style":11821},[1052],"height:1.183em;",[31,11823],{},[31,11825],{"className":11826,"style":1467},[119],[31,11828,1409],{"className":11829,"style":1688},[1460,1461,1687],[31,11831],{"className":11832,"style":1467},[119],[31,11834,8059],{"className":11835},[75,76],[31,11837],{"className":11838,"style":1467},[119],[31,11840,1435],{"className":11841},[75,76],[31,11843,211],{"className":11844},[264],[31,11846,852],{"className":11847},[75,76],[31,11849],{"className":11850,"style":120},[119],[31,11852,97],{"className":11853},[124],[31,11855],{"className":11856,"style":120},[119],[31,11858,11860,11863,11866,11869],{"className":11859},[66],[31,11861],{"className":11862,"style":260},[70],[31,11864,884],{"className":11865,"style":900},[75,76],[31,11867,220],{"className":11868},[289],[31,11870,495],{"className":11871},[75],[11,11873,11874],{},"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。",[11,11876,11877,11878,12068,12069,12097,12098,12126,12127,12184,12185,12213,12214,2372],{},"二次代价 ",[31,11879,11881,11919],{"className":11880,"translate":35},[34],[31,11882,11884],{"className":11883},[39],[41,11885,11886],{"xmlns":43},[45,11887,11888,11916],{},[48,11889,11890,11892,11894,11902,11904,11906,11908,11910],{},[51,11891,1412],{},[95,11893,223],{},[2071,11895,11896],{"scriptlevel":1977,"displaystyle":210},[2074,11897,11898,11900],{},[91,11899,205],{},[91,11901,152],{},[51,11903,4572],{"mathvariant":494},[51,11905,53],{},[95,11907,97],{},[51,11909,3003],{},[1180,11911,11912,11914],{},[51,11913,4572],{"mathvariant":494},[91,11915,152],{},[55,11917,11918],{"encoding":57},"c=\\tfrac12\\|x-y\\|^2",[31,11920,11922,11940,12029],{"className":11921,"ariaHidden":62},[61],[31,11923,11925,11928,11931,11934,11937],{"className":11924},[66],[31,11926],{"className":11927,"style":71},[70],[31,11929,1412],{"className":11930},[75,76],[31,11932],{"className":11933,"style":293},[119],[31,11935,223],{"className":11936},[297],[31,11938],{"className":11939,"style":293},[119],[31,11941,11943,11946,12014,12017,12020,12023,12026],{"className":11942},[66],[31,11944],{"className":11945,"style":2132},[70],[31,11947,11949,11952,12011],{"className":11948},[75],[31,11950],{"className":11951},[264,2139],[31,11953,11955],{"className":11954},[2074],[31,11956,11958,12003],{"className":11957},[1043,1044],[31,11959,11961,12000],{"className":11960},[1048],[31,11962,11964,11978,11986],{"className":11963,"style":2152},[1052],[31,11965,11966,11969],{"style":2155},[31,11967],{"className":11968,"style":1637},[1060],[31,11970,11972],{"className":11971},[1065,1066,1067,1068],[31,11973,11975],{"className":11974},[75,1068],[31,11976,152],{"className":11977},[75,1068],[31,11979,11980,11983],{"style":2170},[31,11981],{"className":11982,"style":1637},[1060],[31,11984],{"className":11985,"style":2178},[2177],[31,11987,11988,11991],{"style":2181},[31,11989],{"className":11990,"style":1637},[1060],[31,11992,11994],{"className":11993},[1065,1066,1067,1068],[31,11995,11997],{"className":11996},[75,1068],[31,11998,205],{"className":11999},[75,1068],[31,12001,1076],{"className":12002},[1075],[31,12004,12006],{"className":12005},[1048],[31,12007,12009],{"className":12008,"style":2203},[1052],[31,12010],{},[31,12012],{"className":12013},[289,2139],[31,12015,4572],{"className":12016},[75],[31,12018,53],{"className":12019},[75,76],[31,12021],{"className":12022,"style":120},[119],[31,12024,97],{"className":12025},[124],[31,12027],{"className":12028,"style":120},[119],[31,12030,12032,12036,12039],{"className":12031},[66],[31,12033],{"className":12034,"style":12035},[70],"height:1.0641em;vertical-align:-0.25em;",[31,12037,3003],{"className":12038,"style":2666},[75,76],[31,12040,12042,12045],{"className":12041},[75],[31,12043,4572],{"className":12044},[75],[31,12046,12048],{"className":12047},[1039],[31,12049,12051],{"className":12050},[1043],[31,12052,12054],{"className":12053},[1048],[31,12055,12057],{"className":12056,"style":5135},[1052],[31,12058,12059,12062],{"style":3631},[31,12060],{"className":12061,"style":1061},[1060],[31,12063,12065],{"className":12064},[1065,1066,1067,1068],[31,12066,152],{"className":12067},[75,1068]," 时还有一个 Brenier 定理：若 ",[31,12070,12072,12085],{"className":12071,"translate":35},[34],[31,12073,12075],{"className":12074},[39],[41,12076,12077],{"xmlns":43},[45,12078,12079,12083],{},[48,12080,12081],{},[51,12082,852],{},[55,12084,855],{"encoding":57},[31,12086,12088],{"className":12087,"ariaHidden":62},[61],[31,12089,12091,12094],{"className":12090},[66],[31,12092],{"className":12093,"style":865},[70],[31,12095,852],{"className":12096},[75,76]," 关于 Lebesgue 测度绝对连续，最优的 ",[31,12099,12101,12114],{"className":12100,"translate":35},[34],[31,12102,12104],{"className":12103},[39],[41,12105,12106],{"xmlns":43},[45,12107,12108,12112],{},[48,12109,12110],{},[51,12111,2515],{},[55,12113,3485],{"encoding":57},[31,12115,12117],{"className":12116,"ariaHidden":62},[61],[31,12118,12120,12123],{"className":12119},[66],[31,12121],{"className":12122,"style":71},[70],[31,12124,2515],{"className":12125,"style":2666},[75,76]," 由映射 ",[31,12128,12130,12151],{"className":12129,"translate":35},[34],[31,12131,12133],{"className":12132},[39],[41,12134,12135],{"xmlns":43},[45,12136,12137,12148],{},[48,12138,12139,12141,12143,12146],{},[51,12140,919],{},[95,12142,223],{},[51,12144,12145],{"mathvariant":494},"∇",[51,12147,5324],{},[55,12149,12150],{"encoding":57},"T=\\nabla u",[31,12152,12154,12172],{"className":12153,"ariaHidden":62},[61],[31,12155,12157,12160,12163,12166,12169],{"className":12156},[66],[31,12158],{"className":12159,"style":945},[70],[31,12161,919],{"className":12162,"style":949},[75,76],[31,12164],{"className":12165,"style":293},[119],[31,12167,223],{"className":12168},[297],[31,12170],{"className":12171,"style":293},[119],[31,12173,12175,12178,12181],{"className":12174},[66],[31,12176],{"className":12177,"style":945},[70],[31,12179,12145],{"className":12180},[75],[31,12182,5324],{"className":12183},[75,76]," 给出，其中 ",[31,12186,12188,12201],{"className":12187,"translate":35},[34],[31,12189,12191],{"className":12190},[39],[41,12192,12193],{"xmlns":43},[45,12194,12195,12199],{},[48,12196,12197],{},[51,12198,5324],{},[55,12200,5324],{"encoding":57},[31,12202,12204],{"className":12203,"ariaHidden":62},[61],[31,12205,12207,12210],{"className":12206},[66],[31,12208],{"className":12209,"style":71},[70],[31,12211,5324],{"className":12212},[75,76]," 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 ",[31,12215,12217,12233],{"className":12216,"translate":35},[34],[31,12218,12220],{"className":12219},[39],[41,12221,12222],{"xmlns":43},[45,12223,12224,12230],{},[48,12225,12226,12228],{},[51,12227,12145],{"mathvariant":494},[51,12229,5324],{},[55,12231,12232],{"encoding":57},"\\nabla u",[31,12234,12236],{"className":12235,"ariaHidden":62},[61],[31,12237,12239,12242,12245],{"className":12238},[66],[31,12240],{"className":12241,"style":945},[70],[31,12243,12145],{"className":12244},[75],[31,12246,5324],{"className":12247},[75,76],[18,12249,12250],{"id":12250},"一维与高斯",[11,12252,12253],{},"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？",[11,12255,12256,12257,12287,12288,12316,12317,12351,12352,12380],{},"“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 ",[31,12258,12260,12274],{"className":12259,"translate":35},[34],[31,12261,12263],{"className":12262},[39],[41,12264,12265],{"xmlns":43},[45,12266,12267,12272],{},[48,12268,12269],{},[51,12270,12271],{},"t",[55,12273,12271],{"encoding":57},[31,12275,12277],{"className":12276,"ariaHidden":62},[61],[31,12278,12280,12284],{"className":12279},[66],[31,12281],{"className":12282,"style":12283},[70],"height:0.6151em;",[31,12285,12271],{"className":12286},[75,76]," 的人从 ",[31,12289,12291,12304],{"className":12290,"translate":35},[34],[31,12292,12294],{"className":12293},[39],[41,12295,12296],{"xmlns":43},[45,12297,12298,12302],{},[48,12299,12300],{},[51,12301,12271],{},[55,12303,12271],{"encoding":57},[31,12305,12307],{"className":12306,"ariaHidden":62},[61],[31,12308,12310,12313],{"className":12309},[66],[31,12311],{"className":12312,"style":12283},[70],[31,12314,12271],{"className":12315},[75,76]," 万变成 ",[31,12318,12320,12336],{"className":12319,"translate":35},[34],[31,12321,12323],{"className":12322},[39],[41,12324,12325],{"xmlns":43},[45,12326,12327,12333],{},[48,12328,12329,12331],{},[91,12330,152],{},[51,12332,12271],{},[55,12334,12335],{"encoding":57},"2t",[31,12337,12339],{"className":12338,"ariaHidden":62},[61],[31,12340,12342,12345,12348],{"className":12341},[66],[31,12343],{"className":12344,"style":326},[70],[31,12346,152],{"className":12347},[75],[31,12349,12271],{"className":12350},[75,76]," 万，涨幅是 ",[31,12353,12355,12368],{"className":12354,"translate":35},[34],[31,12356,12358],{"className":12357},[39],[41,12359,12360],{"xmlns":43},[45,12361,12362,12366],{},[48,12363,12364],{},[51,12365,12271],{},[55,12367,12271],{"encoding":57},[31,12369,12371],{"className":12370,"ariaHidden":62},[61],[31,12372,12374,12377],{"className":12373},[66],[31,12375],{"className":12376,"style":12283},[70],[31,12378,12271],{"className":12379},[75,76]," 万，平均",[31,12382,12384],{"className":12383,"translate":35},[411],[31,12385,12387,12431],{"className":12386,"translate":35},[34],[31,12388,12390],{"className":12389},[39],[41,12391,12392],{"xmlns":43,"display":420},[45,12393,12394,12428],{},[48,12395,12396,12405,12407,12409,12411,12413,12415,12423,12426],{},[12397,12398,12399,12401,12403],"msubsup",{},[95,12400,1409],{},[91,12402,1977],{},[91,12404,205],{},[51,12406,12271],{},[1430,12408,1432],{},[51,12410,1435],{},[51,12412,12271],{},[95,12414,223],{},[2071,12416,12417],{"scriptlevel":1977,"displaystyle":210},[2074,12418,12419,12421],{},[91,12420,205],{},[91,12422,152],{},[1430,12424,12425],{}," 万",[51,12427,495],{"mathvariant":494},[55,12429,12430],{"encoding":57},"\\int_0^1 t\\,dt=\\tfrac12\\ \\text{万}.",[31,12432,12434,12518],{"className":12433,"ariaHidden":62},[61],[31,12435,12437,12441,12494,12497,12500,12503,12506,12509,12512,12515],{"className":12436},[66],[31,12438],{"className":12439,"style":12440},[70],"height:2.476em;vertical-align:-0.9119em;",[31,12442,12444,12447],{"className":12443},[1460],[31,12445,1409],{"className":12446,"style":1688},[1460,1461,1687],[31,12448,12450],{"className":12449},[1039],[31,12451,12453,12486],{"className":12452},[1043,1044],[31,12454,12456,12483],{"className":12455},[1048],[31,12457,12460,12471],{"className":12458,"style":12459},[1052],"height:1.564em;",[31,12461,12462,12465],{"style":1704},[31,12463],{"className":12464,"style":1061},[1060],[31,12466,12468],{"className":12467},[1065,1066,1067,1068],[31,12469,1977],{"className":12470},[75,1068],[31,12472,12474,12477],{"style":12473},"top:-3.8129em;margin-right:0.05em;",[31,12475],{"className":12476,"style":1061},[1060],[31,12478,12480],{"className":12479},[1065,1066,1067,1068],[31,12481,205],{"className":12482},[75,1068],[31,12484,1076],{"className":12485},[1075],[31,12487,12489],{"className":12488},[1048],[31,12490,12492],{"className":12491,"style":1726},[1052],[31,12493],{},[31,12495],{"className":12496,"style":1467},[119],[31,12498,12271],{"className":12499},[75,76],[31,12501],{"className":12502,"style":1467},[119],[31,12504,1435],{"className":12505},[75,76],[31,12507,12271],{"className":12508},[75,76],[31,12510],{"className":12511,"style":293},[119],[31,12513,223],{"className":12514},[297],[31,12516],{"className":12517,"style":293},[119],[31,12519,12521,12524,12592,12595,12603],{"className":12520},[66],[31,12522],{"className":12523,"style":2132},[70],[31,12525,12527,12530,12589],{"className":12526},[75],[31,12528],{"className":12529},[264,2139],[31,12531,12533],{"className":12532},[2074],[31,12534,12536,12581],{"className":12535},[1043,1044],[31,12537,12539,12578],{"className":12538},[1048],[31,12540,12542,12556,12564],{"className":12541,"style":2152},[1052],[31,12543,12544,12547],{"style":2155},[31,12545],{"className":12546,"style":1637},[1060],[31,12548,12550],{"className":12549},[1065,1066,1067,1068],[31,12551,12553],{"className":12552},[75,1068],[31,12554,152],{"className":12555},[75,1068],[31,12557,12558,12561],{"style":2170},[31,12559],{"className":12560,"style":1637},[1060],[31,12562],{"className":12563,"style":2178},[2177],[31,12565,12566,12569],{"style":2181},[31,12567],{"className":12568,"style":1637},[1060],[31,12570,12572],{"className":12571},[1065,1066,1067,1068],[31,12573,12575],{"className":12574},[75,1068],[31,12576,205],{"className":12577},[75,1068],[31,12579,1076],{"className":12580},[1075],[31,12582,12584],{"className":12583},[1048],[31,12585,12587],{"className":12586,"style":2203},[1052],[31,12588],{},[31,12590],{"className":12591},[289,2139],[31,12593,1545],{"className":12594},[119],[31,12596,12598],{"className":12597},[75,1793],[31,12599,12602],{"className":12600},[75,12601],"cjk_fallback","万",[31,12604,495],{"className":12605},[75],[11,12607,12608,12609,12665,12666,12731,12732,2372],{},"这里可以直接用 ",[31,12610,12612,12632],{"className":12611,"translate":35},[34],[31,12613,12615],{"className":12614},[39],[41,12616,12617],{"xmlns":43},[45,12618,12619,12629],{},[48,12620,12621,12623,12625,12627],{},[91,12622,152],{},[51,12624,12271],{},[95,12626,97],{},[51,12628,12271],{},[55,12630,12631],{"encoding":57},"2t-t",[31,12633,12635,12656],{"className":12634,"ariaHidden":62},[61],[31,12636,12638,12641,12644,12647,12650,12653],{"className":12637},[66],[31,12639],{"className":12640,"style":112},[70],[31,12642,152],{"className":12643},[75],[31,12645,12271],{"className":12646},[75,76],[31,12648],{"className":12649,"style":120},[119],[31,12651,97],{"className":12652},[124],[31,12654],{"className":12655,"style":120},[119],[31,12657,12659,12662],{"className":12658},[66],[31,12660],{"className":12661,"style":12283},[70],[31,12663,12271],{"className":12664},[75,76]," 而不是 ",[31,12667,12669,12694],{"className":12668,"translate":35},[34],[31,12670,12672],{"className":12671},[39],[41,12673,12674],{"xmlns":43},[45,12675,12676,12691],{},[48,12677,12678,12681,12683,12685,12687,12689],{},[51,12679,12680],{"mathvariant":494},"∣",[91,12682,152],{},[51,12684,12271],{},[95,12686,97],{},[51,12688,12271],{},[51,12690,12680],{"mathvariant":494},[55,12692,12693],{"encoding":57},"|2t-t|",[31,12695,12697,12719],{"className":12696,"ariaHidden":62},[61],[31,12698,12700,12703,12707,12710,12713,12716],{"className":12699},[66],[31,12701],{"className":12702,"style":260},[70],[31,12704,12706],{"className":12705},[75],"∣2",[31,12708,12271],{"className":12709},[75,76],[31,12711],{"className":12712,"style":120},[119],[31,12714,97],{"className":12715},[124],[31,12717],{"className":12718,"style":120},[119],[31,12720,12722,12725,12728],{"className":12721},[66],[31,12723],{"className":12724,"style":260},[70],[31,12726,12271],{"className":12727},[75,76],[31,12729,12680],{"className":12730},[75],"，因为所有人都在往同一个方向涨。而这个“平均涨幅”正是 ",[31,12733,12735,12752],{"className":12734,"translate":35},[34],[31,12736,12738],{"className":12737},[39],[41,12739,12740],{"xmlns":43},[45,12741,12742,12750],{},[48,12743,12744],{},[1004,12745,12746,12748],{},[51,12747,3117],{},[91,12749,205],{},[55,12751,5220],{"encoding":57},[31,12753,12755],{"className":12754,"ariaHidden":62},[61],[31,12756,12758,12761],{"className":12757},[66],[31,12759],{"className":12760,"style":5230},[70],[31,12762,12764,12767],{"className":12763},[75],[31,12765,3117],{"className":12766,"style":949},[75,76],[31,12768,12770],{"className":12769},[1039],[31,12771,12773,12793],{"className":12772},[1043,1044],[31,12774,12776,12790],{"className":12775},[1048],[31,12777,12779],{"className":12778,"style":2028},[1052],[31,12780,12781,12784],{"style":1056},[31,12782],{"className":12783,"style":1061},[1060],[31,12785,12787],{"className":12786},[1065,1066,1067,1068],[31,12788,205],{"className":12789},[75,1068],[31,12791,1076],{"className":12792},[1075],[31,12794,12796],{"className":12795},[1048],[31,12797,12799],{"className":12798,"style":2050},[1052],[31,12800],{},[11,12802,12803,12804,12863,12864,12921,12922,12991],{},"如果反过来，A 是 ",[31,12805,12807,12832],{"className":12806,"translate":35},[34],[31,12808,12810],{"className":12809},[39],[41,12811,12812],{"xmlns":43},[45,12813,12814,12829],{},[48,12815,12816,12819,12821,12823,12825,12827],{},[51,12817,12818],{},"U",[95,12820,211],{"stretchy":210},[91,12822,1977],{},[95,12824,1210],{"separator":62},[91,12826,152],{},[95,12828,220],{"stretchy":210},[55,12830,12831],{"encoding":57},"U(0,2)",[31,12833,12835],{"className":12834,"ariaHidden":62},[61],[31,12836,12838,12841,12845,12848,12851,12854,12857,12860],{"className":12837},[66],[31,12839],{"className":12840,"style":260},[70],[31,12842,12818],{"className":12843,"style":12844},[75,76],"margin-right:0.109em;",[31,12846,211],{"className":12847},[264],[31,12849,1977],{"className":12850},[75],[31,12852,1210],{"className":12853},[1391],[31,12855],{"className":12856,"style":1467},[119],[31,12858,152],{"className":12859},[75],[31,12861,220],{"className":12862},[289],"、B 是 ",[31,12865,12867,12891],{"className":12866,"translate":35},[34],[31,12868,12870],{"className":12869},[39],[41,12871,12872],{"xmlns":43},[45,12873,12874,12888],{},[48,12875,12876,12878,12880,12882,12884,12886],{},[51,12877,12818],{},[95,12879,211],{"stretchy":210},[91,12881,1977],{},[95,12883,1210],{"separator":62},[91,12885,205],{},[95,12887,220],{"stretchy":210},[55,12889,12890],{"encoding":57},"U(0,1)",[31,12892,12894],{"className":12893,"ariaHidden":62},[61],[31,12895,12897,12900,12903,12906,12909,12912,12915,12918],{"className":12896},[66],[31,12898],{"className":12899,"style":260},[70],[31,12901,12818],{"className":12902,"style":12844},[75,76],[31,12904,211],{"className":12905},[264],[31,12907,1977],{"className":12908},[75],[31,12910,1210],{"className":12911},[1391],[31,12913],{"className":12914,"style":1467},[119],[31,12916,205],{"className":12917},[75],[31,12919,220],{"className":12920},[289],"，那就是平均降 0.5 万，",[31,12923,12925,12942],{"className":12924,"translate":35},[34],[31,12926,12928],{"className":12927},[39],[41,12929,12930],{"xmlns":43},[45,12931,12932,12940],{},[48,12933,12934],{},[1004,12935,12936,12938],{},[51,12937,3117],{},[91,12939,205],{},[55,12941,5220],{"encoding":57},[31,12943,12945],{"className":12944,"ariaHidden":62},[61],[31,12946,12948,12951],{"className":12947},[66],[31,12949],{"className":12950,"style":5230},[70],[31,12952,12954,12957],{"className":12953},[75],[31,12955,3117],{"className":12956,"style":949},[75,76],[31,12958,12960],{"className":12959},[1039],[31,12961,12963,12983],{"className":12962},[1043,1044],[31,12964,12966,12980],{"className":12965},[1048],[31,12967,12969],{"className":12968,"style":2028},[1052],[31,12970,12971,12974],{"style":1056},[31,12972],{"className":12973,"style":1061},[1060],[31,12975,12977],{"className":12976},[1065,1066,1067,1068],[31,12978,205],{"className":12979},[75,1068],[31,12981,1076],{"className":12982},[1075],[31,12984,12986],{"className":12985},[1048],[31,12987,12989],{"className":12988,"style":2050},[1052],[31,12990],{}," 仍然是 0.5——它量的是“搬了多少”，不管涨还是跌。",[11,12993,12994,12995,13064],{},"一般地，一维的 ",[31,12996,12998,13015],{"className":12997,"translate":35},[34],[31,12999,13001],{"className":13000},[39],[41,13002,13003],{"xmlns":43},[45,13004,13005,13013],{},[48,13006,13007],{},[1004,13008,13009,13011],{},[51,13010,3117],{},[51,13012,11],{},[55,13014,4830],{"encoding":57},[31,13016,13018],{"className":13017,"ariaHidden":62},[61],[31,13019,13021,13024],{"className":13020},[66],[31,13022],{"className":13023,"style":1029},[70],[31,13025,13027,13030],{"className":13026},[75],[31,13028,3117],{"className":13029,"style":949},[75,76],[31,13031,13033],{"className":13032},[1039],[31,13034,13036,13056],{"className":13035},[1043,1044],[31,13037,13039,13053],{"className":13038},[1048],[31,13040,13042],{"className":13041,"style":3230},[1052],[31,13043,13044,13047],{"style":1056},[31,13045],{"className":13046,"style":1061},[1060],[31,13048,13050],{"className":13049},[1065,1066,1067,1068],[31,13051,11],{"className":13052},[75,76,1068],[31,13054,1076],{"className":13055},[1075],[31,13057,13059],{"className":13058},[1048],[31,13060,13062],{"className":13061,"style":1083},[1052],[31,13063],{}," 有闭式解：",[31,13066,13068],{"className":13067,"translate":35},[411],[31,13069,13071,13166],{"className":13070,"translate":35},[34],[31,13072,13074],{"className":13073},[39],[41,13075,13076],{"xmlns":43,"display":420},[45,13077,13078,13163],{},[48,13079,13080,13088,13090,13092,13094,13096,13098,13100,13108,13110,13123,13125,13127,13129,13131,13143,13145,13147,13149,13155,13157,13159,13161],{},[12397,13081,13082,13084,13086],{},[51,13083,3117],{},[51,13085,11],{},[51,13087,11],{},[95,13089,211],{"stretchy":210},[51,13091,852],{},[95,13093,1210],{"separator":62},[51,13095,884],{},[95,13097,220],{"stretchy":210},[95,13099,223],{},[12397,13101,13102,13104,13106],{},[95,13103,1409],{},[91,13105,1977],{},[91,13107,205],{},[95,13109,12680],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[12397,13111,13112,13115,13117],{},[51,13113,13114],{},"F",[51,13116,852],{},[48,13118,13119,13121],{},[95,13120,97],{},[91,13122,205],{},[95,13124,211],{"stretchy":210},[51,13126,12271],{},[95,13128,220],{"stretchy":210},[95,13130,97],{},[12397,13132,13133,13135,13137],{},[51,13134,13114],{},[51,13136,884],{},[48,13138,13139,13141],{},[95,13140,97],{},[91,13142,205],{},[95,13144,211],{"stretchy":210},[51,13146,12271],{},[95,13148,220],{"stretchy":210},[1180,13150,13151,13153],{},[95,13152,12680],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,13154,11],{},[1430,13156,1432],{},[51,13158,1435],{},[51,13160,12271],{},[95,13162,1210],{"separator":62},[55,13164,13165],{"encoding":57},"W_p^p(\\mu,\\nu)=\\int_0^1\\bigl|F_\\mu^{-1}(t)-F_\\nu^{-1}(t)\\bigr|^p\\,dt ,",[31,13167,13169,13256,13437],{"className":13168,"ariaHidden":62},[61],[31,13170,13172,13176,13229,13232,13235,13238,13241,13244,13247,13250,13253],{"className":13171},[66],[31,13173],{"className":13174,"style":13175},[70],"height:1.1331em;vertical-align:-0.3831em;",[31,13177,13179,13182],{"className":13178},[75],[31,13180,3117],{"className":13181,"style":949},[75,76],[31,13183,13185],{"className":13184},[1039],[31,13186,13188,13220],{"className":13187},[1043,1044],[31,13189,13191,13217],{"className":13190},[1048],[31,13192,13194,13206],{"className":13193,"style":3987},[1052],[31,13195,13197,13200],{"style":13196},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[31,13198],{"className":13199,"style":1061},[1060],[31,13201,13203],{"className":13202},[1065,1066,1067,1068],[31,13204,11],{"className":13205},[75,76,1068],[31,13207,13208,13211],{"style":1330},[31,13209],{"className":13210,"style":1061},[1060],[31,13212,13214],{"className":13213},[1065,1066,1067,1068],[31,13215,11],{"className":13216},[75,76,1068],[31,13218,1076],{"className":13219},[1075],[31,13221,13223],{"className":13222},[1048],[31,13224,13227],{"className":13225,"style":13226},[1052],"height:0.3831em;",[31,13228],{},[31,13230,211],{"className":13231},[264],[31,13233,852],{"className":13234},[75,76],[31,13236,1210],{"className":13237},[1391],[31,13239],{"className":13240,"style":1467},[119],[31,13242,884],{"className":13243,"style":900},[75,76],[31,13245,220],{"className":13246},[289],[31,13248],{"className":13249,"style":293},[119],[31,13251,223],{"className":13252},[297],[31,13254],{"className":13255,"style":293},[119],[31,13257,13259,13262,13313,13362,13419,13422,13425,13428,13431,13434],{"className":13258},[66],[31,13260],{"className":13261,"style":12440},[70],[31,13263,13265,13268],{"className":13264},[1460],[31,13266,1409],{"className":13267,"style":1688},[1460,1461,1687],[31,13269,13271],{"className":13270},[1039],[31,13272,13274,13305],{"className":13273},[1043,1044],[31,13275,13277,13302],{"className":13276},[1048],[31,13278,13280,13291],{"className":13279,"style":12459},[1052],[31,13281,13282,13285],{"style":1704},[31,13283],{"className":13284,"style":1061},[1060],[31,13286,13288],{"className":13287},[1065,1066,1067,1068],[31,13289,1977],{"className":13290},[75,1068],[31,13292,13293,13296],{"style":12473},[31,13294],{"className":13295,"style":1061},[1060],[31,13297,13299],{"className":13298},[1065,1066,1067,1068],[31,13300,205],{"className":13301},[75,1068],[31,13303,1076],{"className":13304},[1075],[31,13306,13308],{"className":13307},[1048],[31,13309,13311],{"className":13310,"style":1726},[1052],[31,13312],{},[31,13314,13316],{"className":13315},[264],[31,13317,13320],{"className":13318},[1307,13319],"mult",[31,13321,13323,13353],{"className":13322},[1043,1044],[31,13324,13326,13350],{"className":13325},[1048],[31,13327,13330],{"className":13328,"style":13329},[1052],"height:0.85em;",[31,13331,13333,13337],{"style":13332},"top:-2.85em;",[31,13334],{"className":13335,"style":13336},[1060],"height:3.2em;",[31,13338,13340],{"style":13339},"width:0.333em;height:1.2em;",[13341,13342,13346],"svg",{"xmlns":13343,"width":13344,"height":1178,"viewBox":13345},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.333em","0 0 333 1200",[13347,13348],"path",{"d":13349},"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 h43z",[31,13351,1076],{"className":13352},[1075],[31,13354,13356],{"className":13355},[1048],[31,13357,13360],{"className":13358,"style":13359},[1052],"height:0.35em;",[31,13361],{},[31,13363,13365,13368],{"className":13364},[75],[31,13366,13114],{"className":13367,"style":949},[75,76],[31,13369,13371],{"className":13370},[1039],[31,13372,13374,13411],{"className":13373},[1043,1044],[31,13375,13377,13408],{"className":13376},[1048],[31,13378,13380,13391],{"className":13379,"style":1327},[1052],[31,13381,13382,13385],{"style":13196},[31,13383],{"className":13384,"style":1061},[1060],[31,13386,13388],{"className":13387},[1065,1066,1067,1068],[31,13389,852],{"className":13390},[75,76,1068],[31,13392,13393,13396],{"style":1330},[31,13394],{"className":13395,"style":1061},[1060],[31,13397,13399],{"className":13398},[1065,1066,1067,1068],[31,13400,13402,13405],{"className":13401},[75,1068],[31,13403,97],{"className":13404},[75,1068],[31,13406,205],{"className":13407},[75,1068],[31,13409,1076],{"className":13410},[1075],[31,13412,13414],{"className":13413},[1048],[31,13415,13417],{"className":13416,"style":13226},[1052],[31,13418],{},[31,13420,211],{"className":13421},[264],[31,13423,12271],{"className":13424},[75,76],[31,13426,220],{"className":13427},[289],[31,13429],{"className":13430,"style":120},[119],[31,13432,97],{"className":13433},[124],[31,13435],{"className":13436,"style":120},[119],[31,13438,13440,13444,13502,13505,13508,13511,13576,13579,13582,13585],{"className":13439},[66],[31,13441],{"className":13442,"style":13443},[70],"height:1.2543em;vertical-align:-0.35em;",[31,13445,13447,13450],{"className":13446},[75],[31,13448,13114],{"className":13449,"style":949},[75,76],[31,13451,13453],{"className":13452},[1039],[31,13454,13456,13493],{"className":13455},[1043,1044],[31,13457,13459,13490],{"className":13458},[1048],[31,13460,13462,13473],{"className":13461,"style":1327},[1052],[31,13463,13464,13467],{"style":13196},[31,13465],{"className":13466,"style":1061},[1060],[31,13468,13470],{"className":13469},[1065,1066,1067,1068],[31,13471,884],{"className":13472,"style":900},[75,76,1068],[31,13474,13475,13478],{"style":1330},[31,13476],{"className":13477,"style":1061},[1060],[31,13479,13481],{"className":13480},[1065,1066,1067,1068],[31,13482,13484,13487],{"className":13483},[75,1068],[31,13485,97],{"className":13486},[75,1068],[31,13488,205],{"className":13489},[75,1068],[31,13491,1076],{"className":13492},[1075],[31,13494,13496],{"className":13495},[1048],[31,13497,13500],{"className":13498,"style":13499},[1052],"height:0.247em;",[31,13501],{},[31,13503,211],{"className":13504},[264],[31,13506,12271],{"className":13507},[75,76],[31,13509,220],{"className":13510},[289],[31,13512,13514,13551],{"className":13513},[289],[31,13515,13517],{"className":13516},[289],[31,13518,13520],{"className":13519},[1307,13319],[31,13521,13523,13543],{"className":13522},[1043,1044],[31,13524,13526,13540],{"className":13525},[1048],[31,13527,13529],{"className":13528,"style":13329},[1052],[31,13530,13531,13534],{"style":13332},[31,13532],{"className":13533,"style":13336},[1060],[31,13535,13536],{"style":13339},[13341,13537,13538],{"xmlns":13343,"width":13344,"height":1178,"viewBox":13345},[13347,13539],{"d":13349},[31,13541,1076],{"className":13542},[1075],[31,13544,13546],{"className":13545},[1048],[31,13547,13549],{"className":13548,"style":13359},[1052],[31,13550],{},[31,13552,13554],{"className":13553},[1039],[31,13555,13557],{"className":13556},[1043],[31,13558,13560],{"className":13559},[1048],[31,13561,13564],{"className":13562,"style":13563},[1052],"height:0.9043em;",[31,13565,13567,13570],{"style":13566},"top:-3.3029em;margin-right:0.05em;",[31,13568],{"className":13569,"style":1061},[1060],[31,13571,13573],{"className":13572},[1065,1066,1067,1068],[31,13574,11],{"className":13575},[75,76,1068],[31,13577],{"className":13578,"style":1467},[119],[31,13580,1435],{"className":13581},[75,76],[31,13583,12271],{"className":13584},[75,76],[31,13586,1210],{"className":13587},[1391],[11,13589,13590,13591,13642,13643,3641,13671,13768],{},"也就是把两条分位数曲线逐点相减再积分。",[31,13592,13594,13612],{"className":13593,"translate":35},[34],[31,13595,13597],{"className":13596},[39],[41,13598,13599],{"xmlns":43},[45,13600,13601,13609],{},[48,13602,13603,13605,13607],{},[51,13604,11],{},[95,13606,223],{},[91,13608,205],{},[55,13610,13611],{"encoding":57},"p=1",[31,13613,13615,13633],{"className":13614,"ariaHidden":62},[61],[31,13616,13618,13621,13624,13627,13630],{"className":13617},[66],[31,13619],{"className":13620,"style":865},[70],[31,13622,11],{"className":13623},[75,76],[31,13625],{"className":13626,"style":293},[119],[31,13628,223],{"className":13629},[297],[31,13631],{"className":13632,"style":293},[119],[31,13634,13636,13639],{"className":13635},[66],[31,13637],{"className":13638,"style":326},[70],[31,13640,205],{"className":13641},[75]," 时有更漂亮的形式。用上一节的对偶，对绝对连续的 ",[31,13644,13646,13659],{"className":13645,"translate":35},[34],[31,13647,13649],{"className":13648},[39],[41,13650,13651],{"xmlns":43},[45,13652,13653,13657],{},[48,13654,13655],{},[51,13656,8059],{},[55,13658,11503],{"encoding":57},[31,13660,13662],{"className":13661,"ariaHidden":62},[61],[31,13663,13665,13668],{"className":13664},[66],[31,13666],{"className":13667,"style":865},[70],[31,13669,8059],{"className":13670},[75,76],[31,13672,13674,13701],{"className":13673,"translate":35},[34],[31,13675,13677],{"className":13676},[39],[41,13678,13679],{"xmlns":43},[45,13680,13681,13698],{},[48,13682,13683,13685,13692,13694,13696],{},[51,13684,12680],{"mathvariant":494},[1180,13686,13687,13689],{},[51,13688,8059],{},[95,13690,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},"′",[51,13693,12680],{"mathvariant":494},[95,13695,347],{},[91,13697,205],{},[55,13699,13700],{"encoding":57},"|\\varphi'|\\le1",[31,13702,13704,13759],{"className":13703,"ariaHidden":62},[61],[31,13705,13707,13711,13714,13747,13750,13753,13756],{"className":13706},[66],[31,13708],{"className":13709,"style":13710},[70],"height:1.0019em;vertical-align:-0.25em;",[31,13712,12680],{"className":13713},[75],[31,13715,13717,13720],{"className":13716},[75],[31,13718,8059],{"className":13719},[75,76],[31,13721,13723],{"className":13722},[1039],[31,13724,13726],{"className":13725},[1043],[31,13727,13729],{"className":13728},[1048],[31,13730,13733],{"className":13731,"style":13732},[1052],"height:0.7519em;",[31,13734,13735,13738],{"style":3631},[31,13736],{"className":13737,"style":1061},[1060],[31,13739,13741],{"className":13740},[1065,1066,1067,1068],[31,13742,13744],{"className":13743},[75,1068],[31,13745,13691],{"className":13746},[75,1068],[31,13748,12680],{"className":13749},[75],[31,13751],{"className":13752,"style":293},[119],[31,13754,347],{"className":13755},[297],[31,13757],{"className":13758,"style":293},[119],[31,13760,13762,13765],{"className":13761},[66],[31,13763],{"className":13764,"style":326},[70],[31,13766,205],{"className":13767},[75],"）分部积分，",[31,13770,13772],{"className":13771,"translate":35},[411],[31,13773,13775,13909],{"className":13774,"translate":35},[34],[31,13776,13778],{"className":13777},[39],[41,13779,13780],{"xmlns":43,"display":420},[45,13781,13782,13906],{},[48,13783,13784,13786,13788,13790,13792,13794,13796,13798,13800,13802,13804,13806,13808,13810,13812,13814,13816,13824,13826,13828,13830,13832,13840,13842,13844,13846,13848,13850,13852,13854,13856,13858,13864,13866,13868,13870,13872,13878,13880,13882,13884,13886,13892,13894,13896,13898,13900,13902,13904],{},[95,13785,1409],{},[51,13787,8059],{},[1430,13789,1432],{},[51,13791,1435],{},[95,13793,211],{"stretchy":210},[51,13795,852],{},[95,13797,97],{},[51,13799,884],{},[95,13801,220],{"stretchy":210},[95,13803,223],{},[95,13805,1409],{},[51,13807,8059],{},[95,13809,211],{"stretchy":210},[51,13811,53],{},[95,13813,220],{"stretchy":210},[95,13815,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[12397,13817,13818,13820,13822],{},[51,13819,13114],{},[51,13821,852],{},[95,13823,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,13825,211],{"stretchy":210},[51,13827,53],{},[95,13829,220],{"stretchy":210},[95,13831,97],{},[12397,13833,13834,13836,13838],{},[51,13835,13114],{},[51,13837,884],{},[95,13839,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,13841,211],{"stretchy":210},[51,13843,53],{},[95,13845,220],{"stretchy":210},[95,13847,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,13849,1435],{},[51,13851,53],{},[95,13853,223],{},[95,13855,97],{},[95,13857,1409],{},[1180,13859,13860,13862],{},[51,13861,8059],{},[95,13863,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,13865,211],{"stretchy":210},[51,13867,53],{},[95,13869,220],{"stretchy":210},[95,13871,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,13873,13874,13876],{},[51,13875,13114],{},[51,13877,852],{},[95,13879,211],{"stretchy":210},[51,13881,53],{},[95,13883,220],{"stretchy":210},[95,13885,97],{},[1004,13887,13888,13890],{},[51,13889,13114],{},[51,13891,884],{},[95,13893,211],{"stretchy":210},[51,13895,53],{},[95,13897,220],{"stretchy":210},[95,13899,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,13901,1435],{},[51,13903,53],{},[95,13905,1210],{"separator":62},[55,13907,13908],{"encoding":57},"\\int\\varphi\\,d(\\mu-\\nu)=\\int\\varphi(x)\\bigl(F_\\mu'(x)-F_\\nu'(x)\\bigr)dx\n=-\\int\\varphi'(x)\\bigl(F_\\mu(x)-F_\\nu(x)\\bigr)dx,",[31,13910,13912,13948,13969,14072,14162,14285],{"className":13911,"ariaHidden":62},[61],[31,13913,13915,13918,13921,13924,13927,13930,13933,13936,13939,13942,13945],{"className":13914},[66],[31,13916],{"className":13917,"style":10946},[70],[31,13919,1409],{"className":13920,"style":1688},[1460,1461,1687],[31,13922],{"className":13923,"style":1467},[119],[31,13925,8059],{"className":13926},[75,76],[31,13928],{"className":13929,"style":1467},[119],[31,13931,1435],{"className":13932}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",[31,14359,14361,14399],{"className":14360,"translate":35},[34],[31,14362,14364],{"className":14363},[39],[41,14365,14366],{"xmlns":43},[45,14367,14368,14396],{},[48,14369,14370,14372,14374,14380,14382,14388,14390,14392,14394],{},[95,14371,1409],{},[51,14373,12680],{"mathvariant":494},[1004,14375,14376,14378],{},[51,14377,13114],{},[51,14379,852],{},[95,14381,97],{},[1004,14383,14384,14386],{},[51,14385,13114],{},[51,14387,884],{},[51,14389,12680],{"mathvariant":494},[1430,14391,1432],{},[51,14393,1435],{},[51,14395,53],{},[55,14397,14398],{"encoding":57},"\\int|F_\\mu-F_\\nu|\\,dx",[31,14400,14402,14466],{"className":14401,"ariaHidden":62},[61],[31,14403,14405,14408,14411,14414,14417,14457,14460,14463],{"className":14404},[66],[31,14406],{"className":14407,"style":1456},[70],[31,14409,1409],{"className":14410,"style":1463},[1460,1461,1462],[31,14412],{"className":14413,"style":1467},[119],[31,14415,12680],{"className":14416},[75],[31,14418,14420,14423],{"className":14419},[75],[31,14421,13114],{"className":14422,"style":949},[75,76],[31,14424,14426],{"className":14425},[1039],[31,14427,14429,14449],{"className":14428},[1043,1044],[31,14430,14432,14446],{"className":14431},[1048],[31,14433,14435],{"className":14434,"style":3230},[1052],[31,14436,14437,14440],{"style":1056},[31,14438],{"className":14439,"style":1061},[1060],[31,14441,14443],{"className":14442},[1065,1066,1067,1068],[31,14444,852],{"className":14445},[75,76,1068],[31,14447,1076],{"className":14448},[1075],[31,14450,14452],{"className":14451},[1048],[31,14453,14455],{"className":14454,"style":1083},[1052],[31,14456],{},[31,14458],{"className":14459,"style":120},[119],[31,14461,97],{"className":14462},[124],[31,14464],{"className":14465,"style":120},[119],[31,14467,14469,14472,14512,14515,14518,14521],{"className":14468},[66],[31,14470],{"className":14471,"style":260},[70],[31,14473,14475,14478],{"className":14474},[75],[31,14476,13114],{"className":14477,"style":949},[75,76],[31,14479,14481],{"className":14480},[1039],[31,14482,14484,14504],{"className":14483},[1043,1044],[31,14485,14487,14501],{"className":14486},[1048],[31,14488,14490],{"className":14489,"style":3230},[1052],[31,14491,14492,14495],{"style":1056},[31,14493],{"className":14494,"style":1061},[1060],[31,14496,14498],{"className":14497},[1065,1066,1067,1068],[31,14499,884],{"className":14500,"style":900},[75,76,1068],[31,14502,1076],{"className":14503},[1075],[31,14505,14507],{"className":14506},[1048],[31,14508,14510],{"className":14509,"style":2050},[1052],[31,14511],{},[31,14513,12680],{"className":14514},[75],[31,14516],{"className":14517,"style":1467},[119],[31,14519,1435],{"className":14520},[75,76],[31,14522,53],{"className":14523},[75,76],"；取 ",[31,14526,14528,14581],{"className":14527,"translate":35},[34],[31,14529,14531],{"className":14530},[39],[41,14532,14533],{"xmlns":43},[45,14534,14535,14578],{},[48,14536,14537,14543,14545,14547,14560,14562,14568,14570,14576],{},[1180,14538,14539,14541],{},[51,14540,8059],{},[95,14542,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,14544,223],{},[95,14546,97],{},[48,14548,14549,14552,14554,14557],{},[51,14550,14551],{"mathvariant":494},"s",[51,14553,5391],{"mathvariant":494},[51,14555,14556],{"mathvariant":494},"g",[51,14558,14559],{"mathvariant":494},"n",[95,14561,211],{"stretchy":210},[1004,14563,14564,14566],{},[51,14565,13114],{},[51,14567,852],{},[95,14569,97],{},[1004,14571,14572,14574],{},[51,14573,13114],{},[51,14575,884],{},[95,14577,220],{"stretchy":210},[55,14579,14580],{"encoding":57},"\\varphi'=-\\mathrm{sign}(F_\\mu-F_\\nu)",[31,14582,14584,14632,14700],{"className":14583,"ariaHidden":62},[61],[31,14585,14587,14591,14623,14626,14629],{"className":14586},[66],[31,14588],{"className":14589,"style":14590},[70],"height:0.9463em;vertical-align:-0.1944em;",[31,14592,14594,14597],{"className":14593},[75],[31,14595,8059],{"className":14596},[75,76],[31,14598,14600],{"className":14599},[1039],[31,14601,14603],{"className":14602},[1043],[31,14604,14606],{"className":14605},[1048],[31,14607,14609],{"className":14608,"style":13732},[1052],[31,14610,14611,14614],{"style":3631},[31,14612],{"className":14613,"style":1061},[1060],[31,14615,14617],{"className":14616},[1065,1066,1067,1068],[31,14618,14620],{"className":14619},[75,1068],[31,14621,13691],{"className":14622},[75,1068],[31,14624],{"className":14625,"style":293},[119],[31,14627,223],{"className":14628},[297],[31,14630],{"className":14631,"style":293},[119],[31,14633,14635,14638,14641,14648,14651,14691,14694,14697],{"className":14634},[66],[31,14636],{"className":14637,"style":1223},[70],[31,14639,97],{"className":14640},[75],[31,14642,14644],{"className":14643},[75],[31,14645,14647],{"className":14646},[75,2745],"sign",[31,14649,211],{"className":14650},[264],[31,14652,14654,14657],{"className":14653},[75],[31,14655,13114],{"className":14656,"style":949},[75,76],[31,14658,14660],{"className":14659},[1039],[31,14661,14663,14683],{"className":14662},[1043,1044],[31,14664,14666,14680],{"className":14665},[1048],[31,14667,14669],{"className":14668,"style":3230},[1052],[31,14670,14671,14674],{"style":1056},[31,14672],{"className":14673,"style":1061},[1060],[31,14675,14677],{"className":14676},[1065,1066,1067,1068],[31,14678,852],{"className":14679},[75,76,1068],[31,14681,1076],{"className":14682},[1075],[31,14684,14686],{"className":14685},[1048],[31,14687,14689],{"className":14688,"style":1083},[1052],[31,14690],{},[31,14692],{"className":14693,"style":120},[119],[31,14695,97],{"className":14696},[124],[31,14698],{"className":14699,"style":120},[119],[31,14701,14703,14706,14746],{"className":14702},[66],[31,14704],{"className":14705,"style":260},[70],[31,14707,14709,14712],{"className":14708},[75],[31,14710,13114],{"className":14711,"style":949},[75,76],[31,14713,14715],{"className":14714},[1039],[31,14716,14718,14738],{"className":14717},[1043,1044],[31,14719,14721,14735],{"className":14720},[1048],[31,14722,14724],{"className":14723,"style":3230},[1052],[31,14725,14726,14729],{"style":1056},[31,14727],{"className":14728,"style":1061},[1060],[31,14730,14732],{"className":14731},[1065,1066,1067,1068],[31,14733,884],{"className":14734,"style":900},[75,76,1068],[31,14736,1076],{"className":14737},[1075],[31,14739,14741],{"className":14740},[1048],[31,14742,14744],{"className":14743,"style":2050},[1052],[31,14745],{},[31,14747,220],{"className":14748},[289]," 时等号成立。于是",[31,14751,14753],{"className":14752,"translate":35},[411],[31,14754,14756,14832],{"className":14755,"translate":35},[34],[31,14757,14759],{"className":14758},[39],[41,14760,14761],{"xmlns":43,"display":420},[45,14762,14763,14829],{},[48,14764,14765,14771,14773,14775,14777,14779,14781,14783,14791,14793,14799,14801,14803,14805,14807,14813,14815,14817,14819,14821,14823,14825,14827],{},[1004,14766,14767,14769],{},[51,14768,3117],{},[91,14770,205],{},[95,14772,211],{"stretchy":210},[51,14774,852],{},[95,14776,1210],{"separator":62},[51,14778,884],{},[95,14780,220],{"stretchy":210},[95,14782,223],{},[1004,14784,14785,14787],{},[95,14786,1409],{},[51,14788,14790],{"mathvariant":14789},"double-struck","R",[95,14792,12680],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,14794,14795,14797],{},[51,14796,13114],{},[51,14798,852],{},[95,14800,211],{"stretchy":210},[51,14802,53],{},[95,14804,220],{"stretchy":210},[95,14806,97],{},[1004,14808,14809,14811],{},[51,14810,13114],{},[51,14812,884],{},[95,14814,211],{"stretchy":210},[51,14816,53],{},[95,14818,220],{"stretchy":210},[95,14820,12680],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1430,14822,1432],{},[51,14824,1435],{},[51,14826,53],{},[51,14828,495],{"mathvariant":494},[55,14830,14831],{"encoding":57},"W_1(\\mu,\\nu)=\\int_{\\mathbb R}\\bigl|F_\\mu(x)-F_\\nu(x)\\bigr|\\,dx .",[31,14833,14835,14908,15054],{"className":14834,"ariaHidden":62},[61],[31,14836,14838,14841,14881,14884,14887,14890,14893,14896,14899,14902,14905],{"className":14837},[66],[31,14839],{"className":14840,"style":260},[70],[31,14842,14844,14847],{"className":14843},[75],[31,14845,3117],{"className":14846,"style":949},[75,76],[31,14848,14850],{"className":14849},[1039],[31,14851,14853,14873],{"className":14852},[1043,1044],[31,14854,14856,14870],{"className":14855},[1048],[31,14857,14859],{"className":14858,"style":2028},[1052],[31,14860,14861,14864],{"style":1056},[31,14862],{"className":14863,"style":1061},[1060],[31,14865,14867],{"className":14866},[1065,1066,1067,1068],[31,14868,205],{"className":14869},[75,1068],[31,14871,1076],{"className":14872},[1075],[31,14874,14876],{"className":14875},[1048],[31,14877,14879],{"className":14878,"style":2050},[1052],[31,14880],{},[31,14882,211],{"className":14883},[264],[31,14885,852],{"className":14886},[75,76],[31,14888,1210],{"className":14889},[1391],[31,14891],{"className":14892,"style":1467},[119],[31,14894,884],{"className":14895,"style":900},[75,76],[31,14897,220],{"className":14898},[289],[31,14900],{"className":14901,"style":293},[119],[31,14903,223],{"className":14904},[297],[31,14906],{"className":14907,"style":293},[119],[31,14909,14911,14914,14959,14996,15036,15039,15042,15045,15048,15051],{"className":14910},[66],[31,14912],{"className":14913,"style":1616},[70],[31,14915,14917,14920],{"className":14916},[1460],[31,14918,1409],{"className":14919,"style":1688},[1460,1461,1687],[31,14921,14923],{"className":14922},[1039],[31,14924,14926,14951],{"className":14925},[1043,1044],[31,14927,14929,14948],{"className":14928},[1048],[31,14930,14933],{"className":14931,"style":14932},[1052],"height:-0.4297em;",[31,14934,14935,14938],{"style":1704},[31,14936],{"className":14937,"style":1061},[1060],[31,14939,14941],{"className":14940},[1065,1066,1067,1068],[31,14942,14944],{"className":14943},[75,1068],[31,14945,14790],{"className":14946},[75,14947,1068],"mathbb",[31,14949,1076],{"className":14950},[1075],[31,14952,14954],{"className":14953},[1048],[31,14955,14957],{"className":14956,"style":1726},[1052],[31,14958],{},[31,14960,14962],{"className":14961},[264],[31,14963,14965],{"className":14964},[1307,13319],[31,14966,14968,14988],{"className":14967},[1043,1044],[31,14969,14971,14985],{"className":14970},[1048],[31,14972,14974],{"className":14973,"style":13329},[1052],[31,14975,14976,14979],{"style":13332},[31,14977],{"className":14978,"style":13336},[1060],[31,14980,14981],{"style":13339},[13341,14982,14983],{"xmlns":13343,"width":13344,"height":1178,"viewBox":13345},[13347,14984],{"d":13349},[31,14986,1076],{"className":14987},[1075],[31,14989,14991],{"className":14990},[1048],[31,14992,14994],{"className":14993,"style":13359},[1052],[31,14995],{},[31,14997,14999,15002],{"className":14998},[75],[31,15000,13114],{"className":15001,"style":949},[75,76],[31,15003,15005],{"className":15004},[1039],[31,15006,15008,15028],{"className":15007},[1043,1044],[31,15009,15011,15025],{"className":15010},[1048],[31,15012,15014],{"className":15013,"style":3230},[1052],[31,15015,15016,15019],{"style":1056},[31,15017],{"className":15018,"style":1061},[1060],[31,15020,15022],{"className":15021},[1065,1066,1067,1068],[31,15023,852],{"className":15024},[75,76,1068],[31,15026,1076],{"className":15027},[1075],[31,15029,15031],{"className":15030},[1048],[31,15032,15034],{"className":15033,"style":1083},[1052],[31,15035],{},[31,15037,211],{"className":15038},[264],[31,15040,53],{"className":15041},[75,76],[31,15043,220],{"className":15044},[289],[31,15046],{"className":15047,"style":120},[119],[31,15049,97],{"className":15050},[124],[31,15052],{"className":15053,"style":120},[119],[31,15055,15057,15060,15100,15103,15106,15109,15146,15149,15152,15155],{"className":15056},[66],[31,15058],{"className":15059,"style":4988},[70],[31,15061,15063,15066],{"className":15062},[75],[31,15064,13114],{"className":15065,"style":949},[75,76],[31,15067,15069],{"className":15068},[1039],[31,15070,15072,15092],{"className":15071},[1043,1044],[31,15073,15075,15089],{"className":15074},[1048],[31,15076,15078],{"className":15077,"style":3230},[1052],[31,15079,15080,15083],{"style":1056},[31,15081],{"className":15082,"style":1061},[1060],[31,15084,15086],{"className":15085},[1065,1066,1067,1068],[31,15087,884],{"className":15088,"style":900},[75,76,1068],[31,15090,1076],{"className":15091},[1075],[31,15093,15095],{"className":15094},[1048],[31,15096,15098],{"className":15097,"style":2050},[1052],[31,15099],{},[31,15101,211],{"className":15102},[264],[31,15104,53],{"className":15105},[75,76],[31,15107,220],{"className":15108},[289],[31,15110,15112],{"className":15111},[289],[31,15113,15115],{"className":15114},[1307,13319],[31,15116,15118,15138],{"className":15117},[1043,1044],[31,15119,15121,15135],{"className":15120},[1048],[31,15122,15124],{"className":15123,"style":13329},[1052],[31,15125,15126,15129],{"st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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 ",[31,15162,15164,15188],{"className":15163,"translate":35},[34],[31,15165,15167],{"className":15166},[39],[41,15168,15169],{"xmlns":43},[45,15170,15171,15185],{},[48,15172,15173,15176,15178,15180,15182],{},[95,15174,15175],{"stretchy":210},"[",[91,15177,1977],{},[95,15179,1210],{"separator":62},[91,15181,205],{},[95,15183,15184],{"stretchy":210},"]",[55,15186,15187],{"encoding":57},"[0,1]",[31,15189,15191],{"className":15190,"ariaHidden":62},[61],[31,15192,15194,15197,15200,15203,15206,15209,15212],{"className":15193},[66],[31,15195],{"className":15196,"style":260},[70],[31,15198,15175],{"className":15199},[264],[31,15201,1977],{"className":15202},[75],[31,15204,1210],{"className":15205},[1391],[31,15207],{"className":15208,"style":1467},[119],[31,15210,205],{"className":15211},[75],[31,15213,15184],{"className":15214},[289]," 上 ",[31,15217,15219,15243],{"className":15218,"translate":35},[34],[31,15220,15222],{"className":15221},[39],[41,15223,15224],{"xmlns":43},[45,15225,15226,15240],{},[48,15227,15228,15230,15232,15234,15236,15238],{},[51,15229,13114],{},[95,15231,211],{"stretchy":210},[51,15233,53],{},[95,15235,220],{"stretchy":210},[95,15237,223],{},[51,15239,53],{},[55,15241,15242],{"encoding":57},"F(x)=x",[31,15244,15246,15273],{"className":15245,"ariaHidden":62},[61],[31,15247,15249,15252,15255,15258,15261,15264,15267,15270],{"className":15248},[66],[31,15250],{"className":15251,"style":260},[70],[31,15253,13114],{"className":15254,"style":949},[75,76],[31,15256,211],{"className":15257},[264],[31,15259,53],{"className":15260},[75,76],[31,15262,220],{"className":15263},[289],[31,15265],{"className":15266,"style":293},[119],[31,15268,223],{"className":15269},[297],[31,15271],{"className":15272,"style":293},[119],[31,15274,15276,15279],{"className":15275},[66],[31,15277],{"className":15278,"style":71},[70],[31,15280,53],{"className":15281},[75,76],"、",[31,15284,15286,15315],{"className":15285,"translate":35},[34],[31,15287,15289],{"className":15288},[39],[41,15290,15291],{"xmlns":43},[45,15292,15293,15312],{},[48,15294,15295,15298,15300,15302,15304,15306,15308,15310],{},[51,15296,15297],{},"G",[95,15299,211],{"stretchy":210},[51,15301,53],{},[95,15303,220],{"stretchy":210},[95,15305,223],{},[51,15307,53],{},[51,15309,3768],{"mathvariant":494},[91,15311,152],{},[55,15313,15314],{"encoding":57},"G(x)=x\u002F2",[31,15316,15318,15345],{"className":15317,"ariaHidden":62},[61],[31,15319,15321,15324,15327,15330,15333,15336,15339,15342],{"className":15320},[66],[31,15322],{"className":15323,"style":260},[70],[31,15325,15297],{"className":15326},[75,76],[31,15328,211],{"className":15329},[264],[31,15331,53],{"className":15332},[75,76],[31,15334,220],{"className":15335},[289],[31,15337],{"className":15338,"style":293},[119],[31,15340,223],{"className":15341},[297],[31,15343],{"className":15344,"style":293},[119],[31,15346,15348,15351,15354],{"className":15347},[66],[31,15349],{"className":15350,"style":260},[70],[31,15352,53],{"className":15353},[75,76],[31,15355,15357],{"className":15356},[75],"\u002F2","，差是 ",[31,15360,15362,15380],{"className":15361,"translate":35},[34],[31,15363,15365],{"className":15364},[39],[41,15366,15367],{"xmlns":43},[45,15368,15369,15377],{},[48,15370,15371,15373,15375],{},[51,15372,53],{},[51,15374,3768],{"mathvariant":494},[91,15376,152],{},[55,15378,15379],{"encoding":57},"x\u002F2",[31,15381,15383],{"className":15382,"ariaHidden":62},[61],[31,15384,15386,15389,15392],{"className":15385},[66],[31,15387],{"className":15388,"style":260},[70],[31,15390,53],{"className":15391},[75,76],[31,15393,15357],{"className":15394},[75],"，积出 ",[31,15397,15399,15420],{"className":15398,"translate":35},[34],[31,15400,15402],{"className":15401},[39],[41,15403,15404],{"xmlns":43},[45,15405,15406,15417],{},[48,15407,15408],{},[2071,15409,15410],{"scriptlevel":1977,"displaystyle":210},[2074,15411,15412,15414],{},[91,15413,205],{},[91,15415,15416],{},"4",[55,15418,15419],{"encoding":57},"\\tfrac14",[31,15421,15423],{"className":15422,"ariaHidden":62},[61],[31,15424,15426,15429],{"className":15425},[66],[31,15427],{"className":15428,"style":2132},[70],[31,15430,15432,15435,15494],{"className":15431},[75],[31,15433],{"className":15434},[264,2139],[31,15436,15438],{"className":15437},[2074],[31,15439,15441,15486],{"className":15440},[1043,1044],[31,15442,15444,15483],{"className":15443},[1048],[31,15445,15447,15461,15469],{"className":15446,"style":2152},[1052],[31,15448,15449,15452],{"style":2155},[31,15450],{"className":15451,"style":1637},[1060],[31,15453,15455],{"className":15454},[1065,1066,1067,1068],[31,15456,15458],{"className":15457},[75,1068],[31,15459,15416],{"className":15460},[75,1068],[31,15462,15463,15466],{"style":2170},[31,15464],{"className":15465,"style":1637},[1060],[31,15467],{"className":15468,"style":2178},[2177],[31,15470,15471,15474],{"style":2181},[31,15472],{"className":15473,"style":1637},[1060],[31,15475,15477],{"className":15476},[1065,1066,1067,1068],[31,15478,15480],{"className":15479},[75,1068],[31,15481,205],{"className":15482},[75,1068],[31,15484,1076],{"className":15485},[1075],[31,15487,15489],{"className":15488},[1048],[31,15490,15492],{"className":15491,"style":2203},[1052],[31,15493],{},[31,15495],{"className":15496},[289,2139],"；在 ",[31,15499,15501,15523],{"className":15500,"translate":35},[34],[31,15502,15504],{"className":15503},[39],[41,15505,15506],{"xmlns":43},[45,15507,15508,15520],{},[48,15509,15510,15512,15514,15516,15518],{},[95,15511,15175],{"stretchy":210},[91,15513,205],{},[95,15515,1210],{"separator":62},[91,15517,152],{},[95,15519,15184],{"stretchy":210},[55,15521,15522],{"encoding":57},"[1,2]",[31,15524,15526],{"className":15525,"ariaHidden":62},[61],[31,15527,15529,15532,15535,15538,15541,15544,15547],{"className":15528},[66],[31,15530],{"className":15531,"style":260},[70],[31,15533,15175],{"className":15534},[264],[31,15536,205],{"className":15537},[75],[31,15539,1210],{"className":15540},[1391],[31,15542],{"className":15543,"style":1467},[119],[31,15545,152],{"className":15546},[75],[31,15548,15184],{"className":15549},[289],[31,15551,15553,15577],{"className":15552,"translate":35},[34],[31,15554,15556],{"className":15555},[39],[41,15557,15558],{"xmlns":43},[45,15559,15560,15574],{},[48,15561,15562,15564,15566,15568,15570,15572],{},[51,15563,13114],{},[95,15565,211],{"stretchy":210},[51,15567,53],{},[95,15569,220],{"stretchy":210},[95,15571,223],{},[91,15573,205],{},[55,15575,15576],{"encoding":57},"F(x)=1",[31,15578,15580,15607],{"className":15579,"ariaHidden":62},[61],[31,15581,15583,15586,15589,15592,15595,15598,15601,15604],{"className":15582},[66],[31,15584],{"className":15585,"style":260},[70],[31,15587,13114],{"className":15588,"style":949},[75,76],[31,15590,211],{"className":15591},[264],[31,15593,53],{"className":15594},[75,76],[31,15596,220],{"className":15597},[289],[31,15599],{"className":15600,"style":293},[119],[31,15602,223],{"className":15603},[297],[31,15605],{"className":15606,"style":293},[119],[31,15608,15610,15613],{"className":15609},[66],[31,15611],{"className":15612,"style":326},[70],[31,15614,205],{"className":15615},[75],[31,15617,15619,15641],{"className":15618,"translate":35},[34],[31,15620,15622],{"className":15621},[39],[41,15623,15624],{"xmlns":43},[45,15625,15626,15638],{},[48,15627,15628,15630,15632,15634,15636],{},[91,15629,205],{},[95,15631,97],{},[51,15633,53],{},[51,15635,3768],{"mathvariant":494},[91,15637,152],{},[55,15639,15640],{"encoding":57},"1-x\u002F2",[31,15642,15644,15662],{"className":15643,"ariaHidden":62},[61],[31,15645,15647,15650,15653,15656,15659],{"className":15646},[66],[31,15648],{"className":15649,"style":112},[70],[31,15651,205],{"className":15652},[75],[31,15654],{"className":15655,"style":120},[119],[31,15657,97],{"className":15658},[124],[31,15660],{"className":15661,"style":120},[119],[31,15663,15665,15668,15671],{"className":15664},[66],[31,15666],{"className":15667,"style":260},[70],[31,15669,53],{"className":15670},[75,76],[31,15672,15357],{"className":15673},[75],"，也积出 ",[31,15676,15678,15697],{"className":15677,"translate":35},[34],[31,15679,15681],{"className":15680},[39],[41,15682,15683],{"xmlns":43},[45,15684,15685,15695],{},[48,15686,15687],{},[2071,15688,15689],{"scriptlevel":1977,"displaystyle":210},[2074,15690,15691,15693],{},[91,15692,205],{},[91,15694,15416],{},[55,15696,15419],{"encoding":57},[31,15698,15700],{"className":15699,"ariaHidden":62},[61],[31,15701,15703,15706],{"className":15702},[66],[31,15704],{"className":15705,"style":2132},[70],[31,15707,15709,15712,15771],{"className":15708},[75],[31,15710],{"className":15711},[264,2139],[31,15713,15715],{"className":15714},[2074],[31,15716,15718,15763],{"className":15717},[1043,1044],[31,15719,15721,15760],{"className":15720},[1048],[31,15722,15724,15738,15746],{"className":15723,"style":2152},[1052],[31,15725,15726,15729],{"style":2155},[31,15727],{"className":15728,"style":1637},[1060],[31,15730,15732],{"className":15731},[1065,1066,1067,1068],[31,15733,15735],{"className":15734},[75,1068],[31,15736,15416],{"className":15737},[75,1068],[31,15739,15740,15743],{"style":2170},[31,15741],{"className":15742,"style":1637},[1060],[31,15744],{"className":15745,"style":2178},[2177],[31,15747,15748,15751],{"style":2181},[31,15749],{"className":15750,"style":1637},[1060],[31,15752,15754],{"className":15753},[1065,1066,1067,1068],[31,15755,15757],{"className":15756},[75,1068],[31,15758,205],{"className":15759},[75,1068],[31,15761,1076],{"className":15762},[1075],[31,15764,15766],{"className":15765},[1048],[31,15767,15769],{"className":15768,"style":2203},[1052],[31,15770],{},[31,15772],{"className":15773},[289,2139],"。合计 ",[31,15776,15778,15798],{"className":15777,"translate":35},[34],[31,15779,15781],{"className":15780},[39],[41,15782,15783],{"xmlns":43},[45,15784,15785,15795],{},[48,15786,15787],{},[2071,15788,15789],{"scriptlevel":1977,"displaystyle":210},[2074,15790,15791,15793],{},[91,15792,205],{},[91,15794,152],{},[55,15796,15797],{"encoding":57},"\\tfrac12",[31,15799,15801],{"className":15800,"ariaHidden":62},[61],[31,15802,15804,15807],{"className":15803},[66],[31,15805],{"className":15806,"style":2132},[70],[31,15808,15810,15813,15872],{"className":15809},[75],[31,15811],{"className":15812},[264,2139],[31,15814,15816],{"className":15815},[2074],[31,15817,15819,15864],{"className":15818},[1043,1044],[31,15820,15822,15861],{"className":15821},[1048],[31,15823,15825,15839,15847],{"className":15824,"style":2152},[1052],[31,15826,15827,15830],{"style":2155},[31,15828],{"className":15829,"style":1637},[1060],[31,15831,15833],{"className":15832},[1065,1066,1067,1068],[31,15834,15836],{"className":15835},[75,1068],[31,15837,152],{"className":15838},[75,1068],[31,15840,15841,15844],{"style":2170},[31,15842],{"className":15843,"style":1637},[1060],[31,15845],{"className":15846,"style":2178},[2177],[31,15848,15849,15852],{"style":2181},[31,15850],{"className":15851,"style":1637},[1060],[31,15853,15855],{"className":15854},[1065,1066,1067,1068],[31,15856,15858],{"className":15857},[75,1068],[31,15859,205],{"className":15860},[75,1068],[31,15862,1076],{"className":15863},[1075],[31,15865,15867],{"className":15866},[1048],[31,15868,15870],{"className":15869,"style":2203},[1052],[31,15871],{},[31,15873],{"className":15874},[289,2139],"，和分位数算法的结果一致。",[11,15877,15878,15879,16013,16014,16178,16179,16359],{},"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 ",[31,15880,15882,15908],{"className":15881,"translate":35},[34],[31,15883,15885],{"className":15884},[39],[41,15886,15887],{"xmlns":43},[45,15888,15889,15905],{},[48,15890,15891,15897,15899],{},[1004,15892,15893,15895],{},[51,15894,53],{},[91,15896,205],{},[95,15898,9758],{},[1004,15900,15901,15903],{},[51,15902,53],{},[91,15904,152],{},[55,15906,15907],{"encoding":57},"x_1\u003Cx_2",[31,15909,15911,15967],{"className":15910,"ariaHidden":62},[61],[31,15912,15914,15918,15958,15961,15964],{"className":15913},[66],[31,15915],{"className":15916,"style":15917},[70],"height:0.6891em;vertical-align:-0.15em;",[31,15919,15921,15924],{"className":15920},[75],[31,15922,53],{"className":15923},[75,76],[31,15925,15927],{"className":15926},[1039],[31,15928,15930,15950],{"className":15929},[1043,1044],[31,15931,15933,15947],{"className":15932},[1048],[31,15934,15936],{"className":15935,"style":2028},[1052],[31,15937,15938,15941],{"style":2807},[31,15939],{"className":15940,"style":1061},[1060],[31,15942,15944],{"className":15943},[1065,1066,1067,1068],[31,15945,205],{"className":15946},[75,1068],[31,15948,1076],{"className":15949},[1075],[31,15951,15953],{"className":15952},[1048],[31,15954,15956],{"className":15955,"style":2050},[1052],[31,15957],{},[31,15959],{"className":15960,"style":293},[119],[31,15962,9758],{"className":15963},[297],[31,15965],{"className":15966,"style":293},[119],[31,15968,15970,15973],{"className":15969},[66],[31,15971],{"className":15972,"style":7090},[70],[31,15974,15976,15979],{"className":15975},[75],[31,15977,53],{"className":15978},[75,76],[31,15980,15982],{"className":15981},[1039],[31,15983,15985,16005],{"className":15984},[1043,1044],[31,15986,15988,16002],{"className":15987},[1048],[31,15989,15991],{"className":15990,"style":2028},[1052],[31,15992,15993,15996],{"style":2807},[31,15994],{"className":15995,"style":1061},[1060],[31,15997,15999],{"className":15998},[1065,1066,1067,1068],[31,16000,152],{"className":16001},[75,1068],[31,16003,1076],{"className":16004},[1075],[31,16006,16008],{"className":16007},[1048],[31,16009,16011],{"className":16010,"style":2050},[1052],[31,16012],{}," 却 ",[31,16015,16017,16056],{"className":16016,"translate":35},[34],[31,16018,16020],{"className":16019},[39],[41,16021,16022],{"xmlns":43},[45,16023,16024,16053],{},[48,16025,16026,16028,16030,16036,16038,16041,16043,16045,16051],{},[51,16027,919],{},[95,16029,211],{"stretchy":210},[1004,16031,16032,16034],{},[51,16033,53],{},[91,16035,205],{},[95,16037,220],{"stretchy":210},[95,16039,16040],{},">",[51,16042,919],{},[95,16044,211],{"stretchy":210},[1004,16046,16047,16049],{},[51,16048,53],{},[91,16050,152],{},[95,16052,220],{"stretchy":210},[55,16054,16055],{"encoding":57},"T(x_1)>T(x_2)",[31,16057,16059,16123],{"className":16058,"ariaHidden":62},[61],[31,16060,16062,16065,16068,16071,16111,16114,16117,16120],{"className":16061},[66],[31,16063],{"className":16064,"style":260},[70],[31,16066,919],{"className":16067,"style":949},[75,76],[31,16069,211],{"className":16070},[264],[31,16072,16074,16077],{"className":16073},[75],[31,16075,53],{"className":16076},[75,76],[31,16078,16080],{"className":16079},[1039],[31,16081,16083,16103],{"className":16082},[1043,1044],[31,16084,16086,16100],{"className":16085},[1048],[31,16087,16089],{"className":16088,"style":2028},[1052],[31,16090,16091,16094],{"style":2807},[31,16092],{"className":16093,"style":1061},[1060],[31,16095,16097],{"className":16096},[1065,1066,1067,1068],[31,16098,205],{"className":16099},[75,1068],[31,16101,1076],{"className":16102},[1075],[31,16104,16106],{"className":16105},[1048],[31,16107,16109],{"className":16108,"style":2050},[1052],[31,16110],{},[31,16112,220],{"className":16113},[289],[31,16115],{"className":16116,"style":293},[119],[31,16118,16040],{"className":16119},[297],[31,16121],{"className":16122,"style":293},[119],[31,16124,16126,16129,16132,16135,16175],{"className":16125},[66],[31,16127],{"className":16128,"style":260},[70],[31,16130,919],{"className":16131,"style":949},[75,76],[31,16133,211],{"className":16134},[264],[31,16136,16138,16141],{"className":16137},[75],[31,16139,53],{"className":16140},[75,76],[31,16142,16144],{"className":16143},[1039],[31,16145,16147,16167],{"className":16146},[1043,1044],[31,16148,16150,16164],{"className":16149},[1048],[31,16151,16153],{"className":16152,"style":2028},[1052],[31,16154,16155,16158],{"style":2807},[31,16156],{"className":16157,"style":1061},[1060],[31,16159,16161],{"className":16160},[1065,1066,1067,1068],[31,16162,152],{"className":16163},[75,1068],[31,16165,1076],{"className":16166},[1075],[31,16168,16170],{"className":16169},[1048],[31,16171,16173],{"className":16172,"style":2050},[1052],[31,16174],{},[31,16176,220],{"className":16177},[289],"，否则交换一下就更省。所以 ",[31,16180,16182,16219],{"className":16181,"translate":35},[34],[31,16183,16185],{"className":16184},[39],[41,16186,16187],{"xmlns":43},[45,16188,16189,16216],{},[48,16190,16191,16193,16195,16207,16210],{},[51,16192,919],{},[95,16194,223],{},[12397,16196,16197,16199,16201],{},[51,16198,13114],{},[51,16200,884],{},[48,16202,16203,16205],{},[95,16204,97],{},[91,16206,205],{},[95,16208,16209],{},"∘",[1004,16211,16212,16214],{},[51,16213,13114],{},[51,16215,852],{},[55,16217,16218],{"encoding":57},"T=F_\\nu^{-1}\\circ F_\\mu",[31,16220,16222,16240,16313],{"className":16221,"ariaHidden":62},[61],[31,16223,16225,16228,16231,16234,16237],{"className":16224},[66],[31,16226],{"className":16227,"style":945},[70],[31,16229,919],{"className":16230,"style":949},[75,76],[31,16232],{"className":16233,"style":293},[119],[31,16235,223],{"className":16236},[297],[31,16238],{"className":16239,"style":293},[119],[31,16241,16243,16247,16304,16307,16310],{"className":16242},[66],[31,16244],{"className":16245,"style":16246},[70],"height:1.0611em;vertical-align:-0.247em;",[31,16248,16250,16253],{"className":16249},[75],[31,16251,13114],{"className":16252,"style":949},[75,76],[31,16254,16256],{"className":16255},[1039],[31,16257,16259,16296],{"className":16258},[1043,1044],[31,16260,16262,16293],{"className":16261},[1048],[31,16263,16265,16276],{"className":16264,"style":5135},[1052],[31,16266,16267,16270],{"style":13196},[31,16268],{"className":16269,"style":1061},[1060],[31,16271,16273],{"className":16272},[1065,1066,1067,1068],[31,16274,884],{"className":16275,"style":900},[75,76,1068],[31,16277,16278,16281],{"style":3631},[31,16279],{"className":16280,"style":1061},[1060],[31,16282,16284],{"className":16283},[1065,1066,1067,1068],[31,16285,16287,16290],{"className":16286},[75,1068],[31,16288,97],{"className":16289},[75,1068],[31,16291,205],{"className":16292},[75,1068],[31,16294,1076],{"className":16295},[1075],[31,16297,16299],{"className":16298},[1048],[31,16300,16302],{"className":16301,"style":13499},[1052],[31,16303],{},[31,16305],{"className":16306,"style":120},[119],[31,16308,16209],{"className":16309},[124],[31,16311],{"className":16312,"style":120},[119],[31,16314,16316,16319],{"className":16315},[66],[31,16317],{"className":16318,"style":1029},[70],[31,16320,16322,16325],{"className":16321},[75],[31,16323,13114],{"className":16324,"style":949},[75,76],[31,16326,16328],{"className":16327},[1039],[31,16329,16331,16351],{"className":16330},[1043,1044],[31,16332,16334,16348],{"className":16333},[1048],[31,16335,16337],{"className":16336,"style":3230},[1052],[31,16338,16339,16342],{"style":1056},[31,16340],{"className":16341,"style":1061},[1060],[31,16343,16345],{"className":16344},[1065,1066,1067,1068],[31,16346,852],{"className":16347},[75,76,1068],[31,16349,1076],{"className":16350},[1075],[31,16352,16354],{"className":16353},[1048],[31,16355,16357],{"className":16356,"style":1083},[1052],[31,16358],{}," 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。",[11,16361,16362],{},"高斯之间有一个著名的闭式解：",[31,16364,16366],{"className":16365,"translate":35},[411],[31,16367,16369,16541],{"className":16368,"translate":35},[34],[31,16370,16372],{"className":16371},[39],[41,16373,16374],{"xmlns":43,"display":420},[45,16375,16376,16538],{},[48,16377,16378,16386,16388,16390,16392,16399,16401,16408,16410,16412,16414,16416,16422,16424,16430,16432,16434,16436,16438,16444,16446,16452,16458,16460,16466,16468,16474,16476,16482,16484,16486,16488,16502,16508,16522,16534,16536],{},[12397,16379,16380,16382,16384],{},[51,16381,3117],{},[91,16383,152],{},[91,16385,152],{},[95,16387,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,16389,4935],{"mathvariant":925},[95,16391,211],{"stretchy":210},[1004,16393,16394,16397],{},[51,16395,16396],{},"m",[91,16398,205],{},[95,16400,1210],{"separator":62},[1004,16402,16403,16406],{},[51,16404,16405],{"mathvariant":494},"Σ",[91,16407,205],{},[95,16409,220],{"stretchy":210},[95,16411,1210],{"separator":62},[51,16413,4935],{"mathvariant":925},[95,16415,211],{"stretchy":210},[1004,16417,16418,16420],{},[51,16419,16396],{},[91,16421,152],{},[95,16423,1210],{"separator":62},[1004,16425,16426,16428],{},[51,16427,16405],{"mathvariant":494},[91,16429,152],{},[95,16431,220],{"stretchy":210},[95,16433,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,16435,223],{},[51,16437,4572],{"mathvariant":494},[1004,16439,16440,16442],{},[51,16441,16396],{},[91,16443,205],{},[95,16445,97],{},[1004,16447,16448,16450],{},[51,16449,16396],{},[91,16451,152],{},[1180,16453,16454,16456],{},[51,16455,4572],{"mathvariant":494},[91,16457,152],{},[95,16459,434],{},[48,16461,16462,16464],{},[51,16463,12271],{"mathvariant":494},[51,16465,2549],{"mathvariant":494},[95,16467,211],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[1004,16469,16470,16472],{},[51,16471,16405],{"mathvariant":494},[91,16473,205],{},[95,16475,434],{},[1004,16477,16478,16480],{},[51,16479,16405],{"mathvariant":494},[91,16481,152],{},[95,16483,97],{},[91,16485,152],{},[95,16487,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[12397,16489,16490,16492,16494],{},[51,16491,16405],{"mathvariant":494},[91,16493,205],{},[48,16495,16496,16498,16500],{},[91,16497,205],{},[51,16499,3768],{"mathvariant":494},[91,16501,152],{},[1004,16503,16504,16506],{},[51,16505,16405],{"mathvariant":494},[91,16507,152],{},[12397,16509,16510,16512,16514],{},[51,16511,16405],{"mathvariant":494},[91,16513,205],{},[48,16515,16516,16518,16520],{},[91,16517,205],{},[51,16519,3768],{"mathvariant":494},[91,16521,152],{},[1180,16523,16524,16526],{},[95,16525,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[48,16527,16528,16530,16532],{},[91,16529,205],{},[51,16531,3768],{"mathvariant":494},[91,16533,152],{},[95,16535,220],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[51,16537,495],{"mathvariant":494},[55,16539,16540],{"encoding":57},"W_2^2\\bigl(\\mathcal N(m_1,\\Sigma_1),\\mathcal N(m_2,\\Sigma_2)\\bigr)\n=\\|m_1-m_2\\|^2+\\mathrm{tr}\\Bigl(\\Sigma_1+\\Sigma_2-2\\bigl(\\Sigma_1^{1\u002F2}\\Sigma_2\\Sigma_1^{1\u002F2}\\bigr)^{1\u002F2}\\Bigr).",[31,16542,16544,16818,16876,16961,17029,17084],{"className":16543,"ariaHidden":62},[61],[31,16545,16547,16550,16601,16607,16610,16613,16653,16656,16659,16699,16702,16705,16708,16711,16714,16754,16757,16760,16800,16803,16809,16812,16815],{"className":16546},[66],[31,16548],{"className":16549,"style":1297},[70],[31,16551,16553,16556],{"className":16552},[75],[31,16554,3117],{"className":16555,"style":949},[75,76],[31,16557,16559],{"className":16558},[1039],[31,16560,16562,16593],{"className":16561},[1043,1044],[31,16563,16565,16590],{"className":16564},[1048],[31,16566,16568,16579],{"className":16567,"style":1327},[1052],[31,16569,16570,16573],{"style":13196},[31,16571],{"className":16572,"style":1061},[1060],[31,16574,16576],{"className":16575},[1065,1066,1067,1068],[31,16577,152],{"className":16578},[75,1068],[31,16580,16581,16584],{"style":1330},[31,16582],{"className":16583,"style":1061},[1060],[31,16585,16587],{"className":16586},[1065,1066,1067,1068],[31,16588,152],{"className":16589},[75,1068],[31,16591,1076],{"className":16592},[1075],[31,16594,16596],{"className":16595},[1048],[31,16597,16599],{"className":16598,"style":13499},[1052],[31,16600],{},[31,16602,16604],{"className":16603},[264],[31,16605,211],{"className":16606},[1307,1308],[31,16608,4935],{"className":16609,"style":5004},[75,968],[31,16611,211],{"className":16612},[264],[31,16614,16616,16619],{"className":16615},[75],[31,16617,16396],{"className":16618},[75,76],[31,16620,16622],{"className":16621},[1039],[31,16623,16625,16645],{"className":16624},[1043,1044],[31,16626,16628,16642],{"className":16627},[1048],[31,16629,16631],{"className":16630,"style":2028},[1052],[31,16632,16633,16636],{"style":2807},[31,16634],{"className":16635,"style":1061},[1060],[31,16637,16639],{"className":16638},[1065,1066,1067,1068],[31,16640,205],{"className":16641},[75,1068],[31,16643,1076],{"className":16644},[1075],[31,16646,16648],{"className":16647},[1048],[31,16649,16651],{"className":16650,"style":2050},[1052],[31,16652],{},[31,16654,1210],{"className":16655},[1391],[31,16657],{"className":16658,"style":1467},[119],[31,16660,16662,16665],{"className":16661},[75],[31,16663,16405],{"className":16664},[75],[31,16666,16668],{"className":16667},[1039],[31,16669,16671,16691],{"className":16670},[1043,1044],[31,16672,16674,16688],{"className":16673},[1048],[31,16675,16677],{"className":16676,"style":2028},[1052],[31,16678,16679,16682],{"style":2807},[31,16680],{"className":16681,"style":1061},[1060],[31,16683,16685],{"className":16684},[1065,1066,1067,1068],[31,16686,205],{"className":16687},[75,1068],[31,16689,1076],{"className":16690},[1075],[31,16692,16694],{"className":16693},[1048],[31,16695,16697],{"className":16696,"style":2050},[1052],[31,16698],{},[31,16700,220],{"className":16701},[289],[31,16703,1210],{"className":16704},[1391],[31,16706],{"className":16707,"style":1467},[119],[31,16709,4935],{"className":16710,"style":5004},[75,968],[31,16712,211],{"className":16713},[264],[31,16715,16717,16720],{"className":16716},[75],[31,16718,16396],{"className":16719},[75,76],[31,16721,16723],{"className":16722},[1039],[31,16724,16726,16746],{"className":16725},[1043,1044],[31,16727,16729,16743],{"className":16728},[1048],[31,16730,16732],{"className":16731,"style":2028},[1052],[31,16733,16734,16737],{"style":2807},[31,16735],{"className":16736,"style":1061},[1060],[31,16738,16740],{"className":16739},[1065,1066,1067,1068],[31,16741,152],{"className":16742},[75,1068],[31,16744,1076],{"className":16745},[1075],[31,16747,16749],{"className":16748},[1048],[31,16750,16752],{"className":16751,"style":2050},[1052],[31,16753],{},[31,16755,1210],{"className":16756},[1391],[31,16758],{"className":16759,"style":1467},[119],[31,16761,16763,16766],{"className":16762},[75],[31,16764,16405],{"className":16765},[75],[31,16767,16769],{"className":16768},[1039],[31,16770,16772,16792],{"className":16771},[1043,1044],[31,16773,16775,16789],{"className":16774},[1048],[31,16776,16778],{"className":16777,"style":2028},[1052],[31,16779,16780,16783],{"style":2807},[31,16781],{"className":16782,"style":1061},[1060],[31,16784,16786],{"className":16785},[1065,1066,1067,1068],[31,16787,152],{"className":16788},[75,1068],[31,16790,1076],{"className":16791},[1075],[31,16793,16795],{"className":16794},[1048],[31,16796,16798],{"className":16797,"style":2050},[1052],[31,16799],{},[31,16801,220],{"className":16802},[289],[31,16804,16806],{"className":16805},[289],[31,16807,220],{"className":16808},[1307,1308],[31,16810],{"className":16811,"style":293},[119],[31,16813,223],{"className":16814},[297],[31,16816],{"className":16817,"style":293},[119],[31,16819,16821,16824,16827,16867,16870,16873],{"className":16820},[66],[31,16822],{"className":16823,"style":260},[70],[31,16825,4572],{"className":16826},[75],[31,16828,16830,16833],{"cla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",[31,17301,17303,17362],{"className":17302,"translate":35},[34],[31,17304,17306],{"className":17305},[39],[41,17307,17308],{"xmlns":43},[45,17309,17310,17359],{},[48,17311,17312,17314,17320,17322,17328,17334,17336,17338,17345,17347,17353],{},[95,17313,211],{"stretchy":210},[1004,17315,17316,17318],{},[51,17317,16396],{},[91,17319,205],{},[95,17321,97],{},[1004,17323,17324,17326],{},[51,17325,16396],{},[91,17327,152],{},[1180,17329,17330,17332],{},[95,17331,220],{"stretchy":210},[91,17333,152],{},[95,17335,434],{},[95,17337,211],{"stretchy":210},[1004,17339,17340,17343],{},[51,17341,17342],{},"σ",[91,17344,205],{},[95,17346,97],{},[1004,17348,17349,17351],{},[51,17350,17342],{},[91,17352,152],{},[1180,17354,17355,17357],{},[95,17356,220],{"stretchy":210},[91,17358,152],{},[55,17360,17361],{"encoding":57},"(m_1-m_2)^2+(\\sigma_1-\\sigma_2)^2",[31,17363,17365,17423,17507,17565],{"className":17364,"ariaHidden":62},[61],[31,17366,17368,17371,17374,17414,17417,17420],{"className":17367},[66],[31,17369],{"className":17370,"style":260},[70],[31,17372,211],{"className":17373},[264],[31,17375,17377,17380],{"className":17376},[75],[31,17378,16396],{"className":17379},[75,76],[31,17381,17383],{"className":17382},[1039],[31,17384,17386,17406],{"className":17385},[1043,1044],[31,17387,17389,17403],{"className":17388},[1048],[31,17390,17392],{"className":17391,"style":2028},[1052],[31,17393,17394,17397],{"style":2807},[31,17395],{"className":17396,"style":1061},[1060],[31,17398,17400],{"className":17399},[1065,1066,1067,1068],[31,17401,205],{"className":17402},[75,1068],[31,17404,1076],{"className":17405},[1075],[31,17407,17409],{"className":17408},[1048],[31,17410,17412],{"className":17411,"style":2050},[1052],[31,17413],{},[31,17415],{"className":17416,"style":120},[119],[31,17418,97],{"className":17419},[124],[31,17421],{"className":17422,"style":120},[119],[31,17424,17426,17429,17469,17498,17501,17504],{"className":17425},[66],[31,17427],{"className":17428,"style":12035},[70],[31,17430,17432,17435],{"className":17431},[75],[31,17433,16396],{"className":17434},[75,76],[31,17436,17438],{"className":17437},[1039],[31,17439,17441,17461],{"className":17440},[1043,1044],[31,17442,17444,17458],{"className":17443},[1048],[31,17445,17447],{"className":17446,"style":2028},[1052],[31,17448,17449,17452],{"style":2807},[31,17450],{"className":17451,"style":1061},[1060],[31,17453,17455],{"className":17454},[1065,1066,1067,1068],[31,17456,152],{"className":17457},[75,1068],[31,17459,1076],{"className":17460},[1075],[31,17462,17464],{"className":17463},[1048],[31,17465,17467],{"className":17466,"style":2050},[1052],[31,17468],{},[31,17470,17472,17475],{"className":17471},[289],[31,17473,220],{"className":17474},[289],[31,17476,17478],{"className":17477},[1039],[31,17479,17481],{"className":17480},[1043],[31,17482,17484],{"className":17483},[1048],[31,17485,17487],{"className":17486,"style":5135},[1052],[31,17488,17489,17492],{"style":3631},[31,17490],{"className":17491,"style":1061},[1060],[31,17493,17495],{"className":17494},[1065,1066,1067,1068],[31,17496,152],{"className":17497},[75,1068],[31,17499],{"className":17500,"style":120},[119],[31,17502,434],{"className":17503},[124],[31,17505],{"className":17506,"style":120},[119],[31,17508,17510,17513,17516,17556,17559,17562],{"className":17509},[66],[31,17511],{"className":17512,"style":260},[70],[31,17514,211],{"className":17515},[264],[31,17517,17519,17522],{"className":17518},[75],[31,17520,17342],{"className":17521,"style":2666},[75,76],[31,17523,17525],{"className":17524},[1039],[31,17526,17528,17548],{"className":17527},[1043,1044],[31,17529,17531,17545],{"className":17530},[1048],[31,17532,17534],{"className":17533,"style":2028},[1052],[31,17535,17536,17539],{"style":5504},[31,17537],{"className":17538,"style":1061},[1060],[31,17540,17542],{"className":17541},[1065,1066,1067,1068],[31,17543,205],{"className":17544},[75,1068],[31,17546,1076],{"className":17547},[1075],[31,17549,17551],{"className":17550},[1048],[31,17552,17554],{"className":17553,"style":2050},[1052],[31,17555],{},[31,17557],{"className":17558,"style":120},[119],[31,17560,97],{"className":17561},[124],[31,17563],{"className":17564,"style":120},[119],[31,17566,17568,17571,17611],{"className":17567},[66],[31,17569],{"className":17570,"style":12035},[70],[31,17572,17574,17577],{"className":17573},[75],[31,17575,17342],{"className":17576,"style":2666},[75,76],[31,17578,17580],{"className":17579},[1039],[31,17581,17583,17603],{"className":17582},[1043,1044],[31,17584,17586,17600],{"className":17585},[1048],[31,17587,17589],{"className":17588,"style":2028},[1052],[31,17590,17591,17594],{"style":5504},[31,17592],{"className":17593,"style":1061},[1060],[31,17595,17597],{"className":17596},[1065,1066,1067,1068],[31,17598,152],{"className":17599},[75,1068],[31,17601,1076],{"className":17602},[1075],[31,17604,17606],{"className":17605},[1048],[31,17607,17609],{"className":17608,"style":2050},[1052],[31,17610],{},[31,17612,17614,17617],{"className":17613},[289],[31,17615,220],{"className":17616},[289],[31,17618,17620],{"className":17619},[1039],[31,17621,17623],{"className":17622},[1043],[31,17624,17626],{"className":17625},[1048],[31,17627,17629],{"className":17628,"style":5135},[1052],[31,17630,17631,17634],{"style":3631},[31,17632],{"className":17633,"style":1061},[1060],[31,17635,17637],{"className":17636},[1065,1066,1067,1068],[31,17638,152],{"className":17639},[75,1068],"，验算一下：",[31,17642,17644,17728],{"className":17643,"translate":35},[34],[31,17645,17647],{"className":17646},[39],[41,17648,17649],{"xmlns":43},[45,17650,17651,17725],{},[48,17652,17653,17659,17661,17669,17671,17679,17681,17683,17685,17691,17697,17699,17701,17703,17705,17711,17713,17719],{},[48,17654,17655,17657],{},[51,17656,12271],{"mathvariant":494},[51,17658,2549],{"mathvariant":494},[95,17660,211],{"stretchy":210},[12397,17662,17663,17665,17667],{},[51,17664,17342],{},[91,17666,205],{},[91,17668,152],{},[95,17670,434],{},[12397,17672,17673,17675,17677],{},[51,17674,17342],{},[91,17676,152],{},[91,17678,152],{},[95,17680,97],{},[91,17682,152],{},[51,17684,12680],{"mathvariant":494},[1004,17686,17687,17689],{},[51,17688,17342],{},[91,17690,205],{},[1004,17692,17693,17695],{},[51,17694,17342],{},[91,17696,152],{},[51,17698,12680],{"mathvariant":494},[95,17700,220],{"stretchy":210},[95,17702,223],{},[95,17704,211],{"stretchy":210},[1004,17706,17707,17709],{},[51,17708,17342],{},[91,17710,205],{},[95,17712,97],{},[1004,17714,17715,17717],{},[51,17716,17342],{},[91,17718,152],{},[1180,17720,17721,17723],{},[95,17722,220],{"stretchy":210},[91,17724,152],{},[55,17726,17727],{"encoding":57},"\\mathrm{tr}(\\sigma_1^2+\\sigma_2^2-2|\\sigma_1\\sigma_2|)=(\\sigma_1-\\sigma_2)^2",[31,17729,17731,17808,17875,17980,18038],{"className":17730,"ariaHidden":62},[61],[31,17732,17734,17737,17743,17746,17799,17802,17805],{"className":17733},[66],[31,17735],{"className":17736,"style":12035},[70],[31,17738,17740],{"className":17739},[75],[31,17741,16973],{"className":17742},[75,2745],[31,17744,211],{"className":17745},[264],[31,17747,17749,17752],{"className":17748},[75],[31,17750,17342],{"className":17751,"style":2666},[75,76],[31,17753,17755],{"className":17754},[1039],[31,17756,17758,17790],{"className":17757},[1043,1044],[31,17759,17761,17787],{"className":17760},[1048],[31,17762,17764,17776],{"className":17763,"style":5135},[1052],[31,17765,17767,17770],{"style":17766},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[31,17768],{"className":17769,"style":1061},[1060],[31,17771,17773],{"className":17772},[1065,1066,1067,1068],[31,17774,205],{"className":17775},[75,1068],[31,17777,17778,17781],{"style":3631},[31,17779],{"className":17780,"style":1061},[1060],[31,17782,17784],{"className":17783},[1065,1066,1067,1068],[31,17785,152],{"className":17786},[75,1068],[31,17788,1076],{"className":17789},[1075],[31,17791,17793],{"className":17792},[1048],[31,17794,17797],{"className":17795,"style":17796},[1052],"height:0.2481em;",[31,17798],{},[31,17800],{"className":17801,"style":120},[119],[31,17803,434],{"className":17804},[124],[31,17806],{"className":17807,"style":120},[119],[31,17809,17811,17815,17866,17869,17872],{"className":17810},[66],[31,17812],{"className":17813,"style":17814},[70],"height:1.0622em;vertical-align:-0.2481em;",[31,17816,17818,17821],{"className":17817},[75],[31,17819,17342],{"className":17820,"style":2666},[75,76],[31,17822,17824],{"className":17823},[1039],[31,17825,17827,17858],{"className":17826},[1043,1044],[31,17828,17830,17855],{"className":17829},[1048],[31,17831,17833,17844],{"className":17832,"style":5135},[1052],[31,17834,17835,17838],{"style":17766},[31,17836],{"className":17837,"style":1061},[1060],[31,17839,17841],{"className":17840},[1065,1066,1067,1068],[31,17842,152],{"className":17843},[75,1068],[31,17845,17846,17849],{"style":3631},[31,17847],{"className":17848,"style":1061},[1060],[31,17850,17852],{"className":17851},[1065,1066,1067,1068],[31,17853,152],{"className":17854},[75,1068],[31,17856,1076],{"className":17857},[1075],[31,17859,17861],{"className":17860},[1048],[31,17862,17864],{"className":17863,"style":17796},[1052],[31,17865],{},[31,17867],{"className":17868,"style":120},[119],[31,17870,97],{"className":17871},[124],[31,17873],{"className":17874,"style":120},[119],[31,17876,17878,17881,17885,17925,17965,17968,17971,17974,17977],{"className":17877},[66],[31,17879],{"className":17880,"style":260},[70],[31,17882,17884],{"className":17883},[75],"2∣",[31,17886,17888,17891],{"className":17887},[75],[31,17889,17342],{"className":17890,"style":2666},[75,76],[31,17892,17894],{"className":17893},[1039],[31,17895,17897,17917],{"className":17896},[1043,1044],[31,17898,17900,17914],{"className":17899},[1048],[31,17901,17903],{"className":17902,"style":2028},[1052],[31,17904,17905,17908],{"style":5504},[31,17906],{"className":17907,"style":1061},[1060],[31,17909,17911],{"className":17910},[1065,1066,1067,1068],[31,17912,205],{"className":17913},[75,1068],[31,17915,1076],{"className":17916},[1075],[31,17918,17920],{"className":17919},[1048],[31,17921,17923],{"className":17922,"style":2050},[1052],[31,17924],{},[31,17926,17928,17931],{"className":17927},[75],[31,17929,17342],{"className":17930,"style":2666},[75,76],[31,17932,17934],{"className":17933},[1039],[31,17935,17937,17957],{"className":17936},[1043,1044],[31,17938,17940,17954],{"className":17939},[1048],[31,17941,17943],{"className":17942,"style":2028},[1052],[31,17944,17945,17948],{"style":5504},[31,17946],{"className":17947,"style":1061},[1060],[31,17949,17951],{"className":17950},[1065,1066,1067,1068],[31,17952,152],{"className":17953},[75,1068],[31,17955,1076],{"className":17956},[1075],[31,17958,17960],{"className":17959},[1048],[31,17961,17963],{"className":17962,"style":2050},[1052],[31,17964],{},[31,17966,12680],{"className":17967},[75],[31,17969,220],{"className":17970},[289],[31,17972],{"className":17973,"style":293},[119],[31,17975,223],{"className":17976},[297],[31,17978],{"className":17979,"style":293},[119],[31,17981,17983,17986,17989,18029,18032,18035],{"className":17982},[66],[31,17984],{"className":17985,"style":260},[70],[31,17987,211],{"className":17988},[264],[31,17990,17992,17995],{"className":17991},[75],[31,17993,17342],{"className":17994,"style":2666},[75,76],[31,17996,17998],{"className":17997},[1039],[31,17999,18001,18021],{"className":18000},[1043,1044],[31,18002,18004,18018],{"className":18003},[1048],[31,18005,18007],{"className":18006,"style":2028},[1052],[31,18008,18009,18012],{"style":5504},[31,18010],{"className":18011,"style":1061},[1060],[31,18013,18015],{"className":18014},[1065,1066,1067,1068],[31,18016,205],{"className":18017},[75,1068],[31,18019,1076],{"className":18020},[1075],[31,18022,18024],{"className":18023},[1048],[31,18025,18027],{"className":18026,"style":2050},[1052],[31,18028],{},[31,18030],{"className":18031,"style":120},[119],[31,18033,97],{"className":18034},[124],[31,18036],{"className":18037,"style":120},[119],[31,18039,18041,18044,18084],{"className":18040},[66],[31,18042],{"className":18043,"style":12035},[70],[31,18045,18047,18050],{"className":18046},[75],[31,18048,17342],{"className":18049,"style":2666},[75,76],[31,18051,18053],{"className":18052},[1039],[31,18054,18056,18076],{"className":18055},[1043,1044],[31,18057,18059,18073],{"className":18058},[1048],[31,18060,18062],{"className":18061,"style":2028},[1052],[31,18063,18064,18067],{"style":5504},[31,18065],{"className":18066,"style":1061},[1060],[31,18068,18070],{"className":18069},[1065,1066,1067,1068],[31,18071,152],{"className":18072},[75,1068],[31,18074,1076],{"className":18075},[1075],[31,18077,18079],{"className":18078},[1048],[31,18080,18082],{"className":18081,"style":2050},[1052],[31,18083],{},[31,18085,18087,18090],{"className":18086},[289],[31,18088,220],{"className":18089},[289],[31,18091,18093],{"className":18092},[1039],[31,18094,18096],{"className":18095},[1043],[31,18097,18099],{"className":18098},[1048],[31,18100,18102],{"className":18101,"style":5135},[1052],[31,18103,18104,18107],{"style":3631},[31,18105],{"className":18106,"style":1061},[1060],[31,18108,18110],{"className":18109},[1065,1066,1067,1068],[31,18111,152],{"className":18112},[75,1068],"，对得上。取 ",[31,18115,18117,18141],{"className":18116,"translate":35},[34],[31,18118,18120],{"className":18119},[39],[41,18121,18122],{"xmlns":43},[45,18123,18124,18138],{},[48,18125,18126,18128,18130,18132,18134,18136],{},[51,18127,4935],{"mathvariant":925},[95,18129,211],{"stretchy":210},[91,18131,1977],{},[95,18133,1210],{"separator":62},[91,18135,205],{},[95,18137,220],{"stretchy":210},[55,18139,18140],{"encoding":57},"\\mathcal N(0,1)",[31,18142,18144],{"className":18143,"ariaHidden":62},[61],[31,18145,18147,18150,18153,18156,18159,18162,18165,18168],{"className":18146},[66],[31,18148],{"className":18149,"style":260},[70],[31,18151,4935],{"className":18152,"style":5004},[75,968],[31,18154,211],{"className":18155},[264],[31,18157,1977],{"className":18158},[75],[31,18160,1210],{"className":18161},[1391],[31,18163],{"className":18164,"style":1467},[119],[31,18166,205],{"className":18167},[75],[31,18169,220],{"className":18170},[289]," 与 ",[31,18173,18175,18199],{"className":18174,"translate":35},[34],[31,18176,18178],{"className":18177},[39],[41,18179,18180],{"xmlns":43},[45,18181,18182,18196],{},[48,18183,18184,18186,18188,18190,18192,18194],{},[51,18185,4935],{"mathvariant":925},[95,18187,211],{"stretchy":210},[91,18189,1977],{},[95,18191,1210],{"separator":62},[91,18193,15416],{},[95,18195,220],{"stretchy":210},[55,18197,18198],{"encoding":57},"\\mathcal N(0,4)",[31,18200,18202],{"className":18201,"ariaHidden":62},[61],[31,18203,18205,18208,18211,18214,18217,18220,18223,18226],{"className":18204},[66],[31,18206],{"className":18207,"style":260},[70],[31,18209,4935],{"className":18210,"style":5004},[75,968],[31,18212,211],{"className":18213},[264],[31,18215,1977],{"className":18216},[75],[31,18218,1210],{"className":18219},[1391],[31,18221],{"className":18222,"style":1467},[119],[31,18224,15416],{"className":18225},[75],[31,18227,220],{"className":18228},[289],"，得 ",[31,18231,18233,18255],{"className":18232,"translate":35},[34],[31,18234,18236],{"className":18235},[39],[41,18237,18238],{"xmlns":43},[45,18239,18240,18252],{},[48,18241,18242,18248,18250],{},[1004,18243,18244,18246],{},[51,18245,3117],{},[91,18247,152],{},[95,18249,223],{},[91,18251,205],{},[55,18253,18254],{"encoding":57},"W_2=1",[31,18256,18258,18313],{"className":18257,"ariaHidden":62},[61],[31,18259,18261,18264,18304,18307,18310],{"className":18260},[66],[31,18262],{"className":18263,"style":5230},[70],[31,18265,18267,18270],{"className":18266},[75],[31,18268,3117],{"className":18269,"style":949},[75,76],[31,18271,18273],{"className":18272},[1039],[31,18274,18276,18296],{"className":18275},[1043,1044],[31,18277,18279,18293],{"className":18278},[1048],[31,18280,18282],{"className":18281,"style":2028},[1052],[31,18283,18284,18287],{"style":1056},[31,18285],{"className":18286,"style":1061},[1060],[31,18288,18290],{"className":18289},[1065,1066,1067,1068],[31,18291,152],{"className":18292},[75,1068],[31,18294,1076],{"className":18295},[1075],[31,18297,18299],{"className":18298},[1048],[31,18300,18302],{"className":18301,"style":2050},[1052],[31,18303],{},[31,18305],{"className":18306,"style":293},[119],[31,18308,223],{"className":18309},[297],[31,18311],{"className":18312,"style":293},[119],[31,18314,18316,18319],{"className":18315},[66],[31,18317],{"className":18318,"style":326},[70],[31,18320,205],{"className":18321},[75],"，正好是标准差之差；而最优映射是 ",[31,18324,18326,18347],{"className":18325,"translate":35},[34],[31,18327,18329],{"className":18328},[39],[41,18330,18331],{"xmlns":43},[45,18332,18333,18344],{},[48,18334,18335,18337,18340,18342],{},[51,18336,53],{},[95,18338,18339],{},"↦",[91,18341,152],{},[51,18343,53],{},[55,18345,18346],{"encoding":57},"x\\mapsto2x",[31,18348,18350,18369],{"className":18349,"ariaHidden":62},[61],[31,18351,18353,18357,18360,18363,18366],{"className":18352},[66],[31,18354],{"className":18355,"style":18356},[70],"height:0.522em;vertical-align:-0.011em;",[31,18358,53],{"className":18359},[75,76],[31,18361],{"className":18362,"style":293},[119],[31,18364,18339],{"className":18365},[297],[31,18367],{"className":18368,"style":293},[119],[31,18370,18372,18375,18378],{"className":18371},[66],[31,18373],{"className":18374,"style":326},[70],[31,18376,152],{"className":18377},[75],[31,18379,53],{"className":18380},[75,76],"，位移的均方根也正好是 ",[31,18383,18385,18398],{"className":18384,"translate":35},[34],[31,18386,18388],{"className":18387},[39],[41,18389,18390],{"xmlns":43},[45,18391,18392,18396],{},[48,18393,18394],{},[91,18395,205],{},[55,18397,205],{"encoding":57},[31,18399,18401],{"className":18400,"ariaHidden":62},[61],[31,18402,18404,18407],{"className":18403},[66],[31,18405],{"className":18406,"style":326},[70],[31,18408,205],{"className":18409},[75],"。两个数在这里是同一个数。",[11,18412,18413,18414,18622],{},"最优映射还是仿射的：",[31,18415,18417,18464],{"className":18416,"translate":35},[34],[31,18418,18420],{"className":18419},[39],[41,18421,18422],{"xmlns":43},[45,18423,18424,18461],{},[48,18425,18426,18428,18430,18432,18434,18436,18442,18444,18447,18449,18451,18453,18459],{},[51,18427,919],{},[95,18429,211],{"stretchy":210},[51,18431,53],{},[95,18433,220],{"stretchy":210},[95,18435,223],{},[1004,18437,18438,18440],{},[51,18439,16396],{},[91,18441,152],{},[95,18443,434],{},[51,18445,18446],{},"A",[95,18448,211],{"stretchy":210},[51,18450,53],{},[95,18452,97],{},[1004,18454,18455,18457],{},[51,18456,16396],{},[91,18458,205],{},[95,18460,220],{"stretchy":210},[55,18462,18463],{"encoding":57},"T(x)=m_2+A(x-m_1)",[31,18465,18467,18494,18549,18573],{"className":18466,"ariaHidden":62},[61],[31,18468,18470,18473,18476,18479,18482,18485,18488,18491],{"className":18469},[66],[31,18471],{"className":18472,"style":260},[70],[31,18474,919],{"className":18475,"style":949},[75,76],[31,18477,211],{"className":18478},[264],[31,18480,53],{"className":18481},[75,76],[31,18483,220],{"className":18484},[289],[31,18486],{"className":18487,"style":293},[119],[31,18489,223],{"className":18490},[297],[31,18492],{"className":18493,"style":293},[119],[31,18495,18497,18500,18540,18543,18546],{"className":18496},[66],[31,18498],{"className":18499,"style":5426},[70],[31,18501,18503,18506],{"className":18502},[75],[31,18504,16396],{"className":18505},[75,76],[31,18507,18509],{"className":18508},[1039],[31,18510,18512,18532],{"className":18511},[1043,1044],[31,18513,18515,18529],{"className":18514},[1048],[31,18516,18518],{"className":18517,"style":2028},[1052],[31,18519,18520,18523],{"style":2807},[31,18521],{"className":18522,"style":1061},[1060],[31,18524,18526],{"className":18525},[1065,1066,1067,1068],[31,18527,152],{"className":18528},[75,1068],[31,18530,1076],{"className":18531},[1075],[31,18533,18535],{"className":18534},[1048],[31,18536,18538],{"className":18537,"style":2050},[1052],[31,18539],{},[31,18541],{"className":18542,"style":120},[119],[31,18544,434],{"className":18545},[124],[31,18547],{"className":18548,"style":120},[119],[31,18550,18552,18555,18558,18561,18564,18567,18570],{"className":18551},[66],[31,18553],{"className":18554,"style":260},[70],[31,18556,18446],{"className":18557},[75,76],[31,18559,211],{"className":18560},[264],[31,18562,53],{"className":18563},[75,76],[31,18565],{"className":18566,"style":120},[119],[31,18568,97],{"className":18569},[124],[31,18571],{"className":18572,"style":120},[119],[31,18574,18576,18579,18619],{"className":18575},[66],[31,18577],{"className":18578,"style":260},[70],[31,18580,18582,18585],{"className":18581},[75],[31,18583,16396],{"className":18584},[75,76],[31,18586,18588],{"className":18587},[1039],[31,18589,18591,18611],{"className":18590},[1043,1044],[31,18592,18594,18608],{"className":18593},[1048],[31,18595,18597],{"className":18596,"style":2028},[1052],[31,18598,18599,18602],{"style":2807},[31,18600],{"className":18601,"style":1061},[1060],[31,18603,18605],{"className":18604},[1065,1066,1067,1068],[31,18606,205],{"className":18607},[75,1068],[31,18609,1076],{"className":18610},[1075],[31,18612,18614],{"className":18613},[1048],[31,18615,18617],{"className":18616,"style":2050},[1052],[31,18618],{},[31,18620,220],{"className":18621},[289],"，其中",[31,18624,18626],{"className":18625,"translate":35},[411],[31,18627,18629,18727],{"className":18628,"translate":35},[34],[31,18630,18632],{"className":18631},[39],[41,18633,18634],{"xmlns":43,"display":420},[45,18635,18636,18724],{},[48,18637,18638,18640,18642,18658,18660,18674,18680,18694,18706,18722],{},[51,18639,18446],{},[95,18641,223],{},[12397,18643,18644,18646,18648],{},[51,18645,16405],{"mathvariant":494},[91,18647,205],{},[48,18649,18650,18652,18654,18656],{},[95,18651,97],{},[91,18653,205],{},[51,18655,3768],{"mathvariant":494},[91,18657,152],{},[95,18659,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[12397,18661,18662,18664,18666],{},[51,18663,16405],{"mathvariant":494},[91,18665,205],{},[48,18667,18668,18670,18672],{},[91,18669,205],{},[51,18671,3768],{"mathvariant":494},[91,18673,152],{},[1004,18675,18676,18678],{},[51,18677,16405],{"mathvariant":494},[91,18679,152],{},[12397,18681,18682,18684,18686],{},[51,18683,16405],{"mathvariant":494},[91,18685,205],{},[48,18687,18688,18690,18692],{},[91,18689,205],{},[51,18691,3768],{"mathvariant":494},[91,18693,152],{},[1180,18695,18696,18698],{},[95,18697,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[48,18699,18700,18702,18704],{},[91,18701,205],{},[51,18703,3768],{"mathvariant":494},[91,18705,152],{},[12397,18707,18708,18710,18712],{},[51,18709,16405],{"mathvariant":494},[91,18711,205],{},[48,18713,18714,18716,18718,18720],{},[95,18715,97],{},[91,18717,205],{},[51,18719,3768],{"mathvariant":494},[91,18721,152],{},[51,18723,495],{"mathvariant":494},[55,18725,18726],{"encoding":57},"A=\\Sigma_1^{-1\u002F2}\\bigl(\\Sigma_1^{1\u002F2}\\Sigma_2\\Sigma_1^{1\u002F2}\\bigr)^{1\u002F2}\\Sigma_1^{-1\u002F2}.",[31,18728,18730,18748],{"className":18729,"ariaHidden":62},[61],[31,18731,18733,18736,18739,18742,18745],{"className":18732},[66],[31,18734],{"className":18735,"style":945},[70],[31,18737,18446],{"className":18738},[75,76],[31,18740],{"className":18741,"style":293},[119],[31,18743,223],{"className":18744},[297],[31,18746],{"className":18747,"style":293},[119],[31,18749,18751,18755,18812,18818,18872,18912,18966,19001,19058],{"className":18750},[66],[31,18752],{"className":18753,"style":18754},[70],"height:1.4779em;vertical-align:-0.35em;",[31,18756,18758,18761],{"className":18757},[75],[31,18759,16405],{"className":18760},[75],[31,18762,18764],{"className":18763},[1039],[31,18765,18767,18804],{"className":18766},[1043,1044],[31,18768,18770,18801],{"className":18769},[1048],[31,18771,18773,18784],{"className":18772,"style":17117},[1052],[31,18774,18775,18778],{"style":17120},[31,18776],{"className":18777,"style":1061},[1060],[31,18779,18781],{"className":18780},[1065,1066,1067,1068],[31,18782,205],{"className":18783},[75,1068],[31,18785,18786,18789],{"style":17132},[31,18787],{"className":18788,"style":1061},[1060],[31,18790,18792],{"className":18791},[1065,1066,1067,1068],[31,18793,18795,18798],{"className":18794},[75,1068],[31,18796,97],{"className":18797},[75,1068],[31,18799,17145],{"className":18800},[75,1068],[31,18802,1076],{"className":18803},[1075],[31,18805,18807],{"className":18806},[1048],[31,18808,18810],{"className":18809,"style":17155},[1052],[31,18811],{},[31,18813,18815],{"className":18814},[264],[31,18816,211],{"className":18817},[1307,1308],[31,18819,18821,18824],{"className":18820},[75],[31,18822,16405],{"className":18823},[75],[31,18825,18827],{"className":18826},[1039],[31,18828,18830,18864],{"className":18829},[1043,1044],[31,18831,18833,18861],{"className":18832},[1048],[31,18834,18836,18847],{"className":18835,"style":17117},[1052],[31,18837,18838,18841],{"style":17120},[31,18839],{"className":18840,"style":1061},[1060],[31,18842,18844],{"className":18843},[1065,1066,1067,1068],[31,18845,205],{"className":18846},[75,1068],[31,18848,18849,18852],{"style":17132},[31,18850],{"className":18851,"style":1061},[1060],[31,18853,18855],{"className":18854},[1065,1066,1067,1068],[31,18856,18858],{"className":18857},[75,1068],[31,18859,17145],{"className":18860},[75,1068],[31,18862,1076],{"className":18863},[1075],[31,18865,18867],{"className":18866},[1048],[31,18868,18870],{"className":18869,"style":17155},[1052],[31,18871],{},[31,18873,18875,18878],{"className":18874},[75],[31,18876,16405],{"className":18877},[75],[31,18879,18881],{"className":18880},[1039],[31,18882,18884,18904],{"className":18883},[1043,1044],[31,18885,18887,18901],{"className":18886},[1048],[31,18888,18890],{"className":18889,"style":2028},[1052],[31,18891,18892,18895],{"style":2807},[31,18893],{"className":18894,"style":1061},[1060],[31,18896,18898],{"className":18897},[1065,1066,1067,1068],[31,18899,152],{"className":18900},[75,1068],[31,18902,1076],{"className":18903},[1075],[31,18905,18907],{"className":18906},[1048],[31,18908,18910],{"className":18909,"style":2050},[1052],[31,18911],{},[31,18913,18915,18918],{"className":18914},[75],[31,18916,16405],{"className":18917},[75],[31,18919,18921],{"className":18920},[1039],[31,18922,18924,18958],{"className":18923},[1043,1044],[31,18925,18927,18955],{"className":18926},[1048],[31,18928,18930,18941],{"className":18929,"style":17117},[1052],[31,18931,18932,18935],{"style":17120},[31,18933],{"className":18934,"style":1061},[1060],[31,18936,18938],{"className":18937},[1065,1066,1067,1068],[31,18939,205],{"className":18940},[75,1068],[31,18942,18943,18946],{"style":17132},[31,18944],{"className":18945,"style":1061},[1060],[31,18947,18949],{"className":18948},[1065,1066,1067,1068],[31,18950,18952],{"className":18951},[75,1068],[31,18953,17145],{"className":18954},[75,1068],[31,18956,1076],{"className":18957},[1075],[31,18959,18961],{"className":18960},[1048],[31,18962,18964],{"className":18963,"style":17155},[1052],[31,18965],{},[31,18967,18969,18975],{"className":18968},[289],[31,18970,18972],{"className":18971},[289],[31,18973,220],{"className":18974},[1307,1308],[31,18976,18978],{"className":18977},[1039],[31,18979,18981],{"className":18980},[1043],[31,18982,18984],{"className":18983},[1048],[31,18985,18987],{"className":18986,"style":17273},[1052],[31,18988,18989,18992],{"style":13566},[31,18990],{"className":18991,"style":1061},[1060],[31,18993,18995],{"className":18994},[1065,1066,1067,1068],[31,18996,18998],{"className":18997},[75,1068],[31,18999,17145],{"className":19000},[75,1068],[31,19002,19004,19007],{"className":19003},[75],[31,19005,16405],{"className":19006},[75],[31,19008,19010],{"className":19009},[1039],[31,19011,19013,19050],{"className":19012},[1043,1044],[31,19014,19016,19047],{"className":19015},[1048],[31,19017,19019,19030],{"className":19018,"style":17117},[1052],[31,19020,19021,19024],{"style":17120},[31,19022],{"className":19023,"style":1061},[1060],[31,19025,19027],{"className":19026},[1065,1066,1067,1068],[31,19028,205],{"className":19029},[75,1068],[31,19031,19032,19035],{"style":17132},[31,19033],{"className":19034,"style":1061},[1060],[31,19036,19038],{"className":19037},[1065,1066,1067,1068],[31,19039,19041,19044],{"className":19040},[75,1068],[31,19042,97],{"className":19043},[75,1068],[31,19045,17145],{"className":19046},[75,1068],[31,19048,1076],{"className":19049},[1075],[31,19051,19053],{"className":19052},[1048],[31,19054,19056],{"className":19055,"style":17155},[1052],[31,19057],{},[31,19059,495],{"className":19060},[75],[11,19062,19063],{},"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。",[11,19065,19066,19067,19137,19138,19367,19368,19437],{},"高斯集合配上 ",[31,19068,19070,19088],{"className":19069,"translate":35},[34],[31,19071,19073],{"className":19072},[39],[41,19074,19075],{"xmlns":43},[45,19076,19077,19085],{},[48,19078,19079],{},[1004,19080,19081,19083],{},[51,19082,3117],{},[91,19084,152],{},[55,19086,19087],{"encoding":57},"W_2",[31,19089,19091],{"className":19090,"ariaHidden":62},[61],[31,19092,19094,19097],{"className":19093},[66],[31,19095],{"className":19096,"style":5230},[70],[31,19098,19100,19103],{"className":19099},[75],[31,19101,3117],{"className":19102,"style":949},[75,76],[31,19104,19106],{"className":19105},[1039],[31,19107,19109,19129],{"className":19108},[1043,1044],[31,19110,19112,19126],{"className":19111},[1048],[31,19113,19115],{"className":19114,"style":2028},[1052],[31,19116,19117,19120],{"style":1056},[31,19118],{"className":19119,"style":1061},[1060],[31,19121,19123],{"className":19122},[1065,1066,1067,1068],[31,19124,152],{"className":19125},[75,1068],[31,19127,1076],{"className":19128},[1075],[31,19130,19132],{"className":19131},[1048],[31,19133,19135],{"className":19134,"style":2050},[1052],[31,19136],{}," 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 ",[31,19139,19141,19193],{"className":19140,"translate":35},[34],[31,19142,19144],{"className":19143},[39],[41,19145,19146],{"xmlns":43},[45,19147,19148,19190],{},[48,19149,19150,19156,19158,19160,19162,19164,19166,19168,19170,19176,19178,19180,19182,19188],{},[1004,19151,19152,19154],{},[51,19153,852],{},[51,19155,12271],{},[95,19157,223],{},[95,19159,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,19161,211],{"stretchy":210},[91,19163,205],{},[95,19165,97],{},[51,19167,12271],{},[95,19169,220],{"stretchy":210},[48,19171,19172,19174],{},[51,19173,5391],{"mathvariant":494},[51,19175,1435],{"mathvariant":494},[95,19177,434],{},[51,19179,12271],{},[51,19181,919],{},[1004,19183,19184,19186],{},[95,19185,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,19187,1010],{"mathvariant":494},[51,19189,852],{},[55,19191,19192],{"encoding":57},"\\mu_t=\\bigl((1-t)\\mathrm{id}+tT\\bigr)_\\#\\mu",[31,19194,19196,19252,19279,19307],{"className":19195,"ariaHidden":62},[61],[31,19197,19199,19202,19243,19246,19249],{"className":19198},[66],[31,19200],{"className":19201,"style":865},[70],[31,19203,19205,19208],{"className":19204},[75],[31,19206,852],{"className":19207},[75,76],[31,19209,19211],{"className":19210},[1039],[31,19212,19214,19235],{"className":19213},[1043,1044],[31,19215,19217,19232],{"className":19216},[1048],[31,19218,19221],{"className":19219,"style":19220},[1052],"height:0.2806em;",[31,19222,19223,19226],{"style":2807},[31,19224],{"className":19225,"style":1061},[1060],[31,19227,19229],{"className":19228},[1065,1066,1067,1068],[31,19230,12271],{"className":19231},[75,76,1068],[31,19233,1076],{"className":19234},[1075],[31,19236,19238],{"className":19237},[1048],[31,19239,19241],{"className":19240,"style":2050},[1052],[31,19242],{},[31,19244],{"className":19245,"style":293},[119],[31,19247,223],{"className":19248},[297],[31,19250],{"className":19251,"style":293},[119],[31,19253,19255,19258,19264,19267,19270,19273,19276],{"className":19254},[66],[31,19256],{"className":19257,"style":4988},[70],[31,19259,19261],{"className":19260},[264],[31,19262,211],{"className":19263},[1307,1308],[31,19265,211],{"className":19266},[264],[31,19268,205],{"className":19269},[75],[31,19271],{"className":19272,"style":120},[119],[31,19274,97],{"className":19275},[124],[31,19277],{"className":19278,"style":120},[119],[31,19280,19282,19285,19288,19291,19298,19301,19304],{"className":19281},[66],[31,19283],{"className":19284,"style":260},[70],[31,19286,12271],{"className":19287},[75,76],[31,19289,220],{"className":19290},[289],[31,19292,19294],{"className":19293},[75],[31,19295,19297],{"className":19296},[75,2745],"id",[31,19299],{"className":19300,"style":120},[119],[31,19302,434],{"className":19303},[124],[31,19305],{"className":19306,"style":120},[119],[31,19308,19310,19314,19318,19364],{"className":19309},[66],[31,19311],{"className":19312,"style":19313},[70],"height:1.3858em;vertical-align:-0.5358em;",[31,19315,19317],{"className":19316,"style":949},[75,76],"tT",[31,19319,19321,19327],{"className":19320},[289],[31,19322,19324],{"className":19323},[289],[31,19325,220],{"className":19326},[1307,1308],[31,19328,19330],{"className":19329},[1039],[31,19331,19333,19355],{"className":19332},[1043,1044],[31,19334,19336,19352],{"className":19335},[1048],[31,19337,19340],{"className":19338,"style":19339},[1052],"height:0.0864em;",[31,19341,19343,19346],{"style":19342},"top:-2.3003em;margin-right:0.05em;",[31,19344],{"className":19345,"style":1061},[1060],[31,19347,19349],{"className":19348},[1065,1066,1067,1068],[31,19350,1010],{"className":19351},[75,1068],[31,19353,1076],{"className":19354},[1075],[31,19356,19358],{"className":19357},[1048],[31,19359,19362],{"className":19360,"style":19361},[1052],"height:0.5358em;",[31,19363],{},[31,19365,852],{"className":19366},[75,76],"。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；",[31,19369,19371,19388],{"className":19370,"translate":35},[34],[31,19372,19374],{"className":19373},[39],[41,19375,19376],{"xmlns":43},[45,19377,19378,19386],{},[48,19379,19380],{},[1004,19381,19382,19384],{},[51,19383,3117],{},[91,19385,152],{},[55,19387,19087],{"encoding":57},[31,19389,19391],{"className":19390,"ariaHidden":62},[61],[31,19392,19394,19397],{"className":19393},[66],[31,19395],{"className":19396,"style":5230},[70],[31,19398,19400,19403],{"className":19399},[75],[31,19401,3117],{"className":19402,"style":949},[75,76],[31,19404,19406],{"className":19405},[1039],[31,19407,19409,19429],{"className":19408},[1043,1044],[31,19410,19412,19426],{"className":19411},[1048],[31,19413,19415],{"className":19414,"style":2028},[1052],[31,19416,19417,19420],{"style":1056},[31,19418],{"className":19419,"style":1061},[1060],[31,19421,19423],{"className":19422},[1065,1066,1067,1068],[31,19424,152],{"className":19425},[75,1068],[31,19427,1076],{"className":19428},[1075],[31,19430,19432],{"className":19431},[1048],[31,19433,19435],{"className":19434,"style":2050},[1052],[31,19436],{}," 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。",[18,19439,19440],{"id":19440},"计算的代价",[11,19442,19443,19471],{},[31,19444,19446,19459],{"className":19445,"translate":35},[34],[31,19447,19449],{"className":19448},[39],[41,19450,19451],{"xmlns":43},[45,19452,19453,19457],{},[48,19454,19455],{},[51,19456,3117],{},[55,19458,3117],{"encoding":57},[31,19460,19462],{"className":19461,"ariaHidden":62},[61],[31,19463,19465,19468],{"className":19464},[66],[31,19466],{"className":19467,"style":945},[70],[31,19469,3117],{"className":19470,"style":949},[75,76]," 定义了“是什么”，但没告诉我“怎么算”。",[11,19473,19474,19559,19560,19622,19623,19686,19687,19715],{},[31,19475,19477,19500],{"className":19476,"translate":35},[34],[31,19478,19480],{"className":19479},[39],[41,19481,19482],{"xmlns":43},[45,19483,19484,19497],{},[48,19485,19486,19488,19490],{},[51,19487,14559],{},[95,19489,223],{},[1180,19491,19492,19495],{},[91,19493,19494],{},"10",[91,19496,15416],{},[55,19498,19499],{"encoding":57},"n=10^4",[31,19501,19503,19521],{"className":19502,"ariaHidden":62},[61],[31,19504,19506,19509,19512,19515,19518],{"className":19505},[66],[31,19507],{"className":19508,"style":71},[70],[31,19510,14559],{"className":19511},[75,76],[31,19513],{"className":19514,"style":293},[119],[31,19516,223],{"className":19517},[297],[31,19519],{"className":19520,"style":293},[119],[31,19522,19524,19527,19530],{"className":19523},[66],[31,19525],{"className":19526,"style":5135},[70],[31,19528,205],{"className":19529},[75],[31,19531,19533,19536],{"className":19532},[75],[31,19534,1977],{"className":19535},[75],[31,19537,19539],{"className":19538},[1039],[31,19540,19542],{"className":19541},[1043],[31,19543,19545],{"className":19544},[1048],[31,19546,19548],{"className":19547,"style":5135},[1052],[31,19549,19550,19553],{"style":3631},[31,19551],{"className":19552,"style":1061},[1060],[31,19554,19556],{"className":19555},[1065,1066,1067,1068],[31,19557,15416],{"className":19558},[75,1068]," 个样本对 ",[31,19561,19563,19581],{"className":19562,"translate":35},[34],[31,19564,19566],{"className":19565},[39],[41,19567,19568],{"xmlns":43},[45,19569,19570,19578],{},[48,19571,19572],{},[1180,19573,19574,19576],{},[91,19575,19494],{},[91,19577,15416],{},[55,19579,19580],{"encoding":57},"10^4",[31,19582,19584],{"className":19583,"ariaHidden":62},[61],[31,19585,19587,19590,19593],{"className":19586},[66],[31,19588],{"className":19589,"style":5135},[70],[31,19591,205],{"className":19592},[75],[31,19594,19596,19599],{"className":19595},[75],[31,19597,1977],{"className":19598},[75],[31,19600,19602],{"className":19601},[1039],[31,19603,19605],{"className":19604},[1043],[31,19606,19608],{"className":19607},[1048],[31,19609,19611],{"className":19610,"style":5135},[1052],[31,19612,19613,19616],{"style":3631},[31,19614],{"className":19615,"style":1061},[1060],[31,19617,19619],{"className":19618},[1065,1066,1067,1068],[31,19620,15416],{"className":19621},[75,1068]," 个样本，代价矩阵就是 ",[31,19624,19626,19645],{"className":19625,"translate":35},[34],[31,19627,19629],{"className":19628},[39],[41,19630,19631],{"xmlns":43},[45,19632,19633,19642],{},[48,19634,19635],{},[1180,19636,19637,19639],{},[91,19638,19494],{},[91,19640,19641],{},"8",[55,19643,19644],{"encoding":57},"10^8",[31,19646,19648],{"className":19647,"ariaHidden":62},[61],[31,19649,19651,19654,19657],{"className":19650},[66],[31,19652],{"className":19653,"style":5135},[70],[31,19655,205],{"className":19656},[75],[31,19658,19660,19663],{"className":19659},[75],[31,19661,1977],{"className":19662},[75],[31,19664,19666],{"className":19665},[1039],[31,19667,19669],{"className":19668},[1043],[31,19670,19672],{"className":19671},[1048],[31,19673,19675],{"className":19674,"style":5135},[1052],[31,19676,19677,19680],{"style":3631},[31,19678],{"className":19679,"style":1061},[1060],[31,19681,19683],{"className":19682},[1065,1066,1067,1068],[31,19684,19641],{"className":19685},[75,1068]," 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 ",[31,19688,19690,19703],{"className":19689,"translate":35},[34],[31,19691,19693],{"className":19692},[39],[41,19694,19695],{"xmlns":43},[45,19696,19697,19701],{},[48,19698,19699],{},[51,19700,14559],{},[55,19702,14559],{"encoding":57},[31,19704,19706],{"className":19705,"ariaHidden":62},[61],[31,19707,19709,19712],{"className":19708},[66],[31,19710],{"className":19711,"style":71},[70],[31,19713,14559],{"className":19714},[75,76]," 开到 5000 就被内存劝退了。",[11,19717,19718,19719,19795,19796,19824,19825,19853],{},"离散 OT 是线性规划，实践上是 ",[31,19720,19722,19747],{"className":19721,"translate":35},[34],[31,19723,19725],{"className":19724},[39],[41,19726,19727],{"xmlns":43},[45,19728,19729,19744],{},[48,19730,19731,19734,19736,19742],{},[51,19732,19733],{},"O",[95,19735,211],{"stretchy":210},[1180,19737,19738,19740],{},[51,19739,14559],{},[91,19741,93],{},[95,19743,220],{"stretchy":210},[55,19745,19746],{"encoding":57},"O(n^3)",[31,19748,19750],{"className":19749,"ariaHidden":62},[61],[31,19751,19753,19756,19760,19763,19792],{"className":19752},[66],[31,19754],{"className":19755,"style":12035},[70],[31,19757,19733],{"className":19758,"style":19759},[75,76],"margin-right:0.0278em;",[31,19761,211],{"className":19762},[264],[31,19764,19766,19769],{"className":19765},[75],[31,19767,14559],{"className":19768},[75,76],[31,19770,19772],{"className":19771},[1039],[31,19773,19775],{"className":19774},[1043],[31,19776,19778],{"className":19777},[1048],[31,19779,19781],{"className":19780,"style":5135},[1052],[31,19782,19783,19786],{"style":3631},[31,19784],{"className":19785,"style":1061},[1060],[31,19787,19789],{"className":19788},[1065,1066,1067,1068],[31,19790,93],{"className":19791},[75,1068],[31,19793,220],{"className":19794},[289]," 量级的算法。更麻烦的是解落在多面体的顶点上，",[31,19797,19799,19812],{"className":19798,"translate":35},[34],[31,19800,19802],{"className":19801},[39],[41,19803,19804],{"xmlns":43},[45,19805,19806,19810],{},[48,19807,19808],{},[51,19809,2515],{},[55,19811,3485],{"encoding":57},[31,19813,19815],{"className":19814,"ariaHidden":62},[61],[31,19816,19818,19821],{"className":19817},[66],[31,19819],{"className":19820,"style":71},[70],[31,19822,2515],{"className":19823,"style":2666},[75,76]," 很稀疏，而 ",[31,19826,19828,19841],{"className":19827,"translate":35},[34],[31,19829,19831],{"className":19830},[39],[41,19832,19833],{"xmlns":43},[45,19834,19835,19839],{},[48,19836,19837],{},[51,19838,3117],{},[55,19840,3117],{"encoding":57},[31,19842,19844],{"className":19843,"ariaHidden":62},[61],[31,19845,19847,19850],{"className":19846},[66],[31,19848],{"className":19849,"style":945},[70],[31,19851,3117],{"className":19852,"style":949},[75,76]," 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。",[11,19855,19856],{},"Cuturi 在 2013 年的做法是加熵正则：",[31,19858,19860],{"className":19859,"translate":35},[411],[31,19861,19863,19957],{"className":19862,"translate":35},[34],[31,19864,19866],{"className":19865},[39],[41,19867,19868],{"xmlns":43,"display":420},[45,19869,19870,19954],{},[48,19871,19872,19899,19901,19904,19906,19908,19910,19913,19915,19917,19927,19937,19940,19942,19952],{},[1531,19873,19874,19881],{},[48,19875,19876,19879],{},[51,19877,19878],{},"min",[95,19880,1540],{},[48,19882,19883,19885,19887,19889,19891,19893,19895,19897],{},[51,19884,2515],{},[95,19886,2518],{},[51,19888,2496],{"mathvariant":494},[95,19890,211],{"stretchy":210},[51,19892,852],{},[95,19894,1210],{"separator":62},[51,19896,884],{},[95,19898,220],{"stretchy":210},[1430,19900,1545],{},[95,19902,19903],{"stretchy":210},"⟨",[51,19905,8101],{},[95,19907,1210],{"separator":62},[51,19909,2515],{},[95,19911,19912],{"stretchy":210},"⟩",[95,19914,434],{},[51,19916,4793],{},[1531,19918,19919,19921],{},[95,19920,5625],{},[48,19922,19923,19925],{},[51,19924,5391],{},[51,19926,2555],{},[1004,19928,19929,19931],{},[51,19930,2515],{},[48,19932,19933,19935],{},[51,19934,5391],{},[51,19936,2555],{},[51,19938,19939],{},"log",[95,19941,1540],{},[1004,19943,19944,19946],{},[51,19945,2515],{},[48,19947,19948,19950],{},[51,19949,5391],{},[51,19951,2555],{},[51,19953,495],{"mathvariant":494},[55,19955,19956],{"encoding":57},"\\min_{\\pi\\in\\Pi(\\mu,\\nu)}\\ \\langle C,\\pi\\rangle+\\varepsilon\\sum_{ij}\\pi_{ij}\\log\\pi_{ij}.",[31,19958,19960,20066],{"className":19959,"ariaHidden":62},[61],[31,19961,19963,19967,20036,20039,20042,20045,20048,20051,20054,20057,20060,20063],{"className":19962},[66],[31,19964],{"className":19965,"style":19966},[70],"height:1.716em;vertical-align:-0.966em;",[31,19968,19970],{"className":19969},[1460,1620],[31,19971,19973,20028],{"className":19972},[1043,1044],[31,19974,19976,20025],{"className":19975},[1048],[31,19977,19980,20015],{"className":19978,"style":19979},[1052],"height:0.6679em;",[31,19981,19982,19985],{"style":3301},[31,19983],{"className":19984,"style":1637},[1060],[31,19986,19988],{"className":19987},[1065,1066,1067,1068],[31,19989,19991,19994,19997,20000,20003,20006,20009,20012],{"className":19990},[75,1068],[31,19992,2515],{"className":19993,"style":2666},[75,76,1068],[31,19995,2518],{"className":19996},[297,1068],[31,19998,2496],{"className":19999},[75,1068],[31,20001,211],{"className":20002},[264,1068],[31,20004,852],{"className":20005},[75,76,1068],[31,20007,1210],{"className":20008},[1391,1068],[31,20010,884],{"className":20011,"style":900},[75,76,1068],[31,20013,220],{"className":20014},[289,1068],[31,20016,20017,20020],{"style":1649},[31,20018],{"className":20019,"style":1637},[1060],[31,20021,20022],{},[31,20023,19878],{"className":20024},[1460],[31,20026,1076],{"className":20027},[1075],[31,20029,20031],{"className":20030},[1048],[31,20032,20034],{"className":20033,"style":3356},[1052],[31,20035],{},[31,20037,1545],{"className":20038},[119],[31,20040,19903],{"className":20041},[264],[31,20043,8101],{"className":20044,"style":8525},[75,76],[31,20046,1210],{"className":20047},[1391],[31,20049],{"className":20050,"style":1467},[119],[31,20052,2515],{"className":20053,"style":2666},[75,76],[31,20055,19912],{"className":20056},[289],[31,20058],{"className":20059,"style":120},[119],[31,20061,434],{"className":20062},[124],[31,20064],{"className":20065,"style":120},[119],[31,20067,20069,20072,20075,20078,20125,20128,20171,20174,20181,20184,20227],{"className":20068},[66],[31,20070],{"className":20071,"style":10455},[70],[31,20073,4793],{"className":20074},[75,76],[31,20076],{"className":20077,"style":1467},[119],[31,20079,20081],{"className":20080},[1460,1620],[31,20082,20084,20117],{"className":20083},[1043,1044],[31,20085,20087,20114],{"className":20086},[1048],[31,20088,20090,20104],{"className":20089,"style":5900},[1052],[31,20091,20092,20095],{"style":5903},[31,20093],{"className":20094,"style":5907},[1060],[31,20096,20098],{"className":20097},[1065,1066,1067,1068],[31,20099,20101],{"className":20100},[75,1068],[31,20102,5574],{"className":20103,"style":5514},[75,76,1068],[31,20105,20106,20109],{"style":5919},[31,20107],{"className":20108,"style":5907},[1060],[31,20110,20111],{},[31,20112,5625],{"className":20113},[1460,1461,1687],[31,20115,1076],{"className":20116},[1075],[31,20118,20120],{"className":20119},[1048],[31,20121,20123],{"className":20122,"style":5937},[1052],[31,20124],{},[31,20126],{"className":20127,"style":1467},[119],[31,20129,20131,20134],{"className":20130},[75],[31,20132,2515],{"className":20133,"style":2666},[75,76],[31,20135,20137],{"className":20136},[1039],[31,20138,20140,20163],{"className":20139},[1043,1044],[31,20141,20143,20160],{"className":20142},[1048],[31,20144,20146],{"className":20145,"style":5445},[1052],[31,20147,20148,20151],{"style":5504},[31,20149],{"className":20150,"style":1061},[1060],[31,20152,20154],{"className":20153},[1065,1066,1067,1068],[31,20155,20157],{"className":20156},[75,1068],[31,20158,5574],{"className":20159,"style":5514},[75,76,1068],[31,20161,1076],{"className":20162},[1075],[31,20164,20166],{"className":20165},[1048],[31,20167,20169],{"className":20168,"style":1083},[1052],[31,20170],{},[31,20172],{"className":20173,"style":1467},[119],[31,20175,20177,20178],{"className":20176},[1460],"lo",[31,20179,14556],{"style":20180},"margin-right:0.0139em;",[31,20182],{"className":20183,"style":1467},[119],[31,20185,20187,20190],{"className":20186},[75],[31,20188,2515],{"className":20189,"style":2666},[75,76],[31,20191,20193],{"className":20192},[1039],[31,20194,20196,20219],{"className":20195},[1043,1044],[31,20197,20199,20216],{"className":20198},[1048],[31,20200,20202],{"className":20201,"style":5445},[1052],[31,20203,20204,20207],{"style":5504},[31,20205],{"className":20206,"style":1061},[1060],[31,20208,20210],{"className":20209},[1065,1066,1067,1068],[31,20211,20213],{"className":20212},[75,1068],[31,20214,5574],{"className":20215,"style":5514},[75,76,1068],[31,20217,1076],{"className":20218},[1075],[31,20220,20222],{"className":20221},[1048],[31,20223,20225],{"className":20224,"style":1083},[1052],[31,20226],{},[31,20228,495],{"className":20229},[75],[11,20231,20232],{},"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",[31,20234,20236],{"className":20235,"translate":35},[411],[31,20237,20239,20329],{"className":20238,"translate":35},[34],[31,20240,20242],{"className":20241},[39],[41,20243,20244],{"xmlns":43,"display":420},[45,20245,20246,20326],{},[48,20247,20248,20258,20260,20266,20276,20282,20284,20287,20297,20299,20302,20304,20306,20308,20318,20320,20322,20324],{},[1004,20249,20250,20252],{},[51,20251,2515],{},[48,20253,20254,20256],{},[51,20255,5391],{},[51,20257,2555],{},[95,20259,223],{},[1004,20261,20262,20264],{},[51,20263,5324],{},[51,20265,5391],{},[1004,20267,20268,20270],{},[51,20269,4558],{},[48,20271,20272,20274],{},[51,20273,5391],{},[51,20275,2555],{},[1004,20277,20278,20280],{},[51,20279,5354],{},[51,20281,2555],{},[95,20283,1210],{"separator":62},[119,20285],{"width":20286},"2em",[1004,20288,20289,20291],{},[51,20290,4558],{},[48,20292,20293,20295],{},[51,20294,5391],{},[51,20296,2555],{},[95,20298,223],{},[51,20300,20301],{},"exp",[95,20303,1540],{},[95,20305,211],{"stretchy":210},[95,20307,97],{},[1004,20309,20310,20312],{},[51,20311,8101],{},[48,20313,20314,20316],{},[51,20315,5391],{},[51,20317,2555],{},[51,20319,3768],{"mathvariant":494},[51,20321,4793],{},[95,20323,220],{"stretchy":210},[95,20325,1210],{"separator":62},[55,20327,20328],{"encoding":57},"\\pi_{ij}=u_iK_{ij}v_j,\\qquad K_{ij}=\\exp(-C_{ij}\u002F\\varepsilon),",[31,20330,20332,20390,20581],{"className":20331,"ariaHidden":62},[61],[31,20333,20335,20338,20381,20384,20387],{"className":20334},[66],[31,20336],{"className":20337,"style":5541},[70],[31,20339,20341,20344],{"className":20340},[75],[31,20342,2515],{"className":20343,"style":2666},[75,76],[31,20345,20347],{"className":20346},[1039],[31,20348,20350,20373],{"className":20349},[1043,1044],[31,20351,20353,20370],{"className":20352},[1048],[31,20354,20356],{"className":20355,"style":5445},[1052],[31,20357,20358,20361],{"style":5504},[31,20359],{"className":20360,"style":1061},[1060],[31,20362,20364],{"className":20363},[1065,1066,1067,1068],[31,20365,20367],{"className":20366},[75,1068],[31,20368,5574],{"className":20369,"style":5514},[75,76,1068],[31,20371,1076],{"className":20372},[1075],[31,20374,20376],{"className":20375},[1048],[31,20377,20379],{"className":20378,"style":1083},[1052],[31,20380],{},[31,20382],{"className":20383,"style":293},[119],[31,20385,223],{"className":20386},[297],[31,20388],{"className":20389,"style":293},[119],[31,20391,20393,20396,20436,20479,20519,20522,20526,20529,20572,20575,20578],{"className":20392},[66],[31,20394],{"className":20395,"style":1029},[70],[31,20397,20399,20402],{"className":20398},[75],[31,20400,5324],{"className":20401},[75,76],[31,20403,20405],{"className":20404},[1039],[31,20406,20408,20428],{"className":20407},[1043,1044],[31,20409,20411,20425],{"className":20410},[1048],[31,20412,20414],{"className":20413,"style":5445},[1052],[31,20415,20416,20419],{"style":2807},[31,20417],{"className":20418,"style":1061},[1060],[31,20420,20422],{"className":20421},[1065,1066,1067,1068],[31,20423,5391],{"className":20424},[75,76,1068],[31,20426,1076],{"className":20427},[1075],[31,20429,20431],{"className":20430},[1048],[31,20432,20434],{"className":20433,"style":2050},[1052],[31,20435],{},[31,20437,20439,20442],{"className":20438},[75],[31,20440,4558],{"className":20441,"style":8525},[75,76],[31,20443,20445],{"className":20444},[1039],[31,20446,20448,20471],{"className":20447},[1043,1044],[31,20449,20451,20468],{"className":20450},[1048],[31,20452,20454],{"className":20453,"style":5445},[1052],[31,20455,20456,20459],{"style":8540},[31,20457],{"className":20458,"style":1061},[1060],[31,20460,20462],{"className":20461},[1065,1066,1067,1068],[31,20463,20465],{"className":20464},[75,1068],[31,20466,5574],{"className":20467,"style":5514},[75,76,1068],[31,20469,1076],{"className":20470},[1075],[31,20472,20474],{"className":20473},[1048],[31,20475,20477],{"className":20476,"style":1083},[1052],[31,20478],{},[31,20480,20482,20485],{"className":20481},[75],[31,20483,5354],{"className":20484,"style":2666},[75,76],[31,20486,20488],{"className":20487},[1039],[31,20489,20491,20511],{"className":20490},[1043,1044],[31,20492,20494,20508],{"className":20493},[1048],[31,20495,20497],{"className":20496,"style":5445},[1052],[31,20498,20499,20502],{"style":5504},[31,20500],{"className":20501,"style":1061},[1060],[31,20503,20505],{"className":20504},[1065,1066,1067,1068],[31,20506,2555],{"className":20507,"style":5514},[75,76,1068],[31,20509,1076],{"className":20510},[1075],[31,20512,20514],{"className":20513},[1048],[31,20515,20517],{"className":20516,"style":1083},[1052],[31,20518],{},[31,20520,1210],{"className":20521},[1391],[31,20523],{"className":20524,"style":20525},[119],"margin-right:2em;",[31,20527],{"className":20528,"style":1467},[119],[31,20530,20532,20535],{"className":20531},[75],[31,20533,4558],{"className":20534,"style":8525},[75,76],[31,20536,20538],{"className":20537},[1039],[31,20539,20541,20564],{"className":20540},[1043,1044],[31,20542,20544,20561],{"className":20543},[1048],[31,20545,20547],{"className":20546,"style":5445},[1052],[31,20548,20549,20552],{"style":8540},[31,20550],{"className":20551,"style":1061},[1060],[31,20553,20555],{"className":20554},[1065,1066,1067,1068],[31,20556,20558],{"className":20557},[75,1068],[31,20559,5574],{"className":20560,"style":5514},[75,76,1068],[31,20562,1076],{"className":20563},[1075],[31,20565,20567],{"className":20566},[1048],[31,20568,20570],{"className":20569,"style":1083},[1052],[31,20571],{},[31,20573],{"className":20574,"style":293},[119],[31,20576,223],{"className":20577},[297],[31,20579],{"className":20580,"style":293},[119],[31,20582,20584,20587,20590,20593,20596,20639,20642,20645,20648],{"className":20583},[66],[31,20585],{"className":20586,"style":1223},[70],[31,20588,20301],{"className":20589},[1460],[31,20591,211],{"className":20592},[264],[31,20594,97],{"className":20595},[75],[31,20597,20599,20602],{"className":20598},[75],[31,20600,8101],{"className":20601,"style":8525},[75,76],[31,20603,20605],{"className":20604},[1039],[31,20606,20608,20631],{"className":20607},[1043,1044],[31,20609,20611,20628],{"className":20610},[1048],[31,20612,20614],{"className":20613,"style":5445},[1052],[31,20615,20616,20619],{"style":8540},[31,20617],{"className":20618,"style":1061},[1060],[31,20620,20622],{"className":20621},[1065,1066,1067,1068],[31,20623,20625],{"className":20624},[75,1068],[31,20626,5574],{"className":20627,"style":5514},[75,76,1068],[31,20629,1076],{"className":20630},[1075],[31,20632,20634],{"className":20633},[1048],[31,20635,20637],{"className":20636,"style":1083},[1052],[31,20638],{},[31,20640,3768],{"className":20641},[75],[31,20643,4793],{"className":20644},[75,76],[31,20646,220],{"className":20647},[289],[31,20649,1210],{"className":20650},[1391],[11,20652,20653,20694,20695,15282,20772,20881],{},[31,20654,20656,20673],{"className":20655,"translate":35},[34],[31,20657,20659],{"className":20658},[39],[41,20660,20661],{"xmlns":43},[45,20662,20663,20671],{},[48,20664,20665,20667,20669],{},[51,20666,5324],{},[95,20668,1210],{"separator":62},[51,20670,5354],{},[55,20672,7008],{"encoding":57},[31,20674,20676],{"className":20675,"ariaHidden":62},[61],[31,20677,20679,20682,20685,20688,20691],{"className":20678},[66],[31,20680],{"className":20681,"style":865},[70],[31,20683,5324],{"className":20684},[75,76],[31,20686,1210],{"className":20687},[1391],[31,20689],{"className":20690,"style":1467},[119],[31,20692,5354],{"className":20693,"style":2666},[75,76]," 由边缘约束定出，交替更新 ",[31,20696,20698,20727],{"className":20697,"translate":35},[34],[31,20699,20701],{"className":20700},[39],[41,20702,20703],{"xmlns":43},[45,20704,20705,20724],{},[48,20706,20707,20709,20712,20714,20716,20718,20720,20722],{},[51,20708,5324],{},[95,20710,20711],{},"←",[51,20713,5813],{},[51,20715,3768],{"mathvariant":494},[95,20717,211],{"stretchy":210},[51,20719,4558],{},[51,20721,5354],{},[95,20723,220],{"stretchy":210},[55,20725,20726],{"encoding":57},"u\\leftarrow a\u002F(Kv)",[31,20728,20730,20748],{"className":20729,"ariaHidden":62},[61],[31,20731,20733,20736,20739,20742,20745],{"className":20732},[66],[31,20734],{"className":20735,"style":71},[70],[31,20737,5324],{"className":20738},[75,76],[31,20740],{"className":20741,"style":293},[119],[31,20743,20711],{"className":20744},[297],[31,20746],{"className":20747,"style":293},[119],[31,20749,20751,20754,20757,20760,20763,20766,20769],{"className":20750},[66],[31,20752],{"className":20753,"style":260},[70],[31,20755,5813],{"className":20756},[75,76],[31,20758,3768],{"className":20759},[75],[31,20761,211],{"className":20762},[264],[31,20764,4558],{"className":20765,"style":8525},[75,76],[31,20767,5354],{"className":20768,"style":2666},[75,76],[31,20770,220],{"className":20771},[289],[31,20773,20775,20808],{"className":20774,"translate":35},[34],[31,20776,20778],{"className":20777},[39],[41,20779,20780],{"xmlns":43},[45,20781,20782,20805],{},[48,20783,20784,20786,20788,20790,20792,20794,20801,20803],{},[51,20785,5354],{},[95,20787,20711],{},[51,20789,5834],{},[51,20791,3768],{"mathvariant":494},[95,20793,211],{"stretchy":210},[1180,20795,20796,20798],{},[51,20797,4558],{},[51,20799,20800],{"mathvariant":494},"⊤",[51,20802,5324],{},[95,20804,220],{"stretchy":210},[55,20806,20807],{"encoding":57},"v\\leftarrow b\u002F(K^\\top u)",[31,20809,20811,20829],{"className":20810,"ariaHidden":62},[61],[31,20812,20814,20817,20820,20823,20826],{"className":20813},[66],[31,20815],{"className":20816,"style":71},[70],[31,20818,5354],{"className":20819,"style":2666},[75,76],[31,20821],{"className":20822,"style":293},[119],[31,20824,20711],{"className":20825},[297],[31,20827],{"className":20828,"style":293},[119],[31,20830,20832,20836,20839,20842,20845,20875,20878],{"className":20831},[66],[31,20833],{"className":20834,"style":20835},[70],"height:1.0991em;vertical-align:-0.25em;",[31,20837,5834],{"className":20838},[75,76],[31,20840,3768],{"className":20841},[75],[31,20843,211],{"className":20844},[264],[31,20846,20848,20851],{"className":20847},[75],[31,20849,4558],{"className":20850,"style":8525},[75,76],[31,20852,20854],{"className":20853},[1039],[31,20855,20857],{"className":20856},[1043],[31,20858,20860],{"className":20859},[1048],[31,20861,20864],{"className":20862,"style":20863},[1052],"height:0.8491em;",[31,20865,20866,20869],{"style":3631},[31,20867],{"className":20868,"style":1061},[1060],[31,20870,20872],{"className":20871},[1065,1066,1067,1068],[31,20873,20800],{"className":20874},[75,1068],[31,20876,5324],{"className":20877},[75,76],[31,20879,220],{"className":20880},[289],"，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。",[11,20883,20884,20912,20913,20941,20942,20970,20971,20999,21000,21051,21052,21080],{},[31,20885,20887,20900],{"className":20886,"translate":35},[34],[31,20888,20890],{"className":20889},[39],[41,20891,20892],{"xmlns":43},[45,20893,20894,20898],{},[48,20895,20896],{},[51,20897,4793],{},[55,20899,4796],{"encoding":57},[31,20901,20903],{"className":20902,"ariaHidden":62},[61],[31,20904,20906,20909],{"className":20905},[66],[31,20907],{"className":20908,"style":71},[70],[31,20910,4793],{"className":20911},[75,76]," 可以读作一个旋钮。取三个点搬到三个点：",[31,20914,20916,20929],{"className":20915,"translate":35},[34],[31,20917,20919],{"className":20918},[39],[41,20920,20921],{"xmlns":43},[45,20922,20923,20927],{},[48,20924,20925],{},[51,20926,4793],{},[55,20928,4796],{"encoding":57},[31,20930,20932],{"className":20931,"ariaHidden":62},[61],[31,20933,20935,20938],{"className":20934},[66],[31,20936],{"className":20937,"style":71},[70],[31,20939,4793],{"className":20940},[75,76]," 很大时，",[31,20943,20945,20958],{"className":20944,"translate":35},[34],[31,20946,20948],{"className":20947},[39],[41,20949,20950],{"xmlns":43},[45,20951,20952,20956],{},[48,20953,20954],{},[51,20955,4558],{},[55,20957,4558],{"encoding":57},[31,20959,20961],{"className":20960,"ariaHidden":62},[61],[31,20962,20964,20967],{"className":20963},[66],[31,20965],{"className":20966,"style":945},[70],[31,20968,4558],{"className":20969,"style":8525},[75,76]," 接近全 ",[31,20972,20974,20987],{"className":20973,"translate":35},[34],[31,20975,20977],{"className":20976},[39],[41,20978,20979],{"xmlns":43},[45,20980,20981,20985],{},[48,20982,20983],{},[91,20984,205],{},[55,20986,205],{"encoding":57},[31,20988,20990],{"className":20989,"ariaHidden":62},[61],[31,20991,20993,20996],{"className":20992},[66],[31,20994],{"className":20995,"style":326},[70],[31,20997,205],{"className":20998},[75]," 矩阵，每个点都往每处送一点，解是稠密的、模糊的；",[31,21001,21003,21021],{"className":21002,"translate":35},[34],[31,21004,21006],{"className":21005},[39],[41,21007,21008],{"xmlns":43},[45,21009,21010,21018],{},[48,21011,21012,21014,21016],{},[51,21013,4793],{},[95,21015,929],{},[91,21017,1977],{},[55,21019,21020],{"encoding":57},"\\varepsilon\\to0",[31,21022,21024,21042],{"className":21023,"ariaHidden":62},[61],[31,21025,21027,21030,21033,21036,21039],{"className":21026},[66],[31,21028],{"className":21029,"style":71},[70],[31,21031,4793],{"className":21032},[75,76],[31,21034],{"className":21035,"style":293},[119],[31,21037,929],{"className":21038},[297],[31,21040],{"className":21041,"style":293},[119],[31,21043,21045,21048],{"className":21044},[66],[31,21046],{"className":21047,"style":326},[70],[31,21049,1977],{"className":21050},[75]," 时 ",[31,21053,21055,21068],{"className":21054,"translate":35},[34],[31,21056,21058],{"className":21057},[39],[41,21059,21060],{"xmlns":43},[45,21061,21062,21066],{},[48,21063,21064],{},[51,21065,4558],{},[55,21067,4558],{"encoding":57},[31,21069,21071],{"className":21070,"ariaHidden":62},[61],[31,21072,21074,21077],{"className":21073},[66],[31,21075],{"className":21076,"style":945},[70],[31,21078,4558],{"className":21079,"style":8525},[75,76]," 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。",[11,21082,1394,21083,21135,21136,21214,21215,21562],{},[31,21084,21086,21104],{"className":21085,"translate":35},[34],[31,21087,21089],{"className":21088},[39],[41,21090,21091],{"xmlns":43},[45,21092,21093,21101],{},[48,21094,21095,21097,21099],{},[51,21096,4793],{},[95,21098,16040],{},[91,21100,1977],{},[55,21102,21103],{"encoding":57},"\\varepsilon>0",[31,21105,21107,21126],{"className":21106,"ariaHidden":62},[61],[31,21108,21110,21114,21117,21120,21123],{"className":21109},[66],[31,21111],{"className":21112,"style":21113},[70],"height:0.5782em;vertical-align:-0.0391em;",[31,21115,4793],{"className":21116},[75,76],[31,21118],{"className":21119,"style":293},[119],[31,21121,16040],{"className":21122},[297],[31,21124],{"className":21125,"style":293},[119],[31,21127,21129,21132],{"className":21128},[66],[31,21130],{"className":21131,"style":326},[70],[31,21133,1977],{"className":21134},[75]," 时它不再是度量。最直接的一点：把 ",[31,21137,21139,21171],{"className":21138,"translate":35},[34],[31,21140,21142],{"className":21141},[39],[41,21143,21144],{"xmlns":43},[45,21145,21146,21168],{},[48,21147,21148,21150,21152,21162,21164,21166],{},[51,21149,2515],{},[95,21151,223],{},[48,21153,21154,21156,21158,21160],{},[51,21155,1435],{"mathvariant":494},[51,21157,5391],{"mathvariant":494},[51,21159,5813],{"mathvariant":494},[51,21161,14556],{"mathvariant":494},[95,21163,211],{"stretchy":210},[51,21165,5813],{},[95,21167,220],{"stretchy":210},[55,21169,21170],{"encoding":57},"\\pi=\\mathrm{diag}(a)",[31,21172,21174,21192],{"className":21173,"ariaHidden":62},[61],[31,21175,21177,21180,21183,21186,21189],{"className":21176},[66],[31,21178],{"className":21179,"style":71},[70],[31,21181,2515],{"className":21182,"style":2666},[75,76],[31,21184],{"className":21185,"style":293},[119],[31,21187,223],{"className":21188},[297],[31,21190],{"className":21191,"style":293},[119],[31,21193,21195,21198,21205,21208,21211],{"className":21194},[66],[31,21196],{"className":21197,"style":260},[70],[31,21199,21201],{"className":21200},[75],[31,21202,21204],{"className":21203,"style":20180},[75,2745],"diag",[31,21206,211],{"className":21207},[264],[31,21209,5813],{"className":21210},[75,76],[31,21212,220],{"className":21213},[289]," 代入，",[31,21216,21218,21277],{"className":21217,"translate":35},[34],[31,21219,21221],{"className":21220},[39],[41,21222,21223],{"xmlns":43},[45,21224,21225,21274],{},[48,21226,21227,21233,21235,21237,21239,21241,21243,21245,21247,21253,21259,21261,21263,21269,21272],{},[1004,21228,21229,21231],{},[51,21230,3117],{},[51,21232,4793],{},[95,21234,211],{"stretchy":210},[51,21236,852],{},[95,21238,1210],{"separator":62},[51,21240,852],{},[95,21242,220],{"stretchy":210},[95,21244,223],{},[51,21246,4793],{},[1004,21248,21249,21251],{},[95,21250,5625],{},[51,21252,5391],{},[1004,21254,21255,21257],{},[51,21256,5813],{},[51,21258,5391],{},[51,21260,19939],{},[95,21262,1540],{},[1004,21264,21265,21267],{},[51,21266,5813],{},[51,21268,5391],{},[95,21270,21271],{"mathvariant":494},"≠",[91,21273,1977],{},[55,21275,21276],{"encoding":57},"W_\\varepsilon(\\mu,\\mu)=\\varepsilon\\sum_i a_i\\log a_i\\ne0",[31,21278,21280,21353,21553],{"className":21279,"ariaHidden":62},[61],[31,21281,21283,21286,21326,21329,21332,21335,21338,21341,21344,21347,21350],{"className":21282},[66],[31,21284],{"className":21285,"style":260},[70],[31,21287,21289,21292],{"className":21288},[75],[31,21290,3117],{"className":21291,"style":949},[75,76],[31,21293,21295],{"className":21294},[1039],[31,21296,21298,21318],{"className":21297},[1043,1044],[31,21299,21301,21315],{"className":21300},[1048],[31,21302,21304],{"className":21303,"style":3230},[1052],[31,21305,21306,21309],{"style":1056},[31,21307],{"className":21308,"style":1061},[1060],[31,21310,21312],{"className":21311},[1065,1066,1067,1068],[31,21313,4793],{"className":21314},[75,76,1068],[31,21316,1076],{"className":21317},[1075],[31,21319,21321],{"className":21320},[1048],[31,21322,21324],{"className":21323,"style":2050},[1052],[31,21325],{},[31,21327,211],{"className":21328},[264],[31,21330,852],{"className":21331},[75,76],[31,21333,1210],{"className":21334},[1391],[31,21336],{"className":21337,"style":1467},[119],[31,21339,852],{"className":21340},[75,76],[31,21342,220],{"className":21343},[289],[31,21345],{"className":21346,"style":293},[119],[31,21348,223],{"className":21349},[297],[31,21351],{"className":21352,"style":293},[119],[31,21354,21356,21360,21363,21366,21410,21413,21453,21456,21461,21464,21504,21507,21550],{"className":21355},[66],[31,21357],{"className":21358,"style":21359},[70],"height:1.0497em;vertical-align:-0.2997em;",[31,21361,4793],{"className":21362},[75,76],[31,21364],{"className":21365,"style":1467},[119],[31,21367,21369,21373],{"className":21368},[1460],[31,21370,5625],{"className":21371,"style":21372},[1460,1461,1462],"position:relative;top:0em;",[31,21374,21376],{"className":21375},[1039],[31,21377,21379,21401],{"className":21378},[1043,1044],[31,21380,21382,21398],{"className":21381},[1048],[31,21383,21386],{"className":21384,"style":21385},[1052],"height:0.162em;",[31,21387,21389,21392],{"style":21388},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[31,21390],{"className":21391,"style":1061},[1060],[31,21393,21395],{"className":21394},[1065,1066,1067,1068],[31,21396,5391],{"className":21397},[75,76,1068],[31,21399,1076],{"className":21400},[1075],[31,21402,21404],{"className":21403},[1048],[31,21405,21408],{"className":21406,"style":21407},[1052],"height:0.2997em;",[31,21409],{},[31,21411],{"className":21412,"style":1467},[119],[31,21414,21416,21419],{"className":21415},[75],[31,21417,5813],{"className":21418},[75,76],[31,21420,21422],{"className":21421},[1039],[31,21423,21425,21445],{"className":21424},[1043,1044],[31,21426,21428,21442],{"className":21427},[1048],[31,21429,21431],{"className":21430,"style":5445},[1052],[31,21432,21433,21436],{"style":2807},[31,21434],{"className":21435,"style":1061},[1060],[31,21437,21439],{"className":21438},[1065,1066,1067,1068],[31,21440,5391],{"className":21441},[75,76,1068],[31,21443,1076],{"className":21444},[1075],[31,21446,21448],{"className":21447},[1048],[31,21449,21451],{"className":21450,"style":2050},[1052],[31,21452],{},[31,21454],{"className":21455,"style":1467},[119],[31,21457,20177,21459],{"className":21458},[1460],[31,21460,14556],{"style":20180},[31,21462],{"className":21463,"style":1467},[119],[31,21465,21467,21470],{"className":21466},[75],[31,21468,5813],{"className":21469},[75,76],[31,21471,21473],{"className":21472},[1039],[31,21474,21476,21496],{"className":21475},[1043,1044],[31,21477,21479,21493],{"className":21478},[1048],[31,21480,21482],{"className":21481,"style":5445},[1052],[31,21483,21484,21487],{"style":2807},[31,21485],{"className":21486,"style":1061},[1060],[31,21488,21490],{"className":21489},[1065,1066,1067,1068],[31,21491,5391],{"className":21492},[75,76,1068],[31,21494,1076],{"className":21495},[1075],[31,21497,21499],{"className":21498},[1048],[31,21500,21502],{"className":21501,"style":2050},[1052],[31,21503],{},[31,21505],{"className":21506,"style":293},[119],[31,21508,21510,21543,21547],{"className":21509},[297],[31,21511,21513],{"className":21512},[297],[31,21514,21517],{"className":21515},[75,21516],"vbox",[31,21518,21521],{"className":21519},[21520],"thinbox",[31,21522,21525,21528,21539],{"className":21523},[21524],"rlap",[31,21526],{"className":21527,"style":4202},[70],[31,21529,21532],{"className":21530},[21531],"inner",[31,21533,21535],{"className":21534},[75],[31,21536,21538],{"className":21537},[297],"",[31,21540],{"className":21541},[21542],"fix",[31,21544],{"className":21545},[119,21546],"nobreak",[31,21548,223],{"className":21549},[297],[31,21551],{"className":21552,"style":293},[119],[31,21554,21556,21559],{"className":21555},[66],[31,21557],{"className":21558,"style":326},[70],[31,21560,1977],{"className":21561},[75],"。所以工程上常用的 Sinkhorn 散度要先去偏：",[31,21564,21566],{"className":21565,"translate":35},[411],[31,21567,21569,21682],{"className":21568,"translate":35},[34],[31,21570,21572],{"className":21571},[39],[41,21573,21574],{"xmlns":43,"display":420},[45,21575,21576,21679],{},[48,21577,21578,21585,21587,21589,21591,21593,21595,21597,21607,21609,21611,21613,21615,21617,21619,21627,21637,21639,21641,21643,21645,21647,21649,21657,21667,21669,21671,21673,21675,21677],{},[1004,21579,21580,21583],{},[51,21581,21582],{},"S",[51,21584,4793],{},[95,21586,211],{"stretchy":210},[51,21588,852],{},[95,21590,1210],{"separator":62},[51,21592,884],{},[95,21594,220],{"stretchy":210},[95,21596,223],{},[1004,21598,21599,21605],{},[48,21600,21601,21603],{},[51,21602,19733],{"mathvariant":494},[51,21604,919],{"mathvariant":494},[51,21606,4793],{},[95,21608,211],{"stretchy":210},[51,21610,852],{},[95,21612,1210],{"separator":62},[51,21614,884],{},[95,21616,220],{"stretchy":210},[95,21618,97],{},[2071,21620,21621],{"scriptlevel":1977,"displaystyle":210},[2074,21622,21623,21625],{},[91,21624,205],{},[91,21626,152],{},[1004,21628,21629,21635],{},[48,21630,21631,21633],{},[51,21632,19733],{"mathvariant":494},[51,21634,919],{"mathvariant":494},[51,21636,4793],{},[95,21638,211],{"stretchy":210},[51,21640,852],{},[95,21642,1210],{"separator":62},[51,21644,852],{},[95,21646,220],{"stretchy":210},[95,21648,97],{},[2071,21650,21651],{"scriptlevel":1977,"displaystyle":210},[2074,21652,21653,21655],{},[91,21654,205],{},[91,21656,152],{},[1004,21658,21659,21665],{},[48,21660,21661,21663],{},[51,21662,19733],{"mathvariant":494},[51,21664,919],{"mathvariant":494},[51,21666,4793],{},[95,21668,211],{"stretchy":210},[51,21670,884],{},[95,21672,1210],{"separator":62},[51,21674,884],{},[95,21676,220],{"stretchy":210},[51,21678,495],{"mathvariant":494},[55,21680,21681],{"encoding":57},"S_\\varepsilon(\\mu,\\nu)=\\mathrm{OT}_\\varepsilon(\\mu,\\nu)-\\tfrac12\\mathrm{OT}_\\varepsilon(\\mu,\\mu)-\\tfrac12\\mathrm{OT}_\\varepsilon(\\nu,\\nu).",[31,21683,21685,21760,21838,21982],{"className":21684,"ariaHidden":62},[61],[31,21686,21688,21691,21733,21736,21739,21742,21745,21748,21751,21754,21757],{"className":21687},[66],[31,21689],{"className":21690,"style":260},[70],[31,21692,21694,21698],{"className":21693},[75],[31,21695,21582],{"className":21696,"style":21697},[75,76],"margin-right:0.0576em;",[31,21699,21701],{"className":21700},[1039],[31,21702,21704,21725],{"className":21703},[1043,1044],[31,21705,21707,21722],{"className":21706},[1048],[31,21708,21710],{"className":21709,"style":3230},[1052],[31,21711,21713,21716],{"style":21712},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[31,21714],{"className":21715,"style":1061},[1060],[31,21717,21719],{"className":21718},[1065,1066,1067,1068],[31,21720,4793],{"className":21721},[75,76,1068],[31,21723,1076],{"className":21724},[1075],[31,21726,21728],{"className":21727},[1048],[31,21729,21731],{"className":21730,"style":2050},[1052],[31,21732],{},[31,21734,211],{"className":21735},[264],[31,21737,852],{"className":21738},[75,76],[31,21740,1210],{"className":21741},[1391],[31,21743],{"className":21744,"style":1467},[119],[31,21746,884],{"className":21747,"style":900},[75,76],[31,21749,220],{"className":21750},[289],[31,21752],{"className":21753,"style":293},[119],[31,21755,223],{"className":21756},[297],[31,21758],{"className":21759,"style":293},[119],[31,21761,21763,21766,21811,21814,21817,21820,21823,21826,21829,21832,21835],{"className":21762},[66],[31,21764],{"className":21765,"style":260},[70],[31,21767,21769,21776],{"className":21768},[75],[31,21770,21772],{"className":21771},[75],[31,21773,21775],{"className":21774},[75,2745],"OT",[31,21777,21779],{"className":21778},[1039],[31,21780,21782,21803],{"className":21781},[1043,1044],[31,21783,21785,21800],{"className":21784},[1048],[31,21786,21788],{"className":21787,"style":3230},[1052],[31,21789,21791,21794],{"style":21790},"top:-2.55em;margin-right:0.05em;",[31,21792],{"className":21793,"style":1061},[1060],[31,21795,21797],{"className":21796},[1065,1066,1067,1068],[31,21798,4793],{"className":21799},[75,76,1068],[31,21801,1076],{"className":21802},[1075],[31,21804,21806],{"className":21805},[1048],[31,21807,21809],{"className":21808,"style":2050},[1052],[31,21810],{},[31,21812,211],{"className":21813},[264],[31,21815,852],{"className":21816},[75,76],[31,21818,1210],{"className":21819},[1391],[31,21821],{"className":21822,"style":1467},[119],[31,21824,884],{"className":21825,"style":900},[75,76],[31,21827,220],{"className":21828},[289],[31,21830],{"className":21831,"style":120},[119],[31,21833,97],{"className":21834},[124],[31,21836],{"className":21837,"style":120},[119],[31,21839,21841,21844,21912,21955,21958,21961,21964,21967,21970,21973,21976,21979],{"className":21840},[66],[31,21842],{"className":21843,"style":2132},[70],[31,21845,21847,21850,21909],{"className":21846},[75],[31,21848],{"className":21849},[264,2139],[31,21851,21853],{"className":21852},[2074],[31,21854,21856,21901],{"className":21855},[1043,1044],[31,21857,21859,21898],{"className":21858},[1048],[31,21860,21862,21876,21884],{"className":21861,"style":2152},[1052],[31,21863,21864,21867],{"style":2155},[31,21865],{"className":21866,"style":1637},[1060],[31,21868,21870],{"className":21869},[1065,1066,1067,1068],[31,21871,21873],{"className":21872},[75,1068],[31,21874,152],{"className":21875},[75,1068],[31,21877,21878,21881],{"style":2170},[31,21879],{"className":21880,"style":1637},[1060],[31,21882],{"className":21883,"style":2178},[2177],[31,21885,21886,21889],{"style":2181},[31,21887],{"className":21888,"style":1637},[1060],[31,21890,21892],{"className":21891},[1065,1066,1067,1068],[31,21893,21895],{"className":21894},[75,1068],[31,21896,205],{"className":21897},[75,1068],[31,21899,1076],{"className":21900},[1075],[31,21902,21904],{"className":21903},[1048],[31,21905,21907],{"className":21906,"style":2203},[1052],[31,21908],{},[31,21910],{"className":21911},[289,2139],[31,21913,21915,21921],{"className":21914},[75],[31,21916,21918],{"className":21917},[75],[31,21919,21775],{"className":21920},[75,2745],[31,21922,21924],{"className":21923},[1039],[31,21925,21927,21947],{"className":21926},[1043,1044],[31,21928,21930,21944],{"className":21929},[1048],[31,21931,21933],{"className":21932,"style":3230},[1052],[31,21934,21935,21938],{"style":21790},[31,21936],{"className":21937,"style":1061},[1060],[31,21939,21941],{"className":21940},[1065,1066,1067,1068],[31,21942,4793],{"className":21943},[75,76,1068],[31,21945,1076],{"className":21946},[1075],[31,21948,21950],{"className":21949},[1048],[31,21951,21953],{"className":21952,"style":2050},[1052],[31,21954],{},[31,21956,211],{"className":21957},[264],[31,21959,852],{"className":21960},[75,76],[31,21962,1210],{"className":21963},[1391],[31,21965],{"className":21966,"style":1467},[119],[31,21968,852],{"className":21969},[75,76],[31,21971,220],{"className":21972},[289],[31,21974],{"className":21975,"style":120},[119],[31,21977,97],{"className":21978},[124],[31,21980],{"className":21981,"style":120},[119],[31,21983,21985,21988,22056,22099,22102,22105,22108,22111,22114,22117],{"className":21984},[66],[31,21986],{"className":21987,"style":2132},[70],[31,21989,21991,21994,22053],{"className":21990},[75],[31,21992],{"className":21993},[264,2139],[31,21995,21997],{"className":21996},[2074],[31,21998,22000,22045],{"className":21999},[1043,1044],[31,22001,22003,22042],{"className":22002},[1048],[31,22004,22006,22020,22028],{"className":22005,"style":2152},[1052],[31,22007,22008,22011],{"style":2155},[31,22009],{"className":22010,"style":1637},[1060],[31,22012,22014],{"className":22013},[1065,1066,1067,1068],[31,22015,22017],{"className":22016},[75,1068],[31,22018,152],{"className":22019},[75,1068],[31,22021,22022,22025],{"style":2170},[31,22023],{"className":22024,"style":1637},[1060],[31,22026],{"className":22027,"style":2178},[2177],[31,22029,22030,22033],{"style":2181},[31,22031],{"className":22032,"style":1637},[1060],[31,22034,22036],{"className":22035},[1065,1066,1067,1068],[31,22037,22039],{"className":22038},[75,1068],[31,22040,205],{"className":22041},[75,1068],[31,22043,1076],{"className":22044},[1075],[31,22046,22048],{"className":22047},[1048],[31,22049,22051],{"className":22050,"style":2203},[1052],[31,22052],{},[31,22054],{"className":22055},[289,2139],[31,22057,22059,22065],{"className":22058},[75],[31,22060,22062],{"className":22061},[75],[31,22063,21775],{"className":22064},[75,2745],[31,22066,22068],{"className":22067},[1039],[31,22069,22071,22091],{"className":22070},[1043,1044],[31,22072,22074,22088],{"className":22073},[1048],[31,22075,22077],{"className":22076,"style":3230},[1052],[31,22078,22079,22082],{"style":21790},[31,22080],{"className":22081,"style":1061},[1060],[31,22083,22085],{"className":22084},[1065,1066,1067,1068],[31,22086,4793],{"className":22087},[75,76,1068],[31,22089,1076],{"className":22090},[1075],[31,22092,22094],{"className":22093},[1048],[31,22095,22097],{"className":22096,"style":2050},[1052],[31,22098],{},[31,22100,211],{"className":22101},[264],[31,22103,884],{"className":22104,"style":900},[75,76],[31,22106,1210],{"className":22107},[1391],[31,22109],{"className":22110,"style":1467},[119],[31,22112,884],{"className":22113,"style":900},[75,76],[31,22115,220],{"className":22116},[289],[31,22118,495],{"className":22119},[75],[11,22121,22122,22123,22174],{},"去偏之后才非负、才在 ",[31,22124,22126,22144],{"className":22125,"translate":35},[34],[31,22127,22129],{"className":22128},[39],[41,22130,22131],{"xmlns":43},[45,22132,22133,22141],{},[48,22134,22135,22137,22139],{},[51,22136,852],{},[95,22138,223],{},[51,22140,884],{},[55,22142,22143],{"encoding":57},"\\mu=\\nu",[31,22145,22147,22165],{"className":22146,"ariaHidden":62},[61],[31,22148,22150,22153,22156,22159,22162],{"className":22149},[66],[31,22151],{"className":22152,"style":865},[70],[31,22154,852],{"className":22155},[75,76],[31,22157],{"className":22158,"style":293},[119],[31,22160,223],{"className":22161},[297],[31,22163],{"className":22164,"style":293},[119],[31,22166,22168,22171],{"className":22167},[66],[31,22169],{"className":22170,"style":71},[70],[31,22172,884],{"className":22173,"style":900},[75,76]," 处为零。",[11,22176,22177,22178,22248,22249,22277,22278,22306],{},"还有一个更根本的代价，来自样本。经验分布 ",[31,22179,22181,22199],{"className":22180,"translate":35},[34],[31,22182,22184],{"className":22183},[39],[41,22185,22186],{"xmlns":43},[45,22187,22188,22196],{},[48,22189,22190],{},[1004,22191,22192,22194],{},[51,22193,852],{},[51,22195,14559],{},[55,22197,22198],{"encoding":57},"\\mu_n",[31,22200,22202],{"className":22201,"ariaHidden":62},[61],[31,22203,22205,22208],{"className":22204},[66],[31,22206],{"className":22207,"style":865},[70],[31,22209,22211,22214],{"className":22210},[75],[31,22212,852],{"className":22213},[75,76],[31,22215,22217],{"className":22216},[1039],[31,22218,22220,22240],{"className":22219},[1043,1044],[31,22221,22223,22237],{"className":22222},[1048],[31,22224,22226],{"className":22225,"style":3230},[1052],[31,22227,22228,22231],{"style":2807},[31,22229],{"className":22230,"style":1061},[1060],[31,22232,22234],{"className":22233},[1065,1066,1067,1068],[31,22235,14559],{"className":22236},[75,76,1068],[31,22238,1076],{"className":22239},[1075],[31,22241,22243],{"className":22242},[1048],[31,22244,22246],{"className":22245,"style":2050},[1052],[31,22247],{}," 到真分布 ",[31,22250,22252,22265],{"className":22251,"translate":35},[34],[31,22253,22255],{"className":22254},[39],[41,22256,22257],{"xmlns":43},[45,22258,22259,22263],{},[48,22260,22261],{},[51,22262,852],{},[55,22264,855],{"encoding":57},[31,22266,22268],{"className":22267,"ariaHidden":62},[61],[31,22269,22271,22274],{"className":22270},[66],[31,22272],{"className":22273,"style":865},[70],[31,22275,852],{"className":22276},[75,76]," 的收敛速度，在 ",[31,22279,22281,22294],{"className":22280,"translate":35},[34],[31,22282,22284],{"className":22283},[39],[41,22285,22286],{"xmlns":43},[45,22287,22288,22292],{},[48,22289,22290],{},[51,22291,1435],{},[55,22293,1435],{"encoding":57},[31,22295,22297],{"className":22296,"ariaHidden":62},[61],[31,22298,22300,22303],{"className":22299},[66],[31,22301],{"className":22302,"style":1630},[70],[31,22304,1435],{"className":22305},[75,76]," 维空间里大致是",[31,22308,22310],{"className":22309,"translate":35},[411],[31,22311,22313,22387],{"className":22312,"translate":35},[34],[31,22314,22316],{"className":22315},[39],[41,22317,22318],{"xmlns":43,"display":420},[45,22319,22320,22384],{},[48,22321,22322,22325,22327,22333,22335,22341,22343,22345,22347,22349,22352,22354,22368,22370,22372,22374,22376,22378,22380,22382],{},[51,22323,22324],{"mathvariant":14789},"E",[1430,22326,1432],{},[1004,22328,22329,22331],{},[51,22330,3117],{},[51,22332,11],{},[95,22334,211],{"stretchy":210},[1004,22336,22337,22339],{},[51,22338,852],{},[51,22340,14559],{},[95,22342,1210],{"separator":62},[51,22344,852],{},[95,22346,220],{"stretchy":210},[1430,22348,1545],{},[95,22350,22351],{},"≲",[1430,22353,1545],{},[1180,22355,22356,22358],{},[51,22357,14559],{},[48,22359,22360,22362,22364,22366],{},[95,22361,97],{},[91,22363,205],{},[51,22365,3768],{"mathvariant":494},[51,22367,1435],{},[119,22369],{"width":20286},[95,22371,211],{"stretchy":210},[51,22373,1435],{},[95,22375,16040],{},[91,22377,152],{},[51,22379,11],{},[95,22381,220],{"stretchy":210},[51,22383,495],{"mathvariant":494},[55,22385,22386],{"encoding":57},"\\mathbb E\\,W_p(\\mu_n,\\mu)\\ \\lesssim\\ n^{-1\u002Fd}\\qquad(d>2p).",[31,22388,22390,22513,22577],{"className":22389,"ariaHidden":62},[61],[31,22391,22393,22396,22399,22402,22442,22445,22485,22488,22491,22494,22497,22500,22503,22507,22510],{"className":22392},[66],[31,22394],{"className":22395,"style":1223},[70],[31,22397,22324],{"className":22398},[75,14947],[31,22400],{"className":22401,"style":1467},[119],[31,22403,22405,22408],{"className":22404},[75],[31,22406,3117],{"className":22407,"style":949},[75,76],[31,22409,22411],{"className":22410},[1039],[31,22412,22414,22434],{"className":22413},[1043,1044],[31,22415,22417,22431],{"className":22416},[1048],[31,22418,22420],{"className":22419,"style":3230},[1052],[31,22421,22422,22425],{"style":1056},[31,22423],{"className":22424,"style":1061},[1060],[31,22426,22428],{"className":22427},[1065,1066,1067,1068],[31,22429,11],{"className":22430},[75,76,1068],[31,22432,1076],{"className":22433},[1075],[31,22435,22437],{"className":22436},[1048],[31,22438,22440],{"className":22439,"style":1083},[1052],[31,22441],{},[31,22443,211],{"className":22444},[264],[31,22446,22448,22451],{"className":22447},[75],[31,22449,852],{"className":22450},[75,76],[31,22452,22454],{"className":22453},[1039],[31,22455,22457,22477],{"className":22456},[1043,1044],[31,22458,22460,22474],{"className":22459},[1048],[31,22461,22463],{"className":22462,"style":3230},[1052],[31,22464,22465,22468],{"style":2807},[31,22466],{"className":22467,"style":1061},[1060],[31,22469,22471],{"className":22470},[1065,1066,1067,1068],[31,22472,14559],{"className":22473},[75,76,1068],[31,22475,1076],{"className":22476},[1075],[31,22478,22480],{"className":22479},[1048],[31,22481,22483],{"className":22482,"style":2050},[1052],[31,22484],{},[31,22486,1210],{"className":22487},[1391],[31,22489],{"className":22490,"style":1467},[119],[31,22492,852],{"className":22493},[75,76],[31,22495,220],{"className":22496},[289],[31,22498,1545],{"className":22499},[119],[31,22501],{"className":22502,"style":293},[119],[31,22504,22351],{"className":22505},[297,22506],"amsrm",[31,22508,1545],{"className":22509},[119],[31,22511],{"className":22512,"style":293},[119],[31,22514,22516,22520,22559,22562,22565,22568,22571,22574],{"className":22515},[66],[31,22517],{"className":22518,"style":22519},[70],"height:1.188em;vertical-align:-0.25em;",[31,22521,22523,22526],{"className":22522},[75],[31,22524,14559],{"className":22525},[75,76],[31,22527,22529],{"className":22528},[1039],[31,22530,22532],{"className":22531},[1043],[31,22533,22535],{"className":22534},[1048],[31,22536,22539],{"className":22537,"style":22538},[1052],"height:0.938em;",[31,22540,22541,22544],{"style":1330},[31,22542],{"className":22543,"style":1061},[1060],[31,22545,22547],{"className":22546},[1065,1066,1067,1068],[31,22548,22550,22553,22556],{"className":22549},[75,1068],[31,22551,97],{"className":22552},[75,1068],[31,22554,4042],{"className":22555},[75,1068],[31,22557,1435],{"className":22558},[75,76,1068],[31,22560],{"className":22561,"style":20525},[119],[31,22563,211],{"className":22564},[264],[31,22566,1435],{"className":22567},[75,76],[31,22569],{"className":22570,"style":293},[119],[31,22572,16040],{"className":22573},[297],[31,22575],{"className":22576,"style":293},[119],[31,22578,22580,22583,22586,22589,22592],{"className":22579},[66],[31,22581],{"className":22582,"style":260},[70],[31,22584,152],{"className":22585},[75],[31,22587,11],{"className":22588},[75,76],[31,22590,220],{"className":22591},[289],[31,22593,495],{"className":22594},[75],[11,22596,22597,22598,22650,22651,22721,22722,22784],{},"指数太糟糕了。",[31,22599,22601,22620],{"className":22600,"translate":35},[34],[31,22602,22604],{"className":22603},[39],[41,22605,22606],{"xmlns":43},[45,22607,22608,22617],{},[48,22609,22610,22612,22614],{},[51,22611,1435],{},[95,22613,223],{},[91,22615,22616],{},"100",[55,22618,22619],{"encoding":57},"d=100",[31,22621,22623,22641],{"className":22622,"ariaHidden":62},[61],[31,22624,22626,22629,22632,22635,22638],{"className":22625},[66],[31,22627],{"className":22628,"style":1630},[70],[31,22630,1435],{"className":22631},[75,76],[31,22633],{"className":22634,"style":293},[119],[31,22636,223],{"className":22637},[297],[31,22639],{"className":22640,"style":293},[119],[31,22642,22644,22647],{"className":22643},[66],[31,22645],{"className":22646,"style":326},[70],[31,22648,22616],{"className":22649},[75]," 时误差按 ",[31,22652,22654,22677],{"className":22653,"translate":35},[34],[31,22655,22657],{"className":22656},[39],[41,22658,22659],{"xmlns":43},[45,22660,22661,22674],{},[48,22662,22663],{},[1180,22664,22665,22667],{},[51,22666,14559],{},[48,22668,22669,22671],{},[95,22670,97],{},[91,22672,22673],{},"0.01",[55,22675,22676],{"encoding":57},"n^{-0.01}",[31,22678,22680],{"className":22679,"ariaHidden":62},[61],[31,22681,22683,22686],{"className":22682},[66],[31,22684],{"className":22685,"style":5135},[70],[31,22687,22689,22692],{"className":22688},[75],[31,22690,14559],{"className":22691},[75,76],[31,22693,22695],{"className":22694},[1039],[31,22696,22698],{"className":22697},[1043],[31,22699,22701],{"className":22700},[1048],[31,22702,22704],{"className":22703,"style":5135},[1052],[31,22705,22706,22709],{"style":3631},[31,22707],{"className":22708,"style":1061},[1060],[31,22710,22712],{"className":22711},[1065,1066,1067,1068],[31,22713,22715,22718],{"className":22714},[75,1068],[31,22716,97],{"className":22717},[75,1068],[31,22719,22673],{"className":22720},[75,1068]," 衰减，想把误差减半，样本量要乘 ",[31,22723,22725,22743],{"className":22724,"translate":35},[34],[31,22726,22728],{"className":22727},[39],[41,22729,22730],{"xmlns":43},[45,22731,22732,22740],{},[48,22733,22734],{},[1180,22735,22736,22738],{},[91,22737,152],{},[91,22739,22616],{},[55,22741,22742],{"encoding":57},"2^{100}",[31,22744,22746],{"className":22745,"ariaHidden":62},[61],[31,22747,22749,22752],{"className":22748},[66],[31,22750],{"className":22751,"style":5135},[70],[31,22753,22755,22758],{"className":22754},[75],[31,22756,152],{"className":22757},[75],[31,22759,22761],{"className":22760},[1039],[31,22762,22764],{"className":22763},[1043],[31,22765,22767],{"className":22766},[1048],[31,22768,22770],{"className":22769,"style":5135},[1052],[31,22771,22772,22775],{"style":3631},[31,22773],{"className":22774,"style":1061},[1060],[31,22776,22778],{"className":22777},[1065,1066,1067,1068],[31,22779,22781],{"className":22780},[75,1068],[31,22782,22616],{"className":22783},[75,1068],"。所以“直接在高维空间算 Wasserstein”在绝大多数真实问题上不可行。",[11,22786,22787,22788,22816,22817,22891],{},"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 ",[31,22789,22791,22804],{"className":22790,"translate":35},[34],[31,22792,22794],{"className":22793},[39],[41,22795,22796],{"xmlns":43},[45,22797,22798,22802],{},[48,22799,22800],{},[51,22801,3117],{},[55,22803,3117],{"encoding":57},[31,22805,22807],{"className":22806,"ariaHidden":62},[61],[31,22808,22810,22813],{"className":22809},[66],[31,22811],{"className":22812,"style":945},[70],[31,22814,3117],{"className":22815,"style":949},[75,76],"，再对方向平均（一维的收敛是 ",[31,22818,22820,22846],{"className":22819,"translate":35},[34],[31,22821,22823],{"className":22822},[39],[41,22824,22825],{"xmlns":43},[45,22826,22827,22843],{},[48,22828,22829],{},[1180,22830,22831,22833],{},[51,22832,14559],{},[48,22834,22835,22837,22839,22841],{},[95,22836,97],{},[91,22838,205],{},[51,22840,3768],{"mathvariant":494},[91,22842,152],{},[55,22844,22845],{"encoding":57},"n^{-1\u002F2}",[31,22847,22849],{"className":22848,"ariaHidden":62},[61],[31,22850,22852,22856],{"className":22851},[66],[31,22853],{"className":22854,"style":22855},[70],"height:0.888em;",[31,22857,22859,22862],{"className":22858},[75],[31,22860,14559],{"className":22861},[75,76],[31,22863,22865],{"className":22864},[1039],[31,22866,22868],{"className":22867},[1043],[31,22869,22871],{"className":22870},[1048],[31,22872,22874],{"className":22873,"style":22855},[1052],[31,22875,22876,22879],{"style":3631},[31,22877],{"className":22878,"style":1061},[1060],[31,22880,22882],{"className":22881},[1065,1066,1067,1068],[31,22883,22885,22888],{"className":22884},[75,1068],[31,22886,97],{"className":22887},[75,1068],[31,22889,17145],{"className":22890},[75,1068],"，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。",[18,22893,22894],{"id":22894},"单纯形上的尺子",[11,22896,22897,22898,2446,22952,23005,23006,2446,23059,2372],{},"我拿两个模型对同一张图跑一遍，输出分别是 ",[31,22899,22901,22925],{"className":22900,"translate":35},[34],[31,22902,22904],{"className":22903},[39],[41,22905,22906],{"xmlns":43},[45,22907,22908,22922],{},[48,22909,22910,22912,22915,22917,22920],{},[95,22911,211],{"stretchy":210},[91,22913,22914],{},"0.90",[95,22916,1210],{"separator":62},[91,22918,22919],{},"0.10",[95,22921,220],{"stretchy":210},[55,22923,22924],{"encoding":57},"(0.90,0.10)",[31,22926,22928],{"className":22927,"ariaHidden":62},[61],[31,22929,22931,22934,22937,22940,22943,22946,22949],{"className":22930},[66],[31,22932],{"className":22933,"style":260},[70],[31,22935,211],{"className":22936},[264],[31,22938,22914],{"className":22939},[75],[31,22941,1210],{"className":22942},[1391],[31,22944],{"className":22945,"style":1467},[119],[31,22947,22919],{"className":22948},[75],[31,22950,220],{"className":22951},[289],[31,22953,22955,22978],{"className":22954,"translate":35},[34],[31,22956,22958],{"className":22957},[39],[41,22959,22960],{"xmlns":43},[45,22961,22962,22975],{},[48,22963,22964,22966,22969,22971,22973],{},[95,22965,211],{"stretchy":210},[91,22967,22968],{},"0.99",[95,22970,1210],{"separator":62},[91,22972,22673],{},[95,22974,220],{"stretchy":210},[55,22976,22977],{"encoding":57},"(0.99,0.01)",[31,22979,22981],{"className":22980,"ariaHidden":62},[61],[31,22982,22984,22987,22990,22993,22996,22999,23002],{"className":22983},[66],[31,22985],{"className":22986,"style":260},[70],[31,22988,211],{"className":22989},[264],[31,22991,22968],{"className":22992},[75],[31,22994,1210],{"className":22995},[1391],[31,22997],{"className":22998,"style":1467},[119],[31,23000,22673],{"className":23001},[75],[31,23003,220],{"className":23004},[289],"。另一张图上，两个模型的输出是 ",[31,23007,23009,23032],{"className":23008,"translate":35},[34],[31,23010,23012],{"className":23011},[39],[41,23013,23014],{"xmlns":43},[45,23015,23016,23029],{},[48,23017,23018,23020,23023,23025,23027],{},[95,23019,211],{"stretchy":210},[91,23021,23022],{},"0.50",[95,23024,1210],{"separator":62},[91,23026,23022],{},[95,23028,220],{"stretchy":210},[55,23030,23031],{"encoding":57},"(0.50,0.50)",[31,23033,23035],{"className":23034,"ariaHidden":62},[61],[31,23036,23038,23041,23044,23047,23050,23053,23056],{"className":23037},[66],[31,23039],{"className":23040,"style":260},[70],[31,23042,211],{"className":23043},[264],[31,23045,23022],{"className":23046},[75],[31,23048,1210],{"className":23049},[1391],[31,23051],{"className":23052,"style":1467},[119],[31,23054,23022],{"className":23055},[75],[31,23057,220],{"className":23058},[289],[31,23060,23062,23086],{"className":23061,"translate":35},[34],[31,23063,23065],{"className":23064},[39],[41,23066,23067],{"xmlns":43},[45,23068,23069,23083],{},[48,23070,23071,23073,23076,23078,23081],{},[95,23072,211],{"stretchy":210},[91,23074,23075],{},"0.59",[95,23077,1210],{"separator":62},[91,23079,23080],{},"0.41",[95,23082,220],{"stretchy":210},[55,23084,23085],{"encoding":57},"(0.59,0.41)",[31,23087,23089],{"className":23088,"ariaHidden":62},[61],[31,23090,23092,23095,23098,23101,23104,23107,23110],{"className":23091},[66],[31,23093],{"className":23094,"style":260},[70],[31,23096,211],{"className":23097},[264],[31,23099,23075],{"className":23100},[75],[31,23102,1210],{"className":23103},[1391],[31,23105],{"className":23106,"style":1467},[119],[31,23108,23080],{"className":23109},[75],[31,23111,220],{"className":23112},[289],[11,23114,23115,23116,23145],{},"两组变化的欧氏距离都是 ",[31,23117,23119,23133],{"className":23118,"translate":35},[34],[31,23120,23122],{"className":23121},[39],[41,23123,23124],{"xmlns":43},[45,23125,23126,23131],{},[48,23127,23128],{},[91,23129,23130],{},"0.09",[55,23132,23130],{"encoding":57},[31,23134,23136],{"className":23135,"ariaHidden":62},[61],[31,23137,23139,23142],{"className":23138},[66],[31,23140],{"className":23141,"style":326},[70],[31,23143,23130],{"className":23144},[75],"。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。",[11,23147,23148,23149,4880,23202,1138,23231,4880,23284,23313,23314,23342,23343,23371],{},"这种直觉能不能写成数？能。把 Bernoulli 分布按 Fisher–Rao 距离量一下：",[31,23150,23152,23175],{"className":23151,"translate":35},[34],[31,23153,23155],{"className":23154},[39],[41,23156,23157],{"xmlns":43},[45,23158,23159,23172],{},[48,23160,23161,23163,23166,23168,23170],{},[95,23162,211],{"stretchy":210},[91,23164,23165],{},"0.5",[95,23167,1210],{"separator":62},[91,23169,23075],{},[95,23171,220],{"stretchy":210},[55,23173,23174],{"encoding":57},"(0.5,0.59)",[31,23176,23178],{"className":23177,"ariaHidden":62},[61],[31,23179,23181,23184,23187,23190,23193,23196,23199],{"className":23180},[66],[31,23182],{"className":23183,"style":260},[70],[31,23185,211],{"className":23186},[264],[31,23188,23165],{"className":23189},[75],[31,23191,1210],{"className":23192},[1391],[31,23194],{"className":23195,"style":1467},[119],[31,23197,23075],{"className":23198},[75],[31,23200,220],{"className":23201},[289],[31,23203,23205,23219],{"className":23204,"translate":35},[34],[31,23206,23208],{"className":23207},[39],[41,23209,23210],{"xmlns":43},[45,23211,23212,23217],{},[48,23213,23214],{},[91,23215,23216],{},"0.18",[55,23218,23216],{"encoding":57},[31,23220,23222],{"className":23221,"ariaHidden":62},[61],[31,23223,23225,23228],{"className":23224},[66],[31,23226],{"className":23227,"style":326},[70],[31,23229,23216],{"className":23230},[75],[31,23232,23234,23257],{"className":23233,"translate":35},[34],[31,23235,23237],{"className":23236},[39],[41,23238,23239],{"xmlns":43},[45,23240,23241,23254],{},[48,23242,23243,23245,23248,23250,23252],{},[95,23244,211],{"stretchy":210},[91,23246,23247],{},"0.9",[95,23249,1210],{"separator":62},[91,23251,22968],{},[95,23253,220],{"stretchy":210},[55,23255,23256],{"encoding":57},"(0.9,0.99)",[31,23258,23260],{"className":23259,"ariaHidden":62},[61],[31,23261,23263,23266,23269,23272,23275,23278,23281],{"className":23262},[66],[31,23264],{"className":23265,"style":260},[70],[31,23267,211],{"className":23268},[264],[31,23270,23247],{"className":23271},[75],[31,23273,1210],{"className":23274},[1391],[31,23276],{"className":23277,"style":1467},[119],[31,23279,22968],{"className":23280},[75],[31,23282,220],{"className":23283},[289],[31,23285,23287,23301],{"className":23286,"translate":35},[34],[31,23288,23290],{"className":23289},[39],[41,23291,23292],{"xmlns":43},[45,23293,23294,23299],{},[48,23295,23296],{},[91,23297,23298],{},"0.44",[55,23300,23298],{"encoding":57},[31,23302,23304],{"className":23303,"ariaHidden":62},[61],[31,23305,23307,23310],{"className":23306},[66],[31,23308],{"className":23309,"style":326},[70],[31,23311,23298],{"className":23312},[75],"，后者是前者的两倍半。同样的坐标变化，在概率接近 ",[31,23315,23317,23330],{"className":23316,"translate":35},[34],[31,23318,23320],{"className":23319},[39],[41,23321,23322],{"xmlns":43},[45,23323,23324,23328],{},[48,23325,23326],{},[91,23327,1977],{},[55,23329,1977],{"encoding":57},[31,23331,23333],{"className":23332,"ariaHidden":62},[61],[31,23334,23336,23339],{"className":23335},[66],[31,23337],{"className":23338,"style":326},[70],[31,23340,1977],{"className":23341},[75]," 或 ",[31,23344,23346,23359],{"className":23345,"translate":35},[34],[31,23347,23349],{"className":23348},[39],[41,23350,23351],{"xmlns":43},[45,23352,23353,23357],{},[48,23354,23355],{},[91,23356,205],{},[55,23358,205],{"encoding":57},[31,23360,23362],{"className":23361,"ariaHidden":62},[61],[31,23363,23365,23368],{"className":23364},[66],[31,23366],{"className":23367,"style":326},[70],[31,23369,205],{"className":23370},[75]," 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。",[11,23373,23374],{},"那么这把尺子是怎么来的？",[11,23376,23377],{},"模型的输出住在单纯形",[31,23379,23381],{"className":23380,"translate":35},[411],[31,23382,23384,23459],{"className":23383,"translate":35},[34],[31,23385,23387],{"className":23386},[39],[41,23388,23389],{"xmlns":43,"display":420},[45,23390,23391,23456],{},[48,23392,23393,23406,23408,23410,23412,23414,23420,23422,23424,23430,23432,23434,23436,23438,23444,23450,23452,23454],{},[1180,23394,23395,23398],{},[51,23396,23397],{"mathvariant":494},"Δ",[48,23399,23400,23402,23404],{},[51,23401,4558],{},[95,23403,97],{},[91,23405,205],{},[95,23407,223],{},[95,23409,2512],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[51,23411,11],{},[95,23413,2518],{},[1180,23415,23416,23418],{},[51,23417,14790],{"mathvariant":14789},[51,23419,4558],{},[95,23421,922],{},[1430,23423,1545],{},[1004,23425,23426,23428],{},[51,23427,11],{},[51,23429,5391],{},[95,23431,5662],{},[91,23433,1977],{},[95,23435,1210],{"separator":62},[1430,23437,1545],{},[1531,23439,23440,23442],{},[95,23441,5625],{},[51,23443,5391],{},[1004,23445,23446,23448],{},[51,23447,11],{},[51,23449,5391],{},[95,23451,223],{},[91,23453,205],{},[95,23455,2604],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[55,23457,23458],{"encoding":57},"\\Delta^{K-1}=\\Bigl\\{p\\in\\mathbb R^K:\\ p_i\\ge0,\\ \\sum_i p_i=1\\Bigr\\}",[31,23460,23462,23516,23540,23587,23643,23757],{"className":23461,"ariaHidden":62},[61],[31,23463,23465,23469,23507,23510,23513],{"className":23464},[66],[31,23466],{"className":23467,"style":23468},[70],"height:0.8913em;",[31,23470,23472,23475],{"className":23471},[75],[31,23473,23397],{"className":23474},[75],[31,23476,23478],{"className":23477},[1039],[31,23479,23481],{"className":23480},[1043],[31,23482,23484],{"className":23483},[1048],[31,23485,23487],{"className":23486,"style":23468},[1052],[31,23488,23489,23492],{"style":1330},[31,23490],{"className":23491,"style":1061},[1060],[31,23493,23495],{"className":23494},[1065,1066,1067,1068],[31,23496,23498,23501,23504],{"className":23497},[75,1068],[31,23499,4558],{"className":23500,"style":8525},[75,76,1068],[31,23502,97],{"className":23503},[124,1068],[31,23505,205],{"className":23506},[75,1068],[31,23508],{"className":23509,"style":293},[119],[31,23511,223],{"className":23512},[297],[31,23514],{"className":23515,"style":293},[119],[31,23517,23519,23522,23528,23531,23534,23537],{"className":23518},[66],[31,23520],{"className":23521,"style":2655},[70],[31,23523,23525],{"className":23524},[264],[31,23526,2512],{"className":23527},[1307,2662],[31,23529,11],{"className":23530},[75,76],[31,23532],{"className":23533,"style":293},[119],[31,23535,2518],{"className":23536},[297],[31,23538],{"className":23539,"style":293},[119],[31,23541,23543,23546,23575,23578,23581,23584],{"className":23542},[66],[31,23544],{"className":23545,"style":23468},[70],[31,23547,23549,23552],{"className":23548},[75],[31,23550,14790],{"className":23551},[75,14947],[31,23553,23555],{"className":23554},[1039],[31,23556,23558],{"className":23557},[1043],[31,23559,23561],{"className":23560},[1048],[31,23562,23564],{"className":23563,"style":23468},[1052],[31,23565,23566,23569],{"style":1330},[31,23567],{"className":23568,"style":1061},[1060],[31,23570,23572],{"className":23571},[1065,1066,1067,1068],[31,23573,4558],{"className":23574,"style":8525},[75,76,1068],[31,23576],{"className":23577,"style":293},[119],[31,23579,922],{"className":23580},[297],[31,23582,1545],{"className":23583},[119],[31,23585],{"className":23586,"style":293},[119],[31,23588,23590,23594,23634,23637,23640],{"className":23589},[66],[31,23591],{"className":23592,"style":23593},[70],"height:0.8304em;vertical-align:-0.1944em;",[31,23595,23597,23600],{"className":23596},[75],[31,23598,11],{"className":23599},[75,76],[31,23601,23603],{"className":23602},[1039],[31,23604,23606,23626],{"className":23605},[1043,1044],[31,23607,23609,23623],{"className":23608},[1048],[31,23610,23612],{"className":23611,"style":5445},[1052],[31,23613,23614,23617],{"style":2807},[31,23615],{"className":23616,"style":1061},[1060],[31,23618,23620],{"className":23619},[1065,1066,1067,1068],[31,23621,5391],{"className":23622},[75,76,1068],[31,23624,1076],{"className":23625},[1075],[31,23627,23629],{"className":23628},[1048],[31,23630,23632],{"className":23631,"style":2050},[1052],[31,23633],{},[31,23635],{"className":23636,"style":293},[119],[31,23638,5662],{"className":23639},[297],[31,23641],{"className":23642,"style":293},[119],[31,23644,23646,23649,23652,23655,23658,23661,23705,23708,23748,23751,23754],{"className":23645},[66],[31,23647],{"className":23648,"style":10251},[70],[31,23650,1977],{"className":23651},[75],[31,23653,1210],{"className":23654},[1391],[31,23656,1545],{"className":23657},[119],[31,23659],{"className":23660,"style":1467},[119],[31,23662,23664],{"className":23663},[1460,1620],[31,23665,23667,23697],{"className":23666},[1043,1044],[31,23668,23670,23694],{"className":23669},[1048],[31,23671,23673,23684],{"className":23672,"style":5900},[1052],[31,23674,23675,23678],{"style":5903},[31,23676],{"className":23677,"style":5907},[1060],[31,23679,23681],{"className":23680},[1065,1066,1067,1068],[31,23682,5391],{"className":23683},[75,76,1068],[31,23685,23686,23689],{"style":5919},[31,23687],{"className":23688,"style":5907},[1060],[31,23690,23691],{},[31,23692,5625],{"className":23693},[1460,1461,1687],[31,23695,1076],{"className":23696},[1075],[31,23698,23700],{"className":23699},[1048],[31,23701,23703],{"className":23702,"style":6342},[1052],[31,23704],{},[31,23706],{"className":23707,"style":1467},[119],[31,23709,23711,23714],{"className":23710},[75],[31,23712,11],{"className":23713},[75,76],[31,23715,23717],{"className":23716},[1039],[31,23718,23720,23740],{"className":23719},[1043,1044],[31,23721,23723,23737],{"className":23722},[1048],[31,23724,23726],{"className":23725,"style":5445},[1052],[31,23727,23728,23731],{"style":2807},[31,23729],{"className":23730,"style":1061},[1060],[31,23732,23734],{"className":23733},[1065,1066,1067,1068],[31,23735,5391],{"className":23736},[75,76,1068],[31,23738,1076],{"className":23739},[1075],[31,23741,23743],{"className":23742},[1048],[31,23744,23746],{"className":23745,"style":2050},[1052],[31,23747],{},[31,23749],{"className":23750,"style":293},[119],[31,23752,223],{"className":23753},[297],[31,23755],{"className":23756,"style":293},[119],[31,23758,23760,23763,23766],{"className":23759},[66],[31,23761],{"className":23762,"style":2655},[70],[31,23764,205],{"className":23765},[75],[31,23767,23769],{"className":23768},[289],[31,23770,2604],{"className":23771},[1307,2662],[11,23773,23774],{},"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。",[11,23776,23777,23778,23859],{},"Rao 在 1945 年的想法是：把参数化的分布族 ",[31,23779,23781,23804],{"className":23780,"translate":35},[34],[31,23782,23784],{"className":23783},[39],[41,23785,23786],{"xmlns":43},[45,23787,23788,23801],{},[48,23789,23790,23792,23799],{},[95,23791,2512],{"stretchy":210},[1004,23793,23794,23796],{},[51,23795,11],{},[51,23797,23798],{},"θ",[95,23800,2604],{"stretchy":210},[55,23802,23803],{"encoding":57},"\\{p_\\theta\\}",[31,23805,23807],{"className":23806,"ariaHidden":62},[61],[31,23808,23810,23813,23816,23856],{"className":23809},[66],[31,23811],{"className":23812,"style":260},[70],[31,23814,2512],{"className":23815},[264],[31,23817,23819,23822],{"className":23818},[75],[31,23820,11],{"className":23821},[75,76],[31,23823,23825],{"className":23824},[1039],[31,23826,23828,23848],{"className":23827},[1043,1044],[31,23829,23831,23845],{"className":23830},[1048],[31,23832,23834],{"className":23833,"style":1053},[1052],[31,23835,23836,23839],{"style":2807},[31,23837],{"className":23838,"style":1061},[1060],[31,23840,23842],{"className":23841},[1065,1066,1067,1068],[31,23843,23798],{"className":23844,"style":19759},[75,76,1068],[31,23846,1076],{"className":23847},[1075],[31,23849,23851],{"className":23850},[1048],[31,23852,23854],{"className":23853,"style":2050},[1052],[31,23855],{},[31,23857,2604],{"className":23858},[289]," 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",[31,23861,23863],{"className":23862,"translate":35},[411],[31,23864,23866,23987],{"className":23865,"translate":35},[34],[31,23867,23869],{"className":23868},[39],[41,23870,23871],{"xmlns":43,"display":420},[45,23872,23873,23984],{},[48,23874,23875,23885,23887,23889,23891,23893,23903,23905,23912,23914,23916,23922,23924,23930,23932,23934,23940,23942,23944,23946,23956,23958,23964,23970,23972,23974,23980,23982],{},[1004,23876,23877,23879],{},[51,23878,14556],{},[48,23880,23881,23883],{},[51,23882,5391],{},[51,23884,2555],{},[95,23886,211],{"stretchy":210},[51,23888,23798],{},[95,23890,220],{"stretchy":210},[95,23892,223],{},[1004,23894,23895,23897],{},[51,23896,22324],{"mathvariant":14789},[1004,23898,23899,23901],{},[51,23900,11],{},[51,23902,23798],{},[95,23904,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,23906,23907,23910],{},[51,23908,23909],{"mathvariant":494},"∂",[51,23911,5391],{},[51,23913,19939],{},[95,23915,1540],{},[1004,23917,23918,23920],{},[51,23919,11],{},[51,23921,23798],{},[95,23923,429],{},[1004,23925,23926,23928],{},[51,23927,23909],{"mathvariant":494},[51,23929,2555],{},[51,23931,19939],{},[95,23933,1540],{},[1004,23935,23936,23938],{},[51,23937,11],{},[51,23939,23798],{},[95,23941,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,23943,223],{},[95,23945,97],{},[1004,23947,23948,23950],{},[51,23949,22324],{"mathvariant":14789},[1004,23951,23952,23954],{},[51,23953,11],{},[51,23955,23798],{},[95,23957,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,23959,23960,23962],{},[51,23961,23909],{"mathvariant":494},[51,23963,5391],{},[1004,23965,23966,23968],{},[51,23967,23909],{"mathvariant":494},[51,23969,2555],{},[51,23971,19939],{},[95,23973,1540],{},[1004,23975,23976,23978],{},[51,23977,11],{},[51,23979,23798],{},[95,23981,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,23983,1210],{"separator":62},[55,23985,23986],{"encoding":57},"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],",[31,23988,23990,24057,24253,24365],{"className":23989,"ariaHidden":62},[61],[31,23991,23993,23996,24039,24042,24045,24048,24051,24054],{"className":23992},[66],[31,23994],{"className":23995,"style":1223},[70],[31,23997,23999,24002],{"className":23998},[75],[31,24000,14556],{"className":24001,"style":2666},[75,76],[31,24003,24005],{"className":24004},[1039],[31,24006,24008,24031],{"className":24007},[1043,1044],[31,24009,24011,24028],{"className":24010},[1048],[31,24012,24014],{"className":24013,"style":5445},[1052],[31,24015,24016,24019],{"style":5504},[31,24017],{"className":24018,"style":1061},[1060],[31,24020,24022],{"className":24021},[1065,1066,1067,1068],[31,24023,24025],{"className":24024},[75,1068],[31,24026,5574],{"className":24027,"style":5514},[75,76,1068],[31,24029,1076],{"className":24030},[1075],[31,24032,24034],{"className":24033},[1048],[31,24035,24037],{"className":24036,"style":1083},[1052],[31,24038],{},[31,24040,211],{"className":24041},[264],[31,24043,23798],{"className":24044,"style":19759},[75,76],[31,24046,220],{"className":24047},[289],[31,24049],{"className":24050,"style":293},[119],[31,24052,223],{"className":24053},[297],[31,24055],{"className":24056,"style":293},[119],[31,24058,24060,24063,24145,24151,24193,24196,24201,24204,24244,24247,24250],{"className":24059},[66],[31,24061],{"className":24062,"style":4988},[70],[31,24064,24066,24069],{"className":24065},[75],[31,24067,22324],{"className":24068},[75,14947],[31,24070,24072],{"className":24071},[1039],[31,24073,24075,24137],{"className":24074},[1043,1044],[31,24076,24078,24134],{"className":24077},[1048],[31,24079,24081],{"className":24080,"style":3230},[1052],[31,24082,24083,24086],{"style":2807},[31,24084],{"className":24085,"style":1061},[1060],[31,24087,24089],{"className":24088},[1065,1066,1067,1068],[31,24090,24092],{"className":24091},[75,1068],[31,24093,24095,24098],{"className":24094},[75,1068],[31,24096,11],{"className":24097},[75,76,1068],[31,24099,24101],{"className":24100},[1039],[31,24102,24104,24125],{"className":24103},[1043,1044],[31,24105,24107,24122],{"className":24106},[1048],[31,24108,24110],{"className":24109,"style":11762},[1052],[31,24111,24113,24116],{"style":24112},"top:-2.3488em;margin-left:0em;margin-right:0.0714em;",[31,24114],{"className":24115,"style":11769},[1060],[31,24117,24119],{"className":24118},[1065,11773,1308,1068],[31,24120,23798],{"className":24121,"style":19759},[75,76,1068],[31,24123,1076],{"className":24124},[1075],[31,24126,24128],{"className":24127},[1048],[31,24129,24132],{"className":24130,"style":24131},[1052],"height:0.1512em;",[31,24133],{},[31,24135,1076],{"className":24136},[1075],[31,24138,24140],{"className":24139},[1048],[31,24141,24143],{"className":24142,"style":1083},[1052],[31,24144],{},[31,24146,24148],{"className":24147},[264],[31,24149,15175],{"className":24150},[1307,1308],[31,24152,24154,24158],{"className":24153},[75],[31,24155,23909],{"className":24156,"style":24157},[75],"margin-right:0.0556em;",[31,24159,24161],{"className":24160},[1039],[31,24162,24164,24185],{"className":24163},[1043,1044],[31,24165,24167,24182],{"className":24166},[1048],[31,24168,24170],{"className":24169,"style":5445},[1052],[31,24171,24173,24176],{"style":24172},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[31,24174],{"className":24175,"style":1061},[1060],[31,24177,24179],{"className":24178},[1065,1066,1067,1068],[31,24180,5391],{"className":24181},[75,76,1068],[31,24183,1076],{"className":24184},[1075],[31,24186,24188],{"className":24187},[1048],[31,24189,24191],{"className":24190,"style":2050},[1052],[31,24192],{},[31,24194],{"className":24195,"style":1467},[119],[31,24197,20177,24199],{"className":24198},[1460],[31,24200,14556],{"style":20180},[31,24202],{"className":24203,"style":1467},[119],[31,24205,24207,24210],{"className":24206},[75],[31,24208,11],{"className":24209},[75,76],[31,24211,24213],{"className":24212},[1039],[31,24214,24216,24236],{"className":24215},[1043,1044],[31,24217,24219,24233],{"className":24218},[1048],[31,24220,24222],{"className":24221,"style":1053},[1052],[31,24223,24224,24227],{"style":2807},[31,24225],{"className":24226,"style":1061},[1060],[31,24228,24230],{"className":24229},[1065,1066,1067,1068],[31,24231,23798],{"className":24232,"style":19759},[75,76,1068],[31,24234,1076],{"className":24235},[1075],[31,24237,24239],{"className":24238},[1048],[31,24240,24242],{"className":24241,"style":2050},[1052],[31,24243],{},[31,24245],{"className":24246,"style":120},[119],[31,24248,429],{"className":24249},[124],[31,24251],{"className":24252,"style":120},[119],[31,24254,24256,24259,24299,24302,24307,24310,24350,24356,24359,24362],{"className":24255},[66],[31,24257],{"className":24258,"style":4988},[70],[31,24260,24262,24265],{"className":24261},[75],[31,24263,23909],{"className":24264,"style":24157},[75],[31,24266,24268],{"className":24267},[1039],[31,24269,24271,24291],{"className":24270},[1043,1044],[31,24272,24274,24288],{"className":24273},[1048],[31,24275,24277],{"className":24276,"style":5445},[1052],[31,24278,24279,24282],{"style":24172},[31,24280],{"className":24281,"style":1061},[1060],[31,24283,24285],{"className":24284},[1065,1066,1067,1068],[31,24286,2555],{"className":24287,"style":5514},[75,76,1068],[31,24289,1076],{"className":24290},[1075],[31,24292,24294],{"className":24293},[1048],[31,24295,24297],{"className":24296,"style":1083},[1052],[31,24298],{},[31,24300],{"className":24301,"style":1467},[119],[31,24303,20177,24305],{"className":24304},[1460],[31,24306,14556],{"style":20180},[31,24308],{"className":24309,"style":1467},[119],[31,24311,24313,24316],{"className":24312},[75],[31,24314,11],{"className":24315},[75,76],[31,24317,24319],{"className":24318},[1039],[31,24320,24322,24342],{"className":24321},[1043,1044],[31,24323,24325,24339],{"className":24324},[1048],[31,24326,24328],{"className":24327,"style":1053},[1052],[31,24329,24330,24333],{"style":2807},[31,24331],{"className":24332,"style":1061},[1060],[31,24334,24336],{"className":24335},[1065,1066,1067,1068],[31,24337,23798],{"className":24338,"style":19759},[75,76,1068],[31,24340,1076],{"className":24341},[1075],[31,24343,24345],{"className":24344},[1048],[31,24346,24348],{"className":24347,"style":2050},[1052],[31,24349],{},[31,24351,24353],{"className":24352},[289],[31,24354,15184],{"className":24355},[1307,1308],[31,24357],{"className":24358,"style":293},[119],[31,24360,223],{"className":24361},[297],[31,24363],{"className":24364,"style":293},[119],[31,24366,24368,24371,24374,24454,24460,24500,24540,24543,24548,24551,24591,24597],{"className":24367},[66],[31,24369],{"className":24370,"style":4988},[70],[31,24372,97],{"className":24373},[75],[31,24375,24377,24380],{"className":24376},[75],[31,24378,22324],{"className":24379},[75,14947],[31,24381,24383],{"className":24382},[1039],[31,24384,24386,24446],{"className":24385},[1043,1044],[31,24387,24389,24443],{"className":24388},[1048],[31,24390,24392],{"className":24391,"style":3230},[1052],[31,24393,24394,24397],{"style":2807},[31,24395],{"className":24396,"style":1061},[1060],[31,24398,24400],{"className":24399},[1065,1066,1067,1068],[31,24401,24403],{"className":24402},[75,1068],[31,24404,24406,24409],{"className":24405},[75,1068],[31,24407,11],{"className":24408},[75,76,1068],[31,24410,24412],{"className":24411},[1039],[31,24413,24415,24435],{"className":24414},[1043,1044],[31,24416,24418,24432],{"className":24417},[1048],[31,24419,24421],{"className":24420,"style":11762},[1052],[31,24422,24423,24426],{"style":24112},[31,24424],{"className":24425,"style":11769},[1060],[31,24427,24429],{"className":24428},[1065,11773,1308,1068],[31,24430,23798],{"className":24431,"style":19759},[75,76,1068],[31,24433,1076],{"className":24434},[1075],[31,24436,24438],{"className":24437},[1048],[31,24439,24441],{"className":24440,"style":24131},[1052],[31,24442],{},[31,24444,1076],{"className":24445},[1075],[31,24447,24449],{"className":24448},[1048],[31,24450,24452],{"className":24451,"style":1083},[1052],[31,24453],{},[31,24455,24457],{"className":24456},[264],[31,24458,15175],{"className":24459},[1307,1308],[31,24461,24463,24466],{"className":24462},[75],[31,24464,23909],{"className":24465,"style":24157},[75],[31,24467,24469],{"className":24468},[1039],[31,24470,24472,24492],{"className":24471},[1043,1044],[31,24473,24475,24489],{"className":24474},[1048],[31,24476,24478],{"className":24477,"style":5445},[1052],[31,24479,24480,24483],{"style":24172},[31,24481],{"className":24482,"style":1061},[1060],[31,24484,24486],{"className":24485},[1065,1066,1067,1068],[31,24487,5391],{"className":24488},[75,76,1068],[31,24490,1076],{"className":24491},[1075],[31,24493,24495],{"className":24494},[1048],[31,24496,24498],{"className":24497,"style":2050},[1052],[31,24499],{},[31,24501,24503,24506],{"className":24502},[75],[31,24504,23909],{"className":24505,"style":24157},[75],[31,24507,24509],{"className":24508},[1039],[31,24510,24512,24532],{"className":24511},[1043,1044],[31,24513,24515,24529],{"className":24514},[1048],[31,24516,24518],{"className":24517,"style":5445},[1052],[31,24519,24520,24523],{"style":24172},[31,24521],{"className":24522,"style":1061},[1060],[31,24524,24526],{"className":24525},[1065,1066,1067,1068],[31,24527,2555],{"className":24528,"style":5514},[75,76,1068],[31,24530,1076],{"className":24531},[1075],[31,24533,24535],{"className":24534},[1048],[31,24536,24538],{"className":24537,"style":1083},[1052],[31,24539],{},[31,24541],{"className":24542,"style":1467},[119],[31,24544,20177,24546],{"className":24545},[1460],[31,24547,14556],{"style":20180},[31,24549],{"className":24550,"style":1467},[119],[31,24552,24554,24557],{"className":24553},[75],[31,24555,11],{"className":24556},[75,76],[31,24558,24560],{"className":24559},[1039],[31,24561,24563,24583],{"className":24562},[1043,1044],[31,24564,24566,24580],{"className":24565},[1048],[31,24567,24569],{"className":24568,"style":1053},[1052],[31,24570,24571,24574],{"style":2807},[31,24572],{"className":24573,"style":1061},[1060],[31,24575,24577],{"className":24576},[1065,1066,1067,1068],[31,24578,23798],{"className":24579,"style":19759},[75,76,1068],[31,24581,1076],{"className":24582},[1075],[31,24584,24586],{"className":24585},[1048],[31,24587,24589],{"className":24588,"style":2050},[1052],[31,24590],{},[31,24592,24594],{"className":24593},[289],[31,24595,15184],{"className":24596},[1307,1308],[31,24598,1210],{"className":24599},[1391],[11,24601,24602,24603,24730,24731,24760,24761,2372],{},"两个表达式的相等用到了 ",[31,24604,24606,24640],{"className":24605,"translate":35},[34],[31,24607,24609],{"className":24608},[39],[41,24610,24611],{"xmlns":43},[45,24612,24613,24637],{},[48,24614,24615,24617,24619,24625,24627,24629,24631,24633,24635],{},[51,24616,22324],{"mathvariant":14789},[95,24618,15175],{"stretchy":210},[1004,24620,24621,24623],{},[51,24622,23909],{"mathvariant":494},[51,24624,5391],{},[51,24626,19939],{},[95,24628,1540],{},[51,24630,11],{},[95,24632,15184],{"stretchy":210},[95,24634,223],{},[91,24636,1977],{},[55,24638,24639],{"encoding":57},"\\mathbb E[\\partial_i\\log p]=0",[31,24641,24643,24721],{"className":24642,"ariaHidden":62},[61],[31,24644,24646,24649,24652,24655,24695,24698,24703,24706,24709,24712,24715,24718],{"className":24645},[66],[31,24647],{"className":24648,"style":260},[70],[31,24650,22324],{"className":24651},[75,14947],[31,24653,15175],{"className":24654},[264],[31,24656,24658,24661],{"className":24657},[75],[31,24659,23909],{"className":24660,"style":24157},[75],[31,24662,24664],{"className":24663},[1039],[31,24665,24667,24687],{"className":24666},[1043,1044],[31,24668,24670,24684],{"className":24669},[1048],[31,24671,24673],{"className":24672,"style":5445},[1052],[31,24674,24675,24678],{"style":24172},[31,24676],{"className":24677,"style":1061},[1060],[31,24679,24681],{"className":24680},[1065,1066,1067,1068],[31,24682,5391],{"className":24683},[75,76,1068],[31,24685,1076],{"className":24686},[1075],[31,24688,24690],{"className":24689},[1048],[31,24691,24693],{"className":24692,"style":2050},[1052],[31,24694],{},[31,24696],{"className":24697,"style":1467},[119],[31,24699,20177,24701],{"className":24700},[1460],[31,24702,14556],{"style":20180},[31,24704],{"className":24705,"style":1467},[119],[31,24707,11],{"className":24708},[75,76],[31,24710,15184],{"className":24711},[289],[31,24713],{"className":24714,"style":293},[119],[31,24716,223],{"className":24717},[297],[31,24719],{"className":24720,"style":293},[119],[31,24722,24724,24727],{"className":24723},[66],[31,24725],{"className":24726,"style":326},[70],[31,24728,1977],{"className":24729},[75],"，因为对 ",[31,24732,24734,24748],{"className":24733,"translate":35},[34],[31,24735,24737],{"className":24736},[39],[41,24738,24739],{"xmlns":43},[45,24740,24741,24745],{},[48,24742,24743],{},[51,24744,23798],{},[55,24746,24747],{"encoding":57},"\\theta",[31,24749,24751],{"className":24750,"ariaHidden":62},[61],[31,24752,24754,24757],{"className":24753},[66],[31,24755],{"className":24756,"style":1630},[70],[31,24758,23798],{"className":24759,"style":19759},[75,76]," 求导再积分，",[31,24762,24764,24814],{"className":24763,"translate":35},[34],[31,24765,24767],{"className":24766},[39],[41,24768,24769],{"xmlns":43},[45,24770,24771,24811],{},[48,24772,24773,24775,24781,24783,24785,24787,24789,24791,24797,24799,24801,24803,24805,24807,24809],{},[95,24774,1409],{},[1004,24776,24777,24779],{},[51,24778,23909],{"mathvariant":494},[51,24780,5391],{},[51,24782,11],{},[1430,24784,1432],{},[51,24786,1435],{},[51,24788,53],{},[95,24790,223],{},[1004,24792,24793,24795],{},[51,24794,23909],{"mathvariant":494},[51,24796,5391],{},[95,24798,1409],{},[51,24800,11],{},[1430,24802,1432],{},[51,24804,1435],{},[51,24806,53],{},[95,24808,223],{},[91,24810,1977],{},[55,24812,24813],{"encoding":57},"\\int\\partial_i p\\,dx=\\partial_i\\int p\\,dx=0",[31,24815,24817,24890,24966],{"className":24816,"ariaHidden":62},[61],[31,24818,24820,24823,24826,24829,24869,24872,24875,24878,24881,24884,24887],{"className":24819},[66],[31,24821],{"className":24822,"style":1456},[70],[31,24824,1409],{"className":24825,"style":1463},[1460,1461,1462],[31,24827],{"className":24828,"style":1467},[119],[31,24830,24832,24835],{"className":24831},[75],[31,24833,23909],{"className":24834,"style":24157},[75],[31,24836,24838],{"className":24837},[1039],[31,24839,24841,24861],{"className":24840},[1043,1044],[31,24842,24844,24858],{"className":24843},[1048],[31,24845,24847],{"className":24846,"style":5445},[1052],[31,24848,24849,24852],{"style":24172},[31,24850],{"className":24851,"style":1061},[1060],[31,24853,24855],{"className":24854},[1065,1066,1067,1068],[31,24856,5391],{"className":24857},[75,76,1068],[31,24859,1076],{"className":24860},[1075],[31,24862,24864],{"className":24863},[1048],[31,24865,24867],{"className":24866,"style":2050},[1052],[31,24868],{},[31,24870,11],{"className":24871},[75,76],[31,24873],{"className":24874,"style":1467},[119],[31,24876,1435],{"className":24877},[75,76],[31,24879,53],{"className":24880},[75,76],[31,24882],{"className":24883,"style":293},[119],[31,24885,223],{"className":24886},[297],[31,24888],{"className":24889,"style":293},[119],[31,24891,24893,24896,24936,24939,24942,24945,24948,24951,24954,24957,24960,24963],{"className":24892},[66],[31,24894],{"className":24895,"style":1456},[70],[31,24897,24899,24902],{"className":24898},[75],[31,24900,23909],{"className":24901,"style":24157},[75],[31,24903,24905],{"className":24904},[1039],[31,24906,24908,24928],{"className":24907},[1043,1044],[31,24909,24911,24925],{"className":24910},[1048],[31,24912,24914],{"className":24913,"style":5445},[1052],[31,24915,24916,24919],{"style":24172},[31,24917],{"className":24918,"style":1061},[1060],[31,24920,24922],{"className":24921},[1065,1066,1067,1068],[31,24923,5391],{"className":24924},[75,76,1068],[31,24926,1076],{"className":24927},[1075],[31,24929,24931],{"className":24930},[1048],[31,24932,24934],{"className":24933,"style":2050},[1052],[31,24935],{},[31,24937],{"className":24938,"style":1467},[119],[31,24940,1409],{"className":24941,"style":1463},[1460,1461,1462],[31,24943],{"className":24944,"style":1467},[119],[31,24946,11],{"className":24947},[75,76],[31,24949],{"className":24950,"style":1467},[119],[31,24952,1435],{"className":24953},[75,76],[31,24955,53],{"className":24956},[75,76],[31,24958],{"className":24959,"style":293},[119],[31,24961,223],{"className":24962},[297],[31,24964],{"className":24965,"style":293},[119],[31,24967,24969,24972],{"className":24968},[66],[31,24970],{"className":24971,"style":326},[70],[31,24973,1977],{"className":24974},[75],[11,24976,24977],{},"为什么是它？有三个我觉得足够有说服力的理由。",[11,24979,24980,24981,25085,25086,25114],{},"第一，它是 KL 的二阶展开。把 ",[31,24982,24984,25014],{"className":24983,"translate":35},[34],[31,24985,24987],{"className":24986},[39],[41,24988,24989],{"xmlns":43},[45,24990,24991,25011],{},[48,24992,24993,24995,24997],{},[51,24994,19939],{},[95,24996,1540],{},[1004,24998,24999,25001],{},[51,25000,11],{},[48,25002,25003,25005,25007,25009],{},[51,25004,23798],{},[95,25006,434],{},[51,25008,1435],{},[51,25010,23798],{},[55,25012,25013],{"encoding":57},"\\log p_{\\theta+d\\theta}",[31,25015,25017],{"className":25016,"ariaHidden":62},[61],[31,25018,25020,25024,25029,25032],{"className":25019},[66],[31,25021],{"className":25022,"style":25023},[70],"height:0.9028em;vertical-align:-0.2083em;",[31,25025,20177,25027],{"className":25026},[1460],[31,25028,14556],{"style":20180},[31,25030],{"className":25031,"style":1467},[119],[31,25033,25035,25038],{"className":25034},[75],[31,25036,11],{"className":25037},[75,76],[31,25039,25041],{"className":25040},[1039],[31,25042,25044,25076],{"className":25043},[1043,1044],[31,25045,25047,25073],{"className":25046},[1048],[31,25048,25050],{"className":25049,"style":1053},[1052],[31,25051,25052,25055],{"style":2807},[31,25053],{"className":25054,"style":1061},[1060],[31,25056,25058],{"className":25057},[1065,1066,1067,1068],[31,25059,25061,25064,25067,25070],{"className":25060},[75,1068],[31,25062,23798],{"className":25063,"style":19759},[75,76,1068],[31,25065,434],{"className":25066},[124,1068],[31,25068,1435],{"className":25069},[75,76,1068],[31,25071,23798],{"className":25072,"style":19759},[75,76,1068],[31,25074,1076],{"className":25075},[1075],[31,25077,25079],{"className":25078},[1048],[31,25080,25083],{"className":25081,"style":25082},[1052],"height:0.2083em;",[31,25084],{}," 在 ",[31,25087,25089,25102],{"className":25088,"translate":35},[34],[31,25090,25092],{"className":25091},[39],[41,25093,25094],{"xmlns":43},[45,25095,25096,25100],{},[48,25097,25098],{},[51,25099,23798],{},[55,25101,24747],{"encoding":57},[31,25103,25105],{"className":25104,"ariaHidden":62},[61],[31,25106,25108,25111],{"className":25107},[66],[31,25109],{"className":25110,"style":1630},[70],[31,25112,23798],{"className":25113,"style":19759},[75,76]," 处展开到二阶，一阶项的期望为零，剩下",[31,25116,25118],{"className":25117,"translate":35},[411],[31,25119,25121,25311],{"className":25120,"translate":35},[34],[31,25122,25124],{"className":25123},[39],[41,25125,25126],{"xmlns":43,"display":420},[45,25127,25128,25308],{},[48,25129,25130,25136,25138,25144,25146,25160,25162,25164,25166,25174,25176,25178,25180,25186,25192,25194,25196,25202,25204,25206,25208,25214,25216,25218,25224,25226,25228,25230,25232,25234,25236,25242,25244,25246,25254,25256,25266,25268,25270,25276,25278,25280,25286,25288,25290,25292,25294,25296,25298,25304,25306],{},[48,25131,25132,25134],{},[51,25133,4558],{"mathvariant":494},[51,25135,4561],{"mathvariant":494},[95,25137,211],{"stretchy":210},[1004,25139,25140,25142],{},[51,25141,11],{},[51,25143,23798],{},[51,25145,4572],{"mathvariant":494},[1004,25147,25148,25150],{},[51,25149,11],{},[48,25151,25152,25154,25156,25158],{},[51,25153,23798],{},[95,25155,434],{},[51,25157,1435],{},[51,25159,23798],{},[95,25161,220],{"stretchy":210},[95,25163,223],{},[95,25165,97],{},[2071,25167,25168],{"scriptlevel":1977,"displaystyle":210},[2074,25169,25170,25172],{},[91,25171,205],{},[91,25173,152],{},[1430,25175,1432],{},[51,25177,22324],{"mathvariant":14789},[95,25179,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,25181,25182,25184],{},[51,25183,23909],{"mathvariant":494},[51,25185,5391],{},[1004,25187,25188,25190],{},[51,25189,23909],{"mathvariant":494},[51,25191,2555],{},[51,25193,19939],{},[95,25195,1540],{},[1004,25197,25198,25200],{},[51,25199,11],{},[51,25201,23798],{},[95,25203,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1430,25205,1432],{},[51,25207,1435],{},[1180,25209,25210,25212],{},[51,25211,23798],{},[51,25213,5391],{},[1430,25215,1432],{},[51,25217,1435],{},[1180,25219,25220,25222],{},[51,25221,23798],{},[51,25223,2555],{},[95,25225,434],{},[51,25227,19733],{},[95,25229,211],{"stretchy":210},[51,25231,4572],{"mathvariant":494},[51,25233,1435],{},[51,25235,23798],{},[1180,25237,25238,25240],{},[51,25239,4572],{"mathvariant":494},[91,25241,93],{},[95,25243,220],{"stretchy":210},[95,25245,223],{},[2071,25247,25248],{"scriptlevel":1977,"displaystyle":210},[2074,25249,25250,25252],{},[91,25251,205],{},[91,25253,152],{},[1430,25255,1432],{},[1004,25257,25258,25260],{},[51,25259,14556],{},[48,25261,25262,25264],{},[51,25263,5391],{},[51,25265,2555],{},[1430,25267,1432],{},[51,25269,1435],{},[1180,25271,25272,25274],{},[51,25273,23798],{},[51,25275,5391],{},[1430,25277,1432],{},[51,25279,1435],{},[1180,25281,25282,25284],{},[51,25283,23798],{},[51,25285,2555],{},[95,25287,434],{},[51,25289,19733],{},[95,25291,211],{"stretchy":210},[51,25293,4572],{"mathvariant":494},[51,25295,1435],{},[51,25297,23798],{},[1180,25299,25300,25302],{},[51,25301,4572],{"mathvariant":494},[91,25303,93],{},[95,25305,220],{"stretchy":210},[51,25307,495],{"mathvariant":494},[55,25309,25310],{"encoding":57},"\\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).",[31,25312,25314,25436,25743,25805,26005],{"className":25313,"ariaHidden":62},[61],[31,25315,25317,25320,25326,25329,25369,25372,25424,25427,25430,25433],{"className":25316},[66],[31,25318],{"className":25319,"style":260},[70],[31,25321,25323],{"className":25322},[75],[31,25324,4606],{"className":25325},[75,2745],[31,25327,211],{"className":25328},[264],[31,25330,25332,25335],{"className":25331},[75],[31,25333,11],{"className":25334},[75,76],[31,25336,25338],{"className":25337},[1039],[31,25339,25341,25361],{"className":25340},[1043,1044],[31,25342,25344,25358],{"className":25343},[1048],[31,25345,25347],{"className":25346,"style":1053},[1052],[31,25348,25349,25352],{"style":2807},[31,25350],{"className":25351,"style":1061},[1060],[31,25353,25355],{"className":25354},[1065,1066,1067,1068],[31,25356,23798],{"className":25357,"style":19759},[75,76,1068],[31,25359,1076],{"className":25360},[1075],[31,25362,25364],{"className":25363},[1048],[31,25365,25367],{"className":25366,"style":2050},[1052],[31,25368],{},[31,25370,4572],{"className":25371},[75],[31,25373,25375,25378],{"className":25374},[75],[31,25376,11],{"className":25377},[75,76],[31,25379,25381],{"className":25380},[1039],[31,25382,25384,25416],{"className":25383},[1043,1044],[31,25385,25387,25413],{"className":25386},[1048],[31,25388,25390],{"className":25389,"style":1053},[1052],[31,25391,25392,25395],{"style":2807},[31,25393],{"className":25394,"style":1061},[1060],[31,25396,25398],{"className":25397},[1065,1066,1067,1068],[31,25399,25401,25404,25407,25410],{"className":25400},[75,1068],[31,25402,23798],{"className":25403,"style":19759},[75,76,1068],[31,25405,434],{"className":25406},[124,1068],[31,25408,1435],{"className":25409},[75,76,1068],[31,25411,23798],{"className":25412,"style":19759},[75,76,1068],[31,25414,1076],{"className":25415},[1075],[31,25417,25419],{"className":25418},[1048],[31,25420,25422],{"className":25421,"style":25082},[1052],[31,25423],{},[31,25425,220],{"className":25426},[289],[31,25428],{"className":25429,"style":293},[119],[31,25431,223],{"className":25432},[297],[31,25434],{"className":25435,"style":293},[119],[31,25437,25439,25443,25446,25514,25517,25520,25526,25566,25606,25609,25614,25617,25657,25663,25666,25669,25699,25702,25705,25734,25737,25740],{"className":25438},[66],[31,25440],{"className":25441,"style":25442},[70],"height:1.2247em;vertical-align:-0.35em;",[31,25444,97],{"className":25445},[75],[31,25447,25449,25452,25511],{"className":25448},[75],[31,25450],{"className":25451},[264,2139],[31,25453,25455],{"className":25454},[2074],[31,25456,25458,25503],{"className":25457},[1043,1044],[31,25459,25461,25500],{"className":25460},[1048],[31,25462,25464,25478,25486],{"className":25463,"style":2152},[1052],[31,25465,25466,25469],{"style":2155},[31,25467],{"className":25468,"style":1637},[1060],[31,25470,25472],{"className":25471},[1065,1066,1067,1068],[31,25473,25475],{"className":25474},[75,1068],[31,25476,152],{"className":25477},[75,1068],[31,25479,25480,25483],{"style":2170},[31,25481],{"className":25482,"style":1637},[1060],[31,25484],{"className":25485,"style":2178},[2177],[31,25487,25488,25491],{"style":2181},[31,25489],{"className":25490,"style":1637},[1060],[31,25492,25494],{"className":25493},[1065,1066,1067,1068],[31,25495,25497],{"className":25496},[75,1068],[31,25498,205],{"className":25499},[75,1068],[31,25501,1076],{"className":25502},[1075],[31,25504,25506],{"className":25505},[1048],[31,25507,25509],{"className":25508,"style":2203},[1052],[31,25510],{},[31,25512],{"className":25513},[289,2139],[31,25515],{"className":25516,"style":1467},[119],[31,25518,22324],{"className":25519},[75,14947],[31,25521,25523],{"className":25522},[264],[31,25524,15175],{"className":25525},[1307,1308],[31,25527,25529,25532],{"className":25528},[75],[31,25530,23909],{"className":25531,"style":24157},[75],[31,25533,25535],{"className":25534},[1039],[31,25536,25538,25558],{"className":25537},[1043,1044],[31,25539,25541,25555],{"className":25540},[1048],[31,25542,25544],{"className":25543,"style":5445},[1052],[31,25545,25546,25549],{"style":24172},[31,25547],{"className":25548,"style":1061},[1060],[31,25550,25552],{"className":25551},[1065,1066,1067,1068],[31,25553,5391],{"className":25554},[75,76,1068],[31,25556,1076],{"className":25557},[1075],[31,25559,25561],{"className":25560},[1048],[31,25562,25564],{"className":25563,"style":2050},[1052],[31,25565],{},[31,25567,25569,25572],{"className":25568},[75],[31,25570,23909],{"className":25571,"style":24157},[75],[31,25573,25575],{"className":25574},[1039],[31,25576,25578,25598],{"className":25577},[1043,1044],[31,25579,25581,25595],{"className":25580},[1048],[31,25582,25584],{"className":25583,"style":5445},[1052],[31,25585,25586,25589],{"style":24172},[31,25587],{"className":25588,"style":1061},[1060],[31,25590,25592],{"className":25591},[1065,106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,[75],[31,25887,14556],{"className":25888,"style":2666},[75,76],[31,25890,25892],{"className":25891},[1039],[31,25893,25895,25918],{"className":25894},[1043,1044],[31,25896,25898,25915],{"className":25897},[1048],[31,25899,25901],{"className":25900,"style":5445},[1052],[31,25902,25903,25906],{"style":5504},[31,25904],{"className":25905,"style":1061},[1060],[31,25907,25909],{"className":25908},[1065,1066,1067,1068],[31,25910,25912],{"className":25911},[75,1068],[31,25913,5574],{"className":25914,"style":5514},[75,76,1068],[31,25916,1076],{"className":25917},[1075],[31,25919,25921],{"className":25920},[1048],[31,25922,25924],{"className":25923,"style":1083},[1052],[31,25925],{},[31,25927],{"className":25928,"style":1467},[119],[31,25930,1435],{"className":25931},[75,76],[31,25933,25935,25938],{"className":25934},[75],[31,25936,23798],{"className":25937,"style":19759},[75,76],[31,25939,25941],{"className":25940},[1039],[31,25942,25944],{"className":25943},[1043],[31,25945,25947],{"className":25946},[1048],[31,25948,25950],{"className":25949,"style":25687},[1052],[31,25951,25952,25955],{"style":1330},[31,25953],{"className":25954,"style":1061},[1060],[31,25956,25958],{"className":25957},[1065,1066,1067,1068],[31,25959,5391],{"className":25960},[75,76,1068],[31,25962],{"className":25963,"style":1467},[119],[31,25965,1435],{"className":25966},[75,76],[31,25968,25970,25973],{"className":25969},[75],[31,25971,23798],{"className":25972,"style":19759},[75,76],[31,25974,25976],{"className":25975},[1039],[31,25977,25979],{"className":25978},[1043],[31,25980,25982],{"className":25981},[1048],[31,25983,25985],{"className":25984,"style":25687},[1052],[31,25986,25987,25990],{"style":1330},[31,25988],{"className":25989,"style":1061},[1060],[31,25991,25993],{"className":25992},[1065,1066,1067,1068],[31,25994,2555],{"className":25995,"style":5514},[75,76,1068],[31,25997],{"className":25998,"style":120},[119],[31,26000,434],{"className":26001},[124],[31,26003],{"className":26004,"style":120},[119],[31,26006,26008,26011,26014,26017,26020,26023,26026,26055,26058],{"className":26007},[66],[31,26009],{"className":26010,"style":16882},[70],[31,26012,19733],{"className":26013,"style":19759},[75,76],[31,26015,211],{"className":26016},[264],[31,26018,4572],{"className":26019},[75],[31,26021,1435],{"className":26022},[75,76],[31,26024,23798],{"className":26025,"style":19759},[75,76],[31,26027,26029,26032],{"className":26028},[75],[31,26030,4572],{"className":26031},[75],[31,26033,26035],{"className":26034},[1039],[31,26036,26038],{"className":26037},[1043],[31,26039,26041],{"className":26040},[1048],[31,26042,26044],{"className":26043,"style":1327},[1052],[31,26045,26046,26049],{"style":1330},[31,26047],{"className":26048,"style":1061},[1060],[31,26050,26052],{"className":26051},[1065,1066,1067,1068],[31,26053,93],{"className":26054},[75,1068],[31,26056,220],{"className":26057},[289],[31,26059,495],{"className":26060},[75],[11,26062,26063],{},"所以 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。",[11,26065,26066],{},"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。",[11,26068,26069],{},"第三，对指数族它恰好是势函数的 Hessian，下一节说。",[11,26071,26072,26073,1138],{},"把 Bernoulli 族算一遍。",[31,26074,26076,26128],{"className":26075,"translate":35},[34],[31,26077,26079],{"className":26078},[39],[41,26080,26081],{"xmlns":43},[45,26082,26083,26125],{},[48,26084,26085,26087,26089,26091,26093,26095,26097,26099,26105,26107,26109,26111,26113],{},[51,26086,11],{},[95,26088,211],{"stretchy":210},[51,26090,53],{},[95,26092,12680],{},[51,26094,23798],{},[95,26096,220],{"stretchy":210},[95,26098,223],{},[1180,26100,26101,26103],{},[51,26102,23798],{},[51,26104,53],{},[95,26106,211],{"stretchy":210},[91,26108,205],{},[95,26110,97],{},[51,26112,23798],{},[1180,26114,26115,26117],{},[95,26116,220],{"stretchy":210},[48,26118,26119,26121,26123],{},[91,26120,205],{},[95,26122,97],{},[51,26124,53],{},[55,26126,26127],{"encoding":57},"p(x\\mid\\theta)=\\theta^x(1-\\theta)^{1-x}",[31,26129,26131,26155,26176,26226],{"className":26130,"ariaHidden":62},[61],[31,26132,26134,26137,26140,26143,26146,26149,26152],{"className":26133},[66],[31,26135],{"className":26136,"style":260},[70],[31,26138,11],{"className":26139},[75,76],[31,26141,211],{"className":26142},[264],[31,26144,53],{"className":26145},[75,76],[31,26147],{"className":26148,"style":293},[119],[31,26150,12680],{"className":26151},[297],[31,26153],{"className":26154,"style":293},[119],[31,26156,26158,26161,26164,26167,26170,26173],{"className":26157},[66],[31,26159],{"className":26160,"style":260},[70],[31,26162,23798],{"className":26163,"style":19759},[75,76],[31,26165,220],{"className":26166},[289],[31,26168],{"className":26169,"style":293},[119],[31,26171,223],{"className":26172},[297],[31,26174],{"className":26175,"style":293},[119],[31,26177,26179,26182,26211,26214,26217,26220,26223],{"className":26178},[66],[31,26180],{"className":26181,"style":260},[70],[31,26183,26185,26188],{"className":26184},[75],[31,26186,23798],{"className":26187,"style":19759},[75,76],[31,26189,26191],{"className":26190},[1039],[31,26192,26194],{"className":26193},[1043],[31,26195,26197],{"className":26196},[1048],[31,26198,26200],{"className":26199,"style":3628},[1052],[31,26201,26202,26205],{"style":3631},[31,26203],{"className":26204,"style":1061},[1060],[31,26206,26208],{"className":26207},[1065,1066,1067,1068],[31,26209,53],{"className":26210},[75,76,1068],[31,26212,211],{"className":26213},[264],[31,26215,205],{"className":26216},[75],[31,26218],{"className":26219,"style":120},[119],[31,26221,97],{"className":26222},[124],[31,26224],{"className":26225,"style":120},[119],[31,26227,26229,26232,26235],{"className":26228},[66],[31,26230],{"className":26231,"style":12035},[70],[31,26233,23798],{"className":26234,"style":19759},[75,76],[31,26236,26238,26241],{"className":26237},[289],[31,26239,220],{"className":26240},[289],[31,26242,26244],{"className":26243},[1039],[31,26245,26247],{"className":26246},[1043],[31,26248,26250],{"className":26249},[1048],[31,26251,26253],{"className":26252,"style":5135},[1052],[31,26254,26255,26258],{"style":3631},[31,26256],{"className":26257,"style":1061},[1060],[31,26259,26261],{"className":26260},[1065,1066,1067,1068],[31,26262,26264,26267,26270],{"className":26263},[75,1068],[31,26265,205],{"className":26266},[75,1068],[31,26268,97],{"className":26269},[124,1068],[31,26271,53],{"className":26272},[75,76,1068],[31,26274,26276],{"className":26275,"translate":35},[411],[31,26277,26279,26321],{"className":26278,"translate":35},[34],[31,26280,26282],{"className":26281},[39],[41,26283,26284],{"xmlns":43,"display":420},[45,26285,26286,26318],{},[48,26287,26288,26290,26292,26294,26296,26298,26316],{},[51,26289,14556],{},[95,26291,211],{"stretchy":210},[51,26293,23798],{},[95,26295,220],{"stretchy":210},[95,26297,223],{},[2074,26299,26300,26302],{},[91,26301,205],{},[48,26303,26304,26306,26308,26310,26312,26314],{},[51,26305,23798],{},[95,26307,211],{"stretchy":210},[91,26309,205],{},[95,26311,97],{},[51,26313,23798],{},[95,26315,220],{"stretchy":210},[51,26317,495],{"mathvariant":494},[55,26319,26320],{"encoding":57},"g(\\theta)=\\frac{1}{\\theta(1-\\theta)} .",[31,26322,26324,26351],{"className":26323,"ariaHidden":62},[61],[31,26325,26327,26330,26333,26336,26339,26342,26345,26348],{"className":26326},[66],[31,26328],{"className":26329,"style":260},[70],[31,26331,14556],{"className":26332,"style":2666},[75,76],[31,26334,211],{"className":26335},[264],[31,26337,23798],{"className":26338,"style":19759},[75,76],[31,26340,220],{"className":26341},[289],[31,26343],{"className":26344,"style":293},[119],[31,26346,223],{"className":26347},[297],[31,26349],{"className":26350,"style":293},[119],[31,26352,26354,26358,26443],{"className":26353},[66],[31,26355],{"className":26356,"style":26357},[70],"height:2.2574em;vertical-align:-0.936em;",[31,26359,26361,26364,26440],{"className":26360},[75],[31,26362],{"className":26363},[264,2139],[31,26365,26367],{"className":26366},[2074],[31,26368,26370,26431],{"className":26369},[1043,1044],[31,26371,26373,26428],{"className":26372},[1048],[31,26374,26377,26409,26417],{"className":26375,"style":26376},[1052],"height:1.3214em;",[31,26378,26379,26382],{"style":5090},[31,26380],{"className":26381,"style":1637},[1060],[31,26383,26385,26388,26391,26394,26397,26400,26403,26406],{"className":26384},[75],[31,26386,23798],{"className":26387,"style":19759},[75,76],[31,26389,211],{"className":26390},[264],[31,26392,205],{"className":26393},[75],[31,26395],{"className":26396,"style":120},[119],[31,26398,97],{"className":26399},[124],[31,26401],{"className":26402,"style":120},[119],[31,26404,23798],{"className":26405,"style":19759},[75,76],[31,26407,220],{"className":26408},[289],[31,26410,26411,26414],{"style":2170},[31,26412],{"className":26413,"style":1637},[1060],[31,26415],{"className":26416,"style":2178},[2177],[31,26418,26419,26422],{"style":5110},[31,26420],{"className":26421,"style":1637},[1060],[31,26423,26425],{"className":26424},[75],[31,26426,205],{"className":26427},[75],[31,26429,1076],{"className":26430},[1075],[31,26432,26434],{"className":26433},[1048],[31,26435,26438],{"className":26436,"style":26437},[1052],"height:0.936em;",[31,26439],{},[31,26441],{"className":26442},[289,2139],[31,26444,495],{"className":26445},[75],[11,26447,26448,26449,26615],{},"弧长 ",[31,26450,26452,26493],{"className":26451,"translate":35},[34],[31,26453,26455],{"className":26454},[39],[41,26456,26457],{"xmlns":43},[45,26458,26459,26490],{},[48,26460,26461,26463,26465,26467,26469,26471,26473],{},[51,26462,1435],{},[51,26464,14551],{},[95,26466,223],{},[51,26468,1435],{},[51,26470,23798],{},[51,26472,3768],{"mathvariant":494},[26474,26475,26476],"msqrt",{},[48,26477,26478,26480,26482,26484,26486,26488],{},[51,26479,23798],{},[95,26481,211],{"stretchy":210},[91,26483,205],{},[95,26485,97],{},[51,26487,23798],{},[95,26489,220],{"stretchy":210},[55,26491,26492],{"encoding":57},"ds=d\\theta\u002F\\sqrt{\\theta(1-\\theta)}",[31,26494,26496,26517],{"className":26495,"ariaHidden":62},[61],[31,26497,26499,26502,26505,26508,26511,26514],{"className":26498},[66],[31,26500],{"className":26501,"style":1630},[70],[31,26503,1435],{"className":26504},[75,76],[31,26506,14551],{"className":26507},[75,76],[31,26509],{"className":26510,"style":293},[119],[31,26512,223],{"className":26513},[297],[31,26515],{"className":26516,"style":293},[119],[31,26518,26520,26524,26527,26530,26533],{"className":26519},[66],[31,26521],{"className":26522,"style":26523},[70],"height:1.24em;vertical-align:-0.305em;",[31,26525,1435],{"className":26526},[75,76],[31,26528,23798],{"className":26529,"style":19759},[75,76],[31,26531,3768],{"className":26532},[75],[31,26534,26537],{"className":26535},[75,26536],"sqrt",[31,26538,26540,26606],{"className":26539},[1043,1044],[31,26541,26543,26603],{"className":26542},[1048],[31,26544,26547,26583],{"className":26545,"style":26546},[1052],"height:0.935em;",[31,26548,26552,26555],{"className":26549,"style":26551},[26550],"svg-align","top:-3.2em;",[31,26553],{"className":26554,"style":13336},[1060],[31,26556,26559,26562,26565,26568,26571,26574,26577,26580],{"className":26557,"style":26558},[75],"padding-left:1em;",[31,26560,23798],{"className":26561,"style":19759},[75,76],[31,26563,211],{"className":26564},[264],[31,26566,205],{"className":26567},[75],[31,26569],{"className":26570,"style":120},[119],[31,26572,97],{"className":26573},[124],[31,26575],{"className":26576,"style":120},[119],[31,26578,23798],{"className":26579,"style":19759},[75,76],[31,26581,220],{"className":26582},[289],[31,26584,26586,26589],{"style":26585},"top:-2.895em;",[31,26587],{"className":26588,"style":13336},[1060],[31,26590,26594],{"className":26591,"style":26593},[26592],"hide-tail","min-width:1.02em;height:1.28em;",[13341,26595,26600],{"xmlns":13343,"width":26596,"height":26597,"viewBox":26598,"preserveAspectRatio":26599},"400em","1.28em","0 0 400000 1296","xMinYMin slice",[13347,26601],{"d":26602},"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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",[31,27289,27291,27309],{"className":27290,"translate":35},[34],[31,27292,27294],{"className":27293},[39],[41,27295,27296],{"xmlns":43},[45,27297,27298,27306],{},[48,27299,27300,27302,27304],{},[51,27301,23798],{},[95,27303,929],{},[91,27305,1977],{},[55,27307,27308],{"encoding":57},"\\theta\\to0",[31,27310,27312,27330],{"className":27311,"ariaHidden":62},[61],[31,27313,27315,27318,27321,27324,27327],{"className":27314},[66],[31,27316],{"className":27317,"style":1630},[70],[31,27319,23798],{"className":27320,"style":19759},[75,76],[31,27322],{"className":27323,"style":293},[119],[31,27325,929],{"className":27326},[297],[31,27328],{"className":27329,"style":293},[119],[31,27331,27333,27336],{"className":27332},[66],[31,27334],{"className":27335,"style":326},[70],[31,27337,1977],{"className":27338},[75],[31,27340,27342,27360],{"className":27341,"translate":35},[34],[31,27343,27345],{"className":27344},[39],[41,27346,27347],{"xmlns":43},[45,27348,27349,27357],{},[48,27350,27351,27353,27355],{},[51,27352,23798],{},[95,27354,929],{},[91,27356,205],{},[55,27358,27359],{"encoding":57},"\\theta\\to1",[31,27361,27363,27381],{"className":27362,"ariaHidden":62},[61],[31,27364,27366,27369,27372,27375,27378],{"className":27365},[66],[31,27367],{"className":27368,"style":1630},[70],[31,27370,23798],{"className":27371,"style":19759},[75,76],[31,27373],{"className":27374,"style":293},[119],[31,27376,929],{"className":27377},[297],[31,27379],{"className":27380,"style":293},[119],[31,27382,27384,27387],{"className":27383},[66],[31,27385],{"className":27386,"style":326},[70],[31,27388,205],{"className":27389},[75],[31,27391,27393,27406],{"className":27392,"translate":35},[34],[31,27394,27396],{"className":27395},[39],[41,27397,27398],{"xmlns":43},[45,27399,27400,27404],{},[48,27401,27402],{},[51,27403,2515],{},[55,27405,3485],{"encoding":57},[31,27407,27409],{"className":27408,"ariaHidden":62},[61],[31,27410,27412,27415],{"className":27411},[66],[31,27413],{"className":27414,"style":71},[70],[31,27416,2515],{"className":27417,"style":2666},[75,76],"，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 ",[31,27420,27422,27435],{"className":27421,"translate":35},[34],[31,27423,27425],{"className":27424},[39],[41,27426,27427],{"xmlns":43},[45,27428,27429,27433],{},[48,27430,27431],{},[51,27432,2515],{},[55,27434,3485],{"encoding":57},[31,27436,27438],{"className":27437,"ariaHidden":62},[61],[31,27439,27441,27444],{"className":27440},[66],[31,27442],{"className":27443,"style":71},[70],[31,27445,2515],{"className":27446,"style":2666},[75,76],"；而 ",[31,27449,27451,27486],{"className":27450,"translate":35},[34],[31,27452,27454],{"className":27453},[39],[41,27455,27456],{"xmlns":43},[45,27457,27458,27483],{},[48,27459,27460,27466,27468,27471,27473,27476,27478,27480],{},[48,27461,27462,27464],{},[51,27463,4558],{"mathvariant":494},[51,27465,4561],{"mathvariant":494},[95,27467,211],{"stretchy":210},[91,27469,27470],{},"0.999",[51,27472,4572],{"mathvariant":494},[91,27474,27475],{},"0.001",[95,27477,220],{"stretchy":210},[95,27479,27223],{},[91,27481,27482],{},"6.9",[55,27484,27485],{"encoding":57},"\\mathrm{KL}(0.999\\|0.001)\\approx6.9",[31,27487,27489,27520],{"className":27488,"ariaHidden":62},[61],[31,27490,27492,27495,27501,27504,27508,27511,27514,27517],{"className":27491},[66],[31,27493],{"className":27494,"style":260},[70],[31,27496,27498],{"className":27497},[75],[31,27499,4606],{"className":27500},[75,2745],[31,27502,211],{"className":27503},[264],[31,27505,27507],{"className":27506},[75],"0.999∥0.001",[31,27509,220],{"className":27510},[289],[31,27512],{"className":27513,"style":293},[119],[31,27515,27223],{"className":27516},[297],[31,27518],{"className":27519,"style":293},[119],[31,27521,27523,27526],{"className":27522},[66],[31,27524],{"className":27525,"style":326},[70],[31,27527,27482],{"className":27528},[75]," 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。",[11,27531,27532,27533,27632],{},"那个 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0 400000 1080",[13347,27618],{"d":27619},"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",[31,27621,1076],{"className":27622},[1075],[31,27624,27626],{"className":27625},[1048],[31,27627,27630],{"className":27628,"style":27629},[1052],"height:0.1078em;",[31,27631],{}," 不是巧合。对一般的离散分布，",[31,27634,27636],{"className":27635,"translate":35},[411],[31,27637,27639,27709],{"className":27638,"translate":35},[34],[31,27640,27642],{"className":27641},[39],[41,27643,27644],{"xmlns":43,"display":420},[45,27645,27646,27706],{},[48,27647,27648,27658,27660,27662,27664,27667,27669,27671,27673,27676,27678,27680,27686,27702,27704],{},[1004,27649,27650,27652],{},[51,27651,1435],{},[48,27653,27654,27656],{},[51,27655,13114],{"mathvariant":494},[51,27657,14790],{"mathvariant":494},[95,27659,211],{"stretchy":210},[51,27661,11],{},[95,27663,1210],{"separator":62},[51,27665,27666],{},"q",[95,27668,220],{"stretchy":210},[95,27670,223],{},[91,27672,152],{},[51,27674,27675],{},"arccos",[95,27677,1540],{},[95,27679,211],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[1531,27681,27682,27684],{},[95,27683,5625],{},[51,27685,5391],{},[26474,27687,27688],{},[48,27689,27690,27696],{},[1004,27691,27692,27694],{},[51,27693,11],{},[51,27695,5391],{},[1004,27697,27698,27700],{},[51,27699,27666],{},[51,27701,5391],{},[95,27703,220],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[95,27705,1210],{"separator":62},[55,27707,27708],{"encoding":57},"d_{\\mathrm{FR}}(p,q)=2\\arccos\\Bigl(\\sum_i\\sqrt{p_iq_i}\\Bigr),",[31,27710,27712,27793],{"className":27711,"ariaHidden":62},[61],[31,27713,27715,27718,27766,27769,27772,27775,27778,27781,27784,27787,27790],{"className":27714},[66],[31,27716],{"className":27717,"style":260},[70],[31,27719,27721,27724],{"className":27720},[75],[31,27722,1435],{"className":27723},[75,76],[31,27725,27727],{"className":27726},[1039],[31,27728,27730,27758],{"className":27729},[1043,1044],[31,27731,27733,27755],{"className":27732},[1048],[31,27734,27737],{"className":27735,"style":27736},[1052],"height:0.3283em;",[31,27738,27739,27742],{"style":2807},[31,27740],{"className":27741,"style":1061},[1060],[31,27743,27745],{"className":27744},[1065,1066,1067,1068],[31,27746,27748],{"className":27747},[75,1068],[31,27749,27751],{"className":27750},[75,1068],[31,27752,27754],{"className":27753},[75,2745,1068],"FR",[31,27756,1076],{"className":27757},[1075],[31,27759,27761],{"className":27760},[1048],[31,27762,27764],{"className":27763,"style":2050},[1052],[31,27765],{},[31,27767,211],{"className":27768},[264],[31,27770,11],{"className":27771},[75,76],[31,27773,1210],{"className":27774},[1391],[31,27776],{"className":27777,"style":1467},[119],[31,27779,27666],{"className":27780,"style":2666},[75,76],[31,27782,220],{"className":27783},[289],[31,27785],{"className":27786,"style":293},[119],[31,27788,223],{"className":27789},[297],[31,27791],{"className":27792,"style":293},[119],[31,27794,27796,27800,27803,27806,27809,27815,27859,27862,27989,27995],{"className":27795},[66],[31,27797],{"className":27798,"style":27799},[70],"height:2.4277em;vertical-align:-1.2777em;",[31,27801,152],{"className":27802},[75],[31,27804],{"className":27805,"style":1467},[119],[31,27807,27675],{"className":27808},[1460],[31,27810,27812],{"className":27811},[264],[31,27813,211],{"className":27814},[1307,2662],[31,27816,27818],{"className":27817},[1460,1620],[31,27819,27821,27851],{"className":27820},[1043,1044],[31,27822,27824,27848],{"className":27823},[1048],[31,27825,27827,27838],{"className":27826,"style":5900},[1052],[31,27828,27829,27832],{"style":5903},[31,27830],{"className":27831,"style":5907},[1060],[31,27833,27835],{"className":27834},[1065,1066,1067,1068],[31,27836,5391],{"className":27837},[75,76,1068],[31,27839,27840,27843],{"style":5919},[31,27841],{"className":27842,"style":5907},[1060],[31,27844,27845],{},[31,27846,5625],{"className":27847},[1460,1461,1687],[31,27849,1076],{"className":27850},[1075],[31,27852,27854],{"className":27853},[1048],[31,27855,27857],{"className":27856,"style":6342},[1052],[31,27858],{},[31,27860],{"className":27861,"style":1467},[119],[31,27863,27865],{"className":27864},[75,26536],[31,27866,27868,27980],{"className":27867},[1043,1044],[31,27869,27871,27977],{"className":27870},[1048],[31,27872,27874,27964],{"className":27873,"style":13732},[1052],[31,27875,27877,27880],{"className":27876,"style":1649},[26550],[31,27878],{"className":27879,"style":1637},[1060],[31,27881,27884,27924],{"className":27882,"style":27883},[75],"padding-left:0.833em;",[31,27885,27887,27890],{"className":27886},[75],[31,27888,11],{"className":27889},[75,76],[31,27891,27893],{"className":27892},[1039],[31,27894,27896,27916],{"className":27895},[1043,1044],[31,27897,27899,27913],{"className":27898},[1048],[31,27900,27902],{"className":27901,"style":5445},[1052],[31,27903,27904,27907],{"style":2807},[31,27905],{"className":27906,"style":1061},[1060],[31,27908,27910],{"className":27909},[1065,1066,1067,1068],[31,27911,5391],{"className":27912},[75,76,1068],[31,27914,1076],{"className":27915},[1075],[31,27917,27919],{"className":27918},[1048],[31,27920,27922],{"className":27921,"style":2050},[1052],[31,27923],{},[31,27925,27927,27930],{"className":27926},[75],[31,27928,27666],{"className":27929,"style":2666},[75,76],[31,27931,27933],{"className":27932},[1039],[31,27934,27936,27956],{"className":27935},[1043,1044],[31,27937,27939,27953],{"className":27938},[1048],[31,27940,27942],{"className":27941,"style":5445},[1052],[31,27943,27944,27947],{"style":5504},[31,27945],{"className":27946,"style":1061},[1060],[31,27948,27950],{"className":27949},[1065,1066,1067,1068],[31,27951,5391],{"className":27952},[75,76,1068],[31,27954,1076],{"className":27955},[1075],[31,27957,27959],{"className":27958},[1048],[31,27960,27962],{"className":27961,"style":2050},[1052],[31,27963],{},[31,27965,27967,27970],{"style":27966},"top:-2.7119em;",[31,27968],{"className":27969,"style":1637},[1060],[31,27971,27973],{"className":27972,"style":27612},[26592],[13341,27974,27975],{"xmlns":13343,"width":26596,"height":27615,"viewBox":27616,"preserveAspectRatio":26599},[13347,27976],{"d":27619},[31,27978,1076],{"className":27979},[1075],[31,27981,27983],{"className":27982},[1048],[31,27984,27987],{"className":27985,"style":27986},[1052],"height:0.2881em;",[31,27988],{},[31,27990,27992],{"className":27991},[289],[31,27993,220],{"className":27994},[1307,2662],[31,27996,1210],{"className":27997},[1391],[11,27999,28000,28001,28127,28128,28188,28189,28217,28218,28364,28365,28579],{},"而映射 ",[31,28002,28004,28030],{"className":28003,"translate":35},[34],[31,28005,28007],{"className":28006},[39],[41,28008,28009],{"xmlns":43},[45,28010,28011,28027],{},[48,28012,28013,28015,28017,28019,28021,28023],{},[51,28014,11],{},[95,28016,18339],{},[51,28018,5324],{},[95,28020,223],{},[91,28022,152],{},[26474,28024,28025],{},[51,28026,11],{},[55,28028,28029],{"encoding":57},"p\\mapsto u=2\\sqrt p",[31,28031,28033,28052,28070],{"className":28032,"ariaHidden":62},[61],[31,28034,28036,28040,28043,28046,28049],{"className":28035},[66],[31,28037],{"className":28038,"style":28039},[70],"height:0.7054em;vertical-align:-0.1944em;",[31,28041,11],{"className":28042},[75,76],[31,28044],{"className":28045,"style":293},[119],[31,28047,18339],{"className":28048},[297],[31,28050],{"className":28051,"style":293},[119],[31,28053,28055,28058,28061,28064,28067],{"className":28054},[66],[31,28056],{"className":28057,"style":71},[70],[31,28059,5324],{"className":28060},[75,76],[31,28062],{"className":28063,"style":293},[119],[31,28065,223],{"className":28066},[297],[31,28068],{"className":28069,"style":293},[119],[31,28071,28073,28077,28080],{"className":28072},[66],[31,28074],{"className":28075,"style":28076},[70],"height:1.04em;vertical-align:-0.3369em;",[31,28078,152],{"className":28079},[75],[31,28081,28083],{"className":28082},[75,26536],[31,28084,28086,28118],{"className":28085},[1043,1044],[31,28087,28089,28115],{"className":28088},[1048],[31,28090,28093,28102],{"className":28091,"style":28092},[1052],"height:0.7031em;",[31,28094,28096,28099],{"className":28095,"style":1649},[26550],[31,28097],{"className":28098,"style":1637},[1060],[31,28100,11],{"className":28101,"style":27883},[75,76],[31,28103,28105,28108],{"style":28104},"top:-2.6631em;",[31,28106],{"className":28107,"style":1637},[1060],[31,28109,28111],{"className":28110,"style":27612},[26592],[13341,28112,28113],{"xmlns":13343,"width":26596,"height":27615,"viewBox":27616,"preserveAspectRatio":26599},[13347,28114],{"d":27619},[31,28116,1076],{"className":28117},[1075],[31,28119,28121],{"className":28120},[1048],[31,28122,28125],{"className":28123,"style":28124},[1052],"height:0.3369em;",[31,28126],{}," 把单纯形嵌入到 ",[31,28129,28131,28149],{"className":28130,"translate":35},[34],[31,28132,28134],{"className":28133},[39],[41,28135,28136],{"xmlns":43},[45,28137,28138,28146],{},[48,28139,28140],{},[1180,28141,28142,28144],{},[51,28143,14790],{"mathvariant":14789},[51,28145,4558],{},[55,28147,28148],{"encoding":57},"\\mathbb R^K",[31,28150,28152],{"className":28151,"ariaHidden":62},[61],[31,28153,28155,28159],{"className":28154},[66],[31,28156],{"className":28157,"style":28158},[70],"height:0.8413em;",[31,28160,28162,28165],{"className":28161},[75],[31,28163,14790],{"className":28164},[75,14947],[31,28166,28168],{"className":28167},[1039],[31,28169,28171],{"className":28170},[1043],[31,28172,28174],{"className":28173},[1048],[31,28175,28177],{"className":28176,"style":28158},[1052],[31,28178,28179,28182],{"style":3631},[31,28180],{"className":28181,"style":1061},[1060],[31,28183,28185],{"className":28184},[1065,1066,1067,1068],[31,28186,4558],{"className":28187,"style":8525},[75,76,1068]," 中半径为 ",[31,28190,28192,28205],{"className":28191,"translate":35},[34],[31,28193,28195],{"className":28194},[39],[41,28196,28197],{"xmlns":43},[45,28198,28199,28203],{},[48,28200,28201],{},[91,28202,152],{},[55,28204,152],{"encoding":57},[31,28206,28208],{"className":28207,"ariaHidden":62},[61],[31,28209,28211,28214],{"className":28210},[66],[31,28212],{"className":28213,"style":326},[70],[31,28215,152],{"className":28216},[75]," 的球面上（因为 ",[31,28219,28221,28251],{"className":28220,"translate":35},[34],[31,28222,28224],{"className":28223},[39],[41,28225,28226],{"xmlns":43},[45,28227,28228,28248],{},[48,28229,28230,28236,28238,28244,28246],{},[1004,28231,28232,28234],{},[95,28233,5625],{},[51,28235,5391],{},[91,28237,15416],{},[1004,28239,28240,28242],{},[51,28241,11],{},[51,28243,5391],{},[95,28245,223],{},[91,28247,15416],{},[55,28249,28250],{"encoding":57},"\\sum_i4p_i=4",[31,28252,28254,28355],{"className":28253,"ariaHidden":62},[61],[31,28255,28257,28260,28300,28303,28306,28346,28349,28352],{"className":28256},[66],[31,28258],{"className":28259,"style":21359},[70],[31,28261,28263,28266],{"className":28262},[1460],[31,28264,5625],{"className":28265,"style":21372},[1460,1461,1462],[31,28267,28269],{"className":28268},[1039],[31,28270,28272,28292],{"className":28271},[1043,1044],[31,28273,28275,28289],{"className":28274},[1048],[31,28276,28278],{"className":28277,"style":21385},[1052],[31,28279,28280,28283],{"style":21388},[31,28281],{"className":28282,"style":1061},[1060],[31,28284,28286],{"className":28285},[1065,1066,1067,1068],[31,28287,5391],{"className":28288},[75,76,1068],[31,28290,1076],{"className":28291},[1075],[31,28293,28295],{"className":28294},[1048],[31,28296,28298],{"className":28297,"style":21407},[1052],[31,28299],{},[31,28301],{"className":28302,"style":1467},[119],[31,28304,15416],{"className":28305},[75],[31,28307,28309,28312],{"className":28308},[75],[31,28310,11],{"className":28311},[75,76],[31,28313,28315],{"className":28314},[1039],[31,28316,28318,28338],{"className":28317},[1043,1044],[31,28319,28321,28335],{"className":28320},[1048],[31,28322,28324],{"className":28323,"style":5445},[1052],[31,28325,28326,28329],{"style":2807},[31,28327],{"className":28328,"style":1061},[1060],[31,28330,28332],{"className":28331},[1065,1066,1067,1068],[31,28333,5391],{"className":28334},[75,76,1068],[31,28336,1076],{"className":28337},[1075],[31,28339,28341],{"className":28340},[1048],[31,28342,28344],{"className":28343,"style":2050},[1052],[31,28345],{},[31,28347],{"className":28348,"style":293},[119],[31,28350,223],{"className":28351},[297],[31,28353],{"className":28354,"style":293},[119],[31,28356,28358,28361],{"className":28357},[66],[31,28359],{"className":28360,"style":326},[70],[31,28362,15416],{"className":28363},[75],"），Fisher–Rao 距离正好是球面上的大圆距离。这个嵌入叫 Hellinger 嵌入，",[31,28366,28368,28402],{"className":28367,"translate":35},[34],[31,28369,28371],{"className":28370},[39],[41,28372,28373],{"xmlns":43},[45,28374,28375,28399],{},[48,28376,28377,28383],{},[1004,28378,28379,28381],{},[95,28380,5625],{},[51,28382,5391],{},[26474,28384,28385],{},[48,28386,28387,28393],{},[1004,28388,28389,28391],{},[51,28390,11],{},[51,28392,5391],{},[1004,28394,28395,28397],{},[51,28396,27666],{},[51,28398,5391],{},[55,28400,28401],{"encoding":57},"\\sum_i\\sqrt{p_iq_i}",[31,28403,28405],{"className":28404,"ariaHidden":62},[61],[31,28406,28408,28412,28452,28455],{"className":28407},[66],[31,28409],{"className":28410,"style":28411},[70],"height:1.0869em;vertical-align:-0.3369em;",[31,28413,28415,28418],{"className":28414},[1460],[31,28416,5625],{"className":28417,"style":21372},[1460,1461,1462],[31,28419,28421],{"className":28420},[1039],[31,28422,28424,28444],{"className":28423},[1043,1044],[31,28425,28427,28441],{"className":28426},[1048],[31,28428,28430],{"className":28429,"style":21385},[1052],[31,28431,28432,28435],{"style":21388},[31,28433],{"className":28434,"style":1061},[1060],[31,28436,28438],{"className":28437},[1065,1066,1067,1068],[31,28439,5391],{"className":28440},[75,76,1068],[31,28442,1076],{"className":28443},[1075],[31,28445,28447],{"className":28446},[1048],[31,28448,28450],{"className":28449,"style":21407},[1052],[31,28451],{},[31,28453],{"className":28454,"style":1467},[119],[31,28456,28458],{"className":28457},[75,26536],[31,28459,28461,28571],{"className":28460},[1043,1044],[31,28462,28464,28568],{"className":28463},[1048],[31,28465,28467,28556],{"className":28466,"style":28092},[1052],[31,28468,28470,28473],{"className":28469,"style":1649},[26550],[31,28471],{"className":28472,"style":1637},[1060],[31,28474,28476,28516],{"className":28475,"style":27883},[75],[31,28477,28479,28482],{"className":28478},[75],[31,28480,11],{"className":28481},[75,76],[31,28483,28485],{"className":28484},[1039],[31,28486,28488,28508],{"className":28487},[1043,1044],[31,28489,28491,28505],{"className":28490},[1048],[31,28492,28494],{"className":28493,"style":5445},[1052],[31,28495,28496,28499],{"style":2807},[31,28497],{"className":28498,"style":1061},[1060],[31,28500,28502],{"className":28501},[1065,1066,1067,1068],[31,28503,5391],{"className":28504},[75,76,1068],[31,28506,1076],{"className":28507},[1075],[31,28509,28511],{"className":28510},[1048],[31,28512,28514],{"className":28513,"style":2050},[1052],[31,28515],{},[31,28517,28519,28522],{"className":28518},[75],[31,28520,27666],{"className":28521,"style":2666},[75,76],[31,28523,28525],{"className":28524},[1039],[31,28526,28528,28548],{"className":28527},[1043,1044],[31,28529,28531,28545],{"className":28530},[1048],[31,28532,28534],{"className":28533,"style":5445},[1052],[31,28535,28536,28539],{"style":5504},[31,28537],{"className":28538,"style":1061},[1060],[31,28540,28542],{"className":28541},[1065,1066,1067,1068],[31,28543,5391],{"className":28544},[75,76,1068],[31,28546,1076],{"className":28547},[1075],[31,28549,28551],{"className":28550},[1048],[31,28552,28554],{"className":28553,"style":2050},[1052],[31,28555],{},[31,28557,28558,28561],{"style":28104},[31,28559],{"className":28560,"style":1637},[1060],[31,28562,28564],{"className":28563,"style":27612},[26592],[13341,28565,28566],{"xmlns":13343,"width":26596,"height":27615,"viewBox":27616,"preserveAspectRatio":26599},[13347,28567],{"d":27619},[31,28569,1076],{"className":28570},[1075],[31,28572,28574],{"className":28573},[1048],[31,28575,28577],{"className":28576,"style":28124},[1052],[31,28578],{}," 叫 Bhattacharyya 系数。",[11,28581,28582],{},"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",[31,28584,28586],{"className":28585,"translate":35},[411],[31,28587,28589,28708],{"className":28588,"translate":35},[34],[31,28590,28592],{"className":28591},[39],[41,28593,28594],{"xmlns":43,"display":420},[45,28595,28596,28705],{},[48,28597,28598,28604,28606,28647,28649,28651,28657,28659,28665,28667,28669,28671,28673,28675,28677,28685,28687,28689,28697,28703],{},[1004,28599,28600,28602],{},[51,28601,11],{},[51,28603,12271],{},[95,28605,223],{},[2074,28607,28608,28640],{},[48,28609,28610,28612,28614,28616,28618,28620,28622,28626,28628,28630,28634],{},[95,28611,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,28613,211],{"stretchy":210},[91,28615,205],{},[95,28617,97],{},[51,28619,12271],{},[95,28621,220],{"stretchy":210},[26474,28623,28624],{},[51,28625,11],{},[95,28627,434],{},[51,28629,12271],{},[26474,28631,28632],{},[51,28633,27666],{},[1180,28635,28636,28638],{},[95,28637,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[91,28639,152],{},[1004,28641,28642,28645],{},[51,28643,28644],{},"Z",[51,28646,12271],{},[95,28648,1210],{"separator":62},[119,28650],{"width":20286},[1004,28652,28653,28655],{},[51,28654,28644],{},[51,28656,12271],{},[95,28658,223],{},[1531,28660,28661,28663],{},[95,28662,5625],{},[51,28664,5391],{},[95,28666,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,28668,211],{"stretchy":210},[91,28670,205],{},[95,28672,97],{},[51,28674,12271],{},[95,28676,220],{"stretchy":210},[26474,28678,28679],{},[1004,28680,28681,28683],{},[51,28682,11],{},[51,28684,5391],{},[95,28686,434],{},[51,28688,12271],{},[26474,28690,28691],{},[1004,28692,28693,28695],{},[51,28694,27666],{},[51,28696,5391],{},[1180,28698,28699,28701],{},[95,28700,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[91,28702,152],{},[51,28704,495],{"mathvariant":494},[55,28706,28707],{"encoding":57},"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 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29284],{"className":29283},[75,26536],[31,29285,29287,29357],{"className":29286},[1043,1044],[31,29288,29290,29354],{"className":29289},[1048],[31,29291,29293,29342],{"className":29292,"style":13732},[1052],[31,29294,29296,29299],{"className":29295,"style":1649},[26550],[31,29297],{"className":29298,"style":1637},[1060],[31,29300,29302],{"className":29301,"style":27883},[75],[31,29303,29305,29308],{"className":29304},[75],[31,29306,27666],{"className":29307,"style":2666},[75,76],[31,29309,29311],{"className":29310},[1039],[31,29312,29314,29334],{"className":29313},[1043,1044],[31,29315,29317,29331],{"className":29316},[1048],[31,29318,29320],{"className":29319,"style":5445},[1052],[31,29321,29322,29325],{"style":5504},[31,29323],{"className":29324,"style":1061},[1060],[31,29326,29328],{"className":29327},[1065,1066,1067,1068],[31,29329,5391],{"className":29330},[75,76,1068],[31,29332,1076],{"className":29333},[1075],[31,29335,29337],{"className":29336},[1048],[31,29338,29340],{"className":29339,"style":2050},[1052],[31,29341],{},[31,29343,29344,29347],{"style":27966},[31,29345],{"className":29346,"style":1637},[1060],[31,29348,29350],{"className":29349,"style":27612},[26592],[13341,29351,29352],{"xmlns":13343,"width":26596,"height":27615,"viewBox":27616,"preserveAspectRatio":26599},[13347,29353],{"d":27619},[31,29355,1076],{"className":29356},[1075],[31,29358,29360],{"className":29359},[1048],[31,29361,29363],{"className":29362,"style":27986},[1052],[31,29364],{},[31,29366,29368,29374],{"className":29367},[289],[31,29369,29371],{"className":29370},[289],[31,29372,220],{"className":29373},[1307,1308],[31,29375,29377],{"className":29376},[1039],[31,29378,29380],{"className":29379},[1043],[31,29381,29383],{"className":29382},[1048],[31,29384,29386],{"className":29385,"style":29009},[1052],[31,29387,29388,29391],{"style":13566},[31,29389],{"className":29390,"style":1061},[1060],[31,29392,29394],{"className":29393},[1065,1066,1067,1068],[31,29395,152],{"className":29396},[75,1068],[31,29398,495],{"className":29399},[75],[11,29401,29402],{},"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。",[18,29404,29405],{"id":29405},"指数族",[11,29407,29408,29409,29437,29438,29466],{},"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 ",[31,29410,29412,29425],{"className":29411,"translate":35},[34],[31,29413,29415],{"className":29414},[39],[41,29416,29417],{"xmlns":43},[45,29418,29419,29423],{},[48,29420,29421],{},[51,29422,23798],{},[55,29424,24747],{"encoding":57},[31,29426,29428],{"className":29427,"ariaHidden":62},[61],[31,29429,29431,29434],{"className":29430},[66],[31,29432],{"className":29433,"style":1630},[70],[31,29435,23798],{"className":29436,"style":19759},[75,76],"，正面次数服从二项分布，而样本对 ",[31,29439,29441,29454],{"className":29440,"translate":35},[34],[31,29442,29444],{"className":29443},[39],[41,29445,29446],{"xmlns":43},[45,29447,29448,29452],{},[48,29449,29450],{},[51,29451,23798],{},[55,29453,24747],{"encoding":57},[31,29455,29457],{"className":29456,"ariaHidden":62},[61],[31,29458,29460,29463],{"className":29459},[66],[31,29461],{"className":29462,"style":1630},[70],[31,29464,23798],{"className":29465,"style":19759},[75,76]," 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。",[11,29468,29469],{},"指数族把这个结构写成一个模板：",[31,29471,29473],{"className":29472,"translate":35},[411],[31,29474,29476,29545],{"className":29475,"translate":35},[34],[31,29477,29479],{"className":29478},[39],[41,29480,29481],{"xmlns":43,"display":420},[45,29482,29483,29542],{},[48,29484,29485,29487,29489,29491,29493,29495,29497,29499,29502,29504,29506,29508,29510,29512,29514,29520,29522,29524,29526,29528,29530,29532,29534,29536,29538,29540],{},[51,29486,11],{},[95,29488,211],{"stretchy":210},[51,29490,53],{},[95,29492,12680],{},[51,29494,23798],{},[95,29496,220],{"stretchy":210},[95,29498,223],{},[51,29500,29501],{},"h",[95,29503,211],{"stretchy":210},[51,29505,53],{},[95,29507,220],{"stretchy":210},[51,29509,20301],{},[95,29511,1540],{},[95,29513,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1180,29515,29516,29518],{},[51,29517,23798],{},[51,29519,20800],{"mathvariant":494},[51,29521,919],{},[95,29523,211],{"stretchy":210},[51,29525,53],{},[95,29527,220],{"stretchy":210},[95,29529,97],{},[51,29531,8064],{},[95,29533,211],{"stretchy":210},[51,29535,23798],{},[95,29537,220],{"stretchy":210},[95,29539,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,29541,495],{"mathvariant":494},[55,29543,29544],{"encoding":57},"p(x\\mid\\theta)=h(x)\\exp\\bigl(\\theta^\\top T(x)-\\psi(\\theta)\\bigr).",[31,29546,29548,29572,29593,29675],{"className":29547,"ariaHidden":62},[61],[31,29549,29551,29554,29557,29560,29563,29566,29569],{"className":29550},[66],[31,29552],{"className":29553,"style":260},[70],[31,29555,11],{"className":29556},[75,76],[31,29558,211],{"className":29559},[264],[31,29561,53],{"className":29562},[75,76],[31,29564],{"className":29565,"style":293},[119],[31,29567,12680],{"className":29568},[297],[31,29570],{"className":29571,"style":293},[119],[31,29573,29575,29578,29581,29584,29587,29590],{"className":29574},[66],[31,29576],{"className":29577,"style":260},[70],[31,29579,23798],{"className":29580,"style":19759},[75,76],[31,29582,220],{"className":29583},[289],[31,29585],{"className":29586,"style":293},[119],[31,29588,223],{"className":29589},[297],[31,29591],{"className":29592,"style":293},[119],[31,29594,29596,29600,29603,29606,29609,29612,29615,29618,29624,29654,29657,29660,29663,29666,29669,29672],{"className":29595},[66],[31,29597],{"className":29598,"style":29599},[70],"height:1.2491em;vertical-align:-0.35em;",[31,29601,29501],{"className":29602},[75,76],[31,29604,211],{"className":29605},[264],[31,29607,53],{"className":29608},[75,76],[31,29610,220],{"className":29611},[289],[31,29613],{"className":29614,"style":1467},[119],[31,29616,20301],{"className":29617},[1460],[31,29619,29621],{"className":29620},[264],[31,29622,211],{"className":29623},[1307,1308],[31,29625,29627,29630],{"className":29626},[75],[31,29628,23798],{"className":29629,"style":19759},[75,76],[31,29631,29633],{"className":29632},[1039],[31,29634,29636],{"className":29635},[1043],[31,29637,29639],{"className":29638},[1048],[31,29640,29643],{"className":29641,"style":29642},[1052],"height:0.8991em;",[31,29644,29645,29648],{"style":1330},[31,29646],{"className":29647,"style":1061},[1060],[31,29649,29651],{"className":29650},[1065,1066,1067,1068],[31,29652,20800],{"className":29653},[75,1068],[31,29655,919],{"className":29656,"style":949},[75,76],[31,29658,211],{"className":29659},[264],[31,29661,53],{"className":29662},[75,76],[31,29664,220],{"className":29665},[289],[31,29667],{"className":29668,"style":120},[119],[31,29670,97],{"className":29671},[124],[31,29673],{"className":29674,"style":120},[119],[31,29676,29678,29681,29684,29687,29690,29693,29699],{"className":29677},[66],[31,29679],{"className":29680,"style":4988},[70],[31,29682,8064],{"className":29683,"style":2666},[75,76],[31,29685,211],{"className":29686},[264],[31,29688,23798],{"className":29689,"style":19759},[75,76],[31,29691,220],{"className":29692},[289],[31,29694,29696],{"className":29695},[289],[31,29697,220],{"className":29698},[1307,1308],[31,29700,495],{"className":29701},[75],[11,29703,29704,29732,29733,29761,29762,29791,29792,15282,29858,15282,30006,2372],{},[31,29705,29707,29720],{"className":29706,"translate":35},[34],[31,29708,29710],{"className":29709},[39],[41,29711,29712],{"xmlns":43},[45,29713,29714,29718],{},[48,29715,29716],{},[51,29717,919],{},[55,29719,919],{"encoding":57},[31,29721,29723],{"className":29722,"ariaHidden":62},[61],[31,29724,29726,29729],{"className":29725},[66],[31,29727],{"className":29728,"style":945},[70],[31,29730,919],{"className":29731,"style":949},[75,76]," 是充分统计量，",[31,29734,29736,29749],{"className":29735,"translate":35},[34],[31,29737,29739],{"className":29738},[39],[41,29740,29741],{"xmlns":43},[45,29742,29743,29747],{},[48,29744,29745],{},[51,29746,23798],{},[55,29748,24747],{"encoding":57},[31,29750,29752],{"className":29751,"ariaHidden":62},[61],[31,29753,29755,29758],{"className":29754},[66],[31,29756],{"className":29757,"style":1630},[70],[31,29759,23798],{"className":29760,"style":19759},[75,76]," 是自然参数，",[31,29763,29765,29779],{"className":29764,"translate":35},[34],[31,29766,29768],{"className":29767},[39],[41,29769,29770],{"xmlns":43},[45,29771,29772,29776],{},[48,29773,29774],{},[51,29775,8064],{},[55,29777,29778],{"encoding":57},"\\psi",[31,29780,29782],{"className":29781,"ariaHidden":62},[61],[31,29783,29785,29788],{"className":29784},[66],[31,29786],{"className":29787,"style":4202},[70],[31,29789,8064],{"className":29790,"style":2666},[75,76]," 是对数配分函数，负责把总质量归一化成 1。抛硬币就是 ",[31,29793,29795,29819],{"className":29794,"translate":35},[34],[31,29796,29798],{"className":29797},[39],[41,29799,29800],{"xmlns":43},[45,29801,29802,29816],{},[48,29803,29804,29806,29808,29810,29812,29814],{},[51,29805,919],{},[95,29807,211],{"stretchy":210},[51,29809,53],{},[95,29811,220],{"stretchy":210},[95,29813,223],{},[51,29815,53],{},[55,29817,29818],{"encoding":57},"T(x)=x",[31,29820,29822,29849],{"className":29821,"ariaHidden":62},[61],[31,29823,29825,29828,29831,29834,29837,29840,29843,29846],{"className":29824},[66],[31,29826],{"className":29827,"style":260},[70],[31,29829,919],{"className":29830,"style":949},[75,76],[31,29832,211],{"className":29833},[264],[31,29835,53],{"className":29836},[75,76],[31,29838,220],{"className":29839},[289],[31,29841],{"className":29842,"style":293},[119],[31,29844,223],{"className":29845},[297],[31,29847],{"className":29848,"style":293},[119],[31,29850,29852,29855],{"className":29851},[66],[31,29853],{"className":29854,"style":71},[70],[31,29856,53],{"className":29857},[75,76],[31,29859,29861,29893],{"className":29860,"translate":35},[34],[31,29862,29864],{"className":29863},[39],[41,29865,29866],{"xmlns":43},[45,29867,29868,29890],{},[48,29869,29870,29872,29874,29876,29878],{},[51,29871,23798],{},[95,29873,223],{},[51,29875,19939],{},[95,29877,1540],{},[2074,29879,29880,29882],{},[51,29881,11],{},[48,29883,29884,29886,29888],{},[91,29885,205],{},[95,29887,97],{},[51,29889,11],{},[55,29891,29892],{"encoding":57},"\\theta=\\log\\frac{p}{1-p}",[31,29894,29896,29914],{"className":29895,"ariaHidden":62},[61],[31,29897,29899,29902,29905,29908,29911],{"className":29898},[66],[31,29900],{"className":29901,"style":1630},[70],[31,29903,23798],{"className":29904,"style":19759},[75,76],[31,29906],{"className":29907,"style":293},[119],[31,29909,223],{"className":29910},[297],[31,29912],{"className":29913,"style":293},[119],[31,29915,29917,29921,29926,29929],{"className":29916},[66],[31,29918],{"className":29919,"style":29920},[70],"height:1.2286em;vertical-align:-0.4811em;",[31,29922,20177,29924],{"className":29923},[1460],[31,29925,14556],{"style":20180},[31,29927],{"className":29928,"style":1467},[119],[31,29930,29932,29935,30003],{"className":29931},[75],[31,29933],{"className":29934},[264,2139],[31,29936,29938],{"className":29937},[2074],[31,29939,29941,29994],{"className":29940},[1043,1044],[31,29942,29944,29991],{"className":29943},[1048],[31,29945,29948,29968,29976],{"className":29946,"style":29947},[1052],"height:0.7475em;",[31,29949,29950,29953],{"style":2155},[31,29951],{"className":29952,"style":1637},[1060],[31,29954,29956],{"className":29955},[1065,1066,1067,1068],[31,29957,29959,29962,29965],{"className":29958},[75,1068],[31,29960,205],{"className":29961},[75,1068],[31,29963,97],{"className":29964},[124,1068],[31,29966,11],{"className":29967},[75,76,1068],[31,29969,29970,29973],{"style":2170},[31,29971],{"className":29972,"style":1637},[1060],[31,29974],{"className":29975,"style":2178},[2177],[31,29977,29979,29982],{"style":29978},"top:-3.4461em;",[31,29980],{"className":29981,"style":1637},[1060],[31,29983,29985],{"className":29984},[1065,1066,1067,1068],[31,29986,29988],{"className":29987},[75,1068],[31,29989,11],{"className":29990},[75,76,1068],[31,29992,1076],{"className":29993},[1075],[31,29995,29997],{"className":29996},[1048],[31,29998,30001],{"className":29999,"style":30000},[1052],"height:0.4811em;",[31,30002],{},[31,30004],{"className":30005},[289,2139],[31,30007,30009,30050],{"className":30008,"translate":35},[34],[31,30010,30012],{"className":30011},[39],[41,30013,30014],{"xmlns":43},[45,30015,30016,30047],{},[48,30017,30018,30020,30022,30024,30026,30028,30030,30032,30034,30036,30038,30045],{},[51,30019,8064],{},[95,30021,211],{"stretchy":210},[51,30023,23798],{},[95,30025,220],{"stretchy":210},[95,30027,223],{},[51,30029,19939],{},[95,30031,1540],{},[95,30033,211],{"stretchy":210},[91,30035,205],{},[95,30037,434],{},[1180,30039,30040,30043],{},[51,30041,30042],{},"e",[51,30044,23798],{},[95,30046,220],{"stretchy":210},[55,30048,30049],{"encoding":57},"\\psi(\\theta)=\\log(1+e^\\theta)",[31,30051,30053,30080,30106],{"className":30052,"ariaHidden":62},[61],[31,30054,30056,30059,30062,30065,30068,30071,30074,30077],{"className":30055},[66],[31,30057],{"className":30058,"style":260},[70],[31,30060,8064],{"className":30061,"style":2666},[75,76],[31,30063,211],{"className":30064},[264],[31,30066,23798],{"className":30067,"style":19759},[75,76],[31,30069,220],{"className":30070},[289],[31,30072],{"className":30073,"style":293},[119],[31,30075,223],{"className":30076},[297],[31,30078],{"className":30079,"style":293},[119],[31,30081,30083,30086,30091,30094,30097,30100,30103],{"className":30082},[66],[31,30084],{"className":30085,"style":260},[70],[31,30087,20177,30089],{"className":30088},[1460],[31,30090,14556],{"style":20180},[31,30092,211],{"className":30093},[264],[31,30095,205],{"className":30096},[75],[31,30098],{"className":30099,"style":120},[119],[31,30101,434],{"className":30102},[124],[31,30104],{"className":30105,"style":120},[119],[31,30107,30109,30112,30141],{"className":30108},[66],[31,30110],{"className":30111,"style":20835},[70],[31,30113,30115,30118],{"className":30114},[75],[31,30116,30042],{"className":30117},[75,76],[31,30119,30121],{"className":30120},[1039],[31,30122,30124],{"className":30123},[1043],[31,30125,30127],{"className":30126},[1048],[31,30128,30130],{"className":30129,"style":20863},[1052],[31,30131,30132,30135],{"style":3631},[31,30133],{"className":30134,"style":1061},[1060],[31,30136,30138],{"className":30137},[1065,1066,1067,1068],[31,30139,23798],{"className":30140,"style":19759},[75,76,1068],[31,30142,220],{"className":30143},[289],[11,30145,30146,30174,30175,1138,30316,30495,30496,30524],{},[31,30147,30149,30162],{"className":30148,"translate":35},[34],[31,30150,30152],{"className":30151},[39],[41,30153,30154],{"xmlns":43},[45,30155,30156,30160],{},[48,30157,30158],{},[51,30159,8064],{},[55,30161,29778],{"encoding":57},[31,30163,30165],{"className":30164,"ariaHidden":62},[61],[31,30166,30168,30171],{"className":30167},[66],[31,30169],{"className":30170,"style":4202},[70],[31,30172,8064],{"className":30173,"style":2666},[75,76]," 的性质很好：",[31,30176,30178,30220],{"className":30177,"translate":35},[34],[31,30179,30181],{"className":30180},[39],[41,30182,30183],{"xmlns":43},[45,30184,30185,30217],{},[48,30186,30187,30189,30191,30193,30195,30197,30199,30205,30207,30209,30211,30213,30215],{},[51,30188,12145],{"mathvariant":494},[51,30190,8064],{},[95,30192,211],{"stretchy":210},[51,30194,23798],{},[95,30196,220],{"stretchy":210},[95,30198,223],{},[1004,30200,30201,30203],{},[51,30202,22324],{"mathvariant":14789},[51,30204,23798],{},[95,30206,15175],{"stretchy":210},[51,30208,919],{},[95,30210,211],{"stretchy":210},[51,30212,926],{},[95,30214,220],{"stretchy":210},[95,30216,15184],{"stretchy":210},[55,30218,30219],{"encoding":57},"\\nabla\\psi(\\theta)=\\mathbb E_\\theta[T(X)]",[31,30221,30223,30253],{"className":30222,"ariaHidden":62},[61],[31,30224,30226,30229,30232,30235,30238,30241,30244,30247,30250],{"className":30225},[66],[31,30227],{"className":30228,"style":260},[70],[31,30230,12145],{"className":30231},[75],[31,30233,8064],{"className":30234,"style":2666},[75,76],[31,30236,211],{"className":30237},[264],[31,30239,23798],{"className":30240,"style":19759},[75,76],[31,30242,220],{"className":30243},[289],[31,30245],{"className":30246,"style":293},[119],[31,30248,223],{"className":30249},[297],[31,30251],{"className":30252,"style":293},[119],[31,30254,30256,30259,30299,30302,30305,30308,30312],{"className":30255},[66],[31,30257],{"className":30258,"style":260},[70],[31,30260,30262,30265],{"className":30261},[75],[31,30263,22324],{"className":30264},[75,14947],[31,30266,30268],{"className":30267},[1039],[31,30269,30271,30291],{"className":30270},[1043,1044],[31,30272,30274,30288],{"className":30273},[1048],[31,30275,30277],{"className":30276,"style":1053},[1052],[31,30278,30279,30282],{"style":2807},[31,30280],{"className":30281,"style":1061},[1060],[31,30283,30285],{"className":30284},[1065,1066,1067,1068],[31,30286,23798],{"className":30287,"style":19759},[75,76,1068],[31,30289,1076],{"className":30290},[1075],[31,30292,30294],{"className":30293},[1048],[31,30295,30297],{"className":30296,"style":2050},[1052],[31,30298],{},[31,30300,15175],{"className":30301},[264],[31,30303,919],{"className":30304,"style":949},[75,76],[31,30306,211],{"className":30307},[264],[31,30309,926],{"className":30310,"style":30311},[75,76],"margin-right:0.0785em;",[31,30313,30315],{"className":30314},[289],")]",[31,30317,30319,30371],{"className":30318,"translate":35},[34],[31,30320,30322],{"className":30321},[39],[41,30323,30324],{"xmlns":43},[45,30325,30326,30368],{},[48,30327,30328,30334,30336,30338,30340,30342,30344,30356,30358,30360,30362,30364,30366],{},[1180,30329,30330,30332],{},[51,30331,12145],{"mathvariant":494},[91,30333,152],{},[51,30335,8064],{},[95,30337,211],{"stretchy":210},[51,30339,23798],{},[95,30341,220],{"stretchy":210},[95,30343,223],{},[1004,30345,30346,30354],{},[48,30347,30348,30350,30352],{},[51,30349,8101],{"mathvariant":494},[51,30351,2552],{"mathvariant":494},[51,30353,5354],{"mathvariant":494},[51,30355,23798],{},[95,30357,15175],{"stretchy":210},[51,30359,919],{},[95,30361,211],{"stretchy":210},[51,30363,926],{},[95,30365,220],{"stretchy":210},[95,30367,15184],{"stretchy":210},[55,30369,30370],{"encoding":57},"\\nabla^2\\psi(\\theta)=\\mathrm{Cov}_\\theta[T(X)]",[31,30372,30374,30430],{"className":30373,"ariaHidden":62},[61],[31,30375,30377,30380,30409,30412,30415,30418,30421,30424,30427],{"className":30376},[66],[31,30378],{"className":30379,"style":12035},[70],[31,30381,30383,30386],{"className":30382},[75],[31,30384,12145],{"className":30385},[75],[31,30387,30389],{"className":30388},[1039],[31,30390,30392],{"className":30391},[1043],[31,30393,30395],{"className":30394},[1048],[31,30396,30398],{"className":30397,"style":5135},[1052],[31,30399,30400,30403],{"style":3631},[31,30401],{"className":30402,"style":1061},[1060],[31,30404,30406],{"className":30405},[1065,1066,1067,1068],[31,30407,152],{"className":30408},[75,1068],[31,30410,8064],{"className":30411,"style":2666},[75,76],[31,30413,211],{"className":30414},[264],[31,30416,23798],{"className":30417,"style":19759},[75,76],[31,30419,220],{"className":30420},[289],[31,30422],{"className":30423,"style":293},[119],[31,30425,223],{"className":30426},[297],[31,30428],{"className":30429,"style":293},[119],[31,30431,30433,30436,30480,30483,30486,30489,30492],{"className":30432},[66],[31,30434],{"className":30435,"style":260},[70],[31,30437,30439,30446],{"className":30438},[75],[31,30440,30442],{"className":30441},[75],[31,30443,30445],{"className":30444,"style":20180},[75,2745],"Cov",[31,30447,30449],{"className":30448},[1039],[31,30450,30452,30472],{"className":30451},[1043,1044],[31,30453,30455,30469],{"className":30454},[1048],[31,30456,30458],{"className":30457,"style":1053},[1052],[31,30459,30460,30463],{"style":21790},[31,30461],{"className":30462,"style":1061},[1060],[31,30464,30466],{"className":30465},[1065,1066,1067,1068],[31,30467,23798],{"className":30468,"style":19759},[75,76,1068],[31,30470,1076],{"className":30471},[1075],[31,30473,30475],{"className":30474},[1048],[31,30476,30478],{"className":30477,"style":2050},[1052],[31,30479],{},[31,30481,15175],{"className":30482},[264],[31,30484,919],{"className":30485,"style":949},[75,76],[31,30487,211],{"className":30488},[264],[31,30490,926],{"className":30491,"style":30311},[75,76],[31,30493,30315],{"className":30494},[289],"。第二个等式说明 Fisher 度量就是 ",[31,30497,30499,30512],{"className":30498,"translate":35},[34],[31,30500,30502],{"className":30501},[39],[41,30503,30504],{"xmlns":43},[45,30505,30506,30510],{},[48,30507,30508],{},[51,30509,8064],{},[55,30511,29778],{"encoding":57},[31,30513,30515],{"className":30514,"ariaHidden":62},[61],[31,30516,30518,30521],{"className":30517},[66],[31,30519],{"className":30520,"style":4202},[70],[31,30522,8064],{"className":30523,"style":2666},[75,76]," 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。",[11,30526,30527,30528,30600,30601,30629,30630,30858,30859,30960,30961,30989,30990,31019],{},"令 ",[31,30529,30531,30558],{"className":30530,"translate":35},[34],[31,30532,30534],{"className":30533},[39],[41,30535,30536],{"xmlns":43},[45,30537,30538,30555],{},[48,30539,30540,30543,30545,30547,30549,30551,30553],{},[51,30541,30542],{},"η",[95,30544,223],{},[51,30546,12145],{"mathvariant":494},[51,30548,8064],{},[95,30550,211],{"stretchy":210},[51,30552,23798],{},[95,30554,220],{"stretchy":210},[55,30556,30557],{"encoding":57},"\\eta=\\nabla\\psi(\\theta)",[31,30559,30561,30579],{"className":30560,"ariaHidden":62},[61],[31,30562,30564,30567,30570,30573,30576],{"className":30563},[66],[31,30565],{"className":30566,"style":865},[70],[31,30568,30542],{"className":30569,"style":2666},[75,76],[31,30571],{"className":30572,"style":293},[119],[31,30574,223],{"className":30575},[297],[31,30577],{"className":30578,"style":293},[119],[31,30580,30582,30585,30588,30591,30594,30597],{"className":30581},[66],[31,30583],{"className":30584,"style":260},[70],[31,30586,12145],{"className":30587},[75],[31,30589,8064],{"className":30590,"style":2666},[75,76],[31,30592,211],{"className":30593},[264],[31,30595,23798],{"className":30596,"style":19759},[75,76],[31,30598,220],{"className":30599},[289]," 为期望参数，它与 ",[31,30602,30604,30617],{"className":30603,"translate":35},[34],[31,30605,30607],{"className":30606},[39],[41,30608,30609],{"xmlns":43},[45,30610,30611,30615],{},[48,30612,30613],{},[51,30614,23798],{},[55,30616,24747],{"encoding":57},[31,30618,30620],{"className":30619,"ariaHidden":62},[61],[31,30621,30623,30626],{"className":30622},[66],[31,30624],{"className":30625,"style":1630},[70],[31,30627,23798],{"className":30628,"style":19759},[75,76]," 之间是 Legendre 变换的关系：",[31,30631,30633,30692],{"className":30632,"translate":35},[34],[31,30634,30636],{"className":30635},[39],[41,30637,30638],{"xmlns":43},[45,30639,30640,30689],{},[48,30641,30642,30649,30651,30653,30655,30657,30667,30669,30675,30677,30679,30681,30683,30685,30687],{},[1180,30643,30644,30646],{},[51,30645,8064],{},[95,30647,30648],{},"∗",[95,30650,211],{"stretchy":210},[51,30652,30542],{},[95,30654,220],{"stretchy":210},[95,30656,223],{},[1004,30658,30659,30665],{},[48,30660,30661,30663],{},[51,30662,10792],{},[95,30664,1540],{},[51,30666,23798],{},[95,30668,211],{"stretchy":210},[1180,30670,30671,30673],{},[51,30672,23798],{},[51,30674,20800],{"mathvariant":494},[51,30676,30542],{},[95,30678,97],{},[51,30680,8064],{},[95,30682,211],{"stretchy":210},[51,30684,23798],{},[95,30686,220],{"stretchy":210},[95,30688,220],{"stretchy":210},[55,30690,30691],{"encoding":57},"\\psi^*(\\eta)=\\sup_\\theta(\\theta^\\top\\eta-\\psi(\\theta))",[31,30693,30695,30749,30840],{"className":30694,"ariaHidden":62},[61],[31,30696,30698,30701,30731,30734,30737,30740,30743,30746],{"className":30697},[66],[31,30699],{"className":30700,"style":260},[70],[31,30702,30704,30707],{"className":30703},[75],[31,30705,8064],{"className":30706,"style":2666},[75,76],[31,30708,30710],{"className":30709},[1039],[31,30711,30713],{"className":30712},[1043],[31,30714,30716],{"className":30715},[1048],[31,30717,30720],{"className":30718,"style":30719},[1052],"height:0.6887em;",[31,30721,30722,30725],{"style":3631},[31,30723],{"className":30724,"style":1061},[1060],[31,30726,30728],{"className":30727},[1065,1066,1067,1068],[31,30729,30648],{"className":30730},[124,1068],[31,30732,211],{"className":30733},[264],[31,30735,30542],{"className":30736,"style":2666},[75,76],[31,30738,220],{"className":30739},[289],[31,30741],{"className":30742,"style":293},[119],[31,30744,223],{"className":30745},[297],[31,30747],{"className":30748,"style":293},[119],[31,30750,30752,30755,30796,30799,30828,30831,30834,30837],{"className":30751},[66],[31,30753],{"className":30754,"style":20835},[70],[31,30756,30758,30761],{"className":30757},[1460],[31,30759,10792],{"className":30760},[1460],[31,30762,30764],{"className":30763},[1039],[31,30765,30767,30788],{"className":30766},[1043,1044],[31,30768,30770,30785],{"className":30769},[1048],[31,30771,30774],{"className":30772,"style":30773},[1052],"height:0.242em;",[31,30775,30776,30779],{"style":2762},[31,30777],{"className":30778,"style":1061},[1060],[31,30780,30782],{"className":30781},[1065,1066,1067,1068],[31,30783,23798],{"className":30784,"style":19759},[75,76,1068],[31,30786,1076],{"className":30787},[1075],[31,30789,30791],{"className":30790},[1048],[31,30792,30794],{"className":30793,"style":2784},[1052],[31,30795],{},[31,30797,211],{"className":30798},[264],[31,30800,30802,30805],{"className":30801},[75],[31,30803,23798],{"className":30804,"style":19759},[75,76],[31,30806,30808],{"className":30807},[1039],[31,30809,30811],{"className":30810},[1043],[31,30812,30814],{"className":30813},[1048],[31,30815,30817],{"className":30816,"style":20863},[1052],[31,30818,30819,30822],{"style":3631},[31,30820],{"className":30821,"style":1061},[1060],[31,30823,30825],{"className":30824},[1065,1066,1067,1068],[31,30826,20800],{"className":30827},[75,1068],[31,30829,30542],{"className":30830,"style":2666},[75,76],[31,30832],{"className":30833,"style":120},[119],[31,30835,97],{"className":30836},[124],[31,30838],{"className":30839,"style":120},[119],[31,30841,30843,30846,30849,30852,30855],{"className":30842},[66],[31,30844],{"className":30845,"style":260},[70],[31,30847,8064],{"className":30848,"style":2666},[75,76],[31,30850,211],{"className":30851},[264],[31,30853,23798],{"className":30854,"style":19759},[75,76],[31,30856,1495],{"className":30857},[289],"，且 ",[31,30860,30862,30892],{"className":30861,"translate":35},[34],[31,30863,30865],{"className":30864},[39],[41,30866,30867],{"xmlns":43},[45,30868,30869,30889],{},[48,30870,30871,30873,30879,30881,30883,30885,30887],{},[51,30872,12145],{"mathvariant":494},[1180,30874,30875,30877],{},[51,30876,8064],{},[95,30878,30648],{},[95,30880,211],{"stretchy":210},[51,30882,30542],{},[95,30884,220],{"stretchy":210},[95,30886,223],{},[51,30888,23798],{},[55,30890,30891],{"encoding":57},"\\nabla\\psi^*(\\eta)=\\theta",[31,30893,30895,30951],{"className":30894,"ariaHidden":62},[61],[31,30896,30898,30901,30904,30933,30936,30939,30942,30945,30948],{"className":30897},[66],[31,30899],{"className":30900,"style":260},[70],[31,30902,12145],{"className":30903},[75],[31,30905,30907,30910],{"className":30906},[75],[31,30908,8064],{"className":30909,"style":2666},[75,76],[31,30911,30913],{"className":30912},[1039],[31,30914,30916],{"className":30915},[1043],[31,30917,30919],{"className":30918},[1048],[31,30920,30922],{"className":30921,"style":30719},[1052],[31,30923,30924,30927],{"style":3631},[31,30925],{"className":30926,"style":1061},[1060],[31,30928,30930],{"className":30929},[1065,1066,1067,1068],[31,30931,30648],{"className":30932},[124,1068],[31,30934,211],{"className":30935},[264],[31,30937,30542],{"className":30938,"style":2666},[75,76],[31,30940,220],{"className":30941},[289],[31,30943],{"className":30944,"style":293},[119],[31,30946,223],{"className":30947},[297],[31,30949],{"className":30950,"style":293},[119],[31,30952,30954,30957],{"className":30953},[66],[31,30955],{"className":30956,"style":1630},[70],[31,30958,23798],{"className":30959,"style":19759},[75,76],"。所以同一个分布有两套坐标，自然参数 ",[31,30962,30964,30977],{"className":30963,"translate":35},[34],[31,30965,30967],{"className":30966},[39],[41,30968,30969],{"xmlns":43},[45,30970,30971,30975],{},[48,30972,30973],{},[51,30974,23798],{},[55,30976,24747],{"encoding":57},[31,30978,30980],{"className":30979,"ariaHidden":62},[61],[31,30981,30983,30986],{"className":30982},[66],[31,30984],{"className":30985,"style":1630},[70],[31,30987,23798],{"className":30988,"style":19759},[75,76]," 和期望参数 ",[31,30991,30993,31007],{"className":30992,"translate":35},[34],[31,30994,30996],{"className":30995},[39],[41,30997,30998],{"xmlns":43},[45,30999,31000,31004],{},[48,31001,31002],{},[51,31003,30542],{},[55,31005,31006],{"encoding":57},"\\eta",[31,31008,31010],{"className":31009,"ariaHidden":62},[61],[31,31011,31013,31016],{"className":31012},[66],[31,31014],{"className":31015,"style":865},[70],[31,31017,30542],{"className":31018,"style":2666},[75,76],"，互为对偶。",[11,31021,31022],{},"现在算两个成员之间的 KL：",[31,31024,31026],{"className":31025,"translate":35},[411],[31,31027,31029,31341],{"className":31028,"translate":35},[34],[31,31030,31032],{"className":31031},[39],[41,31033,31034],{"xmlns":43,"display":420},[45,31035,31036,31338],{},[5606,31037,31038,31120,31196,31266],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,31039,31040,31074],{},[5615,31041,31042],{},[2071,31043,31044],{"scriptlevel":1977,"displaystyle":62},[48,31045,31046,31052,31054,31060,31062,31072],{},[48,31047,31048,31050],{},[51,31049,4558],{"mathvariant":494},[51,31051,4561],{"mathvariant":494},[95,31053,211],{"stretchy":210},[1004,31055,31056,31058],{},[51,31057,11],{},[51,31059,23798],{},[51,31061,4572],{"mathvariant":494},[1004,31063,31064,31066],{},[51,31065,11],{},[1180,31067,31068,31070],{},[51,31069,23798],{},[95,31071,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31073,220],{"stretchy":210},[5615,31075,31076],{},[2071,31077,31078],{"scriptlevel":1977,"displaystyle":62},[48,31079,31080,31082,31084,31090,31092,31094,31096,31102,31104,31106,31108,31118],{},[48,31081],{},[95,31083,223],{},[1004,31085,31086,31088],{},[51,31087,22324],{"mathvariant":14789},[51,31089,23798],{},[95,31091,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,31093,19939],{},[95,31095,1540],{},[1004,31097,31098,31100],{},[51,31099,11],{},[51,31101,23798],{},[95,31103,97],{},[51,31105,19939],{},[95,31107,1540],{},[1004,31109,31110,31112],{},[51,31111,11],{},[1180,31113,31114,31116],{},[51,31115,23798],{},[95,31117,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31119,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[5612,31121,31122,31128],{},[5615,31123,31124],{},[2071,31125,31126],{"scriptlevel":1977,"displaystyle":62},[48,31127],{},[5615,31129,31130],{},[2071,31131,31132],{"scriptlevel":1977,"displaystyle":62},[48,31133,31134,31136,31138,31144,31146,31148,31150,31152,31158,31164,31166,31168,31170,31172,31174,31176,31178,31180,31182,31184,31186,31192,31194],{},[48,31135],{},[95,31137,223],{},[1004,31139,31140,31142],{},[51,31141,22324],{"mathvariant":14789},[51,31143,23798],{},[95,31145,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,31147,211],{"stretchy":210},[51,31149,23798],{},[95,31151,97],{},[1180,31153,31154,31156],{},[51,31155,23798],{},[95,31157,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[1180,31159,31160,31162],{},[95,31161,220],{"stretchy":210},[51,31163,20800],{"mathvariant":494},[51,31165,919],{},[95,31167,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,31169,97],{},[95,31171,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,31173,8064],{},[95,31175,211],{"stretchy":210},[51,31177,23798],{},[95,31179,220],{"stretchy":210},[95,31181,97],{},[51,31183,8064],{},[95,31185,211],{"stretchy":210},[1180,31187,31188,31190],{},[51,31189,23798],{},[95,31191,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31193,220],{"stretchy":210},[95,31195,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[5612,31197,31198,31204],{},[5615,31199,31200],{},[2071,31201,31202],{"scriptlevel":1977,"displaystyle":62},[48,31203],{},[5615,31205,31206],{},[2071,31207,31208],{"scriptlevel":1977,"displaystyle":62},[48,31209,31210,31212,31214,31216,31218,31220,31226,31232,31234,31236,31238,31240,31242,31244,31246,31248,31250,31252,31254,31256,31258,31264],{},[48,31211],{},[95,31213,223],{},[95,31215,211],{"stretchy":210},[51,31217,23798],{},[95,31219,97],{},[1180,31221,31222,31224],{},[51,31223,23798],{},[95,31225,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[1180,31227,31228,31230],{},[95,31229,220],{"stretchy":210},[51,31231,20800],{"mathvariant":494},[51,31233,12145],{"mathvariant":494},[51,31235,8064],{},[95,31237,211],{"stretchy":210},[51,31239,23798],{},[95,31241,220],{"stretchy":210},[95,31243,97],{},[51,31245,8064],{},[95,31247,211],{"stretchy":210},[51,31249,23798],{},[95,31251,220],{"stretchy":210},[95,31253,434],{},[51,31255,8064],{},[95,31257,211],{"stretchy":210},[1180,31259,31260,31262],{},[51,31261,23798],{},[95,31263,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31265,220],{"stretchy":210},[5612,31267,31268,31274],{},[5615,31269,31270],{},[2071,31271,31272],{"scriptlevel":1977,"displaystyle":62},[48,31273],{},[5615,31275,31276],{},[2071,31277,31278],{"scriptlevel":1977,"displaystyle":62},[48,31279,31280,31282,31284,31286,31288,31294,31296,31298,31300,31302,31304,31306,31308,31310,31312,31314,31316,31322,31324,31330,31332,31334,31336],{},[48,31281],{},[95,31283,223],{},[51,31285,8064],{},[95,31287,211],{"stretchy":210},[1180,31289,31290,31292],{},[51,31291,23798],{},[95,31293,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31295,220],{"stretchy":210},[95,31297,97],{},[51,31299,8064],{},[95,31301,211],{"stretchy":210},[51,31303,23798],{},[95,31305,220],{"stretchy":210},[95,31307,97],{},[51,31309,12145],{"mathvariant":494},[51,31311,8064],{},[95,31313,211],{"stretchy":210},[51,31315,23798],{},[1180,31317,31318,31320],{},[95,31319,220],{"stretchy":210},[51,31321,20800],{"mathvariant":494},[95,31323,211],{"stretchy":210},[1180,31325,31326,31328],{},[51,31327,23798],{},[95,31329,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31331,97],{},[51,31333,23798],{},[95,31335,220],{"stretchy":210},[51,31337,495],{"mathvariant":494},[55,31339,31340],{"encoding":57},"\\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}",[31,31342,31344],{"className":31343,"ariaHidden":62},[61],[31,31345,31347,31351],{"className":31346},[66],[31,31348],{"className":31349,"style":31350},[70],"height:5.8873em;vertical-align:-2.6937em;",[31,31352,31354],{"className":31353},[75],[31,31355,31357,31547],{"className":31356},[5606],[31,31358,31360],{"className":31359},[5867],[31,31361,31363,31538],{"className":31362},[1043,1044],[31,31364,31366,31535],{"className":31365},[1048],[31,31367,31370,31508,31517,31526],{"className":31368,"style":31369},[1052],"height:3.1937em;",[31,31371,31373,31376],{"style":31372},"top:-5.3437em;",[31,31374],{"className":31375,"style":1637},[1060],[31,31377,31379,31385,31388,31428,31431,31505],{"className":31378},[75],[31,31380,31382],{"className":31381},[75],[31,31383,4606],{"className":31384},[75,2745],[31,31386,211],{"className":31387},[264],[31,31389,31391,31394],{"className":31390},[75],[31,31392,11],{"className":31393},[75,76],[31,31395,31397],{"className":31396},[1039],[31,31398,31400,31420],{"className":31399},[1043,1044],[31,31401,31403,31417],{"className":31402},[1048],[31,31404,31406],{"className":31405,"style":1053},[1052],[31,31407,31408,31411],{"style":2807},[31,31409],{"className":31410,"style":1061},[1060],[31,31412,31414],{"className":31413},[1065,1066,1067,1068],[31,31415,23798],{"className":31416,"style":19759},[75,76,1068],[31,31418,1076],{"className":31419},[1075],[31,31421,31423],{"className":31422},[1048],[31,31424,31426],{"className":31425,"style":2050},[1052],[31,31427],{},[31,31429,4572],{"className":31430},[75],[31,31432,31434,31437],{"className":31433},[75],[31,31435,11],{"className":31436},[75,76],[31,31438,31440],{"className":31439},[1039],[31,31441,31443,31497],{"className":31442},[1043,1044],[31,31444,31446,31494],{"className":31445},[1048],[31,31447,31449],{"className":31448,"style":1053},[1052],[31,31450,31451,31454],{"style":2807},[31,31452],{"className":31453,"style":1061},[1060],[31,31455,31457],{"className":31456},[1065,1066,1067,1068],[31,31458,31460],{"className":31459},[75,1068],[31,31461,31463,31466],{"className":31462},[75,1068],[31,31464,23798],{"className":31465,"style":19759},[75,76,1068],[31,31467,31469],{"className":31468},[1039],[31,31470,31472],{"className":31471},[1043],[31,31473,31475],{"className":31474},[1048],[31,31476,31479],{"className":31477,"style":31478},[1052],"height:0.6828em;",[31,31480,31482,31485],{"style":31481},"top:-2.786em;margin-right:0.0714em;",[31,31483],{"className":31484,"style":11769},[1060],[31,31486,31488],{"className":31487},[1065,11773,1308,1068],[31,31489,31491],{"className":31490},[75,1068],[31,31492,13691],{"className":31493},[75,1068],[31,31495,1076],{"className":31496},[1075],[31,31498,31500],{"className":31499},[1048],[31,31501,31503],{"className":31502,"style":2050},[1052],[31,31504],{},[31,31506,220],{"className":31507},[289],[31,31509,31511,31514],{"style":31510},"top:-3.7846em;",[31,31512],{"className":31513,"style":1637},[1060],[31,31515],{"className":31516},[75],[31,31518,31520,31523],{"style":31519},"top:-2.2254em;",[31,31521],{"className":31522,"style":1637},[1060],[31,31524],{"className":31525},[75],[31,31527,31529,31532],{"style":31528},"top:-0.6663em;",[31,31530],{"className":31531,"style":1637},[1060],[31,31533],{"className":31534},[75],[31,31536,1076],{"className":31537},[1075],[31,31539,31541],{"className":31540},[1048],[31,31542,31545],{"className":31543,"style":31544},[1052],"height:2.6937em;",[31,31546],{},[31,31548,31550],{"className":31549},[6062],[31,31551,31553,32372],{"className":31552},[1043,1044],[31,31554,31556,32369],{"className":31555},[1048],[31,31557,31559,31768,32002,32184],{"className":31558,"style":31369},[1052],[31,31560,31561,31564],{"style":31372},[31,31562],{"className":31563,"style":1637},[1060],[31,31565,31567,31570,31573,31576,31579,31619,31625,31630,31633,31673,31676,31679,31682,31687,31690,31762],{"className":31566},[75],[31,31568],{"className":31569},[75],[31,31571],{"className":31572,"style":293},[119],[31,31574,223],{"className":31575},[297],[31,31577],{"className":31578,"style":293},[119],[31,31580,31582,31585],{"className":31581},[75],[31,31583,22324],{"className":31584},[75,14947],[31,31586,31588],{"className":31587},[1039],[31,31589,31591,31611],{"className":31590},[1043,1044],[31,31592,31594,31608],{"className":31593},[1048],[31,31595,31597],{"className":31596,"style":1053},[1052],[31,31598,31599,31602],{"style":2807},[31,31600],{"className":31601,"style":1061},[1060],[31,31603,31605],{"className":31604},[1065,1066,1067,1068],[31,31606,23798],{"className":31607,"style":19759},[75,76,1068],[31,31609,1076],{"className":31610},[1075],[31,31612,31614],{"className":31613},[1048],[31,31615,31617],{"className":31616,"style":2050},[1052],[31,31618],{},[31,31620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",[31,32384,32386,32399],{"className":32385,"translate":35},[34],[31,32387,32389],{"className":32388},[39],[41,32390,32391],{"xmlns":43},[45,32392,32393,32397],{},[48,32394,32395],{},[51,32396,8064],{},[55,32398,29778],{"encoding":57},[31,32400,32402],{"className":32401,"ariaHidden":62},[61],[31,32403,32405,32408],{"className":32404},[66],[31,32406],{"className":32407,"style":4202},[70],[31,32409,8064],{"className":32410,"style":2666},[75,76],[31,32412,32414,32427],{"className":32413,"translate":35},[34],[31,32415,32417],{"className":32416},[39],[41,32418,32419],{"xmlns":43},[45,32420,32421,32425],{},[48,32422,32423],{},[51,32424,23798],{},[55,32426,24747],{"encoding":57},[31,32428,32430],{"className":32429,"ariaHidden":62},[61],[31,32431,32433,32436],{"className":32432},[66],[31,32434],{"className":32435,"style":1630},[70],[31,32437,23798],{"className":32438,"style":19759},[75,76]," 处的切平面在 ",[31,32441,32443,32461],{"className":32442,"translate":35},[34],[31,32444,32446],{"className":32445},[39],[41,32447,32448],{"xmlns":43},[45,32449,32450,32458],{},[48,32451,32452],{},[1180,32453,32454,32456],{},[51,32455,23798],{},[95,32457,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[55,32459,32460],{"encoding":57},"\\theta'",[31,32462,32464],{"className":32463,"ariaHidden":62},[61],[31,32465,32467,32470],{"className":32466},[66],[31,32468],{"className":32469,"style":13732},[70],[31,32471,32473,32476],{"className":32472},[75],[31,32474,23798],{"className":32475,"style":19759},[75,76],[31,32477,32479],{"className":32478},[1039],[31,32480,32482],{"className":32481},[1043],[31,32483,32485],{"className":32484},[1048],[31,32486,32488],{"className":32487,"style":13732},[1052],[31,32489,32490,32493],{"style":3631},[31,32491],{"className":32492,"style":1061},[1060],[31,32494,32496],{"className":32495},[1065,1066,1067,1068],[31,32497,32499],{"className":32498},[75,1068],[31,32500,13691],{"className":32501},[75,1068]," 处高出的那部分。它有名字：Bregman 散度。",[11,32504,32505,32506,1138,32557,32660],{},"拿方差固定为 1 的高斯族验一下。此时 ",[31,32507,32509,32527],{"className":32508,"translate":35},[34],[31,32510,32512],{"className":32511},[39],[41,32513,32514],{"xmlns":43},[45,32515,32516,32524],{},[48,32517,32518,32520,32522],{},[51,32519,23798],{},[95,32521,223],{},[51,32523,852],{},[55,32525,32526],{"encoding":57},"\\theta=\\mu",[31,32528,32530,32548],{"className":32529,"ariaHidden":62},[61],[31,32531,32533,32536,32539,32542,32545],{"className":32532},[66],[31,32534],{"className":32535,"style":1630},[70],[31,32537,23798],{"className":32538,"style":19759},[75,76],[31,32540],{"className":32541,"style":293},[119],[31,32543,223],{"className":32544},[297],[31,32546],{"className":32547,"style":293},[119],[31,32549,32551,32554],{"className":32550},[66],[31,32552],{"className":32553,"style":865},[70],[31,32555,852],{"className":32556},[75,76],[31,32558,32560,32592],{"className":32559,"translate":35},[34],[31,32561,32563],{"className":32562},[39],[41,32564,32565],{"xmlns":43},[45,32566,32567,32589],{},[48,32568,32569,32571,32573,32575,32577,32579,32585,32587],{},[51,32570,8064],{},[95,32572,211],{"stretchy":210},[51,32574,23798],{},[95,32576,220],{"stretchy":210},[95,32578,223],{},[1180,32580,32581,32583],{},[51,32582,23798],{},[91,32584,152],{},[51,32586,3768],{"mathvariant":494},[91,32588,152],{},[55,32590,32591],{"encoding":57},"\\psi(\\theta)=\\theta^2\u002F2",[31,32593,32595,32622],{"className":32594,"ariaHidden":62},[61],[31,32596,32598,32601,32604,32607,32610,32613,32616,32619],{"className":32597},[66],[31,32599],{"className":32600,"style":260},[70],[31,32602,8064],{"className":32603,"style":2666},[75,76],[31,32605,211],{"className":32606},[264],[31,32608,23798],{"className":32609,"style":19759},[75,76],[31,32611,220],{"className":32612},[289],[31,32614],{"className":32615,"style":293},[119],[31,32617,223],{"className":32618},[297],[31,32620],{"className":32621,"style":293},[119],[31,32623,32625,32628,32657],{"className":32624},[66],[31,32626],{"className":32627,"style":12035},[70],[31,32629,32631,32634],{"className":32630},[75],[31,32632,23798],{"className":32633,"style":19759},[75,76],[31,32635,32637],{"className":32636},[1039],[31,32638,32640],{"className":32639},[1043],[31,32641,32643],{"className":32642},[1048],[31,32644,32646],{"className":32645,"style":5135},[1052],[31,32647,32648,32651],{"style":3631},[31,32649],{"className":32650,"style":1061},[1060],[31,32652,32654],{"className":32653},[1065,1066,1067,1068],[31,32655,152],{"className":32656},[75,1068],[31,32658,15357],{"className":32659},[75],"，手算",[31,32662,32664],{"className":32663,"translate":35},[411],[31,32665,32667,32755],{"className":32666,"translate":35},[34],[31,32668,32670],{"className":32669},[39],[41,32671,32672],{"xmlns":43,"display":420},[45,32673,32674,32752],{},[48,32675,32676,32682,32684,32686,32688,32694,32696,32698,32700,32702,32704,32706,32712,32714,32716,32718,32720,32722,32750],{},[48,32677,32678,32680],{},[51,32679,4558],{"mathvariant":494},[51,32681,4561],{"mathvariant":494},[95,32683,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,32685,4935],{"mathvariant":925},[95,32687,211],{"stretchy":210},[1004,32689,32690,32692],{},[51,32691,23798],{},[91,32693,205],{},[95,32695,1210],{"separator":62},[91,32697,205],{},[95,32699,220],{"stretchy":210},[95,32701,4572],{"fence":210,"stretchy":62,"minsize":1178,"maxsize":1178},[51,32703,4935],{"mathvariant":925},[95,32705,211],{"stretchy":210},[1004,32707,32708,32710],{},[51,32709,23798],{},[91,32711,152],{},[95,32713,1210],{"separator":62},[91,32715,205],{},[95,32717,220],{"stretchy":210},[95,32719,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,32721,223],{},[2074,32723,32724,32748],{},[48,32725,32726,32728,32734,32736,32742],{},[95,32727,211],{"stretchy":210},[1004,32729,32730,32732],{},[51,32731,23798],{},[91,32733,205],{},[95,32735,97],{},[1004,32737,32738,32740],{},[51,32739,23798],{},[91,32741,152],{},[1180,32743,32744,32746],{},[95,32745,220],{"stretchy":210},[91,32747,152],{},[91,32749,152],{},[95,32751,1210],{"separator":62},[55,32753,32754],{"encoding":57},"\\mathrm{KL}\\bigl(\\mathcal N(\\theta_1,1)\\big\\|\\mathcal N(\\theta_2,1)\\bigr)=\\frac{(\\theta_1-\\theta_2)^2}{2},",[31,32756,32758,32948],{"className":32757,"ariaHidden":62},[61],[31,32759,32761,32764,32770,32776,32779,32782,32822,32825,32828,32831,32834,32875,32878,32881,32921,32924,32927,32930,32933,32939,32942,32945],{"className":32760},[66],[31,32762],{"className":32763,"style":4988},[70],[31,32765,32767],{"className":32766},[75],[31,32768,4606],{"className":32769},[75,2745],[31,32771,32773],{"className":32772},[264],[31,32774,211],{"className":32775},[1307,1308],[31,32777,4935],{"className":32778,"style":5004},[75,968],[31,32780,211],{"className":32781},[264],[31,32783,32785,32788],{"className":32784},[75],[31,32786,23798],{"className":32787,"style":19759},[75,76],[31,32789,32791],{"className":32790},[1039],[31,32792,32794,32814],{"className":32793},[1043,1044],[31,32795,32797,32811],{"className":32796},[1048],[31,32798,32800],{"className":32799,"style":2028},[1052],[31,32801,32802,32805],{"style":26726},[31,32803],{"className":32804,"style":1061},[1060],[31,32806,32808],{"className":32807},[1065,1066,1067,1068],[31,32809,205],{"className":32810},[75,1068],[31,32812,1076],{"className":32813},[1075],[31,32815,32817],{"className":32816},[1048],[31,32818,32820],{"className":32819,"style":2050},[1052],[31,32821],{},[31,32823,1210],{"className":32824},[1391],[31,32826],{"className":32827,"style":1467},[119],[31,32829,205],{"className":32830},[75],[31,32832,220],{"className":32833},[289],[31,32835,32837],{"className":32836},[75],[31,32838,32840],{"className":32839},[1307,13319],[31,32841,32843,32867],{"className":32842},[1043,1044],[31,32844,32846,32864],{"className":32845},[1048],[31,32847,32849],{"className":32848,"style":13329},[1052],[31,32850,32851,32854],{"style":13332},[31,32852],{"className":32853,"style":13336},[1060],[31,32855,32857],{"style":32856},"width:0.556em;height:1.2em;",[13341,32858,32861],{"xmlns":13343,"width":32859,"height":1178,"viewBox":32860},"0.556em","0 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h43z",[31,32865,1076],{"className":32866},[1075],[31,32868,32870],{"className":32869},[1048],[31,32871,32873],{"className":32872,"style":13359},[1052],[31,32874],{},[31,32876,4935],{"className":32877,"style":5004},[75,968],[31,32879,211],{"className":32880},[264],[31,32882,32884,32887],{"className":32883},[75],[31,32885,23798],{"className":32886,"style":19759},[75,76],[31,32888,32890],{"className":32889},[1039],[31,32891,32893,32913],{"className":32892},[1043,1044],[31,32894,32896,32910],{"className":32895},[1048],[31,32897,32899],{"className":32898,"style":2028},[1052],[31,32900,32901,32904],{"style":26726},[31,32902],{"className":32903,"style":1061},[1060],[31,32905,32907],{"className":32906},[1065,1066,1067,1068],[31,32908,152],{"className":32909},[75,1068],[31,32911,1076],{"className":32912},[1075],[31,32914,32916],{"className":32915},[1048],[31,32917,32919],{"className":32918,"style":2050},[1052],[31,32920],{},[31,32922,1210],{"className":32923},[1391],[31,32925],{"className":32926,"style":1467},[119],[31,32928,205],{"className":32929},[75],[31,32931,220],{"className":32932},[289],[31,32934,32936],{"className":32935},[289],[31,32937,220],{"className":32938},[1307,1308],[31,32940],{"className":32941,"style":293},[119],[31,32943,223],{"className":32944},[297],[31,32946],{"className":32947,"style":293},[119],[31,32949,32951,32954,33134],{"className":32950},[66],[31,32952],{"className":32953,"style":5068},[70],[31,32955,32957,32960,33131],{"className":32956},[75],[31,32958],{"className":32959},[264,2139],[31,32961,32963],{"className":32962},[2074],[31,32964,32966,33123],{"className":32965},[1043,1044],[31,32967,32969,33120],{"className":32968},[1048],[31,32970,32972,32983,32991],{"className":32971,"style":5087},[1052],[31,32973,32974,32977],{"style":5090},[31,32975],{"className":32976,"style":1637},[1060],[31,32978,32980],{"className":32979},[75],[31,32981,152],{"className":32982},[75],[31,32984,32985,32988],{"style":2170},[31,32986],{"className":32987,"style":1637},[1060],[31,32989],{"className":32990,"style":2178},[2177],[31,32992,32993,32996],{"style":5110},[31,32994],{"className":32995,"style":1637},[1060],[31,32997,32999,33002,33042,33045,33048,33051,33091],{"className":32998},[75],[31,33000,211],{"className":33001},[264],[31,33003,33005,33008],{"className":33004},[75],[31,33006,23798],{"className":33007,"style":19759},[75,76],[31,33009,33011],{"className":33010},[1039],[31,33012,33014,33034],{"className":33013},[1043,1044],[31,33015,33017,33031],{"className":33016},[1048],[31,33018,33020],{"className":33019,"style":2028},[1052],[31,33021,33022,33025],{"style":26726},[31,33023],{"className":33024,"style":1061},[1060],[31,33026,33028],{"className":33027},[1065,1066,1067,1068],[31,33029,205],{"className":33030},[75,1068],[31,33032,1076],{"className":33033},[1075],[31,33035,33037],{"className":33036},[1048],[31,33038,33040],{"className":33039,"style":2050},[1052],[31,33041],{},[31,33043],{"className":33044,"style":120},[119],[31,33046,97],{"className":33047},[124],[31,33049],{"className":33050,"style":120},[119],[31,33052,33054,33057],{"className":33053},[75],[31,33055,23798],{"className":33056,"style":19759},[75,76],[31,33058,33060],{"className":33059},[1039],[31,33061,33063,33083],{"className":33062},[1043,1044],[31,33064,33066,33080],{"className":33065},[1048],[31,33067,33069],{"className":33068,"style":2028},[1052],[31,33070,33071,33074],{"style":26726},[31,33072],{"className":33073,"style":1061},[1060],[31,33075,33077],{"className":33076},[1065,1066,1067,1068],[31,33078,152],{"className":33079},[75,1068],[31,33081,1076],{"className":33082},[1075],[31,33084,33086],{"className":33085},[1048],[31,33087,33089],{"className":33088,"style":2050},[1052],[31,33090],{},[31,33092,33094,33097],{"className":33093},[289],[31,33095,220],{"className":33096},[289],[31,33098,33100],{"className":33099},[1039],[31,33101,33103],{"className":33102},[1043],[31,33104,33106],{"className":33105},[1048],[31,33107,33109],{"className":33108,"style":5135},[1052],[31,33110,33111,33114],{"style":3631},[31,33112],{"className":33113,"style":1061},[1060],[31,33115,33117],{"className":33116},[1065,1066,1067,1068],[31,33118,152],{"className":33119},[75,1068],[31,33121,1076],{"className":33122},[1075],[31,33124,33126],{"className":33125},[1048],[31,33127,33129],{"className":33128,"style":5156},[1052],[31,33130],{},[31,33132],{"className":33133},[289,2139],[31,33135,1210],{"className":33136},[1391],[11,33138,33139,33140,33168,33169,33905],{},"而 ",[31,33141,33143,33156],{"className":33142,"translate":35},[34],[31,33144,33146],{"className":33145},[39],[41,33147,33148],{"xmlns":43},[45,33149,33150,33154],{},[48,33151,33152],{},[51,33153,8064],{},[55,33155,29778],{"encoding":57},[31,33157,33159],{"className":33158,"ariaHidden":62},[61],[31,33160,33162,33165],{"className":33161},[66],[31,33163],{"className":33164,"style":4202},[70],[31,33166,8064],{"className":33167,"style":2666},[75,76]," 的 Bregman 散度是 ",[31,33170,33172,33276],{"className":33171,"translate":35},[34],[31,33173,33175],{"className":33174},[39],[41,33176,33177],{"xmlns":43},[45,33178,33179,33273],{},[48,33180,33181,33189,33197,33199,33207,33215,33217,33223,33225,33231,33233,33239,33241,33243,33251,33253,33259,33261,33267],{},[2071,33182,33183],{"scriptlevel":1977,"displaystyle":210},[2074,33184,33185,33187],{},[91,33186,205],{},[91,33188,152],{},[12397,33190,33191,33193,33195],{},[51,33192,23798],{},[91,33194,152],{},[91,33196,152],{},[95,33198,97],{},[2071,33200,33201],{"scriptlevel":1977,"displaystyle":210},[2074,33202,33203,33205],{},[91,33204,205],{},[91,33206,152],{},[12397,33208,33209,33211,33213],{},[51,33210,23798],{},[91,33212,205],{},[91,33214,152],{},[95,33216,97],{},[1004,33218,33219,33221],{},[51,33220,23798],{},[91,33222,205],{},[95,33224,211],{"stretchy":210},[1004,33226,33227,33229],{},[51,33228,23798],{},[91,33230,152],{},[95,33232,97],{},[1004,33234,33235,33237],{},[51,33236,23798],{},[91,33238,205],{},[95,33240,220],{"stretchy":210},[95,33242,223],{},[2071,33244,33245],{"scriptlevel":1977,"displaystyle":210},[2074,33246,33247,33249],{},[91,33248,205],{},[91,33250,152],{},[95,33252,211],{"stretchy":210},[1004,33254,33255,33257],{},[51,33256,23798],{},[91,33258,152],{},[95,33260,97],{},[1004,33262,33263,33265],{},[51,33264,23798],{},[91,33266,205],{},[1180,33268,33269,33271],{},[95,33270,220],{"stretchy":210},[91,33272,152],{},[55,33274,33275],{"encoding":57},"\\tfrac12\\theta_2^2-\\tfrac12\\theta_1^2-\\theta_1(\\theta_2-\\theta_1)=\\tfrac12(\\theta_2-\\theta_1)^2",[31,33277,33279,33414,33548,33646,33704,33830],{"className":33278,"ariaHidden":62},[61],[31,33280,33282,33285,33353,33405,33408,33411],{"className":33281},[66],[31,33283],{"className":33284,"style":2132},[70],[31,33286,33288,33291,33350],{"className":33287},[75],[31,33289],{"className":33290},[264,2139],[31,33292,33294],{"className":33293},[2074],[31,33295,33297,33342],{"className":33296},[1043,1044],[31,33298,33300,33339],{"className":33299},[1048],[31,33301,33303,33317,33325],{"className":33302,"style":2152},[1052],[31,33304,33305,33308],{"style":2155},[31,33306],{"className":33307,"style":1637},[1060],[31,33309,33311],{"className":33310},[1065,1066,1067,1068],[31,33312,33314],{"className":33313},[75,1068],[31,33315,152],{"className":33316},[75,1068],[31,33318,33319,33322],{"style":2170},[31,33320],{"className":33321,"style":1637},[1060],[31,33323],{"className":33324,"style":2178},[2177],[31,33326,33327,33330],{"style":2181},[31,33328],{"className":33329,"style":1637},[1060],[31,33331,33333],{"className":33332},[1065,1066,1067,1068],[31,33334,33336],{"className":33335},[75,1068],[31,33337,205],{"className":33338},[75,1068],[31,33340,1076],{"className":33341},[1075],[31,33343,33345],{"className":33344},[1048],[31,33346,33348],{"className":33347,"style":2203},[1052],[31,33349],{},[31,33351],{"className":33352},[289,2139],[31,33354,33356,33359],{"className":33355},[75],[31,33357,23798],{"className":33358,"style":19759},[75,76],[31,33360,33362],{"className":33361},[1039],[31,33363,33365,33397],{"className":33364},[1043,1044],[31,33366,33368,33394],{"className":33367},[1048],[31,33369,33371,33383],{"className":33370,"style":5135},[1052],[31,33372,33374,33377],{"style":33373},"top:-2.4519em;margin-left:-0.0278em;margin-right:0.05em;",[31,33375],{"className":33376,"style":1061},[1060],[31,33378,33380],{"className":33379},[1065,1066,1067,1068],[31,33381,152],{"className":33382},[75,1068],[31,33384,33385,33388],{"style":3631},[31,33386],{"className":33387,"style":1061},[1060],[31,33389,33391],{"className":33390},[1065,1066,1067,1068],[31,33392,152],{"className":33393},[75,1068],[31,33395,1076],{"className":33396},[1075],[31,33398,33400],{"className":33399},[1048],[31,33401,33403],{"className":33402,"style":17796},[1052],[31,33404],{},[31,33406],{"className":33407,"style":120},[119],[31,33409,97],{"className":33410},[124],[31,33412],{"className":33413,"style":120},[119],[31,33415,33417,33420,33488,33539,33542,33545],{"className":33416},[66],[31,33418],{"className":33419,"style":2132},[70],[31,33421,33423,33426,33485],{"className":33422},[75],[31,33424],{"className":33425},[264,2139],[31,33427,33429],{"className":33428},[2074],[31,33430,33432,33477],{"className":33431},[1043,1044],[31,33433,33435,33474],{"className":33434},[1048],[31,33436,33438,33452,33460],{"className":33437,"style":2152},[1052],[31,33439,33440,33443],{"style":2155},[31,33441],{"className":33442,"style":1637},[1060],[31,33444,33446],{"className":33445},[1065,1066,1067,1068],[31,33447,33449],{"className":33448},[75,1068],[31,33450,152],{"className":33451},[75,1068],[31,33453,33454,33457],{"style":2170},[31,33455],{"className":33456,"style":1637},[1060],[31,33458],{"className":33459,"style":2178},[2177],[31,33461,33462,33465],{"style":2181},[31,33463],{"className":33464,"style":1637},[1060],[31,33466,33468],{"className":33467},[1065,1066,1067,1068],[31,33469,33471],{"className":33470},[75,1068],[31,33472,20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,{"className":33772,"style":2203},[1052],[31,33774],{},[31,33776],{"className":33777},[289,2139],[31,33779,211],{"className":33780},[264],[31,33782,33784,33787],{"className":33783},[75],[31,33785,23798],{"className":33786,"style":19759},[75,76],[31,33788,33790],{"className":33789},[1039],[31,33791,33793,33813],{"className":33792},[1043,1044],[31,33794,33796,33810],{"className":33795},[1048],[31,33797,33799],{"className":33798,"style":2028},[1052],[31,33800,33801,33804],{"style":26726},[31,33802],{"className":33803,"style":1061},[1060],[31,33805,33807],{"className":33806},[1065,1066,1067,1068],[31,33808,152],{"className":33809},[75,1068],[31,33811,1076],{"className":33812},[1075],[31,33814,33816],{"className":33815},[1048],[31,33817,33819],{"className":33818,"style":2050},[1052],[31,33820],{},[31,33822],{"className":33823,"style":120},[119],[31,33825,97],{"className":33826},[124],[31,33828],{"className":33829,"style":120},[119],[31,33831,33833,33836,33876],{"className":33832},[66],[31,33834],{"className":33835,"style":12035},[70],[31,33837,33839,33842],{"className":33838},[75],[31,33840,23798],{"className":33841,"style":19759},[75,76],[31,33843,33845],{"className":33844},[1039],[31,33846,33848,33868],{"className":33847},[1043,1044],[31,33849,33851,33865],{"className":33850},[1048],[31,33852,33854],{"className":33853,"style":2028},[1052],[31,33855,33856,33859],{"style":26726},[31,33857],{"className":33858,"style":1061},[1060],[31,33860,33862],{"className":33861},[1065,1066,1067,1068],[31,33863,205],{"className":33864},[75,1068],[31,33866,1076],{"className":33867},[1075],[31,33869,33871],{"className":33870},[1048],[31,33872,33874],{"className":33873,"style":2050},[1052],[31,33875],{},[31,33877,33879,33882],{"className":33878},[289],[31,33880,220],{"className":33881},[289],[31,33883,33885],{"className":33884},[1039],[31,33886,33888],{"className":33887},[1043],[31,33889,33891],{"className":33890},[1048],[31,33892,33894],{"className":33893,"style":5135},[1052],[31,33895,33896,33899],{"style":3631},[31,33897],{"className":33898,"style":1061},[1060],[31,33900,33902],{"className":33901},[1065,1066,1067,1068],[31,33903,152],{"className":33904},[75,1068],"。同一个数。",[11,33907,33908],{},"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。",[18,33910,33912],{"id":33911},"bregman","Bregman",[11,33914,33915,33916,33990,33991,34065],{},"先看一个会让人不舒服的例子。",[31,33917,33919,33945],{"className":33918,"translate":35},[34],[31,33920,33922],{"className":33921},[39],[41,33923,33924],{"xmlns":43},[45,33925,33926,33942],{},[48,33927,33928,33930,33932,33934,33936,33938,33940],{},[51,33929,11],{},[95,33931,223],{},[95,33933,211],{"stretchy":210},[91,33935,205],{},[95,33937,1210],{"separator":62},[91,33939,1977],{},[95,33941,220],{"stretchy":210},[55,33943,33944],{"encoding":57},"p=(1,0)",[31,33946,33948,33966],{"className":33947,"ariaHidden":62},[61],[31,33949,33951,33954,33957,33960,33963],{"className":33950},[66],[31,33952],{"className":33953,"style":865},[70],[31,33955,11],{"className":33956},[75,76],[31,33958],{"className":33959,"style":293},[119],[31,33961,223],{"className":33962},[297],[31,33964],{"className":33965,"style":293},[119],[31,33967,33969,33972,33975,33978,33981,33984,33987],{"className":33968},[66],[31,33970],{"className":33971,"style":260},[70],[31,33973,211],{"className":33974},[264],[31,33976,205],{"className":33977},[75],[31,33979,1210],{"className":33980},[1391],[31,33982],{"className":33983,"style":1467},[119],[31,33985,1977],{"className":33986},[75],[31,33988,220],{"className":33989},[289]," 是“真相”，",[31,33992,33994,34020],{"className":33993,"translate":35},[34],[31,33995,33997],{"className":33996},[39],[41,33998,33999],{"xmlns":43},[45,34000,34001,34017],{},[48,34002,34003,34005,34007,34009,34011,34013,34015],{},[51,34004,27666],{},[95,34006,223],{},[95,34008,211],{"stretchy":210},[91,34010,23165],{},[95,34012,1210],{"separator":62},[91,34014,23165],{},[95,34016,220],{"stretchy":210},[55,34018,34019],{"encoding":57},"q=(0.5,0.5)",[31,34021,34023,34041],{"className":34022,"ariaHidden":62},[61],[31,34024,34026,34029,34032,34035,34038],{"className":34025},[66],[31,34027],{"className":34028,"style":865},[70],[31,34030,27666],{"className":34031,"style":2666},[75,76],[31,34033],{"className":34034,"style":293},[119],[31,34036,223],{"className":34037},[297],[31,34039],{"className":34040,"style":293},[119],[31,34042,34044,34047,34050,34053,34056,34059,34062],{"className":34043},[66],[31,34045],{"className":34046,"style":260},[70],[31,34048,211],{"className":34049},[264],[31,34051,23165],{"className":34052},[75],[31,34054,1210],{"className":34055},[1391],[31,34057],{"className":34058,"style":1467},[119],[31,34060,23165],{"className":34061},[75],[31,34063,220],{"className":34064},[289]," 是“模型的预测”。",[31,34067,34069],{"className":34068,"translate":35},[411],[31,34070,34072,34141],{"className":34071,"translate":35},[34],[31,34073,34075],{"className":34074},[39],[41,34076,34077],{"xmlns":43,"display":420},[45,34078,34079,34138],{},[48,34080,34081,34087,34089,34091,34093,34095,34097,34099,34101,34103,34105,34107,34110,34112,34114,34120,34122,34124,34126,34128,34130,34132,34134,34136],{},[48,34082,34083,34085],{},[51,34084,4558],{"mathvariant":494},[51,34086,4561],{"mathvariant":494},[95,34088,211],{"stretchy":210},[51,34090,11],{},[51,34092,4572],{"mathvariant":494},[51,34094,27666],{},[95,34096,220],{"stretchy":210},[95,34098,223],{},[51,34100,19939],{},[95,34102,1540],{},[91,34104,152],{},[95,34106,27223],{},[91,34108,34109],{},"0.693",[95,34111,1210],{"separator":62},[119,34113],{"width":20286},[48,34115,34116,34118],{},[51,34117,4558],{"mathvariant":494},[51,34119,4561],{"mathvariant":494},[95,34121,211],{"stretchy":210},[51,34123,27666],{},[51,34125,4572],{"mathvariant":494},[51,34127,11],{},[95,34129,220],{"stretchy":210},[95,34131,223],{},[95,34133,434],{},[51,34135,4587],{"mathvariant":494},[51,34137,495],{"mathvariant":494},[55,34139,34140],{"encoding":57},"\\mathrm{KL}(p\\|q)=\\log2\\approx0.693,\n\\qquad\n\\mathrm{KL}(q\\|p)=+\\infty .",[31,34142,34144,34180,34206,34254],{"className":34143,"ariaHidden":62},[61],[31,34145,34147,34150,34156,34159,34162,34165,34168,34171,34174,34177],{"className":34146},[66],[31,34148],{"className":34149,"style":260},[70],[31,34151,34153],{"className":34152},[75],[31,34154,4606],{"className":34155},[75,2745],[31,34157,211],{"className":34158},[264],[31,34160,11],{"className":34161},[75,76],[31,34163,4572],{"className":34164},[75],[31,34166,27666],{"className":34167,"style":2666},[75,76],[31,34169,220],{"className":34170},[289],[31,34172],{"className":34173,"style":293},[119],[31,34175,223],{"className":34176},[297],[31,34178],{"className":34179,"style":293},[119],[31,34181,34183,34186,34191,34194,34197,34200,34203],{"className":34182},[66],[31,34184],{"className":34185,"style":4202},[70],[31,34187,20177,34189],{"className":34188},[1460],[31,34190,14556],{"style":20180},[31,34192],{"className":34193,"style":1467},[119],[31,34195,152],{"className":34196},[75],[31,34198],{"className":34199,"style":293},[119],[31,34201,27223],{"className":34202},[297],[31,34204],{"className":34205,"style":293},[119],[31,34207,34209,34212,34215,34218,34221,34224,34230,34233,34236,34239,34242,34245,34248,34251],{"className":34208},[66],[31,34210],{"className":34211,"style":260},[70],[31,34213,34109],{"className":34214},[75],[31,34216,1210],{"className":34217},[1391],[31,34219],{"className":34220,"style":20525},[119],[31,34222],{"className":34223,"style":1467},[119],[31,34225,34227],{"className":34226},[75],[31,34228,4606],{"className":34229},[75,2745],[31,34231,211],{"className":34232},[264],[31,34234,27666],{"className":34235,"style":2666},[75,76],[31,34237,4572],{"className":34238},[75],[31,34240,11],{"className":34241},[75,76],[31,34243,220],{"className":34244},[289],[31,34246],{"className":34247,"style":293},[119],[31,34249,223],{"className":34250},[297],[31,34252],{"className":34253,"style":293},[119],[31,34255,34257,34260,34263],{"className":34256},[66],[31,34258],{"className":34259,"style":307},[70],[31,34261,434],{"className":34262},[75],[31,34264,34266],{"className":34265},[75],"∞.",[11,34268,34269,34270,34298,34299,34327,34328,34356,34357,34385,34386,34356,34414,34442,34443,34471],{},"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：",[31,34271,34273,34286],{"className":34272,"translate":35},[34],[31,34274,34276],{"className":34275},[39],[41,34277,34278],{"xmlns":43},[45,34279,34280,34284],{},[48,34281,34282],{},[51,34283,11],{},[55,34285,11],{"encoding":57},[31,34287,34289],{"className":34288,"ariaHidden":62},[61],[31,34290,34292,34295],{"className":34291},[66],[31,34293],{"className":34294,"style":865},[70],[31,34296,11],{"className":34297},[75,76]," 在第二位上放了 0，而 ",[31,34300,34302,34315],{"className":34301,"translate":35},[34],[31,34303,34305],{"className":34304},[39],[41,34306,34307],{"xmlns":43},[45,34308,34309,34313],{},[48,34310,34311],{},[51,34312,27666],{},[55,34314,27666],{"encoding":57},[31,34316,34318],{"className":34317,"ariaHidden":62},[61],[31,34319,34321,34324],{"className":34320},[66],[31,34322],{"className":34323,"style":865},[70],[31,34325,27666],{"className":34326,"style":2666},[75,76]," 在那里放了 0.5，用 ",[31,34329,34331,34344],{"className":34330,"translate":35},[34],[31,34332,34334],{"className":34333},[39],[41,34335,34336],{"xmlns":43},[45,34337,34338,34342],{},[48,34339,34340],{},[51,34341,11],{},[55,34343,11],{"encoding":57},[31,34345,34347],{"className":34346,"ariaHidden":62},[61],[31,34348,34350,34353],{"className":34349},[66],[31,34351],{"className":34352,"style":865},[70],[31,34354,11],{"className":34355},[75,76]," 去看 ",[31,34358,34360,34373],{"className":34359,"translate":35},[34],[31,34361,34363],{"className":34362},[39],[41,34364,34365],{"xmlns":43},[45,34366,34367,34371],{},[48,34368,34369],{},[51,34370,27666],{},[55,34372,27666],{"encoding":57},[31,34374,34376],{"className":34375,"ariaHidden":62},[61],[31,34377,34379,34382],{"className":34378},[66],[31,34380],{"className":34381,"style":865},[70],[31,34383,27666],{"className":34384,"style":2666},[75,76],"，只是多花了 0.69 nat；用 ",[31,34387,34389,34402],{"className":34388,"translate":35},[34],[31,34390,34392],{"className":34391},[39],[41,34393,34394],{"xmlns":43},[45,34395,34396,34400],{},[48,34397,34398],{},[51,34399,27666],{},[55,34401,27666],{"encoding":57},[31,34403,34405],{"className":34404,"ariaHidden":62},[61],[31,34406,34408,34411],{"className":34407},[66],[31,34409],{"className":34410,"style":865},[70],[31,34412,27666],{"className":34413,"style":2666},[75,76],[31,34415,34417,34430],{"className":34416,"translate":35},[34],[31,34418,34420],{"className":34419},[39],[41,34421,34422],{"xmlns":43},[45,34423,34424,34428],{},[48,34425,34426],{},[51,34427,11],{},[55,34429,11],{"encoding":57},[31,34431,34433],{"className":34432,"ariaHidden":62},[61],[31,34434,34436,34439],{"className":34435},[66],[31,34437],{"className":34438,"style":865},[70],[31,34440,11],{"className":34441},[75,76],"，等于说“我给了一个 ",[31,34444,34446,34459],{"className":34445,"translate":35},[34],[31,34447,34449],{"className":34448},[39],[41,34450,34451],{"xmlns":43},[45,34452,34453,34457],{},[48,34454,34455],{},[51,34456,11],{},[55,34458,11],{"encoding":57},[31,34460,34462],{"className":34461,"ariaHidden":62},[61],[31,34463,34465,34468],{"className":34464},[66],[31,34466],{"className":34467,"style":865},[70],[31,34469,11],{"className":34470},[75,76]," 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。",[11,34473,34474,34475,34536,34537,34565,34566,34594],{},"这正是交叉熵写成 ",[31,34476,34478,34506],{"className":34477,"translate":35},[34],[31,34479,34481],{"className":34480},[39],[41,34482,34483],{"xmlns":43},[45,34484,34485,34503],{},[48,34486,34487,34493,34495,34497,34499,34501],{},[48,34488,34489,34491],{},[51,34490,4558],{"mathvariant":494},[51,34492,4561],{"mathvariant":494},[95,34494,211],{"stretchy":210},[51,34496,3003],{},[51,34498,4572],{"mathvariant":494},[51,34500,27666],{},[95,34502,220],{"stretchy":210},[55,34504,34505],{"encoding":57},"\\mathrm{KL}(y\\|q)",[31,34507,34509],{"className":34508,"ariaHidden":62},[61],[31,34510,34512,34515,34521,34524,34527,34530,34533],{"className":34511},[66],[31,34513],{"className":34514,"style":260},[70],[31,34516,34518],{"className":34517},[75],[31,34519,4606],{"className":34520},[75,2745],[31,34522,211],{"className":34523},[264],[31,34525,3003],{"className":34526,"style":2666},[75,76],[31,34528,4572],{"className":34529},[75],[31,34531,27666],{"className":34532,"style":2666},[75,76],[31,34534,220],{"className":34535},[289]," 而不是反过来的原因：",[31,34538,34540,34553],{"className":34539,"translate":35},[34],[31,34541,34543],{"className":34542},[39],[41,34544,34545],{"xmlns":43},[45,34546,34547,34551],{},[48,34548,34549],{},[51,34550,3003],{},[55,34552,3003],{"encoding":57},[31,34554,34556],{"className":34555,"ariaHidden":62},[61],[31,34557,34559,34562],{"className":34558},[66],[31,34560],{"className":34561,"style":865},[70],[31,34563,3003],{"className":34564,"style":2666},[75,76]," 是真相，模型 ",[31,34567,34569,34582],{"className":34568,"translate":35},[34],[31,34570,34572],{"className":34571},[39],[41,34573,34574],{"xmlns":43},[45,34575,34576,34580],{},[48,34577,34578],{},[51,34579,27666],{},[55,34581,27666],{"encoding":57},[31,34583,34585],{"className":34584,"ariaHidden":62},[61],[31,34586,34588,34591],{"className":34587},[66],[31,34589],{"className":34590,"style":865},[70],[31,34592,27666],{"className":34593,"style":2666},[75,76]," 不该在真答案上给 0 概率。",[11,34596,34597,34598,34811,34812,34993,34994,35074,35075,35103,35104,2446,35133,35161],{},"再看一个尺度上的例子。Itakura–Saito 散度 ",[31,34599,34601,34663],{"className":34600,"translate":35},[34],[31,34602,34604],{"className":34603},[39],[41,34605,34606],{"xmlns":43},[45,34607,34608,34660],{},[48,34609,34610,34622,34624,34626,34628,34630,34632,34634,34636,34638,34640,34642,34644,34646,34648,34650,34652,34654,34656,34658],{},[1004,34611,34612,34615],{},[51,34613,34614],{},"D",[48,34616,34617,34620],{},[51,34618,34619],{"mathvariant":494},"I",[51,34621,21582],{"mathvariant":494},[95,34623,211],{"stretchy":210},[51,34625,53],{},[51,34627,4572],{"mathvariant":494},[51,34629,3003],{},[95,34631,220],{"stretchy":210},[95,34633,223],{},[51,34635,53],{},[51,34637,3768],{"mathvariant":494},[51,34639,3003],{},[95,34641,97],{},[51,34643,19939],{},[95,34645,1540],{},[95,34647,211],{"stretchy":210},[51,34649,53],{},[51,34651,3768],{"mathvariant":494},[51,34653,3003],{},[95,34655,220],{"stretchy":210},[95,34657,97],{},[91,34659,205],{},[55,34661,34662],{"encoding":57},"D_{\\mathrm{IS}}(x\\|y)=x\u002Fy-\\log(x\u002Fy)-1",[31,34664,34666,34743,34767,34802],{"className":34665,"ariaHidden":62},[61],[31,34667,34669,34672,34719,34722,34725,34728,34731,34734,34737,34740],{"className":34668},[66],[31,34670],{"className":34671,"style":260},[70],[31,34673,34675,34678],{"className":34674},[75],[31,34676,34614],{"className":34677,"style":19759},[75,76],[31,34679,34681],{"className":34680},[1039],[31,34682,34684,34711],{"className":34683},[1043,1044],[31,34685,34687,34708],{"className":34686},[1048],[31,34688,34690],{"className":34689,"style":27736},[1052],[31,34691,34692,34695],{"style":26726},[31,34693],{"className":34694,"style":1061},[1060],[31,34696,34698],{"className":34697},[1065,1066,1067,1068],[31,34699,34701],{"className":34700},[75,1068],[31,34702,34704],{"className":34703},[75,1068],[31,34705,34707],{"className":34706},[75,2745,1068],"IS",[31,34709,1076],{"className":34710},[1075],[31,34712,34714],{"className":34713},[1048],[31,34715,34717],{"className":34716,"style":2050},[1052],[31,34718],{},[31,34720,211],{"className":34721},[264],[31,34723,53],{"className":34724},[75,76],[31,34726,4572],{"className":34727},[75],[31,34729,3003],{"className":34730,"style":2666},[75,76],[31,34732,220],{"className":34733},[289],[31,34735],{"className":34736,"style":293},[119],[31,34738,223],{"className":34739},[297],[31,34741],{"className":34742,"style":293},[119],[31,34744,34746,34749,34752,34755,34758,34761,34764],{"className":34745},[66],[31,34747],{"className":34748,"style":260},[70],[31,34750,53],{"className":34751},[75,76],[31,34753,3768],{"className":34754},[75],[31,34756,3003],{"className":34757,"style":2666},[75,76],[31,34759],{"className":34760,"style":120},[119],[31,34762,97],{"className":34763},[124],[31,34765],{"className":34766,"style":120},[119],[31,34768,34770,34773,34778,34781,34784,34787,34790,34793,34796,34799],{"className":34769},[66],[31,34771],{"className":34772,"style":260},[70],[31,34774,20177,34776],{"className":34775},[1460],[31,34777,14556],{"style":20180},[31,34779,211],{"className":34780},[264],[31,34782,53],{"className":34783},[75,76],[31,34785,3768],{"className":34786},[75],[31,34788,3003],{"className":34789,"style":2666},[75,76],[31,34791,220],{"className":34792},[289],[31,34794],{"className":34795,"style":120},[119],[31,34797,97],{"className":34798},[124],[31,34800],{"className":34801,"style":120},[119],[31,34803,34805,34808],{"className":34804},[66],[31,34806],{"className":34807,"style":326},[70],[31,34809,205],{"className":34810},[75],"：把音量从 1 减半到 0.5，",[31,34813,34815,34858],{"className":34814,"translate":35},[34],[31,34816,34818],{"className":34817},[39],[41,34819,34820],{"xmlns":43},[45,34821,34822,34855],{},[48,34823,34824,34834,34836,34838,34840,34842,34844,34846,34848,34850,34852],{},[1004,34825,34826,34828],{},[51,34827,34614],{},[48,34829,34830,34832],{},[51,34831,34619],{"mathvariant":494},[51,34833,21582],{"mathvariant":494},[95,34835,223],{},[91,34837,152],{},[95,34839,97],{},[51,34841,19939],{},[95,34843,1540],{},[91,34845,152],{},[95,34847,97],{},[91,34849,205],{},[95,34851,223],{},[91,34853,34854],{},"0.307",[55,34856,34857],{"encoding":57},"D_{\\mathrm{IS}}=2-\\log2-1=0.307",[31,34859,34861,34922,34940,34966,34984],{"className":34860,"ariaHidden":62},[61],[31,34862,34864,34867,34913,34916,34919],{"className":34863},[66],[31,34865],{"className":34866,"style":5230},[70],[31,34868,34870,34873],{"className":34869},[75],[31,34871,34614],{"className":34872,"style":19759},[75,76],[31,34874,34876],{"className":34875},[1039],[31,34877,34879,34905],{"className":34878},[1043,1044],[31,34880,34882,34902],{"className":34881},[1048],[31,34883,34885],{"className":34884,"style":27736},[1052],[31,34886,34887,34890],{"style":26726},[31,34888],{"className":34889,"style":1061},[1060],[31,34891,34893],{"className":34892},[1065,1066,1067,1068],[31,34894,34896],{"className":34895},[75,1068],[31,34897,34899],{"className":34898},[75,1068],[31,34900,34707],{"className":34901},[75,2745,1068],[31,34903,1076],{"className":34904},[1075],[31,34906,34908],{"className":34907},[1048],[31,34909,34911],{"className":34910,"style":2050},[1052],[31,34912],{},[31,34914],{"className":34915,"style":293},[119],[31,34917,223],{"className":34918},[297],[31,34920],{"className":34921,"style":293},[119],[31,34923,34925,34928,34931,34934,34937],{"className":34924},[66],[31,34926],{"className":34927,"style":112},[70],[31,34929,152],{"className":34930},[75],[31,34932],{"className":34933,"style":120},[119],[31,34935,97],{"className":34936},[124],[31,34938],{"className":34939,"style":120},[119],[31,34941,34943,34946,34951,34954,34957,34960,34963],{"className":34942},[66],[31,34944],{"className":34945,"style":4202},[70],[31,34947,20177,34949],{"className":34948},[1460],[31,34950,14556],{"style":20180},[31,34952],{"className":34953,"style":1467},[119],[31,34955,152],{"className":34956},[75],[31,34958],{"className":34959,"style":120},[119],[31,34961,97],{"className":34962},[124],[31,34964],{"className":34965,"style":120},[119],[31,34967,34969,34972,34975,34978,34981],{"className":34968},[66],[31,34970],{"className":34971,"style":326},[70],[31,34973,205],{"className":34974},[75],[31,34976],{"className":34977,"style":293},[119],[31,34979,223],{"className":34980},[297],[31,34982],{"className":34983,"style":293},[119],[31,34985,34987,34990],{"className":34986},[66],[31,34988],{"className":34989,"style":326},[70],[31,34991,34854],{"className":34992},[75],"；从 2 减半到 1，",[31,34995,34997,35019],{"className":34996,"translate":35},[34],[31,34998,35000],{"className":34999},[39],[41,35001,35002],{"xmlns":43},[45,35003,35004,35016],{},[48,35005,35006],{},[1004,35007,35008,35010],{},[51,35009,34614],{},[48,35011,35012,35014],{},[51,35013,34619],{"mathvariant":494},[51,35015,21582],{"mathvariant":494},[55,35017,35018],{"encoding":57},"D_{\\mathrm{IS}}",[31,35020,35022],{"className":35021,"ariaHidden":62},[61],[31,35023,35025,35028],{"className":35024},[66],[31,35026],{"className":35027,"style":5230},[70],[31,35029,35031,35034],{"className":35030},[75],[31,35032,34614],{"className":35033,"style":19759},[75,76],[31,35035,35037],{"className":35036},[1039],[31,35038,35040,35066],{"className":35039},[1043,1044],[31,35041,35043,35063],{"className":35042},[1048],[31,35044,35046],{"className":35045,"style":27736},[1052],[31,35047,35048,35051],{"style":26726},[31,35049],{"className":35050,"style":1061},[1060],[31,35052,35054],{"className":35053},[1065,1066,1067,1068],[31,35055,35057],{"className":35056},[75,1068],[31,35058,35060],{"className":35059},[75,1068],[31,35061,34707],{"className":35062},[75,2745,1068],[31,35064,1076],{"className":35065},[1075],[31,35067,35069],{"className":35068},[1048],[31,35070,35072],{"className":35071,"style":2050},[1052],[31,35073],{}," 还是 ",[31,35076,35078,35091],{"className":35077,"translate":35},[34],[31,35079,35081],{"className":35080},[39],[41,35082,35083],{"xmlns":43},[45,35084,35085,35089],{},[48,35086,35087],{},[91,35088,34854],{},[55,35090,34854],{"encoding":57},[31,35092,35094],{"className":35093,"ariaHidden":62},[61],[31,35095,35097,35100],{"className":35096},[66],[31,35098],{"className":35099,"style":326},[70],[31,35101,34854],{"className":35102},[75],"。同样的比例变化，距离一样。而欧氏距离给的是 ",[31,35105,35107,35121],{"className":35106,"translate":35},[34],[31,35108,35110],{"className":35109},[39],[41,35111,35112],{"xmlns":43},[45,35113,35114,35119],{},[48,35115,35116],{},[91,35117,35118],{},"0.25",[55,35120,35118],{"encoding":57},[31,35122,35124],{"className":35123,"ariaHidden":62},[61],[31,35125,35127,35130],{"className":35126},[66],[31,35128],{"className":35129,"style":326},[70],[31,35131,35118],{"className":35132},[75],[31,35134,35136,35149],{"className":35135,"translate":35},[34],[31,35137,35139],{"className":35138},[39],[41,35140,35141],{"xmlns":43},[45,35142,35143,35147],{},[48,35144,35145],{},[91,35146,205],{},[55,35148,205],{"encoding":57},[31,35150,35152],{"className":35151,"ariaHidden":62},[61],[31,35153,35155,35158],{"className":35154},[66],[31,35156],{"className":35157,"style":326},[70],[31,35159,205],{"className":35160},[75],"，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。",[11,35163,35164],{},"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。",[11,35166,35167,35168,35197,35198,1138],{},"一般地，给定凸函数 ",[31,35169,35171,35184],{"className":35170,"translate":35},[34],[31,35172,35174],{"className":35173},[39],[41,35175,35176],{"xmlns":43},[45,35177,35178,35182],{},[48,35179,35180],{},[51,35181,1662],{},[55,35183,1662],{"encoding":57},[31,35185,35187],{"className":35186,"ariaHidden":62},[61],[31,35188,35190,35193],{"className":35189},[66],[31,35191],{"className":35192,"style":4202},[70],[31,35194,1662],{"className":35195,"style":35196},[75,76],"margin-right:0.1076em;"," 和可微点 ",[31,35199,35201,35214],{"className":35200,"translate":35},[34],[31,35202,35204],{"className":35203},[39],[41,35205,35206],{"xmlns":43},[45,35207,35208,35212],{},[48,35209,35210],{},[51,35211,3003],{},[55,35213,3003],{"encoding":57},[31,35215,35217],{"className":35216,"ariaHidden":62},[61],[31,35218,35220,35223],{"className":35219},[66],[31,35221],{"className":35222,"style":865},[70],[31,35224,3003],{"className":35225,"style":2666},[75,76],[31,35227,35229],{"className":35228,"translate":35},[411],[31,35230,35232,35308],{"className":35231,"translate":35},[34],[31,35233,35235],{"className":35234},[39],[41,35236,35237],{"xmlns":43,"display":420},[45,35238,35239,35305],{},[48,35240,35241,35247,35249,35251,35253,35255,35257,35259,35261,35263,35265,35267,35269,35271,35273,35275,35277,35279,35281,35283,35285,35287,35293,35295,35297,35299,35301,35303],{},[1004,35242,35243,35245],{},[51,35244,34614],{},[51,35246,1662],{},[95,35248,211],{"stretchy":210},[51,35250,53],{},[51,35252,4572],{"mathvariant":494},[51,35254,3003],{},[95,35256,220],{"stretchy":210},[95,35258,223],{},[51,35260,1662],{},[95,35262,211],{"stretchy":210},[51,35264,53],{},[95,35266,220],{"stretchy":210},[95,35268,97],{},[51,35270,1662],{},[95,35272,211],{"stretchy":210},[51,35274,3003],{},[95,35276,220],{"stretchy":210},[95,35278,97],{},[51,35280,12145],{"mathvariant":494},[51,35282,1662],{},[95,35284,211],{"stretchy":210},[51,35286,3003],{},[1180,35288,35289,35291],{},[95,35290,220],{"stretchy":210},[51,35292,20800],{"mathvariant":494},[95,35294,211],{"stretchy":210},[51,35296,53],{},[95,35298,97],{},[51,35300,3003],{},[95,35302,220],{"stretchy":210},[51,35304,495],{"mathvariant":494},[55,35306,35307],{"encoding":57},"D_f(x\\|y)=f(x)-f(y)-\\nabla f(y)^\\top(x-y).",[31,35309,35311,35381,35408,35435,35498],{"className":35310,"ariaHidden":62},[61],[31,35312,35314,35317,35357,35360,35363,35366,35369,35372,35375,35378],{"className":35313},[66],[31,35315],{"className":35316,"style":1223},[70],[31,35318,35320,35323],{"className":35319},[75],[31,35321,34614],{"className":35322,"style":19759},[75,76],[31,35324,35326],{"className":35325},[1039],[31,35327,35329,35349],{"className":35328},[1043,1044],[31,35330,35332,35346],{"className":35331},[1048],[31,35333,35335],{"className":35334,"style":1053},[1052],[31,35336,35337,35340],{"style":26726},[31,35338],{"className":35339,"style":1061},[1060],[31,35341,35343],{"className":35342},[1065,1066,1067,1068],[31,35344,1662],{"className":35345,"style":35196},[75,76,1068],[31,35347,1076],{"className":35348},[1075],[31,35350,35352],{"className":35351},[1048],[31,35353,35355],{"className":35354,"style":1083},[1052],[31,35356],{},[31,35358,211],{"className":35359},[264],[31,35361,53],{"className":35362},[75,76],[31,35364,4572],{"className":35365},[75],[31,35367,3003],{"className":35368,"style":2666},[75,76],[31,35370,220],{"className":35371},[289],[31,35373],{"className":35374,"style":293},[119],[31,35376,223],{"className":35377},[297],[31,35379],{"className":35380,"style":293},[119],[31,35382,35384,35387,35390,35393,35396,35399,35402,35405],{"className":35383},[66],[31,35385],{"className":35386,"style":260},[70],[31,35388,1662],{"className":35389,"style":35196},[75,76],[31,35391,211],{"className":35392},[264],[31,35394,53],{"className":35395},[75,76],[31,35397,220],{"className":35398},[289],[31,35400],{"className":35401,"style":120},[119],[31,35403,97],{"className":35404},[124],[31,35406],{"className":35407,"style":120},[119],[31,35409,35411,35414,35417,35420,35423,35426,35429,35432],{"className":35410},[66],[31,35412],{"className":35413,"style":260},[70],[31,35415,1662],{"className":35416,"style":35196},[75,76],[31,35418,211],{"className":35419},[264],[31,35421,3003],{"className":35422,"style":2666},[75,76],[31,35424,220],{"className":35425},[289],[31,35427],{"className":35428,"style":120},[119],[31,35430,97],{"className":35431},[124],[31,35433],{"className":35434,"style":120},[119],[31,35436,35438,35442,35445,35448,35451,35454,35483,35486,35489,35492,35495],{"className":35437},[66],[31,35439],{"className":35440,"style":35441},[70],"height:1.1491em;vertical-align:-0.25em;",[31,35443,12145],{"className":35444},[75],[31,35446,1662],{"className":35447,"style":35196},[75,76],[31,35449,211],{"className":35450},[264],[31,35452,3003],{"className":35453,"style":2666},[75,76],[31,35455,35457,35460],{"className":35456},[289],[31,35458,220],{"className":35459},[289],[31,35461,35463],{"className":35462},[1039],[31,35464,35466],{"className":35465},[1043],[31,35467,35469],{"className":35468},[1048],[31,35470,35472],{"className":35471,"style":29642},[1052],[31,35473,35474,35477],{"style":1330},[31,35475],{"className":35476,"style":1061},[1060],[31,35478,35480],{"className":35479},[1065,1066,1067,1068],[31,35481,20800],{"className":35482},[75,1068],[31,35484,211],{"className":35485},[264],[31,35487,53],{"className":35488},[75,76],[31,35490],{"className":35491,"style":120},[119],[31,35493,97],{"className":35494},[124],[31,35496],{"className":35497,"style":120},[119],[31,35499,35501,35504,35507,35510],{"className":35500},[66],[31,35502],{"className":35503,"style":260},[70],[31,35505,3003],{"className":35506,"style":2666},[75,76],[31,35508,220],{"className":35509},[289],[31,35511,495],{"className":35512},[75],[11,35514,35515,35516,35544,35545,35573,35574,35602,35603,35631,35632,35683],{},"几何上，这是 ",[31,35517,35519,35532],{"className":35518,"translate":35},[34],[31,35520,35522],{"className":35521},[39],[41,35523,35524],{"xmlns":43},[45,35525,35526,35530],{},[48,35527,35528],{},[51,35529,1662],{},[55,35531,1662],{"encoding":57},[31,35533,35535],{"className":35534,"ariaHidden":62},[61],[31,35536,35538,35541],{"className":35537},[66],[31,35539],{"className":35540,"style":4202},[70],[31,35542,1662],{"className":35543,"style":35196},[75,76]," 的图像在 ",[31,35546,35548,35561],{"className":35547,"translate":35},[34],[31,35549,35551],{"className":35550},[39],[41,35552,35553],{"xmlns":43},[45,35554,35555,35559],{},[48,35556,35557],{},[51,35558,53],{},[55,35560,53],{"encoding":57},[31,35562,35564],{"className":35563,"ariaHidden":62},[61],[31,35565,35567,35570],{"className":35566},[66],[31,35568],{"className":35569,"style":71},[70],[31,35571,53],{"className":35572},[75,76]," 处比 ",[31,35575,35577,35590],{"className":35576,"translate":35},[34],[31,35578,35580],{"className":35579},[39],[41,35581,35582],{"xmlns":43},[45,35583,35584,35588],{},[48,35585,35586],{},[51,35587,3003],{},[55,35589,3003],{"encoding":57},[31,35591,35593],{"className":35592,"ariaHidden":62},[61],[31,35594,35596,35599],{"className":35595},[66],[31,35597],{"className":35598,"style":865},[70],[31,35600,3003],{"className":35601,"style":2666},[75,76]," 处的切平面高出多少。",[31,35604,35606,35619],{"className":35605,"translate":35},[34],[31,35607,35609],{"className":35608},[39],[41,35610,35611],{"xmlns":43},[45,35612,35613,35617],{},[48,35614,35615],{},[51,35616,1662],{},[55,35618,1662],{"encoding":57},[31,35620,35622],{"className":35621,"ariaHidden":62},[61],[31,35623,35625,35628],{"className":35624},[66],[31,35626],{"className":35627,"style":4202},[70],[31,35629,1662],{"className":35630,"style":35196},[75,76]," 凸所以非负，严格凸所以只在 ",[31,35633,35635,35653],{"className":35634,"translate":35},[34],[31,35636,35638],{"className":35637},[39],[41,35639,35640],{"xmlns":43},[45,35641,35642,35650],{},[48,35643,35644,35646,35648],{},[51,35645,53],{},[95,35647,223],{},[51,35649,3003],{},[55,35651,35652],{"encoding":57},"x=y",[31,35654,35656,35674],{"className":35655,"ariaHidden":62},[61],[31,35657,35659,35662,35665,35668,35671],{"className":35658},[66],[31,35660],{"className":35661,"style":71},[70],[31,35663,53],{"className":35664},[75,76],[31,35666],{"className":35667,"style":293},[119],[31,35669,223],{"className":35670},[297],[31,35672],{"className":35673,"style":293},[119],[31,35675,35677,35680],{"className":35676},[66],[31,35678],{"className":35679,"style":865},[70],[31,35681,3003],{"className":35682,"style":2666},[75,76]," 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。",[11,35685,35686,35687,35869,35870,36037,36038,36255,36256,36037,36317,36493,36494,36677],{},"几个我常用的：",[31,35688,35690,35730],{"className":35689,"translate":35},[34],[31,35691,35693],{"className":35692},[39],[41,35694,35695],{"xmlns":43},[45,35696,35697,35727],{},[48,35698,35699,35701,35703,35705,35707,35709,35717,35719,35721],{},[51,35700,1662],{},[95,35702,211],{"stretchy":210},[51,35704,53],{},[95,35706,220],{"stretchy":210},[95,35708,223],{},[2071,35710,35711],{"scriptlevel":1977,"displaystyle":210},[2074,35712,35713,35715],{},[91,35714,205],{},[91,35716,152],{},[51,35718,4572],{"mathvariant":494},[51,35720,53],{},[1180,35722,35723,35725],{},[51,35724,4572],{"mathvariant":494},[91,35726,152],{},[55,35728,35729],{"encoding":57},"f(x)=\\tfrac12\\|x\\|^2",[31,35731,35733,35760],{"className":35732,"ariaHidden":62},[61],[31,35734,35736,35739,35742,35745,35748,35751,35754,35757],{"className":35735},[66],[31,35737],{"className":35738,"style":260},[70],[31,35740,1662],{"className":35741,"style":35196},[75,76],[31,35743,211],{"className":35744},[264],[31,35746,53],{"className":35747},[75,76],[31,35749,220],{"className":35750},[289],[31,35752],{"className":35753,"style":293},[119],[31,35755,223],{"className":35756},[297],[31,35758],{"className":35759,"style":293},[119],[31,35761,35763,35766,35834,35837,35840],{"className":35762},[66],[31,35764],{"className":35765,"style":2132},[70],[31,35767,35769,35772,35831],{"className":35768},[75],[31,35770],{"className":35771},[264,2139],[31,35773,35775],{"className":35774},[2074],[31,35776,35778,35823],{"className":35777},[1043,1044],[31,35779,35781,35820],{"className":35780},[1048],[31,35782,35784,35798,35806],{"className":35783,"style":2152},[1052],[31,35785,35786,35789],{"style":2155},[31,35787],{"className":35788,"style":1637},[1060],[31,35790,35792],{"className":35791},[1065,1066,1067,1068],[31,35793,35795],{"className":35794},[75,1068],[31,35796,152],{"className":35797},[75,1068],[31,35799,35800,35803],{"style":2170},[31,35801],{"className":35802,"style":1637},[1060],[31,35804],{"className":35805,"style":2178},[2177],[31,35807,35808,35811],{"style":2181},[31,35809],{"className":35810,"style":1637},[1060],[31,35812,35814],{"className":35813},[1065,1066,1067,1068],[31,35815,35817],{"className":35816},[75,1068],[31,35818,205],{"className":35819},[75,1068],[31,35821,1076],{"className":35822},[1075],[31,35824,35826],{"className":35825},[1048],[31,35827,35829],{"className":35828,"style":2203},[1052],[31,35830],{},[31,35832],{"className":35833},[289,2139],[31,35835,4572],{"className":35836},[75],[31,35838,53],{"className":35839},[75,76],[31,35841,35843,35846],{"className":35842},[75],[31,35844,4572],{"className":35845},[75],[31,35847,35849],{"className":35848},[1039],[31,35850,35852],{"className":35851},[1043],[31,35853,35855],{"className":35854},[1048],[31,35856,35858],{"className":35857,"style":5135},[1052],[31,35859,35860,35863],{"style":3631},[31,35861],{"className":35862,"style":1061},[1060],[31,35864,35866],{"className":35865},[1065,1066,1067,1068],[31,35867,152],{"className":35868},[75,1068]," 给 ",[31,35871,35873,35907],{"className":35872,"translate":35},[34],[31,35874,35876],{"className":35875},[39],[41,35877,35878],{"xmlns":43},[45,35879,35880,35904],{},[48,35881,35882,35890,35892,35894,35896,35898],{},[2071,35883,35884],{"scriptlevel":1977,"displaystyle":210},[2074,35885,35886,35888],{},[91,35887,205],{},[91,35889,152],{},[51,35891,4572],{"mathvariant":494},[51,35893,53],{},[95,35895,97],{},[51,35897,3003],{},[1180,35899,35900,35902],{},[51,35901,4572],{"mathvariant":494},[91,35903,152],{},[55,35905,35906],{"encoding":57},"\\tfrac12\\|x-y\\|^2",[31,35908,35910,35999],{"className":35909,"ariaHidden":62},[61],[31,35911,35913,35916,35984,35987,35990,35993,35996],{"className":35912},[66],[31,35914],{"className":35915,"style":2132},[70],[31,35917,35919,35922,35981],{"className":35918},[75],[31,35920],{"className":35921},[264,2139],[31,35923,35925],{"className":35924},[2074],[31,35926,35928,35973],{"className":35927},[1043,1044],[31,35929,35931,35970],{"className":35930},[1048],[31,35932,35934,35948,35956],{"className":35933,"style":2152},[1052],[31,35935,35936,35939],{"style":2155},[31,35937],{"className":35938,"style":1637},[1060],[31,35940,35942],{"className":35941},[1065,1066,1067,1068],[31,35943,35945],{"className":35944},[75,1068],[31,35946,152],{"className":35947},[75,1068],[31,35949,35950,35953],{"style":2170},[31,35951],{"className":35952,"style":1637},[1060],[31,35954],{"className":35955,"style":2178},[2177],[31,35957,35958,35961],{"style":2181},[31,35959],{"className":35960,"style":1637},[1060],[31,35962,35964],{"className":35963},[1065,1066,1067,1068],[31,35965,35967],{"className":35966},[75,1068],[31,35968,205],{"className":35969},[75,1068],[31,35971,1076],{"className":35972},[1075],[31,35974,35976],{"className":35975},[1048],[31,35977,35979],{"className":35978,"style":2203},[1052],[31,35980],{},[31,35982],{"className":35983},[289,2139],[31,35985,4572],{"className":35986},[75],[31,35988,53],{"className":35989},[75,76],[31,35991],{"className":35992,"style":120},[119],[31,35994,97],{"className":35995},[124],[31,35997],{"className":35998,"style":120},[119],[31,36000,36002,36005,36008],{"className":36001},[66],[31,36003],{"className":36004,"style":12035},[70],[31,36006,3003],{"className":36007,"style":2666},[75,76],[31,36009,36011,36014],{"className":36010},[75],[31,36012,4572],{"className":36013},[75],[31,36015,36017],{"className":36016},[1039],[31,36018,36020],{"className":36019},[1043],[31,36021,36023],{"className":36022},[1048],[31,36024,36026],{"className":36025,"style":5135},[1052],[31,36027,36028,36031],{"style":3631},[31,36029],{"className":36030,"style":1061},[1060],[31,36032,36034],{"className":36033},[1065,1066,1067,1068],[31,36035,152],{"className":36036},[75,1068],"；",[31,36039,36041,36085],{"className":36040,"translate":35},[34],[31,36042,36044],{"className":36043},[39],[41,36045,36046],{"xmlns":43},[45,36047,36048,36082],{},[48,36049,36050,36052,36054,36056,36058,36060,36066,36072,36074,36076],{},[51,36051,1662],{},[95,36053,211],{"stretchy":210},[51,36055,11],{},[95,36057,220],{"stretchy":210},[95,36059,223],{},[1004,36061,36062,36064],{},[95,36063,5625],{},[51,36065,5391],{},[1004,36067,36068,36070],{},[51,36069,11],{},[51,36071,5391],{},[51,36073,19939],{},[95,36075,1540],{},[1004,36077,36078,36080],{},[51,36079,11],{},[51,36081,5391],{},[55,36083,36084],{"encoding":57},"f(p)=\\sum_ip_i\\log p_i",[31,36086,36088,36115],{"className":36087,"ariaHidden":62},[61],[31,36089,36091,36094,36097,36100,36103,36106,36109,36112],{"className":36090},[66],[31,36092],{"className":36093,"style":260},[70],[31,36095,1662],{"className":36096,"style":35196},[75,76],[31,36098,211],{"className":36099},[264],[31,36101,11],{"className":36102},[75,76],[31,36104,220],{"className":36105},[289],[31,36107],{"className":36108,"style":293},[119],[31,36110,223],{"className":36111},[297],[31,36113],{"className":36114,"style":293},[119],[31,36116,36118,36121,36161,36164,36204,36207,36212,36215],{"className":36117},[66],[31,36119],{"className":36120,"style":21359},[70],[31,36122,36124,36127],{"className":36123},[1460],[31,36125,5625],{"className":36126,"style":21372},[1460,1461,1462],[31,36128,36130],{"className":36129},[1039],[31,36131,36133,36153],{"className":36132},[1043,1044],[31,36134,36136,36150],{"className":36135},[1048],[31,36137,36139],{"className":36138,"style":21385},[1052],[31,36140,36141,36144],{"style":21388},[31,36142],{"className":36143,"style":1061},[1060],[31,36145,36147],{"className":36146},[1065,1066,1067,1068],[31,36148,5391],{"className":36149},[75,76,1068],[31,36151,1076],{"className":36152},[1075],[31,36154,36156],{"className":36155},[1048],[31,36157,36159],{"className":36158,"style":21407},[1052],[31,36160],{},[31,36162],{"className":36163,"style":1467},[119],[31,36165,36167,36170],{"className":36166},[75],[31,36168,11],{"className":36169},[75,76],[31,36171,36173],{"className":36172},[1039],[31,36174,36176,36196],{"className":36175},[1043,1044],[31,36177,36179,36193],{"className":36178},[1048],[31,36180,36182],{"className":36181,"style":5445},[1052],[31,36183,36184,36187],{"style":2807},[31,36185],{"className":36186,"style":1061},[1060],[31,36188,36190],{"className":36189},[1065,1066,1067,1068],[31,36191,5391],{"className":36192},[75,76,1068],[31,36194,1076],{"className":36195},[1075],[31,36197,36199],{"className":36198},[1048],[31,36200,36202],{"className":36201,"style":2050},[1052],[31,36203],{},[31,36205],{"className":36206,"style":1467},[119],[31,36208,20177,36210],{"className":36209},[1460],[31,36211,14556],{"style":20180},[31,36213],{"className":36214,"style":1467},[119],[31,36216,36218,36221],{"className":36217},[75],[31,36219,11],{"className":36220},[75,76],[31,36222,36224],{"className":36223},[1039],[31,36225,36227,36247],{"className":36226},[1043,1044],[31,36228,36230,36244],{"className":36229},[1048],[31,36231,36233],{"className":36232,"style":5445},[1052],[31,36234,36235,36238],{"style":2807},[31,36236],{"className":36237,"style":1061},[1060],[31,36239,36241],{"className":36240},[1065,1066,1067,1068],[31,36242,5391],{"className":36243},[75,76,1068],[31,36245,1076],{"className":36246},[1075],[31,36248,36250],{"className":36249},[1048],[31,36251,36253],{"className":36252,"style":2050},[1052],[31,36254],{},"（负熵）给 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给 Itakura–Saito；",[31,36495,36497,36537],{"className":36496,"translate":35},[34],[31,36498,36500],{"className":36499},[39],[41,36501,36502],{"xmlns":43},[45,36503,36504,36534],{},[48,36505,36506,36508,36510,36512,36514,36516,36524,36530,36532],{},[51,36507,1662],{},[95,36509,211],{"stretchy":210},[51,36511,53],{},[95,36513,220],{"stretchy":210},[95,36515,223],{},[2071,36517,36518],{"scriptlevel":1977,"displaystyle":210},[2074,36519,36520,36522],{},[91,36521,205],{},[91,36523,152],{},[1180,36525,36526,36528],{},[51,36527,53],{},[51,36529,20800],{"mathvariant":494},[51,36531,18446],{},[51,36533,53],{},[55,36535,36536],{"encoding":57},"f(x)=\\tfrac12x^\\top 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给 Mahalanobis 距离。",[11,36679,36680],{},"它在机器学习里出现的频率比多数人以为的高。交叉熵",[31,36682,36684],{"className":36683,"translate":35},[411],[31,36685,36687,36804],{"className":36686,"translate":35},[34],[31,36688,36690],{"className":36689},[39],[41,36691,36692],{"xmlns":43,"display":420},[45,36693,36694,36801],{},[48,36695,36696,36702,36704,36706,36708,36710,36712,36714,36716,36722,36728,36730,36732,36738,36740,36746,36748,36750,36752,36754,36756,36758,36761,36763,36765,36767,36769,36779,36781,36783,36785,36787,36789,36791,36793,36795,36797,36799],{},[48,36697,36698,36700],{},[51,36699,8101],{"mathvariant":494},[51,36701,22324],{"mathvariant":494},[95,36703,211],{"stretchy":210},[51,36705,3003],{},[95,36707,1210],{"separator":62},[51,36709,27666],{},[95,36711,220],{"stretchy":210},[95,36713,223],{},[95,36715,97],{},[1531,36717,36718,36720],{},[95,36719,5625],{},[51,36721,5391],{},[1004,36723,36724,36726],{},[51,36725,3003],{},[51,36727,5391],{},[51,36729,19939],{},[95,36731,1540],{},[1004,36733,36734,36736],{},[51,36735,27666],{},[51,36737,5391],{},[95,36739,223],{},[48,36741,36742,36744],{},[51,36743,4558],{"mathvariant":494},[51,36745,4561],{"mathvariant":494},[95,36747,211],{"stretchy":210},[51,36749,3003],{},[51,36751,4572],{"mathvariant":494},[51,36753,27666],{},[95,36755,220],{"stretchy":210},[95,36757,434],{},[51,36759,36760],{},"H",[95,36762,211],{"stretchy":210},[51,36764,3003],{},[95,36766,220],{"stretchy":210},[95,36768,223],{},[1004,36770,36771,36773],{},[51,36772,34614],{},[48,36774,36775,36777],{},[95,36776,97],{},[51,36778,36760],{},[95,36780,211],{"stretchy":210},[51,36782,3003],{},[51,36784,4572],{"mathvariant":494},[51,36786,27666],{},[95,36788,220],{"stretchy":210},[95,36790,434],{},[51,36792,36760],{},[95,36794,211],{"stretchy":210},[51,36796,3003],{},[95,36798,220],{"stretchy":210},[95,36800,1210],{"separator":62},[55,36802,36803],{"encoding":57},"\\mathrm{CE}(y,q)=-\\sum_iy_i\\log q_i=\\mathrm{KL}(y\\|q)+H(y)=D_{-H}(y\\|q)+H(y),",[31,36805,36807,36847,37006,37042,37070,37146],{"className":36806,"ariaHidden":62},[61],[31,36808,36810,36813,36820,36823,36826,36829,36832,36835,36838,36841,36844],{"className":36809},[66],[31,36811],{"className":36812,"style":260},[70],[31,36814,36816],{"className":36815},[75],[31,36817,36819],{"className":36818},[75,2745],"CE",[31,36821,211],{"className":36822},[264],[31,36824,3003],{"className":36825,"style":2666},[75,76],[31,36827,1210],{"className":36828},[1391],[31,36830],{"className":36831,"style":1467},[119],[31,36833,27666],{"className":36834,"style":2666},[75,76],[31,36836,220],{"className":36837},[289],[31,36839],{"className":36840,"style":293},[119],[31,36842,223],{"className":36843},[297],[31,36845],{"className":36846,"style":293},[119],[31,36848,36850,36853,36856,36859,36903,36906,36946,36949,36954,36957,36997,37000,37003],{"className":36849},[66],[31,36851],{"className":36852,"style":10251},[70],[31,36854,97],{"className":36855},[75],[31,36857],{"className":36858,"style":1467},[119],[31,36860,36862],{"className":36861},[1460,1620],[31,36863,36865,36895],{"className":36864},[1043,1044],[31,36866,36868,36892],{"className":36867},[1048],[31,36869,36871,36882],{"className":36870,"style":5900},[1052],[31,36872,36873,36876],{"style":5903},[31,36874],{"className":36875,"style":5907},[1060],[31,36877,36879],{"className":36878},[1065,1066,1067,1068],[31,36880,5391],{"className":36881},[75,76,1068],[31,36883,36884,36887],{"style":5919},[31,36885],{"className":36886,"style":5907},[1060],[31,36888,36889],{},[31,36890,5625],{"className":36891},[1460,1461,1687],[31,36893,1076],{"className":36894},[1075],[31,36896,36898],{"className":36897},[1048],[31,36899,36901],{"className":36900,"style":6342},[1052],[31,36902],{},[31,36904],{"className":36905,"style":1467},[119],[31,36907,36909,36912],{"className":36908},[75],[31,36910,3003],{"className":36911,"style":2666},[75,76],[31,36913,36915],{"className":36914},[1039],[31,36916,36918,36938],{"className":36917},[1043,1044],[31,36919,36921,36935],{"className":36920},[1048],[31,36922,36924],{"className":36923,"style":5445},[1052],[31,36925,36926,36929],{"style":5504},[31,36927],{"className":36928,"style":1061},[1060],[31,36930,36932],{"className":36931},[1065,1066,1067,1068],[31,36933,5391],{"className":36934},[75,76,1068],[31,36936,1076],{"className":36937},[1075],[31,36939,36941],{"className":36940},[1048],[31,36942,36944],{"className":36943,"style":2050},[1052],[31,36945],{},[31,36947],{"className":36948,"style":1467},[119],[31,36950,20177,36952],{"className":36951},[1460],[31,36953,14556],{"style":20180},[31,36955],{"className":36956,"style":1467},[119],[31,36958,36960,36963],{"className":36959},[75],[31,36961,27666],{"className":36962,"style":2666},[75,76],[31,36964,36966],{"className":36965},[1039],[31,36967,36969,36989],{"className":36968},[1043,1044],[31,36970,36972,36986],{"className":36971},[1048],[31,36973,36975],{"className":36974,"style":5445},[1052],[31,36976,36977,36980],{"style":5504},[31,36978],{"className":36979,"style":1061},[1060],[31,36981,36983],{"className":36982},[1065,1066,1067,1068],[31,36984,5391],{"className":36985},[75,76,1068],[31,36987,1076],{"className":36988},[1075],[31,36990,36992],{"className":36991},[1048],[31,36993,36995],{"className":36994,"style":2050},[1052],[31,36996],{},[31,36998],{"className":36999,"style":293},[119],[31,37001,223],{"className":37002},[297],[31,37004],{"className":37005,"style":293},[119],[31,37007,37009,37012,37018,37021,37024,37027,37030,37033,37036,37039],{"className":37008},[66],[31,37010],{"className":37011,"style":260},[70],[31,37013,37015],{"className":37014},[75],[31,37016,4606],{"className":37017},[75,2745],[31,37019,211],{"className":37020},[264],[31,37022,3003],{"className":37023,"style":2666},[75,76],[31,37025,4572],{"className":37026},[75],[31,37028,27666],{"className":37029,"style":2666},[75,76],[31,37031,220],{"className":37032},[289],[31,37034],{"className":37035,"style":120},[119],[31,37037,434],{"className":37038},[124],[31,37040],{"className":37041,"style":120},[119],[31,37043,37045,37048,37052,37055,37058,37061,37064,37067],{"className":37044},[66],[31,37046],{"className":37047,"style":260},[70],[31,37049,36760],{"className":37050,"style":37051},[75,76],"margin-right:0.0813em;",[31,37053,211],{"className":37054},[264],[31,37056,3003],{"className":37057,"style":2666},[75,76],[31,37059,220],{"className":37060},[289],[31,37062],{"className":37063,"style":293},[119],[31,37065,223],{"className":37066},[297],[31,37068],{"className":37069,"style":293},[119],[31,37071,37073,37076,37122,37125,37128,37131,37134,37137,37140,37143],{"className":37072},[66],[31,37074],{"className":37075,"style":260},[70],[31,37077,37079,37082],{"className":37078},[75],[31,37080,34614],{"className":37081,"style":19759},[75,76],[31,37083,37085],{"className":37084},[1039],[31,37086,37088,37114],{"className":37087},[1043,1044],[31,37089,37091,37111],{"className":37090},[1048],[31,37092,37094],{"className":37093,"style":27736},[1052],[31,37095,37096,37099],{"style":26726},[31,37097],{"className":37098,"style":1061},[1060],[31,37100,37102],{"className":37101},[1065,1066,1067,1068],[31,37103,37105,37108],{"className":37104},[75,1068],[31,37106,97],{"className":37107},[75,1068],[31,37109,36760],{"className":37110,"style":37051},[75,76,1068],[31,37112,1076],{"className":37113},[1075],[31,37115,37117],{"className":37116},[1048],[31,37118,37120],{"className":37119,"style":25082},[1052],[31,37121],{},[31,37123,211],{"className":37124},[264],[31,37126,3003],{"className":37127,"style":2666},[75,76],[31,37129,4572],{"className":37130},[75],[31,37132,27666],{"className":37133,"style":2666},[75,76],[31,37135,220],{"className":37136},[289],[31,37138],{"className":37139,"style":120},[119],[31,37141,434],{"className":37142},[124],[31,37144],{"className":37145,"style":120},[119],[31,37147,37149,37152,37155,37158,37161,37164],{"className":37148},[66],[31,37150],{"className":37151,"style":260},[70],[31,37153,36760],{"className":37154,"style":37051},[75,76],[31,37156,211],{"className":37157},[264],[31,37159,3003],{"className":37160,"style":2666},[75,76],[31,37162,220],{"className":37163},[289],[31,37165,1210],{"className":37166},[1391],[11,37168,37169,37170,37214,37215,2372],{},"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 ",[31,37171,37173,37193],{"className":37172,"translate":35},[34],[31,37174,37176],{"className":37175},[39],[41,37177,37178],{"xmlns":43},[45,37179,37180,37190],{},[48,37181,37182,37184,37186,37188],{},[51,37183,36760],{},[95,37185,211],{"stretchy":210},[51,37187,3003],{},[95,37189,220],{"stretchy":210},[55,37191,37192],{"encoding":57},"H(y)",[31,37194,37196],{"className":37195,"ariaHidden":62},[61],[31,37197,37199,37202,37205,37208,37211],{"className":37198},[66],[31,37200],{"className":37201,"style":260},[70],[31,37203,36760],{"className":37204,"style":37051},[75,76],[31,37206,211],{"className":37207},[264],[31,37209,3003],{"className":37210,"style":2666},[75,76],[31,37212,220],{"className":37213},[289]," 是常数，梯度完全来自 ",[31,37216,37218,37231],{"className":37217,"translate":35},[34],[31,37219,37221],{"className":37220},[39],[41,37222,37223],{"xmlns":43},[45,37224,37225,37229],{},[48,37226,37227],{},[51,37228,34614],{},[55,37230,34614],{"encoding":57},[31,37232,37234],{"className":37233,"ariaHidden":62},[61],[31,37235,37237,37240],{"className":37236},[66],[31,37238],{"className":37239,"style":945},[70],[31,37241,34614],{"className":37242,"style":19759},[75,76],[11,37244,37245],{},"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，",[31,37247,37249],{"className":37248,"translate":35},[411],[31,37250,37252,37329],{"className":37251,"translate":35},[34],[31,37253,37255],{"className":37254},[39],[41,37256,37257],{"xmlns":43,"display":420},[45,37258,37259,37326],{},[48,37260,37261,37264,37266,37276,37278,37284,37290,37292,37298,37300,37302,37304,37306,37312,37318,37324],{},[51,37262,37263],{},"arg",[95,37265,1540],{},[1531,37267,37268,37274],{},[48,37269,37270,37272],{},[51,37271,19878],{},[95,37273,1540],{},[51,37275,1412],{},[1430,37277,1545],{},[1531,37279,37280,37282],{},[95,37281,5625],{},[51,37283,5391],{},[1004,37285,37286,37288],{},[51,37287,34614],{},[51,37289,1662],{},[95,37291,211],{"stretchy":210},[1004,37293,37294,37296],{},[51,37295,53],{},[51,37297,5391],{},[51,37299,4572],{"mathvariant":494},[51,37301,1412],{},[95,37303,220],{"stretchy":210},[95,37305,223],{},[2074,37307,37308,37310],{},[91,37309,205],{},[51,37311,14559],{},[1531,37313,37314,37316],{},[95,37315,5625],{},[51,37317,5391],{},[1004,37319,37320,37322],{},[51,37321,53],{},[51,37323,5391],{},[51,37325,495],{"mathvariant":494},[55,37327,37328],{"encoding":57},"\\arg\\min_c\\ \\sum_iD_f(x_i\\|c)=\\frac1n\\sum_ix_i .",[31,37330,37332,37547],{"className":37331,"ariaHidden":62},[61],[31,37333,37335,37338,37344,37347,37393,37396,37399,37443,37446,37486,37489,37529,37532,37535,37538,37541,37544],{"className":37334},[66],[31,37336],{"className":37337,"style":10251},[70],[31,37339,37341,37342],{"className":37340},[1460],"ar",[31,37343,14556],{"style":20180},[31,37345],{"className":37346,"style":1467},[119],[31,37348,37350],{"className":37349},[1460,1620],[31,37351,37353,37384],{"className":37352},[1043,1044],[31,37354,37356,37381],{"className":37355},[1048],[31,37357,37359,37371],{"className":37358,"style":19979},[1052],[31,37360,37362,37365],{"style":37361},"top:-2.4em;margin-left:0em;",[31,37363],{"className":37364,"style":1637},[1060],[31,37366,37368],{"className":37367},[1065,1066,1067,1068],[31,37369,1412],{"className":37370},[75,76,1068],[31,37372,37373,37376],{"style":1649},[31,37374],{"className":37375,"style":1637},[1060],[31,37377,37378],{},[31,37379,19878],{"className":37380},[1460],[31,37382,1076],{"className":37383},[1075],[31,37385,37387],{"className":37386},[1048],[31,37388,37391],{"className":37389,"style":37390},[1052],"height:0.7em;",[31,37392],{},[31,37394,1545],{"className":37395},[119],[31,37397],{"className":37398,"style":1467},[119],[31,37400,37402],{"className":37401},[1460,1620],[31,37403,37405,37435],{"className":37404},[1043,1044],[31,37406,37408,37432],{"className":37407},[1048],[31,37409,37411,37422],{"className":37410,"style":5900},[1052],[31,37412,37413,37416],{"style":5903},[31,37414],{"className":37415,"style":5907},[1060],[31,37417,37419],{"className":37418},[1065,1066,1067,1068],[31,37420,5391],{"className":37421},[75,76,1068],[31,37423,37424,37427],{"style":5919},[31,37425],{"className":37426,"style":5907},[1060],[31,37428,37429],{},[31,37430,5625],{"className":37431},[1460,1461,1687],[31,37433,1076],{"className":37434},[1075],[31,37436,37438],{"className":37437},[1048],[31,37439,37441],{"className":37440,"style":6342},[1052],[31,37442],{},[31,37444],{"className":37445,"style":1467},[119],[31,37447,37449,37452],{"className":37448},[75],[31,37450,34614],{"className":37451,"style":19759},[75,76],[31,37453,37455],{"className":37454},[1039],[31,37456,37458,37478],{"className":37457},[1043,1044],[31,37459,37461,37475],{"className":37460},[1048],[31,37462,37464],{"className":37463,"style":1053},[1052],[31,37465,37466,37469],{"style":26726},[31,37467],{"className":37468,"style":1061},[1060],[31,37470,37472],{"className":37471},[1065,1066,1067,1068],[31,37473,1662],{"className":37474,"style":35196},[75,76,1068],[31,37476,1076],{"className":37477},[1075],[31,37479,37481],{"className":37480},[1048],[31,37482,37484],{"className":37483,"style":1083},[1052],[31,37485],{},[31,37487,211],{"className":37488},[264],[31,37490,37492,37495],{"className":37491},[75],[31,37493,53],{"className":37494},[75,76],[31,37496,37498],{"className":37497},[1039],[31,37499,37501,37521],{"className":37500},[1043,1044],[31,37502,37504,37518],{"className":37503},[1048],[31,37505,37507],{"className":37506,"style":5445},[1052],[31,37508,37509,37512],{"style":2807},[31,37510],{"className":37511,"style":1061},[1060],[31,37513,37515],{"className":37514},[1065,1066,1067,1068],[31,37516,5391],{"className":37517},[75,76,1068],[31,37519,1076],{"className":37520},[1075],[31,37522,37524],{"className":37523},[1048],[31,37525,37527],{"className":37526,"style":2050},[1052],[31,37528],{},[31,37530,4572],{"className":37531},[75],[31,37533,1412],{"className":37534},[75,76],[31,37536,220],{"className":37537},[289],[31,37539],{"className":37540,"style":293},[119],[31,37542,223],{"className":37543},[297],[31,37545],{"className":37546,"style":293},[119],[31,37548,37550,37554,37616,37619,37663,37666,37706],{"className":37549},[66],[31,37551],{"className":37552,"style":37553},[70],"height:2.5991em;vertical-align:-1.2777em;",[31,37555,37557,37560,37613],{"className":37556},[75],[31,37558],{"className":37559},[264,2139],[31,37561,37563],{"className":37562},[2074],[31,37564,37566,37605],{"className":37565},[1043,1044],[31,37567,37569,37602],{"className":37568},[1048],[31,37570,37572,37583,37591],{"className":37571,"style":26376},[1052],[31,37573,37574,37577],{"style":5090},[31,37575],{"className":37576,"style":1637},[1060],[31,37578,37580],{"className":37579},[75],[31,37581,14559],{"className":37582},[75,76],[31,37584,37585,37588],{"style":2170},[31,37586],{"className":37587,"style":1637},[1060],[31,37589],{"className":37590,"style":2178},[2177],[31,37592,37593,37596],{"style":5110},[31,37594],{"className":37595,"style":1637},[1060],[31,37597,37599],{"className":37598},[75],[31,37600,205],{"className":37601},[75],[31,37603,1076],{"className":37604},[1075],[31,37606,37608],{"className":37607},[1048],[31,37609,37611],{"className":37610,"style":5156},[1052],[31,37612],{},[31,37614],{"className":37615},[289,2139],[31,37617],{"className":37618,"style":1467},[119],[31,37620,37622],{"className":37621},[1460,1620],[31,37623,37625,37655],{"className":37624},[1043,1044],[31,37626,37628,37652],{"className":37627},[1048],[31,37629,37631,37642],{"className":37630,"style":5900},[1052],[31,37632,37633,37636],{"style":5903},[31,37634],{"className":37635,"style":5907},[1060],[31,37637,37639],{"className":37638},[1065,1066,1067,1068],[31,37640,5391],{"className":37641},[75,76,1068],[31,37643,37644,37647],{"style":5919},[31,37645],{"className":37646,"style":5907},[1060],[31,37648,37649],{},[31,37650,5625],{"className":37651},[1460,1461,1687],[31,37653,1076],{"className":37654},[1075],[31,37656,37658],{"className":37657},[1048],[31,37659,37661],{"className":37660,"style":6342},[1052],[31,37662],{},[31,37664],{"className":37665,"style":1467},[119],[31,37667,37669,37672],{"className":37668},[75],[31,37670,53],{"className":37671},[75,76],[31,37673,37675],{"className":37674},[1039],[31,37676,37678,37698],{"className":37677},[1043,1044],[31,37679,37681,37695],{"className":37680},[1048],[31,37682,37684],{"className":37683,"style":5445},[1052],[31,37685,37686,37689],{"style":2807},[31,37687],{"className":37688,"style":1061},[1060],[31,37690,37692],{"className":37691},[1065,1066,1067,1068],[31,37693,5391],{"className":37694},[75,76,1068],[31,37696,1076],{"className":37697},[1075],[31,37699,37701],{"className":37700},[1048],[31,37702,37704],{"className":37703,"style":2050},[1052],[31,37705],{},[31,37707,495],{"className":37708},[75],[11,37710,37711,37712,37941],{},"求导得到 ",[31,37713,37715,37765],{"className":37714,"translate":35},[34],[31,37716,37718],{"className":37717},[39],[41,37719,37720],{"xmlns":43},[45,37721,37722,37762],{},[48,37723,37724,37730,37736,37738,37740,37742,37744,37746,37752,37754,37756,37758,37760],{},[1004,37725,37726,37728],{},[95,37727,5625],{},[51,37729,5391],{},[1180,37731,37732,37734],{},[51,37733,12145],{"mathvariant":494},[91,37735,152],{},[51,37737,1662],{},[95,37739,211],{"stretchy":210},[51,37741,1412],{},[95,37743,220],{"stretchy":210},[95,37745,211],{"stretchy":210},[1004,37747,37748,37750],{},[51,37749,53],{},[51,37751,5391],{},[95,37753,97],{},[51,37755,1412],{},[95,37757,220],{"stretchy":210},[95,37759,223],{},[91,37761,1977],{},[55,37763,37764],{"encoding":57},"\\sum_i\\nabla^2f(c)(x_i-c)=0",[31,37766,37768,37911,37932],{"className":37767,"ariaHidden":62},[61],[31,37769,37771,37775,37815,37818,37847,37850,37853,37856,37859,37862,37902,37905,37908],{"className":37770},[66],[31,37772],{"className":37773,"style":37774},[70],"height:1.1138em;vertical-align:-0.2997em;",[31,37776,37778,37781],{"className":37777},[1460],[31,37779,5625],{"className":37780,"style":21372},[1460,1461,1462],[31,37782,37784],{"className":37783},[1039],[31,37785,37787,37807],{"className":37786},[1043,1044],[31,37788,37790,37804],{"className":37789},[1048],[31,37791,37793],{"className":37792,"style":21385},[1052],[31,37794,37795,37798],{"style":21388},[31,37796],{"className":37797,"style":1061},[1060],[31,37799,37801],{"className":37800},[1065,1066,1067,1068],[31,37802,5391],{"className":37803},[75,76,1068],[31,37805,1076],{"className":37806},[1075],[31,37808,37810],{"className":37809},[1048],[31,37811,37813],{"className":37812,"style":21407},[1052],[31,37814],{},[31,37816],{"className":37817,"style":1467},[119],[31,37819,37821,37824],{"className":37820},[75],[31,37822,12145],{"className":37823},[75],[31,37825,37827],{"className":37826},[1039],[31,37828,37830],{"className":37829},[1043],[31,37831,37833],{"className":37832},[1048],[31,37834,37836],{"className":37835,"style":5135},[1052],[31,37837,37838,37841],{"style":3631},[31,37839],{"className":37840,"style":1061},[1060],[31,37842,37844],{"className":37843},[1065,1066,1067,1068],[31,37845,152],{"className":37846},[75,1068],[31,37848,1662],{"className":37849,"style":35196},[75,76],[31,37851,211],{"className":37852},[264],[31,37854,1412],{"className":37855},[75,76],[31,37857,220],{"className":37858},[289],[31,37860,211],{"className":37861},[264],[31,37863,37865,37868],{"className":37864},[75],[31,37866,53],{"className":37867},[75,76],[31,37869,37871],{"className":37870},[1039],[31,37872,37874,37894],{"className":37873},[1043,1044],[31,37875,37877,37891],{"className":37876},[1048],[31,37878,37880],{"className":37879,"style":5445},[1052],[31,37881,37882,37885],{"style":2807},[31,37883],{"className":37884,"style":1061},[1060],[31,37886,37888],{"className":37887},[1065,1066,1067,1068],[31,37889,5391],{"className":37890},[75,76,1068],[31,37892,1076],{"className":37893},[1075],[31,37895,37897],{"className":37896},[1048],[31,37898,37900],{"className":37899,"style":2050},[1052],[31,37901],{},[31,37903],{"className":37904,"style":120},[119],[31,37906,97],{"className":37907},[124],[31,37909],{"className":37910,"style":120},[119],[31,37912,37914,37917,37920,37923,37926,37929],{"className":37913},[66],[31,37915],{"className":37916,"style":260},[70],[31,37918,1412],{"className":37919},[75,76],[31,37921,220],{"className":37922},[289],[31,37924],{"className":37925,"style":293},[119],[31,37927,223],{"className":37928},[297],[31,37930],{"className":37931,"style":293},[119],[31,37933,37935,37938],{"className":37934},[66],[31,37936],{"className":37937,"style":326},[70],[31,37939,1977],{"className":37940},[75],"，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。",[11,37943,37944,37945,38069,38070,18171,38206,38342],{},"这一点值得展开，因为它关系到分类边界。设两个类原型 ",[31,37946,37948,37974],{"className":37947,"translate":35},[34],[31,37949,37951],{"className":37950},[39],[41,37952,37953],{"xmlns":43},[45,37954,37955,37971],{},[48,37956,37957,37963,37965],{},[1004,37958,37959,37961],{},[51,37960,1412],{},[91,37962,205],{},[95,37964,1210],{"separator":62},[1004,37966,37967,37969],{},[51,37968,1412],{},[91,37970,152],{},[55,37972,37973],{"encoding":57},"c_1,c_2",[31,37975,37977],{"className":37976,"ariaHidden":62},[61],[31,37978,37980,37983,38023,38026,38029],{"className":37979},[66],[31,37981],{"className":37982,"style":865},[70],[31,37984,37986,37989],{"className":37985},[75],[31,37987,1412],{"className":37988},[75,76],[31,37990,37992],{"className":37991},[1039],[31,37993,37995,38015],{"className":37994},[1043,1044],[31,37996,37998,38012],{"className":37997},[1048],[31,37999,38001],{"className":38000,"style":2028},[1052],[31,38002,38003,38006],{"style":2807},[31,38004],{"className":38005,"style":1061},[1060],[31,38007,38009],{"className":38008},[1065,1066,1067,1068],[31,38010,205],{"className":38011},[75,1068],[31,38013,1076],{"className":38014},[1075],[31,38016,38018],{"className":38017},[1048],[31,38019,38021],{"className":38020,"style":2050},[1052],[31,38022],{},[31,38024,1210],{"className":38025},[1391],[31,38027],{"className":38028,"style":1467},[119],[31,38030,38032,38035],{"className":38031},[75],[31,38033,1412],{"className":38034},[75,76],[31,38036,38038],{"className":38037},[1039],[31,38039,38041,38061],{"className":38040},[1043,1044],[31,38042,38044,38058],{"className":38043},[1048],[31,38045,38047],{"className":38046,"style":2028},[1052],[31,38048,38049,38052],{"style":2807},[31,38050],{"className":38051,"style":1061},[1060],[31,38053,38055],{"className":38054},[1065,1066,1067,1068],[31,38056,152],{"className":38057},[75,1068],[31,38059,1076],{"className":38060},[1075],[31,38062,38064],{"className":38063},[1048],[31,38065,38067],{"className":38066,"style":2050},[1052],[31,38068],{},"，最近原型判类就是比较 ",[31,38071,38073,38105],{"className":38072,"translate":35},[34],[31,38074,38076],{"className":38075},[39],[41,38077,38078],{"xmlns":43},[45,38079,38080,38102],{},[48,38081,38082,38088,38090,38092,38094,38100],{},[1004,38083,38084,38086],{},[51,38085,34614],{},[51,38087,1662],{},[95,38089,211],{"stretchy":210},[51,38091,53],{},[51,38093,4572],{"mathvariant":494},[1004,38095,38096,38098],{},[51,38097,1412],{},[91,38099,205],{},[95,38101,220],{"stretchy":210},[55,38103,38104],{"encoding":57},"D_f(x\\|c_1)",[31,38106,38108],{"className":38107,"ariaHidden":62},[61],[31,38109,38111,38114,38154,38157,38160,38163,38203],{"className":38110},[66],[31,38112],{"className":38113,"style":1223},[70],[31,38115,38117,38120],{"className":38116},[75],[31,38118,34614],{"className":38119,"style":19759},[75,76],[31,38121,38123],{"className":38122},[1039],[31,38124,38126,38146],{"className":38125},[1043,1044],[31,38127,38129,38143],{"className":38128},[1048],[31,38130,38132],{"className":38131,"style":1053},[1052],[31,38133,38134,38137],{"style":26726},[31,38135],{"className":38136,"style":1061},[1060],[31,38138,38140],{"className":38139},[1065,1066,1067,1068],[31,38141,1662],{"className":38142,"style":35196},[75,76,1068],[31,38144,1076],{"className":38145},[1075],[31,38147,38149],{"className":38148},[1048],[31,38150,38152],{"className":38151,"style":1083},[1052],[31,38153],{},[31,38155,211],{"className":38156},[264],[31,38158,53],{"className":38159},[75,76],[31,38161,4572],{"className":38162},[75],[31,38164,38166,38169],{"className":38165},[75],[31,38167,1412],{"className":38168},[75,76],[31,38170,38172],{"className":38171},[1039],[31,38173,38175,38195],{"className":38174},[1043,1044],[31,38176,38178,38192],{"className":38177},[1048],[31,38179,38181],{"className":38180,"style":2028},[1052],[31,38182,38183,38186],{"style":2807},[31,38184],{"className":38185,"style":1061},[1060],[31,38187,38189],{"className":38188},[1065,1066,1067,1068],[31,38190,205],{"className":38191},[75,1068],[31,38193,1076],{"className":38194},[1075],[31,38196,38198],{"className":38197},[1048],[31,38199,38201],{"className":38200,"style":2050},[1052],[31,38202],{},[31,38204,220],{"className":38205},[289],[31,38207,38209,38241],{"className":38208,"translate":35},[34],[31,38210,38212],{"className":38211},[39],[41,38213,38214],{"xmlns":43},[45,38215,38216,38238],{},[48,38217,38218,38224,38226,38228,38230,38236],{},[1004,38219,38220,38222],{},[51,38221,34614],{},[51,38223,1662],{},[95,38225,211],{"stretchy":210},[51,38227,53],{},[51,38229,4572],{"mathvariant":494},[1004,38231,38232,38234],{},[51,38233,1412],{},[91,38235,152],{},[95,38237,220],{"stretchy":210},[55,38239,38240],{"encoding":57},"D_f(x\\|c_2)",[31,38242,38244],{"className":38243,"ariaHidden":62},[61],[31,38245,38247,38250,38290,38293,38296,38299,38339],{"className":38246},[66],[31,38248],{"className":38249,"style":1223},[70],[31,38251,38253,38256],{"className":38252},[75],[31,38254,34614],{"className":38255,"style":19759},[75,76],[31,38257,38259],{"className":38258},[1039],[31,38260,38262,38282],{"className":38261},[1043,1044],[31,38263,38265,38279],{"className":38264},[1048],[31,38266,38268],{"className":38267,"style":1053},[1052],[31,38269,38270,38273],{"style":26726},[31,38271],{"className":38272,"style":1061},[1060],[31,38274,38276],{"className":38275},[1065,1066,1067,1068],[31,38277,1662],{"className":38278,"style":35196},[75,76,1068],[31,38280,1076],{"className":38281},[1075],[31,38283,38285],{"className":38284},[1048],[31,38286,38288],{"className":38287,"style":1083},[1052],[31,38289],{},[31,38291,211],{"className":38292},[264],[31,38294,53],{"className":38295},[75,76],[31,38297,4572],{"className":38298},[75],[31,38300,38302,38305],{"className":38301},[75],[31,38303,1412],{"className":38304},[75,76],[31,38306,38308],{"className":38307},[1039],[31,38309,38311,38331],{"className":38310},[1043,1044],[31,38312,38314,38328],{"className":38313},[1048],[31,38315,38317],{"className":38316,"style":2028},[1052],[31,38318,38319,38322],{"style":2807},[31,38320],{"className":38321,"style":1061},[1060],[31,38323,38325],{"className":38324},[1065,1066,1067,1068],[31,38326,152],{"className":38327},[75,1068],[31,38329,1076],{"className":38330},[1075],[31,38332,38334],{"className":38333},[1048],[31,38335,38337],{"className":38336,"style":2050},[1052],[31,38338],{},[31,38340,220],{"className":38341},[289],"。把定义代进去，",[31,38344,38346],{"className":38345,"translate":35},[411],[31,38347,38349,38713],{"className":38348,"translate":35},[34],[31,38350,38352],{"className":38351},[39],[41,38353,38354],{"xmlns":43,"display":420},[45,38355,38356,38710],{},[5606,38357,38358,38416,38556],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,38359,38360,38366],{},[5615,38361,38362],{},[2071,38363,38364],{"scriptlevel":1977,"displaystyle":62},[48,38365],{},[5615,38367,38368],{},[2071,38369,38370],{"scriptlevel":1977,"displaystyle":62},[48,38371,38372,38374,38380,38382,38384,38386,38392,38394,38396,38402,38404,38406,38408,38414],{},[48,38373],{},[1004,38375,38376,38378],{},[51,38377,34614],{},[51,38379,1662],{},[95,38381,211],{"stretchy":210},[51,38383,53],{},[51,38385,4572],{"mathvariant":494},[1004,38387,38388,38390],{},[51,38389,1412],{},[91,38391,205],{},[95,38393,220],{"stretchy":210},[95,38395,97],{},[1004,38397,38398,38400],{},[51,38399,34614],{},[51,38401,1662],{},[95,38403,211],{"stretchy":210},[51,38405,53],{},[51,38407,4572],{"mathvariant":494},[1004,38409,38410,38412],{},[51,38411,1412],{},[91,38413,152],{},[95,38415,220],{"stretchy":210},[5612,38417,38418,38424],{},[5615,38419,38420],{},[2071,38421,38422],{"scriptlevel":1977,"displaystyle":62},[48,38423],{},[5615,38425,38426],{},[2071,38427,38428],{"scriptlevel":1977,"displaystyle":62},[48,38429,38430,38432,38434,38436,38438,38440,38442,38444,38446,38448,38450,38456,38458,38460,38462,38464,38466,38472,38478,38480,38482,38484,38490,38492,38494,38496,38498,38500,38502,38504,38506,38508,38510,38512,38518,38520,38522,38524,38526,38528,38534,38540,38542,38544,38546,38552,38554],{},[48,38431],{},[95,38433,223],{},[95,38435,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,38437,1662],{},[95,38439,211],{"stretchy":210},[51,38441,53],{},[95,38443,220],{"stretchy":210},[95,38445,97],{},[51,38447,1662],{},[95,38449,211],{"stretchy":210},[1004,38451,38452,38454],{},[51,38453,1412],{},[91,38455,205],{},[95,38457,220],{"stretchy":210},[95,38459,97],{},[51,38461,12145],{"mathvariant":494},[51,38463,1662],{},[95,38465,211],{"stretchy":210},[1004,38467,38468,38470],{},[51,38469,1412],{},[91,38471,205],{},[1180,38473,38474,38476],{},[95,38475,220],{"stretchy":210},[51,38477,20800],{"mathvariant":494},[95,38479,211],{"stretchy":210},[51,38481,53],{},[95,38483,97],{},[1004,38485,38486,38488],{},[51,38487,1412],{},[91,38489,205],{},[95,38491,220],{"stretchy":210},[95,38493,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,38495,97],{},[95,38497,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,38499,1662],{},[95,38501,211],{"stretchy":210},[51,38503,53],{},[95,38505,220],{"stretchy":210},[95,38507,97],{},[51,38509,1662],{},[95,38511,211],{"stretchy":210},[1004,38513,38514,38516],{},[51,38515,1412],{},[91,38517,152],{},[95,38519,220],{"stretchy":210},[95,38521,97],{},[51,38523,12145],{"mathvariant":494},[51,38525,1662],{},[95,38527,211],{"stretchy":210},[1004,38529,38530,38532],{},[51,38531,1412],{},[91,38533,152],{},[1180,38535,38536,38538],{},[95,38537,220],{"stretchy":210},[51,38539,20800],{"mathvariant":494},[95,38541,211],{"stretchy":210},[51,38543,53],{},[95,38545,97],{},[1004,38547,38548,38550],{},[51,38549,1412],{},[91,38551,152],{},[95,38553,220],{"stretchy":210},[95,38555,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[5612,38557,38558,38564],{},[5615,38559,38560],{},[2071,38561,38562],{"scriptlevel":1977,"displaystyle":62},[48,38563],{},[5615,38565,38566],{},[2071,38567,38568],{"scriptlevel":1977,"displaystyle":62},[48,38569,38570,38572,38574,38666,38668,38670,38672,38674,38676,38682,38684,38686,38688,38690,38692,38698,38700,38706,38708],{},[48,38571],{},[95,38573,223],{},[1531,38575,38576,38663],{},[1531,38577,38578,38660],{},[48,38579,38580,38582,38584,38586,38592,38594,38596,38598,38600,38606,38608,38610,38612,38614,38616,38622,38628,38634,38636,38638,38640,38642,38648,38654],{},[95,38581,97],{},[51,38583,1662],{},[95,38585,211],{"stretchy":210},[1004,38587,38588,38590],{},[51,38589,1412],{},[91,38591,205],{},[95,38593,220],{"stretchy":210},[95,38595,434],{},[51,38597,1662],{},[95,38599,211],{"stretchy":210},[1004,38601,38602,38604],{},[51,38603,1412],{},[91,38605,152],{},[95,38607,220],{"stretchy":210},[95,38609,434],{},[51,38611,12145],{"mathvariant":494},[51,38613,1662],{},[95,38615,211],{"stretchy":210},[1004,38617,38618,38620],{},[51,38619,1412],{},[91,38621,205],{},[1180,38623,38624,38626],{},[95,38625,220],{"stretchy":210},[51,38627,20800],{"mathvariant":494},[1004,38629,38630,38632],{},[51,38631,1412],{},[91,38633,205],{},[95,38635,97],{},[51,38637,12145],{"mathvariant":494},[51,38639,1662],{},[95,38641,211],{"stretchy":210},[1004,38643,38644,38646],{},[51,38645,1412],{},[91,38647,152],{},[1180,38649,38650,38652],{},[95,38651,220],{"stretchy":210},[51,38653,20800],{"mathvariant":494},[1004,38655,38656,38658],{},[51,38657,1412],{},[91,38659,152],{},[95,38661,38662],{"stretchy":62},"⏟",[1430,38664,38665],{},"常数",[95,38667,434],{},[95,38669,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,38671,12145],{"mathvariant":494},[51,38673,1662],{},[95,38675,211],{"stretchy":210},[1004,38677,38678,38680],{},[51,38679,1412],{},[91,38681,152],{},[95,38683,220],{"stretchy":210},[95,38685,97],{},[51,38687,12145],{"mathvariant":494},[51,38689,1662],{},[95,38691,211],{"stretchy":210},[1004,38693,38694,38696],{},[51,38695,1412],{},[91,38697,205],{},[95,38699,220],{"stretchy":210},[1180,38701,38702,38704],{},[95,38703,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,38705,20800],{"mathvariant":494},[51,38707,53],{},[51,38709,495],{"mathvariant":494},[55,38711,38712],{"encoding":57},"\\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}",[31,38714,38716],{"className":38715,"ariaHidden":62},[61],[31,38717,38719,38723],{"className":38718},[66],[31,38720],{"className":38721,"style":38722},[70],"height:5.7244em;vertical-align:-2.6122em;",[31,38724,38726],{"className":38725},[75],[31,38727,38729,38782],{"className":38728},[5606],[31,38730,38732],{"className":38731},[5867],[31,38733,38735,38773],{"className":38734},[1043,1044],[31,38736,38738,38770],{"className":38737},[1048],[31,38739,38742,38752,38761],{"className":38740,"style":38741},[1052],"height:3.1122em;",[31,38743,38745,38749],{"style":38744},"top:-5.3612em;",[31,38746],{"className":38747,"style":38748},[1060],"height:3.089em;",[31,38750],{"className":38751},[75],[31,38753,38755,38758],{"style":38754},"top:-3.8021em;",[31,38756],{"className":38757,"style":38748},[1060],[31,38759],{"className":38760},[75],[31,38762,38764,38767],{"style":38763},"top:-2.0531em;",[31,38765],{"className":38766,"style":38748},[1060],[31,38768],{"className":38769},[75],[31,38771,1076],{"className":38772},[1075],[31,38774,38776],{"className":38775},[1048],[31,38777,38780],{"className":38778,"style":38779},[1052],"height:2.6122em;",[31,38781],{},[31,38783,38785],{"className":38784},[6062],[31,38786,38788,40158],{"className":38787},[1043,1044],[31,38789,38791,40155],{"className":38790},[1048],[31,38792,38794,38998,39481],{"className":38793,"style":38741},[1052],[31,38795,38796,38799],{"style":38744},[31,38797],{"className":38798,"style":38748},[1060],[31,38800,38802,38805,38845,38848,38851,38854,38894,38897,38900,38903,38906,38946,38949,38952,38955,38995],{"className":38801},[75],[31,38803],{"className":38804},[75],[31,38806,38808,38811],{"className":38807},[75],[31,38809,34614],{"className":38810,"style":19759},[75,76],[31,38812,38814],{"className":38813},[1039],[31,38815,38817,38837],{"className":38816},[1043,1044],[31,38818,38820,38834],{"className":38819},[1048],[31,38821,38823],{"className":38822,"style":1053},[1052],[31,38824,38825,38828],{"style":26726},[31,38826],{"className":38827,"style":1061},[1060],[31,38829,38831],{"className":38830},[1065,1066,1067,1068],[31,38832,1662],{"className":38833,"style":35196},[75,76,1068],[31,38835,1076],{"className":38836},[1075],[31,38838,38840],{"className":38839},[1048],[31,38841,38843],{"className":38842,"style":1083},[1052],[31,38844],{},[31,38846,211],{"className":38847},[264],[31,38849,53],{"className":38850},[75,76],[31,38852,4572],{"className":38853},[75],[31,38855,38857,38860],{"className":38856},[75],[31,38858,1412],{"className":38859},[75,76],[31,38861,38863],{"className":38862},[1039],[31,38864,38866,38886],{"className":38865},[1043,1044],[31,38867,38869,38883],{"className":38868},[1048],[31,38870,38872],{"className":38871,"style":2028},[1052],[31,38873,38874,38877],{"style":2807},[31,38875],{"className":38876,"style":1061},[1060],[31,38878,38880],{"className":38879},[1065,1066,1067,1068],[31,38881,205],{"className":38882},[75,1068],[31,38884,1076],{"className":38885},[1075],[31,38887,38889],{"className":38888},[1048],[31,38890,38892],{"className":38891,"style":2050},[1052],[31,38893],{},[31,38895,220],{"className":38896},[289],[31,38898],{"className":38899,"style":120},[119],[31,38901,97],{"className":38902},[124],[31,38904],{"className":38905,"style":120},[119],[31,38907,38909,38912],{"className":38908},[75],[31,38910,34614],{"className":38911,"style":19759},[75,76],[31,38913,38915],{"className":38914},[1039],[31,38916,38918,38938],{"className":38917},[1043,1044],[31,38919,38921,38935],{"className":38920},[1048],[31,38922,38924],{"className":38923,"style":1053},[1052],[31,38925,38926,38929],{"style":26726},[31,38927],{"className":38928,"style":1061},[1060],[31,38930,38932],{"className":38931},[1065,1066,1067,1068],[31,38933,1662],{"className":38934,"style":35196},[75,76,1068],[31,38936,1076],{"className":38937},[1075],[31,38939,38941],{"className":38940},[1048],[31,38942,38944],{"className":38943,"style":1083},[1052],[31,38945],{},[31,38947,211],{"className":38948},[264],[31,38950,53],{"className":38951},[75,76],[31,38953,4572],{"className":38954},[75],[31,38956,38958,38961],{"className":38957},[75],[31,38959,1412],{"className":38960},[75,76],[31,38962,38964],{"className":38963},[1039],[31,38965,38967,38987],{"className":38966},[1043,1044],[31,38968,38970,38984],{"className":38969},[1048],[31,38971,38973],{"className":38972,"style":2028},[1052],[31,38974,38975,38978],{"style":2807},[31,38976],{"className":38977,"style":1061},[1060],[31,38979,38981],{"className":38980},[1065,1066,1067,1068],[31,38982,152],{"className":38983},[75,1068],[31,38985,1076],{"className":38986},[1075],[31,38988,38990],{"className":38989},[1048],[31,38991,38993],{"className":38992,"style":2050},[1052],[31,38994],{},[31,38996,220],{"className":38997},[289],[31,38999,39000,39003],{"style":38754},[31,39001],{"className":39002,"style":38748},[1060],[31,39004,39006,39009,39012,39015,39018,39024,39027,39030,39033,39036,39039,39042,39045,39048,39051,39091,39094,39097,3910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20-1h17z",[31,39593,39594,39597],{"style":1649},[31,39595],{"className":39596,"style":1637},[1060],[31,39598,39600,39603,39606,39609,39649,39652,39655,39658,39661,39664,39667,39707,39710,39713,39716,39719,39722,39725,39728,39768,39797,39837,39840,39843,39846,39849,39852,39855,39895,39924],{"className":39599},[75],[31,39601,97],{"className":39602},[75],[31,39604,1662],{"className":39605,"style":35196},[75,76],[31,39607,211],{"className":39608},[264],[31,39610,39612,39615],{"className":39611},[75],[31,39613,1412],{"className":39614},[75,76],[31,39616,39618],{"className":39617},[1039],[31,39619,39621,39641],{"className":39620},[1043,1044],[31,39622,39624,39638],{"className":39623},[1048],[31,39625,39627],{"className":39626,"style":2028},[1052],[31,39628,39629,39632],{"style":2807},[31,39630],{"className":39631,"style":1061},[1060],[31,39633,39635],{"className":39634},[1065,1066,1067,1068],[31,39636,205],{"className":39637},[75,1068],[31,39639,1076],{"className":39640},[1075],[31,39642,39644],{"className":39643},[1048],[31,39645,39647],{"className":39646,"style":2050},[1052],[31,39648],{},[31,39650,220],{"className":39651},[289],[31,39653],{"className":39654,"style":120},[119],[31,39656,434],{"className":39657},[124],[31,39659],{"className":39660,"style":120},[119],[31,39662,1662],{"className":39663,"style":35196},[75,76],[31,39665,211],{"className":39666},[264],[31,39668,39670,39673],{"className":39669},[75],[31,39671,1412],{"className":39672},[75,76],[31,39674,39676],{"className":39675},[1039],[31,39677,39679,39699],{"className":39678},[1043,1044],[31,39680,39682,39696],{"className":39681},[1048],[31,39683,39685],{"className":39684,"style":2028},[1052],[31,39686,39687,39690],{"style":2807},[31,39688],{"className":39689,"style":1061},[1060],[31,39691,39693],{"className":39692},[1065,1066,1067,1068],[31,39694,152],{"className":39695},[75,1068],[31,39697,1076],{"className":39698},[1075],[31,39700,39702],{"className":39701},[1048],[31,39703,39705],{"className":39704,"style":2050},[1052],[31,39706],{},[31,39708,220],{"className":39709},[289],[31,39711],{"className":39712,"style":120},[119],[31,39714,434],{"className":39715},[124],[31,39717],{"className":39718,"style":120},[119],[31,39720,12145],{"className":39721},[75],[31,39723,1662],{"className":39724,"style":35196},[75,76],[31,39726,211],{"className":39727},[264],[31,39729,39731,39734],{"className":39730},[75],[31,39732,1412],{"className":39733},[75,76],[31,39735,39737],{"className":39736},[1039],[31,39738,39740,39760],{"className":39739},[1043,1044],[31,39741,39743,39757],{"className":39742},[1048],[31,39744,39746],{"className":39745,"style":2028},[1052],[31,39747,39748,39751],{"style":2807},[31,39749],{"className":39750,"style":1061},[1060],[31,39752,39754],{"className":39753},[1065,1066,1067,1068],[31,39755,205],{"className":39756},[75,1068],[31,39758,1076],{"className":39759},[1075],[31,39761,39763],{"className":39762},[1048],[31,39764,39766],{"className":39765,"style":2050},[1052],[31,39767],{},[31,39769,39771,39774],{"className":39770},[289],[31,39772,220],{"className":39773},[289],[31,39775,39777],{"className":39776},[1039],[31,39778,39780],{"className":39779},[1043],[31,39781,39783],{"className":39782},[1048],[31,39784,39786],{"className":39785,"style":29642},[1052],[31,39787,39788,39791],{"style":1330},[31,39789],{"className":39790,"style":1061},[1060],[31,39792,39794],{"className":39793},[1065,1066,1067,1068],[31,39795,20800],{"className":39796},[75,1068],[31,39798,39800,39803],{"className":39799},[75],[31,39801,1412],{"className":39802},[75,76],[31,39804,39806],{"className":39805},[1039],[31,39807,39809,39829],{"className":39808},[1043,1044],[31,39810,39812,39826],{"className":39811},[1048],[31,39813,39815],{"className":39814,"style":2028},[1052],[31,39816,39817,39820],{"style":2807},[31,39818],{"className":39819,"style":1061},[1060],[31,39821,39823],{"className":39822},[1065,1066,1067,1068],[31,39824,205],{"className":39825},[75,1068],[31,39827,1076],{"className":39828},[1075],[31,39830,39832],{"className":39831},[1048],[31,39833,39835],{"className":39834,"style":2050},[1052],[31,39836],{},[31,39838],{"className":39839,"style":120},[119],[31,39841,97],{"className":39842},[124],[31,39844],{"className":39845,"style":120},[119],[31,39847,12145],{"className":39848},[75],[31,39850,1662],{"className":39851,"style":35196},[75,76],[31,39853,211],{"className":39854},[264],[31,39856,39858,39861],{"className":39857},[75],[31,39859,1412],{"className":39860},[75,76],[31,39862,39864],{"className":39863},[1039],[31,39865,39867,39887],{"className":39866},[1043,1044],[31,39868,39870,39884],{"className":39869},[1048],[31,39871,39873],{"className":39872,"style":2028},[1052],[31,39874,39875,39878],{"style":2807},[31,39876],{"className":39877,"style":1061},[1060],[31,39879,39881],{"className":39880},[1065,1066,1067,1068],[31,39882,152],{"className":39883},[75,1068],[31,39885,1076],{"className":39886},[1075],[31,39888,39890],{"className":39889},[1048],[31,39891,39893],{"className":39892,"style":2050},[1052],[31,39894],{},[31,39896,39898,39901],{"className":39897},[289],[31,39899,220],{"className":39900},[289],[31,39902,39904],{"className":39903},[1039],[31,39905,39907],{"className":39906},[1043],[31,39908,39910],{"className":39909},[1048],[31,39911,39913],{"className":39912,"style":29642},[1052],[31,39914,39915,39918],{"style":1330},[31,39916],{"className":39917,"style":1061},[1060],[31,39919,39921],{"className":39920},[1065,1066,1067,1068],[31,39922,20800],{"className":39923},[75,1068],[31,39925,39927,39930],{"className":39926},[75],[31,39928,1412],{"className":39929},[75,76],[31,39931,39933],{"className":39932},[1039],[31,39934,39936,39956],{"className":39935},[1043,1044],[31,39937,39939,39953],{"className":39938},[1048],[31,39940,39942],{"className":39941,"style":2028},[1052],[31,39943,39944,39947],{"style":2807},[31,39945],{"className":39946,"style":1061},[1060],[31,39948,39950],{"className":39949},[1065,1066,1067,1068],[31,39951,152],{"className":39952},[75,1068],[31,39954,1076],{"className":39955},[1075],[31,39957,39959],{"className":39958},[1048],[31,39960,39962],{"className":39961,"style":2050},[1052],[31,39963],{},[31,39965,1076],{"className":39966},[1075],[31,39968,39970],{"className":39969},[1048],[31,39971,39974],{"className":39972,"style":39973},[1052],"height:0.898em;",[31,39975],{},[31,39977,1076],{"className":39978},[1075],[31,39980,39982],{"className":39981},[1048],[31,39983,39986],{"className":39984,"style":39985},[1052],"height:1.5763em;",[31,39987],{},[31,39989],{"className":39990,"style":120},[119],[31,39992,434],{"className":39993},[124],[31,39995],{"className":39996,"style":120},[119],[31,39998,40000],{"className":39999},[264],[31,40001,211],{"className":40002},[1307,1308],[31,40004,12145],{"className":40005},[75],[31,40007,1662],{"className":40008,"style":35196},[75,76],[31,40010,211],{"className":40011},[264],[31,40013,40015,40018],{"className":40014},[75],[31,40016,1412],{"className":40017},[75,76],[31,40019,40021],{"className":40020},[1039],[31,40022,40024,40044],{"className":40023},[1043,1044],[31,40025,40027,40041],{"className":40026},[1048],[31,40028,40030],{"className":40029,"style":2028},[1052],[31,40031,40032,40035],{"style":2807},[31,40033],{"className":40034,"style":1061},[1060],[31,40036,40038],{"className":40037},[1065,1066,1067,1068],[31,40039,152],{"className":40040},[75,1068],[31,40042,1076],{"className":40043},[1075],[31,40045,40047],{"className":40046},[1048],[31,40048,40050],{"className":40049,"style":2050},[1052],[31,40051],{},[31,40053,220],{"className":40054},[289],[31,40056],{"className":40057,"style":120},[119],[31,40059,97],{"className":40060},[124],[31,40062],{"className":40063,"style":120},[119],[31,40065,12145],{"className":40066},[75],[31,40068,1662],{"className":40069,"style":35196},[75,76],[31,40071,211],{"className":40072},[264],[31,40074,40076,40079],{"className":40075},[75],[31,40077,1412],{"className":40078},[75,76],[31,40080,40082],{"className":40081},[1039],[31,40083,40085,40105],{"className":40084},[1043,1044],[31,40086,40088,40102],{"className":40087},[1048],[31,40089,40091],{"className":40090,"style":2028},[1052],[31,40092,40093,40096],{"style":2807},[31,40094],{"className":40095,"style":1061},[1060],[31,40097,40099],{"className":40098},[1065,1066,1067,1068],[31,40100,205],{"className":40101},[75,1068],[31,40103,1076],{"className":40104},[1075],[31,40106,40108],{"className":40107},[1048],[31,40109,40111],{"className":40110,"style":2050},[1052],[31,40112],{},[31,40114,220],{"className":40115},[289],[31,40117,40119,40125],{"className":40118},[289],[31,40120,40122],{"className":40121},[289],[31,40123,220],{"className":40124},[1307,1308],[31,40126,40128],{"className":40127},[1039],[31,40129,40131],{"className":40130},[1043],[31,40132,40134],{"className":40133},[1048],[31,40135,40138],{"className":40136,"style":40137},[1052],"height:1.089em;",[31,40139,40140,40143],{"style":13566},[31,40141],{"className":40142,"style":1061},[1060],[31,40144,40146],{"className":40145},[1065,1066,1067,1068],[31,40147,20800],{"className":40148},[75,1068],[31,40150,53],{"className":40151},[75,76],[31,40153,495],{"className":40154},[75],[31,40156,1076],{"className":40157},[1075],[31,40159,40161],{"className":40160},[1048],[31,40162,40164],{"className":40163,"style":38779},[1052],[31,40165],{},[11,40167,40168,40212,40213,40241],{},[31,40169,40171,40191],{"className":40170,"translate":35},[34],[31,40172,40174],{"className":40173},[39],[41,40175,40176],{"xmlns":43},[45,40177,40178,40188],{},[48,40179,40180,40182,40184,40186],{},[51,40181,1662],{},[95,40183,211],{"stretchy":210},[51,40185,53],{},[95,40187,220],{"stretchy":210},[55,40189,40190],{"encoding":57},"f(x)",[31,40192,40194],{"className":40193,"ariaHidden":62},[61],[31,40195,40197,40200,40203,40206,40209],{"className":40196},[66],[31,40198],{"className":40199,"style":260},[70],[31,40201,1662],{"className":40202,"style":35196},[75,76],[31,40204,211],{"className":40205},[264],[31,40207,53],{"className":40208},[75,76],[31,40210,220],{"className":40211},[289]," 那一项消掉了，剩下的是 ",[31,40214,40216,40229],{"className":40215,"translate":35},[34],[31,40217,40219],{"className":40218},[39],[41,40220,40221],{"xmlns":43},[45,40222,40223,40227],{},[48,40224,40225],{},[51,40226,53],{},[55,40228,53],{"encoding":57},[31,40230,40232],{"className":40231,"ariaHidden":62},[61],[31,40233,40235,40238],{"className":40234},[66],[31,40236],{"className":40237,"style":71},[70],[31,40239,53],{"className":40240},[75,76]," 的仿射函数。所以边界仍然是一个超平面。",[11,40243,40244],{},"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。",[11,40246,40247,40248,40299,40300,2446,40329,40358,40359,40412,40413,40442,40443,40472],{},"具体到二维单纯形（其实就是线段 ",[31,40249,40251,40272],{"className":40250,"translate":35},[34],[31,40252,40254],{"className":40253},[39],[41,40255,40256],{"xmlns":43},[45,40257,40258,40270],{},[48,40259,40260,40262,40264,40266,40268],{},[95,40261,15175],{"stretchy":210},[91,40263,1977],{},[95,40265,1210],{"separator":62},[91,40267,205],{},[95,40269,15184],{"stretchy":210},[55,40271,15187],{"encoding":57},[31,40273,40275],{"className":40274,"ariaHidden":62},[61],[31,40276,40278,40281,40284,40287,40290,40293,40296],{"className":40277},[66],[31,40279],{"className":40280,"style":260},[70],[31,40282,15175],{"className":40283},[264],[31,40285,1977],{"className":40286},[75],[31,40288,1210],{"className":40289},[1391],[31,40291],{"className":40292,"style":1467},[119],[31,40294,205],{"className":40295},[75],[31,40297,15184],{"className":40298},[289],"）上，两个类原型 ",[31,40301,40303,40317],{"className":40302,"translate":35},[34],[31,40304,40306],{"className":40305},[39],[41,40307,40308],{"xmlns":43},[45,40309,40310,40315],{},[48,40311,40312],{},[91,40313,40314],{},"0.2",[55,40316,40314],{"encoding":57},[31,40318,40320],{"className":40319,"ariaHidden":62},[61],[31,40321,40323,40326],{"className":40322},[66],[31,40324],{"className":40325,"style":326},[70],[31,40327,40314],{"className":40328},[75],[31,40330,40332,40346],{"className":40331,"translate":35},[34],[31,40333,40335],{"className":40334},[39],[41,40336,40337],{"xmlns":43},[45,40338,40339,40344],{},[48,40340,40341],{},[91,40342,40343],{},"0.6",[55,40345,40343],{"encoding":57},[31,40347,40349],{"className":40348,"ariaHidden":62},[61],[31,40350,40352,40355],{"className":40351},[66],[31,40353],{"className":40354,"style":326},[70],[31,40356,40343],{"className":40357},[75],"，用 KL 判类的分界点在 ",[31,40360,40362,40381],{"className":40361,"translate":35},[34],[31,40363,40365],{"className":40364},[39],[41,40366,40367],{"xmlns":43},[45,40368,40369,40378],{},[48,40370,40371,40373,40375],{},[51,40372,11],{},[95,40374,27223],{},[91,40376,40377],{},"0.387",[55,40379,40380],{"encoding":57},"p\\approx0.387",[31,40382,40384,40403],{"className":40383,"ariaHidden":62},[61],[31,40385,40387,40391,40394,40397,40400],{"className":40386},[66],[31,40388],{"className":40389,"style":40390},[70],"height:0.6776em;vertical-align:-0.1944em;",[31,40392,11],{"className":40393},[75,76],[31,40395],{"className":40396,"style":293},[119],[31,40398,27223],{"className":40399},[297],[31,40401],{"className":40402,"style":293},[119],[31,40404,40406,40409],{"className":40405},[66],[31,40407],{"className":40408,"style":326},[70],[31,40410,40377],{"className":40411},[75],"；欧氏中点是 ",[31,40414,40416,40430],{"className":40415,"translate":35},[34],[31,40417,40419],{"className":40418},[39],[41,40420,40421],{"xmlns":43},[45,40422,40423,40428],{},[48,40424,40425],{},[91,40426,40427],{},"0.4",[55,40429,40427],{"encoding":57},[31,40431,40433],{"className":40432,"ariaHidden":62},[61],[31,40434,40436,40439],{"className":40435},[66],[31,40437],{"className":40438,"style":326},[70],[31,40440,40427],{"className":40441},[75],"；而在 logit 坐标里取中点又得到 ",[31,40444,40446,40460],{"className":40445,"translate":35},[34],[31,40447,40449],{"className":40448},[39],[41,40450,40451],{"xmlns":43},[45,40452,40453,40458],{},[48,40454,40455],{},[91,40456,40457],{},"0.380",[55,40459,40457],{"encoding":57},[31,40461,40463],{"className":40462,"ariaHidden":62},[61],[31,40464,40466,40469],{"className":40465},[66],[31,40467],{"className":40468,"style":326},[70],[31,40470,40457],{"className":40471},[75],"。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。",[11,40474,40475,40476,40580,40581,25085,40906,40934,40935,40963],{},"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 ",[31,40477,40479,40525],{"className":40478,"translate":35},[34],[31,40480,40482],{"className":40481},[39],[41,40483,40484],{"xmlns":43},[45,40485,40486,40522],{},[48,40487,40488,40490,40492,40494,40496,40498,40514,40516,40518,40520],{},[51,40489,27666],{},[95,40491,211],{"stretchy":210},[51,40493,53],{},[95,40495,220],{"stretchy":210},[95,40497,223],{},[48,40499,40500,40502,40504,40506,40508,40510,40512],{},[51,40501,14551],{"mathvariant":494},[51,40503,2552],{"mathvariant":494},[51,40505,1662],{"mathvariant":494},[51,40507,12271],{"mathvariant":494},[51,40509,16396],{"mathvariant":494},[51,40511,5813],{"mathvariant":494},[51,40513,53],{"mathvariant":494},[95,40515,211],{"stretchy":210},[51,40517,3117],{},[51,40519,53],{},[95,40521,220],{"stretchy":210},[55,40523,40524],{"encoding":57},"q(x)=\\mathrm{softmax}(Wx)",[31,40526,40528,40555],{"className":40527,"ariaHidden":62},[61],[31,40529,40531,40534,40537,40540,40543,40546,40549,40552],{"className":40530},[66],[31,40532],{"className":40533,"style":260},[70],[31,40535,27666],{"className":40536,"style":2666},[75,76],[31,40538,211],{"className":40539},[264],[31,40541,53],{"className":40542},[75,76],[31,40544,220],{"className":40545},[289],[31,40547],{"className":40548,"style":293},[119],[31,40550,223],{"className":40551},[297],[31,40553],{"className":40554,"style":293},[119],[31,40556,40558,40561,40568,40571,40574,40577],{"className":40557},[66],[31,40559],{"className":40560,"style":260},[70],[31,40562,40564],{"className":40563},[75],[31,40565,40567],{"className":40566},[75,2745],"softmax",[31,40569,211],{"className":40570},[264],[31,40572,3117],{"className":40573,"style":949},[75,76],[31,40575,53],{"className":40576},[75,76],[31,40578,220],{"className":40579},[289],"，那么 ",[31,40582,40584,40658],{"className":40583,"translate":35},[34],[31,40585,40587],{"className":40586},[39],[41,40588,40589],{"xmlns":43},[45,40590,40591,40655],{},[48,40592,40593,40595,40597,40599,40605,40607,40609,40611,40613,40615,40617,40623,40625,40627,40633,40635,40637,40639,40641,40643,40645,40651,40653],{},[95,40594,2512],{"stretchy":210},[51,40596,53],{},[95,40598,922],{},[1004,40600,40601,40603],{},[51,40602,34614],{},[51,40604,1662],{},[95,40606,211],{"stretchy":210},[51,40608,27666],{},[95,40610,211],{"stretchy":210},[51,40612,53],{},[95,40614,220],{"stretchy":210},[51,40616,4572],{"mathvariant":494},[1004,40618,40619,40621],{},[51,40620,1412],{},[91,40622,205],{},[95,40624,220],{"stretchy":210},[95,40626,223],{},[1004,40628,40629,40631],{},[51,40630,34614],{},[51,40632,1662],{},[95,40634,211],{"stretchy":210},[51,40636,27666],{},[95,40638,211],{"stretchy":210},[51,40640,53],{},[95,40642,220],{"stretchy":210},[51,40644,4572],{"mathvariant":494},[1004,40646,40647,40649],{},[51,40648,1412],{},[91,40650,152],{},[95,40652,220],{"stretchy":210},[95,40654,2604],{"stretchy":210},[55,40656,40657],{"encoding":57},"\\{x:D_f(q(x)\\|c_1)=D_f(q(x)\\|c_2)\\}",[31,40659,40661,40682,40798],{"className":40660,"ariaHidden":62},[61],[31,40662,40664,40667,40670,40673,40676,40679],{"className":40663},[66],[31,40665],{"className":40666,"style":260},[70],[31,40668,2512],{"className":40669},[264],[31,40671,53],{"className":40672},[75,76],[31,40674],{"className":40675,"style":293},[119],[31,40677,922],{"className":40678},[297],[31,40680],{"className":40681,"style":293},[119],[31,40683,40685,40688,40728,40731,40734,40737,40740,40743,40746,40786,40789,40792,40795],{"className":40684},[66],[31,40686],{"className":40687,"style":1223},[70],[31,40689,40691,40694],{"className":40690},[75],[31,40692,34614],{"className":40693,"style":19759},[75,76],[31,40695,40697],{"className":40696},[1039],[31,40698,40700,40720],{"className":40699},[1043,1044],[31,40701,40703,40717],{"className":40702},[1048],[31,40704,40706],{"className":40705,"style":1053},[1052],[31,40707,40708,40711],{"style":26726},[31,40709],{"className":40710,"style":1061},[1060],[31,40712,40714],{"className":40713},[1065,1066,1067,1068],[31,40715,1662],{"className":40716,"style":35196},[75,76,1068],[31,40718,1076],{"className":40719},[1075],[31,40721,40723],{"className":40722},[1048],[31,40724,40726],{"className":40725,"style":1083},[1052],[31,40727],{},[31,40729,211],{"className":40730},[264],[31,40732,27666],{"className":40733,"style":2666},[75,76],[31,40735,211],{"className":40736},[264],[31,40738,53],{"className":40739},[75,76],[31,40741,220],{"className":40742},[289],[31,40744,4572],{"className":40745},[75],[31,40747,40749,40752],{"className":40748},[75],[31,40750,1412],{"className":40751},[75,76],[31,40753,40755],{"className":40754},[1039],[31,40756,40758,40778],{"className":40757},[1043,1044],[31,40759,40761,40775],{"className":40760},[1048],[31,40762,40764],{"className":40763,"style":2028},[1052],[31,40765,40766,40769],{"style":2807},[31,40767],{"className":40768,"style":1061},[1060],[31,40770,40772],{"className":40771},[1065,1066,1067,1068],[31,40773,205],{"className":40774},[75,1068],[31,40776,1076],{"className":40777},[1075],[31,40779,40781],{"className":40780},[1048],[31,40782,40784],{"className":40783,"style":2050},[1052],[31,40785],{},[31,40787,220],{"className":40788},[289],[31,40790],{"className":40791,"style":293},[119],[31,40793,223],{"className":40794},[297],[31,40796],{"className":40797,"style":293},[119],[31,40799,40801,40804,40844,40847,40850,40853,40856,40859,40862,40902],{"className":40800},[66],[31,40802],{"className":40803,"style":1223},[70],[31,40805,40807,40810],{"className":40806},[75],[31,40808,34614],{"className":40809,"style":19759},[75,76],[31,40811,40813],{"className":40812},[1039],[31,40814,40816,40836],{"className":40815},[1043,1044],[31,40817,40819,40833],{"className":40818},[1048],[31,40820,40822],{"className":40821,"style":1053},[1052],[31,40823,40824,40827],{"style":26726},[31,40825],{"className":40826,"style":1061},[1060],[31,40828,40830],{"className":40829},[1065,1066,1067,1068],[31,40831,1662],{"className":40832,"style":35196},[75,76,1068],[31,40834,1076],{"className":40835},[1075],[31,40837,40839],{"className":40838},[1048],[31,40840,40842],{"className":40841,"style":1083},[1052],[31,40843],{},[31,40845,211],{"className":40846},[264],[31,40848,27666],{"className":40849,"style":2666},[75,76],[31,40851,211],{"className":40852},[264],[31,40854,53],{"className":40855},[75,76],[31,40857,220],{"className":40858},[289],[31,40860,4572],{"className":40861},[75],[31,40863,40865,40868],{"className":40864},[75],[31,40866,1412],{"className":40867},[75,76],[31,40869,40871],{"className":40870},[1039],[31,40872,40874,40894],{"className":40873},[1043,1044],[31,40875,40877,40891],{"className":40876},[1048],[31,40878,40880],{"className":40879,"style":2028},[1052],[31,40881,40882,40885],{"style":2807},[31,40883],{"className":40884,"style":1061},[1060],[31,40886,40888],{"className":40887},[1065,1066,1067,1068],[31,40889,152],{"className":40890},[75,1068],[31,40892,1076],{"className":40893},[1075],[31,40895,40897],{"className":40896},[1048],[31,40898,40900],{"className":40899,"style":2050},[1052],[31,40901],{},[31,40903,40905],{"className":40904},[289],")}",[31,40907,40909,40922],{"className":40908,"translate":35},[34],[31,40910,40912],{"className":40911},[39],[41,40913,40914],{"xmlns":43},[45,40915,40916,40920],{},[48,40917,40918],{},[51,40919,27666],{},[55,40921,27666],{"encoding":57},[31,40923,40925],{"className":40924,"ariaHidden":62},[61],[31,40926,40928,40931],{"className":40927},[66],[31,40929],{"className":40930,"style":865},[70],[31,40932,27666],{"className":40933,"style":2666},[75,76]," 空间是超平面，在 ",[31,40936,40938,40951],{"className":40937,"translate":35},[34],[31,40939,40941],{"className":40940},[39],[41,40942,40943],{"xmlns":43},[45,40944,40945,40949],{},[48,40946,40947],{},[51,40948,53],{},[55,40950,53],{"encoding":57},[31,40952,40954],{"className":40953,"ariaHidden":62},[61],[31,40955,40957,40960],{"className":40956},[66],[31,40958],{"className":40959,"style":71},[70],[31,40961,53],{"className":40962},[75,76]," 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。",[11,40965,40966],{},"Bregman 还给出两个优化工具。一个是镜像下降：",[31,40968,40970],{"className":40969,"translate":35},[411],[31,40971,40973,41059],{"className":40972,"translate":35},[34],[31,40974,40976],{"className":40975},[39],[41,40977,40978],{"xmlns":43,"display":420},[45,40979,40980,41056],{},[48,40981,40982,40994,40996,40998,41000,41010,41012,41014,41020,41022,41024,41026,41028,41034,41040,41042,41044,41046,41052,41054],{},[1004,40983,40984,40986],{},[51,40985,53],{},[48,40987,40988,40990,40992],{},[51,40989,12271],{},[95,40991,434],{},[91,40993,205],{},[95,40995,223],{},[51,40997,37263],{},[95,40999,1540],{},[1531,41001,41002,41008],{},[48,41003,41004,41006],{},[51,41005,19878],{},[95,41007,1540],{},[51,41009,53],{},[1430,41011,1545],{},[95,41013,19903],{"stretchy":210},[1004,41015,41016,41018],{},[51,41017,14556],{},[51,41019,12271],{},[95,41021,1210],{"separator":62},[51,41023,53],{},[95,41025,19912],{"stretchy":210},[95,41027,434],{},[2074,41029,41030,41032],{},[91,41031,205],{},[51,41033,30542],{},[1004,41035,41036,41038],{},[51,41037,34614],{},[51,41039,1662],{},[95,41041,211],{"stretchy":210},[51,41043,53],{},[51,41045,4572],{"mathvariant":494},[1004,41047,41048,41050],{},[51,41049,53],{},[51,41051,12271],{},[95,41053,220],{"stretchy":210},[95,41055,1210],{"separator":62},[55,41057,41058],{"encoding":57},"x_{t+1}=\\arg\\min_x\\ \\langle g_t,x\\rangle+\\frac1\\eta D_f(x\\|x_t),",[31,41060,41062,41127,41253],{"className":41061,"ariaHidden":62},[61],[31,41063,41065,41069,41118,41121,41124],{"className":41064},[66],[31,41066],{"className":41067,"style":41068},[70],"height:0.6389em;vertical-align:-0.2083em;",[31,41070,41072,41075],{"className":41071},[75],[31,41073,53],{"className":41074},[75,76],[31,41076,41078],{"className":41077},[1039],[31,41079,41081,41110],{"className":41080},[1043,1044],[31,41082,41084,41107],{"className":41083},[1048],[31,41085,41087],{"className":41086,"style":2028},[1052],[31,41088,41089,41092],{"style":2807},[31,41090],{"className":41091,"style":1061},[1060],[31,41093,41095],{"className":41094},[1065,1066,1067,1068],[31,41096,41098,41101,41104],{"className":41097},[75,1068],[31,41099,12271],{"className":41100},[75,76,1068],[31,41102,434],{"className":41103},[124,1068],[31,41105,205],{"className":41106},[75,1068],[31,41108,1076],{"className":41109},[1075],[31,41111,41113],{"className":41112},[1048],[31,41114,41116],{"className":41115,"style":25082},[1052],[31,41117],{},[31,41119],{"className":41120,"style":293},[119],[31,41122,223],{"className":41123},[297],[31,41125],{"className":41126,"style":293},[119],[31,41128,41130,41134,41139,41142,41186,41189,41192,41232,41235,41238,41241,41244,41247,41250],{"className":41129},[66],[31,41131],{"className":41132,"style":41133},[70],"height:1.45em;vertical-align:-0.7em;",[31,41135,37341,41137],{"className":41136},[1460],[31,41138,14556],{"style":20180},[31,41140],{"className":41141,"style":1467},[119],[31,41143,41145],{"className":41144},[1460,1620],[31,41146,41148,41178],{"className":41147},[1043,1044],[31,41149,41151,41175],{"className":41150},[1048],[31,41152,41154,41165],{"className":41153,"style":19979},[1052],[31,41155,41156,41159],{"style":37361},[31,41157],{"className":41158,"style":1637},[1060],[31,41160,41162],{"className":41161},[1065,1066,1067,1068],[31,41163,53],{"className":41164},[75,76,1068],[31,41166,41167,41170],{"style":1649},[31,41168],{"className":41169,"style":1637},[1060],[31,41171,41172],{},[31,41173,19878],{"className":41174},[1460],[31,41176,1076],{"className":41177},[1075],[31,41179,41181],{"className":41180},[1048],[31,41182,41184],{"className":41183,"style":37390},[1052],[31,41185],{},[31,41187,1545],{"className":41188},[119],[31,41190,19903],{"className":41191},[264],[31,41193,41195,41198],{"className":41194},[75],[31,41196,14556],{"className":41197,"style":2666},[75,76],[31,41199,41201],{"className":41200},[1039],[31,41202,41204,41224],{"className":41203},[1043,1044],[31,41205,41207,41221],{"className":41206},[1048],[31,41208,41210],{"className":41209,"style":19220},[1052],[31,41211,41212,41215],{"style":5504},[31,41213],{"className":41214,"style":1061},[1060],[31,41216,41218],{"className":41217},[1065,1066,1067,1068],[31,41219,12271],{"className":41220},[75,76,1068],[31,41222,1076],{"className":41223},[1075],[31,41225,41227],{"className":41226},[1048],[31,41228,41230],{"className":41229,"style":2050},[1052],[31,41231],{},[31,41233,1210],{"className":41234},[1391],[31,41236],{"className":41237,"style":1467},[119],[31,41239,53],{"className":41240},[75,76],[31,41242,19912],{"className":41243},[289],[31,41245],{"className":41246,"style":120},[119],[31,41248,434],{"className":41249},[124],[31,41251],{"className":41252,"style":120},[119],[31,41254,41256,41260,41323,41363,41366,41369,41372,41412,41415],{"className":41255},[66],[31,41257],{"className":41258,"style":41259},[70],"height:2.2019em;vertical-align:-0.8804em;",[31,41261,41263,41266,41320],{"className":41262},[75],[31,41264],{"className":41265},[264,2139],[31,41267,41269],{"className":41268},[2074],[31,41270,41272,41311],{"className":41271},[1043,1044],[31,41273,41275,41308],{"className":41274},[1048],[31,41276,41278,41289,41297],{"className":41277,"style":26376},[1052],[31,41279,41280,41283],{"style":5090},[31,41281],{"className":41282,"style":1637},[1060],[31,41284,41286],{"className":41285},[75],[31,41287,30542],{"className":41288,"style":2666},[75,76],[31,41290,41291,41294],{"style":2170},[31,41292],{"className":41293,"style":1637},[1060],[31,41295],{"className":41296,"style":2178},[2177],[31,41298,41299,41302],{"style":5110},[31,41300],{"className":41301,"style":1637},[1060],[31,41303,41305],{"className":41304},[75],[31,41306,205],{"className":41307},[75],[31,41309,1076],{"className":41310},[1075],[31,41312,41314],{"className":41313},[1048],[31,41315,41318],{"className":41316,"style":41317},[1052],"height:0.8804em;",[31,41319],{},[31,41321],{"className":41322},[289,2139],[31,41324,41326,41329],{"className":41325},[75],[31,41327,34614],{"className":41328,"style":19759},[75,76],[31,41330,41332],{"className":41331},[1039],[31,41333,41335,41355],{"className":41334},[1043,1044],[31,41336,41338,41352],{"className":41337},[1048],[31,41339,41341],{"className":41340,"style":1053},[1052],[31,41342,41343,41346],{"style":26726},[31,41344],{"className":41345,"style":1061},[1060],[31,41347,41349],{"className":41348},[1065,1066,1067,1068],[31,41350,1662],{"className":41351,"style":35196},[75,76,1068],[31,41353,1076],{"className":41354},[1075],[31,41356,41358],{"className":41357},[1048],[31,41359,41361],{"className":41360,"style":1083},[1052],[31,41362],{},[31,41364,211],{"className":41365},[264],[31,41367,53],{"className":41368},[75,76],[31,41370,4572],{"className":41371},[75],[31,41373,41375,41378],{"className":41374},[75],[31,41376,53],{"className":41377},[75,76],[31,41379,41381],{"className":41380},[1039],[31,41382,41384,41404],{"className":41383},[1043,1044],[31,41385,41387,41401],{"className":41386},[1048],[31,41388,41390],{"className":41389,"style":19220},[1052],[31,41391,41392,41395],{"style":2807},[31,41393],{"className":41394,"style":1061},[1060],[31,41396,41398],{"className":41397},[1065,1066,1067,1068],[31,41399,12271],{"className":41400},[75,76,1068],[31,41402,1076],{"className":41403},[1075],[31,41405,41407],{"className":41406},[1048],[31,41408,41410],{"className":41409,"style":2050},[1052],[31,41411],{},[31,41413,220],{"className":41414},[289],[31,41416,1210],{"className":41417},[1391],[11,41419,41420,41421,41449,41450,41617,41618,41646,41647,41892,41893,41921],{},"直觉是“走一步，但别离原来的位置太远”，而“太远”用 ",[31,41422,41424,41437],{"className":41423,"translate":35},[34],[31,41425,41427],{"className":41426},[39],[41,41428,41429],{"xmlns":43},[45,41430,41431,41435],{},[48,41432,41433],{},[51,41434,1662],{},[55,41436,1662],{"encoding":57},[31,41438,41440],{"className":41439,"ariaHidden":62},[61],[31,41441,41443,41446],{"className":41442},[66],[31,41444],{"className":41445,"style":4202},[70],[31,41447,1662],{"className":41448,"style":35196},[75,76]," 来量。取 ",[31,41451,41453,41487],{"className":41452,"translate":35},[34],[31,41454,41456],{"className":41455},[39],[41,41457,41458],{"xmlns":43},[45,41459,41460,41484],{},[48,41461,41462,41464,41466,41474,41476,41478],{},[51,41463,1662],{},[95,41465,223],{},[2071,41467,41468],{"scriptlevel":1977,"displaystyle":210},[2074,41469,41470,41472],{},[91,41471,205],{},[91,41473,152],{},[51,41475,4572],{"mathvariant":494},[51,41477,53],{},[1180,41479,41480,41482],{},[51,41481,4572],{"mathvariant":494},[91,41483,152],{},[55,41485,41486],{"encoding":57},"f=\\tfrac12\\|x\\|^2",[31,41488,41490,41508],{"className":41489,"ariaHidden":62},[61],[31,41491,41493,41496,41499,41502,41505],{"className":41492},[66],[31,41494],{"className":41495,"style":4202},[70],[31,41497,1662],{"className":41498,"style":35196},[75,76],[31,41500],{"className":41501,"style":293},[119],[31,41503,223],{"className":41504},[297],[31,41506],{"className":41507,"style":293},[119],[31,41509,41511,41514,41582,41585,41588],{"className":41510},[66],[31,41512],{"className":41513,"style":2132},[70],[31,41515,41517,41520,41579],{"className":41516},[75],[31,41518],{"className":41519},[264,2139],[31,41521,41523],{"className":41522},[2074],[31,41524,41526,41571],{"className":41525},[1043,1044],[31,41527,41529,41568],{"className":41528},[1048],[31,41530,41532,41546,41554],{"className":41531,"style":2152},[1052],[31,41533,41534,41537],{"style":2155},[31,41535],{"className":41536,"style":1637},[1060],[31,41538,41540],{"className":41539},[1065,1066,1067,1068],[31,41541,41543],{"className":41542},[75,1068],[31,41544,152],{"className":41545},[75,1068],[31,41547,41548,41551],{"style":2170},[31,41549],{"className":41550,"style":1637},[1060],[31,41552],{"className":41553,"style":2178},[2177],[31,41555,41556,41559],{"style":2181},[31,41557],{"className":41558,"style":1637},[1060],[31,41560,41562],{"className":41561},[1065,1066,1067,1068],[31,41563,41565],{"className":41564},[75,1068],[31,41566,205],{"className":41567},[75,1068],[31,41569,1076],{"className":41570},[1075],[31,41572,41574],{"className":41573},[1048],[31,41575,41577],{"className":41576,"style":2203},[1052],[31,41578],{},[31,41580],{"className":41581},[289,2139],[31,41583,4572],{"className":41584},[75],[31,41586,53],{"className":41587},[75,76],[31,41589,41591,41594],{"className":41590},[75],[31,41592,4572],{"className":41593},[75],[31,41595,41597],{"className":41596},[1039],[31,41598,41600],{"className":41599},[1043],[31,41601,41603],{"className":41602},[1048],[31,41604,41606],{"className":41605,"style":5135},[1052],[31,41607,41608,41611],{"style":3631},[31,41609],{"className":41610,"style":1061},[1060],[31,41612,41614],{"className":41613},[1065,1066,1067,1068],[31,41615,152],{"className":41616},[75,1068]," 就是梯度下降；取 ",[31,41619,41621,41634],{"className":41620,"translate":35},[34],[31,41622,41624],{"className":41623},[39],[41,41625,41626],{"xmlns":43},[45,41627,41628,41632],{},[48,41629,41630],{},[51,41631,1662],{},[55,41633,1662],{"encoding":57},[31,41635,41637],{"className":41636,"ariaHidden":62},[61],[31,41638,41640,41643],{"className":41639},[66],[31,41641],{"className":41642,"style":4202},[70],[31,41644,1662],{"className":41645,"style":35196},[75,76]," 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：",[31,41648,41650,41708],{"className":41649,"translate":35},[34],[31,41651,41653],{"className":41652},[39],[41,41654,41655],{"xmlns":43},[45,41656,41657,41705],{},[48,41658,41659,41671,41673,41679,41681,41683,41685,41687,41689,41691,41701,41703],{},[1004,41660,41661,41663],{},[51,41662,23798],{},[48,41664,41665,41667,41669],{},[51,41666,12271],{},[95,41668,434],{},[91,41670,205],{},[95,41672,223],{},[1004,41674,41675,41677],{},[51,41676,23798],{},[51,41678,12271],{},[95,41680,97],{},[51,41682,30542],{},[1430,41684,1432],{},[51,41686,15297],{},[95,41688,211],{"stretchy":210},[51,41690,23798],{},[1180,41692,41693,41695],{},[95,41694,220],{"stretchy":210},[48,41696,41697,41699],{},[95,41698,97],{},[91,41700,205],{},[51,41702,12145],{"mathvariant":494},[51,41704,4561],{},[55,41706,41707],{"encoding":57},"\\theta_{t+1}=\\theta_t-\\eta\\,G(\\theta)^{-1}\\nabla L",[31,41709,41711,41775,41830],{"className":41710,"ariaHidden":62},[61],[31,41712,41714,41717,41766,41769,41772],{"className":41713},[66],[31,41715],{"className":41716,"style":25023},[70],[31,41718,41720,41723],{"className":41719},[75],[31,41721,23798],{"className":41722,"style":19759},[75,76],[31,41724,41726],{"className":41725},[1039],[31,41727,41729,41758],{"className":41728},[1043,1044],[31,41730,41732,41755],{"className":41731},[1048],[31,41733,41735],{"className":41734,"style":2028},[1052],[31,41736,41737,41740],{"style":26726},[31,41738],{"className":41739,"style":1061},[1060],[31,41741,41743],{"className":41742},[1065,1066,1067,1068],[31,41744,41746,41749,41752],{"className":41745},[75,1068],[31,41747,12271],{"className":41748},[75,76,1068],[31,41750,434],{"className":41751},[124,1068],[31,41753,205],{"className":41754},[75,1068],[31,41756,1076],{"className":41757},[1075],[31,41759,41761],{"className":41760},[1048],[31,41762,41764],{"className":41763,"style":25082},[1052],[31,41765],{},[31,41767],{"className":41768,"style":293},[119],[31,41770,223],{"className":41771},[297],[31,41773],{"className":41774,"style":293},[119],[31,41776,41778,41781,41821,41824,41827],{"className":41777},[66],[31,41779],{"className":41780,"style":2008},[70],[31,41782,41784,41787],{"className":41783},[75],[31,41785,23798],{"className":41786,"style":19759},[75,76],[31,41788,41790],{"className":41789},[1039],[31,41791,41793,41813],{"className":41792},[1043,1044],[31,41794,41796,41810],{"className":41795},[1048],[31,41797,41799],{"className":41798,"style":19220},[1052],[31,41800,41801,41804],{"style":26726},[31,41802],{"className":41803,"style":1061},[1060],[31,41805,41807],{"className":41806},[1065,1066,1067,1068],[31,41808,12271],{"className":41809},[75,76,1068],[31,41811,1076],{"className":41812},[1075],[31,41814,41816],{"className":41815},[1048],[31,41817,41819],{"className":41818,"style":2050},[1052],[31,41820],{},[31,41822],{"className":41823,"style":120},[119],[31,41825,97],{"className":41826},[124],[31,41828],{"className":41829,"style":120},[119],[31,41831,41833,41836,41839,41842,41845,41848,41851,41886,41889],{"className":41832},[66],[31,41834],{"className":41835,"style":12035},[70],[31,41837,30542],{"className":41838,"style":2666},[75,76],[31,41840],{"className":41841,"style":1467},[119],[31,41843,15297],{"className":41844},[75,76],[31,41846,211],{"className":41847},[264],[31,41849,23798],{"className":41850,"style":19759},[75,76],[31,41852,41854,41857],{"className":41853},[289],[31,41855,220],{"className":41856},[289],[31,41858,41860],{"className":41859},[1039],[31,41861,41863],{"className":41862},[1043],[31,41864,41866],{"className":41865},[1048],[31,41867,41869],{"className":41868,"style":5135},[1052],[31,41870,41871,41874],{"style":3631},[31,41872],{"className":41873,"style":1061},[1060],[31,41875,41877],{"className":41876},[1065,1066,1067,1068],[31,41878,41880,41883],{"className":41879},[75,1068],[31,41881,97],{"className":41882},[75,1068],[31,41884,205],{"className":41885},[75,1068],[31,41887,12145],{"className":41888},[75],[31,41890,4561],{"className":41891},[75,76],"，其中 ",[31,41894,41896,41909],{"className":41895,"translate":35},[34],[31,41897,41899],{"className":41898},[39],[41,41900,41901],{"xmlns":43},[45,41902,41903,41907],{},[48,41904,41905],{},[51,41906,15297],{},[55,41908,15297],{"encoding":57},[31,41910,41912],{"className":41911,"ariaHidden":62},[61],[31,41913,41915,41918],{"className":41914},[66],[31,41916],{"className":41917,"style":945},[70],[31,41919,15297],{"className":41920},[75,76]," 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”。",[18,41923,41924],{"id":41924},"点态密度",[11,41926,41927,41928,42034,42035,42063],{},"手上有五个一维样本：",[31,41929,41931,41973],{"className":41930,"translate":35},[34],[31,41932,41934],{"className":41933},[39],[41,41935,41936],{"xmlns":43},[45,41937,41938,41970],{},[48,41939,41940,41942,41944,41946,41949,41951,41953,41956,41958,41960,41963,41965,41967],{},[91,41941,23247],{},[95,41943,1210],{"separator":62},[1430,41945,1545],{},[91,41947,41948],{},"1.0",[95,41950,1210],{"separator":62},[1430,41952,1545],{},[91,41954,41955],{},"1.1",[95,41957,1210],{"separator":62},[1430,41959,1545],{},[91,41961,41962],{},"5.0",[95,41964,1210],{"separator":62},[1430,41966,1545],{},[91,41968,41969],{},"5.1",[55,41971,41972],{"encoding":57},"0.9,\\ 1.0,\\ 1.1,\\ 5.0,\\ 5.1",[31,41974,41976],{"className":41975,"ariaHidden":62},[61],[31,41977,41979,41983,41986,41989,41992,41995,41998,42001,42004,42007,42010,42013,42016,42019,42022,42025,42028,42031],{"className":41978},[66],[31,41980],{"className":41981,"style":41982},[70],"height:0.8389em;vertical-align:-0.1944em;",[31,41984,23247],{"className":41985},[75],[31,41987,1210],{"className":41988},[1391],[31,41990,1545],{"className":41991},[119],[31,41993],{"className":41994,"style":1467},[119],[31,41996,41948],{"className":41997},[75],[31,41999,1210],{"className":42000},[1391],[31,42002,1545],{"className":42003},[119],[31,42005],{"className":42006,"style":1467},[119],[31,42008,41955],{"className":42009},[75],[31,42011,1210],{"className":42012},[1391],[31,42014,1545],{"className":42015},[119],[31,42017],{"className":42018,"style":1467},[119],[31,42020,41962],{"className":42021},[75],[31,42023,1210],{"className":42024},[1391],[31,42026,1545],{"className":42027},[119],[31,42029],{"className":42030,"style":1467},[119],[31,42032,41969],{"className":42033},[75],"。肉眼看出两簇，中间在 ",[31,42036,42038,42051],{"className":42037,"translate":35},[34],[31,42039,42041],{"className":42040},[39],[41,42042,42043],{"xmlns":43},[45,42044,42045,42049],{},[48,42046,42047],{},[91,42048,93],{},[55,42050,93],{"encoding":57},[31,42052,42054],{"className":42053,"ariaHidden":62},[61],[31,42055,42057,42060],{"className":42056},[66],[31,42058],{"className":42059,"style":326},[70],[31,42061,93],{"className":42062},[75]," 附近是空的。",[11,42065,42066,42067,42096,42097,42125,42126,42177,42178,42207,42208,42259,42260,42289,42290,42293],{},"现在用核密度估计去拟合，只用高斯核、只改带宽。带宽取 ",[31,42068,42070,42084],{"className":42069,"translate":35},[34],[31,42071,42073],{"className":42072},[39],[41,42074,42075],{"xmlns":43},[45,42076,42077,42082],{},[48,42078,42079],{},[91,42080,42081],{},"0.1",[55,42083,42081],{"encoding":57},[31,42085,42087],{"className":42086,"ariaHidden":62},[61],[31,42088,42090,42093],{"className":42089},[66],[31,42091],{"className":42092,"style":326},[70],[31,42094,42081],{"className":42095},[75],"，每个点都是一个尖峰，曲线在两个簇内部剧烈起伏；带宽取 ",[31,42098,42100,42113],{"className":42099,"translate":35},[34],[31,42101,42103],{"className":42102},[39],[41,42104,42105],{"xmlns":43},[45,42106,42107,42111],{},[48,42108,42109],{},[91,42110,7853],{},[55,42112,7853],{"encoding":57},[31,42114,42116],{"className":42115,"ariaHidden":62},[61],[31,42117,42119,42122],{"className":42118},[66],[31,42120],{"className":42121,"style":326},[70],[31,42123,7853],{"className":42124},[75],"，曲线被抹平。抹平到什么程度？算一下会发现，",[31,42127,42129,42147],{"className":42128,"translate":35},[34],[31,42130,42132],{"className":42131},[39],[41,42133,42134],{"xmlns":43},[45,42135,42136,42144],{},[48,42137,42138,42140,42142],{},[51,42139,53],{},[95,42141,223],{},[91,42143,93],{},[55,42145,42146],{"encoding":57},"x=3",[31,42148,42150,42168],{"className":42149,"ariaHidden":62},[61],[31,42151,42153,42156,42159,42162,42165],{"className":42152},[66],[31,42154],{"className":42155,"style":71},[70],[31,42157,53],{"className":42158},[75,76],[31,42160],{"className":42161,"style":293},[119],[31,42163,223],{"className":42164},[297],[31,42166],{"className":42167,"style":293},[119],[31,42169,42171,42174],{"className":42170},[66],[31,42172],{"className":42173,"style":326},[70],[31,42175,93],{"className":42176},[75]," 那个空洞处的密度大约是 ",[31,42179,42181,42195],{"className":42180,"translate":35},[34],[31,42182,42184],{"className":42183},[39],[41,42185,42186],{"xmlns":43},[45,42187,42188,42193],{},[48,42189,42190],{},[91,42191,42192],{},"0.073",[55,42194,42192],{"encoding":57},[31,42196,42198],{"className":42197,"ariaHidden":62},[61],[31,42199,42201,42204],{"className":42200},[66],[31,42202],{"className":42203,"style":326},[70],[31,42205,42192],{"className":42206},[75],"，而簇心 ",[31,42209,42211,42229],{"className":42210,"translate":35},[34],[31,42212,42214],{"className":42213},[39],[41,42215,42216],{"xmlns":43},[45,42217,42218,42226],{},[48,42219,42220,42222,42224],{},[51,42221,53],{},[95,42223,223],{},[91,42225,205],{},[55,42227,42228],{"encoding":57},"x=1",[31,42230,42232,42250],{"className":42231,"ariaHidden":62},[61],[31,42233,42235,42238,42241,42244,42247],{"className":42234},[66],[31,42236],{"className":42237,"style":71},[70],[31,42239,53],{"className":42240},[75,76],[31,42242],{"className":42243,"style":293},[119],[31,42245,223],{"className":42246},[297],[31,42248],{"className":42249,"style":293},[119],[31,42251,42253,42256],{"className":42252},[66],[31,42254],{"className":42255,"style":326},[70],[31,42257,205],{"className":42258},[75]," 处是 ",[31,42261,42263,42277],{"className":42262,"translate":35},[34],[31,42264,42266],{"className":42265},[39],[41,42267,42268],{"xmlns":43},[45,42269,42270,42275],{},[48,42271,42272],{},[91,42273,42274],{},"0.071",[55,42276,42274],{"encoding":57},[31,42278,42280],{"className":42279,"ariaHidden":62},[61],[31,42281,42283,42286],{"className":42282},[66],[31,42284],{"className":42285,"style":326},[70],[31,42287,42274],{"className":42288},[75],"——",[7246,42291,42292],{},"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。",[11,42295,42296,42297,42325,42326,42354,42355,42383],{},"这个例子里，带宽就是分辨率。",[31,42298,42300,42313],{"className":42299,"translate":35},[34],[31,42301,42303],{"className":42302},[39],[41,42304,42305],{"xmlns":43},[45,42306,42307,42311],{},[48,42308,42309],{},[51,42310,29501],{},[55,42312,29501],{"encoding":57},[31,42314,42316],{"className":42315,"ariaHidden":62},[61],[31,42317,42319,42322],{"className":42318},[66],[31,42320],{"className":42321,"style":1630},[70],[31,42323,29501],{"className":42324},[75,76]," 太小，方差爆炸；",[31,42327,42329,42342],{"className":42328,"translate":35},[34],[31,42330,42332],{"className":42331},[39],[41,42333,42334],{"xmlns":43},[45,42335,42336,42340],{},[48,42337,42338],{},[51,42339,29501],{},[55,42341,29501],{"encoding":57},[31,42343,42345],{"className":42344,"ariaHidden":62},[61],[31,42346,42348,42351],{"className":42347},[66],[31,42349],{"className":42350,"style":1630},[70],[31,42352,29501],{"className":42353},[75,76]," 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 ",[31,42356,42358,42371],{"className":42357,"translate":35},[34],[31,42359,42361],{"className":42360},[39],[41,42362,42363],{"xmlns":43},[45,42364,42365,42369],{},[48,42366,42367],{},[51,42368,29501],{},[55,42370,29501],{"encoding":57},[31,42372,42374],{"className":42373,"ariaHidden":62},[61],[31,42375,42377,42380],{"className":42376},[66],[31,42378],{"className":42379,"style":1630},[70],[31,42381,29501],{"className":42382},[75,76]," 其实是在调“我允许多大的类间间隙”。",[11,42385,42386],{},"形式上是",[31,42388,42390],{"className":42389,"translate":35},[411],[31,42391,42393,42508],{"className":42392,"translate":35},[34],[31,42394,42396],{"className":42395},[39],[41,42397,42398],{"xmlns":43,"display":420},[45,42399,42400,42505],{},[48,42401,42402,42414,42416,42418,42420,42422,42428,42443,42449,42451,42453,42455,42461,42463,42465,42467,42473,42475,42477,42479,42481,42491,42493,42495,42497,42499,42501,42503],{},[1004,42403,42404,42412],{},[42405,42406,42407,42409],"mover",{"accent":62},[51,42408,11],{},[95,42410,42411],{},"^",[51,42413,29501],{},[95,42415,211],{"stretchy":210},[51,42417,53],{},[95,42419,220],{"stretchy":210},[95,42421,223],{},[2074,42423,42424,42426],{},[91,42425,205],{},[51,42427,14559],{},[42429,42430,42431,42433,42441],"munderover",{},[95,42432,5625],{},[48,42434,42435,42437,42439],{},[51,42436,5391],{},[95,42438,223],{},[91,42440,205],{},[51,42442,14559],{},[1004,42444,42445,42447],{},[51,42446,4558],{},[51,42448,29501],{},[95,42450,211],{"stretchy":210},[51,42452,53],{},[95,42454,97],{},[1004,42456,42457,42459],{},[51,42458,53],{},[51,42460,5391],{},[95,42462,220],{"stretchy":210},[95,42464,1210],{"separator":62},[119,42466],{"width":20286},[1004,42468,42469,42471],{},[51,42470,4558],{},[51,42472,29501],{},[95,42474,211],{"stretchy":210},[51,42476,5324],{},[95,42478,220],{"stretchy":210},[95,42480,223],{},[1180,42482,42483,42485],{},[51,42484,29501],{},[48,42486,42487,42489],{},[95,42488,97],{},[51,42490,1435],{},[51,42492,4558],{},[95,42494,211],{"stretchy":210},[51,42496,5324],{},[51,42498,3768],{"mathvariant":494},[51,42500,29501],{},[95,42502,220],{"stretchy":210},[51,42504,495],{"mathvariant":494},[55,42506,42507],{"encoding":57},"\\hat p_h(x)=\\frac1n\\sum_{i=1}^nK_h(x-x_i),\\qquad K_h(u)=h^{-d}K(u\u002Fh).",[31,42509,42511,42618,42814,42930],{"className":42510,"ariaHidden":62},[61],[31,42512,42514,42517,42600,42603,42606,42609,42612,42615],{"className":42513},[66],[31,42515],{"className":42516,"style":260},[70],[31,42518,42520,42566],{"className":42519},[75],[31,42521,42524],{"className":42522},[75,42523],"accent",[31,42525,42527,42557],{"className":42526},[1043,1044],[31,42528,42530,42554],{"className":42529},[1048],[31,42531,42533,42541],{"className":42532,"style":1630},[1052],[31,42534,42535,42538],{"style":1649},[31,42536],{"className":42537,"style":1637},[1060],[31,42539,11],{"className":42540},[75,76],[31,42542,42543,42546],{"style":1649},[31,42544],{"className":42545,"style":1637},[1060],[31,42547,42551],{"className":42548,"style":42550},[42549],"accent-body","left:-0.1667em;",[31,42552,42411],{"className":42553},[75],[31,42555,1076],{"className":42556},[1075],[31,42558,42560],{"className":42559},[1048],[31,42561,42564],{"className":42562,"style":42563},[1052],"height:0.1944em;",[31,42565],{},[31,42567,42569],{"className":42568},[1039],[31,42570,42572,42592],{"className":42571},[1043,1044],[31,42573,42575,42589],{"className":42574},[1048],[31,42576,42578],{"className":42577,"style":1053},[1052],[31,42579,42580,42583],{"style":2807},[31,42581],{"className":42582,"style":1061},[1060],[31,42584,42586],{"className":42585},[1065,1066,1067,1068],[31,42587,29501],{"className":42588},[75,76,1068],[31,42590,1076],{"className":42591},[1075],[31,42593,42595],{"className":42594},[1048],[31,42596,42598],{"className":42597,"style":2050},[1052],[31,42599],{},[31,42601,211],{"className":42602},[264],[31,42604,53],{"className":42605},[75,76],[31,42607,220],{"className":42608},[289],[31,42610],{"className":42611,"style":293},[119],[31,42613,223],{"className":42614},[297],[31,42616],{"className":42617,"style":293},[119],[31,42619,42621,42625,42687,42690,42756,42759,42799,42802,42805,42808,42811],{"className":42620},[66],[31,42622],{"className":42623,"style":42624},[70],"height:2.9291em;vertical-align:-1.2777em;",[31,42626,42628,42631,42684],{"className":42627},[75],[31,42629],{"className":42630},[264,2139],[31,42632,42634],{"className":42633},[2074],[31,42635,42637,42676],{"className":42636},[1043,1044],[31,42638,42640,42673],{"className":42639},[1048],[31,42641,42643,42654,42662],{"className":42642,"style":26376},[1052],[31,42644,42645,42648],{"style":5090},[31,42646],{"className":42647,"style":1637},[1060],[31,42649,42651],{"className":42650},[75],[31,42652,14559],{"className":42653},[75,76],[31,42655,42656,42659],{"style":2170},[31,42657],{"className":42658,"style":1637},[1060],[31,42660],{"className":42661,"style":2178},[2177],[31,42663,42664,42667],{"style":5110},[31,42665],{"className":42666,"style":1637},[1060],[31,42668,42670],{"className":42669},[75],[31,42671,205],{"className":42672},[75],[31,42674,1076],{"className":42675},[1075],[31,42677,42679],{"className":42678},[1048],[31,42680,42682],{"className":42681,"style":5156},[1052],[31,42683],{},[31,42685],{"className":42686},[289,2139],[31,42688],{"className":42689,"style":1467},[119],[31,42691,42693],{"className":42692},[1460,1620],[31,42694,42696,42748],{"className":42695},[1043,1044],[31,42697,42699,42745],{"className":42698},[1048],[31,42700,42703,42723,42733],{"className":42701,"style":42702},[1052],"height:1.6514em;",[31,42704,42705,42708],{"style":5903},[31,42706],{"className":42707,"style":5907},[1060],[31,42709,42711],{"className":42710},[1065,1066,1067,1068],[31,42712,42714,42717,42720],{"className":42713},[75,1068],[31,42715,5391],{"className":42716},[75,76,1068],[31,42718,223],{"className":42719},[297,1068],[31,42721,205],{"className":42722},[75,1068],[31,42724,42725,42728],{"style":5919},[31,42726],{"className":42727,"style":5907},[1060],[31,42729,42730],{},[31,42731,5625],{"className":42732},[1460,1461,1687],[31,42734,42736,42739],{"style":42735},"top:-4.3em;margin-left:0em;",[31,42737],{"className":42738,"style":5907},[1060],[31,42740,42742],{"className":42741},[1065,1066,1067,1068],[31,42743,14559],{"className":42744},[75,76,1068],[31,42746,1076],{"className":42747},[1075],[31,42749,42751],{"className":42750},[1048],[31,42752,42754],{"className":42753,"style":6342},[1052],[31,42755],{},[31,42757],{"className":42758,"style":1467},[119],[31,42760,42762,42765],{"className":42761},[75],[31,42763,4558],{"className":42764,"style":8525},[75,76],[31,42766,42768],{"className":42767},[1039],[31,42769,42771,42791],{"className":42770},[1043,1044],[31,42772,42774,42788],{"className":42773},[1048],[31,42775,42777],{"className":42776,"style":1053},[1052],[31,42778,42779,42782],{"style":8540},[31,42780],{"className":42781,"style":1061},[1060],[31,42783,42785],{"className":42784},[1065,1066,1067,1068],[31,42786,29501],{"className":42787},[75,76,1068],[31,42789,1076],{"className":42790},[1075],[31,42792,42794],{"className":42793},[1048],[31,42795,42797],{"className":42796,"style":2050},[1052],[31,42798],{},[31,42800,211],{"className":42801},[264],[31,42803,53],{"className":42804},[75,76],[31,42806],{"className":42807,"style":120},[119],[31,42809,97],{"className":42810},[124],[31,42812],{"className":42813,"style":120},[119],[31,42815,42817,42820,42860,42863,42866,42869,42872,42912,42915,42918,42921,42924,42927],{"className":42816},[66],[31,42818],{"className":42819,"style":260},[70],[31,42821,42823,42826],{"className":42822},[75],[31,42824,53],{"className":42825},[75,76],[31,42827,42829],{"className":42828},[1039],[31,42830,42832,42852],{"className":42831},[1043,1044],[31,42833,42835,42849],{"className":42834},[1048],[31,42836,42838],{"className":42837,"style":5445},[1052],[31,42839,42840,42843],{"style":2807},[31,42841],{"className":42842,"style":1061},[1060],[31,42844,42846],{"className":42845},[1065,1066,1067,1068],[31,42847,5391],{"className":42848},[75,76,1068],[31,42850,1076],{"className":42851},[1075],[31,42853,42855],{"className":42854},[1048],[31,42856,42858],{"className":42857,"style":2050},[1052],[31,42859],{},[31,42861,220],{"className":42862},[289],[31,42864,1210],{"className":42865},[1391],[31,42867],{"className":42868,"style":20525},[119],[31,42870],{"className":42871,"style":1467},[119],[31,42873,42875,42878],{"className":42874},[75],[31,42876,4558],{"className":42877,"style":8525},[75,76],[31,42879,42881],{"className":42880},[1039],[31,42882,42884,42904],{"className":42883},[1043,1044],[31,42885,42887,42901],{"className":42886},[1048],[31,42888,42890],{"className":42889,"style":1053},[1052],[31,42891,42892,42895],{"style":8540},[31,42893],{"className":42894,"style":1061},[1060],[31,42896,42898],{"className":42897},[1065,1066,1067,1068],[31,42899,29501],{"className":42900},[75,76,1068],[31,42902,1076],{"className":42903},[1075],[31,42905,42907],{"className":42906},[1048],[31,42908,42910],{"className":42909,"style":2050},[1052],[31,42911],{},[31,42913,211],{"className":42914},[264],[31,42916,5324],{"className":42917},[75,76],[31,42919,220],{"className":42920},[289],[31,42922],{"className":42923,"style":293},[119],[31,42925,223],{"className":42926},[297],[31,42928],{"className":42929,"style":293},[119],[31,42931,42933,42936,42971,42974,42977,42980,42983,42986,42989],{"className":42932},[66],[31,42934],{"className":42935,"style":35441},[70],[31,42937,42939,42942],{"className":42938},[75],[31,42940,29501],{"className":42941},[75,76],[31,42943,42945],{"className":42944},[1039],[31,42946,42948],{"className":42947},[1043],[31,42949,42951],{"className":42950},[1048],[31,42952,42954],{"className":42953,"style":29642},[1052],[31,42955,42956,42959],{"style":1330},[31,42957],{"className":42958,"style":1061},[1060],[31,42960,42962],{"className":42961},[1065,1066,1067,1068],[31,42963,42965,42968],{"className":42964},[75,1068],[31,42966,97],{"className":42967},[75,1068],[31,42969,1435],{"className":42970},[75,76,1068],[31,42972,4558],{"className":42973,"style":8525},[75,76],[31,42975,211],{"className":42976},[264],[31,42978,5324],{"className":42979},[75,76],[31,42981,3768],{"className":42982},[75],[31,42984,29501],{"className":42985},[75,76],[31,42987,220],{"className":42988},[289],[31,42990,495],{"className":42991},[75],[11,42993,42994,42995,1138,43054,1138,43241,43395],{},"它的偏差和方差可以算出来（记 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n^{-1\u002F(d+4)}",[31,44236,44238,44282],{"className":44237,"ariaHidden":62},[61],[31,44239,44241,44244,44273,44276,44279],{"className":44240},[66],[31,44242],{"className":44243,"style":1630},[70],[31,44245,44247,44250],{"className":44246},[75],[31,44248,29501],{"className":44249},[75,76],[31,44251,44253],{"className":44252},[1039],[31,44254,44256],{"className":44255},[1043],[31,44257,44259],{"className":44258},[1048],[31,44260,44262],{"className":44261,"style":30719},[1052],[31,44263,44264,44267],{"style":3631},[31,44265],{"className":44266,"style":1061},[1060],[31,44268,44270],{"className":44269},[1065,1066,1067,1068],[31,44271,30648],{"className":44272},[124,1068],[31,44274],{"className":44275,"style":293},[119],[31,44277,44209],{"className":44278},[297],[31,44280],{"className":44281,"style":293},[119],[31,44283,44285,44288],{"className":44284},[66],[31,44286],{"className":44287,"style":22855},[70],[31,44289,44291,44294],{"className":44290},[75],[31,44292,14559],{"className":44293},[75,76],[31,44295,44297],{"className":44296},[1039],[31,44298,44300],{"className":44299},[1043],[31,44301,44303],{"className":44302},[1048],[31,44304,44306],{"className":44305,"style":22855},[1052],[31,44307,44308,44311],{"style":3631},[31,44309],{"className":44310,"style":1061},[1060],[31,44312,44314],{"className":44313},[1065,1066,1067,1068],[31,44315,44317,44320,44323,44326,44329,44332,44335],{"className":44316},[75,1068],[31,44318,97],{"className":44319},[75,1068],[31,44321,4042],{"className":44322},[75,1068],[31,44324,211],{"className":44325},[264,1068],[31,44327,1435],{"className":44328},[75,76,1068],[31,44330,434],{"className":44331},[124,1068],[31,44333,15416],{"className":44334},[75,1068],[31,44336,220],{"className":44337},[289,1068],"，此时 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n^{-4\u002F(d+4)}",[31,44388,44390,44412],{"className":44389,"ariaHidden":62},[61],[31,44391,44393,44396,44403,44406,44409],{"className":44392},[66],[31,44394],{"className":44395,"style":945},[70],[31,44397,44399],{"className":44398},[75],[31,44400,44402],{"className":44401},[75,2745],"MSE",[31,44404],{"className":44405,"style":293},[119],[31,44407,44209],{"className":44408},[297],[31,44410],{"className":44411,"style":293},[119],[31,44413,44415,44418],{"className":44414},[66],[31,44416],{"className":44417,"style":22855},[70],[31,44419,44421,44424],{"className":44420},[75],[31,44422,14559],{"className":44423},[75,76],[31,44425,44427],{"className":44426},[1039],[31,44428,44430],{"className":44429},[1043],[31,44431,44433],{"className":44432},[1048],[31,44434,44436],{"className":44435,"style":22855},[1052],[31,44437,44438,44441],{"style":3631},[31,44439],{"className":44440,"style":1061},[1060],[31,44442,44444],{"className":44443},[1065,1066,1067,1068],[31,44445,44447,44450,44454,44457,44460,44463,44466],{"className":44446},[75,1068],[31,44448,97],{"className":44449},[75,1068],[31,44451,44453],{"className":44452},[75,1068],"4\u002F",[31,44455,211],{"className":44456},[264,1068],[31,44458,1435],{"className":44459},[75,76,1068],[31,44461,434],{"className":44462},[124,1068],[31,44464,15416],{"className":44465},[75,1068],[31,44467,220],{"className":44468},[289,1068],[11,44470,44471,44522,44523,44597,44598,44522,44649,44724,44725,44810],{},[31,44472,44474,44492],{"className":44473,"translate":35},[34],[31,44475,44477],{"className":44476},[39],[41,44478,44479],{"xmlns":43},[45,44480,44481,44489],{},[48,44482,44483,44485,44487],{},[51,44484,1435],{},[95,44486,223],{},[91,44488,205],{},[55,44490,44491],{"encoding":57},"d=1",[31,44493,44495,44513],{"className":44494,"ariaHidden":62},[61],[31,44496,44498,44501,44504,44507,44510],{"className":44497},[66],[31,44499],{"className":44500,"style":1630},[70],[31,44502,1435],{"className":44503},[75,76],[31,44505],{"className":44506,"style":293},[119],[31,44508,223],{"className":44509},[297],[31,44511],{"className":44512,"style":293},[119],[31,44514,44516,44519],{"className":44515},[66],[31,44517],{"className":44518,"style":326},[70],[31,44520,205],{"className":44521},[75]," 时是 ",[31,44524,44526,44552],{"className":44525,"translate":35},[34],[31,44527,44529],{"className":44528},[39],[41,44530,44531],{"xmlns":43},[45,44532,44533,44549],{},[48,44534,44535],{},[1180,44536,44537,44539],{},[51,44538,14559],{},[48,44540,44541,44543,44545,44547],{},[95,44542,97],{},[91,44544,15416],{},[51,44546,3768],{"mathvariant":494},[91,44548,7853],{},[55,44550,44551],{"encoding":57},"n^{-4\u002F5}",[31,44553,44555],{"className":44554,"ariaHidden":62},[61],[31,44556,44558,44561],{"className":44557},[66],[31,44559],{"className":44560,"style":22855},[70],[31,44562,44564,44567],{"className":44563},[75],[31,44565,14559],{"className":44566},[75,76],[31,44568,44570],{"className":44569},[1039],[31,44571,44573],{"className":44572},[1043],[31,44574,44576],{"className":44575},[1048],[31,44577,44579],{"className":44578,"style":22855},[1052],[31,44580,44581,44584],{"style":3631},[31,44582],{"className":44583,"style":1061},[1060],[31,44585,44587],{"className":44586},[1065,1066,1067,1068],[31,44588,44590,44593],{"className":44589},[75,1068],[31,44591,97],{"className":44592},[75,1068],[31,44594,44596],{"className":44595},[75,1068],"4\u002F5","，很不错。",[31,44599,44601,44619],{"className":44600,"translate":35},[34],[31,44602,44604],{"className":44603},[39],[41,44605,44606],{"xmlns":43},[45,44607,44608,44616],{},[48,44609,44610,44612,44614],{},[51,44611,1435],{},[95,44613,223],{},[91,44615,19494],{},[55,44617,44618],{"encoding":57},"d=10",[31,44620,44622,44640],{"className":44621,"ariaHidden":62},[61],[31,44623,44625,44628,44631,44634,44637],{"className":44624},[66],[31,44626],{"className":44627,"style":1630},[70],[31,44629,1435],{"className":44630},[75,76],[31,44632],{"className":44633,"style":293},[119],[31,44635,223],{"className":44636},[297],[31,44638],{"className":44639,"style":293},[119],[31,44641,44643,44646],{"className":44642},[66],[31,44644],{"className":44645,"style":326},[70],[31,44647,19494],{"className":44648},[75],[31,44650,44652,44679],{"className":44651,"translate":35},[34],[31,44653,44655],{"className":44654},[39],[41,44656,44657],{"xmlns":43},[45,44658,44659,44676],{},[48,44660,44661],{},[1180,44662,44663,44665],{},[51,44664,14559],{},[48,44666,44667,44669,44671,44673],{},[95,44668,97],{},[91,44670,152],{},[51,44672,3768],{"mathvariant":494},[91,44674,44675],{},"7",[55,44677,44678],{"encoding":57},"n^{-2\u002F7}",[31,44680,44682],{"className":44681,"ariaHidden":62},[61],[31,44683,44685,44688],{"className":44684},[66],[31,44686],{"className":44687,"style":22855},[70],[31,44689,44691,44694],{"className":44690},[75],[31,44692,14559],{"className":44693},[75,76],[31,44695,44697],{"className":44696},[1039],[31,44698,44700],{"className":44699},[1043],[31,44701,44703],{"className":44702},[1048],[31,44704,44706],{"className":44705,"style":22855},[1052],[31,44707,44708,44711],{"style":3631},[31,44709],{"className":44710,"style":1061},[1060],[31,44712,44714],{"className":44713},[1065,1066,1067,1068],[31,44715,44717,44720],{"className":44716},[75,1068],[31,44718,97],{"className":44719},[75,1068],[31,44721,44723],{"className":44722},[75,1068],"2\u002F7","，想把误差减半，样本量要乘 ",[31,44726,44728,44751],{"className":44727,"translate":35},[34],[31,44729,44731],{"className":44730},[39],[41,44732,44733],{"xmlns":43},[45,44734,44735,44748],{},[48,44736,44737,44744,44746],{},[1180,44738,44739,44741],{},[91,44740,152],{},[91,44742,44743],{},"3.5",[95,44745,27223],{},[91,44747,485],{},[55,44749,44750],{"encoding":57},"2^{3.5}\\approx11",[31,44752,44754,44801],{"className":44753,"ariaHidden":62},[61],[31,44755,44757,44760,44792,44795,44798],{"className":44756},[66],[31,44758],{"className":44759,"style":5135},[70],[31,44761,44763,44766],{"className":44762},[75],[31,44764,152],{"className":44765},[75],[31,44767,44769],{"className":44768},[1039],[31,44770,44772],{"className":44771},[1043],[31,44773,44775],{"className":44774},[1048],[31,44776,44778],{"className":44777,"style":5135},[1052],[31,44779,44780,44783],{"style":3631},[31,44781],{"className":44782,"style":1061},[1060],[31,44784,44786],{"className":44785},[1065,1066,1067,1068],[31,44787,44789],{"className":44788},[75,1068],[31,44790,44743],{"className":44791},[75,1068],[31,44793],{"className":44794,"style":293},[119],[31,44796,27223],{"className":44797},[297],[31,44799],{"className":44800,"style":293},[119],[31,44802,44804,44807],{"className":44803},[66],[31,44805],{"className":44806,"style":326},[70],[31,44808,485],{"className":44809},[75],"。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”。",[11,44812,44813,44814,44913,44914,45006,45007,45079,45080,45181,45182,45325,45326,45397,45398,44522,45448,2372],{},"但 KDE 在高维还有一层跟带宽无关的麻烦。考虑标准正态 ",[31,44815,44817,44845],{"className":44816,"translate":35},[34],[31,44818,44820],{"className":44819},[39],[41,44821,44822],{"xmlns":43},[45,44823,44824,44842],{},[48,44825,44826,44828,44830,44832,44834,44840],{},[51,44827,4935],{"mathvariant":925},[95,44829,211],{"stretchy":210},[91,44831,1977],{},[95,44833,1210],{"separator":62},[1004,44835,44836,44838],{},[51,44837,34619],{},[51,44839,1435],{},[95,44841,220],{"stretchy":210},[55,44843,44844],{"encoding":57},"\\mathcal N(0,I_d)",[31,44846,44848],{"className":44847,"ariaHidden":62},[61],[31,44849,44851,44854,44857,44860,44863,44866,44869,44910],{"className":44850},[66],[31,44852],{"className":44853,"style":260},[70],[31,44855,4935],{"className":44856,"style":5004},[75,968],[31,44858,211],{"className":44859},[264],[31,44861,1977],{"className":44862},[75],[31,44864,1210],{"className":44865},[1391],[31,44867],{"className":44868,"style":1467},[119],[31,44870,44872,44875],{"className":44871},[75],[31,44873,34619],{"className":44874,"style":30311},[75,76],[31,44876,44878],{"className":44877},[1039],[31,44879,44881,44902],{"className":44880},[1043,1044],[31,44882,44884,44899],{"className":44883},[1048],[31,44885,44887],{"className":44886,"style":1053},[1052],[31,44888,44890,44893],{"style":44889},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[31,44891],{"className":44892,"style":1061},[1060],[31,44894,44896],{"className":44895},[1065,1066,1067,1068],[31,44897,1435],{"className":44898},[75,76,1068],[31,44900,1076],{"className":44901},[1075],[31,44903,44905],{"className":44904},[1048],[31,44906,44908],{"className":44907,"style":2050},[1052],[31,44909],{},[31,44911,220],{"className":44912},[289],"。众数在原点的密度是 ",[31,44915,44917,44949],{"className":44916,"translate":35},[34],[31,44918,44920],{"className":44919},[39],[41,44921,44922],{"xmlns":43},[45,44923,44924,44946],{},[48,44925,44926,44928,44930,44932],{},[95,44927,211],{"stretchy":210},[91,44929,152],{},[51,44931,2515],{},[1180,44933,44934,44936],{},[95,44935,220],{"stretchy":210},[48,44937,44938,44940,44942,44944],{},[95,44939,97],{},[51,44941,1435],{},[51,44943,3768],{"mathvariant":494},[91,44945,152],{},[55,44947,44948],{"encoding":57},"(2\\pi)^{-d\u002F2}",[31,44950,44952],{"className":44951,"ariaHidden":62},[61],[31,44953,44955,44959,44962,44965,44968],{"className":44954},[66],[31,44956],{"className":44957,"style":44958},[70],"height:1.138em;vertical-align:-0.25em;",[31,44960,211],{"className":44961},[264],[31,44963,152],{"className":44964},[75],[31,44966,2515],{"className":44967,"style":2666},[75,76],[31,44969,44971,44974],{"className":44970},[289],[31,44972,220],{"className":44973},[289],[31,44975,44977],{"className":44976},[1039],[31,44978,44980],{"className":44979},[1043],[31,44981,44983],{"className":44982},[1048],[31,44984,44986],{"className":44985,"style":22855},[1052],[31,44987,44988,44991],{"style":3631},[31,44989],{"className":44990,"style":1061},[1060],[31,44992,44994],{"className":44993},[1065,1066,1067,1068],[31,44995,44997,45000,45003],{"className":44996},[75,1068],[31,44998,97],{"className":44999},[75,1068],[31,45001,1435],{"className":45002},[75,76,1068],[31,45004,15357],{"className":45005},[75,1068],"；而一个“典型”样本落在半径 ",[31,45008,45010,45026],{"className":45009,"translate":35},[34],[31,45011,45013],{"className":45012},[39],[41,45014,45015],{"xmlns":43},[45,45016,45017,45023],{},[48,45018,45019],{},[26474,45020,45021],{},[51,45022,1435],{},[55,45024,45025],{"encoding":57},"\\sqrt d",[31,45027,45029],{"className":45028,"ariaHidden":62},[61],[31,45030,45032,45035],{"className":45031},[66],[31,45033],{"className":45034,"style":27567},[70],[31,45036,45038],{"className":45037},[75,26536],[31,45039,45041,45071],{"className":45040},[1043,1044],[31,45042,45044,45068],{"className":45043},[1048],[31,45045,45047,45056],{"className":45046,"style":27592},[1052],[31,45048,45050,45053],{"className":45049,"style":1649},[26550],[31,45051],{"className":45052,"style":1637},[1060],[31,45054,1435],{"className":45055,"style":27883},[75,76],[31,45057,45058,45061],{"style":27605},[31,45059],{"className":45060,"style":1637},[1060],[31,45062,45064],{"className":45063,"style":27612},[26592],[13341,45065,45066],{"xmlns":13343,"width":26596,"height":27615,"viewBox":27616,"preserveAspectRatio":26599},[13347,45067],{"d":27619},[31,45069,1076],{"className":45070},[1075],[31,45072,45074],{"className":45073},[1048],[31,45075,45077],{"className":45076,"style":27629},[1052],[31,45078],{}," 的球壳上（因为 ",[31,45081,45083,45113],{"className":45082,"translate":35},[34],[31,45084,45086],{"className":45085},[39],[41,45087,45088],{"xmlns":43},[45,45089,45090,45110],{},[48,45091,45092,45094,45096,45102,45104,45106,45108],{},[51,45093,4572],{"mathvariant":494},[51,45095,926],{},[1180,45097,45098,45100],{},[51,45099,4572],{"mathvariant":494},[91,45101,152],{},[51,45103,3768],{"mathvariant":494},[51,45105,1435],{},[95,45107,929],{},[91,45109,205],{},[55,45111,45112],{"encoding":57},"\\|X\\|^2\u002Fd\\to1",[31,45114,45116,45172],{"className":45115,"ariaHidden":62},[61],[31,45117,45119,45122,45125,45128,45157,45160,45163,45166,45169],{"className":45118},[66],[31,45120],{"className":45121,"style":12035},[70],[31,45123,4572],{"className":45124},[75],[31,45126,926],{"className":45127,"style":30311},[75,76],[31,45129,45131,45134],{"className":45130},[75],[31,45132,4572],{"className":45133},[75],[31,45135,45137],{"className":45136},[1039],[31,45138,45140],{"className":45139},[1043],[31,45141,45143],{"className":45142},[1048],[31,45144,45146],{"className":45145,"style":5135},[1052],[31,45147,45148,45151],{"style":3631},[31,45149],{"className":45150,"style":1061},[1060],[31,45152,45154],{"className":45153},[1065,1066,1067,1068],[31,45155,152],{"className":45156},[75,1068],[31,45158,3768],{"className":45159},[75],[31,45161,1435],{"className":45162},[75,76],[31,45164],{"className":45165,"style":293},[119],[31,45167,929],{"className":45168},[297],[31,45170],{"className":45171,"style":293},[119],[31,45173,45175,45178],{"className":45174},[66],[31,45176],{"className":45177,"style":326},[70],[31,45179,205],{"className":45180},[75],"），那里的密度是 ",[31,45183,45185,45231],{"className":45184,"translate":35},[34],[31,45186,45188],{"className":45187},[39],[41,45189,45190],{"xmlns":43},[45,45191,45192,45228],{},[48,45193,45194,45196,45198,45200,45214],{},[95,45195,211],{"stretchy":210},[91,45197,152],{},[51,45199,2515],{},[1180,45201,45202,45204],{},[95,45203,220],{"stretchy":210},[48,45205,45206,45208,45210,45212],{},[95,45207,97],{},[51,45209,1435],{},[51,45211,3768],{"mathvariant":494},[91,45213,152],{},[1180,45215,45216,45218],{},[51,45217,30042],{},[48,45219,45220,45222,45224,45226],{},[95,45221,97],{},[51,45223,1435],{},[51,45225,3768],{"mathvariant":494},[91,45227,152],{},[55,45229,45230],{"encoding":57},"(2\\pi)^{-d\u002F2}e^{-d\u002F2}",[31,45232,45234],{"className":45233,"ariaHidden":62},[61],[31,45235,45237,45240,45243,45246,45249,45287],{"className":45236},[66],[31,45238],{"className":45239,"style":44958},[70],[31,45241,211],{"className":45242},[264],[31,45244,152],{"className":45245},[75],[31,45247,2515],{"className":45248,"style":2666},[75,76],[31,45250,45252,45255],{"className":45251},[289],[31,45253,220],{"className":45254},[289],[31,45256,45258],{"className":45257},[1039],[31,45259,45261],{"className":45260},[1043],[31,45262,45264],{"className":45263},[1048],[31,45265,45267],{"className":45266,"style":22855},[1052],[31,45268,45269,45272],{"style":3631},[31,45270],{"className":45271,"style":1061},[1060],[31,45273,45275],{"className":45274},[1065,1066,1067,1068],[31,45276,45278,45281,45284],{"className":45277},[75,1068],[31,45279,97],{"className":45280},[75,1068],[31,45282,1435],{"className":45283},[75,76,1068],[31,45285,15357],{"className":45286},[75,1068],[31,45288,45290,45293],{"className":45289},[75],[31,45291,30042],{"className":45292},[75,76],[31,45294,45296],{"className":45295},[1039],[31,45297,45299],{"className":45298},[1043],[31,45300,45302],{"className":45301},[1048],[31,45303,45305],{"className":45304,"style":22855},[1052],[31,45306,45307,45310],{"style":3631},[31,45308],{"className":45309,"style":1061},[1060],[31,45311,45313],{"className":45312},[1065,1066,1067,1068],[31,45314,45316,45319,45322],{"className":45315},[75,1068],[31,45317,97],{"className":45318},[75,1068],[31,45320,1435],{"className":45321},[75,76,1068],[31,45323,15357],{"className":45324},[75,1068],"。两者差 ",[31,45327,45329,45353],{"className":45328,"translate":35},[34],[31,45330,45332],{"className":45331},[39],[41,45333,45334],{"xmlns":43},[45,45335,45336,45350],{},[48,45337,45338],{},[1180,45339,45340,45342],{},[51,45341,30042],{},[48,45343,45344,45346,45348],{},[51,45345,1435],{},[51,45347,3768],{"mathvariant":494},[91,45349,152],{},[55,45351,45352],{"encoding":57},"e^{d\u002F2}",[31,45354,45356],{"className":45355,"ariaHidden":62},[61],[31,45357,45359,45362],{"className":45358},[66],[31,45360],{"className":45361,"style":22855},[70],[31,45363,45365,45368],{"className":45364},[75],[31,45366,30042],{"className":45367},[75,76],[31,45369,45371],{"className":45370},[1039],[31,45372,45374],{"className":45373},[1043],[31,45375,45377],{"className":45376},[1048],[31,45378,45380],{"className":45379,"style":22855},[1052],[31,45381,45382,45385],{"style":3631},[31,45383],{"className":45384,"style":1061},[1060],[31,45386,45388],{"className":45387},[1065,1066,1067,1068],[31,45389,45391,45394],{"className":45390},[75,1068],[31,45392,1435],{"className":45393},[75,76,1068],[31,45395,15357],{"className":45396},[75,1068]," 倍，",[31,45399,45401,45418],{"className":45400,"translate":35},[34],[31,45402,45404],{"className":45403},[39],[41,45405,45406],{"xmlns":43},[45,45407,45408,45416],{},[48,45409,45410,45412,45414],{},[51,45411,1435],{},[95,45413,223],{},[91,45415,22616],{},[55,45417,22619],{"encoding":57},[31,45419,45421,45439],{"className":45420,"ariaHidden":62},[61],[31,45422,45424,45427,45430,45433,45436],{"className":45423},[66],[31,45425],{"className":45426,"style":1630},[70],[31,45428,1435],{"className":45429},[75,76],[31,45431],{"className":45432,"style":293},[119],[31,45434,223],{"className":45435},[297],[31,45437],{"className":45438,"style":293},[119],[31,45440,45442,45445],{"className":45441},[66],[31,45443],{"className":45444,"style":326},[70],[31,45446,22616],{"className":45447},[75],[31,45449,45451,45483],{"className":45450,"translate":35},[34],[31,45452,45454],{"className":45453},[39],[41,45455,45456],{"xmlns":43},[45,45457,45458,45480],{},[48,45459,45460,45467,45469,45471,45473],{},[1180,45461,45462,45464],{},[51,45463,30042],{},[91,45465,45466],{},"50",[95,45468,27223],{},[91,45470,7853],{},[95,45472,2528],{},[1180,45474,45475,45477],{},[91,45476,19494],{},[91,45478,45479],{},"21",[55,45481,45482],{"encoding":57},"e^{50}\\approx5\\times10^{21}",[31,45484,45486,45533,45551],{"className":45485,"ariaHidden":62},[61],[31,45487,45489,45492,45524,45527,45530],{"className":45488},[66],[31,45490],{"className":45491,"style":5135},[70],[31,45493,45495,45498],{"className":45494},[75],[31,45496,30042],{"className":45497},[75,76],[31,45499,45501],{"className":45500},[1039],[31,45502,45504],{"className":45503},[1043],[31,45505,45507],{"className":45506},[1048],[31,45508,45510],{"className":45509,"style":5135},[1052],[31,45511,45512,45515],{"style":3631},[31,45513],{"className":45514,"style":1061},[1060],[31,45516,45518],{"className":45517},[1065,1066,1067,1068],[31,45519,45521],{"className":45520},[75,1068],[31,45522,45466],{"className":45523},[75,1068],[31,45525],{"className":45526,"style":293},[119],[31,45528,27223],{"className":45529},[297],[31,45531],{"className":45532,"style":293},[119],[31,45534,45536,45539,45542,45545,45548],{"className":45535},[66],[31,45537],{"className":45538,"style":112},[70],[31,45540,7853],{"className":45541},[75],[31,45543],{"className":45544,"style":120},[119],[31,45546,2528],{"className":45547},[124],[31,45549],{"className":45550,"style":120},[119],[31,45552,45554,45557,45560],{"className":45553},[66],[31,45555],{"className":45556,"style":5135},[70],[31,45558,205],{"className":45559},[75],[31,45561,45563,45566],{"className":45562},[75],[31,45564,1977],{"className":45565},[75],[31,45567,45569],{"className":45568},[1039],[31,45570,45572],{"className":45571},[1043],[31,45573,45575],{"className":45574},[1048],[31,45576,45578],{"className":45577,"style":5135},[1052],[31,45579,45580,45583],{"style":3631},[31,45581],{"className":45582,"style":1061},[1060],[31,45584,45586],{"className":45585},[1065,1066,1067,1068],[31,45587,45589],{"className":45588},[75,1068],[31,45590,45479],{"className":45591},[75,1068],[11,45593,45594],{},"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。",[11,45596,45597],{},"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。",[11,45599,45600],{},"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。",[18,45602,45603],{"id":45603},"可逆变换",[11,45605,45606,45607,45611,45612,45668],{},"写采样代码时我总会用到同一招：",[45608,45609,45610],"code",{},"random()"," 只能给 ",[31,45613,45615,45638],{"className":45614,"translate":35},[34],[31,45616,45618],{"className":45617},[39],[41,45619,45620],{"xmlns":43},[45,45621,45622,45636],{},[48,45623,45624,45626,45628,45630,45632,45634],{},[51,45625,12818],{},[95,45627,211],{"stretchy":210},[91,45629,1977],{},[95,45631,1210],{"separator":62},[91,45633,205],{},[95,45635,220],{"stretchy":210},[55,45637,12890],{"encoding":57},[31,45639,45641],{"className":45640,"ariaHidden":62},[61],[31,45642,45644,45647,45650,45653,45656,45659,45662,45665],{"className":45643},[66],[31,45645],{"className":45646,"style":260},[70],[31,45648,12818],{"className":45649,"style":12844},[75,76],[31,45651,211],{"className":45652},[264],[31,45654,1977],{"className":45655},[75],[31,45657,1210],{"className":45658},[1391],[31,45660],{"className":45661,"style":1467},[119],[31,45663,205],{"className":45664},[75],[31,45666,220],{"className":45667},[289],"，可我想要指数分布，于是取",[31,45670,45672],{"className":45671,"translate":35},[411],[31,45673,45675,45716],{"className":45674,"translate":35},[34],[31,45676,45678],{"className":45677},[39],[41,45679,45680],{"xmlns":43,"display":420},[45,45681,45682,45713],{},[48,45683,45684,45686,45688,45690,45711],{},[51,45685,53],{},[95,45687,223],{},[95,45689,97],{},[2074,45691,45692,45708],{},[48,45693,45694,45696,45698,45700,45702,45704,45706],{},[51,45695,19939],{},[95,45697,1540],{},[95,45699,211],{"stretchy":210},[91,45701,205],{},[95,45703,97],{},[51,45705,5324],{},[95,45707,220],{"stretchy":210},[51,45709,45710],{},"λ",[51,45712,495],{"mathvariant":494},[55,45714,45715],{"encoding":57},"x=-\\frac{\\log(1-u)}{\\lambda}.",[31,45717,45719,45737],{"className":45718,"ariaHidden":62},[61],[31,45720,45722,45725,45728,45731,45734],{"className":45721},[66],[31,45723],{"className":45724,"style":71},[70],[31,45726,53],{"className":45727},[75,76],[31,45729],{"className":45730,"style":293},[119],[31,45732,223],{"className":45733},[297],[31,45735],{"className":45736,"style":293},[119],[31,45738,45740,45744,45747,45833],{"className":45739},[66],[31,45741],{"className":45742,"style":45743},[70],"height:2.113em;vertical-align:-0.686em;",[31,45745,97],{"className":45746},[75],[31,45748,45750,45753,45830],{"className":45749},[75],[31,45751],{"className":45752},[264,2139],[31,45754,45756],{"className":45755},[2074],[31,45757,45759,45822],{"className":45758},[1043,1044],[31,45760,45762,45819],{"className":45761},[1048],[31,45763,45766,45777,45785],{"className":45764,"style":45765},[1052],"height:1.427em;",[31,45767,45768,45771],{"style":5090},[31,45769],{"className":45770,"style":1637},[1060],[31,45772,45774],{"className":45773},[75],[31,45775,45710],{"className":45776},[75,76],[31,45778,45779,45782],{"style":2170},[31,45780],{"className":45781,"style":1637},[1060],[31,45783],{"className":45784,"style":2178},[2177],[31,45786,45787,45790],{"style":5110},[31,45788],{"className":45789,"style":1637},[1060],[31,45791,45793,45798,45801,45804,45807,45810,45813,45816],{"className":45792},[75],[31,45794,20177,45796],{"className":45795},[1460],[31,45797,14556],{"style":20180},[31,45799,211],{"className":45800},[264],[31,45802,205],{"className":45803},[75],[31,45805],{"className":45806,"style":120},[119],[31,45808,97],{"className":45809},[124],[31,45811],{"className":45812,"style":120},[119],[31,45814,5324],{"className":45815},[75,76],[31,45817,220],{"className":45818},[289],[31,45820,1076],{"className":45821},[1075],[31,45823,45825],{"className":45824},[1048],[31,45826,45828],{"className":45827,"style":5156},[1052],[31,45829],{},[31,45831],{"className":45832},[289,2139],[31,45834,495],{"className":45835},[75],[11,45837,45838,45839,4880,45957,45985],{},"这一个式子就是一个标准化流。反过来看，",[31,45840,45842,45874],{"className":45841,"translate":35},[34],[31,45843,45845],{"className":45844},[39],[41,45846,45847],{"xmlns":43},[45,45848,45849,45871],{},[48,45850,45851,45853,45855,45857,45859],{},[51,45852,5324],{},[95,45854,223],{},[91,45856,205],{},[95,45858,97],{},[1180,45860,45861,45863],{},[51,45862,30042],{},[48,45864,45865,45867,45869],{},[95,45866,97],{},[51,45868,45710],{},[51,45870,53],{},[55,45872,45873],{"encoding":57},"u=1-e^{-\\lambda x}",[31,45875,45877,45895,45913],{"className":45876,"ariaHidden":62},[61],[31,45878,45880,45883,45886,45889,45892],{"className":45879},[66],[31,45881],{"className":45882,"style":71},[70],[31,45884,5324],{"className":45885},[75,76],[31,45887],{"className":45888,"style":293},[119],[31,45890,223],{"className":45891},[297],[31,45893],{"className":45894,"style":293},[119],[31,45896,45898,45901,45904,45907,45910],{"className":45897},[66],[31,45899],{"className":45900,"style":112},[70],[31,45902,205],{"className":45903},[75],[31,45905],{"className":45906,"style":120},[119],[31,45908,97],{"className":45909},[124],[31,45911],{"className":45912,"style":120},[119],[31,45914,45916,45919],{"className":45915},[66],[31,45917],{"className":45918,"style":20863},[70],[31,45920,45922,45925],{"className":45921},[75],[31,45923,30042],{"className":45924},[75,76],[31,45926,45928],{"className":45927},[1039],[31,45929,45931],{"className":45930},[1043],[31,45932,45934],{"className":45933},[1048],[31,45935,45937],{"className":45936,"style":20863},[1052],[31,45938,45939,45942],{"style":3631},[31,45940],{"className":45941,"style":1061},[1060],[31,45943,45945],{"className":45944},[1065,1066,1067,1068],[31,45946,45948,45951,45954],{"className":45947},[75,1068],[31,45949,97],{"className":45950},[75,1068],[31,45952,45710],{"className":45953},[75,76,1068],[31,45955,53],{"className":45956},[75,76,1068],[31,45958,45960,45973],{"className":45959,"translate":35},[34],[31,45961,45963],{"className":45962},[39],[41,45964,45965],{"xmlns":43},[45,45966,45967,45971],{},[48,45968,45969],{},[51,45970,53],{},[55,45972,53],{"encoding":57},[31,45974,45976],{"className":45975,"ariaHidden":62},[61],[31,45977,45979,45982],{"className":45978},[66],[31,45980],{"className":45981,"style":71},[70],[31,45983,53],{"className":45984},[75,76]," 的 CDF，而密度按变量替换是",[31,45987,45989],{"className":45988,"translate":35},[411],[31,45990,45992,46072],{"className":45991,"translate":35},[34],[31,45993,45995],{"className":45994},[39],[41,45996,45997],{"xmlns":43,"display":420},[45,45998,45999,46069],{},[48,46000,46001,46007,46009,46011,46013,46015,46021,46023,46025,46027,46047,46049,46051,46053,46055,46067],{},[1004,46002,46003,46005],{},[51,46004,11],{},[51,46006,53],{},[95,46008,211],{"stretchy":210},[51,46010,53],{},[95,46012,220],{"stretchy":210},[95,46014,223],{},[1004,46016,46017,46019],{},[51,46018,11],{},[51,46020,5324],{},[95,46022,211],{"stretchy":210},[51,46024,5324],{},[95,46026,220],{"stretchy":210},[48,46028,46029,46031,46045],{},[95,46030,12680],{"fence":62},[2074,46032,46033,46039],{},[48,46034,46035,46037],{},[51,46036,1435],{},[51,46038,5324],{},[48,46040,46041,46043],{},[51,46042,1435],{},[51,46044,53],{},[95,46046,12680],{"fence":62},[95,46048,223],{},[91,46050,205],{},[95,46052,429],{},[51,46054,45710],{},[1180,46056,46057,46059],{},[51,46058,30042],{},[48,46060,46061,46063,46065],{},[95,46062,97],{},[51,46064,45710],{},[51,46066,53],{},[95,46068,1210],{"separator":62},[55,46070,46071],{"encoding":57},"p_x(x)=p_u(u)\\left|\\frac{du}{dx}\\right|=1\\cdot\\lambda e^{-\\lambda 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p_x(x)=\\log p_z(z)-\\log\\bigl|\\det J_f(z)\\bigr|.",[31,47138,47140,47212,47284],{"className":47139,"ariaHidden":62},[61],[31,47141,47143,47146,47151,47154,47194,47197,47200,47203,47206,47209],{"className":47142},[66],[31,47144],{"className":47145,"style":260},[70],[31,47147,20177,47149],{"className":47148},[1460],[31,47150,14556],{"style":20180},[31,47152],{"className":47153,"style":1467},[119],[31,47155,47157,47160],{"className":47156},[75],[31,47158,11],{"className":47159},[75,76],[31,47161,47163],{"className":47162},[1039],[31,47164,47166,47186],{"className":47165},[1043,1044],[31,47167,47169,47183],{"className":47168},[1048],[31,47170,47172],{"className":47171,"style":3230},[1052],[31,47173,47174,47177],{"style":2807},[31,47175],{"className":47176,"style":1061},[1060],[31,47178,47180],{"className":47179},[1065,1066,1067,1068],[31,47181,53],{"className":47182},[75,76,1068],[31,47184,1076],{"className":47185},[1075],[31,47187,47189],{"className":47188},[1048],[31,47190,47192],{"className":47191,"style":2050},[1052],[31,47193],{},[31,47195,211],{"className":47196},[264],[31,47198,53],{"className":47199},[75,76],[31,47201,220],{"className":47202},[289],[31,47204],{"className":47205,"style":293},[119],[31,47207,223],{"className":47208},[297],[31,47210],{"className":47211,"style":293},[119],[31,47213,47215,47218,47223,47226,47266,47269,47272,47275,47278,47281],{"className":47214},[66],[31,47216],{"className":47217,"style":260},[70],[31,47219,20177,47221],{"className":47220},[1460],[31,47222,14556],{"style":20180},[31,47224],{"className":47225,"style":1467},[119],[31,47227,47229,47232],{"className":47228},[75],[31,47230,11],{"className":47231},[75,76],[31,47233,47235],{"className":47234},[1039],[31,47236,47238,47258],{"className":47237},[1043,1044],[31,47239,47241,47255],{"className":47240},[1048],[31,47242,47244],{"className":47243,"style":3230},[1052],[31,47245,47246,47249],{"style":2807},[31,47247],{"className":47248,"style":1061},[1060],[31,47250,47252],{"className":47251},[1065,1066,1067,1068],[31,47253,46458],{"className":47254,"style":46500},[75,76,1068],[31,47256,1076],{"className":47257},[1075],[31,47259,47261],{"className":47260},[1048],[31,47262,47264],{"className":47263,"style":2050},[1052],[31,47265],{},[31,47267,211],{"className":47268},[264],[31,47270,46458],{"className":47271,"style":46500},[75,76],[31,47273,220],{"className":47274},[289],[31,47276],{"className":47277,"style":120},[119],[31,47279,97],{"className":47280},[124],[31,47282],{"className":47283,"style":120},[119],[31,47285,47287,47290,47295,47332,47335,47338,47380,47383,47386,47389,47426],{"className":47286},[66],[31,47288],{"className":47289,"style":4988},[70],[31,47291,20177,47293],{"className":47292},[1460],[31,47294,14556],{"style":20180},[31,47296,47298],{"className":47297},[264],[31,47299,47301],{"className":47300},[1307,13319],[31,47302,47304,47324],{"className":47303},[1043,1044],[31,47305,47307,47321],{"className":47306},[1048],[31,47308,47310],{"className":47309,"style":13329},[1052],[31,47311,47312,47315],{"style":13332},[31,47313],{"className":47314,"style":13336},[1060],[31,47316,47317],{"style":13339},[13341,47318,47319],{"xmlns":13343,"width":13344,"height":1178,"viewBox":13345},[13347,47320],{"d":13349},[31,47322,1076],{"className":47323},[1075],[31,47325,47327],{"className":47326},[1048],[31,47328,47330],{"className":47329,"style":13359},[1052],[31,47331],{},[31,47333,46659],{"className":47334},[1460],[31,47336],{"className":47337,"style":1467},[119],[31,47339,47341,47345],{"className":47340},[75],[31,47342,47121],{"className":47343,"style":47344},[75,76],"margin-right:0.0962em;",[31,47346,47348],{"className":47347},[1039],[31,47349,47351,47372],{"className":47350},[1043,1044],[31,47352,47354,47369],{"className":47353},[1048],[31,47355,47357],{"className":47356,"style":1053},[1052],[31,47358,47360,47363],{"style":47359},"top:-2.55em;margin-left:-0.0962em;margin-right:0.05em;",[31,47361],{"className":47362,"style":1061},[1060],[31,47364,47366],{"className":47365},[1065,1066,1067,1068],[31,47367,1662],{"className":47368,"style":35196},[75,76,1068],[31,47370,1076],{"className":47371},[1075],[31,47373,47375],{"className":47374},[1048],[31,47376,47378],{"className":47377,"style":1083},[1052],[31,47379],{},[31,47381,211],{"className":47382},[264],[31,47384,46458],{"className":47385,"style":46500},[75,76],[31,47387,220],{"className":47388},[289],[31,47390,47392],{"className":47391},[289],[31,47393,47395],{"className":47394},[1307,13319],[31,47396,47398,47418],{"className":47397},[1043,1044],[31,47399,47401,47415],{"className":47400},[1048],[31,47402,47404],{"className":47403,"style":13329},[1052],[31,47405,47406,47409],{"style":13332},[31,47407],{"className":47408,"style":13336},[1060],[31,47410,47411],{"style":13339},[13341,47412,47413],{"xmlns":13343,"width":13344,"height":1178,"viewBox":13345},[13347,47414],{"d":13349},[31,47416,1076],{"className":47417},[1075],[31,47419,47421],{"className":47420},[1048],[31,47422,47424],{"className":47423,"style":13359},[1052],[31,47425],{},[31,47427,495],{"className":47428},[75],[11,47430,27532,47431,47490],{},[31,47432,47434,47458],{"className":47433,"translate":35},[34],[31,47435,47437],{"className":47436},[39],[41,47438,47439],{"xmlns":43},[45,47440,47441,47455],{},[48,47442,47443,47445,47447,47449,47451,47453],{},[51,47444,19939],{},[95,47446,1540],{},[51,47448,12680],{"mathvariant":494},[51,47450,46659],{},[95,47452,1540],{},[51,47454,12680],{"mathvariant":494},[55,47456,47457],{"encoding":57},"\\log|\\det|",[31,47459,47461],{"className":47460,"ariaHidden":62},[61],[31,47462,47464,47467,47472,47475,47478,47481,47484,47487],{"className":47463},[66],[31,47465],{"className":47466,"style":260},[70],[31,47468,20177,47470],{"className":47469},[1460],[31,47471,14556],{"style":20180},[31,47473],{"className":47474,"style":1467},[119],[31,47476,12680],{"className":47477},[75],[31,47479],{"className":47480,"style":1467},[119],[31,47482,46659],{"className":47483},[1460],[31,47485],{"className":47486,"style":1467},[119],[31,47488,12680],{"className":47489},[75]," 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）。",[11,47492,47493,47494,15282,47573,1138,47664,47729,47730,47758,47759,1138,47838,47952,47953,2372],{},"一维的情形可以推到极致：任意一维分布都能写成逆 CDF。取 ",[31,47495,47497,47525],{"className":47496,"translate":35},[34],[31,47498,47500],{"className":47499},[39],[41,47501,47502],{"xmlns":43},[45,47503,47504,47522],{},[48,47505,47506,47508,47510,47512,47514,47516,47518,47520],{},[51,47507,46458],{},[95,47509,46520],{},[51,47511,12818],{},[95,47513,211],{"stretchy":210},[91,47515,1977],{},[95,47517,1210],{"separator":62},[91,47519,205],{},[95,47521,220],{"stretchy":210},[55,47523,47524],{"encoding":57},"z\\sim U(0,1)",[31,47526,47528,47546],{"className":47527,"ariaHidden":62},[61],[31,47529,47531,47534,47537,47540,47543],{"className":47530},[66],[31,47532],{"className":47533,"style":71},[70],[31,47535,46458],{"className":47536,"style":46500},[75,76],[31,47538],{"className":47539,"style":293},[119],[31,47541,46520],{"className":47542},[297],[31,47544],{"className":47545,"style":293},[119],[31,47547,47549,47552,47555,47558,47561,47564,47567,47570],{"className":47548},[66],[31,47550],{"className":47551,"style":260},[70],[31,47553,12818],{"className":47554,"style":12844},[75,76],[31,47556,211],{"className":47557},[264],[31,47559,1977],{"className":47560},[75],[31,47562,1210],{"className":47563},[1391],[31,47565],{"className":47566,"style":1467},[119],[31,47568,205],{"className":47569},[75],[31,47571,220],{"className":47572},[289],[31,47574,47576,47602],{"className":47575,"translate":35},[34],[31,47577,47579],{"className":47578},[39],[41,47580,47581],{"xmlns":43},[45,47582,47583,47599],{},[48,47584,47585,47587,47589],{},[51,47586,1662],{},[95,47588,223],{},[1180,47590,47591,47593],{},[51,47592,13114],{},[48,47594,47595,47597],{},[95,47596,97],{},[91,47598,205],{},[55,47600,47601],{"encoding":57},"f=F^{-1}",[31,47603,47605,47623],{"className":47604,"ariaHidden":62},[61],[31,47606,47608,47611,47614,47617,47620],{"className":47607},[66],[31,47609],{"className":47610,"style":4202},[70],[31,47612,1662],{"className":47613,"style":35196},[75,76],[31,47615],{"className":47616,"style":293},[119],[31,47618,223],{"className":47619},[297],[31,47621],{"className":47622,"style":293},[119],[31,47624,47626,47629],{"className":47625},[66],[31,47627],{"className":47628,"style":5135},[70],[31,47630,47632,47635],{"className":47631},[75],[31,47633,13114],{"className":47634,"style":949},[75,76],[31,47636,47638],{"className":47637},[1039],[31,47639,47641],{"className":47640},[1043],[31,47642,47644],{"className":47643},[1048],[31,47645,47647],{"className":47646,"style":5135},[1052],[31,47648,47649,47652],{"style":3631},[31,47650],{"className":47651,"style":1061},[1060],[31,47653,47655],{"className":47654},[1065,1066,1067,1068],[31,47656,47658,47661],{"className":47657},[75,1068],[31,47659,97],{"className":47660},[75,1068],[31,47662,205],{"className":47663},[75,1068],[31,47665,47667,47690],{"className":47666,"translate":35},[34],[31,47668,47670],{"className":47669},[39],[41,47671,47672],{"xmlns":43},[45,47673,47674,47688],{},[48,47675,47676,47678,47680,47682,47684,47686],{},[51,47677,53],{},[95,47679,223],{},[51,47681,1662],{},[95,47683,211],{"stretchy":210},[51,47685,46458],{},[95,47687,220],{"stretchy":210},[55,47689,46463],{"encoding":57},[31,47691,47693,47711],{"className":47692,"ariaHidden":62},[61],[31,47694,47696,47699,47702,47705,47708],{"className":47695},[66],[31,47697],{"className":47698,"style":71},[70],[31,47700,53],{"className":47701},[75,76],[31,47703],{"className":47704,"style":293},[119],[31,47706,223],{"className":47707},[297],[31,47709],{"className":47710,"style":293},[119],[31,47712,47714,47717,47720,47723,47726],{"className":47713},[66],[31,47715],{"className":47716,"style":260},[70],[31,47718,1662],{"className":47719,"style":35196},[75,76],[31,47721,211],{"className":47722},[264],[31,47724,46458],{"className":47725,"style":46500},[75,76],[31,47727,220],{"className":47728},[289]," 的分布就是 ",[31,47731,47733,47746],{"className":47732,"translate":35},[34],[31,47734,47736],{"className":47735},[39],[41,47737,47738],{"xmlns":43},[45,47739,47740,47744],{},[48,47741,47742],{},[51,47743,13114],{},[55,47745,13114],{"encoding":57},[31,47747,47749],{"className":47748,"ariaHidden":62},[61],[31,47750,47752,47755],{"className":47751},[66],[31,47753],{"className":47754,"style":945},[70],[31,47756,13114],{"className":47757,"style":949},[75,76],"，这是精确的，不是近似。取 ",[31,47760,47762,47790],{"className":47761,"translate":35},[34],[31,47763,47765],{"className":47764},[39],[41,47766,47767],{"xmlns":43},[45,47768,47769,47787],{},[48,47770,47771,47773,47775,47777,47779,47781,47783,47785],{},[51,47772,46458],{},[95,47774,46520],{},[51,47776,4935],{"mathvariant":925},[95,47778,211],{"stretchy":210},[91,47780,1977],{},[95,47782,1210],{"separator":62},[91,47784,205],{},[95,47786,220],{"stretchy":210},[55,47788,47789],{"encoding":57},"z\\sim\\mathcal N(0,1)",[31,47791,47793,47811],{"className":47792,"ariaHidden":62},[61],[31,47794,47796,47799,47802,47805,47808],{"className":47795},[66],[31,47797],{"className":47798,"style":71},[70],[31,47800,46458],{"className":47801,"style":46500},[75,76],[31,47803],{"className":47804,"style":293},[119],[31,47806,46520],{"className":47807},[297],[31,47809],{"className":47810,"style":293},[119],[31,47812,47814,47817,47820,47823,47826,47829,47832,47835],{"className":47813},[66],[31,47815],{"className":47816,"style":260},[70],[31,47818,4935],{"className":47819,"style":5004},[75,968],[31,47821,211],{"className":47822},[264],[31,47824,1977],{"className":47825},[75],[31,47827,1210],{"className":47828},[1391],[31,47830],{"className":47831,"style":1467},[119],[31,47833,205],{"className":47834},[75],[31,47836,220],{"className":47837},[289],[31,47839,47841,47872],{"className":47840,"translate":35},[34],[31,47842,47844],{"className":47843},[39],[41,47845,47846],{"xmlns":43},[45,47847,47848,47869],{},[48,47849,47850,47852,47854,47864,47866],{},[51,47851,1662],{},[95,47853,223],{},[1180,47855,47856,47858],{},[51,47857,13114],{},[48,47859,47860,47862],{},[95,47861,97],{},[91,47863,205],{},[95,47865,16209],{},[51,47867,47868],{"mathvariant":494},"Φ",[55,47870,47871],{"encoding":57},"f=F^{-1}\\circ\\Phi",[31,47873,47875,47893,47943],{"className":47874,"ariaHidden":62},[61],[31,47876,47878,47881,47884,47887,47890],{"className":47877},[66],[31,47879],{"className":47880,"style":4202},[70],[31,47882,1662],{"className":47883,"style":35196},[75,76],[31,47885],{"className":47886,"style":293},[119],[31,47888,223],{"className":47889},[297],[31,47891],{"className":47892,"style":293},[119],[31,47894,47896,47899,47934,47937,47940],{"className":47895},[66],[31,47897],{"className":47898,"style":5135},[70],[31,47900,47902,47905],{"className":47901},[75],[31,47903,13114],{"className":47904,"style":949},[75,76],[31,47906,47908],{"className":47907},[1039],[31,47909,47911],{"className":47910},[1043],[31,47912,47914],{"className":47913},[1048],[31,47915,47917],{"className":47916,"style":5135},[1052],[31,47918,47919,47922],{"style":3631},[31,47920],{"className":47921,"style":1061},[1060],[31,47923,47925],{"className":47924},[1065,1066,1067,1068],[31,47926,47928,47931],{"className":47927},[75,1068],[31,47929,97],{"className":47930},[75,1068],[31,47932,205],{"className":47933},[75,1068],[31,47935],{"className":47936,"style":120},[119],[31,47938,16209],{"className":47939},[124],[31,47941],{"className":47942,"style":120},[119],[31,47944,47946,47949],{"className":47945},[66],[31,47947],{"className":47948,"style":945},[70],[31,47950,47868],{"className":47951},[75]," 就得到任意一维分布。上面那个指数分布的例子就是 ",[31,47954,47956,48006],{"className":47955,"translate":35},[34],[31,47957,47959],{"className":47958},[39],[41,47960,47961],{"xmlns":43},[45,47962,47963,48003],{},[48,47964,47965,47975,47977,47979,47981,47983,47985,47987,47989,47991,47993,47995,47997,47999,48001],{},[1180,47966,47967,47969],{},[51,47968,13114],{},[48,47970,47971,47973],{},[95,47972,97],{},[91,47974,205],{},[95,47976,211],{"stretchy":210},[51,47978,5324],{},[95,47980,220],{"stretchy":210},[95,47982,223],{},[95,47984,97],{},[51,47986,19939],{},[95,47988,1540],{},[95,47990,211],{"stretchy":210},[91,47992,205],{},[95,47994,97],{},[51,47996,5324],{},[95,47998,220],{"stretchy":210},[51,48000,3768],{"mathvariant":494},[51,48002,45710],{},[55,48004,48005],{"encoding":57},"F^{-1}(u)=-\\log(1-u)\u002F\\lambda",[31,48007,48009,48068,48100],{"className":48008,"ariaHidden":62},[61],[31,48010,48012,48015,48050,48053,48056,48059,48062,48065],{"className":48011},[66],[31,48013],{"className":48014,"style":12035},[70],[31,48016,48018,48021],{"className":48017},[75],[31,48019,13114],{"className":48020,"style":949},[75,76],[31,48022,48024],{"className":48023},[1039],[31,48025,48027],{"className":48026},[1043],[31,48028,48030],{"className":48029},[1048],[31,48031,48033],{"className":48032,"style":5135},[1052],[31,48034,48035,48038],{"style":3631},[31,48036],{"className":48037,"style":1061},[1060],[31,48039,48041],{"className":48040},[1065,1066,1067,1068],[31,48042,48044,48047],{"className":48043},[75,1068],[31,48045,97],{"className":48046},[75,1068],[31,48048,205],{"className":48049},[75,1068],[31,48051,211],{"className":48052},[264],[31,48054,5324],{"className":48055},[75,76],[31,48057,220],{"className":48058},[289],[31,48060],{"className":48061,"style":293},[119],[31,48063,223],{"className":48064},[297],[31,48066],{"className":48067,"style":293},[119],[31,48069,48071,48074,48077,48080,48085,48088,48091,48094,48097],{"className":48070},[66],[31,48072],{"className":48073,"style":260},[70],[31,48075,97],{"className":48076},[75],[31,48078],{"className":48079,"style":1467},[119],[31,48081,20177,48083],{"className":48082},[1460],[31,48084,14556],{"style":20180},[31,48086,211],{"className":48087},[264],[31,48089,205],{"className":48090},[75],[31,48092],{"className":48093,"style":120},[119],[31,48095,97],{"className":48096},[124],[31,48098],{"className":48099,"style":120},[119],[31,48101,48103,48106,48109,48112,48115],{"className":48102},[66],[31,48104],{"className":48105,"style":260},[70],[31,48107,5324],{"className":48108},[75,76],[31,48110,220],{"className":48111},[289],[31,48113,3768],{"className":48114},[75],[31,48116,45710],{"className":48117},[75,76],[11,48119,48120,48121,48413,48414,48442,48443,48735,48736,48848,48849,48901,48902,48969,48970,48998,48999,49066],{},"举个能算的例子。",[31,48122,48124,48184],{"className":48123,"translate":35},[34],[31,48125,48127],{"className":48126},[39],[41,48128,48129],{"xmlns":43},[45,48130,48131,48181],{},[48,48132,48133,48135,48137,48139,48141,48143,48151,48153,48155,48157,48159,48161,48169,48171,48173,48175,48177,48179],{},[51,48134,11],{},[95,48136,211],{"stretchy":210},[51,48138,53],{},[95,48140,220],{"stretchy":210},[95,48142,223],{},[2071,48144,48145],{"scriptlevel":1977,"displaystyle":210},[2074,48146,48147,48149],{},[91,48148,205],{},[91,48150,152],{},[51,48152,8059],{},[95,48154,211],{"stretchy":210},[51,48156,53],{},[95,48158,220],{"stretchy":210},[95,48160,434],{},[2071,48162,48163],{"scriptlevel":1977,"displaystyle":210},[2074,48164,48165,48167],{},[91,48166,205],{},[91,48168,152],{},[51,48170,8059],{},[95,48172,211],{"stretchy":210},[51,48174,53],{},[95,48176,97],{},[91,48178,93],{},[95,48180,220],{"stretchy":210},[55,48182,48183],{"encoding":57},"p(x)=\\tfrac12\\varphi(x)+\\tfrac12\\varphi(x-3)",[31,48185,48187,48214,48309,48401],{"className":48186,"ariaHidden":62},[61],[31,48188,48190,48193,48196,48199,48202,48205,48208,48211],{"className":48189},[66],[31,48191],{"className":48192,"style":260},[70],[31,48194,11],{"className":48195},[75,76],[31,48197,211],{"className":48198},[264],[31,48200,53],{"className":48201},[75,76],[31,48203,220],{"className":48204},[289],[31,48206],{"className":48207,"style":293},[119],[31,48209,223],{"className":48210},[297],[31,48212],{"className":48213,"style":293},[119],[31,48215,48217,48220,48288,48291,48294,48297,48300,48303,48306],{"className":48216},[66],[31,48218],{"className":48219,"style":2132},[70],[31,48221,48223,48226,48285],{"className":48222},[75],[31,48224],{"className":48225},[264,2139],[31,48227,48229],{"className":48228},[2074],[31,48230,48232,48277],{"className":48231},[1043,1044],[31,48233,48235,48274],{"className":48234},[1048],[31,48236,48238,48252,48260],{"className":48237,"style":2152},[1052],[31,48239,48240,48243],{"style":2155},[31,48241],{"className":48242,"style":1637},[1060],[31,48244,48246],{"className":48245},[1065,1066,1067,1068],[31,48247,48249],{"className":48248},[75,1068],[31,48250,152],{"className":48251},[75,1068],[31,48253,48254,48257],{"style":2170},[31,48255],{"className":48256,"style":1637},[1060],[31,48258],{"className":48259,"style":2178},[2177],[31,48261,48262,48265],{"style":2181},[31,48263],{"className":48264,"style":1637},[1060],[31,48266,48268],{"className":48267},[1065,1066,1067,1068],[31,48269,48271],{"className":48270},[75,1068],[31,48272,205],{"className":48273},[75,1068],[31,48275,1076],{"className":48276},[1075],[31,48278,48280],{"className":48279},[1048],[31,48281,48283],{"className":48282,"style":2203},[1052],[31,48284],{},[31,48286],{"className":48287},[289,2139],[31,48289,8059],{"className":48290},[75,76],[31,48292,211],{"className":48293},[264],[31,48295,53],{"className":48296},[75,76],[31,48298,220],{"className":48299},[289],[31,48301],{"className":48302,"style":120},[119],[31,48304,434],{"className":48305},[124],[31,48307],{"className":48308,"style":120},[119],[31,48310,48312,48315,48383,48386,48389,48392,48395,48398],{"className":48311},[66],[31,48313],{"className":48314,"style":2132},[70],[31,48316,48318,48321,48380],{"className":48317},[75],[31,48319],{"className":48320},[264,2139],[31,48322,48324],{"className":48323},[2074],[31,48325,48327,48372],{"className":48326},[1043,1044],[31,48328,48330,48369],{"className":48329},[1048],[31,48331,48333,48347,48355],{"className":48332,"style":2152},[1052],[31,48334,48335,48338],{"style":2155},[31,48336],{"className":48337,"style":1637},[1060],[31,48339,48341],{"className":48340},[1065,1066,1067,1068],[31,48342,48344],{"className":48343},[75,1068],[31,48345,152],{"className":48346},[75,1068],[31,48348,48349,48352],{"style":2170},[31,48350],{"className":48351,"style":1637},[1060],[31,48353],{"className":48354,"style":2178},[2177],[31,48356,48357,48360],{"style":2181},[31,48358],{"className":48359,"style":1637},[1060],[31,48361,48363],{"className":48362},[1065,1066,1067,1068],[31,48364,48366],{"className":48365},[75,1068],[31,48367,205],{"className":48368},[75,1068],[31,48370,1076],{"className":48371},[1075],[31,48373,48375],{"className":48374},[1048],[31,48376,48378],{"className":48377,"style":2203},[1052],[31,48379],{},[31,48381],{"className":48382},[289,2139],[31,48384,8059],{"className":48385},[75,76],[31,48387,211],{"className":48388},[264],[31,48390,53],{"className":48391},[75,76],[31,48393],{"className":48394,"style":120},[119],[31,48396,97],{"className":48397},[124],[31,48399],{"className":48400,"style":120},[119],[31,48402,48404,48407,48410],{"className":48403},[66],[31,48405],{"className":48406,"style":260},[70],[31,48408,93],{"className":48409},[75],[31,48411,220],{"className":48412},[289],"，两个单位方差高斯，中心相距 ",[31,48415,48417,48430],{"className":48416,"translate":35},[34],[31,48418,48420],{"className":48419},[39],[41,48421,48422],{"xmlns":43},[45,48423,48424,48428],{},[48,48425,48426],{},[91,48427,93],{},[55,48429,93],{"encoding":57},[31,48431,48433],{"className":48432,"ariaHidden":62},[61],[31,48434,48436,48439],{"className":48435},[66],[31,48437],{"className":48438,"style":326},[70],[31,48440,93],{"className":48441},[75],"。它的 CDF 是 ",[31,48444,48446,48506],{"className":48445,"translate":35},[34],[31,48447,48449],{"className":48448},[39],[41,48450,48451],{"xmlns":43},[45,48452,48453,48503],{},[48,48454,48455,48457,48459,48461,48463,48465,48473,48475,48477,48479,48481,48483,48491,48493,48495,48497,48499,48501],{},[51,48456,13114],{},[95,48458,211],{"stretchy":210},[51,48460,53],{},[95,48462,220],{"stretchy":210},[95,48464,223],{},[2071,48466,48467],{"scriptlevel":1977,"displaystyle":210},[2074,48468,48469,48471],{},[91,48470,205],{},[91,48472,152],{},[51,48474,47868],{"mathvariant":494},[95,48476,211],{"stretchy":210},[51,48478,53],{},[95,48480,220],{"stretchy":210},[95,48482,434],{},[2071,48484,48485],{"scriptlevel":1977,"displaystyle":210},[2074,48486,48487,48489],{},[91,48488,205],{},[91,48490,152],{},[51,48492,47868],{"mathvariant":494},[95,48494,211],{"stretchy":210},[51,48496,53],{},[95,48498,97],{},[91,48500,93],{},[95,48502,220],{"stretchy":210},[55,48504,48505],{"encoding":57},"F(x)=\\tfrac12\\Phi(x)+\\tfrac12\\Phi(x-3)",[31,48507,48509,48536,48631,48723],{"className":48508,"ariaHidden":62},[61],[31,48510,48512,48515,48518,48521,48524,48527,48530,48533],{"className":48511},[66],[31,48513],{"className":48514,"style":260},[70],[31,48516,13114],{"className":48517,"style":949},[75,76],[31,48519,211],{"className":48520},[264],[31,48522,53],{"className":48523},[75,76],[31,48525,220],{"className":48526},[289],[31,48528],{"className":48529,"style":293},[119],[31,48531,223],{"className":48532},[297],[31,48534],{"className":48535,"style":293},[119],[31,48537,48539,48542,48610,48613,48616,48619,48622,48625,48628],{"className":48538},[66],[31,48540],{"className":48541,"style":2132},[70],[31,48543,48545,48548,48607],{"className":48544},[75],[31,48546],{"className":48547},[264,2139],[31,48549,48551],{"className":48550},[2074],[31,48552,48554,48599],{"className":48553},[1043,1044],[31,48555,48557,48596],{"className":48556},[1048],[31,48558,48560,48574,48582],{"className":48559,"style":2152},[1052],[31,48561,48562,48565],{"style":2155},[31,48563],{"className":48564,"style":1637},[1060],[31,48566,48568],{"className":48567},[1065,1066,1067,1068],[31,48569,48571],{"className":48570},[75,1068],[31,48572,152],{"className":48573},[75,1068],[31,48575,48576,48579],{"style":2170},[31,48577],{"className":48578,"style":1637},[1060],[31,48580],{"className":48581,"style":2178},[2177],[31,48583,48584,48587],{"style":2181},[31,48585],{"className":48586,"style":1637},[1060],[31,48588,48590],{"className":48589},[1065,1066,1067,1068],[31,48591,48593],{"className":48592},[75,1068],[31,48594,205],{"className":48595},[75,1068],[31,48597,1076],{"className":48598},[1075],[31,48600,48602],{"className":48601},[1048],[31,48603,48605],{"className":48604,"style":2203},[1052],[31,48606],{},[31,48608],{"className":48609},[289,2139],[31,48611,47868],{"className":48612},[75],[31,48614,211],{"className":48615},[264],[31,48617,53],{"className":48618},[75,76],[31,48620,220],{"className":48621},[289],[31,48623],{"className":48624,"style":120},[119],[31,48626,434],{"className":48627},[124],[31,48629],{"className":48630,"style":120},[119],[31,48632,48634,48637,48705,48708,48711,48714,48717,48720],{"className":48633},[66],[31,48635],{"className":48636,"style":2132},[70],[31,48638,48640,48643,48702],{"className":48639},[75],[31,48641],{"className":48642},[264,2139],[31,48644,48646],{"className":48645},[2074],[31,48647,48649,48694],{"className":48648},[1043,1044],[31,48650,48652,48691],{"className":48651},[1048],[31,48653,48655,48669,48677],{"className":48654,"style":2152},[1052],[31,48656,48657,48660],{"style":2155},[31,48658],{"className":48659,"style":1637},[1060],[31,48661,48663],{"className":48662},[1065,1066,1067,1068],[31,48664,48666],{"className":48665},[75,1068],[31,48667,152],{"className":48668},[75,1068],[31,48670,48671,48674],{"style":2170},[31,48672],{"className":48673,"style":1637},[1060],[31,48675],{"className":48676,"style":2178},[2177],[31,48678,48679,48682],{"style":2181},[31,48680],{"className":48681,"style":1637},[1060],[31,48683,48685],{"className":48684},[1065,1066,1067,1068],[31,48686,48688],{"className":48687},[75,1068],[31,48689,205],{"className":48690},[75,1068],[31,48692,1076],{"className":48693},[1075],[31,48695,48697],{"className":48696},[1048],[31,48698,48700],{"className":48699,"style":2203},[1052],[31,48701],{},[31,48703],{"className":48704},[289,2139],[31,48706,47868],{"className":48707},[75],[31,48709,211],{"className":48710},[264],[31,48712,53],{"className":48713},[75,76],[31,48715],{"className":48716,"style":120},[119],[31,48718,97],{"className":48719},[124],[31,48721],{"className":48722,"style":120},[119],[31,48724,48726,48729,48732],{"className":48725},[66],[31,48727],{"className":48728,"style":260},[70],[31,48730,93],{"className":48731},[75],[31,48733,220],{"className":48734},[289],"，流就是 ",[31,48737,48739,48768],{"className":48738,"translate":35},[34],[31,48740,48742],{"className":48741},[39],[41,48743,48744],{"xmlns":43},[45,48745,48746,48766],{},[48,48747,48748,48750,48752,48762,48764],{},[51,48749,1662],{},[95,48751,223],{},[1180,48753,48754,48756],{},[51,48755,13114],{},[48,48757,48758,48760],{},[95,48759,97],{},[91,48761,205],{},[95,48763,16209],{},[51,48765,47868],{"mathvariant":494},[55,48767,47871],{"encoding":57},[31,48769,48771,48789,48839],{"className":48770,"ariaHidden":62},[61],[31,48772,48774,48777,48780,48783,48786],{"className":48773},[66],[31,48775],{"className":48776,"style":4202},[70],[31,48778,1662],{"className":48779,"style":35196},[75,76],[31,48781],{"className":48782,"style":293},[119],[31,48784,223],{"className":48785},[297],[31,48787],{"className":48788,"style":293},[119],[31,48790,48792,48795,48830,48833,48836],{"className":48791},[66],[31,48793],{"className":48794,"style":5135},[70],[31,48796,48798,48801],{"className":48797},[75],[31,48799,13114],{"className":48800,"style":949},[75,76],[31,48802,48804],{"className":48803},[1039],[31,48805,48807],{"className":48806},[1043],[31,48808,48810],{"className":48809},[1048],[31,48811,48813],{"className":48812,"style":5135},[1052],[31,48814,48815,48818],{"style":3631},[31,48816],{"className":48817,"style":1061},[1060],[31,48819,48821],{"className":48820},[1065,1066,1067,1068],[31,48822,48824,48827],{"className":48823},[75,1068],[31,48825,97],{"className":48826},[75,1068],[31,48828,205],{"className":48829},[75,1068],[31,48831],{"className":48832,"style":120},[119],[31,48834,16209],{"className":48835},[124],[31,48837],{"className":48838,"style":120},[119],[31,48840,48842,48845],{"className":48841},[66],[31,48843],{"className":48844,"style":945},[70],[31,48846,47868],{"className":48847},[75],"。两峰之间 ",[31,48850,48852,48871],{"className":48851,"translate":35},[34],[31,48853,48855],{"className":48854},[39],[41,48856,48857],{"xmlns":43},[45,48858,48859,48868],{},[48,48860,48861,48863,48865],{},[51,48862,53],{},[95,48864,223],{},[91,48866,48867],{},"1.5",[55,48869,48870],{"encoding":57},"x=1.5",[31,48872,48874,48892],{"className":48873,"ariaHidden":62},[61],[31,48875,48877,48880,48883,48886,48889],{"className":48876},[66],[31,48878],{"className":48879,"style":71},[70],[31,48881,53],{"className":48882},[75,76],[31,48884],{"className":48885,"style":293},[119],[31,48887,223],{"className":48888},[297],[31,48890],{"className":48891,"style":293},[119],[31,48893,48895,48898],{"className":48894},[66],[31,48896],{"className":48897,"style":326},[70],[31,48899,48867],{"className":48900},[75]," 处的真实密度是 ",[31,48903,48905,48930],{"className":48904,"translate":35},[34],[31,48906,48908],{"className":48907},[39],[41,48909,48910],{"xmlns":43},[45,48911,48912,48927],{},[48,48913,48914,48916,48918,48920,48922,48924],{},[51,48915,8059],{},[95,48917,211],{"stretchy":210},[91,48919,48867],{},[95,48921,220],{"stretchy":210},[95,48923,27223],{},[91,48925,48926],{},"0.1295",[55,48928,48929],{"encoding":57},"\\varphi(1.5)\\approx0.1295",[31,48931,48933,48960],{"className":48932,"ariaHidden":62},[61],[31,48934,48936,48939,48942,48945,48948,48951,48954,48957],{"className":48935},[66],[31,48937],{"className":48938,"style":260},[70],[31,48940,8059],{"className":48941},[75,76],[31,48943,211],{"className":48944},[264],[31,48946,48867],{"className":48947},[75],[31,48949,220],{"className":48950},[289],[31,48952],{"className":48953,"style":293},[119],[31,48955,27223],{"className":48956},[297],[31,48958],{"className":48959,"style":293},[119],[31,48961,48963,48966],{"className":48962},[66],[31,48964],{"className":48965,"style":326},[70],[31,48967,48926],{"className":48968},[75],"；如果错当成单个高斯（中心在 ",[31,48971,48973,48986],{"className":48972,"translate":35},[34],[31,48974,48976],{"className":48975},[39],[41,48977,48978],{"xmlns":43},[45,48979,48980,48984],{},[48,48981,48982],{},[91,48983,48867],{},[55,48985,48867],{"encoding":57},[31,48987,48989],{"className":48988,"ariaHidden":62},[61],[31,48990,48992,48995],{"className":48991},[66],[31,48993],{"className":48994,"style":326},[70],[31,48996,48867],{"className":48997},[75],"、单峰），会给出 ",[31,49000,49002,49027],{"className":49001,"translate":35},[34],[31,49003,49005],{"className":49004},[39],[41,49006,49007],{"xmlns":43},[45,49008,49009,49024],{},[48,49010,49011,49013,49015,49017,49019,49021],{},[51,49012,8059],{},[95,49014,211],{"stretchy":210},[91,49016,1977],{},[95,49018,220],{"stretchy":210},[95,49020,27223],{},[91,49022,49023],{},"0.3989",[55,49025,49026],{"encoding":57},"\\varphi(0)\\approx0.3989",[31,49028,49030,49057],{"className":49029,"ariaHidden":62},[61],[31,49031,49033,49036,49039,49042,49045,49048,49051,49054],{"className":49032},[66],[31,49034],{"className":49035,"style":260},[70],[31,49037,8059],{"className":49038},[75,76],[31,49040,211],{"className":49041},[264],[31,49043,1977],{"className":49044},[75],[31,49046,220],{"className":49047},[289],[31,49049],{"className":49050,"style":293},[119],[31,49052,27223],{"className":49053},[297],[31,49055],{"className":49056,"style":293},[119],[31,49058,49060,49063],{"className":49059},[66],[31,49061],{"className":49062,"style":326},[70],[31,49064,49023],{"className":49065},[75],"，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。",[11,49068,49069,49070,49169],{},"连续情形就是 Neural ODE：",[31,49071,49073,49109],{"className":49072,"translate":35},[34],[31,49074,49076],{"className":49075},[39],[41,49077,49078],{"xmlns":43},[45,49079,49080,49106],{},[48,49081,49082,49084,49086,49088,49090,49092,49094,49096,49098,49100,49102,49104],{},[51,49083,1435],{},[51,49085,46458],{},[51,49087,3768],{"mathvariant":494},[51,49089,1435],{},[51,49091,12271],{},[95,49093,223],{},[51,49095,5354],{},[95,49097,211],{"stretchy":210},[51,49099,46458],{},[95,49101,1210],{"separator":62},[51,49103,12271],{},[95,49105,220],{"stretchy":210},[55,49107,49108],{"encoding":57},"dz\u002Fdt=v(z,t)",[31,49110,49112,49142],{"className":49111,"ariaHidden":62},[61],[31,49113,49115,49118,49121,49124,49127,49130,49133,49136,49139],{"className":49114},[66],[31,49116],{"className":49117,"style":260},[70],[31,49119,1435],{"className":49120},[75,76],[31,49122,46458],{"className":49123,"style":46500},[75,76],[31,49125,3768],{"className":49126},[75],[31,49128,1435],{"className":49129},[75,76],[31,49131,12271],{"className":49132},[75,76],[31,49134],{"className":49135,"style":293},[119],[31,49137,223],{"className":49138},[297],[31,49140],{"className":49141,"style":293},[119],[31,49143,49145,49148,49151,49154,49157,49160,49163,49166],{"className":49144},[66],[31,49146],{"className":49147,"style":260},[70],[31,49149,5354],{"className":49150,"style":2666},[75,76],[31,49152,211],{"className":49153},[264],[31,49155,46458],{"className":49156,"style":46500},[75,76],[31,49158,1210],{"className":49159},[1391],[31,49161],{"className":49162,"style":1467},[119],[31,49164,12271],{"className":49165},[75,76],[31,49167,220],{"className":49168},[289],"，密度沿轨迹演化，",[31,49171,49173],{"className":49172,"translate":35},[411],[31,49174,49176,49248],{"className":49175,"translate":35},[34],[31,49177,49179],{"className":49178},[39],[41,49180,49181],{"xmlns":43,"display":420},[45,49182,49183,49245],{},[48,49184,49185,49195,49197,49199,49205,49207,49213,49215,49217,49219,49225,49227,49241,49243],{},[2074,49186,49187,49189],{},[51,49188,1435],{},[48,49190,49191,49193],{},[51,49192,1435],{},[51,49194,12271],{},[51,49196,19939],{},[95,49198,1540],{},[1004,49200,49201,49203],{},[51,49202,11],{},[51,49204,12271],{},[95,49206,211],{"stretchy":210},[1004,49208,49209,49211],{},[51,49210,46458],{},[51,49212,12271],{},[95,49214,220],{"stretchy":210},[95,49216,223],{},[95,49218,97],{},[48,49220,49221,49223],{},[51,49222,12271],{"mathvariant":494},[51,49224,2549],{"mathvariant":494},[95,49226,211],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[2074,49228,49229,49235],{},[48,49230,49231,49233],{},[51,49232,23909],{"mathvariant":494},[51,49234,5354],{},[48,49236,49237,49239],{},[51,49238,23909],{"mathvariant":494},[51,49240,46458],{},[95,49242,220],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[95,49244,1210],{"separator":62},[55,49246,49247],{"encoding":57},"\\frac{d}{dt}\\log p_t(z_t)=-\\mathrm{tr}\\Bigl(\\frac{\\partial v}{\\partial z}\\Bigr),",[31,49249,49251,49430],{"className":49250,"ariaHidden":62},[61],[31,49252,49254,49258,49323,49326,49331,49334,49374,49377,49418,49421,49424,49427],{"className":49253},[66],[31,49255],{"className":49256,"style":49257},[70],"height:2.0574em;vertical-align:-0.686em;",[31,49259,49261,49264,49320],{"className":49260},[75],[31,49262],{"className":49263},[264,2139],[31,49265,49267],{"className":49266},[2074],[31,49268,49270,49312],{"className":49269},[1043,1044],[31,49271,49273,49309],{"className":49272},[1048],[31,49274,49276,49290,49298],{"className":49275,"style":46264},[1052],[31,49277,49278,49281],{"style":5090},[31,49279],{"className":49280,"style":1637},[1060],[31,49282,49284,49287],{"className":49283},[75],[31,49285,1435],{"className":49286},[75,76],[31,49288,12271],{"className":49289},[75,76],[31,49291,49292,49295],{"style":2170},[31,49293],{"className":49294,"style":1637},[1060],[31,49296],{"className":49297,"style":2178},[2177],[31,49299,49300,49303],{"style":5110},[31,49301],{"className":49302,"style":1637},[1060],[31,49304,49306],{"className":49305},[75],[31,49307,1435],{"className":49308},[75,76],[31,49310,1076],{"className":49311},[1075],[31,49313,49315],{"className":49314},[1048],[31,49316,49318],{"className":49317,"style":5156},[1052],[31,49319],{},[31,49321],{"className":49322},[289,2139],[31,49324],{"className":49325,"style":1467},[119],[31,49327,20177,49329],{"className":49328},[1460],[31,49330,14556],{"style":20180},[31,49332],{"className":49333,"style":1467},[119],[31,49335,49337,49340],{"className":49336},[75],[31,49338,11],{"className":49339},[75,76],[31,49341,49343],{"className":49342},[1039],[31,49344,49346,49366],{"className":49345},[1043,1044],[31,49347,49349,49363],{"className":49348},[1048],[31,49350,49352],{"className":49351,"style":19220},[1052],[31,49353,49354,49357],{"style":2807},[31,49355],{"className":49356,"style":1061},[1060],[31,49358,49360],{"className":49359},[1065,1066,1067,1068],[31,49361,12271],{"className":49362},[75,76,1068],[31,49364,1076],{"className":49365},[1075],[31,49367,49369],{"className":49368},[1048],[31,49370,49372],{"className":49371,"style":2050},[1052],[31,49373],{},[31,49375,211],{"className":49376},[264],[31,49378,49380,49383],{"className":49379},[75],[31,49381,46458],{"className":49382,"style":46500},[75,76],[31,49384,49386],{"className":49385},[1039],[31,49387,49389,49410],{"className":49388},[1043,1044],[31,49390,49392,49407],{"className":49391},[1048],[31,49393,49395],{"className":49394,"style":19220},[1052],[31,49396,49398,49401],{"style":49397},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[31,49399],{"className":49400,"style":1061},[1060],[31,49402,49404],{"className":49403},[1065,1066,1067,1068],[31,49405,12271],{"className":49406},[75,76,1068],[31,49408,1076],{"className":49409},[1075],[31,49411,49413],{"className":49412},[1048],[31,49414,49416],{"className":49415,"style":2050},[1052],[31,49417],{},[31,49419,220],{"className":49420},[289],[31,49422],{"className":49423,"style":293},[119],[31,49425,223],{"className":49426},[297],[31,49428],{"className":49429,"style":293},[119],[31,49431,49433,49436,49439,49445,49451,49519,49525],{"className":49432},[66],[31,49434],{"className":49435,"style":49257},[70],[31,49437,97],{"className":49438},[75],[31,49440,49442],{"className":49441},[75],[31,49443,16973],{"className":49444},[75,2745],[31,49446,49448],{"className":49447},[264],[31,49449,211],{"className":49450},[1307,2662],[31,49452,49454,49457,49516],{"className":49453},[75],[31,49455],{"className":49456},[264,2139],[31,49458,49460],{"className":49459},[2074],[31,49461,49463,49508],{"className":49462},[1043,1044],[31,49464,49466,49505],{"className":49465},[1048],[31,49467,49469,49483,49491],{"className":49468,"style":46264},[1052],[31,49470,49471,49474],{"style":5090},[31,49472],{"className":49473,"style":1637},[1060],[31,49475,49477,49480],{"className":49476},[75],[31,49478,23909],{"className":49479,"style":24157},[75],[31,49481,46458],{"className":49482,"style":46500},[75,76],[31,49484,49485,49488],{"style":2170},[31,49486],{"className":49487,"style":1637},[1060],[31,49489],{"className":49490,"style":2178},[2177],[31,49492,49493,49496],{"style":5110},[31,49494],{"className":49495,"style":1637},[1060],[31,49497,49499,49502],{"className":49498},[75],[31,49500,23909],{"className":49501,"style":24157},[75],[31,49503,5354],{"className":49504,"style":2666},[75,76],[31,49506,1076],{"className":49507},[1075],[31,49509,49511],{"className":49510},[1048],[31,49512,49514],{"className":49513,"style":5156},[1052],[31,49515],{},[31,49517],{"className":49518},[289,2139],[31,49520,49522],{"className":49521},[289],[31,49523,220],{"className":49524},[1307,2662],[31,49526,1210],{"className":49527},[1391],[11,49529,49530,49531,1138,49700,49728],{},"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 ",[31,49532,49534,49580],{"className":49533,"translate":35},[34],[31,49535,49537],{"className":49536},[39],[41,49538,49539],{"xmlns":43},[45,49540,49541,49577],{},[48,49542,49543,49549,49551,49553,49555,49557,49563,49565,49571,49573,49575],{},[48,49544,49545,49547],{},[51,49546,12271],{"mathvariant":494},[51,49548,2549],{"mathvariant":494},[95,49550,211],{"stretchy":210},[51,49552,18446],{},[95,49554,220],{"stretchy":210},[95,49556,27223],{},[1004,49558,49559,49561],{},[51,49560,22324],{"mathvariant":14789},[51,49562,4793],{},[95,49564,15175],{"stretchy":210},[1180,49566,49567,49569],{},[51,49568,4793],{},[51,49570,20800],{"mathvariant":494},[51,49572,18446],{},[51,49574,4793],{},[95,49576,15184],{"stretchy":210},[55,49578,49579],{"encoding":57},"\\mathrm{tr}(A)\\approx\\mathbb E_\\varepsilon[\\varepsilon^\\top A\\varepsilon]",[31,49581,49583,49613],{"className":49582,"ariaHidden":62},[61],[31,49584,49586,49589,49595,49598,49601,49604,49607,49610],{"className":49585},[66],[31,49587],{"className":49588,"style":260},[70],[31,49590,49592],{"className":49591},[75],[31,49593,16973],{"className":49594},[75,2745],[31,49596,211],{"className":49597},[264],[31,49599,18446],{"className":49600},[75,76],[31,49602,220],{"className":49603},[289],[31,49605],{"className":49606,"style":293},[119],[31,49608,27223],{"className":49609},[297],[31,49611],{"className":49612,"style":293},[119],[31,49614,49616,49619,49659,49662,49691,49694,49697],{"className":49615},[66],[31,49617],{"className":49618,"style":20835},[70],[31,49620,49622,49625],{"className":49621},[75],[31,49623,22324],{"className":49624},[75,14947],[31,49626,49628],{"className":49627},[1039],[31,49629,49631,49651],{"className":49630},[1043,1044],[31,49632,49634,49648],{"className":49633},[1048],[31,49635,49637],{"className":49636,"style":3230},[1052],[31,49638,49639,49642],{"style":2807},[31,49640],{"className":49641,"style":1061},[1060],[31,49643,49645],{"className":49644},[1065,1066,1067,1068],[31,49646,4793],{"className":49647},[75,76,1068],[31,49649,1076],{"className":49650},[1075],[31,49652,49654],{"className":49653},[1048],[31,49655,49657],{"className":49656,"style":2050},[1052],[31,49658],{},[31,49660,15175],{"className":49661},[264],[31,49663,49665,49668],{"className":49664},[75],[31,49666,4793],{"className":49667},[75,76],[31,49669,49671],{"className":49670},[1039],[31,49672,49674],{"className":49673},[1043],[31,49675,49677],{"className":49676},[1048],[31,49678,49680],{"className":49679,"style":20863},[1052],[31,49681,49682,49685],{"style":3631},[31,49683],{"className":49684,"style":1061},[1060],[31,49686,49688],{"className":49687},[1065,1066,1067,1068],[31,49689,20800],{"className":49690},[75,1068],[31,49692,18446],{"className":49693},[75,76],[31,49695,4793],{"className":49696},[75,76],[31,49698,15184],{"className":49699},[289],[31,49701,49703,49716],{"className":49702,"translate":35},[34],[31,49704,49706],{"className":49705},[39],[41,49707,49708],{"xmlns":43},[45,49709,49710,49714],{},[48,49711,49712],{},[51,49713,4793],{},[55,49715,4796],{"encoding":57},[31,49717,49719],{"className":49718,"ariaHidden":62},[61],[31,49720,49722,49725],{"className":49721},[66],[31,49723],{"className":49724,"style":71},[70],[31,49726,4793],{"className":49727},[75,76]," 取 Rademacher 向量，代价是方差。",[11,49730,49731,49732,49765,49766,49799],{},"流与最优传输有一个漂亮的接口。Brenier 说最优映射是 ",[31,49733,49735,49750],{"className":49734,"translate":35},[34],[31,49736,49738],{"className":49737},[39],[41,49739,49740],{"xmlns":43},[45,49741,49742,49748],{},[48,49743,49744,49746],{},[51,49745,12145],{"mathvariant":494},[51,49747,5324],{},[55,49749,12232],{"encoding":57},[31,49751,49753],{"className":49752,"ariaHidden":62},[61],[31,49754,49756,49759,49762],{"className":49755},[66],[31,49757],{"className":49758,"style":945},[70],[31,49760,12145],{"className":49761},[75],[31,49763,5324],{"className":49764},[75,76],"，流说数据由可逆映射得到；当映射取 ",[31,49767,49769,49784],{"className":49768,"translate":35},[34],[31,49770,49772],{"className":49771},[39],[41,49773,49774],{"xmlns":43},[45,49775,49776,49782],{},[48,49777,49778,49780],{},[51,49779,12145],{"mathvariant":494},[51,49781,5324],{},[55,49783,12232],{"encoding":57},[31,49785,49787],{"className":49786,"ariaHidden":62},[61],[31,49788,49790,49793,49796],{"className":49789},[66],[31,49791],{"className":49792,"style":945},[70],[31,49794,12145],{"className":49795},[75],[31,49797,5324],{"className":49798},[75,76]," 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",[31,49801,49803],{"className":49802,"translate":35},[411],[31,49804,49806,49934],{"className":49805,"translate":35},[34],[31,49807,49809],{"className":49808},[39],[41,49810,49811],{"xmlns":43,"display":420},[45,49812,49813,49931],{},[48,49814,49815,49817,49819,49843,49845,49851,49853,49859,49861,49863,49865,49867,49869,49875,49877,49883,49885,49891,49893,49895,49901,49903,49905,49907,49909,49911,49913,49919,49921,49923,49929],{},[51,49816,4561],{"mathvariant":925},[95,49818,223],{},[1004,49820,49821,49823],{},[51,49822,22324],{"mathvariant":14789},[48,49824,49825,49827,49829,49835,49837],{},[51,49826,12271],{},[95,49828,1210],{"separator":62},[1004,49830,49831,49833],{},[51,49832,53],{},[91,49834,1977],{},[95,49836,1210],{"separator":62},[1004,49838,49839,49841],{},[51,49840,53],{},[91,49842,205],{},[95,49844,4572],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,49846,49847,49849],{},[51,49848,5354],{},[51,49850,23798],{},[95,49852,211],{"stretchy":210},[1004,49854,49855,49857],{},[51,49856,53],{},[51,49858,12271],{},[95,49860,1210],{"separator":62},[51,49862,12271],{},[95,49864,220],{"stretchy":210},[95,49866,97],{},[95,49868,211],{"stretchy":210},[1004,49870,49871,49873],{},[51,49872,53],{},[91,49874,205],{},[95,49876,97],{},[1004,49878,49879,49881],{},[51,49880,53],{},[91,49882,1977],{},[95,49884,220],{"stretchy":210},[1180,49886,49887,49889],{},[95,49888,4572],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[91,49890,152],{},[95,49892,1210],{"separator":62},[119,49894],{"width":20286},[1004,49896,49897,49899],{},[51,49898,53],{},[51,49900,12271],{},[95,49902,223],{},[95,49904,211],{"stretchy":210},[91,49906,205],{},[95,49908,97],{},[51,49910,12271],{},[95,49912,220],{"stretchy":210},[1004,49914,49915,49917],{},[51,49916,53],{},[91,49918,1977],{},[95,49920,434],{},[51,49922,12271],{},[1004,49924,49925,49927],{},[51,49926,53],{},[91,49928,205],{},[95,49930,1210],{"separator":62},[55,49932,49933],{"encoding":57},"\\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,",[31,49935,49937,49955,50234,50292,50462,50483,50544],{"className":49936,"ariaHidden":62},[61],[31,49938,49940,49943,49946,49949,49952],{"className":49939},[66],[31,49941],{"className":49942,"style":945},[70],[31,49944,4561],{"className":49945},[75,968],[31,49947],{"className":49948,"style":293},[119],[31,49950,223],{"className":49951},[297],[31,49953],{"className":49954,"style":293},[119],[31,49956,49958,49961,50093,50130,50170,50173,50213,50216,50219,50222,50225,50228,50231],{"className":49957},[66],[31,49959],{"className":49960,"style":4988},[70],[31,49962,49964,49967],{"className":49963},[75],[31,49965,22324],{"className":49966},[75,14947],[31,49968,49970],{"className":49969},[1039],[31,49971,49973,50085],{"className":49972},[1043,1044],[31,49974,49976,50082],{"className":49975},[1048],[31,49977,49979],{"className":49978,"style":19220},[1052],[31,49980,49981,49984],{"style":2807},[31,49982],{"className":49983,"style":1061},[1060],[31,49985,49987],{"className":49986},[1065,1066,1067,1068],[31,49988,49990,49993,49996,50039,50042],{"className":49989},[75,1068],[31,49991,12271],{"className":49992},[75,76,1068],[31,49994,1210],{"className":49995},[1391,1068],[31,49997,49999,50002],{"className":49998},[75,1068],[31,50000,53],{"className":50001},[75,76,1068],[31,50003,50005],{"className":50004},[1039],[31,50006,50008,50030],{"className":50007},[1043,1044],[31,50009,50011,50027],{"className":50010},[1048],[31,50012,50015],{"className":50013,"style":50014},[1052],"height:0.3173em;",[31,50016,50018,50021],{"style":50017},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[31,50019],{"className":50020,"style":11769},[1060],[31,50022,50024],{"className":50023},[1065,11773,1308,1068],[31,50025,1977],{"className":50026},[75,1068],[31,50028,1076],{"className":50029},[1075],[31,50031,50033],{"className":50032},[1048],[31,50034,50037],{"className":50035,"style":50036},[1052],"height:0.143em;",[31,50038],{},[31,50040,1210],{"className":50041},[1391,1068],[31,50043,50045,50048],{"className":50044},[75,1068],[31,50046,53],{"className":50047},[75,76,1068],[31,50049,50051],{"className":50050},[1039],[31,50052,50054,50074],{"className":50053},[1043,1044],[31,50055,50057,50071],{"className":50056},[1048],[31,50058,50060],{"className":50059,"style":50014},[1052],[31,50061,50062,50065],{"style":50017},[31,50063],{"className":50064,"style":11769},[1060],[31,50066,50068],{"className":50067},[1065,11773,1308,1068],[31,50069,205],{"className":50070},[75,1068],[31,50072,1076],{"className":50073},[1075],[31,50075,50077],{"className":50076},[1048],[31,50078,50080],{"className":50079,"style":50036},[1052],[31,50081],{},[31,50083,1076],{"className":50084},[1075],[31,50086,50088],{"className":50087},[1048],[31,50089,50091],{"className":50090,"style":1083},[1052],[31,50092],{},[31,50094,50096],{"className":50095},[264],[31,50097,50099],{"className":50098},[1307,13319],[31,50100,50102,50122],{"className":50101},[1043,1044],[31,50103,50105,50119],{"className":50104},[1048],[31,50106,50108],{"className":50107,"style":13329},[1052],[31,50109,50110,50113],{"style":13332},[31,50111],{"className":50112,"style":13336},[1060],[31,50114,50115],{"style":32856},[13341,50116,50117],{"xmlns":13343,"width":32859,"height":1178,"viewBox":32860},[13347,50118],{"d":32863},[31,50120,1076],{"className":50121},[1075],[31,50123,50125],{"className":50124},[1048],[31,50126,50128],{"className":50127,"style":13359},[1052],[31,50129],{},[31,50131,50133,50136],{"className":50132},[75],[31,50134,5354],{"className":50135,"style":2666},[75,76],[31,50137,50139],{"className":50138},[1039],[31,50140,50142,50162],{"className":50141},[1043,1044],[31,50143,50145,50159],{"className":50144},[1048],[31,50146,50148],{"className":50147,"style":1053},[1052],[31,50149,50150,50153],{"style":5504},[31,50151],{"className":50152,"style":1061},[1060],[31,50154,50156],{"className":50155},[1065,1066,1067,1068],[31,50157,23798],{"className":50158,"style":19759},[75,76,1068],[31,50160,1076],{"className":50161},[1075],[31,50163,50165],{"className":50164},[1048],[31,50166,50168],{"className":50167,"style":2050},[1052],[31,50169],{},[31,50171,211],{"className":50172},[264],[31,50174,50176,50179],{"className":50175},[75],[31,50177,53],{"className":50178},[75,76],[31,50180,50182],{"className":50181},[1039],[31,50183,50185,50205],{"className":50184},[1043,1044],[31,50186,50188,50202],{"className":50187},[1048],[31,50189,50191],{"className":50190,"style":19220},[1052],[31,50192,50193,50196],{"style":2807},[31,50194],{"className":50195,"style":1061},[1060],[31,50197,50199],{"className":50198},[1065,1066,1067,1068],[31,50200,12271],{"className":50201},[75,76,1068],[31,50203,1076],{"className":50204},[1075],[31,50206,50208],{"className":50207},[1048],[31,50209,50211],{"className":50210,"style":2050},[1052],[31,50212],{},[31,50214,1210],{"className":50215},[1391],[31,50217],{"className":50218,"style":1467},[119],[31,50220,12271],{"className":50221},[75,76],[31,50223,220],{"classNam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",[31,50601,50603,50633],{"className":50602,"translate":35},[34],[31,50604,50606],{"className":50605},[39],[41,50607,50608],{"xmlns":43},[45,50609,50610,50630],{},[48,50611,50612,50614,50620,50622,50628],{},[95,50613,211],{"stretchy":210},[1004,50615,50616,50618],{},[51,50617,53],{},[91,50619,1977],{},[95,50621,1210],{"separator":62},[1004,50623,50624,50626],{},[51,50625,53],{},[91,50627,205],{},[95,50629,220],{"stretchy":210},[55,50631,50632],{"encoding":57},"(x_0,x_1)",[31,50634,50636],{"className":50635,"ariaHidden":62},[61],[31,50637,50639,50642,50645,50685,50688,50691,50731],{"className":50638},[66],[31,50640],{"className":50641,"style":260},[70],[31,50643,211],{"className":50644},[264],[31,50646,50648,50651],{"className":50647},[75],[31,50649,53],{"className":50650},[75,76],[31,50652,50654],{"className":50653},[1039],[31,50655,50657,50677],{"className":50656},[1043,1044],[31,50658,50660,50674],{"className":50659},[1048],[31,50661,50663],{"className":50662,"style":2028},[1052],[31,50664,50665,50668],{"style":2807},[31,50666],{"className":50667,"style":1061},[1060],[31,50669,50671],{"className":50670},[1065,1066,1067,1068],[31,50672,1977],{"className":50673},[75,1068],[31,50675,1076],{"className":50676},[1075],[31,50678,50680],{"className":50679},[1048],[31,50681,50683],{"className":50682,"style":2050},[1052],[31,50684],{},[31,50686,1210],{"className":50687},[1391],[31,50689],{"className":50690,"style":1467},[119],[31,50692,50694,50697],{"className":50693},[75],[31,50695,53],{"className":50696},[75,76],[31,50698,50700],{"className":50699},[1039],[31,50701,50703,50723],{"className":50702},[1043,1044],[31,50704,50706,50720],{"className":50705},[1048],[31,50707,50709],{"className":50708,"style":2028},[1052],[31,50710,50711,50714],{"style":2807},[31,50712],{"className":50713,"style":1061},[1060],[31,50715,50717],{"className":50716},[1065,1066,1067,1068],[31,50718,205],{"className":50719},[75,1068],[31,50721,1076],{"className":50722},[1075],[31,50724,50726],{"className":50725},[1048],[31,50727,50729],{"className":50728,"style":2050},[1052],[31,50730],{},[31,50732,220],{"className":50733},[289]," 是噪声与数据的配对。如果配对来自 OT 计划，这条直线就是 ",[31,50736,50738,50755],{"className":50737,"translate":35},[34],[31,50739,50741],{"className":50740},[39],[41,50742,50743],{"xmlns":43},[45,50744,50745,50753],{},[48,50746,50747],{},[1004,50748,50749,50751],{},[51,50750,3117],{},[91,50752,152],{},[55,50754,19087],{"encoding":57},[31,50756,50758],{"className":50757,"ariaHidden":62},[61],[31,50759,50761,50764],{"className":50760},[66],[31,50762],{"className":50763,"style":5230},[70],[31,50765,50767,50770],{"className":50766},[75],[31,50768,3117],{"className":50769,"style":949},[75,76],[31,50771,50773],{"className":50772},[1039],[31,50774,50776,50796],{"className":50775},[1043,1044],[31,50777,50779,50793],{"className":50778},[1048],[31,50780,50782],{"className":50781,"style":2028},[1052],[31,50783,50784,50787],{"style":1056},[31,50785],{"className":50786,"style":1061},[1060],[31,50788,50790],{"className":50789},[1065,1066,1067,1068],[31,50791,152],{"className":50792},[75,1068],[31,50794,1076],{"className":50795},[1075],[31,50797,50799],{"className":50798},[1048],[31,50800,50802],{"className":50801,"style":2050},[1052],[31,50803],{}," 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。",[11,50806,50807],{},"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。",[18,50809,50810],{"id":50810},"落地",[11,50812,50813],{},"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。",[11,50815,50816,50817,50858,50859,50928],{},"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 ",[31,50818,50820,50838],{"className":50819,"translate":35},[34],[31,50821,50823],{"className":50822},[39],[41,50824,50825],{"xmlns":43},[45,50826,50827,50835],{},[48,50828,50829,50831,50833],{},[51,50830,19939],{},[95,50832,1540],{},[91,50834,152],{},[55,50836,50837],{"encoding":57},"\\log2",[31,50839,50841],{"className":50840,"ariaHidden":62},[61],[31,50842,50844,50847,50852,50855],{"className":50843},[66],[31,50845],{"className":50846,"style":4202},[70],[31,50848,20177,50850],{"className":50849},[1460],[31,50851,14556],{"style":20180},[31,50853],{"className":50854,"style":1467},[119],[31,50856,152],{"className":50857},[75],"，梯度为零——这是早期 GAN 训练不稳定的理论解释之一。WGAN 换成 ",[31,50860,50862,50879],{"className":50861,"translate":35},[34],[31,50863,50865],{"className":50864},[39],[41,50866,50867],{"xmlns":43},[45,50868,50869,50877],{},[48,50870,50871],{},[1004,50872,50873,50875],{},[51,50874,3117],{},[91,50876,205],{},[55,50878,5220],{"encoding":57},[31,50880,50882],{"className":50881,"ariaHidden":62},[61],[31,50883,50885,50888],{"className":50884},[66],[31,50886],{"className":50887,"style":5230},[70],[31,50889,50891,50894],{"className":50890},[75],[31,50892,3117],{"className":50893,"style":949},[75,76],[31,50895,50897],{"className":50896},[1039],[31,50898,50900,50920],{"className":50899},[1043,1044],[31,50901,50903,50917],{"className":50902},[1048],[31,50904,50906],{"className":50905,"style":2028},[1052],[31,50907,50908,50911],{"style":1056},[31,50909],{"className":50910,"style":1061},[1060],[31,50912,50914],{"className":50913},[1065,1066,1067,1068],[31,50915,205],{"className":50916},[75,1068],[31,50918,1076],{"className":50919},[1075],[31,50921,50923],{"className":50922},[1048],[31,50924,50926],{"className":50925,"style":2050},[1052],[31,50927],{},"，用对偶式",[31,50930,50932],{"className":50931,"translate":35},[411],[31,50933,50935,51037],{"className":50934,"translate":35},[34],[31,50936,50938],{"className":50937},[39],[41,50939,50940],{"xmlns":43,"display":420},[45,50941,50942,51034],{},[48,50943,50944,50954,50984,50986,50996,50998,51000,51002,51004,51006,51008,51010,51020,51022,51024,51026,51028,51030,51032],{},[1531,50945,50946,50952],{},[48,50947,50948,50950],{},[51,50949,19878],{},[95,50951,1540],{},[51,50953,15297],{},[1531,50955,50956,50962],{},[48,50957,50958,50960],{},[51,50959,10154],{},[95,50961,1540],{},[48,50963,50964,50966,50968,50980,50982],{},[51,50965,4572],{"mathvariant":494},[51,50967,1662],{},[1004,50969,50970,50972],{},[51,50971,4572],{"mathvariant":494},[48,50973,50974,50976,50978],{},[51,50975,4561],{"mathvariant":494},[51,50977,5391],{"mathvariant":494},[51,50979,11],{"mathvariant":494},[95,50981,347],{},[91,50983,205],{},[1430,50985,1545],{},[1004,50987,50988,50990],{},[51,50989,22324],{"mathvariant":14789},[1004,50991,50992,50994],{},[51,50993,2521],{},[51,50995,2549],{},[95,50997,15175],{"stretchy":210},[51,50999,1662],{},[95,51001,211],{"stretchy":210},[51,51003,53],{},[95,51005,220],{"stretchy":210},[95,51007,15184],{"stretchy":210},[95,51009,97],{},[1004,51011,51012,51014],{},[51,51013,22324],{"mathvariant":14789},[1004,51015,51016,51018],{},[51,51017,2521],{},[51,51019,14556],{},[95,51021,15175],{"stretchy":210},[51,51023,1662],{},[95,51025,211],{"stretchy":210},[51,51027,53],{},[95,51029,220],{"stretchy":210},[95,51031,15184],{"stretchy":210},[95,51033,1210],{"separator":62},[55,51035,51036],{"encoding":57},"\\min_G\\max_{\\|f\\|_{\\mathrm{Lip}}\\le1}\\ \\mathbb E_{P_r}[f(x)]-\\mathbb E_{P_g}[f(x)],",[31,51038,51040,51310],{"className":51039,"ariaHidden":62},[61],[31,51041,51043,51047,51091,51094,51197,51200,51203,51286,51289,51292,51295,51298,51301,51304,51307],{"className":51042},[66],[31,51044],{"className":51045,"style":51046},[70],"height:1.7385em;vertical-align:-0.9885em;",[31,51048,51050],{"className":51049},[1460,1620],[31,51051,51053,51083],{"className":51052},[1043,1044],[31,51054,51056,51080],{"className":51055},[1048],[31,51057,51059,51070],{"className":51058,"style":19979},[1052],[31,51060,51061,51064],{"style":1633},[31,51062],{"className":51063,"style":1637},[1060],[31,51065,51067],{"className":51066},[1065,1066,1067,1068],[31,51068,15297],{"className":51069},[75,76,1068],[31,51071,51072,51075],{"style":1649},[31,51073],{"className":51074,"style":1637},[1060],[31,51076,51077],{},[31,51078,19878],{"className":51079},[1460],[31,51081,1076],{"className":51082},[1075],[31,51084,51086],{"className":51085},[1048],[31,51087,51089],{"className":51088,"style":1672},[1052],[31,51090],{},[31,51092],{"className":51093,"style":1467},[119],[31,51095,51097],{"className":51096},[1460,1620],[31,51098,51100,51188],{"className":51099},[1043,1044],[31,51101,51103,51185],{"className":51102},[1048],[31,51104,51106,51175],{"className":51105,"style":71},[1052],[31,51107,51108,51111],{"style":3301},[31,51109],{"className":51110,"style":1637},[1060],[31,51112,51114],{"className":51113},[1065,1066,1067,1068],[31,51115,51117,51120,51123,51169,51172],{"className":51116},[75,1068],[31,51118,4572],{"className":51119},[75,1068],[31,51121,1662],{"className":51122,"style":35196},[75,76,1068],[31,51124,51126,51129],{"className":51125},[75,1068],[31,51127,4572],{"className":51128},[75,1068],[31,51130,51132],{"className":51131},[1039],[31,51133,51135,51161],{"className":51134},[1043,1044],[31,51136,51138,51158],{"className":51137},[1048],[31,51139,51141],{"className":51140,"style":11762},[1052],[31,51142,51143,51146],{"style":11765},[31,51144],{"className":51145,"style":11769},[1060],[31,51147,51149],{"className":51148},[1065,11773,1308,1068],[31,51150,51152],{"className":51151},[75,1068],[31,51153,51155],{"className":51154},[75,1068],[31,51156,11783],{"className":51157},[75,2745,1068],[31,51159,1076],{"className":51160},[1075],[31,51162,51164],{"className":51163},[1048],[31,51165,51167],{"className":51166,"style":11793},[1052],[31,51168],{},[31,51170,347],{"className":51171},[297,1068],[31,51173,205],{"className":51174},[75,1068],[31,51176,51177,51180],{"style":1649},[31,51178],{"className":51179,"style":1637},[1060],[31,51181,51182],{},[31,51183,10154],{"className":51184},[1460],[31,51186,1076],{"className":51187},[1075],[31,51189,51191],{"className":51190},[1048],[31,51192,51195],{"className":51193,"style":51194},[1052],"height:0.9885em;",[31,51196],{},[31,51198,1545],{"className":51199},[119],[31,51201],{"className":51202,"style":1467},[119],[31,51204,51206,51209],{"className":51205},[75],[31,51207,22324],{"className":51208},[75,14947],[31,51210,51212],{"className":51211},[1039],[31,51213,51215,51277],{"className":51214},[1043,1044],[31,51216,51218,51274],{"className":51217},[1048],[31,51219,51221],{"className":51220,"style":27736},[1052],[31,51222,51223,51226],{"style":2807},[31,51224],{"className":51225,"style":1061},[1060],[31,51227,51229],{"className":51228},[1065,1066,1067,1068],[31,51230,51232],{"className":51231},[75,1068],[31,51233,51235,51238],{"className":51234},[75,1068],[31,51236,2521],{"className":51237,"style":949},[75,76,1068],[31,51239,51241],{"className":51240},[1039],[31,51242,51244,51266],{"className":51243},[1043,1044],[31,51245,51247,51263],{"className":51246},[1048],[31,51248,51251],{"className":51249,"style":51250},[1052],"height:0.1645em;",[31,51252,51254,51257],{"style":51253},"top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;",[31,51255],{"className":51256,"style":11769},[1060],[31,51258,51260],{"className":51259},[1065,11773,1308,1068],[31,51261,2549],{"className":51262,"style":19759},[75,76,1068],[31,51264,1076],{"className":51265},[1075],[31,51267,51269],{"className":51268},[1048],[31,51270,51272],{"className":51271,"style":50036},[1052],[31,51273],{},[31,51275,1076],{"className":51276},[1075],[31,51278,51280],{"className":51279},[1048],[31,51281,51284],{"className":51282,"style":51283},[1052],"height:0.2501em;",[31,51285],{},[31,51287,15175],{"className":51288},[264],[31,51290,1662],{"className":51291,"style":35196},[75,76],[31,51293,211],{"className":51294},[264],[31,51296,53],{"className":51297},[75,76],[31,51299,30315],{"className":51300},[289],[31,51302],{"className":51303,"style":120},[119],[31,51305,97],{"className":51306},[124],[31,51308],{"className":51309,"style":120},[119],[31,51311,51313,51317,51399,51402,51405,51408,51411,51414],{"className":51312},[66],[31,51314],{"className":51315,"style":51316},[70],"height:1.0973em;vertical-align:-0.3473em;",[31,51318,51320,51323],{"className":51319},[75],[31,51321,22324],{"className":51322},[75,14947],[31,51324,51326],{"className":51325},[1039],[31,51327,51329,51390],{"className":51328},[1043,1044],[31,51330,51332,51387],{"className":51331},[1048],[31,51333,51335],{"className":51334,"style":27736},[1052],[31,51336,51337,51340],{"style":2807},[31,51338],{"className":51339,"style":1061},[1060],[31,51341,51343],{"className":51342},[1065,1066,1067,1068],[31,51344,51346],{"className":51345},[75,1068],[31,51347,51349,51352],{"className":51348},[75,1068],[31,51350,2521],{"className":51351,"style":949},[75,76,1068],[31,51353,51355],{"className":51354},[1039],[31,51356,51358,51378],{"className":51357},[1043,1044],[31,51359,51361,51375],{"className":51360},[1048],[31,51362,51364],{"className":51363,"style":51250},[1052],[31,51365,51366,51369],{"style":51253},[31,51367],{"className":51368,"style":11769},[1060],[31,51370,51372],{"className":51371},[1065,11773,1308,1068],[31,51373,14556],{"className":51374,"style":2666},[75,76,1068],[31,51376,1076],{"className":51377},[1075],[31,51379,51381],{"className":51380},[1048],[31,51382,51385],{"className":51383,"style":51384},[1052],"height:0.2819em;",[31,51386],{},[31,51388,1076],{"className":51389},[1075],[31,51391,51393],{"className":51392},[1048],[31,51394,51397],{"className":51395,"style":51396},[1052],"height:0.3473em;",[31,51398],{},[31,51400,15175],{"className":51401},[264],[31,51403,1662],{"className":51404,"style":35196},[75,76],[31,51406,211],{"className":51407},[264],[31,51409,53],{"className":51410},[75,76],[31,51412,30315],{"className":51413},[289],[31,51415,1210],{"className":51416},[1391],[11,51418,51419,51420,51448],{},"critic ",[31,51421,51423,51436],{"className":51422,"translate":35},[34],[31,51424,51426],{"className":51425},[39],[41,51427,51428],{"xmlns":43},[45,51429,51430,51434],{},[48,51431,51432],{},[51,51433,1662],{},[55,51435,1662],{"encoding":57},[31,51437,51439],{"className":51438,"ariaHidden":62},[61],[31,51440,51442,51445],{"className":51441},[66],[31,51443],{"className":51444,"style":4202},[70],[31,51446,1662],{"className":51447,"style":35196},[75,76]," 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。",[11,51450,51451],{},"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。",[11,51453,51454,51455,51483,51484,51608,51609,51748,51749,51777,51778,51957,51958,52027],{},"域适应与分布偏移是 ",[31,51456,51458,51471],{"className":51457,"translate":35},[34],[31,51459,51461],{"className":51460},[39],[41,51462,51463],{"xmlns":43},[45,51464,51465,51469],{},[48,51466,51467],{},[51,51468,3117],{},[55,51470,3117],{"encoding":57},[31,51472,51474],{"className":51473,"ariaHidden":62},[61],[31,51475,51477,51480],{"className":51476},[66],[31,51478],{"className":51479,"style":945},[70],[31,51481,3117],{"className":51482,"style":949},[75,76]," 最能发挥的地方。给定源域和目标域的特征分布 ",[31,51485,51487,51513],{"className":51486,"translate":35},[34],[31,51488,51490],{"className":51489},[39],[41,51491,51492],{"xmlns":43},[45,51493,51494,51510],{},[48,51495,51496,51502,51504],{},[1004,51497,51498,51500],{},[51,51499,852],{},[51,51501,21582],{},[95,51503,1210],{"separator":62},[1004,51505,51506,51508],{},[51,51507,852],{},[51,51509,919],{},[55,51511,51512],{"encoding":57},"\\mu_S,\\mu_T",[31,51514,51516],{"className":51515,"ariaHidden":62},[61],[31,51517,51519,51522,51562,51565,51568],{"className":51518},[66],[31,51520],{"className":51521,"style":865},[70],[31,51523,51525,51528],{"className":51524},[75],[31,51526,852],{"className":51527},[75,76],[31,51529,51531],{"className":51530},[1039],[31,51532,51534,51554],{"className":51533},[1043,1044],[31,51535,51537,51551],{"className":51536},[1048],[31,51538,51540],{"className":51539,"style":27736},[1052],[31,51541,51542,51545],{"style":2807},[31,51543],{"className":51544,"style":1061},[1060],[31,51546,51548],{"className":51547},[1065,1066,1067,1068],[31,51549,21582],{"className":51550,"style":21697},[75,76,1068],[31,51552,1076],{"className":51553},[1075],[31,51555,51557],{"className":51556},[1048],[31,51558,51560],{"className":51559,"style":2050},[1052],[31,51561],{},[31,51563,1210],{"className":51564},[1391],[31,51566],{"className":51567,"style":1467},[119],[31,51569,51571,51574],{"className":51570},[75],[31,51572,852],{"className":51573},[75,76],[31,51575,51577],{"className":51576},[1039],[31,51578,51580,51600],{"className":51579},[1043,1044],[31,51581,51583,51597],{"className":51582},[1048],[31,51584,51586],{"className":51585,"style":27736},[1052],[31,51587,51588,51591],{"style":2807},[31,51589],{"className":51590,"style":1061},[1060],[31,51592,51594],{"className":51593},[1065,1066,1067,1068],[31,51595,919],{"className":51596,"style":949},[75,76,1068],[31,51598,1076],{"className":51599},[1075],[31,51601,51603],{"className":51602},[1048],[31,51604,51606],{"className":51605,"style":2050},[1052],[31,51607],{},"，直接最小化 ",[31,51610,51612,51644],{"className":51611,"translate":35},[34],[31,51613,51615],{"className":51614},[39],[41,51616,51617],{"xmlns":43},[45,51618,51619,51641],{},[48,51620,51621,51623,51625,51631,51633,51639],{},[51,51622,3117],{},[95,51624,211],{"stretchy":210},[1004,51626,51627,51629],{},[51,51628,852],{},[51,51630,21582],{},[95,51632,1210],{"separator":62},[1004,51634,51635,51637],{},[51,51636,852],{},[51,51638,919],{},[95,51640,220],{"stretchy":210},[55,51642,51643],{"encoding":57},"W(\\mu_S,\\mu_T)",[31,51645,51647],{"className":51646,"ariaHidden":62},[61],[31,51648,51650,51653,51656,51659,51699,51702,51705,51745],{"className":51649},[66],[31,51651],{"className":51652,"style":260},[70],[31,51654,3117],{"className":51655,"style":949},[75,76],[31,51657,211],{"className":51658},[264],[31,51660,51662,51665],{"className":51661},[75],[31,51663,852],{"className":51664},[75,76],[31,51666,51668],{"className":51667},[1039],[31,51669,51671,51691],{"className":51670},[1043,1044],[31,51672,51674,51688],{"className":51673},[1048],[31,51675,51677],{"className":51676,"style":27736},[1052],[31,51678,51679,51682],{"style":2807},[31,51680],{"className":51681,"style":1061},[1060],[31,51683,51685],{"className":51684},[1065,1066,1067,1068],[31,51686,21582],{"className":51687,"style":21697},[75,76,1068],[31,51689,1076],{"className":51690},[1075],[31,51692,51694],{"className":51693},[1048],[31,51695,51697],{"className":51696,"style":2050},[1052],[31,51698],{},[31,51700,1210],{"className":51701},[1391],[31,51703],{"className":51704,"style":1467},[119],[31,51706,51708,51711],{"className":51707},[75],[31,51709,852],{"className":51710},[75,76],[31,51712,51714],{"className":51713},[1039],[31,51715,51717,51737],{"className":51716},[1043,1044],[31,51718,51720,51734],{"className":51719},[1048],[31,51721,51723],{"className":51722,"style":27736},[1052],[31,51724,51725,51728],{"style":2807},[31,51726],{"className":51727,"style":1061},[1060],[31,51729,51731],{"className":51730},[1065,1066,1067,1068],[31,51732,919],{"className":51733,"style":949},[75,76,1068],[31,51735,1076],{"className":51736},[1075],[31,51738,51740],{"className":51739},[1048],[31,51741,51743],{"className":51742,"style":2050},[1052],[31,51744],{},[31,51746,220],{"className":51747},[289]," 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 ",[31,51750,51752,51765],{"className":51751,"translate":35},[34],[31,51753,51755],{"className":51754},[39],[41,51756,51757],{"xmlns":43},[45,51758,51759,51763],{},[48,51760,51761],{},[51,51762,8059],{},[55,51764,11503],{"encoding":57},[31,51766,51768],{"className":51767,"ariaHidden":62},[61],[31,51769,51771,51774],{"className":51770},[66],[31,51772],{"className":51773,"style":865},[70],[31,51775,8059],{"className":51776},[75,76]," 去最大化 ",[31,51779,51781,51823],{"className":51780,"translate":35},[34],[31,51782,51784],{"className":51783},[39],[41,51785,51786],{"xmlns":43},[45,51787,51788,51820],{},[48,51789,51790,51792,51794,51796,51798,51804,51806,51808,51810,51812,51814],{},[95,51791,1409],{},[51,51793,8059],{},[1430,51795,1432],{},[51,51797,1435],{},[1004,51799,51800,51802],{},[51,51801,852],{},[51,51803,21582],{},[95,51805,97],{},[95,51807,1409],{},[51,51809,8059],{},[1430,51811,1432],{},[51,51813,1435],{},[1004,51815,51816,51818],{},[51,51817,852],{},[51,51819,919],{},[55,51821,51822],{"encoding":57},"\\int\\varphi\\,d\\mu_S-\\int\\varphi\\,d\\mu_T",[31,51824,51826,51896],{"className":51825,"ariaHidden":62},[61],[31,51827,51829,51832,51835,51838,51841,51844,51847,51887,51890,51893],{"className":51828},[66],[31,51830],{"className":51831,"style":1456},[70],[31,51833,1409],{"className":51834,"style":1463},[1460,1461,1462],[31,51836],{"className":51837,"style":1467},[119],[31,51839,8059],{"className":51840},[75,76],[31,51842],{"className":51843,"style":1467},[119],[31,51845,1435],{"className":51846},[75,76],[31,51848,51850,51853],{"className":51849},[75],[31,51851,852],{"className":51852},[75,76],[31,51854,51856],{"className":51855},[1039],[31,51857,51859,51879],{"className":51858},[1043,1044],[31,51860,51862,51876],{"className":51861},[1048],[31,51863,51865],{"className":51864,"style":27736},[1052],[31,51866,51867,51870],{"style":2807},[31,51868],{"className":51869,"style":1061},[1060],[31,51871,51873],{"className":51872},[1065,1066,1067,1068],[31,51874,21582],{"className":51875,"style":21697},[75,76,1068],[31,51877,1076],{"className":51878},[1075],[31,51880,51882],{"className":51881},[1048],[31,51883,51885],{"className":51884,"style":2050},[1052],[31,51886],{},[31,51888],{"className":51889,"style":120},[119],[31,51891,97],{"className":51892},[124],[31,51894],{"className":51895,"style":120},[119],[31,51897,51899,51902,51905,51908,51911,51914,51917],{"className":51898},[66],[31,51900],{"className":51901,"style":1456},[70],[31,51903,1409],{"className":51904,"style":1463},[1460,1461,1462],[31,51906],{"className":51907,"style":1467},[119],[31,51909,8059],{"className":51910},[75,76],[31,51912],{"className":51913,"style":1467},[119],[31,51915,1435],{"className":51916},[75,76],[31,51918,51920,51923],{"className":51919},[75],[31,51921,852],{"className":51922},[75,76],[31,51924,51926],{"className":51925},[1039],[31,51927,51929,51949],{"className":51928},[1043,1044],[31,51930,51932,51946],{"className":51931},[1048],[31,51933,51935],{"className":51934,"style":27736},[1052],[31,51936,51937,51940],{"style":2807},[31,51938],{"className":51939,"style":1061},[1060],[31,51941,51943],{"className":51942},[1065,1066,1067,1068],[31,51944,919],{"className":51945,"style":949},[75,76,1068],[31,51947,1076],{"className":51948},[1075],[31,51950,51952],{"className":51951},[1048],[31,51953,51955],{"className":51954,"style":2050},[1052],[31,51956],{},"，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 ",[31,51959,51961,51978],{"className":51960,"translate":35},[34],[31,51962,51964],{"className":51963},[39],[41,51965,51966],{"xmlns":43},[45,51967,51968,51976],{},[48,51969,51970],{},[1004,51971,51972,51974],{},[51,51973,3117],{},[91,51975,152],{},[55,51977,19087],{"encoding":57},[31,51979,51981],{"className":51980,"ariaHidden":62},[61],[31,51982,51984,51987],{"className":51983},[66],[31,51985],{"className":51986,"style":5230},[70],[31,51988,51990,51993],{"className":51989},[75],[31,51991,3117],{"className":51992,"style":949},[75,76],[31,51994,51996],{"className":51995},[1039],[31,51997,51999,52019],{"className":51998},[1043,1044],[31,52000,52002,52016],{"className":52001},[1048],[31,52003,52005],{"className":52004,"style":2028},[1052],[31,52006,52007,52010],{"style":1056},[31,52008],{"className":52009,"style":1061},[1060],[31,52011,52013],{"className":52012},[1065,1066,1067,1068],[31,52014,152],{"className":52015},[75,1068],[31,52017,1076],{"className":52018},[1075],[31,52020,52022],{"className":52021},[1048],[31,52023,52025],{"className":52024,"style":2050},[1052],[31,52026],{}," 空间里求“平均分布”，这正是 Fréchet 均值在分布空间里的实例。",[11,52029,52030],{},"OOD 检测，也就是我开头那个 0.99 的问题，我会分三层来做。",[11,52032,52033],{},"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。",[11,52035,52036,52037,52187],{},"点态一层比较样本密度：对每个类拟合 KDE 或流，取 ",[31,52038,52040,52075],{"className":52039,"translate":35},[34],[31,52041,52043],{"className":52042},[39],[41,52044,52045],{"xmlns":43},[45,52046,52047,52072],{},[48,52048,52049,52051,52053,52055,52066,52068,52070],{},[95,52050,97],{},[51,52052,19939],{},[95,52054,1540],{},[1004,52056,52057,52063],{},[42405,52058,52059,52061],{"accent":62},[51,52060,11],{},[95,52062,42411],{},[51,52064,52065],{},"k",[95,52067,211],{"stretchy":210},[51,52069,53],{},[95,52071,220],{"stretchy":210},[55,52073,52074],{"encoding":57},"-\\log\\hat p_k(x)",[31,52076,52078],{"className":52077,"ariaHidden":62},[61],[31,52079,52081,52084,52087,52090,52095,52098,52178,52181,52184],{"className":52080},[66],[31,52082],{"className":52083,"style":260},[70],[31,52085,97],{"className":52086},[75],[31,52088],{"className":52089,"style":1467},[119],[31,52091,20177,52093],{"className":52092},[1460],[31,52094,14556],{"style":20180},[31,52096],{"className":52097,"style":1467},[119],[31,52099,52101,52143],{"className":52100},[75],[31,52102,52104],{"className":52103},[75,42523],[31,52105,52107,52135],{"className":52106},[1043,1044],[31,52108,52110,52132],{"className":52109},[1048],[31,52111,52113,52121],{"className":52112,"style":1630},[1052],[31,52114,52115,52118],{"style":1649},[31,52116],{"className":52117,"style":1637},[1060],[31,52119,11],{"className":52120},[75,76],[31,52122,52123,52126],{"style":1649},[31,52124],{"className":52125,"style":1637},[1060],[31,52127,52129],{"className":52128,"style":42550},[42549],[31,52130,42411],{"className":52131},[75],[31,52133,1076],{"className":52134},[1075],[31,52136,52138],{"className":52137},[1048],[31,52139,52141],{"className":52140,"style":42563},[1052],[31,52142],{},[31,52144,52146],{"className":52145},[1039],[31,52147,52149,52170],{"className":52148},[1043,1044],[31,52150,52152,52167],{"className":52151},[1048],[31,52153,52155],{"className":52154,"style":1053},[1052],[31,52156,52157,52160],{"style":2807},[31,52158],{"className":52159,"style":1061},[1060],[31,52161,52163],{"className":52162},[1065,1066,1067,1068],[31,52164,52065],{"className":52165,"style":52166},[75,76,1068],"margin-right:0.0315em;",[31,52168,1076],{"className":52169},[1075],[31,52171,52173],{"className":52172},[1048],[31,52174,52176],{"className":52175,"style":2050},[1052],[31,52177],{},[31,52179,211],{"className":52180},[264],[31,52182,53],{"className":52183},[75,76],[31,52185,220],{"className":52186},[289]," 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。",[11,52189,52190,52191,52260,52261,52337,52338,52366,52367,1138,52395,52439],{},"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 ",[31,52192,52194,52211],{"className":52193,"translate":35},[34],[31,52195,52197],{"className":52196},[39],[41,52198,52199],{"xmlns":43},[45,52200,52201,52209],{},[48,52202,52203],{},[1004,52204,52205,52207],{},[51,52206,3117],{},[91,52208,152],{},[55,52210,19087],{"encoding":57},[31,52212,52214],{"className":52213,"ariaHidden":62},[61],[31,52215,52217,52220],{"className":52216},[66],[31,52218],{"className":52219,"style":5230},[70],[31,52221,52223,52226],{"className":52222},[75],[31,52224,3117],{"className":52225,"style":949},[75,76],[31,52227,52229],{"className":52228},[1039],[31,52230,52232,52252],{"className":52231},[1043,1044],[31,52233,52235,52249],{"className":52234},[1048],[31,52236,52238],{"className":52237,"style":2028},[1052],[31,52239,52240,52243],{"style":1056},[31,52241],{"className":52242,"style":1061},[1060],[31,52244,52246],{"className":52245},[1065,1066,1067,1068],[31,52247,152],{"className":52248},[75,1068],[31,52250,1076],{"className":52251},[1075],[31,52253,52255],{"className":52254},[1048],[31,52256,52258],{"className":52257,"style":2050},[1052],[31,52259],{},"，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 ",[31,52262,52264,52290],{"className":52263,"translate":35},[34],[31,52265,52267],{"className":52266},[39],[41,52268,52269],{"xmlns":43},[45,52270,52271,52287],{},[48,52272,52273],{},[1180,52274,52275,52277],{},[51,52276,14559],{},[48,52278,52279,52281,52283,52285],{},[95,52280,97],{},[91,52282,205],{},[51,52284,3768],{"mathvariant":494},[51,52286,1435],{},[55,52288,52289],{"encoding":57},"n^{-1\u002Fd}",[31,52291,52293],{"className":52292,"ariaHidden":62},[61],[31,52294,52296,52299],{"className":52295},[66],[31,52297],{"className":52298,"style":22855},[70],[31,52300,52302,52305],{"className":52301},[75],[31,52303,14559],{"className":52304},[75,76],[31,52306,52308],{"className":52307},[1039],[31,52309,52311],{"className":52310},[1043],[31,52312,52314],{"className":52313},[1048],[31,52315,52317],{"className":52316,"style":22855},[1052],[31,52318,52319,52322],{"style":3631},[31,52320],{"className":52321,"style":1061},[1060],[31,52323,52325],{"className":52324},[1065,1066,1067,1068],[31,52326,52328,52331,52334],{"className":52327},[75,1068],[31,52329,97],{"className":52330},[75,1068],[31,52332,4042],{"className":52333},[75,1068],[31,52335,1435],{"className":52336},[75,76,1068]," 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 ",[31,52339,52341,52354],{"className":52340,"translate":35},[34],[31,52342,52344],{"className":52343},[39],[41,52345,52346],{"xmlns":43},[45,52347,52348,52352],{},[48,52349,52350],{},[51,52351,8059],{},[55,52353,11503],{"encoding":57},[31,52355,52357],{"className":52356,"ariaHidden":62},[61],[31,52358,52360,52363],{"className":52359},[66],[31,52361],{"className":52362,"style":865},[70],[31,52364,8059],{"className":52365},[75,76],"——一旦在一批数据上解出 ",[31,52368,52370,52383],{"className":52369,"translate":35},[34],[31,52371,52373],{"className":52372},[39],[41,52374,52375],{"xmlns":43},[45,52376,52377,52381],{},[48,52378,52379],{},[51,52380,8059],{},[55,52382,11503],{"encoding":57},[31,52384,52386],{"className":52385,"ariaHidden":62},[61],[31,52387,52389,52392],{"className":52388},[66],[31,52390],{"className":52391,"style":865},[70],[31,52393,8059],{"className":52394},[75,76],[31,52396,52398,52418],{"className":52397,"translate":35},[34],[31,52399,52401],{"className":52400},[39],[41,52402,52403],{"xmlns":43},[45,52404,52405,52415],{},[48,52406,52407,52409,52411,52413],{},[51,52408,8059],{},[95,52410,211],{"stretchy":210},[51,52412,53],{},[95,52414,220],{"stretchy":210},[55,52416,52417],{"encoding":57},"\\varphi(x)",[31,52419,52421],{"className":52420,"ariaHidden":62},[61],[31,52422,52424,52427,52430,52433,52436],{"className":52423},[66],[31,52425],{"className":52426,"style":260},[70],[31,52428,8059],{"className":52429},[75,76],[31,52431,211],{"className":52432},[264],[31,52434,53],{"className":52435},[75,76],[31,52437,220],{"className":52438},[289]," 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。",[11,52441,52442],{},"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。",[11,52444,52445],{},"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。",[18,52447,52448],{"id":52448},"回到开头",[11,52450,52451],{},"回到那个 0.99。",[11,52453,52454,52455,52483],{},"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 ",[31,52456,52458,52471],{"className":52457,"translate":35},[34],[31,52459,52461],{"className":52460},[39],[41,52462,52463],{"xmlns":43},[45,52464,52465,52469],{},[48,52466,52467],{},[91,52468,205],{},[55,52470,205],{"encoding":57},[31,52472,52474],{"className":52473,"ariaHidden":62},[61],[31,52475,52477,52480],{"className":52476},[66],[31,52478],{"className":52479,"style":326},[70],[31,52481,205],{"className":52482},[75],"。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。",[11,52485,52486],{},"如果只留一个式子，我会留 Kantorovich 对偶：",[31,52488,52490],{"className":52489,"translate":35},[411],[31,52491,52493,52573],{"className":52492,"translate":35},[34],[31,52494,52496],{"className":52495},[39],[41,52497,52498],{"xmlns":43,"display":420},[45,52499,52500,52570],{},[48,52501,52502,52508,52510,52512,52514,52516,52518,52520,52550,52552,52554,52556,52558,52560,52562,52564,52566,52568],{},[1004,52503,52504,52506],{},[51,52505,3117],{},[91,52507,205],{},[95,52509,211],{"stretchy":210},[51,52511,852],{},[95,52513,1210],{"separator":62},[51,52515,884],{},[95,52517,220],{"stretchy":210},[95,52519,223],{},[1531,52521,52522,52528],{},[48,52523,52524,52526],{},[51,52525,10792],{},[95,52527,1540],{},[48,52529,52530,52532,52534,52546,52548],{},[51,52531,4572],{"mathvariant":494},[51,52533,8059],{},[1004,52535,52536,52538],{},[51,52537,4572],{"mathvariant":494},[48,52539,52540,52542,52544],{},[51,52541,4561],{"mathvariant":494},[51,52543,5391],{"mathvariant":494},[51,52545,11],{"mathvariant":494},[95,52547,347],{},[91,52549,205],{},[95,52551,1409],{},[51,52553,8059],{},[1430,52555,1432],{},[51,52557,1435],{},[95,52559,211],{"stretchy":210},[51,52561,852],{},[95,52563,97],{},[51,52565,884],{},[95,52567,220],{"stretchy":210},[51,52569,495],{"mathvariant":494},[55,52571,52572],{"encoding":57},"W_1(\\mu,\\nu)=\\sup_{\\|\\varphi\\|_{\\mathrm{Lip}}\\le1}\\int\\varphi\\,d(\\mu-\\nu).",[31,52574,52576,52649,52790],{"className":52575,"ariaHidden":62},[61],[31,52577,52579,52582,52622,52625,52628,52631,52634,52637,52640,52643,52646],{"className":52578},[66],[31,52580],{"className":52581,"style":260},[70],[31,52583,52585,52588],{"className":52584},[75],[31,52586,3117],{"className":52587,"style":949},[75,76],[31,52589,52591],{"className":52590},[1039],[31,52592,52594,52614],{"className":52593},[1043,1044],[31,52595,52597,52611],{"className":52596},[1048],[31,52598,52600],{"className":52599,"style":2028},[1052],[31,52601,52602,52605],{"style":1056},[31,52603],{"className":52604,"style":1061},[1060],[31,52606,52608],{"className":52607},[1065,1066,1067,1068],[31,52609,205],{"className":52610},[75,1068],[31,52612,1076],{"className":52613},[1075],[31,52615,52617],{"className":52616},[1048],[31,52618,52620],{"className":52619,"style":2050},[1052],[31,52621],{},[31,52623,211],{"className":52624},[264],[31,52626,852],{"className":52627},[75,76],[31,52629,1210],{"className":52630},[1391],[31,52632],{"className":52633,"style":1467},[119],[31,52635,884],{"className":52636,"style":900},[75,76],[31,52638,220],{"className":52639},[289],[31,52641],{"className":52642,"style":293},[119],[31,52644,223],{"className":52645},[297],[31,52647],{"className":52648,"style":293},[119],[31,52650,52652,52655,52757,52760,52763,52766,52769,52772,52775,52778,52781,52784,52787],{"className":52651},[66],[31,52653],{"className":52654,"style":11713},[70],[31,52656,52658],{"className":52657},[1460,1620],[31,52659,52661,52749],{"className":52660},[1043,1044],[31,52662,52664,52746],{"className":52663},[1048],[31,52665,52667,52736],{"className":52666,"style":71},[1052],[31,52668,52669,52672],{"style":11728},[31,52670],{"className":52671,"style":1637},[1060],[31,52673,52675],{"className":52674},[1065,1066,1067,1068],[31,52676,52678,52681,52684,52730,52733],{"className":52677},[75,1068],[31,52679,4572],{"className":52680},[75,1068],[31,52682,8059],{"className":52683},[75,76,1068],[31,52685,52687,52690],{"className":52686},[75,1068],[31,52688,4572],{"className":52689},[75,1068],[31,52691,52693],{"className":52692},[1039],[31,52694,52696,52722],{"className":52695},[1043,1044],[31,52697,52699,52719],{"className":52698},[1048],[31,52700,52702],{"className":52701,"style":11762},[1052],[31,52703,52704,52707],{"style":11765},[31,52705],{"className":52706,"style":11769},[1060],[31,52708,52710],{"className":52709},[1065,11773,1308,1068],[31,52711,52713],{"className":52712},[75,1068],[31,52714,52716],{"className":52715},[75,1068],[31,52717,11783],{"className":52718},[75,2745,1068],[31,52720,1076],{"className":52721},[1075],[31,52723,52725],{"className":52724},[1048],[31,52726,52728],{"className":52727,"style":11793},[1052],[31,52729],{},[31,52731,347],{"className":52732},[297,1068],[31,52734,205],{"className":52735},[75,1068],[31,52737,52738,52741],{"style":1649},[31,52739],{"className":52740,"style":1637},[1060],[31,52742,52743],{},[31,52744,10792],{"className":52745},[1460],[31,52747,1076],{"className":52748},[1075],[31,52750,52752],{"className":52751},[1048],[31,52753,52755],{"className":52754,"style":11821},[1052],[31,52756],{},[31,52758],{"className":52759,"style":1467},[119],[31,52761,1409],{"className":52762,"style":1688},[1460,1461,1687],[31,52764],{"className":52765,"style":1467},[119],[31,52767,8059],{"className":52768},[75,76],[31,52770],{"className":52771,"style":1467},[119],[31,52773,1435],{"className":52774},[75,76],[31,52776,211],{"className":52777},[264],[31,52779,852],{"className":52780},[75,76],[31,52782],{"className":52783,"style":120},[119],[31,52785,97],{"className":52786},[124],[31,52788],{"className":52789,"style":120},[119],[31,52791,52793,52796,52799,52802],{"className":52792},[66],[31,52794],{"className":52795,"style":260},[70],[31,52797,884],{"className":52798,"style":900},[75,76],[31,52800,220],{"className":52801},[289],[31,52803,495],{"className":52804},[75],[11,52806,52807],{},"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。",[11,52809,52810],{},"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":52812,"searchDepth":52813,"depth":52813,"links":52814},"",2,[52815,52816,52817,52818,52819,52820,52821,52822,52823,52824,52825],{"id":20,"depth":52813,"text":20},{"id":5303,"depth":52813,"text":5303},{"id":12250,"depth":52813,"text":12250},{"id":19440,"depth":52813,"text":19440},{"id":22894,"depth":52813,"text":22894},{"id":29405,"depth":52813,"text":29405},{"id":33911,"depth":52813,"text":33912},{"id":41924,"depth":52813,"text":41924},{"id":45603,"depth":52813,"text":45603},{"id":50810,"depth":52813,"text":50810},{"id":52448,"depth":52813,"text":52448},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun",{"title":6,"description":13},"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。",[52836,52837,52838],"mathematics","machine-learning","statistics","2026-10-31T20:17:23+08:00","IAT5rYeOtscnEe8Mhh1eCeIF4PJR1Ewyc86Z_G_0ncI",{"id":52842,"title":52843,"body":52844,"description":52848,"draft":52826,"enableCodeHighlight":52827,"enableCodeHightlight":52826,"enableComment":52827,"enableD2":52827,"enableKatex":52827,"extension":52828,"image":52812,"important":52826,"location":52829,"meta":72545,"navigation":52827,"ogImage":52829,"path":72546,"seo":72547,"stem":72548,"summary":72549,"tags":72550,"time":72552,"__hash__":72553},"blog\u002Fblog\u002F2026\u002F0013-satisfiability-modulo-theories.md","可满足性模理论 (Satisfiability Modulo Theories)",{"type":8,"value":52845,"toc":72506},[52846,52849,52852,52994,53005,53008,53156,53159,53167,53189,53195,53199,53204,53239,53242,53245,53368,53430,53601,53605,53608,53934,54481,54484,55163,55529,55588,55599,55605,55609,55612,55820,55826,55858,55861,55864,55868,55872,55879,55925,56055,56678,57094,57097,57100,57128,57236,57454,57457,57461,57464,57467,57658,57661,58036,58226,58229,58232,58236,58239,58242,58413,58416,58843,59103,59239,59246,59250,59256,59259,59296,59299,59908,59911,59914,59918,59921,59935,59941,59945,59949,59956,59959,60058,60090,60109,60112,60116,60122,60306,60309,60479,60482,60916,61016,61023,61154,61158,61164,61171,61174,61344,61789,61792,61795,61979,62079,62085,62274,62278,62284,62360,62366,62370,62377,62380,62393,62396,62400,62404,62415,62422,62425,62446,62449,62453,62456,62459,62675,62678,62932,62935,63031,63238,63245,63401,63481,63789,63793,63796,63803,63806,63826,63829,63833,63869,63872,63877,63880,64020,64023,64249,64470,64473,64476,64481,64617,64619,64812,64936,65338,65341,65343,65348,65351,65550,65553,65560,65821,65964,65966,65974,65985,65987,66288,66508,66511,66517,66521,66528,66531,66686,66689,66918,66921,67032,67093,67096,67100,67103,67318,67771,67899,68055,68149,68272,68275,68282,68285,68322,68326,68332,68360,68363,68366,68370,68374,68377,68383,68386,68810,68896,68905,68911,68915,68918,69007,69024,69035,69166,69169,69361,69364,69396,69399,69405,69541,69544,69584,69587,69780,69786,69800,69838,69855,69859,69952,70030,70033,70141,70144,70535,70599,70602,70644,70647,70653,70656,70664,70860,70946,70956,70960,70963,71028,71039,71130,71136,71283,71289,71292,71358,71448,71505,71515,71522,71526,71600,71603,71712,71722,71725,71765,71768,71931,71936,71947,72267,72358,72362,72365,72372,72378,72385,72389,72492,72502],[11,52847,52848],{},"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。",[11,52850,52851],{},"看一段再普通不过的代码：",[52853,52854,52857],"pre",{"className":52855,"code":52856,"language":1412,"meta":52812,"style":52812},"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",[45608,52858,52859,52892,52918,52938,52975,52982,52988],{"__ignoreMap":52812},[31,52860,52863,52867,52871,52874,52877,52881,52884,52886,52889],{"class":52861,"line":52862},"line",1,[31,52864,52866],{"class":52865},"s8jYJ","void",[31,52868,52870],{"class":52869},"snPdu"," f",[31,52872,211],{"class":52873},"sxrX7",[31,52875,52876],{"class":52865},"int",[31,52878,52880],{"class":52879},"sP4rz"," x",[31,52882,52883],{"class":52873},", ",[31,52885,52876],{"class":52865},[31,52887,52888],{"class":52879}," y",[31,52890,52891],{"class":52873},") {\n",[31,52893,52894,52897,52900,52902,52905,52908,52910,52913,52915],{"class":52861,"line":52813},[31,52895,52896],{"class":52865},"    int",[31,52898,52899],{"class":52879}," a",[31,52901,15175],{"class":52873},[31,52903,22616],{"class":52904},"sBjJW",[31,52906,52907],{"class":52873},"] ",[31,52909,223],{"class":52865},[31,52911,52912],{"class":52873}," {",[31,52914,1977],{"class":52904},[31,52916,52917],{"class":52873},"};\n",[31,52919,52921,52924,52926,52928,52930,52932,52935],{"class":52861,"line":52920},3,[31,52922,52923],{"class":52879},"    a",[31,52925,15175],{"class":52873},[31,52927,1977],{"class":52904},[31,52929,52907],{"class":52873},[31,52931,223],{"class":52865},[31,52933,52934],{"class":52904}," 5",[31,52936,52937],{"class":52873},";\n",[31,52939,52941,52944,52947,52949,52952,52954,52957,52960,52962,52964,52966,52968,52971,52973],{"class":52861,"line":52940},4,[31,52942,52943],{"class":52865},"    if",[31,52945,52946],{"class":52873}," (x ",[31,52948,434],{"class":52865},[31,52950,52951],{"class":52873}," y ",[31,52953,16040],{"class":52865},[31,52955,52956],{"class":52904}," 10",[31,52958,52959],{"class":52865}," &&",[31,52961,52899],{"class":52879},[31,52963,15175],{"class":52873},[31,52965,1977],{"class":52904},[31,52967,52907],{"class":52873},[31,52969,52970],{"class":52865},"==",[31,52972,52934],{"class":52904},[31,52974,52891],{"class":52873},[31,52976,52978],{"class":52861,"line":52977},5,[31,52979,52981],{"class":52980},"sCsY4","        \u002F\u002F 目标分支\n",[31,52983,52985],{"class":52861,"line":52984},6,[31,52986,52987],{"class":52873},"    }\n",[31,52989,52991],{"class":52861,"line":52990},7,[31,52992,52993],{"class":52873},"}\n",[11,52995,52996,52997,15282,52999,53001,53002],{},"假设 ",[45608,52998,53],{},[45608,53000,3003],{}," 是来自外部的整数。我们想知道：",[7246,53003,53004],{},"有没有一组输入，能走进这个分支？",[11,53006,53007],{},"沿着这条路径，把进入分支需要满足的条件收集起来：",[31,53009,53011],{"className":53010,"translate":35},[411],[31,53012,53014,53056],{"className":53013,"translate":35},[34],[31,53015,53017],{"className":53016},[39],[41,53018,53019],{"xmlns":43,"display":420},[45,53020,53021,53053],{},[48,53022,53023,53025,53027,53029,53031,53033,53036,53039,53041,53043,53045,53047,53049,53051],{},[51,53024,53],{},[95,53026,434],{},[51,53028,3003],{},[95,53030,16040],{},[91,53032,19494],{},[1430,53034,53035],{},"  ",[95,53037,53038],{},"∧",[1430,53040,53035],{},[51,53042,5813],{},[95,53044,15175],{"stretchy":210},[91,53046,1977],{},[95,53048,15184],{"stretchy":210},[95,53050,223],{},[91,53052,7853],{},[55,53054,53055],{"encoding":57},"x + y > 10 \\;\\wedge\\; a[0] = 5",[31,53057,53059,53077,53096,53120,53147],{"className":53058,"ariaHidden":62},[61],[31,53060,53062,53065,53068,53071,53074],{"className":53061},[66],[31,53063],{"className":53064,"style":307},[70],[31,53066,53],{"className":53067},[75,76],[31,53069],{"className":53070,"style":120},[119],[31,53072,434],{"className":53073},[124],[31,53075],{"className":53076,"style":120},[119],[31,53078,53080,53084,53087,53090,53093],{"className":53079},[66],[31,53081],{"className":53082,"style":53083},[70],"height:0.7335em;vertical-align:-0.1944em;",[31,53085,3003],{"className":53086,"style":2666},[75,76],[31,53088],{"className":53089,"style":293},[119],[31,53091,16040],{"className":53092},[297],[31,53094],{"className":53095,"style":293},[119],[31,53097,53099,53102,53105,53108,53111,53114,53117],{"className":53098},[66],[31,53100],{"className":53101,"style":326},[70],[31,53103,19494],{"className":53104},[75],[31,53106],{"className":53107,"style":293},[119],[31,53109],{"className":53110,"style":120},[119],[31,53112,53038],{"className":53113},[124],[31,53115],{"className":53116,"style":293},[119],[31,53118],{"className":53119,"style":120},[119],[31,53121,53123,53126,53129,53132,53135,53138,53141,53144],{"className":53122},[66],[31,53124],{"className":53125,"style":260},[70],[31,53127,5813],{"className":53128},[75,76],[31,53130,15175],{"className":53131},[264],[31,53133,1977],{"className":53134},[75],[31,53136,15184],{"className":53137},[289],[31,53139],{"className":53140,"style":293},[119],[31,53142,223],{"className":53143},[297],[31,53145],{"className":53146,"style":293},[119],[31,53148,53150,53153],{"className":53149},[66],[31,53151],{"className":53152,"style":326},[70],[31,53154,7853],{"className":53155},[75],[11,53157,53158],{},"问题于是变成：这组约束有没有解？",[11,53160,53161,53162,15282,53164,53166],{},"纯 SAT 求解器也能做，但你得先把 ",[45608,53163,53],{},[45608,53165,3003],{}," 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。",[11,53168,53169,53170,53173,53174,53177,53178,53181,53182,53184,53185,53188],{},"SMT 求解器则直接读原式：",[45608,53171,53172],{},"x + y > 10"," 交给算术，",[45608,53175,53176],{},"a[0] = 5"," 交给数组。算术说取 ",[45608,53179,53180],{},"x = 6, y = 5"," 就行，数组说 ",[45608,53183,53176],{}," 本来就成立。两边一对，返回 ",[7246,53186,53187],{},"sat","，并给出这组解。",[11,53190,53191,53192],{},"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。",[7246,53193,53194],{},"SMT 要做的，就是后者。",[18,53196,53198],{"id":53197},"一smt-是什么","一、SMT 是什么",[53200,53201,53203],"h3",{"id":53202},"_11-定义","1.1 定义",[11,53205,53206,53207],{},"把上面的过程抽象一下。",[7246,53208,53209,53210,53238],{},"SMT 是「SAT + 背景理论」：在某个理论 ",[31,53211,53213,53226],{"className":53212,"translate":35},[34],[31,53214,53216],{"className":53215},[39],[41,53217,53218],{"xmlns":43},[45,53219,53220,53224],{},[48,53221,53222],{},[51,53223,919],{},[55,53225,919],{"encoding":57},[31,53227,53229],{"className":53228,"ariaHidden":62},[61],[31,53230,53232,53235],{"className":53231},[66],[31,53233],{"className":53234,"style":945},[70],[31,53236,919],{"className":53237,"style":949},[75,76]," 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。",[11,53240,53241],{},"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。",[11,53243,53244],{},"形式化一点，SMT 要回答的是：",[31,53246,53248],{"className":53247,"translate":35},[411],[31,53249,53251,53285],{"className":53250,"translate":35},[34],[31,53252,53254],{"className":53253},[39],[41,53255,53256],{"xmlns":43,"display":420},[45,53257,53258,53282],{},[48,53259,53260,53263,53265,53267,53269,53271,53273,53280],{},[51,53261,53262],{"mathvariant":494},"∃",[1430,53264,1432],{},[51,53266,17342],{},[51,53268,495],{"mathvariant":494},[119,53270],{"width":1584},[51,53272,17342],{},[1004,53274,53275,53278],{},[95,53276,53277],{},"⊨",[51,53279,919],{},[51,53281,8059],{},[55,53283,53284],{"encoding":57},"\\exists\\, \\sigma.\\quad \\sigma \\models_T \\varphi",[31,53286,53288,53359],{"className":53287,"ariaHidden":62},[61],[31,53289,53291,53295,53298,53301,53304,53307,53310,53313,53316,53356],{"className":53290},[66],[31,53292],{"className":53293,"style":53294},[70],"height:0.999em;vertical-align:-0.249em;",[31,53296,53262],{"className":53297},[75],[31,53299],{"className":53300,"style":1467},[119],[31,53302,17342],{"className":53303,"style":2666},[75,76],[31,53305,495],{"className":53306},[75],[31,53308],{"className":53309,"style":1789},[119],[31,53311,17342],{"className":53312,"style":2666},[75,76],[31,53314],{"className":53315,"style":293},[119],[31,53317,53319,53322],{"className":53318},[297],[31,53320,53277],{"className":53321},[297],[31,53323,53325],{"className":53324},[1039],[31,53326,53328,53348],{"className":53327},[1043,1044],[31,53329,53331,53345],{"className":53330},[1048],[31,53332,53334],{"className":53333,"style":27736},[1052],[31,53335,53336,53339],{"style":2807},[31,53337],{"className":53338,"style":1061},[1060],[31,53340,53342],{"className":53341},[1065,1066,1067,1068],[31,53343,919],{"className":53344,"style":949},[75,76,1068],[31,53346,1076],{"className":53347},[1075],[31,53349,53351],{"className":53350},[1048],[31,53352,53354],{"className":53353,"style":2050},[1052],[31,53355],{},[31,53357],{"className":53358,"style":293},[119],[31,53360,53362,53365],{"className":53361},[66],[31,53363],{"className":53364,"style":865},[70],[31,53366,8059],{"className":53367},[75,76],[11,53369,53370,53371,53400,53401,53429],{},"也就是：是否存在一个赋值 ",[31,53372,53374,53388],{"className":53373,"translate":35},[34],[31,53375,53377],{"className":53376},[39],[41,53378,53379],{"xmlns":43},[45,53380,53381,53385],{},[48,53382,53383],{},[51,53384,17342],{},[55,53386,53387],{"encoding":57},"\\sigma",[31,53389,53391],{"className":53390,"ariaHidden":62},[61],[31,53392,53394,53397],{"className":53393},[66],[31,53395],{"className":53396,"style":71},[70],[31,53398,17342],{"className":53399,"style":2666},[75,76],"，把每个变量映射到理论域里的值，使得 ",[31,53402,53404,53417],{"className":53403,"translate":35},[34],[31,53405,53407],{"className":53406},[39],[41,53408,53409],{"xmlns":43},[45,53410,53411,53415],{},[48,53412,53413],{},[51,53414,8059],{},[55,53416,11503],{"encoding":57},[31,53418,53420],{"className":53419,"ariaHidden":62},[61],[31,53421,53423,53426],{"className":53422},[66],[31,53424],{"className":53425,"style":865},[70],[31,53427,8059],{"className":53428},[75,76]," 在这个理论的所有公理下为真。",[11,53431,53432,53433,53505,53506,53571,53572,53600],{},"回到开头的例子：",[31,53434,53436,53457],{"className":53435,"translate":35},[34],[31,53437,53439],{"className":53438},[39],[41,53440,53441],{"xmlns":43},[45,53442,53443,53455],{},[48,53444,53445,53447,53449,53451,53453],{},[51,53446,53],{},[95,53448,434],{},[51,53450,3003],{},[95,53452,16040],{},[91,53454,19494],{},[55,53456,53172],{"encoding":57},[31,53458,53460,53478,53496],{"className":53459,"ariaHidden":62},[61],[31,53461,53463,53466,53469,53472,53475],{"className":53462},[66],[31,53464],{"className":53465,"style":307},[70],[31,53467,53],{"className":53468},[75,76],[31,53470],{"className":53471,"style":120},[119],[31,53473,434],{"className":53474},[124],[31,53476],{"className":53477,"style":120},[119],[31,53479,53481,53484,53487,53490,53493],{"className":53480},[66],[31,53482],{"className":53483,"style":53083},[70],[31,53485,3003],{"className":53486,"style":2666},[75,76],[31,53488],{"className":53489,"style":293},[119],[31,53491,16040],{"className":53492},[297],[31,53494],{"className":53495,"style":293},[119],[31,53497,53499,53502],{"className":53498},[66],[31,53500],{"className":53501,"style":326},[70],[31,53503,19494],{"className":53504},[75]," 是算术原子，",[31,53507,53509,53532],{"className":53508,"translate":35},[34],[31,53510,53512],{"className":53511},[39],[41,53513,53514],{"xmlns":43},[45,53515,53516,53530],{},[48,53517,53518,53520,53522,53524,53526,53528],{},[51,53519,5813],{},[95,53521,15175],{"stretchy":210},[91,53523,1977],{},[95,53525,15184],{"stretchy":210},[95,53527,223],{},[91,53529,7853],{},[55,53531,53176],{"encoding":57},[31,53533,53535,53562],{"className":53534,"ariaHidden":62},[61],[31,53536,53538,53541,53544,53547,53550,53553,53556,53559],{"className":53537},[66],[31,53539],{"className":53540,"style":260},[70],[31,53542,5813],{"className":53543},[75,76],[31,53545,15175],{"className":53546},[264],[31,53548,1977],{"className":53549},[75],[31,53551,15184],{"className":53552},[289],[31,53554],{"className":53555,"style":293},[119],[31,53557,223],{"className":53558},[297],[31,53560],{"className":53561,"style":293},[119],[31,53563,53565,53568],{"className":53564},[66],[31,53566],{"className":53567,"style":326},[70],[31,53569,7853],{"className":53570},[75]," 是数组原子，理论 ",[31,53573,53575,53588],{"className":53574,"translate":35},[34],[31,53576,53578],{"className":53577},[39],[41,53579,53580],{"xmlns":43},[45,53581,53582,53586],{},[48,53583,53584],{},[51,53585,919],{},[55,53587,919],{"encoding":57},[31,53589,53591],{"className":53590,"ariaHidden":62},[61],[31,53592,53594,53597],{"className":53593},[66],[31,53595],{"className":53596,"style":945},[70],[31,53598,919],{"className":53599,"style":949},[75,76]," 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。",[53200,53602,53604],{"id":53603},"_12-公式的样子","1.2 公式的样子",[11,53606,53607],{},"SMT 公式由布尔结构和理论原子组成：",[31,53609,53611],{"className":53610,"translate":35},[411],[31,53612,53614,53701],{"className":53613,"translate":35},[34],[31,53615,53617],{"className":53616},[39],[41,53618,53619],{"xmlns":43,"display":420},[45,53620,53621,53698],{},[48,53622,53623,53625,53627,53629,53631,53633,53635,53637,53639,53642,53644,53646,53648,53650,53652,53654,53656,53659,53661,53663,53665,53667,53669,53671,53673,53675,53677,53679,53681,53684,53686,53688,53690,53692,53694,53696],{},[51,53624,8059],{},[95,53626,922],{},[95,53628,922],{},[95,53630,223],{},[51,53632,20800],{"mathvariant":494},[1430,53634,53035],{},[95,53636,12680],{},[1430,53638,53035],{},[51,53640,53641],{"mathvariant":494},"⊥",[1430,53643,53035],{},[95,53645,12680],{},[1430,53647,53035],{},[51,53649,18446],{},[1430,53651,53035],{},[95,53653,12680],{},[1430,53655,53035],{},[51,53657,53658],{"mathvariant":494},"¬",[51,53660,8059],{},[1430,53662,53035],{},[95,53664,12680],{},[1430,53666,53035],{},[51,53668,8059],{},[95,53670,53038],{},[51,53672,8059],{},[1430,53674,53035],{},[95,53676,12680],{},[1430,53678,53035],{},[51,53680,8059],{},[95,53682,53683],{},"∨",[51,53685,8059],{},[1430,53687,53035],{},[95,53689,12680],{},[1430,53691,53035],{},[51,53693,8059],{},[95,53695,929],{},[51,53697,8059],{},[55,53699,53700],{"encoding":57},"\\varphi ::= \\top \\;\\mid\\; \\bot \\;\\mid\\; A \\;\\mid\\; \\neg\\varphi \\;\\mid\\; \\varphi \\wedge \\varphi \\;\\mid\\; \\varphi \\vee \\varphi \\;\\mid\\; \\varphi \\rightarrow \\varphi",[31,53702,53704,53723,53747,53771,53795,53822,53841,53865,53883,53907,53925],{"className":53703,"ariaHidden":62},[61],[31,53705,53707,53710,53713,53716,53720],{"className":53706},[66],[31,53708],{"className":53709,"style":865},[70],[31,53711,8059],{"className":53712},[75,76],[31,53714],{"className":53715,"style":293},[119],[31,53717,53719],{"className":53718},[297],"::=",[31,53721],{"className":53722,"style":293},[119],[31,53724,53726,53729,53732,53735,53738,53741,53744],{"className":53725},[66],[31,53727],{"className":53728,"style":260},[70],[31,53730,20800],{"className":53731},[75],[31,53733],{"className":53734,"style":293},[119],[31,53736],{"className":53737,"style":293},[119],[31,53739,12680],{"className":53740},[297],[31,53742],{"className":53743,"style":293},[119],[31,53745],{"className":53746,"style":293},[119],[31,53748,53750,53753,53756,53759,53762,53765,53768],{"className":53749},[66],[31,53751],{"className":53752,"style":260},[70],[31,53754,53641],{"className":53755},[75],[31,53757],{"className":53758,"style":293},[119],[31,53760],{"className":53761,"style":293},[119],[31,53763,12680],{"className":53764},[297],[31,53766],{"className":53767,"style":293},[119],[31,53769],{"className":53770,"style":293},[119],[31,53772,53774,53777,53780,53783,53786,53789,53792],{"className":53773},[66],[31,53775],{"className":53776,"style":260},[70],[31,53778,18446],{"className":53779},[75,76],[31,53781],{"className":53782,"style":293},[119],[31,53784],{"className":53785,"style":293},[119],[31,53787,12680],{"className":53788},[297],[31,53790],{"className":53791,"style":293},[119],[31,53793],{"className":53794,"style":293},[119],[31,53796,53798,53801,53804,53807,53810,53813,53816,53819],{"className":53797},[66],[31,53799],{"className":53800,"style":260},[70],[31,53802,53658],{"className":53803},[75],[31,53805,8059],{"className":53806},[75,76],[31,53808],{"className":53809,"style":293},[119],[31,53811],{"className":53812,"style":293},[119],[31,53814,12680],{"className":53815},[297],[31,53817],{"className":53818,"style":293},[119],[31,53820],{"className":53821,"style":293},[119],[31,53823,53825,53829,53832,53835,53838],{"className":53824},[66],[31,53826],{"className":53827,"style":53828},[70],"height:0.75em;vertical-align:-0.1944em;",[31,53830,8059],{"className":53831},[75,76],[31,53833],{"className":53834,"style":120},[119],[31,53836,53038],{"className":53837},[124],[31,53839],{"className":53840,"style":120},[119],[31,53842,53844,53847,53850,53853,53856,53859,53862],{"className":53843},[66],[31,53845],{"className":53846,"style":260},[70],[31,53848,8059],{"className":53849},[75,76],[31,53851],{"className":53852,"style":293},[119],[31,53854],{"className":53855,"style":293},[119],[31,53857,12680],{"className":53858},[297],[31,53860],{"className":53861,"style":293},[119],[31,53863],{"className":53864,"style":293},[119],[31,53866,53868,53871,53874,53877,53880],{"className":53867},[66],[31,53869],{"className":53870,"style":53828},[70],[31,53872,8059],{"className":53873},[75,76],[31,53875],{"className":53876,"style":120},[119],[31,53878,53683],{"className":53879},[124],[31,53881],{"className":53882,"style":120},[119],[31,53884,53886,53889,53892,53895,53898,53901,53904],{"className":53885},[66],[31,53887],{"className":53888,"style":260},[70],[31,53890,8059],{"className":53891},[75,76],[31,53893],{"className":53894,"style":293},[119],[31,53896],{"className":53897,"style":293},[119],[31,53899,12680],{"className":53900},[297],[31,53902],{"className":53903,"style":293},[119],[31,53905],{"className":53906,"style":293},[119],[31,53908,53910,53913,53916,53919,53922],{"className":53909},[66],[31,53911],{"className":53912,"style":865},[70],[31,53914,8059],{"className":53915},[75,76],[31,53917],{"className":53918,"style":293},[119],[31,53920,929],{"className":53921},[297],[31,53923],{"className":53924,"style":293},[119],[31,53926,53928,53931],{"className":53927},[66],[31,53929],{"className":53930,"style":865},[70],[31,53932,8059],{"className":53933},[75,76],[11,53935,50599,53936,53964,53965,53968,53969,15282,54103,15282,54236,15282,54384,54480],{},[31,53937,53939,53952],{"className":53938,"translate":35},[34],[31,53940,53942],{"className":53941},[39],[41,53943,53944],{"xmlns":43},[45,53945,53946,53950],{},[48,53947,53948],{},[51,53949,18446],{},[55,53951,18446],{"encoding":57},[31,53953,53955],{"className":53954,"ariaHidden":62},[61],[31,53956,53958,53961],{"className":53957},[66],[31,53959],{"className":53960,"style":945},[70],[31,53962,18446],{"className":53963},[75,76]," 是",[7246,53966,53967],{},"理论原子","，形如 ",[31,53970,53972,53998],{"className":53971,"translate":35},[34],[31,53973,53975],{"className":53974},[39],[41,53976,53977],{"xmlns":43},[45,53978,53979,53995],{},[48,53980,53981,53987,53989],{},[1004,53982,53983,53985],{},[51,53984,12271],{},[91,53986,205],{},[95,53988,223],{},[1004,53990,53991,53993],{},[51,53992,12271],{},[91,53994,152],{},[55,53996,53997],{"encoding":57},"t_1 = t_2",[31,53999,54001,54057],{"className":54000,"ariaHidden":62},[61],[31,54002,54004,54008,54048,54051,54054],{"className":54003},[66],[31,54005],{"className":54006,"style":54007},[70],"height:0.7651em;vertical-align:-0.15em;",[31,54009,54011,54014],{"className":54010},[75],[31,54012,12271],{"className":54013},[75,76],[31,54015,54017],{"className":54016},[1039],[31,54018,54020,54040],{"className":54019},[1043,1044],[31,54021,54023,54037],{"className":54022},[1048],[31,54024,54026],{"className":54025,"style":2028},[1052],[31,54027,54028,54031],{"style":2807},[31,54029],{"className":54030,"style":1061},[1060],[31,54032,54034],{"className":54033},[1065,1066,1067,1068],[31,54035,205],{"className":54036},[75,1068],[31,54038,1076],{"className":54039},[1075],[31,54041,54043],{"className":54042},[1048],[31,54044,54046],{"className":54045,"style":2050},[1052],[31,54047],{},[31,54049],{"className":54050,"style":293},[119],[31,54052,223],{"className":54053},[297],[31,54055],{"className":54056,"style":293},[119],[31,54058,54060,54063],{"className":54059},[66],[31,54061],{"className":54062,"style":54007},[70],[31,54064,54066,54069],{"className":54065},[75],[31,54067,12271],{"className":54068},[75,76],[31,54070,54072],{"className":54071},[1039],[31,54073,54075,54095],{"className":54074},[1043,1044],[31,54076,54078,54092],{"className":54077},[1048],[31,54079,54081],{"className":54080,"style":2028},[1052],[31,54082,54083,54086],{"style":2807},[31,54084],{"className":54085,"style":1061},[1060],[31,54087,54089],{"className":54088},[1065,1066,1067,1068],[31,54090,152],{"className":54091},[75,1068],[31,54093,1076],{"className":54094},[1075],[31,54096,54098],{"className":54097},[1048],[31,54099,54101],{"className":54100,"style":2050},[1052],[31,54102],{},[31,54104,54106,54132],{"className":54105,"translate":35},[34],[31,54107,54109],{"className":54108},[39],[41,54110,54111],{"xmlns":43},[45,54112,54113,54129],{},[48,54114,54115,54121,54123],{},[1004,54116,54117,54119],{},[51,54118,12271],{},[91,54120,205],{},[95,54122,9758],{},[1004,54124,54125,54127],{},[51,54126,12271],{},[91,54128,152],{},[55,54130,54131],{"encoding":57},"t_1 \u003C 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属于",[7246,55243,55244],{},"线性整数算术","：它知道 ",[31,55247,55249,55262],{"className":55248,"translate":35},[34],[31,55250,55252],{"className":55251},[39],[41,55253,55254],{"xmlns":43},[45,55255,55256,55260],{},[48,55257,55258],{},[95,55259,434],{},[55,55261,434],{"encoding":57},[31,55263,55265],{"className":55264,"ariaHidden":62},[61],[31,55266,55268,55271],{"className":55267},[66],[31,55269],{"className":55270,"style":307},[70],[31,55272,434],{"className":55273},[75]," 是加法、",[31,55276,55278,55291],{"className":55277,"translate":35},[34],[31,55279,55281],{"className":55280},[39],[41,55282,55283],{"xmlns":43},[45,55284,55285,55289],{},[48,55286,55287],{},[95,55288,16040],{},[55,55290,16040],{"encoding":57},[31,55292,55294],{"className":55293,"ariaHidden":62},[61],[31,55295,55297,55300],{"className":55296},[66],[31,55298],{"className":55299,"style":21113},[70],[31,55301,16040],{"className":55302},[297]," 是全序；",[55167,55305,55306,55241,55371,55245,55374,2446,55376,55379],{},[31,55307,55309,55332],{"className":55308,"translate":35},[34],[31,55310,55312],{"className":55311},[39],[41,55313,55314],{"xmlns":43},[45,55315,55316,55330],{},[48,55317,55318,55320,55322,55324,55326,55328],{},[51,55319,5813],{},[95,55321,15175],{"stretchy":210},[91,55323,1977],{},[95,55325,15184],{"stretchy":210},[95,55327,223],{},[91,55329,7853],{},[55,55331,53176],{"encoding":57},[31,55333,55335,55362],{"className":55334,"ariaHidden":62},[61],[31,55336,55338,55341,55344,55347,55350,55353,55356,55359],{"className":55337},[66],[31,55339],{"className":55340,"style":260},[70],[31,55342,5813],{"className":55343},[75,76],[31,55345,15175],{"className":55346},[264],[31,55348,1977],{"className":55349},[75],[31,55351,15184],{"className":55352},[289],[31,55354],{"className":55355,"style":293},[119],[31,55357,223],{"className":55358},[297],[31,55360],{"className":55361,"style":293},[119],[31,55363,55365,55368],{"className":55364},[66],[31,55366],{"className":55367,"style":326},[70],[31,55369,7853],{"className":55370},[75],[7246,55372,55373],{},"数组理论",[45608,55375,54443],{},[45608,55377,55378],{},"store"," 的读写语义；",[55167,55381,55382,55241,55496,55499,55500,55528],{},[31,55383,55385,55415],{"className":55384,"translate":35},[34],[31,55386,55388],{"className":55387},[39],[41,55389,55390],{"xmlns":43},[45,55391,55392,55412],{},[48,55393,55394,55396,55398,55400,55402,55404,55406,55408,55410],{},[51,55395,1662],{},[95,55397,211],{"stretchy":210},[51,55399,53],{},[95,55401,220],{"stretchy":210},[95,55403,21271],{"mathvariant":494},[51,55405,1662],{},[95,55407,211],{"stretchy":210},[51,55409,3003],{},[95,55411,220],{"stretchy":210},[55,55413,55414],{"encoding":57},"f(x) \\ne f(y)",[31,55416,55418,55478],{"className":55417,"ariaHidden":62},[61],[31,55419,55421,55424,55427,55430,55433,55436,55439,55475],{"className":55420},[66],[31,55422],{"className":55423,"style":260},[70],[31,55425,1662],{"className":55426,"style":35196},[75,76],[31,55428,211],{"className":55429},[264],[31,55431,53],{"className":55432},[75,76],[31,55434,220],{"className":55435},[289],[31,55437],{"className":55438,"style":293},[119],[31,55440,55442,55469,55472],{"className":55441},[297],[31,55443,55445],{"className":55444},[297],[31,55446,55448],{"className":55447},[75,21516],[31,55449,55451],{"className":55450},[21520],[31,55452,55454,55457,55466],{"className":55453},[21524],[31,55455],{"className":55456,"style":4202},[70],[31,55458,55460],{"className":55459},[21531],[31,55461,55463],{"className":55462},[75],[31,55464,21538],{"className":55465},[297],[31,55467],{"className":55468},[21542],[31,55470],{"className":55471},[119,21546],[31,55473,223],{"className":55474},[297],[31,55476],{"className":55477,"style":293},[119],[31,55479,55481,55484,55487,55490,55493],{"className":55480},[66],[31,55482],{"className":55483,"style":260},[70],[31,55485,1662],{"className":55486,"style":35196},[75,76],[31,55488,211],{"className":55489},[264],[31,55491,3003],{"className":55492,"style":2666},[75,76],[31,55494,220],{"className":55495},[289],[7246,55497,55498],{},"未解释函数","：它不知道 ",[31,55501,55503,55516],{"className":55502,"translate":35},[34],[31,55504,55506],{"className":55505},[39],[41,55507,55508],{"xmlns":43},[45,55509,55510,55514],{},[48,55511,55512],{},[51,55513,1662],{},[55,55515,1662],{"encoding":57},[31,55517,55519],{"className":55518,"ariaHidden":62},[61],[31,55520,55522,55525],{"className":55521},[66],[31,55523],{"className":55524,"style":4202},[70],[31,55526,1662],{"className":55527,"style":35196},[75,76]," 具体算什么，只知道「同一个输入必须得到同一个输出」。",[11,55530,55531,55532,55587],{},"对 SAT 来说，这三个原子只是三个裸的布尔变量 ",[31,55533,55535,55557],{"className":55534,"translate":35},[34],[31,55536,55538],{"className":55537},[39],[41,55539,55540],{"xmlns":43},[45,55541,55542,55554],{},[48,55543,55544,55546,55548,55550,55552],{},[51,55545,11],{},[95,55547,1210],{"separator":62},[51,55549,27666],{},[95,55551,1210],{"separator":62},[51,55553,2549],{},[55,55555,55556],{"encoding":57},"p, q, r",[31,55558,55560],{"className":55559,"ariaHidden":62},[61],[31,55561,55563,55566,55569,55572,55575,55578,55581,55584],{"className":55562},[66],[31,55564],{"className":55565,"style":865},[70],[31,55567,11],{"className":55568},[75,76],[31,55570,1210],{"className":55571},[1391],[31,55573],{"className":55574,"style":1467},[119],[31,55576,27666],{"className":55577,"style":2666},[75,76],[31,55579,1210],{"className":55580},[1391],[31,55582],{"className":55583,"style":1467},[119],[31,55585,2549],{"className":55586,"style":19759},[75,76],"。对 SMT 来说，它们各自带着一整套内部结构。",[55589,55590,55591],"div",{},[55592,55593],"img",{"height":55594,"width":55595,"style":55596,"data-d2-theme":55597,"alt":52812,"src":55598},614,552,"max-width:100%;height:auto","default","data:image\u002Fsvg+xml;base64,PHN2ZyB3aWR0aD0iNTUyIiBoZWlnaHQ9IjYxNCIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIiBkYXRhLWQyLXZlcnNpb249InYwLjkuMCIgdmlld0JveD0iMCAwIDU1MiA2MTQiPjxzdmcgd2lkdGg9IjU1MiIgaGVpZ2h0PSI2MTQiIGNsYXNzPSJkMi0xNDUwNDQ0NTY2IGQyLXN2ZyIgdmlld0JveD0iLTEgLTEgNTUyIDYxNCI+PHJlY3Qgd2lkdGg9IjU1MiIgaGVpZ2h0PSI2MTQiIHg9Ii0xIiB5PSItMSIgc3Ryb2tlLXdpZHRoPSIwIiByeD0iMCIgc3R5bGU9ImZpbGw6I2ZmZiIvPjxzdHlsZT4uZDItMTQ1MDQ0NDU2NiAudGV4dC1ib2xke2ZvbnQtZmFtaWx5OiZxdW90O2QyLTE0NTA0NDQ1NjYtZm9udC1ib2xkJnF1b3Q7fUBmb250LWZhY2V7Zm9udC1mYW1pbHk6ZDItMTQ1MDQ0NDU2Ni1mb250LWJvbGQ7c3JjOnVybChkYXRhOmFwcGxpY2F0aW9uL2ZvbnQtd29mZjtiYXNlNjQsZDA5R1JnQUJBQUFBQUF4SUFBb0FBQUFBRW5BQUFndUZBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUJQVXk4eUFBQUE5QUFBQUdBQUFBQmdYeEhYcm1OdFlYQUFBQUZVQUFBQXF3QUFBT0lseENiM1oyeDVaZ0FBQWdBQUFBWE9BQUFIRExtU01jbG9aV0ZrQUFBSDBBQUFBRFlBQUFBMkczOGUxR2hvWldFQUFBZ0lBQUFBSkFBQUFDUUtmd1hiYUcxMGVBQUFDQ3dBQUFCd0FBQUFjRGI4QlF4c2IyTmhBQUFJbkFBQUFEb0FBQUE2SEV3YWttMWhlSEFBQUFqWUFBQUFJQUFBQUNBQU5BRDNibUZ0WlFBQUNQZ0FBQU12QUFBSUtnandWa0Z3YjNOMEFBQU1LQUFBQUNBQUFBQWcvOUVBTWdBREFpb0N2QUFGQUFBQ2lnSllBQUFBU3dLS0FsZ0FBQUZlQURJQktRQUFBZ3NIQXdNRUF3SUNCR0FBQXZjQUFBQURBQUFBQUFBQUFBQkJSRUpQQUNBQUlQLy9BdTcvQmdBQUE5Z0JFU0FBQVo4QUFBQUFBZkFDbEFBQUFDQUFBM2ljaE0wN0s0VnhBTWZ4eitQODNZLzczZlIwTXRCWkdNN0l4blltbWFRb21XWGloVmlzQnJ1U1Mza2JCam5KeW92NEtjcXE3L3FwTHlvTkZacUtGNnlvRlVWdFRkdUdqaTA3dXZic08zRGt4Sm1MMW1Ick9FRnRWZHU2amszYnVuYi8xS256WDVXUGZPWXI3M25MYTU1em45czg1aUUzdWM1VkxudFB2YnVmLy85VnBzMllOMkhTbEFXTCtqUVUvUVlNbWpWbnlMQVJvNXJHakZ1MnhEY0FBQUQvL3dNQTZvQXU1QUI0bkZ4VVcyd1QyUm4rLzVPeEozRk13dGcrTTNaOGkyZmlHY2VCWER5ZW1UUkpFNUlsRjlUWThhNEF3MktTSlNxSmdleEZRQlV1b1VnRmdWcThiQ3NRbXhZSnRSRkkxWXBLcVhoSjk2R3RxcjVVbTZkdVYwaFZsYTYyKzlCcTFVQ3JxTzIyc1YyZHdaSFFQdGh6SHY3emY1Zi8rdzg0SUFkQVpzbGRxSU1HYUFZUFVBQmRpQWx4WGRNVTN0SXRTNUhxTEEwRlBrYzhsVWNQdFhhdXZaMUx0aTVITDgzTVlHYWEzQzB2SE12TXp2NXJwcisvOHVBWEgxYmV4WE1mQWlCa0FQQVpLVUdqM1kvR3FFNFZHcU1aWEs3OGIyTURtMGxwNmRybGUwdXdVL3RYVW9LR2wyc3plTC95bjg4K0k2V2w5NWZLc0ZOSGZrWktFR1YxdWxjVUpkMDBMYTh1S0ViYU5DMkY1eFZOVXlLRTBzeFBUcms4THM0bHVPWi9mSU52cU9PTTQ2OGVUM05jUFU5S2xZM1FZQ1F5R0VLNWZQNVo2MVF1ZXYvTEwrOUhjMU90enhpR1V0MGlMcklNU1FDSHJHcVdLT29wMDBpcm10WkpHSXFlRWlWZVZSWFpTWDJpSklraTlUbWQ2TnQzTlhWUU9aem8zS3QzSElvTnFQMm45dmUray94RzZ6NU4zZnUxNU1IK3NiNjMzTjJkSnlPcUhJNkdQVzFOWFdOZDVwSDBudVR4UURBYWlrUUUyWDl3MUN6MEFrSUFnSGhKQ1hpbVVURmlWQkhXbitCL241RGRTMHZsNXk5OE9GRGRJbE9rQklMTjBSQjB3V2V6WkFjblpxL2V1TnRuV1FPM3I3bnZQY1RweXAwM0ppZmZ3RGNyS3cvdkFZRmtkUXYvZ05zUUFBVkFrbFVqYlZxMkhGNnp4VkZCMFJTbjAwcVpsdUZrR24rNVAzZjlEbEhhby92YWpLNHpmVE56RjF4Y2RMdytFUGRtQjZMdS9GRDJTSE5NODlNVDRiYTN6bFkrMTBQS1djbWJkM1dFL1pMTjFRK0E1OG52d1A5Q2ovVVZEMWtzaE5kSFJ0cHlyMFRUdTRPN1d0ekJ5TkdqZU9WTlI5QTRuSFk3Rnh5T21CbzVWN2tHUUdDNE9rRkUzQVl2UkFBa1FXZmpaNVI5VHFjaXE1cEV2V3hVaXN4VFVSeSs0T0xDQjdUWGl3TXpadXRBaTJOS05ROTNKSDJKTmZMVG5oYmx1K2NPWFJnS0JxWitnRzFqa3pmMmZ1UnBZdjIxNmdSdTIvMWJBUnlHWmJmZElheGJ1bERIWE5rWlBwNXlCa2Jrd3R2OU0rbjJ3VERuUHZlNVgvTjRPM3lLMmJYMi9xUS9ldXZpcTk4YURMZjJsSTlnbXkvd2thZnBsZkVEbzdZbnd3RDRIbTZEaDNteW82TG1DQzh3NXNGSmxZWmMvbDJCM2FHdisvQjVQdFhqY0Z6bHVQWlU1Vk5BYUFMQUxYek9zcUo3TlYycUxZS2w4NUtpcVNxYkg4ODNMYi8zWUk5TGRISDFubnA1K2ZzL2ZORHRsdHhjZzY5QlE3S1pveDJVZHRCYzlSK3YwVDJVZG9pdk1WN3U2aUNXOFRrRTdXVFVzbTlaZFM4aDFEV1JDMktzdVlYMzFNY1RMdjVYZDhjYlBTNnVYbWdZZVBleDFEdjFHeWYzRGpyYXdpMzRsMC9rc2JneXJueFNhUnc4bEdTNUpSQ3ZidUVYNUNZME1nU1czSFJ0ZkZyS1pEdkdqR1hPaWxnL2QrWEtIUHNGRXBLVUNQZ1Rmbi9DL2NIS3lxTkhLeXNmbkkxUDUvTUZXUzdrODlOeHhuc01BUDlJTG9NYlFEZFl5a3pUMGdXZGpuMXZNVDBoTHl3dTR0dkhYQ0ZmZVh1UjhVQ1FxbHY0S2JrTTBaZDFHbWtHejJ2ZUdGVjR4c3ZPMWIrekMrcEllSCtpcDY5M1R5Z2VIdkZnOFcrTk1kVTYxanQ4MnAyT0gyK0pwM3E2VTAyZUpBNHZMVFluOC92SHZwbTJNU2FxR1V5UVA4TXVBRWxsU2kzSlhpZHBmV2gwZEtoZ3BWTFdrL21ONjljMzV0VVRUOCtjZmpvTENOM1ZETzZ1M2RGc1I5Z01xTTlaS3ZTbVVyMkZvZEhSSityczA5Tm5ucDVRN2J1QWtLOFdNVWwreTk0TFNXRExwTlA4eHlkUEx0Y1ZzdVY5MlJkNk82dEZ4RnFON1k4dWRNN05mYnljSmIvT2JqOThVU1BETkc0U2s3M1hscUVZdXFHejFhUy9YMTFkV0YyZFhpdXVyUlhYb0ZvRkdWWndFLzlKVkFqQ2VYQkNFRXExKzBYY0pOM3NEYmNNeXpJMHpYQlFlZlhtelZWY0wxN3M2K0s2K2k3K2ZJNWhFWkFCYXJXN0FPSnhJMDU1bmtvU0dqR0ttNVUwcmorK2Rlc3grLzdvOXFYYkZ6dGxUdTZzL0wxWXJPRWtjSk1zZ0JmQXNoUkR0eFFqWnVtOFRoVmVwekUrUFg3MHpvRS9JWjJZbm1DSHloY1RpVXZ6MzVtL1pQL1o5MUhFVGZ3MjFBRjRqUmlWY1IzRlloRUEvZzhBQVAvL0F3QzZqNUVuQUFBQUFRQUFBQUlMaFNSUXM3VmZEenoxQUFFRDZBQUFBQURZWGFDRUFBQUFBTjFtTHpiK04vN0VDRzBEOFFBQkFBTUFBZ0FBQUFBQUFBQUJBQUFEMlA3dkFBQUltUDQzL2pjSWJRQUJBQUFBQUFBQUFBQUFBQUFBQUFBQUhBS3lBRkFBeUFBQUFpUUFUUUlNQUUwQytnQk5BaXdBSXdJc0FCa0NtUUJKQWc4QUtnRlZBQmdDUFFCQkFqMEFKd0dPQUVFQ0FnQU9BZ2tBREFJUUFDVUNFQUJHQWhBQUZ3RllBRWdCV0FBd0FWZ0FWd0ZZQUM0Q0VBQWlBaEFBSWdJUUFDSUNFQUFpQWhBQUlnSVFBQ0lBQUFBc0FDd0FRZ0JXQUlnQXlBRGFBUGdCTUFGV0FZWUJ1Z0hhQWdZQ05nSmlBbm9DckFMR0F1QUM4Z01FQXhnREpBTTZBMWdEZUFPR0FBQUFBUUFBQUJ3QWtBQU1BR01BQndBQkFBQUFBQUFBQUFBQUFBQUFBQVFBQTNpY25KVFBiaHRsRk1WL1RtelRDc0VDUlZXNmliNEZpeUsxWTFNbFZkdXNKcVJXUmtSeDhMZ2dKSVEwdGllMjVmSE15RE4yR3A2QU5XL0JXM1RGUS9BY2lEVzYxOWV1WFJBRks0cDFadWIrT2QrNTUzN0FBWCt5VDZWNkgvaXRuaHF1Y0ZTL05yekh2ZnFGNFgxYTlUM0RWUjdWZmpkY1kxQmJHSzd6ZWExaitDUGVWbjh4ZkkvajZvK0c3M05ZYlJuK21LZlZBOE9mN0R2K01Qd3B4N3hkNGdvODUyZkRGUTdKRGU5eHdBK0c5M21BMWF4VWVVRFRjSTNQT0RKYzV3am9NcVlrWVV6S0VNY05ZNGJNbVJGVEVCSXpZOHdOTVFNY0FUNEpwYjVOaVJRNWh2LzROaUtrWkVha0ZVZVVPQkpDRWlJS1JsYnhKODNLdU5hTzBtZW1TTHI1bEl5STZHbkdoSWdVUjhhUWpKU1lpZFlwS2NsNVNZTUdCWDNsbTFOUzRGRXdKc0VqWThhUUJtMWFYTkpseEpnQ1Iwc3JDYk9RakJ0S2JvbTB2N01JVWFaUFRFcGhyT2FrREpTblUzNnhaZ2RjNG1pVGEreG01Y3V0Q285eGZLdlp3azFpSEYvaTZiL2JZTGJkZDhVbVlxRjZpb1k5RXVWNXF4TWNxZUxTMStjYnhTVXZjVHZwczgza3d4b05sSjNNZWt5UHVjNWY1aWQ1d2lURnVVTjhRblZRNkI3aU9OUG5nRkFWK1k2QUxoZTBlVTFYbjMwNmRQQzVva3ZBSzgxdDA4SHhGVzJ1T05lTVFQSHlXMHNkYzhYM09MNG0wQmlwSFpzK29yazh2U0UzZHd0M2NZYWNZMHF1V3lBemx2T0w4K09kSml3N2xHMjVvMUJYOUhXUEpGTDJRRlNSUFlzWW1pdHlkY1ZVdFZ4NW96RDlCdVlJK1ZycWJOOTlsMjFZMk82dHR2aU9USGZZTVRkT01ya2xvdzlOMVh2UE03ZjY1eEV4SUtPbkVYMHlwalFvdWRPelhSTXhKOEZ4cmo2KzBDMHA4ZGM1MHVkT1hSSXpWUVlCcWRhWmtldHZaTDRKalh0L3kvZk83aExadktudTZHUjlxbDI2U09WMFkwYXZWYjNWdDlCVWpqY20wTENwWnBZakU1Ykt5NU9LOXlYYTIrSWZxdnNMdmQweW5uQkdSc0xnYnpmQVV6eWFISFBDS1NQdElKVlRGblk3TlBXR2FQS0NVejM5aEZpajVMNThvenBKaFJNOG5uSENDUzk0OXA2T0t5Yk91TGcxbDgzZVB1ZWMwZWIwUDUxaUdkL21qRmZyZC8vZTlWZGwydFN6T0o2c241N3ZQTVZILzhPdFgvQjQ2NzdzNkMwZ203T3dhdSsyNG9xSXFYQnhoNXRhdUlkNGZ3RUFBUC8vQXdEMHQwOVJBQUFEQUFBQUFBQUEvODRBTWdBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQSl9LmNvbm5lY3Rpb24sLnNoYXBle3N0cm9rZS1saW5lam9pbjpyb3VuZH0uc2hhcGV7c2hhcGUtcmVuZGVyaW5nOmdlb21ldHJpY1ByZWNpc2lvbn0uY29ubmVjdGlvbntzdHJva2UtbGluZWNhcDpyb3VuZH0uZDItMTQ1MDQ0NDU2NiAuZmlsbC1OMXtmaWxsOiMwYTBmMjV9LmQyLTE0NTA0NDQ1NjYgLnN0cm9rZS1CMXtzdHJva2U6IzBkMzJiMn08L3N0eWxlPjxnIGNsYXNzPSJabTl5YlhWc1lRPT0gbGF5ZXIiPjxwYXRoIGZpbGw9IiNlZWY0ZmYiIHN0cm9rZT0iIzRhNmZhNSIgZD0iTTEzMSAwaDI4M3Y4MkgxMzF6IiBjbGFzcz0ic2hhcGUiIHN0eWxlPSJzdHJva2Utd2lkdGg6MiIvPjx0ZXh0IHg9IjI3Mi41IiB5PSIzOC41IiBmaWxsPSIjMGEwZjI1IiBjbGFzcz0idGV4dC1ib2xkIGZpbGwtTjEiIHN0eWxlPSJ0ZXh0LWFuY2hvcjptaWRkbGU7Zm9udC1zaXplOjE2cHgiPjx0c3BhbiB4PSIyNzIuNSIgZHk9IjAiPlNNVCDlhazlvI88L3RzcGFuPjx0c3BhbiB4PSIyNzIuNSIgZHk9IjE4LjUiPih4ICsgeSAmZ3Q7IDEwKSDiiKcgKGFbMF0gPSA1KSDiiKcgZih4KSDiiaAgZih5KTwvdHNwYW4+PC90ZXh0PjwvZz48ZyBjbGFzcz0iYzJ0bGJHVjBiMjQ9IGxheWVyIj48cGF0aCBmaWxsPSIjZWVmNGZmIiBzdHJva2U9IiM0YTZmYTUiIGQ9Ik0xOTAgMTgyaDE2NnY4MkgxOTB6IiBjbGFzcz0ic2hhcGUiIHN0eWxlPSJzdHJva2Utd2lkdGg6MiIvPjx0ZXh0IHg9IjI3MyIgeT0iMjIwLjUiIGZpbGw9IiMwYTBmMjUiIGNsYXNzPSJ0ZXh0LWJvbGQgZmlsbC1OMSIgc3R5bGU9InRleHQtYW5jaG9yOm1pZGRsZTtmb250LXNpemU6MTZweCI+PHRzcGFuIHg9IjI3MyIgZHk9IjAiPuW4g+WwlOmqqOaetjwvdHNwYW4+PHRzcGFuIHg9IjI3MyIgZHk9IjE4LjUiPnAg4oinIHEg4oinIChyIOKIqCB5IOKJpCAwKTwvdHNwYW4+PC90ZXh0PjwvZz48ZyBjbGFzcz0iWVRFPSBhdG9tIj48cGF0aCBmaWxsPSIjZmZmOGU2IiBzdHJva2U9IiNjOWEyMjciIGQ9Ik00MTYgMzY0aDEyNHY4Mkg0MTZ6IiBjbGFzcz0ic2hhcGUiIHN0eWxlPSJzdHJva2Utd2lkdGg6MiIvPjx0ZXh0IHg9IjQ3OCIgeT0iNDAyLjUiIGZpbGw9IiMwYTBmMjUiIGNsYXNzPSJ0ZXh0LWJvbGQgZmlsbC1OMSIgc3R5bGU9InRleHQtYW5jaG9yOm1pZGRsZTtmb250LXNpemU6MTZweCI+PHRzcGFuIHg9IjQ3OCIgZHk9IjAiPnggKyB5ICZndDsgMTA8L3RzcGFuPjx0c3BhbiB4PSI0NzgiIGR5PSIxOC41Ij7nur\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",[11,55600,55601,55602,2372],{},"这张图就是 SMT 的骨架：",[7246,55603,55604],{},"布尔层负责「怎么组合」，理论层负责「每个原子是什么意思」",[53200,55606,55608],{"id":55607},"_13-背景理论","1.3 背景理论",[11,55610,55611],{},"SMT 里常用的理论有这么几类：",[55164,55613,55614,55624,55700,55712,55781,55798,55807],{},[55167,55615,55616,55623],{},[7246,55617,55618,55619,55622],{},"EUF（",[45608,55620,55621],{},"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。",[55167,55625,55626,55636,55637,15282,55665,55694,55695,55697,55698,2372],{},[7246,55627,55628,55629,55632,55633,55622],{},"线性算术（",[45608,55630,55631],{},"QF_LRA"," \u002F ",[45608,55634,55635],{},"QF_LIA","：处理 ",[31,55638,55640,55653],{"className":55639,"translate":35},[34],[31,55641,55643],{"className":55642},[39],[41,55644,55645],{"xmlns":43},[45,55646,55647,55651],{},[48,55648,55649],{},[95,55650,434],{},[55,55652,434],{"encoding":57},[31,55654,55656],{"className":55655,"ariaHidden":62},[61],[31,55657,55659,55662],{"className":55658},[66],[31,55660],{"className":55661,"style":307},[70],[31,55663,434],{"className":55664},[75],[31,55666,55668,55682],{"className":55667,"translate":35},[34],[31,55669,55671],{"className":55670},[39],[41,55672,55673],{"xmlns":43},[45,55674,55675,55679],{},[48,55676,55677],{},[95,55678,347],{},[55,55680,55681],{"encoding":57},"\\le",[31,55683,55685],{"className":55684,"ariaHidden":62},[61],[31,55686,55688,55691],{"className":55687},[66],[31,55689],{"className":55690,"style":385},[70],[31,55692,347],{"className":55693},[297]," 和常数；实数版是 ",[45608,55696,55631],{},"，整数版是 ",[45608,55699,55635],{},[55167,55701,55702,55711],{},[7246,55703,55704,55705,55632,55708,55622],{},"非线性算术（",[45608,55706,55707],{},"QF_NRA",[45608,55709,55710],{},"QF_NIA","：允许乘法；实数版可判定，整数版不可判定。",[55167,55713,55714,55720,55721,55780],{},[7246,55715,55716,55717,55622],{},"位向量（",[45608,55718,55719],{},"QF_BV","：描述机器整数和硬件电路，加法是模 ",[31,55722,55724,55742],{"className":55723,"translate":35},[34],[31,55725,55727],{"className":55726},[39],[41,55728,55729],{"xmlns":43},[45,55730,55731,55739],{},[48,55732,55733],{},[1180,55734,55735,55737],{},[91,55736,152],{},[51,55738,14559],{},[55,55740,55741],{"encoding":57},"2^n",[31,55743,55745],{"className":55744,"ariaHidden":62},[61],[31,55746,55748,55751],{"className":55747},[66],[31,55749],{"className":55750,"style":3628},[70],[31,55752,55754,55757],{"className":55753},[75],[31,55755,152],{"className":55756},[75],[31,55758,55760],{"className":55759},[1039],[31,55761,55763],{"className":55762},[1043],[31,55764,55766],{"className":55765},[1048],[31,55767,55769],{"className":55768,"style":3628},[1052],[31,55770,55771,55774],{"style":3631},[31,55772],{"className":55773,"style":1061},[1060],[31,55775,55777],{"className":55776},[1065,1066,1067,1068],[31,55778,14559],{"className":55779},[75,76,1068]," 的。",[55167,55782,55783,55792,55793,2446,55795,55797],{},[7246,55784,55785,55786,55632,55789,55622],{},"数组（",[45608,55787,55788],{},"QF_AX",[45608,55790,55791],{},"QF_AUFBV","：用 ",[45608,55794,54443],{},[45608,55796,55378],{}," 描述内存读写。",[55167,55799,55800,55806],{},[7246,55801,55802,55803,55622],{},"浮点（",[45608,55804,55805],{},"QF_FP","：描述 IEEE 754 浮点运算。",[55167,55808,55809,55812,55813,15282,55816,55819],{},[7246,55810,55811],{},"字符串","：描述 ",[45608,55814,55815],{},"concat",[45608,55817,55818],{},"contains"," 等操作，工业实现常常不完整。",[11,55821,55822,55825],{},[45608,55823,55824],{},"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。",[11,55827,55828,55829,55857],{},"可以把 SAT 看成 SMT 的一个特例：当理论 ",[31,55830,55832,55845],{"className":55831,"translate":35},[34],[31,55833,55835],{"className":55834},[39],[41,55836,55837],{"xmlns":43},[45,55838,55839,55843],{},[48,55840,55841],{},[51,55842,919],{},[55,55844,919],{"encoding":57},[31,55846,55848],{"className":55847,"ariaHidden":62},[61],[31,55849,55851,55854],{"className":55850},[66],[31,55852],{"className":55853,"style":945},[70],[31,55855,919],{"className":55856,"style":949},[75,76]," 为空、所有原子都是裸命题时，SMT 就退化成 SAT。",[11,55859,55860],{},"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。",[11,55862,55863],{},"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。",[18,55865,55867],{"id":55866},"二smt-之前","二、SMT 之前",[53200,55869,55871],{"id":55870},"_21-位爆破","2.1 位爆破",[11,55873,55874,55875,55878],{},"SAT 只认布尔变量。要用它处理整数，标准做法叫",[7246,55876,55877],{},"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。",[11,55880,55881,55882,55924],{},"看一个具体的小例子。取两个 4 位无符号整数 ",[31,55883,55885,55903],{"className":55884,"translate":35},[34],[31,55886,55888],{"className":55887},[39],[41,55889,55890],{"xmlns":43},[45,55891,55892,55900],{},[48,55893,55894,55896,55898],{},[51,55895,53],{},[95,55897,1210],{"separator":62},[51,55899,3003],{},[55,55901,55902],{"encoding":57},"x, y",[31,55904,55906],{"className":55905,"ariaHidden":62},[61],[31,55907,55909,55912,55915,55918,55921],{"className":55908},[66],[31,55910],{"className":55911,"style":865},[70],[31,55913,53],{"className":55914},[75,76],[31,55916,1210],{"className":55917},[1391],[31,55919],{"className":55920,"style":1467},[119],[31,55922,3003],{"className":55923,"style":2666},[75,76],"，约束是：",[31,55926,55928],{"className":55927,"translate":35},[411],[31,55929,55931,55965],{"className":55930,"translate":35},[34],[31,55932,55934],{"className":55933},[39],[41,55935,55936],{"xmlns":43,"display":420},[45,55937,55938,55962],{},[48,55939,55940,55942,55944,55946,55948,55950,55952,55954,55956,55958,55960],{},[51,55941,53],{},[95,55943,434],{},[51,55945,3003],{},[95,55947,223],{},[91,55949,19494],{},[1430,55951,53035],{},[95,55953,53038],{},[1430,55955,53035],{},[51,55957,53],{},[95,55959,16040],{},[51,55961,3003],{},[55,55963,55964],{"encoding":57},"x + y = 10 \\;\\wedge\\; x > y",[31,55966,55968,55986,56004,56028,56046],{"className":55967,"ariaHidden":62},[61],[31,55969,55971,55974,55977,55980,55983],{"className":55970},[66],[31,55972],{"className":55973,"style":307},[70],[31,55975,53],{"className":55976},[75,76],[31,55978],{"className":55979,"style":120},[119],[31,55981,434],{"className":55982},[124],[31,55984],{"className":55985,"style":120},[119],[31,55987,55989,55992,55995,55998,56001],{"className":55988},[66],[31,55990],{"className":55991,"style":865},[70],[31,55993,3003],{"className":55994,"style":2666},[75,76],[31,55996],{"className":55997,"style":293},[119],[31,55999,223],{"className":56000},[297],[31,56002],{"className":56003,"style":293},[119],[31,56005,56007,56010,56013,56016,56019,56022,56025],{"className":56006},[66],[31,56008],{"className":56009,"style":326},[70],[31,56011,19494],{"className":56012},[75],[31,56014],{"className":56015,"style":293},[119],[31,56017],{"className":56018,"style":120},[119],[31,56020,53038],{"className":56021},[124],[31,56023],{"className":56024,"style":293},[119],[31,56026],{"className":56027,"style":120},[119],[31,56029,56031,56034,56037,56040,56043],{"className":56030},[66],[31,56032],{"className":56033,"style":21113},[70],[31,56035,53],{"className":56036},[75,76],[31,56038],{"className":56039,"style":293},[119],[31,56041,16040],{"className":56042},[297],[31,56044],{"className":56045,"style":293},[119],[31,56047,56049,56052],{"className":56048},[66],[31,56050],{"className":56051,"style":865},[70],[31,56053,3003],{"className":56054,"style":2666},[75,76],[11,56056,56057,56058,2446,56266,56474,56475,56578,56579,56677],{},"SAT 看到的是 8 个布尔变量 ",[31,56059,56061,56097],{"className":56060,"translate":35},[34],[31,56062,56064],{"className":56063},[39],[41,56065,56066],{"xmlns":43},[45,56067,56068,56094],{},[48,56069,56070,56076,56082,56088],{},[1004,56071,56072,56074],{},[51,56073,53],{},[91,56075,93],{},[1004,56077,56078,56080],{},[51,56079,53],{},[91,56081,152],{},[1004,56083,56084,56086],{},[51,56085,53],{},[91,56087,205],{},[1004,56089,56090,56092],{},[51,56091,53],{},[91,56093,1977],{},[55,56095,56096],{"encoding":57},"x_3 x_2 x_1 x_0",[31,56098,56100],{"className":56099,"ariaHidden":62},[61],[31,56101,56103,56106,56146,56186,56226],{"className":56102},[66],[31,56104],{"className":56105,"style":7090},[70],[31,56107,56109,56112],{"className":56108},[75],[31,56110,53],{"className":56111},[75,76],[31,56113,56115],{"className":56114},[1039],[31,56116,56118,56138],{"className":56117},[1043,1044],[31,56119,56121,56135],{"className":56120},[1048],[31,56122,56124],{"className":56123,"style":2028},[1052],[31,56125,56126,56129],{"style":2807},[31,56127],{"className":56128,"style":1061},[1060],[31,56130,56132],{"className":56131},[1065,1066,1067,1068],[31,56133,93],{"className":56134},[75,1068],[31,56136,1076],{"className":56137},[1075],[31,56139,56141],{"className":56140},[1048],[31,56142,56144],{"className":56143,"style":2050},[1052],[31,56145],{},[31,56147,56149,56152],{"className":56148},[75],[31,56150,53],{"className":56151},[75,76],[31,56153,56155],{"className":56154},[1039],[31,56156,56158,56178],{"className":56157},[1043,1044],[31,56159,56161,56175],{"className":56160},[1048],[31,56162,56164],{"className":56163,"style":2028},[1052],[31,56165,56166,56169],{"style":2807},[31,56167],{"className":56168,"style":1061},[1060],[31,56170,56172],{"className":56171},[1065,1066,1067,1068],[31,56173,152],{"className":56174},[75,1068],[31,56176,1076],{"className":56177},[1075],[31,56179,56181],{"className":56180},[1048],[31,56182,56184],{"className":56183,"style":2050},[1052],[31,56185],{},[31,56187,56189,56192],{"className":56188},[75],[31,56190,53],{"className":56191},[75,76],[31,56193,56195],{"className":56194},[1039],[31,56196,56198,56218],{"className":56197},[1043,1044],[31,56199,56201,56215],{"className":56200},[1048],[31,56202,56204],{"className":56203,"style":2028},[1052],[31,56205,56206,56209],{"style":2807},[31,56207],{"className":56208,"style":1061},[1060],[31,56210,56212],{"className":56211},[1065,1066,1067,1068],[31,56213,205],{"className":56214},[75,1068],[31,56216,1076],{"className":56217},[1075],[31,56219,56221],{"className":56220},[1048],[31,56222,56224],{"className":56223,"style":2050},[1052],[31,56225],{},[31,56227,56229,56232],{"className":56228},[75],[31,56230,53],{"className":56231},[75,76],[31,56233,56235],{"className":56234},[1039],[31,56236,56238,56258],{"className":56237},[1043,1044],[31,56239,56241,56255],{"className":56240},[1048],[31,56242,56244],{"className":56243,"style":2028},[1052],[31,56245,56246,56249],{"style":2807},[31,56247],{"className":56248,"style":1061},[1060],[31,56250,56252],{"className":56251},[1065,1066,1067,1068],[31,56253,1977],{"className":56254},[75,1068],[31,56256,1076],{"className":56257},[1075],[31,56259,56261],{"className":56260},[1048],[31,56262,56264],{"className":56263,"style":2050},[1052],[31,56265],{},[31,56267,56269,56305],{"className":56268,"translate":35},[34],[31,56270,56272],{"className":56271},[39],[41,56273,56274],{"xmlns":43},[45,56275,56276,56302],{},[48,56277,56278,56284,56290,56296],{},[1004,56279,56280,56282],{},[51,56281,3003],{},[91,56283,93],{},[1004,56285,56286,56288],{},[51,56287,3003],{},[91,56289,152],{},[1004,56291,56292,56294],{},[51,56293,3003],{},[91,56295,205],{},[1004,56297,56298,56300],{},[51,56299,3003],{},[91,56301,1977],{},[55,56303,56304],{"encoding":57},"y_3 y_2 y_1 y_0",[31,56306,56308],{"className":56307,"ariaHidden":62},[61],[31,56309,56311,56314,56354,56394,56434],{"className":56310},[66],[31,56312],{"className":56313,"style":865},[70],[31,56315,56317,56320],{"className":56316},[75],[31,56318,3003],{"className":56319,"style":2666},[75,76],[31,56321,56323],{"className":56322},[1039],[31,56324,56326,56346],{"className":56325},[1043,1044],[31,56327,56329,56343],{"className":56328},[1048],[31,56330,56332],{"className":56331,"style":2028},[1052],[31,56333,56334,56337],{"style":5504},[31,56335],{"className":56336,"style":1061},[1060],[31,56338,56340],{"className":56339},[1065,1066,1067,1068],[31,56341,93],{"className":56342},[75,1068],[31,56344,1076],{"className":56345},[1075],[31,56347,56349],{"className":56348},[1048],[31,56350,56352],{"className":56351,"style":2050},[1052],[31,56353],{},[31,56355,56357,56360],{"className":56356},[75],[31,56358,3003],{"className":56359,"style":2666},[75,76],[31,56361,56363],{"className":56362},[1039],[31,56364,56366,56386],{"className":56365},[1043,1044],[31,56367,56369,56383],{"className":56368},[1048],[31,56370,56372],{"className":56371,"style":2028},[1052],[31,56373,56374,56377],{"style":5504},[31,56375],{"className":56376,"style":1061},[1060],[31,56378,56380],{"className":56379},[1065,1066,1067,1068],[31,56381,152],{"className":56382},[75,1068],[31,56384,1076],{"className":56385},[1075],[31,56387,56389],{"className":56388},[1048],[31,56390,56392],{"className":56391,"style":2050},[1052],[31,56393],{},[31,56395,56397,56400],{"className":56396},[75],[31,56398,3003],{"className":56399,"style":2666},[75,76],[31,56401,56403],{"className":56402},[1039],[31,56404,56406,56426],{"className":56405},[1043,1044],[31,56407,56409,56423],{"className":56408},[1048],[31,56410,56412],{"className":56411,"style":2028},[1052],[31,56413,56414,56417],{"style":5504},[31,56415],{"className":56416,"style":1061},[1060],[31,56418,56420],{"className":56419},[1065,1066,1067,1068],[31,56421,205],{"className":56422},[75,1068],[31,56424,1076],{"className":56425},[1075],[31,56427,56429],{"className":56428},[1048],[31,56430,56432],{"className":56431,"style":2050},[1052],[31,56433],{},[31,56435,56437,56440],{"className":56436},[75],[31,56438,3003],{"className":56439,"style":2666},[75,76],[31,56441,56443],{"className":56442},[1039],[31,56444,56446,56466],{"className":56445},[1043,1044],[31,56447,56449,56463],{"className":56448},[1048],[31,56450,56452],{"className":56451,"style":2028},[1052],[31,56453,56454,56457],{"style":5504},[31,56455],{"className":56456,"style":1061},[1060],[31,56458,56460],{"className":56459},[1065,1066,1067,1068],[31,56461,1977],{"className":56462},[75,1068],[31,56464,1076],{"className":56465},[1075],[31,56467,56469],{"className":56468},[1048],[31,56470,56472],{"className":56471,"style":2050},[1052],[31,56473],{},"。为了表达加法，需要 4 个全加器。每个全加器有三个输入 ",[31,56476,56478,56508],{"className":56477,"translate":35},[34],[31,56479,56481],{"className":56480},[39],[41,56482,56483],{"xmlns":43},[45,56484,56485,56505],{},[48,56486,56487,56489,56491,56493,56495],{},[51,56488,5813],{},[95,56490,1210],{"separator":62},[51,56492,5834],{},[95,56494,1210],{"separator":62},[1004,56496,56497,56499],{},[51,56498,1412],{},[48,56500,56501,56503],{},[51,56502,5391],{},[51,56504,14559],{},[55,56506,56507],{"encoding":57},"a, b, c_{in}",[31,56509,56511],{"className":56510,"ariaHidden":62},[61],[31,56512,56514,56517,56520,56523,56526,56529,56532,56535],{"className":56513},[66],[31,56515],{"className":56516,"style":4202},[70],[31,56518,5813],{"className":56519},[75,76],[31,56521,1210],{"className":56522},[1391],[31,56524],{"className":56525,"style":1467},[119],[31,56527,5834],{"className":56528},[75,76],[31,56530,1210],{"className":56531},[1391],[31,56533],{"className":56534,"style":1467},[119],[31,56536,56538,56541],{"className":56537},[75],[31,56539,1412],{"className":56540},[75,76],[31,56542,56544],{"className":56543},[1039],[31,56545,56547,56570],{"className":56546},[1043,1044],[31,56548,56550,56567],{"className":56549},[1048],[31,56551,56553],{"className":56552,"style":5445},[1052],[31,56554,56555,56558],{"style":2807},[31,56556],{"className":56557,"style":1061},[1060],[31,56559,56561],{"className":56560},[1065,1066,1067,1068],[31,56562,56564],{"className":56563},[75,1068],[31,56565,1658],{"className":56566},[75,76,1068],[31,56568,1076],{"className":56569},[1075],[31,56571,56573],{"className":56572},[1048],[31,56574,56576],{"className":56575,"style":2050},[1052],[31,56577],{},"、两个输出 ",[31,56580,56582,56610],{"className":56581,"translate":35},[34],[31,56583,56585],{"className":56584},[39],[41,56586,56587],{"xmlns":43},[45,56588,56589,56607],{},[48,56590,56591,56593,56595],{},[51,56592,14551],{},[95,56594,1210],{"separator":62},[1004,56596,56597,56599],{},[51,56598,1412],{},[48,56600,56601,56603,56605],{},[51,56602,2552],{},[51,56604,5324],{},[51,56606,12271],{},[55,56608,56609],{"encoding":57},"s, c_{out}",[31,56611,56613],{"className":56612,"ariaHidden":62},[61],[31,56614,56616,56619,56622,56625,56628],{"className":56615},[66],[31,56617],{"className":56618,"style":865},[70],[31,56620,14551],{"className":56621},[75,76],[31,56623,1210],{"className":56624},[1391],[31,56626],{"className":56627,"style":1467},[119],[31,56629,56631,56634],{"className":56630},[75],[31,56632,1412],{"className":56633},[75,76],[31,56635,56637],{"className":56636},[1039],[31,56638,56640,56669],{"className":56639},[1043,1044],[31,56641,56643,56666],{"className":56642},[1048],[31,56644,56646],{"className":56645,"style":19220},[1052],[31,56647,56648,56651],{"style":2807},[31,56649],{"className":56650,"style":1061},[1060],[31,56652,56654],{"className":56653},[1065,1066,1067,1068],[31,56655,56657,56660,56663],{"className":56656},[75,1068],[31,56658,2552],{"className":56659},[75,76,1068],[31,56661,5324],{"className":56662},[75,76,1068],[31,56664,12271],{"className":56665},[75,76,1068],[31,56667,1076],{"className":56668},[1075],[31,56670,56672],{"className":56671},[1048],[31,56673,56675],{"className":56674,"style":2050},[1052],[31,56676],{},"，满足：",[31,56679,56681],{"className":56680,"translate":35},[411],[31,56682,56684,56775],{"className":56683,"translate":35},[34],[31,56685,56687],{"className":56686},[39],[41,56688,56689],{"xmlns":43,"display":420},[45,56690,56691,56772],{},[48,56692,56693,56695,56697,56699,56702,56704,56706,56716,56718,56720,56732,56734,56736,56738,56740,56742,56744,56746,56748,56758,56760,56762,56764,56766,56768,56770],{},[51,56694,14551],{},[95,56696,223],{},[51,56698,5813],{},[95,56700,56701],{},"⊕",[51,56703,5834],{},[95,56705,56701],{},[1004,56707,56708,56710],{},[51,56709,1412],{},[48,56711,56712,56714],{},[51,56713,5391],{},[51,56715,14559],{},[95,56717,1210],{"separator":62},[119,56719],{"width":20286},[1004,56721,56722,56724],{},[51,56723,1412],{},[48,56725,56726,56728,56730],{},[51,56727,2552],{},[51,56729,5324],{},[51,56731,12271],{},[95,56733,223],{},[95,56735,211],{"stretchy":210},[51,56737,5813],{},[95,56739,53038],{},[51,56741,5834],{},[95,56743,220],{"stretchy":210},[95,56745,53683],{},[95,56747,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,56749,56750,56752],{},[51,56751,1412],{},[48,56753,56754,56756],{},[51,56755,5391],{},[51,56757,14559],{},[95,56759,53038],{},[95,56761,211],{"stretchy":210},[51,56763,5813],{},[95,56765,56701],{},[51,56767,5834],{},[95,56769,220],{"stretchy":210},[95,56771,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[55,56773,56774],{"encoding":57},"s = a \\oplus b \\oplus c_{in}, \\qquad c_{out} = (a \\wedge b) \\vee \\bigl(c_{in} \\wedge (a \\oplus b)\\bigr)",[31,56776,56778,56796,56814,56833,56949,56970,56991,57055,57076],{"className":56777,"ariaHidden":62},[61],[31,56779,56781,56784,56787,56790,56793],{"className":56780},[66],[31,56782],{"className":56783,"style":71},[70],[31,56785,14551],{"className":56786},[75,76],[31,56788],{"className":56789,"style":293},[119],[31,56791,223],{"className":56792},[297],[31,56794],{"className":56795,"style":293},[119],[31,56797,56799,56802,56805,56808,56811],{"className":56798},[66],[31,56800],{"className":56801,"style":307},[70],[31,56803,5813],{"className":56804},[75,76],[31,56806],{"className":56807,"style":120},[119],[31,56809,56701],{"className":56810},[124],[31,56812],{"className":56813,"style":120},[119],[31,56815,56817,56821,56824,56827,56830],{"className":56816},[66],[31,56818],{"className":56819,"style":56820},[70],"height:0.7778em;vertical-align:-0.0833em;",[31,56822,5834],{"className":56823},[75,76],[31,56825],{"className":56826,"style":120},[119],[31,56828,56701],{"className":56829},[124],[31,56831],{"className":56832,"style":120},[119],[31,56834,56836,56839,56882,56885,56888,56891,56940,56943,56946],{"className":56835},[66],[31,56837],{"className":56838,"style":865},[70],[31,56840,56842,56845],{"className":56841},[75],[31,56843,1412],{"className":56844},[75,76],[31,56846,56848],{"className":56847},[1039],[31,56849,56851,56874],{"className":56850},[1043,1044],[31,56852,56854,56871],{"className":56853},[1048],[31,56855,56857],{"className":56856,"style":5445},[1052],[31,56858,56859,56862],{"style":2807},[31,56860],{"className":56861,"style":1061},[1060],[31,56863,56865],{"className":56864},[1065,1066,1067,1068],[31,56866,56868],{"className":56867},[75,1068],[31,56869,1658],{"className":56870},[75,76,1068],[31,56872,1076],{"className":56873},[1075],[31,56875,56877],{"className":56876},[1048],[31,56878,56880],{"className":56879,"style":2050},[1052],[31,56881],{},[31,56883,1210],{"className":56884},[1391],[31,56886],{"className":56887,"style":20525},[119],[31,56889],{"className":56890,"style":1467},[119],[31,56892,56894,56897],{"className":56893},[75],[31,56895,1412],{"className":56896},[75,76],[31,56898,56900],{"className":56899},[1039],[31,56901,56903,56932],{"className":56902},[1043,1044],[31,56904,56906,56929],{"className":56905},[1048],[31,56907,56909],{"className":56908,"style":19220},[1052],[31,56910,56911,56914],{"style":2807},[31,56912],{"className":56913,"style":1061},[1060],[31,56915,56917],{"className":56916},[1065,1066,1067,1068],[31,56918,56920,56923,56926],{"className":56919},[75,1068],[31,56921,2552],{"className":56922},[75,76,1068],[31,56924,5324],{"className":56925},[75,76,1068],[31,56927,12271],{"className":56928},[75,76,1068],[31,56930,1076],{"className":56931},[1075],[31,56933,56935],{"className":56934},[1048],[31,56936,56938],{"className":56937,"style":2050},[1052],[31,56939],{},[31,56941],{"className":56942,"style":293},[119],[31,56944,223],{"className":56945},[297],[31,56947],{"className":56948,"style":293},[119],[31,56950,56952,56955,56958,56961,56964,56967],{"className":56951},[66],[31,56953],{"className":56954,"style":260},[70],[31,56956,211],{"className":56957},[264],[31,56959,5813],{"className":56960},[75,76],[31,56962],{"className":56963,"style":120},[119],[31,56965,53038],{"className":56966},[124],[31,56968],{"className":56969,"style":120},[119],[31,56971,56973,56976,56979,56982,56985,56988],{"className":56972},[66],[31,56974],{"className":56975,"style":260},[70],[31,56977,5834],{"className":56978},[75,76],[31,56980,220],{"className":56981},[289],[31,56983],{"className":56984,"style":120},[119],[31,56986,53683],{"className":56987},[124],[31,56989],{"className":56990,"style":120},[119],[31,56992,56994,56997,57003,57046,57049,57052],{"className":56993},[66],[31,56995],{"className":56996,"style":4988},[70],[31,56998,57000],{"className":56999},[264],[31,57001,211],{"className":57002},[1307,1308],[31,57004,57006,57009],{"className":57005},[75],[31,57007,1412],{"className":57008},[75,76],[31,57010,57012],{"className":57011},[1039],[31,57013,57015,57038],{"className":57014},[1043,1044],[31,57016,57018,57035],{"className":57017},[1048],[31,57019,57021],{"className":57020,"style":5445},[1052],[31,57022,57023,57026],{"style":2807},[31,57024],{"className":57025,"style":1061},[1060],[31,57027,57029],{"className":57028},[1065,1066,1067,1068],[31,57030,57032],{"className":57031},[75,1068],[31,57033,1658],{"className":57034},[75,76,1068],[31,57036,1076],{"className":57037},[1075],[31,57039,57041],{"className":57040},[1048],[31,57042,57044],{"className":57043,"style":2050},[1052],[31,57045],{},[31,57047],{"className":57048,"style":120},[119],[31,57050,53038],{"className":57051},[124],[31,57053],{"className":57054,"style":120},[119],[31,57056,57058,57061,57064,57067,57070,57073],{"className":57057},[66],[31,57059],{"className":57060,"style":260},[70],[31,57062,211],{"className":57063},[264],[31,57065,5813],{"className":57066},[75,76],[31,57068],{"className":57069,"style":120},[119],[31,57071,56701],{"className":57072},[124],[31,57074],{"className":57075,"style":120},[119],[31,57077,57079,57082,57085,57088],{"className":57078},[66],[31,57080],{"className":57081,"style":4988},[70],[31,57083,5834],{"className":57084},[75,76],[31,57086,220],{"className":57087},[289],[31,57089,57091],{"className":57090},[289],[31,57092,220],{"className":57093},[1307,1308],[11,57095,57096],{},"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。",[11,57098,57099],{},"而原始约束只有一行：",[52853,57101,57103],{"className":52855,"code":57102,"language":1412,"meta":52812,"style":52812},"x + y == 10 && x > y\n",[45608,57104,57105],{"__ignoreMap":52812},[31,57106,57107,57110,57112,57114,57116,57118,57120,57123,57125],{"class":52861,"line":52862},[31,57108,57109],{"class":52873},"x ",[31,57111,434],{"class":52865},[31,57113,52951],{"class":52873},[31,57115,52970],{"class":52865},[31,57117,52956],{"class":52904},[31,57119,52959],{"class":52865},[31,57121,57122],{"class":52873}," x ",[31,57124,16040],{"class":52865},[31,57126,57127],{"class":52873}," y\n",[11,57129,57130,57131,15282,57183,57235],{},"求解器可能给出 ",[31,57132,57134,57153],{"className":57133,"translate":35},[34],[31,57135,57137],{"className":57136},[39],[41,57138,57139],{"xmlns":43},[45,57140,57141,57150],{},[48,57142,57143,57145,57147],{},[51,57144,53],{},[95,57146,223],{},[91,57148,57149],{},"0110",[55,57151,57152],{"encoding":57},"x = 0110",[31,57154,57156,57174],{"className":57155,"ariaHidden":62},[61],[31,57157,57159,57162,57165,57168,57171],{"className":57158},[66],[31,57160],{"className":57161,"style":71},[70],[31,57163,53],{"className":57164},[75,76],[31,57166],{"className":57167,"style":293},[119],[31,57169,223],{"className":57170},[297],[31,57172],{"className":57173,"style":293},[119],[31,57175,57177,57180],{"className":57176},[66],[31,57178],{"className":57179,"style":326},[70],[31,57181,57149],{"className":57182},[75],[31,57184,57186,57205],{"className":57185,"translate":35},[34],[31,57187,57189],{"className":57188},[39],[41,57190,57191],{"xmlns":43},[45,57192,57193,57202],{},[48,57194,57195,57197,57199],{},[51,57196,3003],{},[95,57198,223],{},[91,57200,57201],{},"0100",[55,57203,57204],{"encoding":57},"y = 0100",[31,57206,57208,57226],{"className":57207,"ariaHidden":62},[61],[31,57209,57211,57214,57217,57220,57223],{"className":57210},[66],[31,57212],{"className":57213,"style":865},[70],[31,57215,3003],{"className":57216,"style":2666},[75,76],[31,57218],{"className":57219,"style":293},[119],[31,57221,223],{"className":57222},[297],[31,57224],{"className":57225,"style":293},[119],[31,57227,57229,57232],{"className":57228},[66],[31,57230],{"className":57231,"style":326},[70],[31,57233,57201],{"className":57234},[75],"，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。",[11,57237,57238,57239,57242,57243,57338,57339,57367,57368,57396,57397,57367,57425,57453],{},"规模还不是最麻烦的，",[7246,57240,57241],{},"语义丢失","才是。假设我们想证明 ",[31,57244,57246,57272],{"className":57245,"translate":35},[34],[31,57247,57249],{"className":57248},[39],[41,57250,57251],{"xmlns":43},[45,57252,57253,57269],{},[48,57254,57255,57257,57259,57261,57263,57265,57267],{},[51,57256,53],{},[95,57258,434],{},[51,57260,3003],{},[95,57262,223],{},[51,57264,3003],{},[95,57266,434],{},[51,57268,53],{},[55,57270,57271],{"encoding":57},"x + y = y + x",[31,57273,57275,57293,57311,57329],{"className":57274,"ariaHidden":62},[61],[31,57276,57278,57281,57284,57287,57290],{"className":57277},[66],[31,57279],{"className":57280,"style":307},[70],[31,57282,53],{"className":57283},[75,76],[31,57285],{"className":57286,"style":120},[119],[31,57288,434],{"className":57289},[124],[31,57291],{"className":57292,"style":120},[119],[31,57294,57296,57299,57302,57305,57308],{"className":57295},[66],[31,57297],{"className":57298,"style":865},[70],[31,57300,3003],{"className":57301,"style":2666},[75,76],[31,57303],{"className":57304,"style":293},[119],[31,57306,223],{"className":57307},[297],[31,57309],{"className":57310,"style":293},[119],[31,57312,57314,57317,57320,57323,57326],{"className":57313},[66],[31,57315],{"className":57316,"style":9831},[70],[31,57318,3003],{"className":57319,"style":2666},[75,76],[31,57321],{"className":57322,"style":120},[119],[31,57324,434],{"className":57325},[124],[31,57327],{"className":57328,"style":120},[119],[31,57330,57332,57335],{"className":57331},[66],[31,57333],{"className":57334,"style":71},[70],[31,57336,53],{"className":57337},[75,76],"。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 ",[31,57340,57342,57355],{"className":57341,"translate":35},[34],[31,57343,57345],{"className":57344},[39],[41,57346,57347],{"xmlns":43},[45,57348,57349,57353],{},[48,57350,57351],{},[51,57352,53],{},[55,57354,53],{"encoding":57},[31,57356,57358],{"className":57357,"ariaHidden":62},[61],[31,57359,57361,57364],{"className":57360},[66],[31,57362],{"className":57363,"style":71},[70],[31,57365,53],{"className":57366},[75,76]," 再加 ",[31,57369,57371,57384],{"className":57370,"translate":35},[34],[31,57372,57374],{"className":57373},[39],[41,57375,57376],{"xmlns":43},[45,57377,57378,57382],{},[48,57379,57380],{},[51,57381,3003],{},[55,57383,3003],{"encoding":57},[31,57385,57387],{"className":57386,"ariaHidden":62},[61],[31,57388,57390,57393],{"className":57389},[66],[31,57391],{"className":57392,"style":865},[70],[31,57394,3003],{"className":57395,"style":2666},[75,76],"」的电路，右边是一个「先加 ",[31,57398,57400,57413],{"className":57399,"translate":35},[34],[31,57401,57403],{"className":57402},[39],[41,57404,57405],{"xmlns":43},[45,57406,57407,57411],{},[48,57408,57409],{},[51,57410,3003],{},[55,57412,3003],{"encoding":57},[31,57414,57416],{"className":57415,"ariaHidden":62},[61],[31,57417,57419,57422],{"className":57418},[66],[31,57420],{"className":57421,"style":865},[70],[31,57423,3003],{"className":57424,"style":2666},[75,76],[31,57426,57428,57441],{"className":57427,"translate":35},[34],[31,57429,57431],{"className":57430},[39],[41,57432,57433],{"xmlns":43},[45,57434,57435,57439],{},[48,57436,57437],{},[51,57438,53],{},[55,57440,53],{"encoding":57},[31,57442,57444],{"className":57443,"ariaHidden":62},[61],[31,57445,57447,57450],{"className":57446},[66],[31,57448],{"className":57449,"style":71},[70],[31,57451,53],{"className":57452},[75,76],"」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。",[11,57455,57456],{},"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。",[53200,57458,57460],{"id":57459},"_22-不可判定","2.2 不可判定",[11,57462,57463],{},"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？",[11,57465,57466],{},"不行。看一个具体方程：",[31,57468,57470],{"className":57469,"translate":35},[411],[31,57471,57473,57512],{"className":57472,"translate":35},[34],[31,57474,57476],{"className":57475},[39],[41,57477,57478],{"xmlns":43,"display":420},[45,57479,57480,57509],{},[48,57481,57482,57488,57490,57496,57498,57504,57506],{},[1180,57483,57484,57486],{},[51,57485,53],{},[91,57487,93],{},[95,57489,434],{},[1180,57491,57492,57494],{},[51,57493,3003],{},[91,57495,93],{},[95,57497,434],{},[1180,57499,57500,57502],{},[51,57501,46458],{},[91,57503,93],{},[95,57505,223],{},[91,57507,57508],{},"33",[55,57510,57511],{"encoding":57},"x^3 + y^3 + z^3 = 33",[31,57513,57515,57560,57605,57649],{"className":57514,"ariaHidden":62},[61],[31,57516,57518,57522,57551,57554,57557],{"className":57517},[66],[31,57519],{"className":57520,"style":57521},[70],"height:0.9474em;vertical-align:-0.0833em;",[31,57523,57525,57528],{"className":57524},[75],[31,57526,53],{"className":57527},[75,76],[31,57529,57531],{"className":57530},[1039],[31,57532,57534],{"className":57533},[1043],[31,57535,57537],{"className":57536},[1048],[31,57538,57540],{"className":57539,"style":1327},[1052],[31,57541,57542,57545],{"style":1330},[31,57543],{"className":57544,"style":1061},[1060],[31,57546,57548],{"className":57547},[1065,1066,1067,1068],[31,57549,93],{"className":57550},[75,1068],[31,57552],{"className":57553,"style":120},[119],[31,57555,434],{"className":57556},[124],[31,57558],{"className":57559,"style":120},[119],[31,57561,57563,57567,57596,57599,57602],{"className":57562},[66],[31,57564],{"className":57565,"style":57566},[70],"height:1.0585em;vertical-align:-0.1944em;",[31,57568,57570,57573],{"className":57569},[75],[31,57571,3003],{"className":57572,"style":2666},[75,76],[31,57574,57576],{"className":57575},[1039],[31,57577,57579],{"className":57578},[1043],[31,57580,57582],{"className":57581},[1048],[31,57583,57585],{"className":57584,"style":1327},[1052],[31,57586,57587,57590],{"style":1330},[31,57588],{"className":57589,"style":1061},[1060],[31,57591,57593],{"className":57592},[1065,1066,1067,1068],[31,57594,93],{"className":57595},[75,1068],[31,57597],{"className":57598,"style":120},[119],[31,57600,434],{"className":57601},[124],[31,57603],{"className":57604,"style":120},[119],[31,57606,57608,57611,57640,57643,57646],{"className":57607},[66],[31,57609],{"className":57610,"style":1327},[70],[31,57612,57614,57617],{"className":57613},[75],[31,57615,46458],{"className":57616,"style":46500},[75,76],[31,57618,57620],{"className":57619},[1039],[31,57621,57623],{"className":57622},[1043],[31,57624,57626],{"className":57625},[1048],[31,57627,57629],{"className":57628,"style":1327},[1052],[31,57630,57631,57634],{"style":1330},[31,57632],{"className":57633,"style":1061},[1060],[31,57635,57637],{"className":57636},[1065,1066,1067,1068],[31,57638,93],{"className":57639},[75,1068],[31,57641],{"className":57642,"style":293},[119],[31,57644,223],{"className":57645},[297],[31,57647],{"className":57648,"style":293},[119],[31,57650,57652,57655],{"className":57651},[66],[31,57653],{"className":57654,"style":326},[70],[31,57656,57508],{"className":57657},[75],[11,57659,57660],{},"它看起来很简单。它有整数解，一个已知解是：",[31,57662,57664],{"className":57663,"translate":35},[411],[31,57665,57667,57770],{"className":57666,"translate":35},[34],[31,57668,57670],{"className":57669},[39],[41,57671,57672],{"xmlns":43,"display":420},[45,57673,57674,57767],{},[5606,57675,57676,57705,57736],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,57677,57678,57690],{},[5615,57679,57680],{},[2071,57681,57682],{"scriptlevel":1977,"displaystyle":62},[48,57683,57684,57686,57688],{},[91,57685,57508],{},[95,57687,223],{},[48,57689],{},[5615,57691,57692],{},[2071,57693,57694],{"scriptlevel":1977,"displaystyle":62},[48,57695,57696,57698],{},[48,57697],{},[1180,57699,57700,57703],{},[91,57701,57702],{},"8866128975287528",[91,57704,93],{},[5612,57706,57707,57713],{},[5615,57708,57709],{},[2071,57710,57711],{"scriptlevel":1977,"displaystyle":62},[48,57712],{},[5615,57714,57715],{},[2071,57716,57717],{"scriptlevel":1977,"displaystyle":62},[48,57718,57719,57721,57723,57725,57727,57730],{},[48,57720],{},[95,57722,434],{},[95,57724,211],{"stretchy":210},[95,57726,97],{},[91,57728,57729],{},"8778405442862239",[1180,57731,57732,57734],{},[95,57733,220],{"stretchy":210},[91,57735,93],{},[5612,57737,57738,57744],{},[5615,57739,57740],{},[2071,57741,57742],{"scriptlevel":1977,"displaystyle":62},[48,57743],{},[5615,57745,57746],{},[2071,57747,57748],{"scriptlevel":1977,"displaystyle":62},[48,57749,57750,57752,57754,57756,57758,57761],{},[48,57751],{},[95,57753,434],{},[95,57755,211],{"stretchy":210},[95,57757,97],{},[91,57759,57760],{},"2736111468807040",[1180,57762,57763,57765],{},[95,57764,220],{"stretchy":210},[91,57766,93],{},[55,57768,57769],{"encoding":57},"\\begin{aligned}\n33 ={}& 8866128975287528^3 \\\\\n      &+ (-8778405442862239)^3 \\\\\n      &+ (-2736111468807040)^3\n\\end{aligned}",[31,57771,57773],{"className":57772,"ariaHidden":62},[61],[31,57774,57776,57780],{"className":57775},[66],[31,57777],{"className":57778,"style":57779},[70],"height:4.2723em;vertical-align:-1.8862em;",[31,57781,57783],{"className":57782},[75],[31,57784,57786,57853],{"className":57785},[5606],[31,57787,57789],{"className":57788},[5867],[31,57790,57792,57844],{"className":57791},[1043,1044],[31,57793,57795,57841],{"className":57794},[1048],[31,57796,57799,57823,57832],{"className":57797,"style":57798},[1052],"height:2.3862em;",[31,57800,57802,57805],{"style":57801},"top:-4.5221em;",[31,57803],{"className":57804,"style":1637},[1060],[31,57806,57808,57811,57814,57817,57820],{"className":57807},[75],[31,57809,57508],{"className":57810},[75],[31,57812],{"className":57813,"style":293},[119],[31,57815,223],{"className":57816},[297],[31,57818],{"className":57819,"style":293},[119],[31,57821],{"className":57822},[75],[31,57824,57826,57829],{"style":57825},"top:-2.9979em;",[31,57827],{"className":57828,"style":1637},[1060],[31,57830],{"className":57831},[75],[31,57833,57835,57838],{"style":57834},"top:-1.4738em;",[31,57836],{"className":57837,"style":1637},[1060],[31,57839],{"className":57840},[75],[31,57842,1076],{"className":57843},[1075],[31,57845,57847],{"className":57846},[1048],[31,57848,57851],{"className":57849,"style":57850},[1052],"height:1.8862em;",[31,57852],{},[31,57854,57856],{"className":57855},[6062],[31,57857,57859,58028],{"className":57858},[1043,1044],[31,57860,57862,58025],{"className":57861},[1048],[31,57863,57865,57909,57967],{"className":57864,"style":57798},[1052],[31,57866,57867,57870],{"style":57801},[31,57868],{"className":57869,"style":1637},[1060],[31,57871,57873,57876,57880],{"className":57872},[75],[31,57874],{"className":57875},[75],[31,57877,57879],{"className":57878},[75],"886612897528752",[31,57881,57883,57886],{"className":57882},[75],[31,57884,19641],{"className":57885},[75],[31,57887,57889],{"className":57888},[1039],[31,57890,57892],{"className":57891},[1043],[31,57893,57895],{"className":57894},[1048],[31,57896,57898],{"className":57897,"style":1327},[1052],[31,57899,57900,57903],{"style":1330},[31,57901],{"className":57902,"style":1061},[1060],[31,57904,57906],{"className":57905},[1065,1066,1067,1068],[31,57907,93],{"className":57908},[75,1068],[31,57910,57911,57914],{"style":57825},[31,57912],{"className":57913,"style":1637},[1060],[31,57915,57917,57920,57923,57926,57929,57932,57935,57938],{"className":57916},[75],[31,57918],{"className":57919},[75],[31,57921],{"className":57922,"style":120},[119],[31,57924,434],{"className":57925},[124],[31,57927],{"className":57928,"style":120},[119],[31,57930,211],{"className":57931},[264],[31,57933,97],{"className":57934},[75],[31,57936,57729],{"className":57937},[75],[31,57939,57941,57944],{"className":57940},[289],[31,57942,220],{"className":57943},[289],[31,57945,57947],{"className":57946},[1039],[31,57948,57950],{"className":57949},[1043],[31,57951,57953],{"className":57952},[1048],[31,57954,57956],{"className":57955,"style":1327},[1052],[31,57957,57958,57961],{"style":1330},[31,57959],{"className":57960,"style":1061},[1060],[31,57962,57964],{"className":57963},[1065,1066,1067,1068],[31,57965,93],{"className":57966},[75,1068],[31,57968,57969,57972],{"style":57834},[31,57970],{"className":57971,"style":1637},[1060],[31,57973,57975,57978,57981,57984,57987,57990,57993,57996],{"className":57974},[75],[31,57976],{"className":57977},[75],[31,57979],{"className":57980,"style":120},[119],[31,57982,434],{"className":57983},[124],[31,57985],{"className":57986,"style":120},[119],[31,57988,211],{"className":57989},[264],[31,57991,97],{"className":57992},[75],[31,57994,57760],{"className":57995},[75],[31,57997,57999,58002],{"className":57998},[289],[31,58000,220],{"className":58001},[289],[31,58003,58005],{"className":58004},[1039],[31,58006,58008],{"className":58007},[1043],[31,58009,58011],{"className":58010},[1048],[31,58012,58014],{"className":58013,"style":1327},[1052],[31,58015,58016,58019],{"style":1330},[31,58017],{"className":58018,"style":1061},[1060],[31,58020,58022],{"className":58021},[1065,1066,1067,1068],[31,58023,93],{"className":58024},[75,1068],[31,58026,1076],{"className":58027},[1075],[31,58029,58031],{"className":58030},[1048],[31,58032,58034],{"className":58033,"style":57850},[1052],[31,58035],{},[11,58037,58038,58039,58094,58095,58158,58159,58225],{},"这三个数都是 16 位数，直到 2019 年才被找到。如果把 ",[31,58040,58042,58064],{"className":58041,"translate":35},[34],[31,58043,58045],{"className":58044},[39],[41,58046,58047],{"xmlns":43},[45,58048,58049,58061],{},[48,58050,58051,58053,58055,58057,58059],{},[51,58052,53],{},[95,58054,1210],{"separator":62},[51,58056,3003],{},[95,58058,1210],{"separator":62},[51,58060,46458],{},[55,58062,58063],{"encoding":57},"x, y, z",[31,58065,58067],{"className":58066,"ariaHidden":62},[61],[31,58068,58070,58073,58076,58079,58082,58085,58088,58091],{"className":58069},[66],[31,58071],{"className":58072,"style":865},[70],[31,58074,53],{"className":58075},[75,76],[31,58077,1210],{"className":58078},[1391],[31,58080],{"className":58081,"style":1467},[119],[31,58083,3003],{"className":58084,"style":2666},[75,76],[31,58086,1210],{"className":58087},[1391],[31,58089],{"className":58090,"style":1467},[119],[31,58092,46458],{"className":58093,"style":46500},[75,76]," 都编码成 64 位整数，搜索空间是 ",[31,58096,58098,58117],{"className":58097,"translate":35},[34],[31,58099,58101],{"className":58100},[39],[41,58102,58103],{"xmlns":43},[45,58104,58105,58114],{},[48,58106,58107],{},[1180,58108,58109,58111],{},[91,58110,152],{},[91,58112,58113],{},"192",[55,58115,58116],{"encoding":57},"2^{192}",[31,58118,58120],{"className":58119,"ariaHidden":62},[61],[31,58121,58123,58126],{"className":58122},[66],[31,58124],{"className":58125,"style":5135},[70],[31,58127,58129,58132],{"className":58128},[75],[31,58130,152],{"className":58131},[75],[31,58133,58135],{"className":58134},[1039],[31,58136,58138],{"className":58137},[1043],[31,58139,58141],{"className":58140},[1048],[31,58142,58144],{"className":58143,"style":5135},[1052],[31,58145,58146,58149],{"style":3631},[31,58147],{"className":58148,"style":1061},[1060],[31,58150,58152],{"className":58151},[1065,1066,1067,1068],[31,58153,58155],{"className":58154},[75,1068],[31,58156,58113],{"className":58157},[75,1068],"；按每秒试十亿个赋值来算，也要 ",[31,58160,58162,58181],{"className":58161,"translate":35},[34],[31,58163,58165],{"className":58164},[39],[41,58166,58167],{"xmlns":43},[45,58168,58169,58178],{},[48,58170,58171],{},[1180,58172,58173,58175],{},[91,58174,19494],{},[91,58176,58177],{},"41",[55,58179,58180],{"encoding":57},"10^{41}",[31,58182,58184],{"className":58183,"ariaHidden":62},[61],[31,58185,58187,58190,58193],{"className":58186},[66],[31,58188],{"className":58189,"style":5135},[70],[31,58191,205],{"className":58192},[75],[31,58194,58196,58199],{"className":58195},[75],[31,58197,1977],{"className":58198},[75],[31,58200,58202],{"className":58201},[1039],[31,58203,58205],{"className":58204},[1043],[31,58206,58208],{"className":58207},[1048],[31,58209,58211],{"className":58210,"style":5135},[1052],[31,58212,58213,58216],{"style":3631},[31,58214],{"className":58215,"style":1061},[1060],[31,58217,58219],{"className":58218},[1065,1066,1067,1068],[31,58220,58222],{"className":58221},[75,1068],[31,58223,58177],{"className":58224},[75,1068]," 年量级。位爆破在这里基本不可行。",[11,58227,58228],{},"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。",[11,58230,58231],{},"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。",[53200,58233,58235],{"id":58234},"_23-难以组合","2.3 难以组合",[11,58237,58238],{},"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。",[11,58240,58241],{},"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：",[31,58243,58245],{"className":58244,"translate":35},[411],[31,58246,58248,58290],{"className":58247,"translate":35},[34],[31,58249,58251],{"className":58250},[39],[41,58252,58253],{"xmlns":43,"display":420},[45,58254,58255,58287],{},[48,58256,58257,58259,58261,58263,58265,58267,58269,58271,58273,58275,58277,58279,58281,58283,58285],{},[51,58258,53],{},[95,58260,223],{},[51,58262,3003],{},[1430,58264,53035],{},[95,58266,53038],{},[1430,58268,53035],{},[51,58270,1662],{},[95,58272,211],{"stretchy":210},[51,58274,53],{},[95,58276,220],{"stretchy":210},[95,58278,21271],{"mathvariant":494},[51,58280,1662],{},[95,58282,211],{"stretchy":210},[51,58284,3003],{},[95,58286,220],{"stretchy":210},[55,58288,58289],{"encoding":57},"x = y \\;\\wedge\\; f(x) \\ne f(y)",[31,58291,58293,58311,58335,58395],{"className":58292,"ariaHidden":62},[61],[31,58294,58296,58299,58302,58305,58308],{"className":58295},[66],[31,58297],{"className":58298,"style":71},[70],[31,58300,53],{"className":58301},[75,76],[31,58303],{"className":58304,"style":293},[119],[31,58306,223],{"className":58307},[297],[31,58309],{"className":58310,"style":293},[119],[31,58312,58314,58317,58320,58323,58326,58329,58332],{"className":58313},[66],[31,58315],{"className":58316,"style":53828},[70],[31,58318,3003],{"className":58319,"style":2666},[75,76],[31,58321],{"className":58322,"style":293},[119],[31,58324],{"className":58325,"style":120},[119],[31,58327,53038],{"className":58328},[124],[31,58330],{"className":58331,"style":293},[119],[31,58333],{"className":58334,"style":120},[119],[31,58336,58338,58341,58344,58347,58350,58353,58356,58392],{"className":58337},[66],[31,58339],{"className":58340,"style":260},[70],[31,58342,1662],{"className":58343,"style":35196},[75,76],[31,58345,211],{"className":58346},[264],[31,58348,53],{"className":58349},[75,76],[31,58351,220],{"className":58352},[289],[31,58354],{"className":58355,"style":293},[119],[31,58357,58359,58386,58389],{"className":58358},[297],[31,58360,58362],{"className":58361},[297],[31,58363,58365],{"className":58364},[75,21516],[31,58366,58368],{"className":58367},[21520],[31,58369,58371,58374,58383],{"className":58370},[21524],[31,58372],{"className":58373,"style":4202},[70],[31,58375,58377],{"className":58376},[21531],[31,58378,58380],{"className":58379},[75],[31,58381,21538],{"className":58382},[297],[31,58384],{"className":58385},[21542],[31,58387],{"className":58388},[119,21546],[31,58390,223],{"className":58391},[297],[31,58393],{"className":58394,"style":293},[119],[31,58396,58398,58401,58404,58407,58410],{"className":58397},[66],[31,58399],{"className":58400,"style":260},[70],[31,58402,1662],{"className":58403,"style":35196},[75,76],[31,58405,211],{"className":58406},[264],[31,58408,3003],{"className":58409,"style":2666},[75,76],[31,58411,220],{"className":58412},[289],[11,58414,58415],{},"假设我们分别去问两个求解器：",[55164,58417,58418,58549],{},[55167,58419,58420,58421,58472,58473,58475,58476,36037],{},"算术求解器只看 ",[31,58422,58424,58442],{"className":58423,"translate":35},[34],[31,58425,58427],{"className":58426},[39],[41,58428,58429],{"xmlns":43},[45,58430,58431,58439],{},[48,58432,58433,58435,58437],{},[51,58434,53],{},[95,58436,223],{},[51,58438,3003],{},[55,58440,58441],{"encoding":57},"x = y",[31,58443,58445,58463],{"className":58444,"ariaHidden":62},[61],[31,58446,58448,58451,58454,58457,58460],{"className":58447},[66],[31,58449],{"className":58450,"style":71},[70],[31,58452,53],{"className":58453},[75,76],[31,58455],{"className":58456,"style":293},[119],[31,58458,223],{"className":58459},[297],[31,58461],{"className":58462,"style":293},[119],[31,58464,58466,58469],{"className":58465},[66],[31,58467],{"className":58468,"style":865},[70],[31,58470,3003],{"className":58471,"style":2666},[75,76],"，回答 ",[7246,58474,53187],{},"，比如 ",[31,58477,58479,58501],{"className":58478,"translate":35},[34],[31,58480,58482],{"className":58481},[39],[41,58483,58484],{"xmlns":43},[45,58485,58486,58498],{},[48,58487,58488,58490,58492,58494,58496],{},[51,58489,53],{},[95,58491,223],{},[51,58493,3003],{},[95,58495,223],{},[91,58497,1977],{},[55,58499,58500],{"encoding":57},"x = y = 0",[31,58502,58504,58522,58540],{"className":58503,"ariaHidden":62},[61],[31,58505,58507,58510,58513,58516,58519],{"className":58506},[66],[31,58508],{"className":58509,"style":71},[70],[31,58511,53],{"className":58512},[75,76],[31,58514],{"className":58515,"style":293},[119],[31,58517,223],{"className":58518},[297],[31,58520],{"className":58521,"style":293},[119],[31,58523,58525,58528,58531,58534,58537],{"className":58524},[66],[31,58526],{"className":58527,"style":865},[70],[31,58529,3003],{"className":58530,"style":2666},[75,76],[31,58532],{"className":58533,"style":293},[119],[31,58535,223],{"className":58536},[297],[31,58538],{"className":58539,"style":293},[119],[31,58541,58543,58546],{"className":58542},[66],[31,58544],{"className":58545,"style":326},[70],[31,58547,1977],{"className":58548},[75],[55167,58550,58551,58552,58665,58666,58668,58669,58710,58711,15282,58777,2372],{},"EUF 求解器只看 ",[31,58553,58555,58584],{"className":58554,"translate":35},[34],[31,58556,58558],{"className":58557},[39],[41,58559,58560],{"xmlns":43},[45,58561,58562,58582],{},[48,58563,58564,58566,58568,58570,58572,58574,58576,58578,58580],{},[51,58565,1662],{},[95,58567,211],{"stretchy":210},[51,58569,53],{},[95,58571,220],{"stretchy":210},[95,58573,21271],{"mathvariant":494},[51,58575,1662],{},[95,58577,211],{"stretchy":210},[51,58579,3003],{},[95,58581,220],{"stretchy":210},[55,58583,55414],{"encoding":57},[31,58585,58587,58647],{"className":58586,"ariaHidden":62},[61],[31,58588,58590,58593,58596,58599,58602,58605,58608,58644],{"className":58589},[66],[31,58591],{"className":58592,"style":260},[70],[31,58594,1662],{"className":58595,"style":35196},[75,76],[31,58597,211],{"className":58598},[264],[31,58600,53],{"className":58601},[75,76],[31,58603,220],{"className":58604},[289],[31,58606],{"className":58607,"style":293},[119],[31,58609,58611,58638,58641],{"className":58610},[297],[31,58612,58614],{"className":58613},[297],[31,58615,58617],{"className":58616},[75,21516],[31,58618,58620],{"className":58619},[21520],[31,58621,58623,58626,58635],{"className":58622},[21524],[31,58624],{"className":58625,"style":4202},[70],[31,58627,58629],{"className":58628},[21531],[31,58630,58632],{"className":58631},[75],[31,58633,21538],{"className":58634},[297],[31,58636],{"className":58637},[21542],[31,58639],{"className":58640},[119,21546],[31,58642,223],{"className":58643},[297],[31,58645],{"className":58646,"style":293},[119],[31,58648,58650,58653,58656,58659,58662],{"className":58649},[66],[31,58651],{"className":58652,"style":260},[70],[31,58654,1662],{"className":58655,"style":35196},[75,76],[31,58657,211],{"className":58658},[264],[31,58660,3003],{"className":58661,"style":2666},[75,76],[31,58663,220],{"className":58664},[289],"，也回答 ",[7246,58667,53187],{},"：它把 ",[31,58670,58672,58689],{"className":58671,"translate":35},[34],[31,58673,58675],{"className":58674},[39],[41,58676,58677],{"xmlns":43},[45,58678,58679,58687],{},[48,58680,58681,58683,58685],{},[51,58682,53],{},[95,58684,1210],{"separator":62},[51,58686,3003],{},[55,58688,55902],{"encoding":57},[31,58690,58692],{"className":58691,"ariaHidden":62},[61],[31,58693,58695,58698,58701,58704,58707],{"className":58694},[66],[31,58696],{"className":58697,"style":865},[70],[31,58699,53],{"className":58700},[75,76],[31,58702,1210],{"className":58703},[1391],[31,58705],{"className":58706,"style":1467},[119],[31,58708,3003],{"className":58709,"style":2666},[75,76]," 当成两个不同的常量，令 ",[31,58712,58714,58738],{"className":58713,"translate":35},[34],[31,58715,58717],{"className":58716},[39],[41,58718,58719],{"xmlns":43},[45,58720,58721,58735],{},[48,58722,58723,58725,58727,58729,58731,58733],{},[51,58724,1662],{},[95,58726,211],{"stretchy":210},[51,58728,53],{},[95,58730,220],{"stretchy":210},[95,58732,223],{},[91,58734,1977],{},[55,58736,58737],{"encoding":57},"f(x) = 0",[31,58739,58741,58768],{"className":58740,"ariaHidden":62},[61],[31,58742,58744,58747,58750,58753,58756,58759,58762,58765],{"className":58743},[66],[31,58745],{"className":58746,"style":260},[70],[31,58748,1662],{"className":58749,"style":35196},[75,76],[31,58751,211],{"className":58752},[264],[31,58754,53],{"className":58755},[75,76],[31,58757,220],{"className":58758},[289],[31,58760],{"className":58761,"style":293},[119],[31,58763,223],{"className":58764},[297],[31,58766],{"className":58767,"style":293},[119],[31,58769,58771,58774],{"className":58770},[66],[31,58772],{"className":58773,"style":326},[70],[31,58775,1977],{"className":58776},[75],[31,58778,58780,58804],{"className":58779,"translate":35},[34],[31,58781,58783],{"className":58782},[39],[41,58784,58785],{"xmlns":43},[45,58786,58787,58801],{},[48,58788,58789,58791,58793,58795,58797,58799],{},[51,58790,1662],{},[95,58792,211],{"stretchy":210},[51,58794,3003],{},[95,58796,220],{"stretchy":210},[95,58798,223],{},[91,58800,205],{},[55,58802,58803],{"encoding":57},"f(y) = 1",[31,58805,58807,58834],{"className":58806,"ariaHidden":62},[61],[31,58808,58810,58813,58816,58819,58822,58825,58828,58831],{"className":58809},[66],[31,58811],{"className":58812,"style":260},[70],[31,58814,1662],{"className":58815,"style":35196},[75,76],[31,58817,211],{"className":58818},[264],[31,58820,3003],{"className":58821,"style":2666},[75,76],[31,58823,220],{"className":58824},[289],[31,58826],{"className":58827,"style":293},[119],[31,58829,223],{"className":58830},[297],[31,58832],{"className":58833,"style":293},[119],[31,58835,58837,58840],{"className":58836},[66],[31,58838],{"className":58839,"style":326},[70],[31,58841,205],{"className":58842},[75],[11,58844,58845,58846,58896,58897,58940,58941,58985,58986,59099,59100,2372],{},"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 ",[31,58847,58849,58866],{"className":58848,"translate":35},[34],[31,58850,58852],{"className":58851},[39],[41,58853,58854],{"xmlns":43},[45,58855,58856,58864],{},[48,58857,58858,58860,58862],{},[51,58859,53],{},[95,58861,223],{},[51,58863,3003],{},[55,58865,58441],{"encoding":57},[31,58867,58869,58887],{"className":58868,"ariaHidden":62},[61],[31,58870,58872,58875,58878,58881,58884],{"className":58871},[66],[31,58873],{"className":58874,"style":71},[70],[31,58876,53],{"className":58877},[75,76],[31,58879],{"className":58880,"style":293},[119],[31,58882,223],{"className":58883},[297],[31,58885],{"className":58886,"style":293},[119],[31,58888,58890,58893],{"className":58889},[66],[31,58891],{"className":58892,"style":865},[70],[31,58894,3003],{"className":58895,"style":2666},[75,76]," 意味着 ",[31,58898,58900,58919],{"className":58899,"translate":35},[34],[31,58901,58903],{"className":58902},[39],[41,58904,58905],{"xmlns":43},[45,58906,58907,58917],{},[48,58908,58909,58911,58913,58915],{},[51,58910,1662],{},[95,58912,211],{"stretchy":210},[51,58914,53],{},[95,58916,220],{"stretchy":210},[55,58918,40190],{"encoding":57},[31,58920,58922],{"className":58921,"ariaHidden":62},[61],[31,58923,58925,58928,58931,58934,58937],{"className":58924},[66],[31,58926],{"className":58927,"style":260},[70],[31,58929,1662],{"className":58930,"style":35196},[75,76],[31,58932,211],{"className":58933},[264],[31,58935,53],{"className":58936},[75,76],[31,58938,220],{"className":58939},[289]," 必须等于 ",[31,58942,58944,58964],{"className":58943,"translate":35},[34],[31,58945,58947],{"className":58946},[39],[41,58948,58949],{"xmlns":43},[45,58950,58951,58961],{},[48,58952,58953,58955,58957,58959],{},[51,58954,1662],{},[95,58956,211],{"stretchy":210},[51,58958,3003],{},[95,58960,220],{"stretchy":210},[55,58962,58963],{"encoding":57},"f(y)",[31,58965,58967],{"className":58966,"ariaHidden":62},[61],[31,58968,58970,58973,58976,58979,58982],{"className":58969},[66],[31,58971],{"className":58972,"style":260},[70],[31,58974,1662],{"className":58975,"style":35196},[75,76],[31,58977,211],{"className":58978},[264],[31,58980,3003],{"className":58981,"style":2666},[75,76],[31,58983,220],{"className":58984},[289],"，与 ",[31,58987,58989,59018],{"className":58988,"translate":35},[34],[31,58990,58992],{"className":58991},[39],[41,58993,58994],{"xmlns":43},[45,58995,58996,59016],{},[48,58997,58998,59000,59002,59004,59006,59008,59010,59012,59014],{},[51,58999,1662],{},[95,59001,211],{"stretchy":210},[51,59003,53],{},[95,59005,220],{"stretchy":210},[95,59007,21271],{"mathvariant":494},[51,59009,1662],{},[95,59011,211],{"stretchy":210},[51,59013,3003],{},[95,59015,220],{"stretchy":210},[55,59017,55414],{"encoding":57},[31,59019,59021,59081],{"className":59020,"ariaHidden":62},[61],[31,59022,59024,59027,59030,59033,59036,59039,59042,59078],{"className":59023},[66],[31,59025],{"className":59026,"style":260},[70],[31,59028,1662],{"className":59029,"style":35196},[75,76],[31,59031,211],{"className":59032},[264],[31,59034,53],{"className":59035},[75,76],[31,59037,220],{"className":59038},[289],[31,59040],{"className":59041,"style":293},[119],[31,59043,59045,59072,59075],{"className":59044},[297],[31,59046,59048],{"className":59047},[297],[31,59049,59051],{"className":59050},[75,21516],[31,59052,59054],{"className":59053},[21520],[31,59055,59057,59060,59069],{"className":59056},[21524],[31,59058],{"className":59059,"style":4202},[70],[31,59061,59063],{"className":59062},[21531],[31,59064,59066],{"className":59065},[75],[31,59067,21538],{"className":59068},[297],[31,59070],{"className":59071},[21542],[31,59073],{"className":59074},[119,21546],[31,59076,223],{"className":59077},[297],[31,59079],{"className":59080,"style":293},[119],[31,59082,59084,59087,59090,59093,59096],{"className":59083},[66],[31,59085],{"className":59086,"style":260},[70],[31,59088,1662],{"className":59089,"style":35196},[75,76],[31,59091,211],{"className":59092},[264],[31,59094,3003],{"className":59095,"style":2666},[75,76],[31,59097,220],{"className":59098},[289]," 直接矛盾。整体其实是 ",[7246,59101,59102],{},"unsat",[11,59104,59105,59106,59156,59157,59238],{},"要得到正确答案，两个求解器必须交换信息：算术把等式 ",[31,59107,59109,59126],{"className":59108,"translate":35},[34],[31,59110,59112],{"className":59111},[39],[41,59113,59114],{"xmlns":43},[45,59115,59116,59124],{},[48,59117,59118,59120,59122],{},[51,59119,53],{},[95,59121,223],{},[51,59123,3003],{},[55,59125,58441],{"encoding":57},[31,59127,59129,59147],{"className":59128,"ariaHidden":62},[61],[31,59130,59132,59135,59138,59141,59144],{"className":59131},[66],[31,59133],{"className":59134,"style":71},[70],[31,59136,53],{"className":59137},[75,76],[31,59139],{"className":59140,"style":293},[119],[31,59142,223],{"className":59143},[297],[31,59145],{"className":59146,"style":293},[119],[31,59148,59150,59153],{"className":59149},[66],[31,59151],{"className":59152,"style":865},[70],[31,59154,3003],{"className":59155,"style":2666},[75,76]," 告诉 EUF，EUF 立刻推出 ",[31,59158,59160,59190],{"className":59159,"translate":35},[34],[31,59161,59163],{"className":59162},[39],[41,59164,59165],{"xmlns":43},[45,59166,59167,59187],{},[48,59168,59169,59171,59173,59175,59177,59179,59181,59183,59185],{},[51,59170,1662],{},[95,59172,211],{"stretchy":210},[51,59174,53],{},[95,59176,220],{"stretchy":210},[95,59178,223],{},[51,59180,1662],{},[95,59182,211],{"stretchy":210},[51,59184,3003],{},[95,59186,220],{"stretchy":210},[55,59188,59189],{"encoding":57},"f(x) = f(y)",[31,59191,59193,59220],{"className":59192,"ariaHidden":62},[61],[31,59194,59196,59199,59202,59205,59208,59211,59214,59217],{"className":59195},[66],[31,59197],{"className":59198,"style":260},[70],[31,59200,1662],{"className":59201,"style":35196},[75,76],[31,59203,211],{"className":59204},[264],[31,59206,53],{"className":59207},[75,76],[31,59209,220],{"className":59210},[289],[31,59212],{"className":59213,"style":293},[119],[31,59215,223],{"className":59216},[297],[31,59218],{"className":59219,"style":293},[119],[31,59221,59223,59226,59229,59232,59235],{"className":59222},[66],[31,59224],{"className":59225,"style":260},[70],[31,59227,1662],{"className":59228,"style":35196},[75,76],[31,59230,211],{"className":59231},[264],[31,59233,3003],{"className":59234,"style":2666},[75,76],[31,59236,220],{"className":59237},[289],"，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。",[11,59240,59241,59242,59245],{},"数组和算术之间也有同样的坑。",[45608,59243,59244],{},"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。",[53200,59247,59249],{"id":59248},"_24-门槛太高","2.4 门槛太高",[11,59251,59252,59253,2372],{},"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",[7246,59254,59255],{},"需要人写证明脚本",[11,59257,59258],{},"比如在 Coq 里证明「任何自然数加零都等于它自己」，要写：",[52853,59260,59264],{"className":59261,"code":59262,"language":59263,"meta":52812,"style":52812},"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",[45608,59265,59266,59271,59276,59281,59286,59291],{"__ignoreMap":52812},[31,59267,59268],{"class":52861,"line":52862},[31,59269,59270],{},"Theorem plus_0_r : forall n : nat, n + 0 = n.\n",[31,59272,59273],{"class":52861,"line":52813},[31,59274,59275],{},"Proof.\n",[31,59277,59278],{"class":52861,"line":52920},[31,59279,59280],{},"  induction n as [| n' IH].\n",[31,59282,59283],{"class":52861,"line":52940},[31,59284,59285],{},"  - simpl. reflexivity.\n",[31,59287,59288],{"class":52861,"line":52977},[31,59289,59290],{},"  - simpl. rewrite IH. reflexivity.\n",[31,59292,59293],{"class":52861,"line":52984},[31,59294,59295],{},"Qed.\n",[11,59297,59298],{},"这几行不是装饰：",[55164,59300,59301,59457,59548],{},[55167,59302,59303,59306,59307,2446,59358,36037],{},[45608,59304,59305],{},"induction n"," 把目标拆成两种情况：",[31,59308,59310,59328],{"className":59309,"translate":35},[34],[31,59311,59313],{"className":59312},[39],[41,59314,59315],{"xmlns":43},[45,59316,59317,59325],{},[48,59318,59319,59321,59323],{},[51,59320,14559],{},[95,59322,223],{},[91,59324,1977],{},[55,59326,59327],{"encoding":57},"n = 0",[31,59329,59331,59349],{"className":59330,"ariaHidden":62},[61],[31,59332,59334,59337,59340,59343,59346],{"className":59333},[66],[31,59335],{"className":59336,"style":71},[70],[31,59338,14559],{"className":59339},[75,76],[31,59341],{"className":59342,"style":293},[119],[31,59344,223],{"className":59345},[297],[31,59347],{"className":59348,"style":293},[119],[31,59350,59352,59355],{"className":59351},[66],[31,59353],{"className":59354,"style":326},[70],[31,59356,1977],{"className":59357},[75],[31,59359,59361,59389],{"className":59360,"translate":35},[34],[31,59362,59364],{"className":59363},[39],[41,59365,59366],{"xmlns":43},[45,59367,59368,59386],{},[48,59369,59370,59372,59374,59376,59378,59384],{},[51,59371,14559],{},[95,59373,223],{},[51,59375,21582],{},[95,59377,211],{"stretchy":210},[1180,59379,59380,59382],{},[51,59381,14559],{},[95,59383,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,59385,220],{"stretchy":210},[55,59387,59388],{"encoding":57},"n = S(n')",[31,59390,59392,59410],{"className":59391,"ariaHidden":62},[61],[31,59393,59395,59398,59401,59404,59407],{"className":59394},[66],[31,59396],{"className":59397,"style":71},[70],[31,59399,14559],{"className":59400},[75,76],[31,59402],{"className":59403,"style":293},[119],[31,59405,223],{"className":59406},[297],[31,59408],{"className":59409,"style":293},[119],[31,59411,59413,59416,59419,59422,59454],{"className":59412},[66],[31,59414],{"className":59415,"style":13710},[70],[31,59417,21582],{"className":59418,"style":21697},[75,76],[31,59420,211],{"className":59421},[264],[31,59423,59425,59428],{"className":59424},[75],[31,59426,14559],{"className":59427},[75,76],[31,59429,59431],{"className":59430},[1039],[31,59432,59434],{"className":59433},[1043],[31,59435,59437],{"className":59436},[1048],[31,59438,59440],{"className":59439,"style":13732},[1052],[31,59441,59442,59445],{"style":3631},[31,59443],{"className":59444,"style":1061},[1060],[31,59446,59448],{"className":59447},[1065,1066,1067,1068],[31,59449,59451],{"className":59450},[75,1068],[31,59452,13691],{"className":59453},[75,1068],[31,59455,220],{"className":59456},[289],[55167,59458,59459,59460,59463,59464,59515,59516,1138,59544,59547],{},"第一种情况，",[45608,59461,59462],{},"simpl"," 把 ",[31,59465,59467,59485],{"className":59466,"translate":35},[34],[31,59468,59470],{"className":59469},[39],[41,59471,59472],{"xmlns":43},[45,59473,59474,59482],{},[48,59475,59476,59478,59480],{},[91,59477,1977],{},[95,59479,434],{},[91,59481,1977],{},[55,59483,59484],{"encoding":57},"0 + 0",[31,59486,59488,59506],{"className":59487,"ariaHidden":62},[61],[31,59489,59491,59494,59497,59500,59503],{"className":59490},[66],[31,59492],{"className":59493,"style":112},[70],[31,59495,1977],{"className":59496},[75],[31,59498],{"className":59499,"style":120},[119],[31,59501,434],{"className":59502},[124],[31,59504],{"className":59505,"style":120},[119],[31,59507,59509,59512],{"className":59508},[66],[31,59510],{"className":59511,"style":326},[70],[31,59513,1977],{"className":59514},[75]," 化简成 ",[31,59517,59519,59532],{"className":59518,"translate":35},[34],[31,59520,59522],{"className":59521},[39],[41,59523,59524],{"xmlns":43},[45,59525,59526,59530],{},[48,59527,59528],{},[91,59529,1977],{},[55,59531,1977],{"encoding":57},[31,59533,59535],{"className":59534,"ariaHidden":62},[61],[31,59536,59538,59541],{"className":59537},[66],[31,59539],{"className":59540,"style":326},[70],[31,59542,1977],{"className":59543},[75],[45608,59545,59546],{},"reflexivity"," 确认两边相等；",[55167,59549,59550,59551,59463,59553,59652,59653,59752,59753,59463,59756,59841,59842,59904,59905,59907],{},"第二种情况，",[45608,59552,59462],{},[31,59554,59556,59584],{"className":59555,"translate":35},[34],[31,59557,59559],{"className":59558},[39],[41,59560,59561],{"xmlns":43},[45,59562,59563,59581],{},[48,59564,59565,59567,59569,59575,59577,59579],{},[51,59566,21582],{},[95,59568,211],{"stretchy":210},[1180,59570,59571,59573],{},[51,59572,14559],{},[95,59574,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,59576,220],{"stretchy":210},[95,59578,434],{},[91,59580,1977],{},[55,59582,59583],{"encoding":57},"S(n') + 0",[31,59585,59587,59643],{"className":59586,"ariaHidden":62},[61],[31,59588,59590,59593,59596,59599,59631,59634,59637,59640],{"className":59589},[66],[31,59591],{"className":59592,"style":13710},[70],[31,59594,21582],{"className":59595,"style":21697},[75,76],[31,59597,211],{"className":59598},[264],[31,59600,59602,59605],{"className":59601},[75],[31,59603,14559],{"className":59604},[75,76],[31,59606,59608],{"className":59607},[1039],[31,59609,59611],{"className":59610},[1043],[31,59612,59614],{"className":59613},[1048],[31,59615,59617],{"className":59616,"style":13732},[1052],[31,59618,59619,59622],{"style":3631},[31,59620],{"className":59621,"style":1061},[1060],[31,59623,59625],{"className":59624},[1065,1066,1067,1068],[31,59626,59628],{"className":59627},[75,1068],[31,59629,13691],{"className":59630},[75,1068],[31,59632,220],{"className":59633},[289],[31,59635],{"className":59636,"style":120},[119],[31,59638,434],{"className":59639},[124],[31,59641],{"className":59642,"style":120},[119],[31,59644,59646,59649],{"className":59645},[66],[31,59647],{"className":59648,"style":326},[70],[31,59650,1977],{"className":59651},[75]," 化成 ",[31,59654,59656,59684],{"className":59655,"translate":35},[34],[31,59657,59659],{"className":59658},[39],[41,59660,59661],{"xmlns":43},[45,59662,59663,59681],{},[48,59664,59665,59667,59669,59675,59677,59679],{},[51,59666,21582],{},[95,59668,211],{"stretchy":210},[1180,59670,59671,59673],{},[51,59672,14559],{},[95,59674,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,59676,434],{},[91,59678,1977],{},[95,59680,220],{"stretchy":210},[55,59682,59683],{"encoding":57},"S(n' + 0)",[31,59685,59687,59740],{"className":59686,"ariaHidden":62},[61],[31,59688,59690,59693,59696,59699,59731,59734,59737],{"className":59689},[66],[31,59691],{"className":59692,"style":13710},[70],[31,59694,21582],{"className":59695,"style":21697},[75,76],[31,59697,211],{"className":59698},[264],[31,59700,59702,59705],{"className":59701},[75],[31,59703,14559],{"className":59704},[75,76],[31,59706,59708],{"className":59707},[1039],[31,59709,59711],{"className":59710},[1043],[31,59712,59714],{"className":59713},[1048],[31,59715,59717],{"className":59716,"style":13732},[1052],[31,59718,59719,59722],{"style":3631},[31,59720],{"className":59721,"style":1061},[1060],[31,59723,59725],{"className":59724},[1065,1066,1067,1068],[31,59726,59728],{"className":59727},[75,1068],[31,59729,13691],{"className":59730},[75,1068],[31,59732],{"className":59733,"style":120},[119],[31,59735,434],{"className":59736},[124],[31,59738],{"className":59739,"style":120},[119],[31,59741,59743,59746,59749],{"className":59742},[66],[31,59744],{"className":59745,"style":260},[70],[31,59747,1977],{"className":59748},[75],[31,59750,220],{"className":59751},[289],"，再用归纳假设 ",[45608,59754,59755],{},"IH",[31,59757,59759,59781],{"className":59758,"translate":35},[34],[31,59760,59762],{"className":59761},[39],[41,59763,59764],{"xmlns":43},[45,59765,59766,59778],{},[48,59767,59768,59774,59776],{},[1180,59769,59770,59772],{},[51,59771,14559],{},[95,59773,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,59775,434],{},[91,59777,1977],{},[55,59779,59780],{"encoding":57},"n' + 0",[31,59782,59784,59832],{"className":59783,"ariaHidden":62},[61],[31,59785,59787,59791,59823,59826,59829],{"className":59786},[66],[31,59788],{"className":59789,"style":59790},[70],"height:0.8352em;vertical-align:-0.0833em;",[31,59792,59794,59797],{"className":59793},[75],[31,59795,14559],{"className":59796},[75,76],[31,59798,59800],{"className":59799},[1039],[31,59801,59803],{"className":59802},[1043],[31,59804,59806],{"className":59805},[1048],[31,59807,59809],{"className":59808,"style":13732},[1052],[31,59810,59811,59814],{"style":3631},[31,59812],{"className":59813,"style":1061},[1060],[31,59815,59817],{"className":59816},[1065,1066,1067,1068],[31,59818,59820],{"className":59819},[75,1068],[31,59821,13691],{"className":59822},[75,1068],[31,59824],{"className":59825,"style":120},[119],[31,59827,434],{"className":59828},[124],[31,59830],{"className":59831,"style":120},[119],[31,59833,59835,59838],{"className":59834},[66],[31,59836],{"className":59837,"style":326},[70],[31,59839,1977],{"className":59840},[75]," 换成 ",[31,59843,59845,59863],{"className":59844,"translate":35},[34],[31,59846,59848],{"className":59847},[39],[41,59849,59850],{"xmlns":43},[45,59851,59852,59860],{},[48,59853,59854],{},[1180,59855,59856,59858],{},[51,59857,14559],{},[95,59859,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[55,59861,59862],{"encoding":57},"n'",[31,59864,59866],{"className":59865,"ariaHidden":62},[61],[31,59867,59869,59872],{"className":59868},[66],[31,59870],{"className":59871,"style":13732},[70],[31,59873,59875,59878],{"className":59874},[75],[31,59876,14559],{"className":59877},[75,76],[31,59879,59881],{"className":59880},[1039],[31,59882,59884],{"className":59883},[1043],[31,59885,59887],{"className":59886},[1048],[31,59888,59890],{"className":59889,"style":13732},[1052],[31,59891,59892,59895],{"style":3631},[31,59893],{"className":59894,"style":1061},[1060],[31,59896,59898],{"className":59897},[1065,1066,1067,1068],[31,59899,59901],{"className":59900},[75,1068],[31,59902,13691],{"className":59903},[75,1068],"，最后 ",[45608,59906,59546],{}," 收尾。",[11,59909,59910],{},"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。",[11,59912,59913],{},"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。",[53200,59915,59917],{"id":59916},"_25-小结","2.5 小结",[11,59919,59920],{},"几条路，各有各的难处：",[55164,59922,59923,59926,59929,59932],{},[55167,59924,59925],{},"纯 SAT 快，但把高层语义压成门电路之后，规模爆炸、语义丢失；",[55167,59927,59928],{},"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；",[55167,59930,59931],{},"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；",[55167,59933,59934],{},"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。",[11,59936,59937,59938],{},"SMT 要做的，就是在这几条路的夹缝里找一个中间地带：",[7246,59939,59940],{},"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。",[18,59942,59944],{"id":59943},"三smt-的解法","三、SMT 的解法",[53200,59946,59948],{"id":59947},"_31-收缩片段","3.1 收缩片段",[11,59950,59951,59952,59955],{},"SMT 的第一个策略是",[7246,59953,59954],{},"主动放弃一部分表达力","，只保留可判定的片段。",[11,59957,59958],{},"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。",[11,59960,59961,59962,1138,59991,60057],{},"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 ",[31,59963,59965,59978],{"className":59964,"translate":35},[34],[31,59966,59968],{"className":59967},[39],[41,59969,59970],{"xmlns":43},[45,59971,59972,59976],{},[48,59973,59974],{},[51,59975,5391],{},[55,59977,5391],{"encoding":57},[31,59979,59981],{"className":59980,"ariaHidden":62},[61],[31,59982,59984,59988],{"className":59983},[66],[31,59985],{"className":59986,"style":59987},[70],"height:0.6595em;",[31,59989,5391],{"className":59990},[75,76],[31,59992,59994,60018],{"className":59993,"translate":35},[34],[31,59995,59997],{"className":59996},[39],[41,59998,59999],{"xmlns":43},[45,60000,60001,60015],{},[48,60002,60003,60005,60007,60009,60011,60013],{},[51,60004,5813],{},[95,60006,15175],{"stretchy":210},[51,60008,5391],{},[95,60010,15184],{"stretchy":210},[95,60012,16040],{},[91,60014,1977],{},[55,60016,60017],{"encoding":57},"a[i] > 0",[31,60019,60021,60048],{"className":60020,"ariaHidden":62},[61],[31,60022,60024,60027,60030,60033,60036,60039,60042,60045],{"className":60023},[66],[31,60025],{"className":60026,"style":260},[70],[31,60028,5813],{"className":60029},[75,76],[31,60031,15175],{"className":60032},[264],[31,60034,5391],{"className":60035},[75,76],[31,60037,15184],{"className":60038},[289],[31,60040],{"className":60041,"style":293},[119],[31,60043,16040],{"className":60044},[297],[31,60046],{"className":60047,"style":293},[119],[31,60049,60051,60054],{"className":60050},[66],[31,60052],{"className":60053,"style":326},[70],[31,60055,1977],{"className":60056},[75],"」这种全称陈述。",[11,60059,60060,60061,60089],{},"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，",[31,60062,60064,60077],{"className":60063,"translate":35},[34],[31,60065,60067],{"className":60066},[39],[41,60068,60069],{"xmlns":43},[45,60070,60071,60075],{},[48,60072,60073],{},[51,60074,52065],{},[55,60076,52065],{"encoding":57},[31,60078,60080],{"className":60079,"ariaHidden":62},[61],[31,60081,60083,60086],{"className":60082},[66],[31,60084],{"className":60085,"style":1630},[70],[31,60087,52065],{"className":60088,"style":52166},[75,76],"-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。",[11,60091,60092,60093,15282,60095,15282,60097,60099,60100,60102,60103,60105,60106,60108],{},"当然，「可判定」不等于「容易」。",[45608,60094,55621],{},[45608,60096,55635],{},[45608,60098,55719],{}," 都是 NP-complete；",[45608,60101,55631],{}," 是多项式时间；",[45608,60104,55707],{}," 可判定，但最坏情况是双指数；",[45608,60107,55710],{}," 则直接不可判定。",[11,60110,60111],{},"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。",[53200,60113,60115],{"id":60114},"_32-直接表达","3.2 直接表达",[11,60117,60118,60119,2372],{},"第二个策略是",[7246,60120,60121],{},"让公式直接说人话",[11,60123,60124,60125,60197,60198,60263,60264,60305],{},"同样是「",[31,60126,60128,60149],{"className":60127,"translate":35},[34],[31,60129,60131],{"className":60130},[39],[41,60132,60133],{"xmlns":43},[45,60134,60135,60147],{},[48,60136,60137,60139,60141,60143,60145],{},[51,60138,53],{},[95,60140,434],{},[51,60142,3003],{},[95,60144,16040],{},[91,60146,19494],{},[55,60148,53172],{"encoding":57},[31,60150,60152,60170,60188],{"className":60151,"ariaHidden":62},[61],[31,60153,60155,60158,60161,60164,60167],{"className":60154},[66],[31,60156],{"className":60157,"style":307},[70],[31,60159,53],{"className":60160},[75,76],[31,60162],{"className":60163,"style":120},[119],[31,60165,434],{"className":60166},[124],[31,60168],{"className":60169,"style":120},[119],[31,60171,60173,60176,60179,60182,60185],{"className":60172},[66],[31,60174],{"className":60175,"style":53083},[70],[31,60177,3003],{"className":60178,"style":2666},[75,76],[31,60180],{"className":60181,"style":293},[119],[31,60183,16040],{"className":60184},[297],[31,60186],{"className":60187,"style":293},[119],[31,60189,60191,60194],{"className":60190},[66],[31,60192],{"className":60193,"style":326},[70],[31,60195,19494],{"className":60196},[75]," 且 ",[31,60199,60201,60224],{"className":60200,"translate":35},[34],[31,60202,60204],{"className":60203},[39],[41,60205,60206],{"xmlns":43},[45,60207,60208,60222],{},[48,60209,60210,60212,60214,60216,60218,60220],{},[51,60211,5813],{},[95,60213,15175],{"stretchy":210},[91,60215,1977],{},[95,60217,15184],{"stretchy":210},[95,60219,223],{},[91,60221,7853],{},[55,60223,53176],{"encoding":57},[31,60225,60227,60254],{"className":60226,"ariaHidden":62},[61],[31,60228,60230,60233,60236,60239,60242,60245,60248,60251],{"className":60229},[66],[31,60231],{"className":60232,"style":260},[70],[31,60234,5813],{"className":60235},[75,76],[31,60237,15175],{"className":60238},[264],[31,60240,1977],{"className":60241},[75],[31,60243,15184],{"className":60244},[289],[31,60246],{"className":60247,"style":293},[119],[31,60249,223],{"className":60250},[297],[31,60252],{"className":60253,"style":293},[119],[31,60255,60257,60260],{"className":60256},[66],[31,60258],{"className":60259,"style":326},[70],[31,60261,7853],{"className":60262},[75],"」，SMT 不需要把 ",[31,60265,60267,60284],{"className":60266,"translate":35},[34],[31,60268,60270],{"className":60269},[39],[41,60271,60272],{"xmlns":43},[45,60273,60274,60282],{},[48,60275,60276,60278,60280],{},[51,60277,53],{},[95,60279,1210],{"separator":62},[51,60281,3003],{},[55,60283,55902],{"encoding":57},[31,60285,60287],{"className":60286,"ariaHidden":62},[61],[31,60288,60290,60293,60296,60299,60302],{"className":60289},[66],[31,60291],{"className":60292,"style":865},[70],[31,60294,53],{"className":60295},[75,76],[31,60297,1210],{"className":60298},[1391],[31,60300],{"className":60301,"style":1467},[119],[31,60303,3003],{"className":60304,"style":2666},[75,76]," 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。",[11,60307,60308],{},"关键在于「按需展开」：",[55164,60310,60311,60317,60323],{},[55167,60312,60313,60316],{},[7246,60314,60315],{},"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。",[55167,60318,60319,60322],{},[7246,60320,60321],{},"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。",[55167,60324,60325,60328,60329,60332,60333,15282,60405,60478],{},[7246,60326,60327],{},"位向量","：这一条要诚实一点——大多数求解器内部",[7246,60330,60331],{},"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 ",[31,60334,60336,60363],{"className":60335,"translate":35},[34],[31,60337,60339],{"className":60338},[39],[41,60340,60341],{"xmlns":43},[45,60342,60343,60360],{},[48,60344,60345,60347,60349,60352,60354,60356,60358],{},[51,60346,53],{},[1430,60348,1432],{},[51,60350,60351],{"mathvariant":494},"&",[1430,60353,1432],{},[91,60355,1977],{},[95,60357,223],{},[91,60359,1977],{},[55,60361,60362],{"encoding":57},"x \\,\\&\\, 0 = 0",[31,60364,60366,60396],{"className":60365,"ariaHidden":62},[61],[31,60367,60369,60372,60375,60378,60381,60384,60387,60390,60393],{"className":60368},[66],[31,60370],{"className":60371,"style":1630},[70],[31,60373,53],{"className":60374},[75,76],[31,60376],{"className":60377,"style":1467},[119],[31,60379,60351],{"className":60380},[75],[31,60382],{"className":60383,"style":1467},[119],[31,60385,1977],{"className":60386},[75],[31,60388],{"className":60389,"style":293},[119],[31,60391,223],{"className":60392},[297],[31,60394],{"className":60395,"style":293},[119],[31,60397,60399,60402],{"className":60398},[66],[31,60400],{"className":60401,"style":326},[70],[31,60403,1977],{"className":60404},[75],[31,60406,60408,60430],{"className":60407,"translate":35},[34],[31,60409,60411],{"className":60410},[39],[41,60412,60413],{"xmlns":43},[45,60414,60415,60427],{},[48,60416,60417,60419,60421,60423,60425],{},[51,60418,53],{},[95,60420,97],{},[51,60422,53],{},[95,60424,223],{},[91,60426,1977],{},[55,60428,60429],{"encoding":57},"x - x = 0",[31,60431,60433,60451,60469],{"className":60432,"ariaHidden":62},[61],[31,60434,60436,60439,60442,60445,60448],{"className":60435},[66],[31,60437],{"className":60438,"style":307},[70],[31,60440,53],{"className":60441},[75,76],[31,60443],{"className":60444,"style":120},[119],[31,60446,97],{"className":60447},[124],[31,60449],{"className":60450,"style":120},[119],[31,60452,60454,60457,60460,60463,60466],{"className":60453},[66],[31,60455],{"className":60456,"style":71},[70],[31,60458,53],{"className":60459},[75,76],[31,60461],{"className":60462,"style":293},[119],[31,60464,223],{"className":60465},[297],[31,60467],{"className":60468,"style":293},[119],[31,60470,60472,60475],{"className":60471},[66],[31,60473],{"className":60474,"style":326},[70],[31,60476,1977],{"className":60477},[75],"）和等价类合并。对使用者来说，它仍然是透明的。",[11,60480,60481],{},"数组的 read-over-write 公理长这样：",[31,60483,60485],{"className":60484,"translate":35},[411],[31,60486,60488,60628],{"className":60487,"translate":35},[34],[31,60489,60491],{"className":60490},[39],[41,60492,60493],{"xmlns":43,"display":420},[45,60494,60495,60625],{},[48,60496,60497,60511,60513,60525,60527,60529,60531,60533,60535,60537,60539,60541,60543,60545,60547],{},[48,60498,60499,60501,60503,60505,60507,60509],{},[51,60500,14551],{"mathvariant":494},[51,60502,30042],{"mathvariant":494},[51,60504,54404],{"mathvariant":494},[51,60506,30042],{"mathvariant":494},[51,60508,1412],{"mathvariant":494},[51,60510,12271],{"mathvariant":494},[95,60512,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[48,60514,60515,60517,60519,60521,60523],{},[51,60516,14551],{"mathvariant":494},[51,60518,12271],{"mathvariant":494},[51,60520,2552],{"mathvariant":494},[51,60522,2549],{"mathvariant":494},[51,60524,30042],{"mathvariant":494},[95,60526,211],{"stretchy":210},[51,60528,5813],{},[95,60530,1210],{"separator":62},[51,60532,5391],{},[95,60534,1210],{"separator":62},[51,60536,5354],{},[95,60538,220],{"stretchy":210},[95,60540,1210],{"separator":62},[51,60542,2555],{},[95,60544,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,60546,223],{},[48,60548,60549,60551],{},[95,60550,2512],{"fence":62},[5606,60552,60555,60579],{"rowspacing":60553,"columnalign":60554,"columnspacing":1584},"0.36em","left left",[5612,60556,60557,60567],{},[5615,60558,60559],{},[2071,60560,60561],{"scriptlevel":1977,"displaystyle":210},[48,60562,60563,60565],{},[51,60564,5354],{},[95,60566,1210],{"separator":62},[5615,60568,60569],{},[2071,60570,60571],{"scriptlevel":1977,"displaystyle":210},[48,60572,60573,60575,60577],{},[51,60574,5391],{},[95,60576,223],{},[51,60578,2555],{},[5612,60580,60581,60613],{},[5615,60582,60583],{},[2071,60584,60585],{"scriptlevel":1977,"displaystyle":210},[48,60586,60587,60601,60603,60605,60607,60609,60611],{},[48,60588,60589,60591,60593,60595,60597,60599],{},[51,60590,14551],{"mathvariant":494},[51,60592,30042],{"mathvariant":494},[51,60594,54404],{"mathvariant":494},[51,60596,30042],{"mathvariant":494},[51,60598,1412],{"mathvariant":494},[51,60600,12271],{"mathvariant":494},[95,60602,211],{"stretchy":210},[51,60604,5813],{},[95,60606,1210],{"separator":62},[51,60608,2555],{},[95,60610,220],{"stretchy":210},[95,60612,1210],{"separator":62},[5615,60614,60615],{},[2071,60616,60617],{"scriptlevel":1977,"displaystyle":210},[48,60618,60619,60621,60623],{},[51,60620,5391],{},[95,60622,21271],{"mathvariant":494},[51,60624,2555],{},[55,60626,60627],{"encoding":57},"\\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}",[31,60629,60631,60706],{"className":60630,"ariaHidden":62},[61],[31,60632,60634,60637,60643,60649,60655,60658,60661,60664,60667,60670,60673,60676,60679,60682,60685,60688,60691,60697,60700,60703],{"className":60633},[66],[31,60635],{"className":60636,"style":4988},[70],[31,60638,60640],{"className":60639},[75],[31,60641,54443],{"className":60642},[75,2745],[31,60644,60646],{"className":60645},[264],[31,60647,211],{"className":60648},[1307,1308],[31,60650,60652],{"className":60651},[75],[31,60653,55378],{"className":60654},[75,2745],[31,60656,211],{"className":60657},[264],[31,60659,5813],{"className":60660},[75,76],[31,60662,1210],{"className":60663},[1391],[31,60665],{"className":60666,"style":1467},[119],[31,60668,5391],{"className":60669},[75,76],[31,60671,1210],{"className":60672},[1391],[31,60674],{"className":60675,"style":1467},[119],[31,60677,5354],{"className":60678,"style":2666},[75,76],[31,60680,220],{"className":60681},[289],[31,60683,1210],{"className":60684},[1391],[31,60686],{"className":60687,"style":1467},[119],[31,60689,2555],{"className":60690,"style":5514},[75,76],[31,60692,60694],{"className":60693},[289],[31,60695,220],{"className":60696},[1307,1308],[31,60698],{"className":60699,"style":293},[119],[31,60701,223],{"className":60702},[297],[31,60704],{"className":60705,"style":293},[119],[31,60707,60709,60713],{"className":60708},[66],[31,60710],{"className":60711,"style":60712},[70],"height:3em;vertical-align:-1.25em;",[31,60714,60716,60723,60913],{"className":60715},[3860],[31,60717,60719],{"className":60718,"style":3868},[264,3867],[31,60720,2512],{"className":60721},[1307,60722],"size4",[31,60724,60726],{"className":60725},[75],[31,60727,60729,60806,60811],{"className":60728},[5606],[31,60730,60732],{"className":60731},[6062],[31,60733,60735,60797],{"className":60734},[1043,1044],[31,60736,60738,60794],{"className":60737},[1048],[31,60739,60742,60758],{"className":60740,"style":60741},[1052],"height:1.69em;",[31,60743,60745,60749],{"style":60744},"top:-3.69em;",[31,60746],{"className":60747,"style":60748},[1060],"height:3.008em;",[31,60750,60752,60755],{"className":60751},[75],[31,60753,5354],{"className":60754,"style":2666},[75,76],[31,60756,1210],{"className":60757},[1391],[31,60759,60761,60764],{"style":60760},"top:-2.25em;",[31,60762],{"className":60763,"style":60748},[1060],[31,60765,60767,60773,60776,60779,60782,60785,60788,60791],{"className":60766},[75],[31,60768,60770],{"className":60769},[75],[31,60771,54443],{"className":60772},[75,2745],[31,60774,211],{"className":60775},[264],[31,60777,5813],{"className":60778},[75,76],[31,60780,1210],{"className":60781},[1391],[31,60783],{"className":60784,"style":1467},[119],[31,60786,2555],{"className":60787,"style":5514},[75,76],[31,60789,220],{"className":60790},[289],[31,60792,1210],{"className":60793},[1391],[31,60795,1076],{"className":60796},[1075],[31,60798,60800],{"className":60799},[1048],[31,60801,60804],{"className":60802,"style":60803},[1052],"height:1.19em;",[31,60805],{},[31,60807],{"className":60808,"style":60810},[60809],"arraycolsep","width:1em;",[31,60812,60814],{"className":60813},[6062],[31,60815,60817,60905],{"className":60816},[1043,1044],[31,60818,60820,60902],{"className":60819},[1048],[31,60821,60823,60846],{"className":60822,"style":60741},[1052],[31,60824,60825,60828],{"style":60744},[31,60826],{"className":60827,"style":60748},[1060],[31,60829,60831,60834,60837,60840,60843],{"className":60830},[75],[31,60832,5391],{"className":60833},[75,76],[31,60835],{"className":60836,"style":293},[119],[31,60838,223],{"className":60839},[297],[31,60841],{"className":60842,"style":293},[119],[31,60844,2555],{"className":60845,"style":5514},[75,76],[31,60847,60848,60851],{"style":60760},[31,60849],{"className":60850,"style":60748},[1060],[31,60852,60854,60857,60860,60896,60899],{"className":60853},[75],[31,60855,5391],{"className":60856},[75,76],[31,60858],{"className":60859,"style":293},[119],[31,60861,60863,60890,60893],{"className":60862},[297],[31,60864,60866],{"className":60865},[297],[31,60867,60869],{"className":60868},[75,21516],[31,60870,60872],{"className":60871},[21520],[31,60873,60875,60878,60887],{"className":60874},[21524],[31,60876],{"className":60877,"style":4202},[70],[31,60879,60881],{"className":60880},[21531],[31,60882,60884],{"className":60883},[75],[31,60885,21538],{"className":60886},[297],[31,60888],{"className":60889},[21542],[31,60891],{"className":60892},[119,21546],[31,60894,223],{"className":60895},[297],[31,60897],{"className":60898,"style":293},[119],[31,60900,2555],{"className":60901,"style":5514},[75,76],[31,60903,1076],{"className":60904},[1075],[31,60906,60908],{"className":60907},[1048],[31,60909,60911],{"className":60910,"style":60803},[1052],[31,60912],{},[31,60914],{"className":60915},[289,2139],[11,60917,60918,60919,60963,60964,61015],{},"如果公式里只关心 ",[31,60920,60922,60942],{"className":60921,"translate":35},[34],[31,60923,60925],{"className":60924},[39],[41,60926,60927],{"xmlns":43},[45,60928,60929,60939],{},[48,60930,60931,60933,60935,60937],{},[51,60932,5813],{},[95,60934,15175],{"stretchy":210},[91,60936,1977],{},[95,60938,15184],{"stretchy":210},[55,60940,60941],{"encoding":57},"a[0]",[31,60943,60945],{"className":60944,"ariaHidden":62},[61],[31,60946,60948,60951,60954,60957,60960],{"className":60947},[66],[31,60949],{"className":60950,"style":260},[70],[31,60952,5813],{"className":60953},[75,76],[31,60955,15175],{"className":60956},[264],[31,60958,1977],{"className":60959},[75],[31,60961,15184],{"className":60962},[289],"，求解器就只实例化 ",[31,60965,60967,60985],{"className":60966,"translate":35},[34],[31,60968,60970],{"className":60969},[39],[41,60971,60972],{"xmlns":43},[45,60973,60974,60982],{},[48,60975,60976,60978,60980],{},[51,60977,5391],{},[95,60979,223],{},[91,60981,1977],{},[55,60983,60984],{"encoding":57},"i = 0",[31,60986,60988,61006],{"className":60987,"ariaHidden":62},[61],[31,60989,60991,60994,60997,61000,61003],{"className":60990},[66],[31,60992],{"className":60993,"style":59987},[70],[31,60995,5391],{"className":60996},[75,76],[31,60998],{"className":60999,"style":293},[119],[31,61001,223],{"className":61002},[297],[31,61004],{"className":61005,"style":293},[119],[31,61007,61009,61012],{"className":61008},[66],[31,61010],{"className":61011,"style":326},[70],[31,61013,1977],{"className":61014},[75]," 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。",[11,61017,61018,61019,61022],{},"所以更准确的说法不是「SMT 消灭了底层编码」，而是：",[7246,61020,61021],{},"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。",[11,61024,61025,61026,61067,61068,2372],{},"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 ",[31,61027,61029,61046],{"className":61028,"translate":35},[34],[31,61030,61032],{"className":61031},[39],[41,61033,61034],{"xmlns":43},[45,61035,61036,61044],{},[48,61037,61038,61040,61042],{},[51,61039,53],{},[95,61041,1210],{"separator":62},[51,61043,3003],{},[55,61045,55902],{"encoding":57},[31,61047,61049],{"className":61048,"ariaHidden":62},[61],[31,61050,61052,61055,61058,61061,61064],{"className":61051},[66],[31,61053],{"className":61054,"style":865},[70],[31,61056,53],{"className":61057},[75,76],[31,61059,1210],{"className":61060},[1391],[31,61062],{"className":61063,"style":1467},[119],[31,61065,3003],{"className":61066,"style":2666},[75,76],"，SMT 直接给出 ",[31,61069,61071,61097],{"className":61070,"translate":35},[34],[31,61072,61074],{"className":61073},[39],[41,61075,61076],{"xmlns":43},[45,61077,61078,61095],{},[48,61079,61080,61082,61084,61087,61089,61091,61093],{},[51,61081,53],{},[95,61083,223],{},[91,61085,61086],{},"6",[95,61088,1210],{"separator":62},[51,61090,3003],{},[95,61092,223],{},[91,61094,7853],{},[55,61096,53180],{"encoding":57},[31,61098,61100,61118,61145],{"className":61099,"ariaHidden":62},[61],[31,61101,61103,61106,61109,61112,61115],{"className":61102},[66],[31,61104],{"className":61105,"style":71},[70],[31,61107,53],{"className":61108},[75,76],[31,61110],{"className":61111,"style":293},[119],[31,61113,223],{"className":61114},[297],[31,61116],{"className":61117,"style":293},[119],[31,61119,61121,61124,61127,61130,61133,61136,61139,61142],{"className":61120},[66],[31,61122],{"className":61123,"style":41982},[70],[31,61125,61086],{"className":61126},[75],[31,61128,1210],{"className":61129},[1391],[31,61131],{"className":61132,"style":1467},[119],[31,61134,3003],{"className":61135,"style":2666},[75,76],[31,61137],{"className":61138,"style":293},[119],[31,61140,223],{"className":61141},[297],[31,61143],{"className":61144,"style":293},[119],[31,61146,61148,61151],{"className":61147},[66],[31,61149],{"className":61150,"style":326},[70],[31,61152,7853],{"className":61153},[75],[53200,61155,61157],{"id":61156},"_33-组合理论","3.3 组合理论",[11,61159,61160,61161,2372],{},"第三个策略是",[7246,61162,61163],{},"让不同理论通过共享等式来协作",[11,61165,61166,61167,61170],{},"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",[7246,61168,61169],{},"变量之间的等式","，就能组合成一个完整的判定过程。",[11,61172,61173],{},"看一个例子：",[31,61175,61177],{"className":61176,"translate":35},[411],[31,61178,61180,61221],{"className":61179,"translate":35},[34],[31,61181,61183],{"className":61182},[39],[41,61184,61185],{"xmlns":43,"display":420},[45,61186,61187,61219],{},[48,61188,61189,61191,61193,61195,61197,61199,61201,61203,61205,61207,61209,61211,61213,61215,61217],{},[51,61190,53],{},[95,61192,223],{},[51,61194,3003],{},[1430,61196,53035],{},[95,61198,53038],{},[1430,61200,53035],{},[51,61202,1662],{},[95,61204,211],{"stretchy":210},[51,61206,53],{},[95,61208,220],{"stretchy":210},[95,61210,21271],{"mathvariant":494},[51,61212,1662],{},[95,61214,211],{"stretchy":210},[51,61216,3003],{},[95,61218,220],{"stretchy":210},[55,61220,58289],{"encoding":57},[31,61222,61224,61242,61266,61326],{"className":61223,"ariaHidden":62},[61],[31,61225,61227,61230,61233,61236,61239],{"className":61226},[66],[31,61228],{"className":61229,"style":71},[70],[31,61231,53],{"className":61232},[75,76],[31,61234],{"className":61235,"style":293},[119],[31,61237,223],{"className":61238},[297],[31,61240],{"className":61241,"style":293},[119],[31,61243,61245,61248,61251,61254,61257,61260,61263],{"className":61244},[66],[31,61246],{"className":61247,"style":53828},[70],[31,61249,3003],{"className":61250,"style":2666},[75,76],[31,61252],{"className":61253,"style":293},[119],[31,61255],{"className":61256,"style":120},[119],[31,61258,53038],{"className":61259},[124],[31,61261],{"className":61262,"style":293},[119],[31,61264],{"className":61265,"style":120},[119],[31,61267,61269,61272,61275,61278,61281,61284,61287,61323],{"className":61268},[66],[31,61270],{"className":61271,"style":260},[70],[31,61273,1662],{"className":61274,"style":35196},[75,76],[31,61276,211],{"className":61277},[264],[31,61279,53],{"className":61280},[75,76],[31,61282,220],{"className":61283},[289],[31,61285],{"className":61286,"style":293},[119],[31,61288,61290,61317,61320],{"className":61289},[297],[31,61291,61293],{"className":61292},[297],[31,61294,61296],{"className":61295},[75,21516],[31,61297,61299],{"className":61298},[21520],[31,61300,61302,61305,61314],{"className":61301},[21524],[31,61303],{"className":61304,"style":4202},[70],[31,61306,61308],{"className":61307},[21531],[31,61309,61311],{"className":61310},[75],[31,61312,21538],{"className":61313},[297],[31,61315],{"className":61316},[21542],[31,61318],{"className":61319},[119,21546],[31,61321,223],{"className":61322},[297],[31,61324],{"className":61325,"style":293},[119],[31,61327,61329,61332,61335,61338,61341],{"className":61328},[66],[31,61330],{"className":61331,"style":260},[70],[31,61333,1662],{"className":61334,"style":35196},[75,76],[31,61336,211],{"className":61337},[264],[31,61339,3003],{"className":61340,"style":2666},[75,76],[31,61342,220],{"className":61343},[289],[55164,61345,61346,61400,61541],{},[55167,61347,61348,61349,61399],{},"算术部分看到 ",[31,61350,61352,61369],{"className":61351,"translate":35},[34],[31,61353,61355],{"className":61354},[39],[41,61356,61357],{"xmlns":43},[45,61358,61359,61367],{},[48,61360,61361,61363,61365],{},[51,61362,53],{},[95,61364,223],{},[51,61366,3003],{},[55,61368,58441],{"encoding":57},[31,61370,61372,61390],{"className":61371,"ariaHidden":62},[61],[31,61373,61375,61378,61381,61384,61387],{"className":61374},[66],[31,61376],{"className":61377,"style":71},[70],[31,61379,53],{"className":61380},[75,76],[31,61382],{"className":61383,"style":293},[119],[31,61385,223],{"className":61386},[297],[31,61388],{"className":61389,"style":293},[119],[31,61391,61393,61396],{"className":61392},[66],[31,61394],{"className":61395,"style":865},[70],[31,61397,3003],{"className":61398,"style":2666},[75,76],"，本身没有矛盾；",[55167,61401,61402,61403,61453,61454,2446,61497,61540],{},"EUF 部分如果不知道 ",[31,61404,61406,61423],{"className":61405,"translate":35},[34],[31,61407,61409],{"className":61408},[39],[41,61410,61411],{"xmlns":43},[45,61412,61413,61421],{},[48,61414,61415,61417,61419],{},[51,61416,53],{},[95,61418,223],{},[51,61420,3003],{},[55,61422,58441],{"encoding":57},[31,61424,61426,61444],{"className":61425,"ariaHidden":62},[61],[31,61427,61429,61432,61435,61438,61441],{"className":61428},[66],[31,61430],{"className":61431,"style":71},[70],[31,61433,53],{"className":61434},[75,76],[31,61436],{"className":61437,"style":293},[119],[31,61439,223],{"className":61440},[297],[31,61442],{"className":61443,"style":293},[119],[31,61445,61447,61450],{"className":61446},[66],[31,61448],{"className":61449,"style":865},[70],[31,61451,3003],{"className":61452,"style":2666},[75,76],"，就会认为 ",[31,61455,61457,61476],{"className":61456,"translate":35},[34],[31,61458,61460],{"className":61459},[39],[41,61461,61462],{"xmlns":43},[45,61463,61464,61474],{},[48,61465,61466,61468,61470,61472],{},[51,61467,1662],{},[95,61469,211],{"stretchy":210},[51,61471,53],{},[95,61473,220],{"stretchy":210},[55,61475,40190],{"encoding":57},[31,61477,61479],{"className":61478,"ariaHidden":62},[61],[31,61480,61482,61485,61488,61491,61494],{"className":61481},[66],[31,61483],{"className":61484,"style":260},[70],[31,61486,1662],{"className":61487,"style":35196},[75,76],[31,61489,211],{"className":61490},[264],[31,61492,53],{"className":61493},[75,76],[31,61495,220],{"className":61496},[289],[31,61498,61500,61519],{"className":61499,"translate":35},[34],[31,61501,61503],{"className":61502},[39],[41,61504,61505],{"xmlns":43},[45,61506,61507,61517],{},[48,61508,61509,61511,61513,61515],{},[51,61510,1662],{},[95,61512,211],{"stretchy":210},[51,61514,3003],{},[95,61516,220],{"stretchy":210},[55,61518,58963],{"encoding":57},[31,61520,61522],{"className":61521,"ariaHidden":62},[61],[31,61523,61525,61528,61531,61534,61537],{"className":61524},[66],[31,61526],{"className":61527,"style":260},[70],[31,61529,1662],{"className":61530,"style":35196},[75,76],[31,61532,211],{"className":61533},[264],[31,61535,3003],{"className":61536,"style":2666},[75,76],[31,61538,220],{"className":61539},[289]," 可以不同，也不会报错；",[55167,61542,61543,61544,61594,61595,58985,61675,61788],{},"一旦算术把等式 ",[31,61545,61547,61564],{"className":61546,"translate":35},[34],[31,61548,61550],{"className":61549},[39],[41,61551,61552],{"xmlns":43},[45,61553,61554,61562],{},[48,61555,61556,61558,61560],{},[51,61557,53],{},[95,61559,223],{},[51,61561,3003],{},[55,61563,58441],{"encoding":57},[31,61565,61567,61585],{"className":61566,"ariaHidden":62},[61],[31,61568,61570,61573,61576,61579,61582],{"className":61569},[66],[31,61571],{"className":61572,"style":71},[70],[31,61574,53],{"className":61575},[75,76],[31,61577],{"className":61578,"style":293},[119],[31,61580,223],{"className":61581},[297],[31,61583],{"className":61584,"style":293},[119],[31,61586,61588,61591],{"className":61587},[66],[31,61589],{"className":61590,"style":865},[70],[31,61592,3003],{"className":61593,"style":2666},[75,76]," 传给 EUF，EUF 立刻推出 ",[31,61596,61598,61627],{"className":61597,"translate":35},[34],[31,61599,61601],{"className":61600},[39],[41,61602,61603],{"xmlns":43},[45,61604,61605,61625],{},[48,61606,61607,61609,61611,61613,61615,61617,61619,61621,61623],{},[51,61608,1662],{},[95,61610,211],{"stretchy":210},[51,61612,53],{},[95,61614,220],{"stretchy":210},[95,61616,223],{},[51,61618,1662],{},[95,61620,211],{"stretchy":210},[51,61622,3003],{},[95,61624,220],{"stretchy":210},[55,61626,59189],{"encoding":57},[31,61628,61630,61657],{"className":61629,"ariaHidden":62},[61],[31,61631,61633,61636,61639,61642,61645,61648,61651,61654],{"className":61632},[66],[31,61634],{"className":61635,"style":260},[70],[31,61637,1662],{"className":61638,"style":35196},[75,76],[31,61640,211],{"className":61641},[264],[31,61643,53],{"className":61644},[75,76],[31,61646,220],{"className":61647},[289],[31,61649],{"className":61650,"style":293},[119],[31,61652,223],{"className":61653},[297],[31,61655],{"className":61656,"style":293},[119],[31,61658,61660,61663,61666,61669,61672],{"className":61659},[66],[31,61661],{"className":61662,"style":260},[70],[31,61664,1662],{"className":61665,"style":35196},[75,76],[31,61667,211],{"className":61668},[264],[31,61670,3003],{"className":61671,"style":2666},[75,76],[31,61673,220],{"className":61674},[289],[31,61676,61678,61707],{"className":61677,"translate":35},[34],[31,61679,61681],{"className":61680},[39],[41,61682,61683],{"xmlns":43},[45,61684,61685,61705],{},[48,61686,61687,61689,61691,61693,61695,61697,61699,61701,61703],{},[51,61688,1662],{},[95,61690,211],{"stretchy":210},[51,61692,53],{},[95,61694,220],{"stretchy":210},[95,61696,21271],{"mathvariant":494},[51,61698,1662],{},[95,61700,211],{"stretchy":210},[51,61702,3003],{},[95,61704,220],{"stretchy":210},[55,61706,55414],{"encoding":57},[31,61708,61710,61770],{"className":61709,"ariaHidden":62},[61],[31,61711,61713,61716,61719,61722,61725,61728,61731,61767],{"className":61712},[66],[31,61714],{"className":61715,"style":260},[70],[31,61717,1662],{"className":61718,"style":35196},[75,76],[31,61720,211],{"className":61721},[264],[31,61723,53],{"className":61724},[75,76],[31,61726,220],{"className":61727},[289],[31,61729],{"className":61730,"style":293},[119],[31,61732,61734,61761,61764],{"className":61733},[297],[31,61735,61737],{"className":61736},[297],[31,61738,61740],{"className":61739},[75,21516],[31,61741,61743],{"className":61742},[21520],[31,61744,61746,61749,61758],{"className":61745},[21524],[31,61747],{"className":61748,"style":4202},[70],[31,61750,61752],{"className":61751},[21531],[31,61753,61755],{"className":61754},[75],[31,61756,21538],{"className":61757},[297],[31,61759],{"className":61760},[21542],[31,61762],{"className":61763},[119,21546],[31,61765,223],{"className":61766},[297],[31,61768],{"className":61769,"style":293},[119],[31,61771,61773,61776,61779,61782,61785],{"className":61772},[66],[31,61774],{"className":61775,"style":260},[70],[31,61777,1662],{"className":61778,"style":35196},[75,76],[31,61780,211],{"className":61781},[264],[31,61783,3003],{"className":61784,"style":2666},[75,76],[31,61786,220],{"className":61787},[289]," 冲突。",[11,61790,61791],{},"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。",[11,61793,61794],{},"再看一个反方向的例子：",[31,61796,61798],{"className":61797,"translate":35},[411],[31,61799,61801,61847],{"className":61800,"translate":35},[34],[31,61802,61804],{"className":61803},[39],[41,61805,61806],{"xmlns":43,"display":420},[45,61807,61808,61844],{},[48,61809,61810,61812,61814,61816,61818,61820,61822,61824,61826,61828,61830,61832,61834,61836,61838,61840,61842],{},[51,61811,53],{},[95,61813,434],{},[51,61815,3003],{},[95,61817,16040],{},[91,61819,19494],{},[1430,61821,53035],{},[95,61823,53038],{},[1430,61825,53035],{},[51,61827,53],{},[95,61829,223],{},[91,61831,93],{},[1430,61833,53035],{},[95,61835,53038],{},[1430,61837,53035],{},[51,61839,3003],{},[95,61841,223],{},[91,61843,15416],{},[55,61845,61846],{"encoding":57},"x + y > 10 \\;\\wedge\\; x = 3 \\;\\wedge\\; y = 4",[31,61848,61850,61868,61886,61910,61928,61952,61970],{"className":61849,"ariaHidden":62},[61],[31,61851,61853,61856,61859,61862,61865],{"className":61852},[66],[31,61854],{"className":61855,"style":307},[70],[31,61857,53],{"className":61858},[75,76],[31,61860],{"className":61861,"style":120},[119],[31,61863,434],{"className":61864},[124],[31,61866],{"className":61867,"style":120},[119],[31,61869,61871,61874,61877,61880,61883],{"className":61870},[66],[31,61872],{"className":61873,"style":53083},[70],[31,61875,3003],{"className":61876,"style":2666},[75,76],[31,61878],{"className":61879,"style":293},[119],[31,61881,16040],{"className":61882},[297],[31,61884],{"className":61885,"style":293},[119],[31,61887,61889,61892,61895,61898,61901,61904,61907],{"className":61888},[66],[31,61890],{"className":61891,"style":326},[70],[31,61893,19494],{"className":61894},[75],[31,61896],{"className":61897,"style":293},[119],[31,61899],{"className":61900,"style":120},[119],[31,61902,53038],{"className":61903},[124],[31,61905],{"className":61906,"style":293},[119],[31,61908],{"className":61909,"style":120},[119],[31,61911,61913,61916,61919,61922,61925],{"className":61912},[66],[31,61914],{"className":61915,"style":71},[70],[31,61917,53],{"className":61918},[75,76],[31,61920],{"className":61921,"style":293},[119],[31,61923,223],{"className":61924},[297],[31,61926],{"className":61927,"style":293},[119],[31,61929,61931,61934,61937,61940,61943,61946,61949],{"className":61930},[66],[31,61932],{"className":61933,"style":326},[70],[31,61935,93],{"className":61936},[75],[31,61938],{"className":61939,"style":293},[119],[31,61941],{"className":61942,"style":120},[119],[31,61944,53038],{"className":61945},[124],[31,61947],{"className":61948,"style":293},[119],[31,61950],{"className":61951,"style":120},[119],[31,61953,61955,61958,61961,61964,61967],{"className":61954},[66],[31,61956],{"className":61957,"style":865},[70],[31,61959,3003],{"className":61960,"style":2666},[75,76],[31,61962],{"className":61963,"style":293},[119],[31,61965,223],{"className":61966},[297],[31,61968],{"className":61969,"style":293},[119],[31,61971,61973,61976],{"className":61972},[66],[31,61974],{"className":61975,"style":326},[70],[31,61977,15416],{"className":61978},[75],[11,61980,61981,61982,62078],{},"这里算术部分自己就能算出 ",[31,61983,61985,62011],{"className":61984,"translate":35},[34],[31,61986,61988],{"className":61987},[39],[41,61989,61990],{"xmlns":43},[45,61991,61992,62008],{},[48,61993,61994,61996,61998,62000,62002,62004,62006],{},[91,61995,93],{},[95,61997,434],{},[91,61999,15416],{},[95,62001,223],{},[91,62003,44675],{},[95,62005,9758],{},[91,62007,19494],{},[55,62009,62010],{"encoding":57},"3 + 4 = 7 \u003C 10",[31,62012,62014,62032,62050,62069],{"className":62013,"ariaHidden":62},[61],[31,62015,62017,62020,62023,62026,62029],{"className":62016},[66],[31,62018],{"className":62019,"style":112},[70],[31,62021,93],{"className":62022},[75],[31,62024],{"className":62025,"style":120},[119],[31,62027,434],{"className":62028},[124],[31,62030],{"className":62031,"style":120},[119],[31,62033,62035,62038,62041,62044,62047],{"className":62034},[66],[31,62036],{"className":62037,"style":326},[70],[31,62039,15416],{"className":62040},[75],[31,62042],{"className":62043,"style":293},[119],[31,62045,223],{"className":62046},[297],[31,62048],{"className":62049,"style":293},[119],[31,62051,62053,62057,62060,62063,62066],{"className":62052},[66],[31,62054],{"className":62055,"style":62056},[70],"height:0.6835em;vertical-align:-0.0391em;",[31,62058,44675],{"className":62059},[75],[31,62061],{"className":62062,"style":293},[119],[31,62064,9758],{"className":62065},[297],[31,62067],{"className":62068,"style":293},[119],[31,62070,62072,62075],{"className":62071},[66],[31,62073],{"className":62074,"style":326},[70],[31,62076,19494],{"className":62077},[75],"，直接判 unsat，根本不需要 EUF 参与。",[11,62080,62081,62082],{},"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：",[7246,62083,62084],{},"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。",[11,62086,62087,62088,62172,62173,62273],{},"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，",[31,62089,62091,62109],{"className":62090,"translate":35},[34],[31,62092,62094],{"className":62093},[39],[41,62095,62096],{"xmlns":43},[45,62097,62098,62106],{},[48,62099,62100,62102,62104],{},[51,62101,53],{},[95,62103,21271],{"mathvariant":494},[91,62105,1977],{},[55,62107,62108],{"encoding":57},"x \\ne 0",[31,62110,62112,62163],{"className":62111,"ariaHidden":62},[61],[31,62113,62115,62118,62121,62124,62160],{"className":62114},[66],[31,62116],{"className":62117,"style":4202},[70],[31,62119,53],{"className":62120},[75,76],[31,62122],{"className":62123,"style":293},[119],[31,62125,62127,62154,62157],{"className":62126},[297],[31,62128,62130],{"className":62129},[297],[31,62131,62133],{"className":62132},[75,21516],[31,62134,62136],{"className":62135},[21520],[31,62137,62139,62142,62151],{"className":62138},[21524],[31,62140],{"className":62141,"style":4202},[70],[31,62143,62145],{"className":62144},[21531],[31,62146,62148],{"className":62147},[75],[31,62149,21538],{"className":62150},[297],[31,62152],{"className":62153},[21542],[31,62155],{"className":62156},[119,21546],[31,62158,223],{"className":62159},[297],[31,62161],{"className":62162,"style":293},[119],[31,62164,62166,62169],{"className":62165},[66],[31,62167],{"className":62168,"style":326},[70],[31,62170,1977],{"className":62171},[75]," 等价于 ",[31,62174,62176,62204],{"className":62175,"translate":35},[34],[31,62177,62179],{"className":62178},[39],[41,62180,62181],{"xmlns":43},[45,62182,62183,62201],{},[48,62184,62185,62187,62189,62191,62193,62195,62197,62199],{},[51,62186,53],{},[95,62188,347],{},[95,62190,97],{},[91,62192,205],{},[95,62194,53683],{},[51,62196,53],{},[95,62198,5662],{},[91,62200,205],{},[55,62202,62203],{"encoding":57},"x \\le -1 \\vee x \\ge 1",[31,62205,62207,62225,62246,62264],{"className":62206,"ariaHidden":62},[61],[31,62208,62210,62213,62216,62219,62222],{"className":62209},[66],[31,62211],{"className":62212,"style":385},[70],[31,62214,53],{"className":62215},[75,76],[31,62217],{"className":62218,"style":293},[119],[31,62220,347],{"className":62221},[297],[31,62223],{"className":62224,"style":293},[119],[31,62226,62228,62231,62234,62237,62240,62243],{"className":62227},[66],[31,62229],{"className":62230,"style":112},[70],[31,62232,97],{"className":62233},[75],[31,62235,205],{"className":62236},[75],[31,62238],{"className":62239,"style":120},[119],[31,62241,53683],{"className":62242},[124],[31,62244],{"className":62245,"style":120},[119],[31,62247,62249,62252,62255,62258,62261],{"className":62248},[66],[31,62250],{"className":62251,"style":385},[70],[31,62253,53],{"className":62254},[75,76],[31,62256],{"className":62257,"style":293},[119],[31,62259,5662],{"className":62260},[297],[31,62262],{"className":62263,"style":293},[119],[31,62265,62267,62270],{"className":62266},[66],[31,62268],{"className":62269,"style":326},[70],[31,62271,205],{"className":62272},[75],"，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。",[53200,62275,62277],{"id":62276},"_34-给出解释","3.4 给出解释",[11,62279,62280,62281,2372],{},"第四个策略是",[7246,62282,62283],{},"不只回答是或否，还给出可操作的结果",[55164,62285,62286,62291,62296,62355],{},[55167,62287,62288,62290],{},[7246,62289,53187],{},"：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。",[55167,62292,62293,62295],{},[7246,62294,59102],{},"：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」。",[55167,62297,62298,62301,62302,62354],{},[7246,62299,62300],{},"插值","：从 ",[31,62303,62305,62324],{"className":62304,"translate":35},[34],[31,62306,62308],{"className":62307},[39],[41,62309,62310],{"xmlns":43},[45,62311,62312,62321],{},[48,62313,62314,62316,62319],{},[51,62315,18446],{},[95,62317,62318],{},"⇒",[51,62320,1122],{},[55,62322,62323],{"encoding":57},"A \\Rightarrow B",[31,62325,62327,62345],{"className":62326,"ariaHidden":62},[61],[31,62328,62330,62333,62336,62339,62342],{"className":62329},[66],[31,62331],{"className":62332,"style":945},[70],[31,62334,18446],{"className":62335},[75,76],[31,62337],{"className":62338,"style":293},[119],[31,62340,62318],{"className":62341},[297],[31,62343],{"className":62344,"style":293},[119],[31,62346,62348,62351],{"className":62347},[66],[31,62349],{"className":62350,"style":945},[70],[31,62352,1122],{"className":62353,"style":1137},[75,76]," 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。",[55167,62356,62357,62359],{},[7246,62358,7248],{},"：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。",[11,62361,62362,62363,62365],{},"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 ",[45608,62364,59102],{}," 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。",[53200,62367,62369],{"id":62368},"_35-增量求解","3.5 增量求解",[11,62371,62372,62373,62376],{},"第五个策略是",[7246,62374,62375],{},"增量求解","（incremental solving）。",[11,62378,62379],{},"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。",[11,62381,62382,62383,55632,62386,2446,62389,62392],{},"SMT-LIB 提供了 ",[45608,62384,62385],{},"push",[45608,62387,62388],{},"pop",[45608,62390,62391],{},"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。",[11,62394,62395],{},"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。",[18,62397,62399],{"id":62398},"四dpllt","四、DPLL(T)",[53200,62401,62403],{"id":62402},"_41-总体架构","4.1 总体架构",[11,62405,62406,62407,62410,62411,62414],{},"现代 SMT 求解器的标准架构叫 ",[7246,62408,62409],{},"DPLL(T)","，更准确的名字是 ",[7246,62412,62413],{},"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。",[55589,62416,62417],{},[55592,62418],{"height":62419,"width":62420,"style":55596,"data-d2-theme":55597,"alt":52812,"src":62421},320,2578,"data:image\u002Fsvg+xml;base64,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\u002Fmu6HotrM\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",[11,62423,62424],{},"用一句话概括这个循环：",[62426,62427,62428,62431,62434,62437,62440,62443],"ol",{},[55167,62429,62430],{},"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；",[55167,62432,62433],{},"让 SAT 求解器找一个布尔模型；",[55167,62435,62436],{},"把布尔模型翻译回理论原子的真假，交给理论求解器检查；",[55167,62438,62439],{},"如果理论层说「一致」，就得到 sat 和模型；",[55167,62441,62442],{},"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；",[55167,62444,62445],{},"如果 SAT 层把所有可能性都穷尽了，就得到 unsat。",[11,62447,62448],{},"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。",[53200,62450,62452],{"id":62451},"_42-布尔抽象","4.2 布尔抽象",[11,62454,62455],{},"布尔抽象（也叫 propositional abstraction）是两层之间的接口。",[11,62457,62458],{},"以这个公式为例：",[31,62460,62462],{"className":62461,"translate":35},[411],[31,62463,62465,62519],{"className":62464,"translate":35},[34],[31,62466,62468],{"className":62467},[39],[41,62469,62470],{"xmlns":43,"display":420},[45,62471,62472,62516],{},[48,62473,62474,62476,62478,62480,62482,62484,62486,62488,62490,62492,62494,62496,62498,62500,62502,62504,62506,62508,62510,62512,62514],{},[51,62475,8059],{},[95,62477,223],{},[95,62479,211],{"stretchy":210},[51,62481,53],{},[95,62483,16040],{},[91,62485,1977],{},[95,62487,53683],{},[51,62489,3003],{},[95,62491,16040],{},[91,62493,1977],{},[95,62495,220],{"stretchy":210},[1430,62497,53035],{},[95,62499,53038],{},[1430,62501,53035],{},[95,62503,211],{"stretchy":210},[51,62505,53],{},[95,62507,434],{},[51,62509,3003],{},[95,62511,9758],{},[91,62513,1977],{},[95,62515,220],{"stretchy":210},[55,62517,62518],{"encoding":57},"\\varphi = (x > 0 \\vee y > 0) \\;\\wedge\\; (x + y \u003C 0)",[31,62520,62522,62540,62561,62579,62597,62624,62645,62663],{"className":62521,"ariaHidden":62},[61],[31,62523,62525,62528,62531,62534,62537],{"className":62524},[66],[31,62526],{"className":62527,"style":865},[70],[31,62529,8059],{"className":62530},[75,76],[31,62532],{"className":62533,"style":293},[119],[31,62535,223],{"className":62536},[297],[31,62538],{"className":62539,"style":293},[119],[31,62541,62543,62546,62549,62552,62555,62558],{"className":62542},[66],[31,62544],{"className":62545,"style":260},[70],[31,62547,211],{"className":62548},[264],[31,62550,53],{"className":62551},[75,76],[31,62553],{"className":62554,"style":293},[119],[31,62556,16040],{"className":62557},[297],[31,62559],{"className":62560,"style":293},[119],[31,62562,62564,62567,62570,62573,62576],{"className":62563},[66],[31,62565],{"className":62566,"style":326},[70],[31,62568,1977],{"className":62569},[75],[31,62571],{"className":62572,"style":120},[119],[31,62574,53683],{"className":62575},[124],[31,62577],{"className":62578,"style":120},[119],[31,62580,62582,62585,62588,62591,62594],{"className":62581},[66],[31,62583],{"className":62584,"style":53083},[70],[31,62586,3003],{"className":62587,"style":2666},[75,76],[31,62589],{"className":62590,"style":293},[119],[31,62592,16040],{"className":62593},[297],[31,62595],{"className":62596,"style":293},[119],[31,62598,62600,62603,62606,62609,62612,62615,62618,62621],{"className":62599},[66],[31,62601],{"className":62602,"style":260},[70],[31,62604,1977],{"className":62605},[75],[31,62607,220],{"className":62608},[289],[31,62610],{"className":62611,"style":293},[119],[31,62613],{"className":62614,"style":120},[119],[31,62616,53038],{"className":62617},[124],[31,62619],{"className":62620,"style":293},[119],[31,62622],{"className":62623,"style":120},[119],[31,62625,62627,62630,62633,62636,62639,62642],{"className":62626},[66],[31,62628],{"className":62629,"style":260},[70],[31,62631,211],{"className":62632},[264],[31,62634,53],{"className":62635},[75,76],[31,62637],{"className":62638,"style":120},[119],[31,62640,434],{"className":62641},[124],[31,62643],{"className":62644,"style":120},[119],[31,62646,62648,62651,62654,62657,62660],{"className":62647},[66],[31,62649],{"className":62650,"style":53083},[70],[31,62652,3003],{"className":62653,"style":2666},[75,76],[31,62655],{"className":62656,"style":293},[119],[31,62658,9758],{"className":62659},[297],[31,62661],{"className":62662,"style":293},[119],[31,62664,62666,62669,62672],{"className":62665},[66],[31,62667],{"className":62668,"style":260},[70],[31,62670,1977],{"className":62671},[75],[31,62673,220],{"className":62674},[289],[11,62676,62677],{},"引入三个命题变量：",[31,62679,62681],{"className":62680,"translate":35},[411],[31,62682,62684,62751],{"className":62683,"translate":35},[34],[31,62685,62687],{"className":62686},[39],[41,62688,62689],{"xmlns":43,"display":420},[45,62690,62691,62748],{},[48,62692,62693,62695,62698,62700,62702,62704,62706,62708,62710,62712,62714,62716,62718,62720,62722,62724,62726,62728,62730,62732,62734,62736,62738,62740,62742,62744,62746],{},[51,62694,11],{},[95,62696,62697],{},"≡",[95,62699,211],{"stretchy":210},[51,62701,53],{},[95,62703,16040],{},[91,62705,1977],{},[95,62707,220],{"stretchy":210},[95,62709,1210],{"separator":62},[119,62711],{"width":20286},[51,62713,27666],{},[95,62715,62697],{},[95,62717,211],{"stretchy":210},[51,62719,3003],{},[95,62721,16040],{},[91,62723,1977],{},[95,62725,220],{"stretchy":210},[95,62727,1210],{"separator":62},[119,62729],{"width":20286},[51,62731,2549],{},[95,62733,62697],{},[95,62735,211],{"stretchy":210},[51,62737,53],{},[95,62739,434],{},[51,62741,3003],{},[95,62743,9758],{},[91,62745,1977],{},[95,62747,220],{"stretchy":210},[55,62749,62750],{"encoding":57},"p \\equiv (x > 0), \\qquad q \\equiv (y > 0), \\qquad r \\equiv (x + y \u003C 0)",[31,62752,62754,62773,62794,62827,62848,62881,62902,62920],{"className":62753,"ariaHidden":62},[61],[31,62755,62757,62761,62764,62767,62770],{"className":62756},[66],[31,62758],{"className":62759,"style":62760},[70],"height:0.6582em;vertical-align:-0.1944em;",[31,62762,11],{"className":62763},[75,76],[31,62765],{"className":62766,"style":293},[119],[31,62768,62697],{"className":62769},[297],[31,62771],{"className":62772,"style":293},[119],[31,62774,62776,62779,62782,62785,62788,62791],{"className":62775},[66],[31,62777],{"className":62778,"style":260},[70],[31,62780,211],{"className":62781},[264],[31,62783,53],{"className":62784},[75,76],[31,62786],{"className":62787,"style":293},[119],[31,62789,16040],{"className":62790},[297],[31,62792],{"className":62793,"style":293},[119],[31,62795,62797,62800,62803,62806,62809,62812,62815,62818,62821,62824],{"className":62796},[66],[31,62798],{"className":62799,"style":260},[70],[31,62801,1977],{"className":62802},[75],[31,62804,220],{"className":62805},[289],[31,62807,1210],{"className":62808},[1391],[31,62810],{"className":62811,"style":20525},[119],[31,62813],{"className":62814,"style":1467},[119],[31,62816,27666],{"className":62817,"style":2666},[75,76],[31,62819],{"className":62820,"style":293},[119],[31,62822,62697],{"className":62823},[297],[31,62825],{"className":62826,"style":293},[119],[31,62828,62830,62833,62836,62839,62842,62845],{"className":62829},[66],[31,62831],{"className":62832,"style":260},[70],[31,62834,211],{"className":62835},[264],[31,62837,3003],{"className":62838,"style":2666},[75,76],[31,62840],{"className":62841,"style":293},[119],[31,62843,16040],{"className":62844},[297],[31,62846],{"className":62847,"style":293},[119],[31,62849,62851,62854,62857,62860,62863,62866,62869,62872,62875,62878],{"className":62850},[66],[31,62852],{"className":62853,"style":260},[70],[31,62855,1977],{"className":62856},[75],[31,62858,220],{"className":62859},[289],[31,62861,1210],{"className":62862},[1391],[31,62864],{"className":62865,"style":20525},[119],[31,62867],{"className":62868,"style":1467},[119],[31,62870,2549],{"className":62871,"style":19759},[75,76],[31,62873],{"className":62874,"style":293},[119],[31,62876,62697],{"className":62877},[297],[31,62879],{"className":62880,"style":293},[119],[31,62882,62884,62887,62890,62893,62896,62899],{"className":62883},[66],[31,62885],{"className":62886,"style":260},[70],[31,62888,211],{"className":62889},[264],[31,62891,53],{"className":62892},[75,76],[31,62894],{"className":62895,"style":120},[119],[31,62897,434],{"className":62898},[124],[31,62900],{"className":62901,"style":120},[119],[31,62903,62905,62908,62911,62914,62917],{"className":62904},[66],[31,62906],{"className":62907,"style":53083},[70],[31,62909,3003],{"className":62910,"style":2666},[75,76],[31,62912],{"className":62913,"style":293},[119],[31,62915,9758],{"className":62916},[297],[31,62918],{"className":62919,"style":293},[119],[31,62921,62923,62926,62929],{"className":62922},[66],[31,62924],{"className":62925,"style":260},[70],[31,62927,1977],{"className":62928},[75],[31,62930,220],{"className":62931},[289],[11,62933,62934],{},"布尔骨架变成：",[31,62936,62938],{"className":62937,"translate":35},[411],[31,62939,62941,62971],{"className":62940,"translate":35},[34],[31,62942,62944],{"className":62943},[39],[41,62945,62946],{"xmlns":43,"display":420},[45,62947,62948,62968],{},[48,62949,62950,62952,62954,62956,62958,62960,62962,62964,62966],{},[95,62951,211],{"stretchy":210},[51,62953,11],{},[95,62955,53683],{},[51,62957,27666],{},[95,62959,220],{"stretchy":210},[1430,62961,53035],{},[95,62963,53038],{},[1430,62965,53035],{},[51,62967,2549],{},[55,62969,62970],{"encoding":57},"(p \\vee q) \\;\\wedge\\; r",[31,62972,62974,62995,63022],{"className":62973,"ariaHidden":62},[61],[31,62975,62977,62980,62983,62986,62989,62992],{"className":62976},[66],[31,62978],{"className":62979,"style":260},[70],[31,62981,211],{"className":62982},[264],[31,62984,11],{"className":62985},[75,76],[31,62987],{"className":62988,"style":120},[119],[31,62990,53683],{"className":62991},[124],[31,62993],{"className":62994,"style":120},[119],[31,62996,62998,63001,63004,63007,63010,63013,63016,63019],{"className":62997},[66],[31,62999],{"className":63000,"style":260},[70],[31,63002,27666],{"className":63003,"style":2666},[75,76],[31,63005,220],{"className":63006},[289],[31,63008],{"className":63009,"style":293},[119],[31,63011],{"className":63012,"style":120},[119],[31,63014,53038],{"className":63015},[124],[31,63017],{"className":63018,"style":293},[119],[31,63020],{"className":63021,"style":120},[119],[31,63023,63025,63028],{"className":63024},[66],[31,63026],{"className":63027,"style":71},[70],[31,63029,2549],{"className":63030,"style":19759},[75,76],[11,63032,63033,63034,63115,63116,63237],{},"SAT 求解器在这个骨架上的搜索空间只有 ",[31,63035,63037,63059],{"className":63036,"translate":35},[34],[31,63038,63040],{"className":63039},[39],[41,63041,63042],{"xmlns":43},[45,63043,63044,63056],{},[48,63045,63046,63052,63054],{},[1180,63047,63048,63050],{},[91,63049,152],{},[91,63051,93],{},[95,63053,223],{},[91,63055,19641],{},[55,63057,63058],{"encoding":57},"2^3 = 8",[31,63060,63062,63106],{"className":63061,"ariaHidden":62},[61],[31,63063,63065,63068,63097,63100,63103],{"className":63064},[66],[31,63066],{"className":63067,"style":5135},[70],[31,63069,63071,63074],{"className":63070},[75],[31,63072,152],{"className":63073},[75],[31,63075,63077],{"className":63076},[1039],[31,63078,63080],{"className":63079},[1043],[31,63081,63083],{"className":63082},[1048],[31,63084,63086],{"className":63085,"style":5135},[1052],[31,63087,63088,63091],{"style":3631},[31,63089],{"className":63090,"style":1061},[1060],[31,63092,63094],{"className":63093},[1065,1066,1067,1068],[31,63095,93],{"className":63096},[75,1068],[31,63098],{"className":63099,"style":293},[119],[31,63101,223],{"className":63102},[297],[31,63104],{"className":63105,"style":293},[119],[31,63107,63109,63112],{"className":63108},[66],[31,63110],{"className":63111,"style":326},[70],[31,63113,19641],{"className":63114},[75]," 种赋值。它先试 ",[31,63117,63119,63153],{"className":63118,"translate":35},[34],[31,63120,63122],{"className":63121},[39],[41,63123,63124],{"xmlns":43},[45,63125,63126,63150],{},[48,63127,63128,63130,63132,63134,63136,63138,63140,63142,63144,63146,63148],{},[51,63129,11],{},[95,63131,223],{},[51,63133,20800],{"mathvariant":494},[95,63135,1210],{"separator":62},[51,63137,27666],{},[95,63139,223],{},[51,63141,20800],{"mathvariant":494},[95,63143,1210],{"separator":62},[51,63145,2549],{},[95,63147,223],{},[51,63149,20800],{"mathvariant":494},[55,63151,63152],{"encoding":57},"p = \\top, q = \\top, r = \\top",[31,63154,63156,63174,63201,63228],{"className":63155,"ariaHidden":62},[61],[31,63157,63159,63162,63165,63168,63171],{"className":63158},[66],[31,63160],{"className":63161,"style":865},[70],[31,63163,11],{"className":63164},[75,76],[31,63166],{"className":63167,"style":293},[119],[31,63169,223],{"className":63170},[297],[31,63172],{"className":63173,"style":293},[119],[31,63175,63177,63180,63183,63186,63189,63192,63195,63198],{"className":63176},[66],[31,63178],{"className":63179,"style":4202},[70],[31,63181,20800],{"className":63182},[75],[31,63184,1210],{"className":63185},[1391],[31,63187],{"className":63188,"style":1467},[119],[31,63190,27666],{"className":63191,"style":2666},[75,76],[31,63193],{"className":63194,"style":293},[119],[31,63196,223],{"className":63197},[297],[31,63199],{"className":63200,"style":293},[119],[31,63202,63204,63207,63210,63213,63216,63219,63222,63225],{"className":63203},[66],[31,63205],{"className":63206,"style":4202},[70],[31,63208,20800],{"className":63209},[75],[31,63211,1210],{"className":63212},[1391],[31,63214],{"className":63215,"style":1467},[119],[31,63217,2549],{"className":63218,"style":19759},[75,76],[31,63220],{"className":63221,"style":293},[119],[31,63223,223],{"className":63224},[297],[31,63226],{"className":63227,"style":293},[119],[31,63229,63231,63234],{"className":63230},[66],[31,63232],{"className":63233,"style":1630},[70],[31,63235,20800],{"className":63236},[75],"，把这三个真值交回理论层。",[11,63239,63240,63241,63244],{},"理论层看到的是三个",[7246,63242,63243],{},"文字","（literal）：",[31,63246,63248],{"className":63247,"translate":35},[411],[31,63249,63251,63293],{"className":63250,"translate":35},[34],[31,63252,63254],{"className":63253},[39],[41,63255,63256],{"xmlns":43,"display":420},[45,63257,63258,63290],{},[48,63259,63260,63262,63264,63266,63268,63270,63272,63274,63276,63278,63280,63282,63284,63286,63288],{},[51,63261,53],{},[95,63263,16040],{},[91,63265,1977],{},[95,63267,1210],{"separator":62},[119,63269],{"width":20286},[51,63271,3003],{},[95,63273,16040],{},[91,63275,1977],{},[95,63277,1210],{"separator":62},[119,63279],{"width":20286},[51,63281,53],{},[95,63283,434],{},[51,63285,3003],{},[95,63287,9758],{},[91,63289,1977],{},[55,63291,63292],{"encoding":57},"x > 0, \\qquad y > 0, \\qquad x + y \u003C 0",[31,63294,63296,63314,63344,63374,63392],{"className":63295,"ariaHidden":62},[61],[31,63297,63299,63302,63305,63308,63311],{"className":63298},[66],[31,63300],{"className":63301,"style":21113},[70],[31,63303,53],{"className":63304},[75,76],[31,63306],{"className":63307,"style":293},[119],[31,63309,16040],{"className":63310},[297],[31,63312],{"className":63313,"style":293},[119],[31,63315,63317,63320,63323,63326,63329,63332,63335,63338,63341],{"className":63316},[66],[31,63318],{"className":63319,"style":41982},[70],[31,63321,1977],{"className":63322},[75],[31,63324,1210],{"className":63325},[1391],[31,63327],{"className":63328,"style":20525},[119],[31,63330],{"className":63331,"style":1467},[119],[31,63333,3003],{"className":63334,"style":2666},[75,76],[31,63336],{"className":63337,"style":293},[119],[31,63339,16040],{"className":63340},[297],[31,63342],{"className":63343,"style":293},[119],[31,63345,63347,63350,63353,63356,63359,63362,63365,63368,63371],{"className":63346},[66],[31,63348],{"className":63349,"style":41982},[70],[31,63351,1977],{"className":63352},[75],[31,63354,1210],{"className":63355},[1391],[31,63357],{"className":63358,"style":20525},[119],[31,63360],{"className":63361,"style":1467},[119],[31,63363,53],{"className":63364},[75,76],[31,63366],{"className":63367,"style":120},[119],[31,63369,434],{"className":63370},[124],[31,63372],{"className":63373,"style":120},[119],[31,63375,63377,63380,63383,63386,63389],{"className":63376},[66],[31,63378],{"className":63379,"style":53083},[70],[31,63381,3003],{"className":63382,"style":2666},[75,76],[31,63384],{"className":63385,"style":293},[119],[31,63387,9758],{"className":63388},[297],[31,63390],{"className":63391,"style":293},[119],[31,63393,63395,63398],{"className":63394},[66],[31,63396],{"className":63397,"style":326},[70],[31,63399,1977],{"className":63400},[75],[11,63402,63403,63404,63477,63478,2372],{},"线性算术求解器立刻发现：前两个推出 ",[31,63405,63407,63429],{"className":63406,"translate":35},[34],[31,63408,63410],{"className":63409},[39],[41,63411,63412],{"xmlns":43},[45,63413,63414,63426],{},[48,63415,63416,63418,63420,63422,63424],{},[51,63417,53],{},[95,63419,434],{},[51,63421,3003],{},[95,63423,16040],{},[91,63425,1977],{},[55,63427,63428],{"encoding":57},"x + y > 0",[31,63430,63432,63450,63468],{"className":63431,"ariaHidden":62},[61],[31,63433,63435,63438,63441,63444,63447],{"className":63434},[66],[31,63436],{"className":63437,"style":307},[70],[31,63439,53],{"className":63440},[75,76],[31,63442],{"className":63443,"style":120},[119],[31,63445,434],{"className":63446},[124],[31,63448],{"className":63449,"style":120},[119],[31,63451,63453,63456,63459,63462,63465],{"className":63452},[66],[31,63454],{"className":63455,"style":53083},[70],[31,63457,3003],{"className":63458,"style":2666},[75,76],[31,63460],{"className":63461,"style":293},[119],[31,63463,16040],{"className":63464},[297],[31,63466],{"className":63467,"style":293},[119],[31,63469,63471,63474],{"className":63470},[66],[31,63472],{"className":63473,"style":326},[70],[31,63475,1977],{"className":63476},[75],"，与第三个矛盾。这是一次",[7246,63479,63480],{},"理论冲突",[11,63482,63483,63484,63535,63536,11486,63602,63653,63654,63788],{},"注意，理论层收到的不是「原子为真」，而是「带正负号的文字」。",[31,63485,63487,63505],{"className":63486,"translate":35},[34],[31,63488,63490],{"className":63489},[39],[41,63491,63492],{"xmlns":43},[45,63493,63494,63502],{},[48,63495,63496,63498,63500],{},[51,63497,27666],{},[95,63499,223],{},[51,63501,53641],{"mathvariant":494},[55,63503,63504],{"encoding":57},"q = \\bot",[31,63506,63508,63526],{"className":63507,"ariaHidden":62},[61],[31,63509,63511,63514,63517,63520,63523],{"className":63510},[66],[31,63512],{"className":63513,"style":865},[70],[31,63515,27666],{"className":63516,"style":2666},[75,76],[31,63518],{"className":63519,"style":293},[119],[31,63521,223],{"className":63522},[297],[31,63524],{"className":63525,"style":293},[119],[31,63527,63529,63532],{"className":63528},[66],[31,63530],{"className":63531,"style":1630},[70],[31,63533,53641],{"className":63534},[75]," 对应的是 ",[31,63537,63539,63563],{"className":63538,"translate":35},[34],[31,63540,63542],{"className":63541},[39],[41,63543,63544],{"xmlns":43},[45,63545,63546,63560],{},[48,63547,63548,63550,63552,63554,63556,63558],{},[51,63549,53658],{"mathvariant":494},[95,63551,211],{"stretchy":210},[51,63553,3003],{},[95,63555,16040],{},[91,63557,1977],{},[95,63559,220],{"stretchy":210},[55,63561,63562],{"encoding":57},"\\neg(y > 0)",[31,63564,63566,63590],{"className":63565,"ariaHidden":62},[61],[31,63567,63569,63572,63575,63578,63581,63584,63587],{"className":63568},[66],[31,63570],{"className":63571,"style":260},[70],[31,63573,53658],{"className":63574},[75],[31,63576,211],{"className":63577},[264],[31,63579,3003],{"className":63580,"style":2666},[75,76],[31,63582],{"className":63583,"style":293},[119],[31,63585,16040],{"className":63586},[297],[31,63588],{"className":63589,"style":293},[119],[31,63591,63593,63596,63599],{"className":63592},[66],[31,63594],{"className":63595,"style":260},[70],[31,63597,1977],{"className":63598},[75],[31,63600,220],{"className":63601},[289],[31,63603,63605,63623],{"className":63604,"translate":35},[34],[31,63606,63608],{"className":63607},[39],[41,63609,63610],{"xmlns":43},[45,63611,63612,63620],{},[48,63613,63614,63616,63618],{},[51,63615,3003],{},[95,63617,347],{},[91,63619,1977],{},[55,63621,63622],{"encoding":57},"y \\le 0",[31,63624,63626,63644],{"className":63625,"ariaHidden":62},[61],[31,63627,63629,63632,63635,63638,63641],{"className":63628},[66],[31,63630],{"className":63631,"style":23593},[70],[31,63633,3003],{"className":63634,"style":2666},[75,76],[31,63636],{"className":63637,"style":293},[119],[31,63639,347],{"className":63640},[297],[31,63642],{"className":63643,"style":293},[119],[31,63645,63647,63650],{"className":63646},[66],[31,63648],{"className":63649,"style":326},[70],[31,63651,1977],{"className":63652},[75],"。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 ",[31,63655,63657,63683],{"className":63656,"translate":35},[34],[31,63658,63660],{"className":63659},[39],[41,63661,63662],{"xmlns":43},[45,63663,63664,63680],{},[48,63665,63666,63672,63674],{},[1004,63667,63668,63670],{},[51,63669,12271],{},[91,63671,205],{},[95,63673,347],{},[1004,63675,63676,63678],{},[51,63677,12271],{},[91,63679,152],{},[55,63681,63682],{"encoding":57},"t_1 \\le t_2",[31,63684,63686,63742],{"className":63685,"ariaHidden":62},[61],[31,63687,63689,63693,63733,63736,63739],{"className":63688},[66],[31,63690],{"className":63691,"style":63692},[70],"height:0.786em;vertical-align:-0.15em;",[31,63694,63696,63699],{"className":63695},[75],[31,63697,12271],{"className":63698},[75,76],[31,63700,63702],{"className":63701},[1039],[31,63703,63705,63725],{"className":63704},[1043,1044],[31,63706,63708,63722],{"className":63707},[1048],[31,63709,63711],{"className":63710,"style":2028},[1052],[31,63712,63713,63716],{"style":2807},[31,63714],{"className":63715,"style":1061},[1060],[31,63717,63719],{"className":63718},[1065,1066,1067,1068],[31,63720,205],{"className":63721},[75,1068],[31,63723,1076],{"className":63724},[1075],[31,63726,63728],{"className":63727},[1048],[31,63729,63731],{"className":63730,"style":2050},[1052],[31,63732],{},[31,63734],{"className":63735,"style":293},[119],[31,63737,347],{"className":63738},[297],[31,63740],{"className":63741,"style":293},[119],[31,63743,63745,63748],{"className":63744},[66],[31,63746],{"className":63747,"style":54007},[70],[31,63749,63751,63754],{"className":63750},[75],[31,63752,12271],{"className":63753},[75,76],[31,63755,63757],{"className":63756},[1039],[31,63758,63760,63780],{"className":63759},[1043,1044],[31,63761,63763,63777],{"className":63762},[1048],[31,63764,63766],{"className":63765,"style":2028},[1052],[31,63767,63768,63771],{"style":2807},[31,63769],{"className":63770,"style":1061},[1060],[31,63772,63774],{"className":63773},[1065,1066,1067,1068],[31,63775,152],{"className":63776},[75,1068],[31,63778,1076],{"className":63779},[1075],[31,63781,63783],{"className":63782},[1048],[31,63784,63786],{"className":63785,"style":2050},[1052],[31,63787],{}," 的不等式，而不是只处理等式。",[53200,63790,63792],{"id":63791},"_43-cdcl","4.3 CDCL",[11,63794,63795],{},"SAT 层用的是 CDCL（冲突驱动子句学习）。它的核心循环大致如下：",[52853,63797,63801],{"className":63798,"code":63800,"language":1793},[63799],"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",[45608,63802,63800],{"__ignoreMap":52812},[11,63804,63805],{},"三个关键机制：",[55164,63807,63808,63814,63820],{},[55167,63809,63810,63813],{},[7246,63811,63812],{},"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。",[55167,63815,63816,63819],{},[7246,63817,63818],{},"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。",[55167,63821,63822,63825],{},[7246,63823,63824],{},"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。",[11,63827,63828],{},"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。",[53200,63830,63832],{"id":63831},"_44-理论求解器","4.4 理论求解器",[11,63834,63835,63836,63839,63840,63868],{},"理论层要做的事情叫 ",[7246,63837,63838],{},"T-一致性检查","：给定一组带正负号的原子，判断它们在理论 ",[31,63841,63843,63856],{"className":63842,"translate":35},[34],[31,63844,63846],{"className":63845},[39],[41,63847,63848],{"xmlns":43},[45,63849,63850,63854],{},[48,63851,63852],{},[51,63853,919],{},[55,63855,919],{"encoding":57},[31,63857,63859],{"className":63858,"ariaHidden":62},[61],[31,63860,63862,63865],{"className":63861},[66],[31,63863],{"className":63864,"style":945},[70],[31,63866,919],{"className":63867,"style":949},[75,76]," 下是否可能同时为真。",[11,63870,63871],{},"不同理论有不同的算法。",[63873,63874,63876],"h4",{"id":63875},"euf","EUF",[11,63878,63879],{},"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：",[62426,63881,63882,63885],{},[55167,63883,63884],{},"用并查集维护变量的等价类；",[55167,63886,63887,63888,40580,63939,2372],{},"用同余闭包（congruence closure）保证：如果 ",[31,63889,63891,63909],{"className":63890,"translate":35},[34],[31,63892,63894],{"className":63893},[39],[41,63895,63896],{"xmlns":43},[45,63897,63898,63906],{},[48,63899,63900,63902,63904],{},[51,63901,5813],{},[95,63903,223],{},[51,63905,5834],{},[55,63907,63908],{"encoding":57},"a = b",[31,63910,63912,63930],{"className":63911,"ariaHidden":62},[61],[31,63913,63915,63918,63921,63924,63927],{"className":63914},[66],[31,63916],{"className":63917,"style":71},[70],[31,63919,5813],{"className":63920},[75,76],[31,63922],{"className":63923,"style":293},[119],[31,63925,223],{"className":63926},[297],[31,63928],{"className":63929,"style":293},[119],[31,63931,63933,63936],{"className":63932},[66],[31,63934],{"className":63935,"style":1630},[70],[31,63937,5834],{"className":63938},[75,76],[31,63940,63942,63972],{"className":63941,"translate":35},[34],[31,63943,63945],{"className":63944},[39],[41,63946,63947],{"xmlns":43},[45,63948,63949,63969],{},[48,63950,63951,63953,63955,63957,63959,63961,63963,63965,63967],{},[51,63952,1662],{},[95,63954,211],{"stretchy":210},[51,63956,5813],{},[95,63958,220],{"stretchy":210},[95,63960,223],{},[51,63962,1662],{},[95,63964,211],{"stretchy":210},[51,63966,5834],{},[95,63968,220],{"stretchy":210},[55,63970,63971],{"encoding":57},"f(a) = f(b)",[31,63973,63975,64002],{"className":63974,"ariaHidden":62},[61],[31,63976,63978,63981,63984,63987,63990,63993,63996,63999],{"className":63977},[66],[31,63979],{"className":63980,"style":260},[70],[31,63982,1662],{"className":63983,"style":35196},[75,76],[31,63985,211],{"className":63986},[264],[31,63988,5813],{"className":63989},[75,76],[31,63991,220],{"className":63992},[289],[31,63994],{"className":63995,"style":293},[119],[31,63997,223],{"className":63998},[297],[31,64000],{"className":64001,"style":293},[119],[31,64003,64005,64008,64011,64014,64017],{"className":64004},[66],[31,64006],{"className":64007,"style":260},[70],[31,64009,1662],{"className":64010,"style":35196},[75,76],[31,64012,211],{"className":64013},[264],[31,64015,5834],{"className":64016},[75,76],[31,64018,220],{"className":64019},[289],[11,64021,64022],{},"例子：",[31,64024,64026],{"className":64025,"translate":35},[411],[31,64027,64029,64083],{"className":64028,"translate":35},[34],[31,64030,64032],{"className":64031},[39],[41,64033,64034],{"xmlns":43,"display":420},[45,64035,64036,64080],{},[48,64037,64038,64040,64042,64044,64046,64048,64050,64052,64054,64056,64058,64060,64062,64064,64066,64068,64070,64072,64074,64076,64078],{},[51,64039,5813],{},[95,64041,223],{},[51,64043,5834],{},[1430,64045,53035],{},[95,64047,53038],{},[1430,64049,53035],{},[51,64051,5834],{},[95,64053,223],{},[51,64055,1412],{},[1430,64057,53035],{},[95,64059,53038],{},[1430,64061,53035],{},[51,64063,1662],{},[95,64065,211],{"stretchy":210},[51,64067,5813],{},[95,64069,220],{"stretchy":210},[95,64071,21271],{"mathvariant":494},[51,64073,1662],{},[95,64075,211],{"stretchy":210},[51,64077,1412],{},[95,64079,220],{"stretchy":210},[55,64081,64082],{"encoding":57},"a = b \\;\\wedge\\; b = c \\;\\wedge\\; f(a) \\ne f(c)",[31,64084,64086,64104,64128,64146,64171,64231],{"className":64085,"ariaHidden":62},[61],[31,64087,64089,64092,64095,64098,64101],{"className":64088},[66],[31,64090],{"className":64091,"style":71},[70],[31,64093,5813],{"className":64094},[75,76],[31,64096],{"className":64097,"style":293},[119],[31,64099,223],{"className":64100},[297],[31,64102],{"className":64103,"style":293},[119],[31,64105,64107,64110,64113,64116,64119,64122,64125],{"className":64106},[66],[31,64108],{"className":64109,"style":1630},[70],[31,64111,5834],{"className":64112},[75,76],[31,64114],{"className":64115,"style":293},[119],[31,64117],{"className":64118,"style":120},[119],[31,64120,53038],{"className":64121},[124],[31,64123],{"className":64124,"style":293},[119],[31,64126],{"className":64127,"style":120},[119],[31,64129,64131,64134,64137,64140,64143],{"className":64130},[66],[31,64132],{"className":64133,"style":1630},[70],[31,64135,5834],{"className":64136},[75,76],[31,64138],{"className":64139,"style":293},[119],[31,64141,223],{"className":64142},[297],[31,64144],{"className":64145,"style":293},[119],[31,64147,64149,64153,64156,64159,64162,64165,64168],{"className":64148},[66],[31,64150],{"className":64151,"style":64152},[70],"height:0.5556em;",[31,64154,1412],{"className":64155},[75,76],[31,64157],{"className":64158,"style":293},[119],[31,64160],{"className":64161,"style":120},[119],[31,64163,53038],{"className":64164},[124],[31,64166],{"className":64167,"style":293},[119],[31,64169],{"className":64170,"style":120},[119],[31,64172,64174,64177,64180,64183,64186,64189,64192,64228],{"className":64173},[66],[31,64175],{"className":64176,"style":260},[70],[31,64178,1662],{"className":64179,"style":35196},[75,76],[31,64181,211],{"className":64182},[264],[31,64184,5813],{"className":64185},[75,76],[31,64187,220],{"className":64188},[289],[31,64190],{"className":64191,"style":293},[119],[31,64193,64195,64222,64225],{"className":64194},[297],[31,64196,64198],{"className":64197},[297],[31,64199,64201],{"className":64200},[75,21516],[31,64202,64204],{"className":64203},[21520],[31,64205,64207,64210,64219],{"className":64206},[21524],[31,64208],{"className":64209,"style":4202},[70],[31,64211,64213],{"className":64212},[21531],[31,64214,64216],{"className":64215},[75],[31,64217,21538],{"className":64218},[297],[31,64220],{"className":64221},[21542],[31,64223],{"className":64224},[119,21546],[31,64226,223],{"className":64227},[297],[31,64229],{"className":64230,"style":293},[119],[31,64232,64234,64237,64240,64243,64246],{"className":64233},[66],[31,64235],{"className":64236,"style":260},[70],[31,64238,1662],{"className":64239,"style":35196},[75,76],[31,64241,211],{"className":64242},[264],[31,64244,1412],{"className":64245},[75,76],[31,64247,220],{"className":64248},[289],[11,64250,64251,64252,64307,64308,2446,64352,64396,64397,64469],{},"并查集把 ",[31,64253,64255,64277],{"className":64254,"translate":35},[34],[31,64256,64258],{"className":64257},[39],[41,64259,64260],{"xmlns":43},[45,64261,64262,64274],{},[48,64263,64264,64266,64268,64270,64272],{},[51,64265,5813],{},[95,64267,1210],{"separator":62},[51,64269,5834],{},[95,64271,1210],{"separator":62},[51,64273,1412],{},[55,64275,64276],{"encoding":57},"a, b, c",[31,64278,64280],{"className":64279,"ariaHidden":62},[61],[31,64281,64283,64286,64289,64292,64295,64298,64301,64304],{"className":64282},[66],[31,64284],{"className":64285,"style":4202},[70],[31,64287,5813],{"className":64288},[75,76],[31,64290,1210],{"className":64291},[1391],[31,64293],{"className":64294,"style":1467},[119],[31,64296,5834],{"className":64297},[75,76],[31,64299,1210],{"className":64300},[1391],[31,64302],{"className":64303,"style":1467},[119],[31,64305,1412],{"className":64306},[75,76]," 合并成一个类；同余闭包发现 ",[31,64309,64311,64331],{"className":64310,"translate":35},[34],[31,64312,64314],{"className":64313},[39],[41,64315,64316],{"xmlns":43},[45,64317,64318,64328],{},[48,64319,64320,64322,64324,64326],{},[51,64321,1662],{},[95,64323,211],{"stretchy":210},[51,64325,5813],{},[95,64327,220],{"stretchy":210},[55,64329,64330],{"encoding":57},"f(a)",[31,64332,64334],{"className":64333,"ariaHidden":62},[61],[31,64335,64337,64340,64343,64346,64349],{"className":64336},[66],[31,64338],{"className":64339,"style":260},[70],[31,64341,1662],{"className":64342,"style":35196},[75,76],[31,64344,211],{"className":64345},[264],[31,64347,5813],{"className":64348},[75,76],[31,64350,220],{"className":64351},[289],[31,64353,64355,64375],{"className":64354,"translate":35},[34],[31,64356,64358],{"className":64357},[39],[41,64359,64360],{"xmlns":43},[45,64361,64362,64372],{},[48,64363,64364,64366,64368,64370],{},[51,64365,1662],{},[95,64367,211],{"stretchy":210},[51,64369,1412],{},[95,64371,220],{"stretchy":210},[55,64373,64374],{"encoding":57},"f(c)",[31,64376,64378],{"className":64377,"ariaHidden":62},[61],[31,64379,64381,64384,64387,64390,64393],{"className":64380},[66],[31,64382],{"className":64383,"style":260},[70],[31,64385,1662],{"className":64386,"style":35196},[75,76],[31,64388,211],{"className":64389},[264],[31,64391,1412],{"className":64392},[75,76],[31,64394,220],{"className":64395},[289]," 的参数相等，于是把 ",[31,64398,64400,64430],{"className":64399,"translate":35},[34],[31,64401,64403],{"className":64402},[39],[41,64404,64405],{"xmlns":43},[45,64406,64407,64427],{},[48,64408,64409,64411,64413,64415,64417,64419,64421,64423,64425],{},[51,64410,1662],{},[95,64412,211],{"stretchy":210},[51,64414,5813],{},[95,64416,220],{"stretchy":210},[95,64418,1210],{"separator":62},[51,64420,1662],{},[95,64422,211],{"stretchy":210},[51,64424,1412],{},[95,64426,220],{"stretchy":210},[55,64428,64429],{"encoding":57},"f(a), f(c)",[31,64431,64433],{"className":64432,"ariaHidden":62},[61],[31,64434,64436,64439,64442,64445,64448,64451,64454,64457,64460,64463,64466],{"className":64435},[66],[31,64437],{"className":64438,"style":260},[70],[31,64440,1662],{"className":64441,"style":35196},[75,76],[31,64443,211],{"className":64444},[264],[31,64446,5813],{"className":64447},[75,76],[31,64449,220],{"className":64450},[289],[31,64452,1210],{"className":64453},[1391],[31,64455],{"className":64456,"style":1467},[119],[31,64458,1662],{"className":64459,"style":35196},[75,76],[31,64461,211],{"className":64462},[264],[31,64464,1412],{"className":64465},[75,76],[31,64467,220],{"className":64468},[289]," 也合并；但公式又要求它们不等，冲突，判 unsat。",[11,64471,64472],{},"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」。",[63873,64474,64475],{"id":64475},"线性算术",[11,64477,64478,64480],{},[45608,64479,55631],{}," 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。",[11,64482,64483,64485,64486,64489,64490,2446,64553,64616],{},[45608,64484,55635],{}," 在此基础上加",[7246,64487,64488],{},"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 ",[31,64491,64493,64517],{"className":64492,"translate":35},[34],[31,64494,64496],{"className":64495},[39],[41,64497,64498],{"xmlns":43},[45,64499,64500,64514],{},[48,64501,64502,64504,64506,64509,64511],{},[51,64503,53],{},[95,64505,347],{},[95,64507,64508],{"stretchy":210},"⌊",[51,64510,5354],{},[95,64512,64513],{"stretchy":210},"⌋",[55,64515,64516],{"encoding":57},"x \\le \\lfloor v \\rfloor",[31,64518,64520,64538],{"className":64519,"ariaHidden":62},[61],[31,64521,64523,64526,64529,64532,64535],{"className":64522},[66],[31,64524],{"className":64525,"style":385},[70],[31,64527,53],{"className":64528},[75,76],[31,64530],{"className":64531,"style":293},[119],[31,64533,347],{"className":64534},[297],[31,64536],{"className":64537,"style":293},[119],[31,64539,64541,64544,64547,64550],{"className":64540},[66],[31,64542],{"className":64543,"style":260},[70],[31,64545,64508],{"className":64546},[264],[31,64548,5354],{"className":64549,"style":2666},[75,76],[31,64551,64513],{"className":64552},[289],[31,64554,64556,64580],{"className":64555,"translate":35},[34],[31,64557,64559],{"className":64558},[39],[41,64560,64561],{"xmlns":43},[45,64562,64563,64577],{},[48,64564,64565,64567,64569,64572,64574],{},[51,64566,53],{},[95,64568,5662],{},[95,64570,64571],{"stretchy":210},"⌈",[51,64573,5354],{},[95,64575,64576],{"stretchy":210},"⌉",[55,64578,64579],{"encoding":57},"x \\ge \\lceil v \\rceil",[31,64581,64583,64601],{"className":64582,"ariaHidden":62},[61],[31,64584,64586,64589,64592,64595,64598],{"className":64585},[66],[31,64587],{"className":64588,"style":385},[70],[31,64590,53],{"className":64591},[75,76],[31,64593],{"className":64594,"style":293},[119],[31,64596,5662],{"className":64597},[297],[31,64599],{"className":64600,"style":293},[119],[31,64602,64604,64607,64610,64613],{"className":64603},[66],[31,64605],{"className":64606,"style":260},[70],[31,64608,64571],{"className":64609},[264],[31,64611,5354],{"className":64612,"style":2666},[75,76],[31,64614,64576],{"className":64615},[289],"，分别再解。",[11,64618,64022],{},[31,64620,64622],{"className":64621,"translate":35},[411],[31,64623,64625,64679],{"className":64624,"translate":35},[34],[31,64626,64628],{"className":64627},[39],[41,64629,64630],{"xmlns":43,"display":420},[45,64631,64632,64676],{},[48,64633,64634,64636,64638,64640,64642,64644,64646,64648,64650,64652,64654,64656,64658,64660,64662,64664,64666,64668,64670,64672,64674],{},[91,64635,93],{},[51,64637,53],{},[95,64639,434],{},[91,64641,152],{},[51,64643,3003],{},[95,64645,223],{},[91,64647,7853],{},[95,64649,1210],{"separator":62},[119,64651],{"width":20286},[51,64653,53],{},[95,64655,1210],{"separator":62},[51,64657,3003],{},[95,64659,5662],{},[91,64661,1977],{},[95,64663,1210],{"separator":62},[119,64665],{"width":20286},[51,64667,53],{},[95,64669,1210],{"separator":62},[51,64671,3003],{},[95,64673,2518],{},[51,64675,28644],{"mathvariant":14789},[55,64677,64678],{"encoding":57},"3x + 2y = 5, \\qquad x, y \\ge 0, \\qquad x, y \\in \\mathbb{Z}",[31,64680,64682,64703,64724,64763,64802],{"className":64681,"ariaHidden":62},[61],[31,64683,64685,64688,64691,64694,64697,64700],{"className":64684},[66],[31,64686],{"className":64687,"style":112},[70],[31,64689,93],{"className":64690},[75],[31,64692,53],{"className":64693},[75,76],[31,64695],{"className":64696,"style":120},[119],[31,64698,434],{"className":64699},[124],[31,64701],{"className":64702,"style":120},[119],[31,64704,64706,64709,64712,64715,64718,64721],{"className":64705},[66],[31,64707],{"className":64708,"style":41982},[70],[31,64710,152],{"className":64711},[75],[31,64713,3003],{"className":64714,"style":2666},[75,76],[31,64716],{"className":64717,"style":293},[119],[31,64719,223],{"className":64720},[297],[31,64722],{"className":64723,"style":293},[119],[31,64725,64727,64730,64733,64736,64739,64742,64745,64748,64751,64754,64757,64760],{"className":64726},[66],[31,64728],{"className":64729,"style":41982},[70],[31,64731,7853],{"className":64732},[75],[31,64734,1210],{"className":64735},[1391],[31,64737],{"className":64738,"style":20525},[119],[31,64740],{"className":64741,"style":1467},[119],[31,64743,53],{"className":64744},[75,76],[31,64746,1210],{"className":64747},[1391],[31,64749],{"className":64750,"style":1467},[119],[31,64752,3003],{"className":64753,"style":2666},[75,76],[31,64755],{"className":64756,"style":293},[119],[31,64758,5662],{"className":64759},[297],[31,64761],{"className":64762,"style":293},[119],[31,64764,64766,64769,64772,64775,64778,64781,64784,64787,64790,64793,64796,64799],{"className":64765},[66],[31,64767],{"className":64768,"style":41982},[70],[31,64770,1977],{"className":64771},[75],[31,64773,1210],{"className":64774},[1391],[31,64776],{"className":64777,"style":20525},[119],[31,64779],{"className":64780,"style":1467},[119],[31,64782,53],{"className":64783},[75,76],[31,64785,1210],{"className":64786},[1391],[31,64788],{"className":64789,"style":1467},[119],[31,64791,3003],{"className":64792,"style":2666},[75,76],[31,64794],{"className":64795,"style":293},[119],[31,64797,2518],{"className":64798},[297],[31,64800],{"className":64801,"style":293},[119],[31,64803,64805,64809],{"className":64804},[66],[31,64806],{"className":64807,"style":64808},[70],"height:0.6889em;",[31,64810,28644],{"className":64811},[75,14947],[11,64813,64814,64815,64906,64907,64935],{},"实数松弛下，",[31,64816,64818,64848],{"className":64817,"translate":35},[34],[31,64819,64821],{"className":64820},[39],[41,64822,64823],{"xmlns":43},[45,64824,64825,64845],{},[48,64826,64827,64829,64831,64833,64835,64837,64839,64841,64843],{},[51,64828,53],{},[95,64830,223],{},[91,64832,7853],{},[51,64834,3768],{"mathvariant":494},[91,64836,93],{},[95,64838,1210],{"separator":62},[51,64840,3003],{},[95,64842,223],{},[91,64844,1977],{},[55,64846,64847],{"encoding":57},"x = 5\u002F3, y = 0",[31,64849,64851,64869,64897],{"className":64850,"ariaHidden":62},[61],[31,64852,64854,64857,64860,64863,64866],{"className":64853},[66],[31,64855],{"className":64856,"style":71},[70],[31,64858,53],{"className":64859},[75,76],[31,64861],{"className":64862,"style":293},[119],[31,64864,223],{"className":64865},[297],[31,64867],{"className":64868,"style":293},[119],[31,64870,64872,64875,64879,64882,64885,64888,64891,64894],{"className":64871},[66],[31,64873],{"className":64874,"style":260},[70],[31,64876,64878],{"className":64877},[75],"5\u002F3",[31,64880,1210],{"className":64881},[1391],[31,64883],{"className":64884,"style":1467},[119],[31,64886,3003],{"className":64887,"style":2666},[75,76],[31,64889],{"className":64890,"style":293},[119],[31,64892,223],{"className":64893},[297],[31,64895],{"className":64896,"style":293},[119],[31,64898,64900,64903],{"className":64899},[66],[31,64901],{"className":64902,"style":326},[70],[31,64904,1977],{"className":64905},[75]," 是一个解，但 ",[31,64908,64910,64923],{"className":64909,"translate":35},[34],[31,64911,64913],{"className":64912},[39],[41,64914,64915],{"xmlns":43},[45,64916,64917,64921],{},[48,64918,64919],{},[51,64920,53],{},[55,64922,53],{"encoding":57},[31,64924,64926],{"className":64925,"ariaHidden":62},[61],[31,64927,64929,64932],{"className":64928},[66],[31,64930],{"className":64931,"style":71},[70],[31,64933,53],{"className":64934},[75,76]," 不是整数。分支：",[55164,64937,64938,65283],{},[55167,64939,52996,64940,4809,64991,65019,65020,23342,65048,36037,65076,21051,65127,65179,65180,21051,65231,65282],{},[31,64941,64943,64961],{"className":64942,"translate":35},[34],[31,64944,64946],{"className":64945},[39],[41,64947,64948],{"xmlns":43},[45,64949,64950,64958],{},[48,64951,64952,64954,64956],{},[51,64953,53],{},[95,64955,347],{},[91,64957,205],{},[55,64959,64960],{"encoding":57},"x \\le 1",[31,64962,64964,64982],{"className":64963,"ariaHidden":62},[61],[31,64965,64967,64970,64973,64976,64979],{"className":64966},[66],[31,64968],{"className":64969,"style":385},[70],[31,64971,53],{"className":64972},[75,76],[31,64974],{"className":64975,"style":293},[119],[31,64977,347],{"className":64978},[297],[31,64980],{"className":64981,"style":293},[119],[31,64983,64985,64988],{"className":64984},[66],[31,64986],{"className":64987,"style":326},[70],[31,64989,205],{"className":64990},[75],[31,64992,64994,65007],{"className":64993,"translate":35},[34],[31,64995,64997],{"className":64996},[39],[41,64998,64999],{"xmlns":43},[45,65000,65001,65005],{},[48,65002,65003],{},[51,65004,53],{},[55,65006,53],{"encoding":57},[31,65008,65010],{"className":65009,"ariaHidden":62},[61],[31,65011,65013,65016],{"className":65012},[66],[31,65014],{"className":65015,"style":71},[70],[31,65017,53],{"className":65018},[75,76]," 只能取 ",[31,65021,65023,65036],{"className":65022,"translate":35},[34],[31,65024,65026],{"className":65025},[39],[41,65027,65028],{"xmlns":43},[45,65029,65030,65034],{},[48,65031,65032],{},[91,65033,1977],{},[55,65035,1977],{"encoding":57},[31,65037,65039],{"className":65038,"ariaHidden":62},[61],[31,65040,65042,65045],{"className":65041},[66],[31,65043],{"className":65044,"style":326},[70],[31,65046,1977],{"className":65047},[75],[31,65049,65051,65064],{"className":65050,"translate":35},[34],[31,65052,65054],{"className":65053},[39],[41,65055,65056],{"xmlns":43},[45,65057,65058,65062],{},[48,65059,65060],{},[91,65061,205],{},[55,65063,205],{"encoding":57},[31,65065,65067],{"className":65066,"ariaHidden":62},[61],[31,65068,65070,65073],{"className":65069},[66],[31,65071],{"className":65072,"style":326},[70],[31,65074,205],{"className":65075},[75],[31,65077,65079,65097],{"className":65078,"translate":35},[34],[31,65080,65082],{"className":65081},[39],[41,65083,65084],{"xmlns":43},[45,65085,65086,65094],{},[48,65087,65088,65090,65092],{},[51,65089,53],{},[95,65091,223],{},[91,65093,1977],{},[55,65095,65096],{"encoding":57},"x = 0",[31,65098,65100,65118],{"className":65099,"ariaHidden":62},[61],[31,65101,65103,65106,65109,65112,65115],{"className":65102},[66],[31,65104],{"className":65105,"style":71},[70],[31,65107,53],{"className":65108},[75,76],[31,65110],{"className":65111,"style":293},[119],[31,65113,223],{"className":65114},[297],[31,65116],{"className":65117,"style":293},[119],[31,65119,65121,65124],{"className":65120},[66],[31,65122],{"className":65123,"style":326},[70],[31,65125,1977],{"className":65126},[75],[31,65128,65130,65149],{"className":65129,"translate":35},[34],[31,65131,65133],{"className":65132},[39],[41,65134,65135],{"xmlns":43},[45,65136,65137,65146],{},[48,65138,65139,65141,65143],{},[51,65140,3003],{},[95,65142,223],{},[91,65144,65145],{},"2.5",[55,65147,65148],{"encoding":57},"y = 2.5",[31,65150,65152,65170],{"className":65151,"ariaHidden":62},[61],[31,65153,65155,65158,65161,65164,65167],{"className":65154},[66],[31,65156],{"className":65157,"style":865},[70],[31,65159,3003],{"className":65160,"style":2666},[75,76],[31,65162],{"className":65163,"style":293},[119],[31,65165,223],{"className":65166},[297],[31,65168],{"className":65169,"style":293},[119],[31,65171,65173,65176],{"className":65172},[66],[31,65174],{"className":65175,"style":326},[70],[31,65177,65145],{"className":65178},[75]," 仍不是整数，",[31,65181,65183,65201],{"className":65182,"translate":35},[34],[31,65184,65186],{"className":65185},[39],[41,65187,65188],{"xmlns":43},[45,65189,65190,65198],{},[48,65191,65192,65194,65196],{},[51,65193,53],{},[95,65195,223],{},[91,65197,205],{},[55,65199,65200],{"encoding":57},"x = 1",[31,65202,65204,65222],{"className":65203,"ariaHidden":62},[61],[31,65205,65207,65210,65213,65216,65219],{"className":65206},[66],[31,65208],{"className":65209,"style":71},[70],[31,65211,53],{"className":65212},[75,76],[31,65214],{"className":65215,"style":293},[119],[31,65217,223],{"className":65218},[297],[31,65220],{"className":65221,"style":293},[119],[31,65223,65225,65228],{"className":65224},[66],[31,65226],{"className":65227,"style":326},[70],[31,65229,205],{"className":65230},[75],[31,65232,65234,65252],{"className":65233,"translate":35},[34],[31,65235,65237],{"className":65236},[39],[41,65238,65239],{"xmlns":43},[45,65240,65241,65249],{},[48,65242,65243,65245,65247],{},[51,65244,3003],{},[95,65246,223],{},[91,65248,205],{},[55,65250,65251],{"encoding":57},"y = 1",[31,65253,65255,65273],{"className":65254,"ariaHidden":62},[61],[31,65256,65258,65261,65264,65267,65270],{"className":65257},[66],[31,65259],{"className":65260,"style":865},[70],[31,65262,3003],{"className":65263,"style":2666},[75,76],[31,65265],{"className":65266,"style":293},[119],[31,65268,223],{"className":65269},[297],[31,65271],{"className":65272,"style":293},[119],[31,65274,65276,65279],{"className":65275},[66],[31,65277],{"className":65278,"style":326},[70],[31,65280,205],{"className":65281},[75],"，得到整数解，sat；",[55167,65284,65285,65286,65337],{},"另一支 ",[31,65287,65289,65307],{"className":65288,"translate":35},[34],[31,65290,65292],{"className":65291},[39],[41,65293,65294],{"xmlns":43},[45,65295,65296,65304],{},[48,65297,65298,65300,65302],{},[51,65299,53],{},[95,65301,5662],{},[91,65303,152],{},[55,65305,65306],{"encoding":57},"x \\ge 2",[31,65308,65310,65328],{"className":65309,"ariaHidden":62},[61],[31,65311,65313,65316,65319,65322,65325],{"className":65312},[66],[31,65314],{"className":65315,"style":385},[70],[31,65317,53],{"className":65318},[75,76],[31,65320],{"className":65321,"style":293},[119],[31,65323,5662],{"className":65324},[297],[31,65326],{"className":65327,"style":293},[119],[31,65329,65331,65334],{"className":65330},[66],[31,65332],{"className":65333,"style":326},[70],[31,65335,152],{"className":65336},[75]," 就无须再看了。",[11,65339,65340],{},"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。",[63873,65342,60327],{"id":60327},[11,65344,65345,65347],{},[45608,65346,55719],{}," 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。",[11,65349,65350],{},"现代求解器会先做字级重写，把明显恒真的部分消掉，比如：",[31,65352,65354],{"className":65353,"translate":35},[411],[31,65355,65357,65412],{"className":65356,"translate":35},[34],[31,65358,65360],{"className":65359},[39],[41,65361,65362],{"xmlns":43,"display":420},[45,65363,65364,65409],{},[48,65365,65366,65368,65370,65372,65374,65376,65378,65380,65382,65384,65386,65388,65390,65392,65394,65396,65398,65400,65403,65405,65407],{},[51,65367,53],{},[1430,65369,1432],{},[51,65371,60351],{"mathvariant":494},[1430,65373,1432],{},[91,65375,1977],{},[95,65377,223],{},[91,65379,1977],{},[95,65381,1210],{"separator":62},[119,65383],{"width":20286},[51,65385,53],{},[95,65387,97],{},[51,65389,53],{},[95,65391,223],{},[91,65393,1977],{},[95,65395,1210],{"separator":62},[119,65397],{"width":20286},[51,65399,53],{},[95,65401,65402],{},"≪",[91,65404,1977],{},[95,65406,223],{},[51,65408,53],{},[55,65410,65411],{"encoding":57},"x \\,\\&\\, 0 = 0, \\qquad x - x = 0, \\qquad x \\ll 0 = x",[31,65413,65415,65445,65475,65493,65523,65541],{"className":65414,"ariaHidden":62},[61],[31,65416,65418,65421,65424,65427,65430,65433,65436,65439,65442],{"className":65417},[66],[31,65419],{"className":65420,"style":1630},[70],[31,65422,53],{"className":65423},[75,76],[31,65425],{"className":65426,"style":1467},[119],[31,65428,60351],{"className":65429},[75],[31,65431],{"className":65432,"style":1467},[119],[31,65434,1977],{"className":65435},[75],[31,65437],{"className":65438,"style":293},[119],[31,65440,223],{"className":65441},[297],[31,65443],{"className":65444,"style":293},[119],[31,65446,65448,65451,65454,65457,65460,65463,65466,65469,65472],{"className":65447},[66],[31,65449],{"className":65450,"style":41982},[70],[31,65452,1977],{"className":65453},[75],[31,65455,1210],{"className":65456},[1391],[31,65458],{"className":65459,"style":20525},[119],[31,65461],{"className":65462,"style":1467},[119],[31,65464,53],{"className":65465},[75,76],[31,65467],{"className":65468,"style":120},[119],[31,65470,97],{"className":65471},[124],[31,65473],{"className":65474,"style":120},[119],[31,65476,65478,65481,65484,65487,65490],{"className":65477},[66],[31,65479],{"className":65480,"style":71},[70],[31,65482,53],{"className":65483},[75,76],[31,65485],{"className":65486,"style":293},[119],[31,65488,223],{"className":65489},[297],[31,65491],{"className":65492,"style":293},[119],[31,65494,65496,65499,65502,65505,65508,65511,65514,65517,65520],{"className":65495},[66],[31,65497],{"className":65498,"style":41982},[70],[31,65500,1977],{"className":65501},[75],[31,65503,1210],{"className":65504},[1391],[31,65506],{"className":65507,"style":20525},[119],[31,65509],{"className":65510,"style":1467},[119],[31,65512,53],{"className":65513},[75,76],[31,65515],{"className":65516,"style":293},[119],[31,65518,65402],{"className":65519},[297],[31,65521],{"className":65522,"style":293},[119],[31,65524,65526,65529,65532,65535,65538],{"className":65525},[66],[31,65527],{"className":65528,"style":326},[70],[31,65530,1977],{"className":65531},[75],[31,65533],{"className":65534,"style":293},[119],[31,65536,223],{"className":65537},[297],[31,65539],{"className":65540,"style":293},[119],[31,65542,65544,65547],{"className":65543},[66],[31,65545],{"className":65546,"style":71},[70],[31,65548,53],{"className":65549},[75,76],[11,65551,65552],{},"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。",[11,65554,65555,65556,65559],{},"位向量的语义是",[7246,65557,65558],{},"模运算","，这一点非常重要：",[31,65561,65563],{"className":65562,"translate":35},[411],[31,65564,65566,65628],{"className":65565,"translate":35},[34],[31,65567,65569],{"className":65568},[39],[41,65570,65571],{"xmlns":43,"display":420},[45,65572,65573,65625],{},[48,65574,65575,65577,65579,65581,65583,65585,65587,65589,65591,65599,65602,65609,65611,65613,65615,65617,65619,65621],{},[51,65576,53],{},[95,65578,434],{},[91,65580,205],{},[95,65582,62697],{},[91,65584,1977],{},[119,65586],{},[119,65588],{"width":1584},[95,65590,211],{"stretchy":210},[48,65592,65593,65595,65597],{},[51,65594,16396],{"mathvariant":494},[51,65596,2552],{"mathvariant":494},[51,65598,1435],{"mathvariant":494},[119,65600],{"width":65601},"0.3333em",[1180,65603,65604,65606],{},[91,65605,152],{},[91,65607,65608],{},"32",[95,65610,220],{"stretchy":210},[1430,65612,53035],{},[95,65614,53038],{},[1430,65616,53035],{},[51,65618,53],{},[95,65620,21271],{"mathvariant":494},[1430,65622,65624],{"mathvariant":65623},"monospace","0xFFFFFFFF",[55,65626,65627],{"encoding":57},"x + 1 \\equiv 0 \\pmod{2^{32}} \\;\\wedge\\; x \\ne \\texttt{0xFFFFFFFF}",[31,65629,65631,65649,65667,65683,65756,65807],{"className":65630,"ariaHidden":62},[61],[31,65632,65634,65637,65640,65643,65646],{"className":65633},[66],[31,65635],{"className":65636,"style":307},[70],[31,65638,53],{"className":65639},[75,76],[31,65641],{"className":65642,"style":120},[119],[31,65644,434],{"className":65645},[124],[31,65647],{"className":65648,"style":120},[119],[31,65650,65652,65655,65658,65661,65664],{"className":65651},[66],[31,65653],{"className":65654,"style":326},[70],[31,65656,205],{"className":65657},[75],[31,65659],{"className":65660,"style":293},[119],[31,65662,62697],{"className":65663},[297],[31,65665],{"className":65666,"style":293},[119],[31,65668,65670,65673,65676,65680],{"className":65669},[66],[31,65671],{"className":65672,"style":326},[70],[31,65674,1977],{"className":65675},[75],[31,65677],{"className":65678},[119,65679],"allowbreak",[31,65681],{"className":65682,"style":1789},[119],[31,65684,65686,65689,65692,65702,65706,65738,65741,65744,65747,65750,65753],{"className":65685},[66],[31,65687],{"className":65688,"style":16882},[70],[31,65690,211],{"className":65691},[264],[31,65693,65695],{"className":65694},[75],[31,65696,65698],{"className":65697},[75],[31,65699,65701],{"className":65700},[75,2745],"mod",[31,65703],{"className":65704,"style":65705},[119],"margin-right:0.3333em;",[31,65707,65709,65712],{"className":65708},[75],[31,65710,152],{"className":65711},[75],[31,65713,65715],{"className":65714},[1039],[31,65716,65718],{"className":65717},[1043],[31,65719,65721],{"className":65720},[1048],[31,65722,65724],{"className":65723,"style":1327},[1052],[31,65725,65726,65729],{"style":1330},[31,65727],{"className":65728,"style":1061},[1060],[31,65730,65732],{"className":65731},[1065,1066,1067,1068],[31,65733,65735],{"className":65734},[75,1068],[31,65736,65608],{"className":65737},[75,1068],[31,65739,220],{"className":65740},[289],[31,65742],{"className":65743,"style":293},[119],[31,65745],{"className":65746,"style":120},[119],[31,65748,53038],{"className":65749},[124],[31,65751],{"className":65752,"style":293},[119],[31,65754],{"className":65755,"style":120},[119],[31,65757,65759,65762,65765,65768,65804],{"className":65758},[66],[31,65760],{"className":65761,"style":4202},[70],[31,65763,53],{"className":65764},[75,76],[31,65766],{"className":65767,"style":293},[119],[31,65769,65771,65798,65801],{"className":65770},[297],[31,65772,65774],{"className":65773},[297],[31,65775,65777],{"className":65776},[75,21516],[31,65778,65780],{"className":65779},[21520],[31,65781,65783,65786,65795],{"className":65782},[21524],[31,65784],{"className":65785,"style":4202},[70],[31,65787,65789],{"className":65788},[21531],[31,65790,65792],{"className":65791},[75],[31,65793,21538],{"className":65794},[297],[31,65796],{"className":65797},[21542],[31,65799],{"className":65800},[119,21546],[31,65802,223],{"className":65803},[297],[31,65805],{"className":65806,"style":293},[119],[31,65808,65810,65814],{"className":65809},[66],[31,65811],{"className":65812,"style":65813},[70],"height:0.6111em;",[31,65815,65817],{"className":65816},[75,1793],[31,65818,65624],{"className":65819},[75,65820],"texttt",[11,65822,65823,65824,65897,65898,65900,65901,65957,65958,65960,65961],{},"在 32 位无符号整数里，唯一满足 ",[31,65825,65827,65849],{"className":65826,"translate":35},[34],[31,65828,65830],{"className":65829},[39],[41,65831,65832],{"xmlns":43},[45,65833,65834,65846],{},[48,65835,65836,65838,65840,65842,65844],{},[51,65837,53],{},[95,65839,434],{},[91,65841,205],{},[95,65843,62697],{},[91,65845,1977],{},[55,65847,65848],{"encoding":57},"x + 1 \\equiv 0",[31,65850,65852,65870,65888],{"className":65851,"ariaHidden":62},[61],[31,65853,65855,65858,65861,65864,65867],{"className":65854},[66],[31,65856],{"className":65857,"style":307},[70],[31,65859,53],{"className":65860},[75,76],[31,65862],{"className":65863,"style":120},[119],[31,65865,434],{"className":65866},[124],[31,65868],{"className":65869,"style":120},[119],[31,65871,65873,65876,65879,65882,65885],{"className":65872},[66],[31,65874],{"className":65875,"style":326},[70],[31,65877,205],{"className":65878},[75],[31,65880],{"className":65881,"style":293},[119],[31,65883,62697],{"className":65884},[297],[31,65886],{"className":65887,"style":293},[119],[31,65889,65891,65894],{"className":65890},[66],[31,65892],{"className":65893,"style":326},[70],[31,65895,1977],{"className":65896},[75]," 的值就是 ",[45608,65899,65624],{},"，所以这个公式是 unsat。如果把它当成数学整数，",[31,65902,65904,65924],{"className":65903,"translate":35},[34],[31,65905,65907],{"className":65906},[39],[41,65908,65909],{"xmlns":43},[45,65910,65911,65921],{},[48,65912,65913,65915,65917,65919],{},[51,65914,53],{},[95,65916,223],{},[95,65918,97],{},[91,65920,205],{},[55,65922,65923],{"encoding":57},"x = -1",[31,65925,65927,65945],{"className":65926,"ariaHidden":62},[61],[31,65928,65930,65933,65936,65939,65942],{"className":65929},[66],[31,65931],{"className":65932,"style":71},[70],[31,65934,53],{"className":65935},[75,76],[31,65937],{"className":65938,"style":293},[119],[31,65940,223],{"className":65941},[297],[31,65943],{"className":65944,"style":293},[119],[31,65946,65948,65951,65954],{"className":65947},[66],[31,65949],{"className":65950,"style":112},[70],[31,65952,97],{"className":65953},[75],[31,65955,205],{"className":65956},[75]," 又不等于 ",[45608,65959,65624],{},"，结论就完全不同。",[7246,65962,65963],{},"选错理论，漏洞检测就会漏报或误报。",[63873,65965,60321],{"id":60321},[11,65967,65968,65969,2446,65971,65973],{},"数组理论的核心是 ",[45608,65970,54443],{},[45608,65972,55378],{}," 两条操作，以及前面提到的 read-over-write 公理。",[11,65975,65976,65977,65980,65981,65984],{},"求解器不会一次性展开所有下标，而是采用",[7246,65978,65979],{},"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 ",[45608,65982,65983],{},"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。",[11,65986,64022],{},[31,65988,65990],{"className":65989,"translate":35},[411],[31,65991,65993,66082],{"className":65992,"translate":35},[34],[31,65994,65996],{"className":65995},[39],[41,65997,65998],{"xmlns":43,"display":420},[45,65999,66000,66079],{},[48,66001,66002,66016,66018,66030,66032,66034,66036,66038,66040,66042,66044,66046,66048,66050,66052,66055,66057,66059,66061,66063,66065,66067,66069,66071,66073,66075,66077],{},[48,66003,66004,66006,66008,66010,66012,66014],{},[51,66005,14551],{"mathvariant":494},[51,66007,30042],{"mathvariant":494},[51,66009,54404],{"mathvariant":494},[51,66011,30042],{"mathvariant":494},[51,66013,1412],{"mathvariant":494},[51,66015,12271],{"mathvariant":494},[95,66017,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[48,66019,66020,66022,66024,66026,66028],{},[51,66021,14551],{"mathvariant":494},[51,66023,12271],{"mathvariant":494},[51,66025,2552],{"mathvariant":494},[51,66027,2549],{"mathvariant":494},[51,66029,30042],{"mathvariant":494},[95,66031,211],{"stretchy":210},[51,66033,5813],{},[95,66035,1210],{"separator":62},[51,66037,5391],{},[95,66039,1210],{"separator":62},[51,66041,5354],{},[95,66043,220],{"stretchy":210},[95,66045,1210],{"separator":62},[51,66047,2555],{},[95,66049,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,66051,223],{},[51,66053,66054],{},"w",[1430,66056,53035],{},[95,66058,53038],{},[1430,66060,53035],{},[51,66062,5391],{},[95,66064,223],{},[51,66066,2555],{},[1430,66068,53035],{},[95,66070,53038],{},[1430,66072,53035],{},[51,66074,66054],{},[95,66076,21271],{"mathvariant":494},[51,66078,5354],{},[55,66080,66081],{"encoding":57},"\\mathrm{select}\\bigl(\\mathrm{store}(a, i, v), j\\bigr) = w \\;\\wedge\\; i = j \\;\\wedge\\; w \\ne v",[31,66083,66085,66160,66185,66203,66228,66279],{"className":66084,"ariaHidden":62},[61],[31,66086,66088,66091,66097,66103,66109,66112,66115,66118,66121,66124,66127,66130,66133,66136,66139,66142,66145,66151,66154,66157],{"className":66087},[66],[31,66089],{"className":66090,"style":4988},[70],[31,66092,66094],{"className":66093},[75],[31,66095,54443],{"className":66096},[75,2745],[31,66098,66100],{"className":66099},[264],[31,66101,211],{"className":66102},[1307,1308],[31,66104,66106],{"className":66105},[75],[31,66107,55378],{"className":66108},[75,2745],[31,66110,211],{"className":66111},[264],[31,66113,5813],{"className":66114},[75,76],[31,66116,1210],{"className":66117},[1391],[31,66119],{"className":66120,"style":1467},[119],[31,66122,5391],{"className":66123},[75,76],[31,66125,1210],{"className":66126},[1391],[31,66128],{"className":66129,"style":1467},[119],[31,66131,5354],{"className":66132,"style":2666},[75,76],[31,66134,220],{"className":66135},[289],[31,66137,1210],{"className":66138},[1391],[31,66140],{"className":66141,"style":1467},[119],[31,66143,2555],{"className":66144,"style":5514},[75,76],[31,66146,66148],{"className":66147},[289],[31,66149,220],{"className":66150},[1307,1308],[31,66152],{"className":66153,"style":293},[119],[31,66155,223],{"className":66156},[297],[31,66158],{"className":66159,"style":293},[119],[31,66161,66163,66166,66170,66173,66176,66179,66182],{"className":66162},[66],[31,66164],{"className":66165,"style":64152},[70],[31,66167,66054],{"className":66168,"style":66169},[75,76],"margin-right:0.0269em;",[31,66171],{"className":66172,"style":293},[119],[31,66174],{"className":66175,"style":120},[119],[31,66177,53038],{"className":66178},[124],[31,66180],{"className":66181,"style":293},[119],[31,66183],{"className":66184,"style":120},[119],[31,66186,66188,66191,66194,66197,66200],{"className":66187},[66],[31,66189],{"className":66190,"style":59987},[70],[31,66192,5391],{"className":66193},[75,76],[31,66195],{"className":66196,"style":293},[119],[31,66198,223],{"className":66199},[297],[31,66201],{"className":66202,"style":293},[119],[31,66204,66206,66210,66213,66216,66219,66222,66225],{"className":66205},[66],[31,66207],{"className":66208,"style":66209},[70],"height:0.854em;vertical-align:-0.1944em;",[31,66211,2555],{"className":66212,"style":5514},[75,76],[31,66214],{"className":66215,"style":293},[119],[31,66217],{"className":66218,"style":120},[119],[31,66220,53038],{"className":66221},[124],[31,66223],{"className":66224,"style":293},[119],[31,66226],{"className":66227,"style":120},[119],[31,66229,66231,66234,66237,66240,66276],{"className":66230},[66],[31,66232],{"className":66233,"style":4202},[70],[31,66235,66054],{"className":66236,"style":66169},[75,76],[31,66238],{"className":66239,"style":293},[119],[31,66241,66243,66270,66273],{"className":66242},[297],[31,66244,66246],{"className":66245},[297],[31,66247,66249],{"className":66248},[75,21516],[31,66250,66252],{"className":66251},[21520],[31,66253,66255,66258,66267],{"className":66254},[21524],[31,66256],{"className":66257,"style":4202},[70],[31,66259,66261],{"className":66260},[21531],[31,66262,66264],{"className":66263},[75],[31,66265,21538],{"className":66266},[297],[31,66268],{"className":66269},[21542],[31,66271],{"className":66272},[119,21546],[31,66274,223],{"className":66275},[297],[31,66277],{"className":66278,"style":293},[119],[31,66280,66282,66285],{"className":66281},[66],[31,66283],{"className":66284,"style":71},[70],[31,66286,5354],{"className":66287,"style":2666},[75,76],[11,66289,66290,66291,66342,66343,66371,66372,58985,66423,66507],{},"由 ",[31,66292,66294,66312],{"className":66293,"translate":35},[34],[31,66295,66297],{"className":66296},[39],[41,66298,66299],{"xmlns":43},[45,66300,66301,66309],{},[48,66302,66303,66305,66307],{},[51,66304,5391],{},[95,66306,223],{},[51,66308,2555],{},[55,66310,66311],{"encoding":57},"i = j",[31,66313,66315,66333],{"className":66314,"ariaHidden":62},[61],[31,66316,66318,66321,66324,66327,66330],{"className":66317},[66],[31,66319],{"className":66320,"style":59987},[70],[31,66322,5391],{"className":66323},[75,76],[31,66325],{"className":66326,"style":293},[119],[31,66328,223],{"className":66329},[297],[31,66331],{"className":66332,"style":293},[119],[31,66334,66336,66339],{"className":66335},[66],[31,66337],{"className":66338,"style":66209},[70],[31,66340,2555],{"className":66341,"style":5514},[75,76]," 和 read-over-write 公理，左边等于 ",[31,66344,66346,66359],{"className":66345,"translate":35},[34],[31,66347,66349],{"className":66348},[39],[41,66350,66351],{"xmlns":43},[45,66352,66353,66357],{},[48,66354,66355],{},[51,66356,5354],{},[55,66358,5354],{"encoding":57},[31,66360,66362],{"className":66361,"ariaHidden":62},[61],[31,66363,66365,66368],{"className":66364},[66],[31,66366],{"className":66367,"style":71},[70],[31,66369,5354],{"className":66370,"style":2666},[75,76],"；于是 ",[31,66373,66375,66393],{"className":66374,"translate":35},[34],[31,66376,66378],{"className":66377},[39],[41,66379,66380],{"xmlns":43},[45,66381,66382,66390],{},[48,66383,66384,66386,66388],{},[51,66385,5354],{},[95,66387,223],{},[51,66389,66054],{},[55,66391,66392],{"encoding":57},"v = w",[31,66394,66396,66414],{"className":66395,"ariaHidden":62},[61],[31,66397,66399,66402,66405,66408,66411],{"className":66398},[66],[31,66400],{"className":66401,"style":71},[70],[31,66403,5354],{"className":66404,"style":2666},[75,76],[31,66406],{"className":66407,"style":293},[119],[31,66409,223],{"className":66410},[297],[31,66412],{"className":66413,"style":293},[119],[31,66415,66417,66420],{"className":66416},[66],[31,66418],{"className":66419,"style":71},[70],[31,66421,66054],{"className":66422,"style":66169},[75,76],[31,66424,66426,66444],{"className":66425,"translate":35},[34],[31,66427,66429],{"className":66428},[39],[41,66430,66431],{"xmlns":43},[45,66432,66433,66441],{},[48,66434,66435,66437,66439],{},[51,66436,66054],{},[95,66438,21271],{"mathvariant":494},[51,66440,5354],{},[55,66442,66443],{"encoding":57},"w \\ne v",[31,66445,66447,66498],{"className":66446,"ariaHidden":62},[61],[31,66448,66450,66453,66456,66459,66495],{"className":66449},[66],[31,66451],{"className":66452,"style":4202},[70],[31,66454,66054],{"className":66455,"style":66169},[75,76],[31,66457],{"className":66458,"style":293},[119],[31,66460,66462,66489,66492],{"className":66461},[297],[31,66463,66465],{"className":66464},[297],[31,66466,66468],{"className":66467},[75,21516],[31,66469,66471],{"className":66470},[21520],[31,66472,66474,66477,66486],{"className":66473},[21524],[31,66475],{"className":66476,"style":4202},[70],[31,66478,66480],{"className":66479},[21531],[31,66481,66483],{"className":66482},[75],[31,66484,21538],{"className":66485},[297],[31,66487],{"className":66488},[21542],[31,66490],{"className":66491},[119,21546],[31,66493,223],{"className":66494},[297],[31,66496],{"className":66497,"style":293},[119],[31,66499,66501,66504],{"className":66500},[66],[31,66502],{"className":66503,"style":71},[70],[31,66505,5354],{"className":66506,"style":2666},[75,76]," 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。",[63873,66509,66510],{"id":66510},"其他理论",[11,66512,66513,66514,66516],{},"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 ",[45608,66515,55719],{}," 后端。",[53200,66518,66520],{"id":66519},"_45-冲突学习","4.5 冲突学习",[11,66522,66523,66524,66527],{},"理论冲突不会只让当前搜索失败一次，它会变成一条",[7246,66525,66526],{},"理论引理","（theory lemma），永久加入布尔公式。",[11,66529,66530],{},"前面 4.2 节的例子：",[31,66532,66534],{"className":66533,"translate":35},[411],[31,66535,66537,66578],{"className":66536,"translate":35},[34],[31,66538,66540],{"className":66539},[39],[41,66541,66542],{"xmlns":43,"display":420},[45,66543,66544,66576],{},[48,66545,66546,66548,66550,66552,66554,66556,66558,66560,66562,66564,66566,66568,66570,66572,66574],{},[51,66547,53],{},[95,66549,16040],{},[91,66551,1977],{},[95,66553,1210],{"separator":62},[119,66555],{"width":20286},[51,66557,3003],{},[95,66559,16040],{},[91,66561,1977],{},[95,66563,1210],{"separator":62},[119,66565],{"width":20286},[51,66567,53],{},[95,66569,434],{},[51,66571,3003],{},[95,66573,9758],{},[91,66575,1977],{},[55,66577,63292],{"encoding":57},[31,66579,66581,66599,66629,66659,66677],{"className":66580,"ariaHidden":62},[61],[31,66582,66584,66587,66590,66593,66596],{"className":66583},[66],[31,66585],{"className":66586,"style":21113},[70],[31,66588,53],{"className":66589},[75,76],[31,66591],{"className":66592,"style":293},[119],[31,66594,16040],{"className":66595},[297],[31,66597],{"className":66598,"style":293},[119],[31,66600,66602,66605,66608,66611,66614,66617,66620,66623,66626],{"className":66601},[66],[31,66603],{"className":66604,"style":41982},[70],[31,66606,1977],{"className":66607},[75],[31,66609,1210],{"className":66610},[1391],[31,66612],{"className":66613,"style":20525},[119],[31,66615],{"className":66616,"style":1467},[119],[31,66618,3003],{"className":66619,"style":2666},[75,76],[31,66621],{"className":66622,"style":293},[119],[31,66624,16040],{"className":66625},[297],[31,66627],{"className":66628,"style":293},[119],[31,66630,66632,66635,66638,66641,66644,66647,66650,66653,66656],{"className":66631},[66],[31,66633],{"className":66634,"style":41982},[70],[31,66636,1977],{"className":66637},[75],[31,66639,1210],{"className":66640},[1391],[31,66642],{"className":66643,"style":20525},[119],[31,66645],{"className":66646,"style":1467},[119],[31,66648,53],{"className":66649},[75,76],[31,66651],{"className":66652,"style":120},[119],[31,66654,434],{"className":66655},[124],[31,66657],{"className":66658,"style":120},[119],[31,66660,66662,66665,66668,66671,66674],{"className":66661},[66],[31,66663],{"className":66664,"style":53083},[70],[31,66666,3003],{"className":66667,"style":2666},[75,76],[31,66669],{"className":66670,"style":293},[119],[31,66672,9758],{"className":66673},[297],[31,66675],{"className":66676,"style":293},[119],[31,66678,66680,66683],{"className":66679},[66],[31,66681],{"className":66682,"style":326},[70],[31,66684,1977],{"className":66685},[75],[11,66687,66688],{},"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：",[31,66690,66692],{"className":66691,"translate":35},[411],[31,66693,66695,66759],{"className":66694,"translate":35},[34],[31,66696,66698],{"className":66697},[39],[41,66699,66700],{"xmlns":43,"display":420},[45,66701,66702,66756],{},[48,66703,66704,66706,66708,66710,66712,66714,66716,66718,66720,66722,66724,66726,66728,66730,66732,66734,66736,66738,66740,66742,66744,66746,66748,66750,66752,66754],{},[51,66705,53658],{"mathvariant":494},[95,66707,211],{"stretchy":210},[51,66709,53],{},[95,66711,16040],{},[91,66713,1977],{},[95,66715,220],{"stretchy":210},[1430,66717,53035],{},[95,66719,53683],{},[1430,66721,53035],{},[51,66723,53658],{"mathvariant":494},[95,66725,211],{"stretchy":210},[51,66727,3003],{},[95,66729,16040],{},[91,66731,1977],{},[95,66733,220],{"stretchy":210},[1430,66735,53035],{},[95,66737,53683],{},[1430,66739,53035],{},[51,66741,53658],{"mathvariant":494},[95,66743,211],{"stretchy":210},[51,66745,53],{},[95,66747,434],{},[51,66749,3003],{},[95,66751,9758],{},[91,66753,1977],{},[95,66755,220],{"stretchy":210},[55,66757,66758],{"encoding":57},"\\neg(x > 0) \\;\\vee\\; \\neg(y > 0) \\;\\vee\\; \\neg(x + y \u003C 0)",[31,66760,66762,66786,66813,66837,66864,66888,66906],{"className":66761,"ariaHidden":62},[61],[31,66763,66765,66768,66771,66774,66777,66780,66783],{"className":66764},[66],[31,66766],{"className":66767,"style":260},[70],[31,66769,53658],{"className":66770},[75],[31,66772,211],{"className":66773},[264],[31,66775,53],{"className":66776},[75,76],[31,66778],{"className":66779,"style":293},[119],[31,66781,16040],{"className":66782},[297],[31,66784],{"className":66785,"style":293},[119],[31,66787,66789,66792,66795,66798,66801,66804,66807,66810],{"className":66788},[66],[31,66790],{"className":66791,"style":260},[70],[31,66793,1977],{"className":66794},[75],[31,66796,220],{"className":66797},[289],[31,66799],{"className":66800,"style":293},[119],[31,66802],{"className":66803,"style":120},[119],[31,66805,53683],{"className":66806},[124],[31,66808],{"className":66809,"style":293},[119],[31,66811],{"className":66812,"style":120},[119],[31,66814,66816,66819,66822,66825,66828,66831,66834],{"className":66815},[66],[31,66817],{"className":66818,"style":260},[70],[31,66820,53658],{"className":66821},[75],[31,66823,211],{"className":66824},[264],[31,66826,3003],{"className":66827,"style":2666},[75,76],[31,66829],{"className":66830,"style":293},[119],[31,66832,16040],{"className":66833},[297],[31,66835],{"className":66836,"style":293},[119],[31,66838,66840,66843,66846,66849,66852,66855,66858,66861],{"className":66839},[66],[31,66841],{"className":66842,"style":260},[70],[31,66844,1977],{"className":66845},[75],[31,66847,220],{"className":66848},[289],[31,66850],{"className":66851,"style":293},[119],[31,66853],{"className":66854,"style":120},[119],[31,66856,53683],{"className":66857},[124],[31,66859],{"className":66860,"style":293},[119],[31,66862],{"className":66863,"style":120},[119],[31,66865,66867,66870,66873,66876,66879,66882,66885],{"className":66866},[66],[31,66868],{"className":66869,"style":260},[70],[31,66871,53658],{"className":66872},[75],[31,66874,211],{"className":66875},[264],[31,66877,53],{"className":66878},[75,76],[31,66880],{"className":66881,"style":120},[119],[31,66883,434],{"className":66884},[124],[31,66886],{"className":66887,"style":120},[119],[31,66889,66891,66894,66897,66900,66903],{"className":66890},[66],[31,66892],{"className":66893,"style":53083},[70],[31,66895,3003],{"className":66896,"style":2666},[75,76],[31,66898],{"className":66899,"style":293},[119],[31,66901,9758],{"className":66902},[297],[31,66904],{"className":66905,"style":293},[119],[31,66907,66909,66912,66915],{"className":66908},[66],[31,66910],{"className":66911,"style":260},[70],[31,66913,1977],{"className":66914},[75],[31,66916,220],{"className":66917},[289],[11,66919,66920],{},"把它翻译回命题变量，就是：",[31,66922,66924],{"className":66923,"translate":35},[411],[31,66925,66927,66963],{"className":66926,"translate":35},[34],[31,66928,66930],{"className":66929},[39],[41,66931,66932],{"xmlns":43,"display":420},[45,66933,66934,66960],{},[48,66935,66936,66938,66940,66942,66944,66946,66948,66950,66952,66954,66956,66958],{},[51,66937,53658],{"mathvariant":494},[51,66939,11],{},[1430,66941,53035],{},[95,66943,53683],{},[1430,66945,53035],{},[51,66947,53658],{"mathvariant":494},[51,66949,27666],{},[1430,66951,53035],{},[95,66953,53683],{},[1430,66955,53035],{},[51,66957,53658],{"mathvariant":494},[51,66959,2549],{},[55,66961,66962],{"encoding":57},"\\neg p \\;\\vee\\; \\neg q \\;\\vee\\; \\neg r",[31,66964,66966,66993,67020],{"className":66965,"ariaHidden":62},[61],[31,66967,66969,66972,66975,66978,66981,66984,66987,66990],{"className":66968},[66],[31,66970],{"className":66971,"style":53828},[70],[31,66973,53658],{"className":66974},[75],[31,66976,11],{"className":66977},[75,76],[31,66979],{"className":66980,"style":293},[119],[31,66982],{"className":66983,"style":120},[119],[31,66985,53683],{"className":66986},[124],[31,66988],{"className":66989,"style":293},[119],[31,66991],{"className":66992,"style":120},[119],[31,66994,66996,66999,67002,67005,67008,67011,67014,67017],{"className":66995},[66],[31,66997],{"className":66998,"style":53828},[70],[31,67000,53658],{"className":67001},[75],[31,67003,27666],{"className":67004,"style":2666},[75,76],[31,67006],{"className":67007,"style":293},[119],[31,67009],{"className":67010,"style":120},[119],[31,67012,53683],{"className":67013},[124],[31,67015],{"className":67016,"style":293},[119],[31,67018],{"className":67019,"style":120},[119],[31,67021,67023,67026,67029],{"className":67022},[66],[31,67024],{"className":67025,"style":71},[70],[31,67027,53658],{"className":67028},[75],[31,67030,2549],{"className":67031,"style":19759},[75,76],[11,67033,67034,67035,67089,67090,2372],{},"SAT 层学到这条子句之后，就再也不会同时选 ",[31,67036,67038,67059],{"className":67037,"translate":35},[34],[31,67039,67041],{"className":67040},[39],[41,67042,67043],{"xmlns":43},[45,67044,67045,67057],{},[48,67046,67047,67049,67051,67053,67055],{},[51,67048,11],{},[95,67050,1210],{"separator":62},[51,67052,27666],{},[95,67054,1210],{"separator":62},[51,67056,2549],{},[55,67058,55556],{"encoding":57},[31,67060,67062],{"className":67061,"ariaHidden":62},[61],[31,67063,67065,67068,67071,67074,67077,67080,67083,67086],{"className":67064},[66],[31,67066],{"className":67067,"style":865},[70],[31,67069,11],{"className":67070},[75,76],[31,67072,1210],{"className":67073},[1391],[31,67075],{"className":67076,"style":1467},[119],[31,67078,27666],{"className":67079,"style":2666},[75,76],[31,67081,1210],{"className":67082},[1391],[31,67084],{"className":67085,"style":1467},[119],[31,67087,2549],{"className":67088,"style":19759},[75,76]," 都为真了。这一步是 DPLL(T) 效率的关键：",[7246,67091,67092],{},"理论层的「一次性检查」变成了布尔层的「永久知识」",[11,67094,67095],{},"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。",[53200,67097,67099],{"id":67098},"_46-完整例子","4.6 完整例子",[11,67101,67102],{},"还是用这个公式：",[31,67104,67106],{"className":67105,"translate":35},[411],[31,67107,67109,67162],{"className":67108,"translate":35},[34],[31,67110,67112],{"className":67111},[39],[41,67113,67114],{"xmlns":43,"display":420},[45,67115,67116,67160],{},[48,67117,67118,67120,67122,67124,67126,67128,67130,67132,67134,67136,67138,67140,67142,67144,67146,67148,67150,67152,67154,67156,67158],{},[51,67119,8059],{},[95,67121,223],{},[95,67123,211],{"stretchy":210},[51,67125,53],{},[95,67127,16040],{},[91,67129,1977],{},[95,67131,53683],{},[51,67133,3003],{},[95,67135,16040],{},[91,67137,1977],{},[95,67139,220],{"stretchy":210},[1430,67141,53035],{},[95,67143,53038],{},[1430,67145,53035],{},[95,67147,211],{"stretchy":210},[51,67149,53],{},[95,67151,434],{},[51,67153,3003],{},[95,67155,9758],{},[91,67157,1977],{},[95,67159,220],{"stretchy":210},[55,67161,62518],{"encoding":57},[31,67163,67165,67183,67204,67222,67240,67267,67288,67306],{"className":67164,"ariaHidden":62},[61],[31,67166,67168,67171,67174,67177,67180],{"className":67167},[66],[31,67169],{"className":67170,"style":865},[70],[31,67172,8059],{"className":67173},[75,76],[31,67175],{"className":67176,"style":293},[119],[31,67178,223],{"className":67179},[297],[31,67181],{"className":67182,"style":293},[119],[31,67184,67186,67189,67192,67195,67198,67201],{"className":67185},[66],[31,67187],{"className":67188,"style":260},[70],[31,67190,211],{"className":67191},[264],[31,67193,53],{"className":67194},[75,76],[31,67196],{"className":67197,"style":293},[119],[31,67199,16040],{"className":67200},[297],[31,67202],{"className":67203,"style":293},[119],[31,67205,67207,67210,67213,67216,67219],{"className":67206},[66],[31,67208],{"className":67209,"style":326},[70],[31,67211,1977],{"className":67212},[75],[31,67214],{"className":67215,"style":120},[119],[31,67217,53683],{"className":67218},[124],[31,67220],{"className":67221,"style":120},[119],[31,67223,67225,67228,67231,67234,67237],{"className":67224},[66],[31,67226],{"className":67227,"style":53083},[70],[31,67229,3003],{"className":67230,"style":2666},[75,76],[31,67232],{"className":67233,"style":293},[119],[31,67235,16040],{"className":67236},[297],[31,67238],{"className":67239,"style":293},[119],[31,67241,67243,67246,67249,67252,67255,67258,67261,67264],{"className":67242},[66],[31,67244],{"className":67245,"style":260},[70],[31,67247,1977],{"className":67248},[75],[31,67250,220],{"className":67251},[289],[31,67253],{"className":67254,"style":293},[119],[31,67256],{"className":67257,"style":120},[119],[31,67259,53038],{"className":67260},[124],[31,67262],{"className":67263,"style":293},[119],[31,67265],{"className":67266,"style":120},[119],[31,67268,67270,67273,67276,67279,67282,67285],{"className":67269},[66],[31,67271],{"className":67272,"style":260},[70],[31,67274,211],{"className":67275},[264],[31,67277,53],{"className":67278},[75,76],[31,67280],{"className":67281,"style":120},[119],[31,67283,434],{"className":67284},[124],[31,67286],{"className":67287,"style":120},[119],[31,67289,67291,67294,67297,67300,67303],{"className":67290},[66],[31,67292],{"className":67293,"style":53083},[70],[31,67295,3003],{"className":67296,"style":2666},[75,76],[31,67298],{"className":67299,"style":293},[119],[31,67301,9758],{"className":67302},[297],[31,67304],{"className":67305,"style":293},[119],[31,67307,67309,67312,67315],{"className":67308},[66],[31,67310],{"className":67311,"style":260},[70],[31,67313,1977],{"className":67314},[75],[31,67316,220],{"className":67317},[289],[11,67319,67320,67323,67324,67444,67445,67608,67609,67681,67682,67770],{},[7246,67321,67322],{},"第 1 轮","：SAT 层选择 ",[31,67325,67327,67360],{"className":67326,"translate":35},[34],[31,67328,67330],{"className":67329},[39],[41,67331,67332],{"xmlns":43},[45,67333,67334,67358],{},[48,67335,67336,67338,67340,67342,67344,67346,67348,67350,67352,67354,67356],{},[51,67337,11],{},[95,67339,223],{},[51,67341,20800],{"mathvariant":494},[95,67343,1210],{"separator":62},[51,67345,27666],{},[95,67347,223],{},[51,67349,20800],{"mathvariant":494},[95,67351,1210],{"separator":62},[51,67353,2549],{},[95,67355,223],{},[51,67357,20800],{"mathvariant":494},[55,67359,63152],{"encoding":57},[31,67361,67363,67381,67408,67435],{"className":67362,"ariaHidden":62},[61],[31,67364,67366,67369,67372,67375,67378],{"className":67365},[66],[31,67367],{"className":67368,"style":865},[70],[31,67370,11],{"className":67371},[75,76],[31,67373],{"className":67374,"style":293},[119],[31,67376,223],{"className":67377},[297],[31,67379],{"className":67380,"style":293},[119],[31,67382,67384,67387,67390,67393,67396,67399,67402,67405],{"className":67383},[66],[31,67385],{"className":67386,"style":4202},[70],[31,67388,20800],{"className":67389},[75],[31,67391,1210],{"className":67392},[1391],[31,67394],{"className":67395,"style":1467},[119],[31,67397,27666],{"className":67398,"style":2666},[75,76],[31,67400],{"className":67401,"style":293},[119],[31,67403,223],{"className":67404},[297],[31,67406],{"className":67407,"style":293},[119],[31,67409,67411,67414,67417,67420,67423,67426,67429,67432],{"className":67410},[66],[31,67412],{"className":67413,"style":4202},[70],[31,67415,20800],{"className":67416},[75],[31,67418,1210],{"className":67419},[1391],[31,67421],{"className":67422,"style":1467},[119],[31,67424,2549],{"className":67425,"style":19759},[75,76],[31,67427],{"className":67428,"style":293},[119],[31,67430,223],{"className":67431},[297],[31,67433],{"className":67434,"style":293},[119],[31,67436,67438,67441],{"className":67437},[66],[31,67439],{"className":67440,"style":1630},[70],[31,67442,20800],{"className":67443},[75],"。理论层检查 ",[31,67446,67448,67494],{"className":67447,"translate":35},[34],[31,67449,67451],{"className":67450},[39],[41,67452,67453],{"xmlns":43},[45,67454,67455,67491],{},[48,67456,67457,67459,67461,67463,67465,67467,67469,67471,67473,67475,67477,67479,67481,67483,67485,67487,67489],{},[95,67458,2512],{"stretchy":210},[51,67460,53],{},[95,67462,16040],{},[91,67464,1977],{},[95,67466,1210],{"separator":62},[1430,67468,1545],{},[51,67470,3003],{},[95,67472,16040],{},[91,67474,1977],{},[95,67476,1210],{"separator":62},[1430,67478,1545],{},[51,67480,53],{},[95,67482,434],{},[51,67484,3003],{},[95,67486,9758],{},[91,67488,1977],{},[95,67490,2604],{"stretchy":210},[55,67492,67493],{"encoding":57},"\\{x > 0,\\ y > 0,\\ x + y \u003C 0\\}",[31,67495,67497,67518,67548,67578,67596],{"className":67496,"ariaHidden":62},[61],[31,67498,67500,67503,67506,67509,67512,67515],{"className":67499},[66],[31,67501],{"className":67502,"style":260},[70],[31,67504,2512],{"className":67505},[264],[31,67507,53],{"className":67508},[75,76],[31,67510],{"className":67511,"style":293},[119],[31,67513,16040],{"className":67514},[297],[31,67516],{"className":67517,"style":293},[119],[31,67519,67521,67524,67527,67530,67533,67536,67539,67542,67545],{"className":67520},[66],[31,67522],{"className":67523,"style":41982},[70],[31,67525,1977],{"className":67526},[75],[31,67528,1210],{"className":67529},[1391],[31,67531,1545],{"className":67532},[119],[31,67534],{"className":67535,"style":1467},[119],[31,67537,3003],{"className":67538,"style":2666},[75,76],[31,67540],{"className":67541,"style":293},[119],[31,67543,16040],{"className":67544},[297],[31,67546],{"className":67547,"style":293},[119],[31,67549,67551,67554,67557,67560,67563,67566,67569,67572,67575],{"className":67550},[66],[31,67552],{"className":67553,"style":41982},[70],[31,67555,1977],{"className":67556},[75],[31,67558,1210],{"className":67559},[1391],[31,67561,1545],{"className":67562},[119],[31,67564],{"className":67565,"style":1467},[119],[31,67567,53],{"className":67568},[75,76],[31,67570],{"className":67571,"style":120},[119],[31,67573,434],{"className":67574},[124],[31,67576],{"className":67577,"style":120},[119],[31,67579,67581,67584,67587,67590,67593],{"className":67580},[66],[31,67582],{"className":67583,"style":53083},[70],[31,67585,3003],{"className":67586,"style":2666},[75,76],[31,67588],{"className":67589,"style":293},[119],[31,67591,9758],{"className":67592},[297],[31,67594],{"className":67595,"style":293},[119],[31,67597,67599,67602,67605],{"className":67598},[66],[31,67600],{"className":67601,"style":260},[70],[31,67603,1977],{"className":67604},[75],[31,67606,2604],{"className":67607},[289],"，发现 ",[31,67610,67612,67633],{"className":67611,"translate":35},[34],[31,67613,67615],{"className":67614},[39],[41,67616,67617],{"xmlns":43},[45,67618,67619,67631],{},[48,67620,67621,67623,67625,67627,67629],{},[51,67622,53],{},[95,67624,434],{},[51,67626,3003],{},[95,67628,16040],{},[91,67630,1977],{},[55,67632,63428],{"encoding":57},[31,67634,67636,67654,67672],{"className":67635,"ariaHidden":62},[61],[31,67637,67639,67642,67645,67648,67651],{"className":67638},[66],[31,67640],{"className":67641,"style":307},[70],[31,67643,53],{"className":67644},[75,76],[31,67646],{"className":67647,"style":120},[119],[31,67649,434],{"className":67650},[124],[31,67652],{"className":67653,"style":120},[119],[31,67655,67657,67660,67663,67666,67669],{"className":67656},[66],[31,67658],{"className":67659,"style":53083},[70],[31,67661,3003],{"className":67662,"style":2666},[75,76],[31,67664],{"className":67665,"style":293},[119],[31,67667,16040],{"className":67668},[297],[31,67670],{"className":67671,"style":293},[119],[31,67673,67675,67678],{"className":67674},[66],[31,67676],{"className":67677,"style":326},[70],[31,67679,1977],{"className":67680},[75],"，冲突。生成引理 ",[31,67683,67685,67713],{"className":67684,"translate":35},[34],[31,67686,67688],{"className":67687},[39],[41,67689,67690],{"xmlns":43},[45,67691,67692,67710],{},[48,67693,67694,67696,67698,67700,67702,67704,67706,67708],{},[51,67695,53658],{"mathvariant":494},[51,67697,11],{},[95,67699,53683],{},[51,67701,53658],{"mathvariant":494},[51,67703,27666],{},[95,67705,53683],{},[51,67707,53658],{"mathvariant":494},[51,67709,2549],{},[55,67711,67712],{"encoding":57},"\\neg p \\vee \\neg q \\vee \\neg r",[31,67714,67716,67737,67758],{"className":67715,"ariaHidden":62},[61],[31,67717,67719,67722,67725,67728,67731,67734],{"className":67718},[66],[31,67720],{"className":67721,"style":53828},[70],[31,67723,53658],{"className":67724},[75],[31,67726,11],{"className":67727},[75,76],[31,67729],{"className":67730,"style":120},[119],[31,67732,53683],{"className":67733},[124],[31,67735],{"className":67736,"style":120},[119],[31,67738,67740,67743,67746,67749,67752,67755],{"className":67739},[66],[31,67741],{"className":67742,"style":53828},[70],[31,67744,53658],{"className":67745},[75],[31,67747,27666],{"className":67748,"style":2666},[75,76],[31,67750],{"className":67751,"style":120},[119],[31,67753,53683],{"className":67754},[124],[31,67756],{"className":67757,"style":120},[119],[31,67759,67761,67764,67767],{"className":67760},[66],[31,67762],{"className":67763,"style":71},[70],[31,67765,53658],{"className":67766},[75],[31,67768,2549],{"className":67769,"style":19759},[75,76]," 并回跳。",[11,67772,67773,67776,67777,67898],{},[7246,67774,67775],{},"第 2 轮","：SAT 层在引理约束下重新搜索，可能选择 ",[31,67778,67780,67814],{"className":67779,"translate":35},[34],[31,67781,67783],{"className":67782},[39],[41,67784,67785],{"xmlns":43},[45,67786,67787,67811],{},[48,67788,67789,67791,67793,67795,67797,67799,67801,67803,67805,67807,67809],{},[51,67790,11],{},[95,67792,223],{},[51,67794,20800],{"mathvariant":494},[95,67796,1210],{"separator":62},[51,67798,27666],{},[95,67800,223],{},[51,67802,53641],{"mathvariant":494},[95,67804,1210],{"separator":62},[51,67806,2549],{},[95,67808,223],{},[51,67810,20800],{"mathvariant":494},[55,67812,67813],{"encoding":57},"p = \\top, q = \\bot, r = \\top",[31,67815,67817,67835,67862,67889],{"className":67816,"ariaHidden":62},[61],[31,67818,67820,67823,67826,67829,67832],{"className":67819},[66],[31,67821],{"className":67822,"style":865},[70],[31,67824,11],{"className":67825},[75,76],[31,67827],{"className":67828,"style":293},[119],[31,67830,223],{"className":67831},[297],[31,67833],{"className":67834,"style":293},[119],[31,67836,67838,67841,67844,67847,67850,67853,67856,67859],{"className":67837},[66],[31,67839],{"className":67840,"style":4202},[70],[31,67842,20800],{"className":67843},[75],[31,67845,1210],{"className":67846},[1391],[31,67848],{"className":67849,"style":1467},[119],[31,67851,27666],{"className":67852,"style":2666},[75,76],[31,67854],{"className":67855,"style":293},[119],[31,67857,223],{"className":67858},[297],[31,67860],{"className":67861,"style":293},[119],[31,67863,67865,67868,67871,67874,67877,67880,67883,67886],{"className":67864},[66],[31,67866],{"className":67867,"style":4202},[70],[31,67869,53641],{"className":67870},[75],[31,67872,1210],{"className":67873},[1391],[31,67875],{"className":67876,"style":1467},[119],[31,67878,2549],{"className":67879,"style":19759},[75,76],[31,67881],{"className":67882,"style":293},[119],[31,67884,223],{"className":67885},[297],[31,67887],{"className":67888,"style":293},[119],[31,67890,67892,67895],{"className":67891},[66],[31,67893],{"className":67894,"style":1630},[70],[31,67896,20800],{"className":67897},[75],"。理论层检查：",[31,67900,67902],{"className":67901,"translate":35},[411],[31,67903,67905,67947],{"className":67904,"translate":35},[34],[31,67906,67908],{"className":67907},[39],[41,67909,67910],{"xmlns":43,"display":420},[45,67911,67912,67944],{},[48,67913,67914,67916,67918,67920,67922,67924,67926,67928,67930,67932,67934,67936,67938,67940,67942],{},[51,67915,53],{},[95,67917,16040],{},[91,67919,1977],{},[95,67921,1210],{"separator":62},[119,67923],{"width":20286},[51,67925,3003],{},[95,67927,347],{},[91,67929,1977],{},[95,67931,1210],{"separator":62},[119,67933],{"width":20286},[51,67935,53],{},[95,67937,434],{},[51,67939,3003],{},[95,67941,9758],{},[91,67943,1977],{},[55,67945,67946],{"encoding":57},"x > 0, \\qquad y \\le 0, \\qquad x + y \u003C 0",[31,67948,67950,67968,67998,68028,68046],{"className":67949,"ariaHidden":62},[61],[31,67951,67953,67956,67959,67962,67965],{"className":67952},[66],[31,67954],{"className":67955,"style":21113},[70],[31,67957,53],{"className":67958},[75,76],[31,67960],{"className":67961,"style":293},[119],[31,67963,16040],{"className":67964},[297],[31,67966],{"className":67967,"style":293},[119],[31,67969,67971,67974,67977,67980,67983,67986,67989,67992,67995],{"className":67970},[66],[31,67972],{"className":67973,"style":41982},[70],[31,67975,1977],{"className":67976},[75],[31,67978,1210],{"className":67979},[1391],[31,67981],{"className":67982,"style":20525},[119],[31,67984],{"className":67985,"style":1467},[119],[31,67987,3003],{"className":67988,"style":2666},[75,76],[31,67990],{"className":67991,"style":293},[119],[31,67993,347],{"className":67994},[297],[31,67996],{"className":67997,"style":293},[119],[31,67999,68001,68004,68007,68010,68013,68016,68019,68022,68025],{"className":68000},[66],[31,68002],{"className":68003,"style":41982},[70],[31,68005,1977],{"className":68006},[75],[31,68008,1210],{"className":68009},[1391],[31,68011],{"className":68012,"style":20525},[119],[31,68014],{"className":68015,"style":1467},[119],[31,68017,53],{"className":68018},[75,76],[31,68020],{"className":68021,"style":120},[119],[31,68023,434],{"className":68024},[124],[31,68026],{"className":68027,"style":120},[119],[31,68029,68031,68034,68037,68040,68043],{"className":68030},[66],[31,68032],{"className":68033,"style":53083},[70],[31,68035,3003],{"className":68036,"style":2666},[75,76],[31,68038],{"className":68039,"style":293},[119],[31,68041,9758],{"className":68042},[297],[31,68044],{"className":68045,"style":293},[119],[31,68047,68049,68052],{"className":68048},[66],[31,68050],{"className":68051,"style":326},[70],[31,68053,1977],{"className":68054},[75],[11,68056,3501,68057,68148],{},[31,68058,68060,68088],{"className":68059,"translate":35},[34],[31,68061,68063],{"className":68062},[39],[41,68064,68065],{"xmlns":43},[45,68066,68067,68085],{},[48,68068,68069,68071,68073,68075,68077,68079,68081,68083],{},[51,68070,53],{},[95,68072,223],{},[91,68074,205],{},[95,68076,1210],{"separator":62},[51,68078,3003],{},[95,68080,223],{},[95,68082,97],{},[91,68084,152],{},[55,68086,68087],{"encoding":57},"x = 1, y = -2",[31,68089,68091,68109,68136],{"className":68090,"ariaHidden":62},[61],[31,68092,68094,68097,68100,68103,68106],{"className":68093},[66],[31,68095],{"className":68096,"style":71},[70],[31,68098,53],{"className":68099},[75,76],[31,68101],{"className":68102,"style":293},[119],[31,68104,223],{"className":68105},[297],[31,68107],{"className":68108,"style":293},[119],[31,68110,68112,68115,68118,68121,68124,68127,68130,68133],{"className":68111},[66],[31,68113],{"className":68114,"style":41982},[70],[31,68116,205],{"className":68117},[75],[31,68119,1210],{"className":68120},[1391],[31,68122],{"className":68123,"style":1467},[119],[31,68125,3003],{"className":68126,"style":2666},[75,76],[31,68128],{"className":68129,"style":293},[119],[31,68131,223],{"className":68132},[297],[31,68134],{"className":68135,"style":293},[119],[31,68137,68139,68142,68145],{"className":68138},[66],[31,68140],{"className":68141,"style":112},[70],[31,68143,97],{"className":68144},[75],[31,68146,152],{"className":68147},[75],"，三条同时成立。理论层返回一致，求解器输出：",[31,68150,68152],{"className":68151,"translate":35},[411],[31,68153,68155,68192],{"className":68154,"translate":35},[34],[31,68156,68158],{"className":68157},[39],[41,68159,68160],{"xmlns":43,"display":420},[45,68161,68162,68189],{},[48,68163,68164,68167,68169,68171,68173,68175,68177,68179,68181,68183,68185,68187],{},[1430,68165,53187],{"mathvariant":68166},"bold",[95,68168,1210],{"separator":62},[119,68170],{"width":20286},[51,68172,53],{},[95,68174,223],{},[91,68176,205],{},[95,68178,1210],{"separator":62},[1430,68180,1545],{},[51,68182,3003],{},[95,68184,223],{},[95,68186,97],{},[91,68188,152],{},[55,68190,68191],{"encoding":57},"\\textbf{sat}, \\qquad x = 1,\\ y = -2",[31,68193,68195,68230,68260],{"className":68194,"ariaHidden":62},[61],[31,68196,68198,68202,68209,68212,68215,68218,68221,68224,68227],{"className":68197},[66],[31,68199],{"className":68200,"style":68201},[70],"height:0.8294em;vertical-align:-0.1944em;",[31,68203,68205],{"className":68204},[75,1793],[31,68206,53187],{"className":68207},[75,68208],"textbf",[31,68210,1210],{"className":68211},[1391],[31,68213],{"className":68214,"style":20525},[119],[31,68216],{"className":68217,"style":1467},[119],[31,68219,53],{"className":68220},[75,76],[31,68222],{"className":68223,"style":293},[119],[31,68225,223],{"className":68226},[297],[31,68228],{"className":68229,"style":293},[119],[31,68231,68233,68236,68239,68242,68245,68248,68251,68254,68257],{"className":68232},[66],[31,68234],{"className":68235,"style":41982},[70],[31,68237,205],{"className":68238},[75],[31,68240,1210],{"className":68241},[1391],[31,68243,1545],{"className":68244},[119],[31,68246],{"className":68247,"style":1467},[119],[31,68249,3003],{"className":68250,"style":2666},[75,76],[31,68252],{"className":68253,"style":293},[119],[31,68255,223],{"className":68256},[297],[31,68258],{"className":68259,"style":293},[119],[31,68261,68263,68266,68269],{"className":68262},[66],[31,68264],{"className":68265,"style":112},[70],[31,68267,97],{"className":68268},[75],[31,68270,152],{"className":68271},[75],[11,68273,68274],{},"这就是一个完整的 DPLL(T) 循环：一次冲突、一次学习、一次成功。",[11,68276,68277,68278,68281],{},"在真实求解器里，理论层还会做",[7246,68279,68280],{},"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。",[11,68283,68284],{},"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：",[55164,68286,68287,68293,68298,68304,68310,68316],{},[55167,68288,68289,68292],{},[7246,68290,68291],{},"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。",[55167,68294,68295,68297],{},[7246,68296,68280],{},"：理论层主动推导文字，而不是被动等待完整赋值。",[55167,68299,68300,68303],{},[7246,68301,68302],{},"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。",[55167,68305,68306,68309],{},[7246,68307,68308],{},"子句删除","：学习子句太多会拖慢传播，需要定期清理。",[55167,68311,68312,68315],{},[7246,68313,68314],{},"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。",[55167,68317,68318,68321],{},[7246,68319,68320],{},"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。",[53200,68323,68325],{"id":68324},"_47-复杂度边界","4.7 复杂度边界",[11,68327,68328,68329,2372],{},"必须强调：",[7246,68330,68331],{},"可判定不等于可解",[55164,68333,68334,68337,68346,68351,68355],{},[55167,68335,68336],{},"SAT 是 NP-complete；",[55167,68338,68339,15282,68341,15282,68343,68345],{},[45608,68340,55719],{},[45608,68342,55635],{},[45608,68344,55621],{}," 也都是 NP-complete；",[55167,68347,68348,68350],{},[45608,68349,55631],{}," 虽然是多项式时间，但实际常数很大；",[55167,68352,68353,60105],{},[45608,68354,55707],{},[55167,68356,68357,68359],{},[45608,68358,55710],{}," 直接不可判定。",[11,68361,68362],{},"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」。",[11,68364,68365],{},"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。",[18,68367,68369],{"id":68368},"五漏洞检测","五、漏洞检测",[53200,68371,68373],{"id":68372},"_51-问题转化","5.1 问题转化",[11,68375,68376],{},"软件漏洞检测的核心问题可以写成一句话：",[68378,68379,68380],"blockquote",{},[11,68381,68382],{},"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？",[11,68384,68385],{},"形式化之后，它就是一个存在性查询：",[31,68387,68389],{"className":68388,"translate":35},[411],[31,68390,68392,68506],{"className":68391,"translate":35},[34],[31,68393,68395],{"className":68394},[39],[41,68396,68397],{"xmlns":43,"display":420},[45,68398,68399,68503],{},[48,68400,68401,68403,68405,68417,68419,68421,68454,68456,68458,68460],{},[51,68402,53262],{"mathvariant":494},[1430,68404,1432],{},[48,68406,68407,68409,68411,68413,68415],{},[51,68408,5391],{},[51,68410,14559],{},[51,68412,11],{},[51,68414,5324],{},[51,68416,12271],{},[51,68418,495],{"mathvariant":494},[119,68420],{"width":1584},[1531,68422,68423,68451],{},[1531,68424,68425,68449],{},[48,68426,68427,68433,68435,68447],{},[48,68428,68429,68431],{},[51,68430,2521],{"mathvariant":494},[51,68432,8101],{"mathvariant":494},[95,68434,211],{"stretchy":210},[48,68436,68437,68439,68441,68443,68445],{},[51,68438,5391],{},[51,68440,14559],{},[51,68442,11],{},[51,68444,5324],{},[51,68446,12271],{},[95,68448,220],{"stretchy":210},[95,68450,38662],{"stretchy":62},[1430,68452,68453],{},"路径条件",[1430,68455,53035],{},[95,68457,53038],{},[1430,68459,53035],{},[1531,68461,68462,68500],{},[1531,68463,68464,68498],{},[48,68465,68466,68482,68484,68496],{},[48,68467,68468,68470,68472,68474,68476,68478,68480],{},[51,68469,43481],{"mathvariant":494},[51,68471,5391],{"mathvariant":494},[51,68473,2552],{"mathvariant":494},[51,68475,54404],{"mathvariant":494},[51,68477,5813],{"mathvariant":494},[51,68479,12271],{"mathvariant":494},[51,68481,30042],{"mathvariant":494},[95,68483,211],{"stretchy":210},[48,68485,68486,68488,68490,68492,68494],{},[51,68487,5391],{},[51,68489,14559],{},[51,68491,11],{},[51,68493,5324],{},[51,68495,12271],{},[95,68497,220],{"stretchy":210},[95,68499,38662],{"stretchy":62},[1430,68501,68502],{},"安全性质取反",[55,68504,68505],{"encoding":57},"\\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}}",[31,68507,68509,68679],{"className":68508,"ariaHidden":62},[61],[31,68510,68512,68516,68519,68522,68530,68533,68536,68539,68664,68667,68670,68673,68676],{"className":68511},[66],[31,68513],{"className":68514,"style":68515},[70],"height:2.3263em;vertical-align:-1.5763em;",[31,68517,53262],{"className":68518},[75],[31,68520],{"className":68521,"style":1467},[119],[31,68523,68525],{"className":68524},[75],[31,68526,68529],{"className":68527},[75,68528],"mathit","input",[31,68531,495],{"className":68532},[75],[31,68534],{"className":68535,"style":1789},[119],[31,68537],{"className":68538,"style":1467},[119],[31,68540,68542],{"className":68541},[3860,1531],[31,68543,68545,68656],{"className":68544},[1043,1044],[31,68546,68548,68653],{"className":68547},[1048],[31,68549,68551,68568],{"className":68550,"style":54628},[1052],[31,68552,68553,68556],{"style":39515},[31,68554],{"className":68555,"style":1637},[1060],[31,68557,68559],{"className":68558},[1065,1066,1067,1068],[31,68560,68562],{"className":68561},[75,1068],[31,68563,68565],{"className":68564},[75,1793,1068],[31,68566,68453],{"className":68567},[75,12601,1068],[31,68569,68570,68573],{"style":1649},[31,68571],{"className":68572,"style":1637},[1060],[31,68574,68576],{"className":68575},[3860,1531],[31,68577,68579,68645],{"className":68578},[1043,1044],[31,68580,68582,68642],{"className":68581},[1048],[31,68583,68585,68615],{"className":68584,"style":54628},[1052],[31,68586,68588,68591],{"className":68587,"style":39551},[26550],[31,68589],{"className":68590,"style":1637},[1060],[31,68592,68594,68601,68608],{"className":68593,"style":39559},[39558],[31,68595,68597],{"className":68596,"style":39564},[39563],[13341,68598,68599],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":26599},[13347,68600],{"d":39571},[31,68602,68604],{"className":68603,"style":39564},[39575],[13341,68605,68606],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":39578},[13347,68607],{"d":39581},[31,68609,68611],{"className":68610,"style":39564},[39585],[13341,68612,68613],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":39588},[13347,68614],{"d":39591},[31,68616,68617,68620],{"style":1649},[31,68618],{"className":68619,"style":1637},[1060],[31,68621,68623,68630,68633,68639],{"className":68622},[75],[31,68624,68626],{"className":68625},[75],[31,68627,68629],{"className":68628},[75,2745],"PC",[31,68631,211],{"className":68632},[264],[31,68634,68636],{"className":68635},[75],[31,68637,68529],{"className":68638},[75,68528],[31,68640,220],{"className":68641},[289],[31,68643,1076],{"className":68644},[1075],[31,68646,68648],{"className":68647},[1048],[31,68649,68651],{"className":68650,"style":39973},[1052],[31,68652],{},[31,68654,1076],{"className":68655},[1075],[31,68657,68659],{"className":68658},[1048],[31,68660,68662],{"className":68661,"style":39985},[1052],[31,68663],{},[31,68665],{"className":68666,"style":293},[119],[31,68668],{"className":68669,"style":120},[119],[31,68671,53038],{"className":68672},[124],[31,68674],{"className":68675,"style":293},[119],[31,68677],{"className":68678,"style":120},[119],[31,68680,68682,68685],{"className":68681},[66],[31,68683],{"className":68684,"style":68515},[70],[31,68686,68688],{"className":68687},[3860,1531],[31,68689,68691,68802],{"className":68690},[1043,1044],[31,68692,68694,68799],{"className":68693},[1048],[31,68695,68697,68714],{"className":68696,"style":54628},[1052],[31,68698,68699,68702],{"style":39515},[31,68700],{"className":68701,"style":1637},[1060],[31,68703,68705],{"className":68704},[1065,1066,1067,1068],[31,68706,68708],{"className":68707},[75,1068],[31,68709,68711],{"className":68710},[75,1793,1068],[31,68712,68502],{"className":68713},[75,12601,1068],[31,68715,68716,68719],{"style":1649},[31,68717],{"className":68718,"style":1637},[1060],[31,68720,68722],{"className":68721},[3860,1531],[31,68723,68725,68791],{"className":68724},[1043,1044],[31,68726,68728,68788],{"className":68727},[1048],[31,68729,68731,68761],{"className":68730,"style":54628},[1052],[31,68732,68734,68737],{"className":68733,"style":39551},[26550],[31,68735],{"className":68736,"style":1637},[1060],[31,68738,68740,68747,68754],{"className":68739,"style":39559},[39558],[31,68741,68743],{"className":68742,"style":39564},[39563],[13341,68744,68745],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":26599},[13347,68746],{"d":39571},[31,68748,68750],{"className":68749,"style":39564},[39575],[13341,68751,68752],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":39578},[13347,68753],{"d":39581},[31,68755,68757],{"className":68756,"style":39564},[39585],[13341,68758,68759],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":39588},[13347,68760],{"d":39591},[31,68762,68763,68766],{"style":1649},[31,68764],{"className":68765,"style":1637},[1060],[31,68767,68769,68776,68779,68785],{"className":68768},[75],[31,68770,68772],{"className":68771},[75],[31,68773,68775],{"className":68774},[75,2745],"Violate",[31,68777,211],{"className":68778},[264],[31,68780,68782],{"className":68781},[75],[31,68783,68529],{"className":68784},[75,68528],[31,68786,220],{"className":68787},[289],[31,68789,1076],{"className":68790},[1075],[31,68792,68794],{"className":68793},[1048],[31,68795,68797],{"className":68796,"style":39973},[1052],[31,68798],{},[31,68800,1076],{"className":68801},[1075],[31,68803,68805],{"className":68804},[1048],[31,68806,68808],{"className":68807,"style":39985},[1052],[31,68809],{},[55164,68811,68812,68849],{},[55167,68813,68814,68848],{},[31,68815,68817,68833],{"className":68816,"translate":35},[34],[31,68818,68820],{"className":68819},[39],[41,68821,68822],{"xmlns":43},[45,68823,68824,68830],{},[48,68825,68826,68828],{},[51,68827,2521],{"mathvariant":494},[51,68829,8101],{"mathvariant":494},[55,68831,68832],{"encoding":57},"\\mathrm{PC}",[31,68834,68836],{"className":68835,"ariaHidden":62},[61],[31,68837,68839,68842],{"className":68838},[66],[31,68840],{"className":68841,"style":945},[70],[31,68843,68845],{"className":68844},[75],[31,68846,68629],{"className":68847},[75,2745]," 是路径条件，描述「走到这里需要满足哪些约束」；",[55167,68850,68851,68895],{},[31,68852,68854,68880],{"className":68853,"translate":35},[34],[31,68855,68857],{"className":68856},[39],[41,68858,68859],{"xmlns":43},[45,68860,68861,68877],{},[48,68862,68863,68865,68867,68869,68871,68873,68875],{},[51,68864,43481],{"mathvariant":494},[51,68866,5391],{"mathvariant":494},[51,68868,2552],{"mathvariant":494},[51,68870,54404],{"mathvariant":494},[51,68872,5813],{"mathvariant":494},[51,68874,12271],{"mathvariant":494},[51,68876,30042],{"mathvariant":494},[55,68878,68879],{"encoding":57},"\\mathrm{Violate}",[31,68881,68883],{"className":68882,"ariaHidden":62},[61],[31,68884,68886,68889],{"className":68885},[66],[31,68887],{"className":68888,"style":1630},[70],[31,68890,68892],{"className":68891},[75],[31,68893,68775],{"className":68894},[75,2745]," 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」。",[11,68897,68898,68899,68901,68902,68904],{},"如果这个公式 ",[7246,68900,53187],{},"，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 ",[7246,68903,59102],{},"，说明在这条路径上不可能违反性质，可以安全剪枝。",[11,68906,68907,68908,2372],{},"这个转化是整个应用的基石：",[7246,68909,68910],{},"「找漏洞」变成了「解约束」",[53200,68912,68914],{"id":68913},"_52-缓冲区溢出","5.2 缓冲区溢出",[11,68916,68917],{},"先看一个最直观的例子：",[52853,68919,68921],{"className":52855,"code":68920,"language":1412,"meta":52812,"style":52812},"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",[45608,68922,68923,68949,68964,68988,68999,69003],{"__ignoreMap":52812},[31,68924,68925,68927,68930,68932,68935,68938,68940,68942,68944,68947],{"class":52861,"line":52862},[31,68926,52866],{"class":52865},[31,68928,68929],{"class":52869}," process",[31,68931,211],{"class":52873},[31,68933,68934],{"class":52865},"char",[31,68936,68937],{"class":52865}," *",[31,68939,68529],{"class":52879},[31,68941,52883],{"class":52873},[31,68943,52876],{"class":52865},[31,68945,68946],{"class":52879}," len",[31,68948,52891],{"class":52873},[31,68950,68951,68954,68957,68959,68961],{"class":52861,"line":52813},[31,68952,68953],{"class":52865},"    char",[31,68955,68956],{"class":52879}," buf",[31,68958,15175],{"class":52873},[31,68960,19641],{"class":52904},[31,68962,68963],{"class":52873},"];\n",[31,68965,68966,68968,68971,68973,68976,68978,68981,68983,68986],{"class":52861,"line":52920},[31,68967,52943],{"class":52865},[31,68969,68970],{"class":52873}," (len ",[31,68972,16040],{"class":52865},[31,68974,68975],{"class":52904}," 4",[31,68977,52959],{"class":52865},[31,68979,68980],{"class":52873}," len ",[31,68982,9758],{"class":52865},[31,68984,68985],{"class":52904}," 12",[31,68987,52891],{"class":52873},[31,68989,68990,68993,68996],{"class":52861,"line":52940},[31,68991,68992],{"class":52869},"        memcpy",[31,68994,68995],{"class":52873},"(buf, input, len);",[31,68997,68998],{"class":52980},"   \u002F\u002F 可疑点\n",[31,69000,69001],{"class":52861,"line":52977},[31,69002,52987],{"class":52873},[31,69004,69005],{"class":52861,"line":52984},[31,69006,52993],{"class":52873},[11,69008,69009,69012,69013,69016,69017,69020,69021,69023],{},[45608,69010,69011],{},"buf"," 只有 8 字节，而 ",[45608,69014,69015],{},"memcpy"," 拷贝 ",[45608,69018,69019],{},"len"," 字节。如果 ",[45608,69022,69019],{}," 能大于 8，就会发生栈溢出。",[11,69025,69026,69027,2446,69029,69031,69032,69034],{},"符号执行引擎把 ",[45608,69028,68529],{},[45608,69030,69019],{}," 当作符号变量，沿路径走到 ",[45608,69033,69015],{}," 时，收集到的路径条件是：",[31,69036,69038],{"className":69037,"translate":35},[411],[31,69039,69041,69080],{"className":69040,"translate":35},[34],[31,69042,69044],{"className":69043},[39],[41,69045,69046],{"xmlns":43,"display":420},[45,69047,69048,69077],{},[48,69049,69050,69052,69054,69056,69058,69060,69062,69064,69066,69068,69070,69072,69074],{},[51,69051,54404],{},[51,69053,30042],{},[51,69055,14559],{},[95,69057,16040],{},[91,69059,15416],{},[1430,69061,53035],{},[95,69063,53038],{},[1430,69065,53035],{},[51,69067,54404],{},[51,69069,30042],{},[51,69071,14559],{},[95,69073,9758],{},[91,69075,69076],{},"12",[55,69078,69079],{"encoding":57},"len > 4 \\;\\wedge\\; len \u003C 12",[31,69081,69083,69109,69133,69157],{"className":69082,"ariaHidden":62},[61],[31,69084,69086,69090,69094,69097,69100,69103,69106],{"className":69085},[66],[31,69087],{"className":69088,"style":69089},[70],"height:0.7335em;vertical-align:-0.0391em;",[31,69091,54404],{"className":69092,"style":69093},[75,76],"margin-right:0.0197em;",[31,69095,30042],{"className":69096},[75,76],[31,69098,14559],{"className":69099},[75,76],[31,69101],{"className":69102,"style":293},[119],[31,69104,16040],{"className":69105},[297],[31,69107],{"className":69108,"style":293},[119],[31,69110,69112,69115,69118,69121,69124,69127,69130],{"className":69111},[66],[31,69113],{"className":69114,"style":326},[70],[31,69116,15416],{"className":69117},[75],[31,69119],{"className":69120,"style":293},[119],[31,69122],{"className":69123,"style":120},[119],[31,69125,53038],{"className":69126},[124],[31,69128],{"className":69129,"style":293},[119],[31,69131],{"className":69132,"style":120},[119],[31,69134,69136,69139,69142,69145,69148,69151,69154],{"className":69135},[66],[31,69137],{"className":69138,"style":69089},[70],[31,69140,54404],{"className":69141,"style":69093},[75,76],[31,69143,30042],{"className":69144},[75,76],[31,69146,14559],{"className":69147},[75,76],[31,69149],{"className":69150,"style":293},[119],[31,69152,9758],{"className":69153},[297],[31,69155],{"className":69156,"style":293},[119],[31,69158,69160,69163],{"className":69159},[66],[31,69161],{"className":69162,"style":326},[70],[31,69164,69076],{"className":69165},[75],[11,69167,69168],{},"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」：",[31,69170,69172],{"className":69171,"translate":35},[411],[31,69173,69175,69229],{"className":69174,"translate":35},[34],[31,69176,69178],{"className":69177},[39],[41,69179,69180],{"xmlns":43,"display":420},[45,69181,69182,69226],{},[48,69183,69184,69186,69188,69190,69192,69194,69196,69198,69200,69202,69204,69206,69208,69210,69212,69214,69216,69218,69220,69222,69224],{},[51,69185,54404],{},[51,69187,30042],{},[51,69189,14559],{},[95,69191,16040],{},[91,69193,15416],{},[1430,69195,53035],{},[95,69197,53038],{},[1430,69199,53035],{},[51,69201,54404],{},[51,69203,30042],{},[51,69205,14559],{},[95,69207,9758],{},[91,69209,69076],{},[1430,69211,53035],{},[95,69213,53038],{},[1430,69215,53035],{},[51,69217,54404],{},[51,69219,30042],{},[51,69221,14559],{},[95,69223,16040],{},[91,69225,19641],{},[55,69227,69228],{"encoding":57},"len > 4 \\;\\wedge\\; len \u003C 12 \\;\\wedge\\; len > 8",[31,69230,69232,69256,69280,69304,69328,69352],{"className":69231,"ariaHidden":62},[61],[31,69233,69235,69238,69241,69244,69247,69250,69253],{"className":69234},[66],[31,69236],{"className":69237,"style":69089},[70],[31,69239,54404],{"className":69240,"style":69093},[75,76],[31,69242,30042],{"className":69243},[75,76],[31,69245,14559],{"className":69246},[75,76],[31,69248],{"className":69249,"style":293},[119],[31,69251,16040],{"className":69252},[297],[31,69254],{"className":69255,"style":293},[119],[31,69257,69259,69262,69265,69268,69271,69274,69277],{"className":69258},[66],[31,69260],{"className":69261,"style":326},[70],[31,69263,15416],{"className":69264},[75],[31,69266],{"className":69267,"style":293},[119],[31,69269],{"className":69270,"style":120},[119],[31,69272,53038],{"className":69273},[124],[31,69275],{"className":69276,"style":293},[119],[31,69278],{"className":69279,"style":120},[119],[31,69281,69283,69286,69289,69292,69295,69298,69301],{"className":69282},[66],[31,69284],{"className":69285,"style":69089},[70],[31,69287,54404],{"className":69288,"style":69093},[75,76],[31,69290,30042],{"className":69291},[75,76],[31,69293,14559],{"className":69294},[75,76],[31,69296],{"className":69297,"style":293},[119],[31,69299,9758],{"className":69300},[297],[31,69302],{"className":69303,"style":293},[119],[31,69305,69307,69310,69313,69316,69319,69322,69325],{"className":69306},[66],[31,69308],{"className":69309,"style":326},[70],[31,69311,69076],{"className":69312},[75],[31,69314],{"className":69315,"style":293},[119],[31,69317],{"className":69318,"style":120},[119],[31,69320,53038],{"className":69321},[124],[31,69323],{"className":69324,"style":293},[119],[31,69326],{"className":69327,"style":120},[119],[31,69329,69331,69334,69337,69340,69343,69346,69349],{"className":69330},[66],[31,69332],{"className":69333,"style":69089},[70],[31,69335,54404],{"className":69336,"style":69093},[75,76],[31,69338,30042],{"className":69339},[75,76],[31,69341,14559],{"className":69342},[75,76],[31,69344],{"className":69345,"style":293},[119],[31,69347,16040],{"className":69348},[297],[31,69350],{"className":69351,"style":293},[119],[31,69353,69355,69358],{"className":69354},[66],[31,69356],{"className":69357,"style":326},[70],[31,69359,19641],{"className":69360},[75],[11,69362,69363],{},"写成 SMT-LIB：",[52853,69365,69369],{"className":69366,"code":69367,"language":69368,"meta":52812,"style":52812},"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",[45608,69370,69371,69376,69381,69386,69391],{"__ignoreMap":52812},[31,69372,69373],{"class":52861,"line":52862},[31,69374,69375],{},"(set-logic QF_LIA)\n",[31,69377,69378],{"class":52861,"line":52813},[31,69379,69380],{},"(declare-const len Int)\n",[31,69382,69383],{"class":52861,"line":52920},[31,69384,69385],{},"(assert (and (> len 4) (\u003C len 12) (> len 8)))\n",[31,69387,69388],{"class":52861,"line":52940},[31,69389,69390],{},"(check-sat)\n",[31,69392,69393],{"class":52861,"line":52977},[31,69394,69395],{},"(get-model)\n",[11,69397,69398],{},"求解器返回：",[52853,69400,69403],{"className":69401,"code":69402,"language":1793},[63799],"sat\n(model\n  (define-fun len () Int\n    9)\n)\n",[45608,69404,69402],{"__ignoreMap":52812},[11,69406,69407,69410,69411,69485,69486,69537,69538,69540],{},[45608,69408,69409],{},"len = 9"," 是一个合法解：它满足 ",[31,69412,69414,69437],{"className":69413,"translate":35},[34],[31,69415,69417],{"className":69416},[39],[41,69418,69419],{"xmlns":43},[45,69420,69421,69434],{},[48,69422,69423,69425,69427,69430,69432],{},[91,69424,15416],{},[95,69426,9758],{},[91,69428,69429],{},"9",[95,69431,9758],{},[91,69433,69076],{},[55,69435,69436],{"encoding":57},"4 \u003C 9 \u003C 12",[31,69438,69440,69458,69476],{"className":69439,"ariaHidden":62},[61],[31,69441,69443,69446,69449,69452,69455],{"className":69442},[66],[31,69444],{"className":69445,"style":62056},[70],[31,69447,15416],{"className":69448},[75],[31,69450],{"className":69451,"style":293},[119],[31,69453,9758],{"className":69454},[297],[31,69456],{"className":69457,"style":293},[119],[31,69459,69461,69464,69467,69470,69473],{"className":69460},[66],[31,69462],{"className":69463,"style":62056},[70],[31,69465,69429],{"className":69466},[75],[31,69468],{"className":69469,"style":293},[119],[31,69471,9758],{"className":69472},[297],[31,69474],{"className":69475,"style":293},[119],[31,69477,69479,69482],{"className":69478},[66],[31,69480],{"className":69481,"style":326},[70],[31,69483,69076],{"className":69484},[75],"，同时 ",[31,69487,69489,69507],{"className":69488,"translate":35},[34],[31,69490,69492],{"className":69491},[39],[41,69493,69494],{"xmlns":43},[45,69495,69496,69504],{},[48,69497,69498,69500,69502],{},[91,69499,69429],{},[95,69501,16040],{},[91,69503,19641],{},[55,69505,69506],{"encoding":57},"9 > 8",[31,69508,69510,69528],{"className":69509,"ariaHidden":62},[61],[31,69511,69513,69516,69519,69522,69525],{"className":69512},[66],[31,69514],{"className":69515,"style":62056},[70],[31,69517,69429],{"className":69518},[75],[31,69520],{"className":69521,"style":293},[119],[31,69523,16040],{"className":69524},[297],[31,69526],{"className":69527,"style":293},[119],[31,69529,69531,69534],{"className":69530},[66],[31,69532],{"className":69533,"style":326},[70],[31,69535,19641],{"className":69536},[75],"。这条路径可达，",[45608,69539,69015],{}," 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。",[11,69542,69543],{},"如果把代码改成：",[52853,69545,69547],{"className":52855,"code":69546,"language":1412,"meta":52812,"style":52812},"if (len > 4 && len \u003C= 8) {\n    memcpy(buf, input, len);\n}\n",[45608,69548,69549,69572,69580],{"__ignoreMap":52812},[31,69550,69551,69554,69556,69558,69560,69562,69564,69567,69570],{"class":52861,"line":52862},[31,69552,69553],{"class":52865},"if",[31,69555,68970],{"class":52873},[31,69557,16040],{"class":52865},[31,69559,68975],{"class":52904},[31,69561,52959],{"class":52865},[31,69563,68980],{"class":52873},[31,69565,69566],{"class":52865},"\u003C=",[31,69568,69569],{"class":52904}," 8",[31,69571,52891],{"class":52873},[31,69573,69574,69577],{"class":52861,"line":52813},[31,69575,69576],{"class":52869},"    memcpy",[31,69578,69579],{"class":52873},"(buf, input, len);\n",[31,69581,69582],{"class":52861,"line":52920},[31,69583,52993],{"class":52873},[11,69585,69586],{},"约束就变成：",[31,69588,69590],{"className":69589,"translate":35},[411],[31,69591,69593,69647],{"className":69592,"translate":35},[34],[31,69594,69596],{"className":69595},[39],[41,69597,69598],{"xmlns":43,"display":420},[45,69599,69600,69644],{},[48,69601,69602,69604,69606,69608,69610,69612,69614,69616,69618,69620,69622,69624,69626,69628,69630,69632,69634,69636,69638,69640,69642],{},[51,69603,54404],{},[51,69605,30042],{},[51,69607,14559],{},[95,69609,16040],{},[91,69611,15416],{},[1430,69613,53035],{},[95,69615,53038],{},[1430,69617,53035],{},[51,69619,54404],{},[51,69621,30042],{},[51,69623,14559],{},[95,69625,347],{},[91,69627,19641],{},[1430,69629,53035],{},[95,69631,53038],{},[1430,69633,53035],{},[51,69635,54404],{},[51,69637,30042],{},[51,69639,14559],{},[95,69641,16040],{},[91,69643,19641],{},[55,69645,69646],{"encoding":57},"len > 4 \\;\\wedge\\; len \\le 8 \\;\\wedge\\; len > 8",[31,69648,69650,69674,69698,69723,69747,69771],{"className":69649,"ariaHidden":62},[61],[31,69651,69653,69656,69659,69662,69665,69668,69671],{"className":69652},[66],[31,69654],{"className":69655,"style":69089},[70],[31,69657,54404],{"className":69658,"style":69093},[75,76],[31,69660,30042],{"className":69661},[75,76],[31,69663,14559],{"className":69664},[75,76],[31,69666],{"className":69667,"style":293},[119],[31,69669,16040],{"className":69670},[297],[31,69672],{"className":69673,"style":293},[119],[31,69675,69677,69680,69683,69686,69689,69692,69695],{"className":69676},[66],[31,69678],{"className":69679,"style":326},[70],[31,69681,15416],{"className":69682},[75],[31,69684],{"className":69685,"style":293},[119],[31,69687],{"className":69688,"style":120},[119],[31,69690,53038],{"className":69691},[124],[31,69693],{"className":69694,"style":293},[119],[31,69696],{"className":69697,"style":120},[119],[31,69699,69701,69705,69708,69711,69714,69717,69720],{"className":69700},[66],[31,69702],{"className":69703,"style":69704},[70],"height:0.8304em;vertical-align:-0.136em;",[31,69706,54404],{"className":69707,"style":69093},[75,76],[31,69709,30042],{"className":69710},[75,76],[31,69712,14559],{"className":69713},[75,76],[31,69715],{"className":69716,"style":293},[119],[31,69718,347],{"className":69719},[297],[31,69721],{"className":69722,"style":293},[119],[31,69724,69726,69729,69732,69735,69738,69741,69744],{"className":69725},[66],[31,69727],{"className":69728,"style":326},[70],[31,69730,19641],{"className":69731},[75],[31,69733],{"className":69734,"style":293},[119],[31,69736],{"className":69737,"style":120},[119],[31,69739,53038],{"className":69740},[124],[31,69742],{"className":69743,"style":293},[119],[31,69745],{"className":69746,"style":120},[119],[31,69748,69750,69753,69756,69759,69762,69765,69768],{"className":69749},[66],[31,69751],{"className":69752,"style":69089},[70],[31,69754,54404],{"className":69755,"style":69093},[75,76],[31,69757,30042],{"className":69758},[75,76],[31,69760,14559],{"className":69761},[75,76],[31,69763],{"className":69764,"style":293},[119],[31,69766,16040],{"className":69767},[297],[31,69769],{"className":69770,"style":293},[119],[31,69772,69774,69777],{"className":69773},[66],[31,69775],{"className":69776,"style":326},[70],[31,69778,19641],{"className":69779},[75],[11,69781,69782,69783,69785],{},"求解器返回 ",[7246,69784,59102],{},"。这条溢出路径不可达，工具不会报告漏洞。",[11,69787,69788,69789,69791,69792,69794,69795,69797,69798,4809],{},"注意这里用的是 ",[45608,69790,55635],{},"（整数线性算术）。真实的 C 代码里 ",[45608,69793,69019],{}," 是 32 位 ",[45608,69796,52876],{},"，更精确的建模应该用 ",[45608,69799,55719],{},[52853,69801,69803],{"className":69366,"code":69802,"language":69368,"meta":52812,"style":52812},"(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",[45608,69804,69805,69810,69815,69820,69825,69830,69834],{"__ignoreMap":52812},[31,69806,69807],{"class":52861,"line":52862},[31,69808,69809],{},"(set-logic QF_BV)\n",[31,69811,69812],{"class":52861,"line":52813},[31,69813,69814],{},"(declare-const len (_ BitVec 32))\n",[31,69816,69817],{"class":52861,"line":52920},[31,69818,69819],{},"(assert (and (bvsgt len (_ bv4 32))\n",[31,69821,69822],{"class":52861,"line":52940},[31,69823,69824],{},"             (bvslt len (_ bv12 32))\n",[31,69826,69827],{"class":52861,"line":52977},[31,69828,69829],{},"             (bvsgt len (_ bv8 32))))\n",[31,69831,69832],{"class":52861,"line":52984},[31,69833,69390],{},[31,69835,69836],{"class":52861,"line":52990},[31,69837,69395],{},[11,69839,69840,69841,55632,69844,69847,69848,69850,69851,69854],{},"注意这里用的是有符号比较（",[45608,69842,69843],{},"bvsgt",[45608,69845,69846],{},"bvslt","），因为 C 里的 ",[45608,69849,52876],{}," 是有符号的。一个可能的模型是 ",[45608,69852,69853],{},"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。",[53200,69856,69858],{"id":69857},"_53-整数溢出","5.3 整数溢出",[52853,69860,69862],{"className":52855,"code":69861,"language":1412,"meta":52812,"style":52812},"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",[45608,69863,69864,69901,69914,69932,69944,69948],{"__ignoreMap":52812},[31,69865,69866,69868,69871,69873,69875,69877,69880,69882,69885,69888,69891,69893,69895,69897,69899],{"class":52861,"line":52862},[31,69867,52866],{"class":52865},[31,69869,69870],{"class":52869}," copy",[31,69872,211],{"class":52873},[31,69874,68934],{"class":52865},[31,69876,68937],{"class":52865},[31,69878,69879],{"class":52879},"src",[31,69881,52883],{"class":52873},[31,69883,69884],{"class":52865},"unsigned",[31,69886,69887],{"class":52865}," int",[31,69889,69890],{"class":52879}," off",[31,69892,52883],{"class":52873},[31,69894,69884],{"class":52865},[31,69896,69887],{"class":52865},[31,69898,68946],{"class":52879},[31,69900,52891],{"class":52873},[31,69902,69903,69905,69907,69909,69912],{"class":52861,"line":52813},[31,69904,68953],{"class":52865},[31,69906,68956],{"class":52879},[31,69908,15175],{"class":52873},[31,69910,69911],{"class":52904},"64",[31,69913,68963],{"class":52873},[31,69915,69916,69918,69921,69923,69925,69927,69930],{"class":52861,"line":52920},[31,69917,52943],{"class":52865},[31,69919,69920],{"class":52873}," (off ",[31,69922,434],{"class":52865},[31,69924,68980],{"class":52873},[31,69926,69566],{"class":52865},[31,69928,69929],{"class":52904}," 64",[31,69931,52891],{"class":52873},[31,69933,69934,69936,69939,69941],{"class":52861,"line":52940},[31,69935,68992],{"class":52869},[31,69937,69938],{"class":52873},"(buf ",[31,69940,434],{"class":52865},[31,69942,69943],{"class":52873}," off, src, len);\n",[31,69945,69946],{"class":52861,"line":52977},[31,69947,52987],{"class":52873},[31,69949,69950],{"class":52861,"line":52984},[31,69951,52993],{"class":52873},[11,69953,69954,69955,69958,69959,2446,69962,69964,69965,55780],{},"看起来检查很严：只有 ",[45608,69956,69957],{},"off + len"," 不超过 64 才拷贝。但 ",[45608,69960,69961],{},"off",[45608,69963,69019],{}," 都是 32 位无符号整数，加法是",[7246,69966,69967,69968],{},"模 ",[31,69969,69971,69989],{"className":69970,"translate":35},[34],[31,69972,69974],{"className":69973},[39],[41,69975,69976],{"xmlns":43},[45,69977,69978,69986],{},[48,69979,69980],{},[1180,69981,69982,69984],{},[91,69983,152],{},[91,69985,65608],{},[55,69987,69988],{"encoding":57},"2^{32}",[31,69990,69992],{"className":69991,"ariaHidden":62},[61],[31,69993,69995,69998],{"className":69994},[66],[31,69996],{"className":69997,"style":5135},[70],[31,69999,70001,70004],{"className":70000},[75],[31,70002,152],{"className":70003},[75],[31,70005,70007],{"className":70006},[1039],[31,70008,70010],{"className":70009},[1043],[31,70011,70013],{"className":70012},[1048],[31,70014,70016],{"className":70015,"style":5135},[1052],[31,70017,70018,70021],{"style":3631},[31,70019],{"className":70020,"style":1061},[1060],[31,70022,70024],{"className":70023},[1065,1066,1067,1068],[31,70025,70027],{"className":70026},[75,1068],[31,70028,65608],{"className":70029},[75,1068],[11,70031,70032],{},"攻击者可以取：",[31,70034,70036],{"className":70035,"translate":35},[411],[31,70037,70039,70069],{"className":70038,"translate":35},[34],[31,70040,70042],{"className":70041},[39],[41,70043,70044],{"xmlns":43,"display":420},[45,70045,70046,70066],{},[48,70047,70048,70050,70052,70055,70057,70059,70061,70063],{},[1430,70049,69961],{},[95,70051,223],{},[1430,70053,70054],{"mathvariant":65623},"0xFFFFFFF0",[95,70056,1210],{"separator":62},[119,70058],{"width":20286},[1430,70060,69019],{},[95,70062,223],{},[1430,70064,70065],{"mathvariant":65623},"0x20",[55,70067,70068],{"encoding":57},"\\text{off} = \\texttt{0xFFFFFFF0}, \\qquad \\text{len} = \\texttt{0x20}",[31,70070,70072,70093,70129],{"className":70071,"ariaHidden":62},[61],[31,70073,70075,70078,70084,70087,70090],{"className":70074},[66],[31,70076],{"className":70077,"style":1630},[70],[31,70079,70081],{"className":70080},[75,1793],[31,70082,69961],{"className":70083},[75],[31,70085],{"className":70086,"style":293},[119],[31,70088,223],{"className":70089},[297],[31,70091],{"className":70092,"style":293},[119],[31,70094,70096,70099,70105,70108,70111,70114,70120,70123,70126],{"className":70095},[66],[31,70097],{"className":70098,"style":4202},[70],[31,70100,70102],{"className":70101},[75,1793],[31,70103,70054],{"className":70104},[75,65820],[31,70106,1210],{"className":70107},[1391],[31,70109],{"className":70110,"style":20525},[119],[31,70112],{"className":70113,"style":1467},[119],[31,70115,70117],{"className":70116},[75,1793],[31,70118,69019],{"className":70119},[75],[31,70121],{"className":70122,"style":293},[119],[31,70124,223],{"className":70125},[297],[31,70127],{"className":70128,"style":293},[119],[31,70130,70132,70135],{"className":70131},[66],[31,70133],{"className":70134,"style":65813},[70],[31,70136,70138],{"className":70137},[75,1793],[31,70139,70065],{"className":70140},[75,65820],[11,70142,70143],{},"计算一下：",[31,70145,70147],{"className":70146,"translate":35},[411],[31,70148,70150,70263],{"className":70149,"translate":35},[34],[31,70151,70153],{"className":70152},[39],[41,70154,70155],{"xmlns":43,"display":420},[45,70156,70157,70260],{},[5606,70158,70159,70189,70210],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,70160,70161,70173],{},[5615,70162,70163],{},[2071,70164,70165],{"scriptlevel":1977,"displaystyle":62},[48,70166,70167,70169,70171],{},[1430,70168,69961],{},[95,70170,434],{},[1430,70172,69019],{},[5615,70174,70175],{},[2071,70176,70177],{"scriptlevel":1977,"displaystyle":62},[48,70178,70179,70181,70183,70185,70187],{},[48,70180],{},[95,70182,223],{},[1430,70184,70054],{"mathvariant":65623},[95,70186,434],{},[1430,70188,70065],{"mathvariant":65623},[5612,70190,70191,70197],{},[5615,70192,70193],{},[2071,70194,70195],{"scriptlevel":1977,"displaystyle":62},[48,70196],{},[5615,70198,70199],{},[2071,70200,70201],{"scriptlevel":1977,"displaystyle":62},[48,70202,70203,70205,70207],{},[48,70204],{},[95,70206,223],{},[1430,70208,70209],{"mathvariant":65623},"0x100000010",[5612,70211,70212,70218],{},[5615,70213,70214],{},[2071,70215,70216],{"scriptlevel":1977,"displaystyle":62},[48,70217],{},[5615,70219,70220],{},[2071,70221,70222],{"scriptlevel":1977,"displaystyle":62},[48,70223,70224,70226,70228,70231,70233,70236,70238,70240,70242,70250,70252,70258],{},[48,70225],{},[95,70227,62697],{},[1430,70229,70230],{"mathvariant":65623},"0x10",[95,70232,223],{},[91,70234,70235],{},"16",[119,70237],{},[119,70239],{"width":1584},[95,70241,211],{"stretchy":210},[48,70243,70244,70246,70248],{},[51,70245,16396],{"mathvariant":494},[51,70247,2552],{"mathvariant":494},[51,70249,1435],{"mathvariant":494},[119,70251],{"width":65601},[1180,70253,70254,70256],{},[91,70255,152],{},[91,70257,65608],{},[95,70259,220],{"stretchy":210},[55,70261,70262],{"encoding":57},"\\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}",[31,70264,70266],{"className":70265,"ariaHidden":62},[61],[31,70267,70269,70273],{"className":70268},[66],[31,70270],{"className":70271,"style":70272},[70],"height:4.2241em;vertical-align:-1.8621em;",[31,70274,70276],{"className":70275},[75],[31,70277,70279,70351],{"className":70278},[5606],[31,70280,70282],{"className":70281},[5867],[31,70283,70285,70342],{"className":70284},[1043,1044],[31,70286,70288,70339],{"className":70287},[1048],[31,70289,70292,70321,70330],{"className":70290,"style":70291},[1052],"height:2.3621em;",[31,70293,70294,70297],{"style":57801},[31,70295],{"className":70296,"style":1637},[1060],[31,70298,70300,70306,70309,70312,70315],{"className":70299},[75],[31,70301,70303],{"className":70302},[75,1793],[31,70304,69961],{"className":70305},[75],[31,70307],{"className":70308,"style":120},[119],[31,70310,434],{"className":70311},[124],[31,70313],{"className":70314,"style":120},[119],[31,70316,70318],{"className":70317},[75,1793],[31,70319,69019],{"className":70320},[75],[31,70322,70324,70327],{"style":70323},"top:-3.0221em;",[31,70325],{"className":70326,"style":1637},[1060],[31,70328],{"className":70329},[75],[31,70331,70333,70336],{"style":70332},"top:-1.4979em;",[31,70334],{"className":70335,"style":1637},[1060],[31,70337],{"className":70338},[75],[31,70340,1076],{"className":70341},[1075],[31,70343,70345],{"className":70344},[1048],[31,70346,70349],{"className":70347,"style":70348},[1052],"height:1.8621em;",[31,70350],{},[31,70352,70354],{"className":70353},[6062],[31,70355,70357,70527],{"className":70356},[1043,1044],[31,70358,70360,70524],{"className":70359},[1048],[31,70361,70363,70404,70430],{"className":70362,"style":70291},[1052],[31,70364,70365,70368],{"style":57801},[31,70366],{"className":70367,"style":1637},[1060],[31,70369,70371,70374,70377,70380,70383,70389,70392,70395,70398],{"className":70370},[75],[31,70372],{"className":70373},[75],[31,70375],{"className":70376,"style":293},[119],[31,70378,223],{"className":70379},[297],[31,70381],{"className":70382,"style":293},[119],[31,70384,70386],{"className":70385},[75,1793],[31,70387,70054],{"className":70388},[75,65820],[31,70390],{"className":70391,"style":120},[119],[31,70393,434],{"className":70394},[124],[31,70396],{"className":70397,"style":120},[119],[31,70399,70401],{"className":70400},[75,1793],[31,70402,70065],{"className":70403},[75,65820],[31,70405,70406,70409],{"style":70323},[31,70407],{"className":70408,"style":1637},[1060],[31,70410,70412,70415,70418,70421,70424],{"className":70411},[75],[31,70413],{"className":70414},[75],[31,70416],{"className":70417,"style":293},[119],[31,70419,223],{"className":70420},[297],[31,70422],{"className":70423,"style":293},[119],[31,70425,70427],{"className":70426},[75,1793],[31,70428,70209],{"className":70429},[75,65820],[31,70431,70432,70435],{"style":70332},[31,70433],{"className":70434,"style":1637},[1060],[31,70436,70438,70441,70444,70447,70450,70456,70459,70462,70465,70468,70471,70474,70477,70486,70489,70521],{"className":70437},[75],[31,70439],{"className":70440},[75],[31,70442],{"className":70443,"style":293},[119],[31,70445,62697],{"className":70446},[297],[31,70448],{"className":70449,"style":293},[119],[31,70451,70453],{"className":70452},[75,1793],[31,70454,70230],{"className":70455},[75,65820],[31,70457],{"className":70458,"style":293},[119],[31,70460,223],{"className":70461},[297],[31,70463],{"className":70464,"style":293},[119],[31,70466,70235],{"className":70467},[75],[31,70469],{"className":70470},[119,65679],[31,70472],{"className":70473,"style":1789},[119],[31,70475,211],{"className":70476},[264],[31,70478,70480],{"className":70479},[75],[31,70481,70483],{"className":70482},[75],[31,70484,65701],{"className":70485},[75,2745],[31,70487],{"className":70488,"style":65705},[119],[31,70490,70492,70495],{"className":70491},[75],[31,70493,152],{"className":70494},[75],[31,70496,70498],{"className":70497},[1039],[31,70499,70501],{"className":70500},[1043],[31,70502,70504],{"className":70503},[1048],[31,70505,70507],{"className":70506,"style":1327},[1052],[31,70508,70509,70512],{"style":1330},[31,70510],{"className":70511,"style":1061},[1060],[31,70513,70515],{"className":70514},[1065,1066,1067,1068],[31,70516,70518],{"className":70517},[75,1068],[31,70519,65608],{"className":70520},[75,1068],[31,70522,220],{"className":70523},[289],[31,70525,1076],{"className":70526},[1075],[31,70528,70530],{"className":70529},[1048],[31,70531,70533],{"className":70532,"style":70348},[1052],[31,70534],{},[11,70536,70537,70588,70589,70592,70593,70595,70596,2372],{},[31,70538,70540,70558],{"className":70539,"translate":35},[34],[31,70541,70543],{"className":70542},[39],[41,70544,70545],{"xmlns":43},[45,70546,70547,70555],{},[48,70548,70549,70551,70553],{},[91,70550,70235],{},[95,70552,347],{},[91,70554,69911],{},[55,70556,70557],{"encoding":57},"16 \\le 64",[31,70559,70561,70579],{"className":70560,"ariaHidden":62},[61],[31,70562,70564,70567,70570,70573,70576],{"className":70563},[66],[31,70565],{"className":70566,"style":366},[70],[31,70568,70235],{"className":70569},[75],[31,70571],{"className":70572,"style":293},[119],[31,70574,347],{"className":70575},[297],[31,70577],{"className":70578,"style":293},[119],[31,70580,70582,70585],{"className":70581},[66],[31,70583],{"className":70584,"style":326},[70],[31,70586,69911],{"className":70587},[75],"，检查通过。但 ",[45608,70590,70591],{},"buf + off"," 指向的是缓冲区外大约 4GB 的位置，",[45608,70594,69015],{}," 会往一个完全无关的地址写 32 字节。这是一个典型的",[7246,70597,70598],{},"整数溢出导致的任意地址写",[11,70600,70601],{},"用位向量理论来问：",[52853,70603,70605],{"className":69366,"code":70604,"language":69368,"meta":52812,"style":52812},"(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",[45608,70606,70607,70611,70616,70620,70625,70630,70635,70639],{"__ignoreMap":52812},[31,70608,70609],{"class":52861,"line":52862},[31,70610,69809],{},[31,70612,70613],{"class":52861,"line":52813},[31,70614,70615],{},"(declare-const off (_ BitVec 32))\n",[31,70617,70618],{"class":52861,"line":52920},[31,70619,69814],{},[31,70621,70622],{"class":52861,"line":52940},[31,70623,70624],{},"(assert (bvule (bvadd off len) (_ bv64 32)))\n",[31,70626,70627],{"class":52861,"line":52977},[31,70628,70629],{},"(assert (bvugt off (_ bv64 32)))\n",[31,70631,70632],{"class":52861,"line":52984},[31,70633,70634],{},"(assert (bvule len (_ bv64 32)))\n",[31,70636,70637],{"class":52861,"line":52990},[31,70638,69390],{},[31,70640,70642],{"class":52861,"line":70641},8,[31,70643,69395],{},[11,70645,70646],{},"求解器返回 sat，模型可能是：",[52853,70648,70651],{"className":70649,"code":70650,"language":1793},[63799],"off = #xfffffff0\nlen = #x00000020\n",[45608,70652,70650],{"__ignoreMap":52812},[11,70654,70655],{},"漏洞可达。",[11,70657,70658,70659,2446,70661,70663],{},"如果换一种建模，把 ",[45608,70660,69961],{},[45608,70662,69019],{}," 当成数学整数：",[31,70665,70667],{"className":70666,"translate":35},[411],[31,70668,70670,70716],{"className":70669,"translate":35},[34],[31,70671,70673],{"className":70672},[39],[41,70674,70675],{"xmlns":43,"display":420},[45,70676,70677,70713],{},[48,70678,70679,70681,70683,70685,70687,70689,70691,70693,70695,70697,70699,70701,70703,70705,70707,70709,70711],{},[1430,70680,69961],{},[95,70682,434],{},[1430,70684,69019],{},[95,70686,347],{},[91,70688,69911],{},[1430,70690,53035],{},[95,70692,53038],{},[1430,70694,53035],{},[1430,70696,69961],{},[95,70698,16040],{},[91,70700,69911],{},[1430,70702,53035],{},[95,70704,53038],{},[1430,70706,53035],{},[1430,70708,69019],{},[95,70710,5662],{},[91,70712,1977],{},[55,70714,70715],{"encoding":57},"\\text{off} + \\text{len} \\le 64 \\;\\wedge\\; \\text{off} > 64 \\;\\wedge\\; \\text{len} \\ge 0",[31,70717,70719,70740,70761,70785,70806,70830,70851],{"className":70718,"ariaHidden":62},[61],[31,70720,70722,70725,70731,70734,70737],{"className":70721},[66],[31,70723],{"className":70724,"style":56820},[70],[31,70726,70728],{"className":70727},[75,1793],[31,70729,69961],{"className":70730},[75],[31,70732],{"className":70733,"style":120},[119],[31,70735,434],{"className":70736},[124],[31,70738],{"className":70739,"style":120},[119],[31,70741,70743,70746,70752,70755,70758],{"className":70742},[66],[31,70744],{"className":70745,"style":69704},[70],[31,70747,70749],{"className":70748},[75,1793],[31,70750,69019],{"className":70751},[75],[31,70753],{"className":70754,"style":293},[119],[31,70756,347],{"className":70757},[297],[31,70759],{"className":70760,"style":293},[119],[31,70762,70764,70767,70770,70773,70776,70779,70782],{"className":70763},[66],[31,70765],{"className":70766,"style":326},[70],[31,70768,69911],{"className":70769},[75],[31,70771],{"className":70772,"style":293},[119],[31,70774],{"className":70775,"style":120},[119],[31,70777,53038],{"className":70778},[124],[31,70780],{"className":70781,"style":293},[119],[31,70783],{"className":70784,"style":120},[119],[31,70786,70788,70791,70797,70800,70803],{"className":70787},[66],[31,70789],{"className":70790,"style":69089},[70],[31,70792,70794],{"className":70793},[75,1793],[31,70795,69961],{"className":70796},[75],[31,70798],{"className":70799,"style":293},[119],[31,70801,16040],{"className":70802},[297],[31,70804],{"className":70805,"style":293},[119],[31,70807,70809,70812,70815,70818,70821,70824,70827],{"className":70808},[66],[31,70810],{"className":70811,"style":326},[70],[31,70813,69911],{"className":70814},[75],[31,70816],{"className":70817,"style":293},[119],[31,70819],{"className":70820,"style":120},[119],[31,70822,53038],{"className":70823},[124],[31,70825],{"className":70826,"style":293},[119],[31,70828],{"className":70829,"style":120},[119],[31,70831,70833,70836,70842,70845,70848],{"className":70832},[66],[31,70834],{"className":70835,"style":69704},[70],[31,70837,70839],{"className":70838},[75,1793],[31,70840,69019],{"className":70841},[75],[31,70843],{"className":70844,"style":293},[119],[31,70846,5662],{"className":70847},[297],[31,70849],{"className":70850,"style":293},[119],[31,70852,70854,70857],{"className":70853},[66],[31,70855],{"className":70856,"style":326},[70],[31,70858,1977],{"className":70859},[75],[11,70861,70862,70863,70865,70866,70945],{},"这个公式是 ",[7246,70864,59102],{},"，因为 ",[31,70867,70869,70891],{"className":70868,"translate":35},[34],[31,70870,70872],{"className":70871},[39],[41,70873,70874],{"xmlns":43},[45,70875,70876,70888],{},[48,70877,70878,70880,70882,70884,70886],{},[1430,70879,69961],{},[95,70881,434],{},[1430,70883,69019],{},[95,70885,16040],{},[91,70887,69911],{},[55,70889,70890],{"encoding":57},"\\text{off} + \\text{len} > 64",[31,70892,70894,70915,70936],{"className":70893,"ariaHidden":62},[61],[31,70895,70897,70900,70906,70909,70912],{"className":70896},[66],[31,70898],{"className":70899,"style":56820},[70],[31,70901,70903],{"className":70902},[75,1793],[31,70904,69961],{"className":70905},[75],[31,70907],{"className":70908,"style":120},[119],[31,70910,434],{"className":70911},[124],[31,70913],{"className":70914,"style":120},[119],[31,70916,70918,70921,70927,70930,70933],{"className":70917},[66],[31,70919],{"className":70920,"style":69089},[70],[31,70922,70924],{"className":70923},[75,1793],[31,70925,69019],{"className":70926},[75],[31,70928],{"className":70929,"style":293},[119],[31,70931,16040],{"className":70932},[297],[31,70934],{"className":70935,"style":293},[119],[31,70937,70939,70942],{"className":70938},[66],[31,70940],{"className":70941,"style":326},[70],[31,70943,69911],{"className":70944},[75]," 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。",[11,70947,70948,70951,70952,70955],{},[7246,70949,70950],{},"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 ",[45608,70953,70954],{},"unsigned int"," 是模运算，不是数学整数。",[53200,70957,70959],{"id":70958},"_54-空指针","5.4 空指针",[11,70961,70962],{},"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。",[52853,70964,70966],{"className":52855,"code":70965,"language":1412,"meta":52812,"style":52812},"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",[45608,70967,70968,70990,71013,71024],{"__ignoreMap":52812},[31,70969,70970,70972,70975,70977,70980,70983,70986,70988],{"class":52861,"line":52862},[31,70971,52876],{"class":52865},[31,70973,70974],{"class":52869}," read_flag",[31,70976,211],{"class":52873},[31,70978,70979],{"class":52865},"struct",[31,70981,70982],{"class":52873}," node ",[31,70984,70985],{"class":52865},"*",[31,70987,11],{"class":52879},[31,70989,52891],{"class":52873},[31,70991,70992,70994,70997,70999,71002,71005,71008,71011],{"class":52861,"line":52813},[31,70993,52943],{"class":52865},[31,70995,70996],{"class":52873}," (p ",[31,70998,52970],{"class":52865},[31,71000,71001],{"class":52904}," NULL",[31,71003,71004],{"class":52873},") ",[31,71006,71007],{"class":52865},"return",[31,71009,71010],{"class":52904}," 0",[31,71012,52937],{"class":52873},[31,71014,71015,71018,71021],{"class":52861,"line":52920},[31,71016,71017],{"class":52865},"    return",[31,71019,71020],{"class":52873}," p->flag;",[31,71022,71023],{"class":52980},"     \u002F\u002F 静态分析：p 可能为空？\n",[31,71025,71026],{"class":52861,"line":52940},[31,71027,52993],{"class":52873},[11,71029,71030,71031,71034,71035,71038],{},"工具在 ",[45608,71032,71033],{},"p->flag"," 处收集路径条件。走到这一行，必然经过了 ",[45608,71036,71037],{},"p == NULL"," 为假的分支，所以路径条件里包含：",[31,71040,71042],{"className":71041,"translate":35},[411],[31,71043,71045,71064],{"className":71044,"translate":35},[34],[31,71046,71048],{"className":71047},[39],[41,71049,71050],{"xmlns":43,"display":420},[45,71051,71052,71061],{},[48,71053,71054,71056,71058],{},[51,71055,11],{},[95,71057,21271],{"mathvariant":494},[1430,71059,71060],{"mathvariant":65623},"NULL",[55,71062,71063],{"encoding":57},"p \\ne \\texttt{NULL}",[31,71065,71067,71118],{"className":71066,"ariaHidden":62},[61],[31,71068,71070,71073,71076,71079,71115],{"className":71069},[66],[31,71071],{"className":71072,"style":4202},[70],[31,71074,11],{"className":71075},[75,76],[31,71077],{"className":71078,"style":293},[119],[31,71080,71082,71109,71112],{"className":71081},[297],[31,71083,71085],{"className":71084},[297],[31,71086,71088],{"className":71087},[75,21516],[31,71089,71091],{"className":71090},[21520],[31,71092,71094,71097,71106],{"className":71093},[21524],[31,71095],{"className":71096,"style":4202},[70],[31,71098,71100],{"className":71099},[21531],[31,71101,71103],{"className":71102},[75],[31,71104,21538],{"className":71105},[297],[31,71107],{"className":71108},[21542],[31,71110],{"className":71111},[119,21546],[31,71113,223],{"className":71114},[297],[31,71116],{"className":71117,"style":293},[119],[31,71119,71121,71124],{"className":71120},[66],[31,71122],{"className":71123,"style":65813},[70],[31,71125,71127],{"className":71126},[75,1793],[31,71128,71060],{"className":71129},[75,65820],[11,71131,71132,71133,71135],{},"静态分析如果只做局部启发式，可能会说「",[45608,71134,11],{}," 可能是空指针」。这时问 SMT：",[31,71137,71139],{"className":71138,"translate":35},[411],[31,71140,71142,71172],{"className":71141,"translate":35},[34],[31,71143,71145],{"className":71144},[39],[41,71146,71147],{"xmlns":43,"display":420},[45,71148,71149,71169],{},[48,71150,71151,71153,71155,71157,71159,71161,71163,71165,71167],{},[51,71152,11],{},[95,71154,223],{},[1430,71156,71060],{"mathvariant":65623},[1430,71158,53035],{},[95,71160,53038],{},[1430,71162,53035],{},[51,71164,11],{},[95,71166,21271],{"mathvariant":494},[1430,71168,71060],{"mathvariant":65623},[55,71170,71171],{"encoding":57},"p = \\texttt{NULL} \\;\\wedge\\; p \\ne \\texttt{NULL}",[31,71173,71175,71193,71220,71271],{"className":71174,"ariaHidden":62},[61],[31,71176,71178,71181,71184,71187,71190],{"className":71177},[66],[31,71179],{"className":71180,"style":865},[70],[31,71182,11],{"className":71183},[75,76],[31,71185],{"className":71186,"style":293},[119],[31,71188,223],{"className":71189},[297],[31,71191],{"className":71192,"style":293},[119],[31,71194,71196,71199,71205,71208,71211,71214,71217],{"className":71195},[66],[31,71197],{"className":71198,"style":65813},[70],[31,71200,71202],{"className":71201},[75,1793],[31,71203,71060],{"className":71204},[75,65820],[31,71206],{"className":71207,"style":293},[119],[31,71209],{"className":71210,"style":120},[119],[31,71212,53038],{"className":71213},[124],[31,71215],{"className":71216,"style":293},[119],[31,71218],{"className":71219,"style":120},[119],[31,71221,71223,71226,71229,71232,71268],{"className":71222},[66],[31,71224],{"className":71225,"style":4202},[70],[31,71227,11],{"className":71228},[75,76],[31,71230],{"className":71231,"style":293},[119],[31,71233,71235,71262,71265],{"className":71234},[297],[31,71236,71238],{"className":71237},[297],[31,71239,71241],{"className":71240},[75,21516],[31,71242,71244],{"className":71243},[21520],[31,71245,71247,71250,71259],{"className":71246},[21524],[31,71248],{"className":71249,"style":4202},[70],[31,71251,71253],{"className":71252},[21531],[31,71254,71256],{"className":71255},[75],[31,71257,21538],{"className":71258},[297],[31,71260],{"className":71261},[21542],[31,71263],{"className":71264},[119,21546],[31,71266,223],{"className":71267},[297],[31,71269],{"className":71270,"style":293},[119],[31,71272,71274,71277],{"className":71273},[66],[31,71275],{"className":71276,"style":65813},[70],[31,71278,71280],{"className":71279},[75,1793],[31,71281,71060],{"className":71282},[75,65820],[11,71284,71285,71286,71288],{},"这是明显的矛盾，返回 ",[7246,71287,59102],{},"，告警被排除。",[11,71290,71291],{},"反过来，如果代码是：",[52853,71293,71295],{"className":52855,"code":71294,"language":1412,"meta":52812,"style":52812},"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",[45608,71296,71297,71315,71333,71342,71346,71354],{"__ignoreMap":52812},[31,71298,71299,71301,71303,71305,71307,71309,71311,71313],{"class":52861,"line":52862},[31,71300,52876],{"class":52865},[31,71302,70974],{"class":52869},[31,71304,211],{"class":52873},[31,71306,70979],{"class":52865},[31,71308,70982],{"class":52873},[31,71310,70985],{"class":52865},[31,71312,11],{"class":52879},[31,71314,52891],{"class":52873},[31,71316,71317,71319,71322,71324,71327,71330],{"class":52861,"line":52813},[31,71318,52943],{"class":52865},[31,71320,71321],{"class":52873}," (p->flag ",[31,71323,52970],{"class":52865},[31,71325,71326],{"class":52904}," 1",[31,71328,71329],{"class":52873},") {",[31,71331,71332],{"class":52980},"   \u002F\u002F 这里没有检查 p\n",[31,71334,71335,71338,71340],{"class":52861,"line":52920},[31,71336,71337],{"class":52865},"        return",[31,71339,71326],{"class":52904},[31,71341,52937],{"class":52873},[31,71343,71344],{"class":52861,"line":52940},[31,71345,52987],{"class":52873},[31,71347,71348,71350,71352],{"class":52861,"line":52977},[31,71349,71017],{"class":52865},[31,71351,71010],{"class":52904},[31,71353,52937],{"class":52873},[31,71355,71356],{"class":52861,"line":52984},[31,71357,52993],{"class":52873},[11,71359,71360,71361,71447],{},"路径条件里没有 ",[31,71362,71364,71381],{"className":71363,"translate":35},[34],[31,71365,71367],{"className":71366},[39],[41,71368,71369],{"xmlns":43},[45,71370,71371,71379],{},[48,71372,71373,71375,71377],{},[51,71374,11],{},[95,71376,21271],{"mathvariant":494},[1430,71378,71060],{"mathvariant":65623},[55,71380,71063],{"encoding":57},[31,71382,71384,71435],{"className":71383,"ariaHidden":62},[61],[31,71385,71387,71390,71393,71396,71432],{"className":71386},[66],[31,71388],{"className":71389,"style":4202},[70],[31,71391,11],{"className":71392},[75,76],[31,71394],{"className":71395,"style":293},[119],[31,71397,71399,71426,71429],{"className":71398},[297],[31,71400,71402],{"className":71401},[297],[31,71403,71405],{"className":71404},[75,21516],[31,71406,71408],{"className":71407},[21520],[31,71409,71411,71414,71423],{"className":71410},[21524],[31,71412],{"className":71413,"style":4202},[70],[31,71415,71417],{"className":71416},[21531],[31,71418,71420],{"className":71419},[75],[31,71421,21538],{"className":71422},[297],[31,71424],{"className":71425},[21542],[31,71427],{"className":71428},[119,21546],[31,71430,223],{"className":71431},[297],[31,71433],{"className":71434,"style":293},[119],[31,71436,71438,71441],{"className":71437},[66],[31,71439],{"className":71440,"style":65813},[70],[31,71442,71444],{"className":71443},[75,1793],[31,71445,71060],{"className":71446},[75,65820],"，查询：",[31,71449,71451],{"className":71450,"translate":35},[411],[31,71452,71454,71472],{"className":71453,"translate":35},[34],[31,71455,71457],{"className":71456},[39],[41,71458,71459],{"xmlns":43,"display":420},[45,71460,71461,71469],{},[48,71462,71463,71465,71467],{},[51,71464,11],{},[95,71466,223],{},[1430,71468,71060],{"mathvariant":65623},[55,71470,71471],{"encoding":57},"p = \\texttt{NULL}",[31,71473,71475,71493],{"className":71474,"ariaHidden":62},[61],[31,71476,71478,71481,71484,71487,71490],{"className":71477},[66],[31,71479],{"className":71480,"style":865},[70],[31,71482,11],{"className":71483},[75,76],[31,71485],{"className":71486,"style":293},[119],[31,71488,223],{"className":71489},[297],[31,71491],{"className":71492,"style":293},[119],[31,71494,71496,71499],{"className":71495},[66],[31,71497],{"className":71498,"style":65813},[70],[31,71500,71502],{"className":71501},[75,1793],[31,71503,71060],{"className":71504},[75,65820],[11,71506,71507,71508,71510,71511,71514],{},"返回 ",[7246,71509,53187],{},"，模型给出 ",[45608,71512,71513],{},"p = NULL","。这是一条真实可达的空指针路径，告警保留。",[11,71516,71517,71518,71521],{},"这个用法体现了一个重要原则：",[7246,71519,71520],{},"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。",[53200,71523,71525],{"id":71524},"_55-数组越界","5.5 数组越界",[52853,71527,71529],{"className":52855,"code":71528,"language":1412,"meta":52812,"style":52812},"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",[45608,71530,71531,71555,71569,71581,71585,71596],{"__ignoreMap":52812},[31,71532,71533,71535,71538,71540,71542,71544,71546,71548,71550,71553],{"class":52861,"line":52862},[31,71534,52876],{"class":52865},[31,71536,71537],{"class":52869}," get",[31,71539,211],{"class":52873},[31,71541,52876],{"class":52865},[31,71543,68937],{"class":52865},[31,71545,5813],{"class":52879},[31,71547,52883],{"class":52873},[31,71549,52876],{"class":52865},[31,71551,71552],{"class":52879}," i",[31,71554,52891],{"class":52873},[31,71556,71557,71559,71562,71564,71567],{"class":52861,"line":52813},[31,71558,52943],{"class":52865},[31,71560,71561],{"class":52873}," (i ",[31,71563,9758],{"class":52865},[31,71565,71566],{"class":52904}," 100",[31,71568,52891],{"class":52873},[31,71570,71571,71573,71575,71578],{"class":52861,"line":52920},[31,71572,71337],{"class":52865},[31,71574,52899],{"class":52879},[31,71576,71577],{"class":52873},"[i];",[31,71579,71580],{"class":52980},"     \u002F\u002F i 可能为负\n",[31,71582,71583],{"class":52861,"line":52940},[31,71584,52987],{"class":52873},[31,71586,71587,71589,71592,71594],{"class":52861,"line":52977},[31,71588,71017],{"class":52865},[31,71590,71591],{"class":52865}," -",[31,71593,205],{"class":52904},[31,71595,52937],{"class":52873},[31,71597,71598],{"class":52861,"line":52984},[31,71599,52993],{"class":52873},[11,71601,71602],{},"只检查了上界，没检查下界。问 SMT：",[31,71604,71606],{"className":71605,"translate":35},[411],[31,71607,71609,71639],{"className":71608,"translate":35},[34],[31,71610,71612],{"className":71611},[39],[41,71613,71614],{"xmlns":43,"display":420},[45,71615,71616,71636],{},[48,71617,71618,71620,71622,71624,71626,71628,71630,71632,71634],{},[51,71619,5391],{},[95,71621,9758],{},[91,71623,22616],{},[1430,71625,53035],{},[95,71627,53038],{},[1430,71629,53035],{},[51,71631,5391],{},[95,71633,9758],{},[91,71635,1977],{},[55,71637,71638],{"encoding":57},"i \u003C 100 \\;\\wedge\\; i \u003C 0",[31,71640,71642,71661,71685,71703],{"className":71641,"ariaHidden":62},[61],[31,71643,71645,71649,71652,71655,71658],{"className":71644},[66],[31,71646],{"className":71647,"style":71648},[70],"height:0.6986em;vertical-align:-0.0391em;",[31,71650,5391],{"className":71651},[75,76],[31,71653],{"className":71654,"style":293},[119],[31,71656,9758],{"className":71657},[297],[31,71659],{"className":71660,"style":293},[119],[31,71662,71664,71667,71670,71673,71676,71679,71682],{"className":71663},[66],[31,71665],{"className":71666,"style":326},[70],[31,71668,22616],{"className":71669},[75],[31,71671],{"className":71672,"style":293},[119],[31,71674],{"className":71675,"style":120},[119],[31,71677,53038],{"className":71678},[124],[31,71680],{"className":71681,"style":293},[119],[31,71683],{"className":71684,"style":120},[119],[31,71686,71688,71691,71694,71697,71700],{"className":71687},[66],[31,71689],{"className":71690,"style":71648},[70],[31,71692,5391],{"className":71693},[75,76],[31,71695],{"className":71696,"style":293},[119],[31,71698,9758],{"className":71699},[297],[31,71701],{"className":71702,"style":293},[119],[31,71704,71706,71709],{"className":71705},[66],[31,71707],{"className":71708,"style":326},[70],[31,71710,1977],{"className":71711},[75],[11,71713,71714,71715,2372,71718,71721],{},"返回 sat，例如 ",[45608,71716,71717],{},"i = -1",[45608,71719,71720],{},"a[-1]"," 是一次越界读。",[11,71723,71724],{},"如果加上下界检查：",[52853,71726,71728],{"className":52855,"code":71727,"language":1412,"meta":52812,"style":52812},"if (i >= 0 && i \u003C 100) {\n    return a[i];\n}\n",[45608,71729,71730,71752,71761],{"__ignoreMap":52812},[31,71731,71732,71734,71736,71739,71741,71743,71746,71748,71750],{"class":52861,"line":52862},[31,71733,69553],{"class":52865},[31,71735,71561],{"class":52873},[31,71737,71738],{"class":52865},">=",[31,71740,71010],{"class":52904},[31,71742,52959],{"class":52865},[31,71744,71745],{"class":52873}," i ",[31,71747,9758],{"class":52865},[31,71749,71566],{"class":52904},[31,71751,52891],{"class":52873},[31,71753,71754,71756,71758],{"class":52861,"line":52813},[31,71755,71017],{"class":52865},[31,71757,52899],{"class":52879},[31,71759,71760],{"class":52873},"[i];\n",[31,71762,71763],{"class":52861,"line":52920},[31,71764,52993],{"class":52873},[11,71766,71767],{},"约束变成：",[31,71769,71771],{"className":71770,"translate":35},[411],[31,71772,71774,71816],{"className":71773,"translate":35},[34],[31,71775,71777],{"className":71776},[39],[41,71778,71779],{"xmlns":43,"display":420},[45,71780,71781,71813],{},[48,71782,71783,71785,71787,71789,71791,71793,71795,71797,71799,71801,71803,71805,71807,71809,71811],{},[51,71784,5391],{},[95,71786,5662],{},[91,71788,1977],{},[1430,71790,53035],{},[95,71792,53038],{},[1430,71794,53035],{},[51,71796,5391],{},[95,71798,9758],{},[91,71800,22616],{},[1430,71802,53035],{},[95,71804,53038],{},[1430,71806,53035],{},[51,71808,5391],{},[95,71810,9758],{},[91,71812,1977],{},[55,71814,71815],{"encoding":57},"i \\ge 0 \\;\\wedge\\; i \u003C 100 \\;\\wedge\\; i \u003C 0",[31,71817,71819,71838,71862,71880,71904,71922],{"className":71818,"ariaHidden":62},[61],[31,71820,71822,71826,71829,71832,71835],{"className":71821},[66],[31,71823],{"className":71824,"style":71825},[70],"height:0.7955em;vertical-align:-0.136em;",[31,71827,5391],{"className":71828},[75,76],[31,71830],{"className":71831,"style":293},[119],[31,71833,5662],{"className":71834},[297],[31,71836],{"className":71837,"style":293},[119],[31,71839,71841,71844,71847,71850,71853,71856,71859],{"className":71840},[66],[31,71842],{"className":71843,"style":326},[70],[31,71845,1977],{"className":71846},[75],[31,71848],{"className":71849,"style":293},[119],[31,71851],{"className":71852,"style":120},[119],[31,71854,53038],{"className":71855},[124],[31,71857],{"className":71858,"style":293},[119],[31,71860],{"className":71861,"style":120},[119],[31,71863,71865,71868,71871,71874,71877],{"className":71864},[66],[31,71866],{"className":71867,"style":71648},[70],[31,71869,5391],{"className":71870},[75,76],[31,71872],{"className":71873,"style":293},[119],[31,71875,9758],{"className":71876},[297],[31,71878],{"className":71879,"style":293},[119],[31,71881,71883,71886,71889,71892,71895,71898,71901],{"className":71882},[66],[31,71884],{"className":71885,"style":326},[70],[31,71887,22616],{"className":71888},[75],[31,71890],{"className":71891,"style":293},[119],[31,71893],{"className":71894,"style":120},[119],[31,71896,53038],{"className":71897},[124],[31,71899],{"className":71900,"style":293},[119],[31,71902],{"className":71903,"style":120},[119],[31,71905,71907,71910,71913,71916,71919],{"className":71906},[66],[31,71908],{"className":71909,"style":71648},[70],[31,71911,5391],{"className":71912},[75,76],[31,71914],{"className":71915,"style":293},[119],[31,71917,9758],{"className":71918},[297],[31,71920],{"className":71921,"style":293},[119],[31,71923,71925,71928],{"className":71924},[66],[31,71926],{"className":71927,"style":326},[70],[31,71929,1977],{"className":71930},[75],[11,71932,71507,71933,71935],{},[7246,71934,59102],{},"，越界路径被排除。",[11,71937,71938,71939,71942,71943,71946],{},"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 ",[45608,71940,71941],{},"a[j]"," 不会影响对 ",[45608,71944,71945],{},"a[i]"," 的读取」，可以把「影响了」写成公式，再问它是否可满足：",[31,71948,71950],{"className":71949,"translate":35},[411],[31,71951,71953,72051],{"className":71952,"translate":35},[34],[31,71954,71956],{"className":71955},[39],[41,71957,71958],{"xmlns":43,"display":420},[45,71959,71960,72048],{},[48,71961,71962,71964,71966,71968,71970,71972,71974,71988,71990,72002,72004,72006,72008,72010,72012,72014,72016,72018,72020,72022,72024,72038,72040,72042,72044,72046],{},[51,71963,5391],{},[95,71965,21271],{"mathvariant":494},[51,71967,2555],{},[1430,71969,53035],{},[95,71971,53038],{},[1430,71973,53035],{},[48,71975,71976,71978,71980,71982,71984,71986],{},[51,71977,14551],{"mathvariant":494},[51,71979,30042],{"mathvariant":494},[51,71981,54404],{"mathvariant":494},[51,71983,30042],{"mathvariant":494},[51,71985,1412],{"mathvariant":494},[51,71987,12271],{"mathvariant":494},[95,71989,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[48,71991,71992,71994,71996,71998,72000],{},[51,71993,14551],{"mathvariant":494},[51,71995,12271],{"mathvariant":494},[51,71997,2552],{"mathvariant":494},[51,71999,2549],{"mathvariant":494},[51,72001,30042],{"mathvariant":494},[95,72003,211],{"stretchy":210},[51,72005,5813],{},[95,72007,1210],{"separator":62},[51,72009,2555],{},[95,72011,1210],{"separator":62},[51,72013,5354],{},[95,72015,220],{"stretchy":210},[95,72017,1210],{"separator":62},[51,72019,5391],{},[95,72021,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,72023,21271],{"mathvariant":494},[48,72025,72026,72028,72030,72032,72034,72036],{},[51,72027,14551],{"mathvariant":494},[51,72029,30042],{"mathvariant":494},[51,72031,54404],{"mathvariant":494},[51,72033,30042],{"mathvariant":494},[51,72035,1412],{"mathvariant":494},[51,72037,12271],{"mathvariant":494},[95,72039,211],{"stretchy":210},[51,72041,5813],{},[95,72043,1210],{"separator":62},[51,72045,5391],{},[95,72047,220],{"stretchy":210},[55,72049,72050],{"encoding":57},"i \\ne j \\;\\wedge\\; \\mathrm{select}\\bigl(\\mathrm{store}(a, j, v), i\\bigr) \\ne \\mathrm{select}(a, i)",[31,72052,72054,72105,72129,72237],{"className":72053,"ariaHidden":62},[61],[31,72055,72057,72060,72063,72066,72102],{"className":72056},[66],[31,72058],{"className":72059,"style":4202},[70],[31,72061,5391],{"className":72062},[75,76],[31,72064],{"className":72065,"style":293},[119],[31,72067,72069,72096,72099],{"className":72068},[297],[31,72070,72072],{"className":72071},[297],[31,72073,72075],{"className":72074},[75,21516],[31,72076,72078],{"className":72077},[21520],[31,72079,72081,72084,72093],{"className":72080},[21524],[31,72082],{"className":72083,"style":4202},[70],[31,72085,72087],{"className":72086},[21531],[31,72088,72090],{"className":72089},[75],[31,72091,21538],{"className":72092},[297],[31,72094],{"className":72095},[21542],[31,72097],{"className":72098},[119,21546],[31,72100,223],{"className":72101},[297],[31,72103],{"className":72104,"style":293},[119],[31,72106,72108,72111,72114,72117,72120,72123,72126],{"className":72107},[66],[31,72109],{"className":72110,"style":66209},[70],[31,72112,2555],{"className":72113,"style":5514},[75,76],[31,72115],{"className":72116,"style":293},[119],[31,72118],{"className":72119,"style":120},[119],[31,72121,53038],{"className":72122},[124],[31,72124],{"className":72125,"style":293},[119],[31,72127],{"className":72128,"style":120},[119],[31,72130,72132,72135,72141,72147,72153,72156,72159,72162,72165,72168,72171,72174,72177,72180,72183,72186,72189,72195,72198,72234],{"className":72131},[66],[31,72133],{"className":72134,"style":4988},[70],[31,72136,72138],{"className":72137},[75],[31,72139,54443],{"className":72140},[75,2745],[31,72142,72144],{"className":72143},[264],[31,72145,211],{"className":72146},[1307,1308],[31,72148,72150],{"className":72149},[75],[31,72151,55378],{"className":72152},[75,2745],[31,72154,211],{"className":72155},[264],[31,72157,5813],{"className":72158},[75,76],[31,72160,1210],{"className":72161},[1391],[31,72163],{"className":72164,"style":1467},[119],[31,72166,2555],{"className":72167,"style":5514},[75,76],[31,72169,1210],{"className":72170},[1391],[31,72172],{"className":72173,"style":1467},[119],[31,72175,5354],{"className":72176,"style":2666},[75,76],[31,72178,220],{"className":72179},[289],[31,72181,1210],{"className":72182},[1391],[31,72184],{"className":72185,"style":1467},[119],[31,72187,5391],{"className":72188},[75,76],[31,72190,72192],{"className":72191},[289],[31,72193,220],{"className":72194},[1307,1308],[31,72196],{"className":72197,"style":293},[119],[31,72199,72201,72228,72231],{"className":72200},[297],[31,72202,72204],{"className":72203},[297],[31,72205,72207],{"className":72206},[75,21516],[31,72208,72210],{"className":72209},[21520],[31,72211,72213,72216,72225],{"className":72212},[21524],[31,72214],{"className":72215,"style":4202},[70],[31,72217,72219],{"className":72218},[21531],[31,72220,72222],{"className":72221},[75],[31,72223,21538],{"className":72224},[297],[31,72226],{"className":72227},[21542],[31,72229],{"className":72230},[119,21546],[31,72232,223],{"className":72233},[297],[31,72235],{"className":72236,"style":293},[119],[31,72238,72240,72243,72249,72252,72255,72258,72261,72264],{"className":72239},[66],[31,72241],{"className":72242,"style":260},[70],[31,72244,72246],{"className":72245},[75],[31,72247,54443],{"className":72248},[75,2745],[31,72250,211],{"className":72251},[264],[31,72253,5813],{"className":72254},[75,76],[31,72256,1210],{"className":72257},[1391],[31,72259],{"className":72260,"style":1467},[119],[31,72262,5391],{"className":72263},[75,76],[31,72265,220],{"className":72266},[289],[11,72268,72269,72270,72354,72355,72357],{},"数组理论会展开 read-over-write 公理，算术理论会处理 ",[31,72271,72273,72291],{"className":72272,"translate":35},[34],[31,72274,72276],{"className":72275},[39],[41,72277,72278],{"xmlns":43},[45,72279,72280,72288],{},[48,72281,72282,72284,72286],{},[51,72283,5391],{},[95,72285,21271],{"mathvariant":494},[51,72287,2555],{},[55,72289,72290],{"encoding":57},"i \\ne j",[31,72292,72294,72345],{"className":72293,"ariaHidden":62},[61],[31,72295,72297,72300,72303,72306,72342],{"className":72296},[66],[31,72298],{"className":72299,"style":4202},[70],[31,72301,5391],{"className":72302},[75,76],[31,72304],{"className":72305,"style":293},[119],[31,72307,72309,72336,72339],{"className":72308},[297],[31,72310,72312],{"className":72311},[297],[31,72313,72315],{"className":72314},[75,21516],[31,72316,72318],{"className":72317},[21520],[31,72319,72321,72324,72333],{"className":72320},[21524],[31,72322],{"className":72323,"style":4202},[70],[31,72325,72327],{"className":72326},[21531],[31,72328,72330],{"className":72329},[75],[31,72331,21538],{"className":72332},[297],[31,72334],{"className":72335},[21542],[31,72337],{"className":72338},[119,21546],[31,72340,223],{"className":72341},[297],[31,72343],{"className":72344,"style":293},[119],[31,72346,72348,72351],{"className":72347},[66],[31,72349],{"className":72350,"style":66209},[70],[31,72352,2555],{"className":72353,"style":5514},[75,76],"，两个理论协作给出 ",[7246,72356,59102],{},"——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。",[53200,72359,72361],{"id":72360},"_56-完整流程","5.6 完整流程",[11,72363,72364],{},"把上面的步骤串起来，一个典型的漏洞检测流程是这样的：",[55589,72366,72367],{},[55592,72368],{"height":72369,"width":72370,"style":55596,"data-d2-theme":55597,"alt":52812,"src":72371},240,2053,"data:image\u002Fsvg+xml;base64,PHN2ZyB3aWR0aD0iMjA1MyIgaGVpZ2h0PSIyNDAiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgZGF0YS1kMi12ZXJzaW9uPSJ2MC45LjAiIHZpZXdCb3g9IjAgMCAyMDUzIDI0MCI+PHN2ZyB3aWR0aD0iMjA1MyIgaGVpZ2h0PSIyNDAiIGNsYXNzPSJkMi0zNzYyNjY4MTczIGQyLXN2ZyIgdmlld0JveD0iLTEgLTEgMjA1MyAyNDAiPjxyZWN0IHdpZHRoPSIyMDUzIiBoZWlnaHQ9IjI0MCIgeD0iLTEiIHk9Ii0xIiBzdHJva2Utd2lkdGg9IjAiIHJ4PSIwIiBzdHlsZT0iZmlsbDojZmZmIi8+PHN0eWxlPi5kMi0zNzYyNjY4MTczIC50ZXh0LWJvbGR7Zm9udC1mYW1pbHk6JnF1b3Q7ZDItMzc2MjY2ODE3My1mb250LWJvbGQmcXVvdDt9QGZvbnQtZmFjZXtmb250LWZhbWlseTpkMi0zNzYyNjY4MTczLWZvbnQtYm9sZDtzcmM6dXJsKGRhdGE6YXBwbGljYXRpb24vZm9udC13b2ZmO2Jhc2U2NCxkMDlHUmdBQkFBQUFBQTRvQUFvQUFBQUFGWHdBQWd1RkFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQlBVeTh5QUFBQTlBQUFBR0FBQUFCZ1h4SFhybU50WVhBQUFBRlVBQUFBckFBQUFPd0VPQVU4WjJ4NVpnQUFBZ0FBQUFlTUFBQUo3QkhWK0RSb1pXRmtBQUFKakFBQUFEWUFBQUEyRzM4ZTFHaG9aV0VBQUFuRUFBQUFKQUFBQUNRS2Z3WGhhRzEwZUFBQUNlZ0FBQUNJQUFBQWlFRmZCYjlzYjJOaEFBQUtjQUFBQUVZQUFBQkdMUzRxMkcxaGVIQUFBQXE0QUFBQUlBQUFBQ0FBT2dEM2JtRnRaUUFBQ3RnQUFBTXZBQUFJS2dqd1ZrRndiM04wQUFBT0NBQUFBQ0FBQUFBZy85RUFNZ0FEQWlvQ3ZBQUZBQUFDaWdKWUFBQUFTd0tLQWxnQUFBRmVBRElCS1FBQUFnc0hBd01FQXdJQ0JHQUFBdmNBQUFBREFBQUFBQUFBQUFCQlJFSlBBQ0FBSVAvL0F1Ny9CZ0FBQTlnQkVTQUFBWjhBQUFBQUFmQUNsQUFBQUNBQUEzaWNoTTA3SzhWeEFJZnh6OS81dVIvMysvME1Ob05CM2dDZFNlb1VrMDJTMUVreWVFY0lrMFd5OEZMWW1DejJyenFUN2ZTc24zcFFxYWxRVjN4Z1hVTlJOR3pZdEdYYnJxWjlMWWVPSER0MXJ1M1NkY0kvdGFOcFQ4dEJSNTA0MDNiaEtzbG5mdk9UNzN6bFBXOTV6VXVlODVUSFBPUStkN25OVGVmZnZjcWFWY3RXOUtncGV2WHBOMkRRa0dGMUkwYU5HVGRoMHBScE0yYk5tYmRnMFJKL0FBQUEvLzhEQUtaYkwxbDRuR1JXU1d3YjF4biszeVBGaVNqSzBuQTRISkxpTm56a0RCZUpzamdjamlSS3BpaFIxR0xSMm14WmpyVTRncHNva1d3SHRsekxpWUFjYWlSb1NxTW82Qlp1QWpRTFdyUUZHdFNGVWRRMTRCNEtkREdTVXgwMHA5aHUwVXR5cUlXVVhkSlF3MktHbEd5cnAzY1l2di8vTnY3L2d3YVlBc0FyK0JvWW9CRmF3QW9zZ0VUemRFZ1NSVUlwa3FJUXpxQ0lpS2Ftc0ZYOTBRL0ZpREVTTVViOTEzMnZMQytqd2hLK3RyTitzckN5OHEvbGRGcjl3YTl2cTFmUmhkc0FCaWdBNEQ1Y0JCcGNFTkJxU2dtN25iV1pLRlkvVE1RZ0pWSnlVaUNFbGhMNldYaVFXeC9vQ0NlR2N1ZEhsb2RUWFlsa2Z2WnlYLzhzTG5yeW1kaHNpN0g1Y0hib2FBUzlFU1dDWDUyZmo0VUFFTVNyWlh3UVg0YzJnSWFBSU1qSlZFcEsyRGxLRUVqQVpHSnRkaW1SVWpnVFdweDVjL2JZMVpuTWFmNklVeUh0WTdHNTBYREdjV1RHTXZIZE0rdmZuNVlDUzV3bnNUUjQrdVdnYytFVUlDZ0FvRWU0Q0UyNkhpelBTaXhoZWJhQXJxdGYzYitQV25CeDZ4dXZmbThMTkF3YTEzVmNCSXZPazVZWXljQVFBOFVXU3NiZnZ2Kzd6OTU3WndJWDFYK2lKcldpYmlMbTlNL3JkOUJmY1JFYWFuZDR0bEJDR0JkM3RyZGdyK1lIdUFnKy9UdGp0M05TS3FVd0VrMDBpZ3FoS0NLS3hJdFp0dkRlaTJhcjJXaW16Uys4K3pyVmFEREtpOU9MU2FQeEdRb1gxZnZ1UTE3dklUY0s3R3c4OGs5TytkNys4c3UzZlZPVC9rY0FXTzl4VU1kdGU4SWhFeUhzbmljUFJ5L204eHZEMDZPYkEzMDVYQlFYSmlkV09qOUZNNnRTdElhVFZNdllqSzlEVk5kZlZPeWE0SEpTRU1VNGZ0b00xbWJudUZvSFpCdDRMWEdVeklYakhWTHNHTjhucEYvTWRiOGNQZXdmRUlXT251alJkTDczck9WZy9HdGVJZUR4ZWF6QkE1MzV6dFI4c2oyNjZHenp1YjFlT3VBNE9weGE2QVlFVGdETTRDSlFHZ01pOHl5aFA3cUovbnNUdDI1dDdXelhNQVlCVUFVWG9SRkFraEV2OHl6aTJTRHlxMStoMitwRFJFNmdYMjJkVU1lMk5FMmkxVEw2R0ZYQUNRU0FDMmlCVXZRc1VhSk9ocVdKU0V3bUpaRlNaRDFmdjhsTlhTbGhFdkVOQk9YT3RkN2w1emZOUnQvSU04NFFjNlRQWnptZU9UTGZ3b3NPOWpsUDhPeDU5VytTbTV6bm1PUG1tTWZCMWJCVnkrZ09xb0JyZjM1SjRIRjZUY2c1ZkM0Nyt2VmNmTVE5VFB4eUpuUFFFV2Q2UTNPVy9vc3pzeHY5WG03Wk01RWRLTEF0cC94dEdtY01ZcldNS3ZnT01PRGY1YUhCNTBSWmVvTEJyakZmTEp4TEx5Y2ozVTVUYWROc2RPV3hRN1F5TVJ0SmRWcStkWG42NGlHM1krS25PME5kTHJKcGMzNW9QVEEwTWpZTVdNZitGMVFCQi9pZVFxOTVUUEZhRWpUc0JpbXBkVUcra2ZPRFErdnBrY1ZPSTFZL01lZTc1RlNYc1BUV1RiRTlrTEljMnBpWjNzaGsxbkpNcURFbDhTZGNYdFFia1RzMUxnWUlWRHN3aFNyUUNXa1kxOWtJY2xLUjlYNzFJeVVsT0lrbDlRQUhSSTJVcE5sbE01a01laUJyUkpuNjBBa0kraysrNkYzcUhtSGEvQTVYcEhkSmJ1ZC9PVWsxSnVjVmo4OGFpRXd0UEpmYkd2ZUlvc2NqaXBIRWdCaVNuTHlscmYrZXE3dTlMMnhzRHZ2YUVxMUdheTdXTnhtMnJEVUZiRDNqUVhPTG5iR21oNlRwT0xvYmpZaVJjRGdTVlV0Qko5ZHFNRGljYm8vR0IwRldNd2pmcWYvcldHbzNXTFF1RkVWblM1VDdjR0o2ck9UeHU4TU9mT2RuSjV5eHRVWDFJOFNud2s1Ty9RVlVxNkFBd0tmNEhoWTBmNEVDSHQ2czFhNldrUlhmZ1JaZEo1bVc2TDBRL1hFaVhhSWJHeWlUMVJLeW5EeU15YzRubkJXaE13M1VMaVpVcVdQaXBQL0R0R2syK2d0N29OQjJ4dHZ4RktaYUhuU2ZXdlpOWTkwRjhRa1hrRDF6THBjN2w4bWN6ZVhPWmpyaThZNTRSMGM5eS8wYnN6TVgreThWQnJJVFdxUTFXTm5xS0xhakNqRGdCZUFlbzlPdEZrU09aYlRhSktEdGx1eW0yZWdaRTU5ZDdWdE8rZnRjRFpOQ2FpNFd0WVZ2NFo5MHVjZzNMeHpiekxRNUo3K0Rndm1KMXpzK3RCNm8rNEcralNwZ2ZaSjdmV3JWbUxkTkNLemI3R2gydHJyN2JXajdlS0tyb2VFMW96R1NVQjhDQXJaYVJ1K2dDb2k2NW85bm9GQ2JnWHZGdEFub3hhek5kSy9yQldFd2tQSHhYay9jNVUySFh6eldjOXczNkVxNmVub0VmMzlrMVNMNEZweHRIRVBiR2JNbDJCTVpuaE1kOHphNzZIQWVhQ0k5OGFIRldvN29haG1keFJ2QTZXckxNcEVWUmRJWDFlUEJBUXVUdVFuNmxVdVhpTWZpTkhPTVlubHA3dTRaMDVVckYvNFFEWm1NYXlaTHJWWmZ0WXorZzdiQnRpODNkSDFjL0hsNnJPVDF1d1Y3YWJQSjRCdTNyQzJpcFBwQWpyZzhhRlJ0SFE2MUE5TDJINnFpYldnR2tBd1NWOTlaaW1TNCtlTnJBMmJHYkd4a3pObXI3NlB0ejBNRlVTeUVQbGRiOWQ3MmFoazl3RzlBWkw5K1JONE56ZTcwZmZ5QStFZmhETWw1OHVIT2J2ZjQ4TnhBV0FnbzN2SDJsZDZWeTRxa2pHVFhMSW53b2pzb0J0MFIrMnFud0llOHJtZUYyTW5acnJ6ZDJGbzRsSjZOMVhoejFUSjZpRjhGMzc3ZW12VWl3N09FMmt2d3Y0K3NDNE9lWExpcnQ3dmRIZklNV3RIcVowMjhvSnpzenI1a1NZWVdYYUZFMThIRUFXc1VaYmN1dFVTUDUvS25rM3FQNXVvcGxNSy9Cd01BeDBpRzVydW43cjVyZUw3eWx2WXRBRXZvN3pnRlRRQ0tUR1JKcmhuNHB4czMxbS9jV0xxMWV1dlc2cTE2UnVGanRLM1YwTjRMMlJMYVZsc0JWVC9BUFRDTDcybjNhWDFiMVZRS3hlT2hVRHlPZTZLRVJLT0VST0YvQUFBQS8vOERBTHloQ3FVQUFRQUFBQUlMaFlCTE1HVmZEenoxQUFFRDZBQUFBQURZWGFDRUFBQUFBTjFtTHpiK04vN0VDRzBEOFFBQkFBTUFBZ0FBQUFBQUFBQUJBQUFEMlA3dkFBQUltUDQzL2pjSWJRQUJBQUFBQUFBQUFBQUFBQUFBQUFBQUlnS3lBRkFBeUFBQUFsMEFUUUpHQUM0Q0pBQk5BbVlBVFFJR0FFMEMrZ0JOQWxRQVRRSXNBQ01DTEFBWkFoMEFKQUlQQUNvQjB3QWtBajBBSndJR0FDUUNGZ0FpQWpzQVFRRVVBRGNCSGdCQkFqd0FRUUlyQUNRQ1BRQkJBWTRBUVFHN0FCVUJmd0FSQWpnQVBBSUxBQXdDRUFBV0FoQUFGd0ZUQUEwQ0VBQWlBUlFBUVFBQS82MEFBQUFzQUN3QVlBQ01BS0lBdkFETUFQNEJJQUZnQVhJQmlBSEFBZXdDSGdKU0Fyb0MzQUxvQXdRREpnTlNBNElEb2dQZUJBUUVKZ1JDQklBRXNnVEFCTlFFNEFUMkFBQUFBUUFBQUNJQWtBQU1BR01BQndBQkFBQUFBQUFBQUFBQUFBQUFBQVFBQTNpY25KVFBiaHRsRk1WL1RtelRDc0VDUlZXNmliNEZpeUsxWTFNbFZkdXNKcVJXUmtSeDhMZ2dKSVEwdGllMjVmSE15RE4yR3A2QU5XL0JXM1RGUS9BY2lEVzYxOWV1WFJBRks0cDFadWIrT2QrNTUzN0FBWCt5VDZWNkgvaXRuaHF1Y0ZTL05yekh2ZnFGNFgxYTlUM0RWUjdWZmpkY1kxQmJHSzd6ZWExaitDUGVWbjh4ZkkvajZvK0c3M05ZYlJuK21LZlZBOE9mN0R2K01Qd3B4N3hkNGdvODUyZkRGUTdKRGU5eHdBK0c5M21BMWF4VWVVRFRjSTNQT0RKYzV3am9NcVlrWVV6S0VNY05ZNGJNbVJGVEVCSXpZOHdOTVFNY0FUNEpwYjVOaVJRNWh2LzROaUtrWkVha0ZVZVVPQkpDRWlJS1JsYnhKODNLdU5hTzBtZW1TTHI1bEl5STZHbkdoSWdVUjhhUWpKU1lpZFlwS2NsNVNZTUdCWDNsbTFOUzRGRXdKc0VqWThhUUJtMWFYTkpseEpnQ1Iwc3JDYk9RakJ0S2JvbTB2N01JVWFaUFRFcGhyT2FrREpTblUzNnhaZ2RjNG1pVGEreG01Y3V0Q285eGZLdlp3azFpSEYvaTZiL2JZTGJkZDhVbVlxRjZpb1k5RXVWNXF4TWNxZUxTMStjYnhTVXZjVHZwczgza3d4b05sSjNNZWt5UHVjNWY1aWQ1d2lURnVVTjhRblZRNkI3aU9OUG5nRkFWK1k2QUxoZTBlVTFYbjMwNmRQQzVva3ZBSzgxdDA4SHhGVzJ1T05lTVFQSHlXMHNkYzhYM09MNG0wQmlwSFpzK29yazh2U0UzZHd0M2NZYWNZMHF1V3lBemx2T0w4K09kSml3N2xHMjVvMUJYOUhXUEpGTDJRRlNSUFlzWW1pdHlkY1ZVdFZ4NW96RDlCdVlJK1ZycWJOOTlsMjFZMk82dHR2aU9USGZZTVRkT01ya2xvdzlOMVh2UE03ZjY1eEV4SUtPbkVYMHlwalFvdWRPelhSTXhKOEZ4cmo2KzBDMHA4ZGM1MHVkT1hSSXpWUVlCcWRhWmtldHZaTDRKalh0L3kvZk83aExadktudTZHUjlxbDI2U09WMFkwYXZWYjNWdDlCVWpqY20wTENwWnBZakU1Ykt5NU9LOXlYYTIrSWZxdnNMdmQweW5uQkdSc0xnYnpmQVV6eWFISFBDS1NQdElKVlRGblk3TlBXR2FQS0NVejM5aEZpajVMNThvenBKaFJNOG5uSENDUzk0OXA2T0t5Yk91TGcxbDgzZVB1ZWMwZWIwUDUxaUdkL21qRmZyZC8vZTlWZGwydFN6T0o2c241N3ZQTVZILzhPdFgvQjQ2NzdzNkMwZ203T3dhdSsyNG9xSXFYQnhoNXRhdUlkNGZ3RUFBUC8vQXdEMHQwOVJBQUFEQUFBQUFBQUEvODRBTWdBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQSl9LmQyLTM3NjI2NjgxNzMgLnRleHQtaXRhbGlje2ZvbnQtZmFtaWx5OiZxdW90O2QyLTM3NjI2NjgxNzMtZm9udC1pdGFsaWMmcXVvdDt9QGZvbnQtZmFjZXtmb250LWZhbWlseTpkMi0zNzYyNjY4MTczLWZvbnQtaXRhbGljO3NyYzp1cmwoZGF0YTphcHBsaWNhdGlvbi9mb250LXdvZmY7YmFzZTY0LGQwOUdSZ0FCQUFBQUFBNUFBQW9BQUFBQUZmUUFBUmhSQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFCUFV5OHlBQUFBOUFBQUFHQUFBQUJnVzFTVmVHTnRZWEFBQUFGVUFBQUFyQUFBQU93RU9BVThaMng1WmdBQUFnQUFBQWVvQUFBS1hCOUkxQWRvWldGa0FBQUpxQUFBQURZQUFBQTJHN1VyMm1ob1pXRUFBQW5nQUFBQUpBQUFBQ1FMZUFqR2FHMTBlQUFBQ2dRQUFBQ0lBQUFBaUR4TkErVnNiMk5oQUFBS2pBQUFBRVlBQUFCR0x2Z3NlbTFoZUhBQUFBclVBQUFBSUFBQUFDQUFPZ0QyYm1GdFpRQUFDdlFBQUFNckFBQUlNZ250VnpOd2IzTjBBQUFPSUFBQUFDQUFBQUFnLzhZQU1nQURBZUVCa0FBRkFBQUNpZ0pZLy9FQVN3S0tBbGdBUkFGZUFESUJJd0FBQWdzRkF3TUVBd2tDQkNBQUFIY0FBQUFEQUFBQUFBQUFBQUJCUkVKUEFBRUFJUC8vQXU3L0JnQUFBOWdCRVNBQUFaTUFBQUFBQWVZQ2xBQUFBQ0FBQTNpY2hNMDdLOFZ4QUlmeHo5LzV1Ui8zKy8wTU5vTkIzZ0NkU2VvVWswMlMxRWt5ZUVjSWswV3k4RkxZbUN6MnJ6cVQ3ZlNzbjNwUXFhbFFWM3hnWFVOUk5Hell0R1hicnFaOUxZZU9IRHQxcnUzU2RjSS90YU5wVDh0QlI1MDQwM2JoS3NsbmZ2T1Q3M3psUFc5NXpVdWU4NVRIUE9RK2Q3bk5UZWZmdmNxYVZjdFc5S2dwZXZYcE4yRFFrR0YxSTBhTkdUZGgwcFJwTTJiTm1iZGcwUkovQUFBQS8vOERBS1piTDFsNG5IeFdiV3diWngzL1A4OWQ3cExXU2VPY1gybzNmbjNzcytPY25mZ2UyeGZIc1IzYlNaejRwVW02cHUyYU9GbXJwVzhySldxM1ViUjFZNjFVZGZzd1RGVStNRTB3OFNhbVNoU2xYNWlvTnFFaVZvWXE4YUdnb1UwSUJHU3dJZzJpcUl5cE9hT3ozU1NORUYvT0ovbnUvL3YvZnYvZi8vY2N0SUFIQUovQjE0Q0JOdGdGWFdBRW9JS0xZYWlpRURORGZUN0M4NHBQRUhqUFJYVG40aHRzN3ZCZi9XLzlSM0t3K1ZmZUx2N2pxZXY0MnZwcDlMWEt5eStyczFjV0Z3L2V2NjhHME8vdUF3QXdRQUN3RTFkQkQxYnRuZ3BVTmhrTkhNZnpwdm92WWFnY2kwWkVzbmxETHQxWU9OMmI4eUE2bG45eDcrRDgvT0hSd3V3elorZlBsQ2VleGRWQ1hocVJXbGxkWm1DaUlxSG44MHBRWHY5MHRDUW5OVHdFOGRvYUR1STN3UUhRNGhiRmFDU0ZxV3d5ODZKSTNCM1lhRENacUJ4VHpCeUgzTVVUc2Y3REYwb0QwN3RqUWt3Y1hNaDYzSVdFUCtja25vb3VkMzV2K2RwWDhrcWd4K2xMUG4xK0tGR0pPdmZJam1BZFErUFVpYXV3VStQajRsMDg1UW52NHNrbGRLcGQvWFBnWHgyZlVTUjI0R3JtdDlrSDJlYnpCM0VWZE5yemxLR0k4Z0poZUo1YzJwdGgwTVNoQjkrY2Z1blZJSzZxdDlESVEvVTBPbnI1NDBjNDZDcXVRa3ZqUFExaDcvUEkwSTZyNjh1UDZ0N0NWYkRVL3hmTVZCRW9RNFJZVENFOFF4Z2Y0VGllSVpjcWNSTTdkcnR5cVZocXMrcll5WjlMU1JQTGRiUVdjRlg5OXBVcjZPajZFam9ybmVxOXFuNGZ6VjJWVGtycTY0QTFiQnlzY3pSc21Sb2hqTEE1cG5mbXpoWmUyWDh5a2xsWVBGVWFYOFRWd29HcDQySDFjNVNmbW96VHhqeDB0VFdrNGpjaEFHQjJpejZscm44MEl2cDgybkJpc1kzaGNKelJZREtiRzY3NEpMZmtqOXRtbEtIcG9MY1VTRVRuRW9tbkhOUXlGdkpHYldGUHFTK1NPS1liSE96dGxVY0dQTElwWkoxUTVIMXl4Qit5OXpqNjk0aDlwbUIzWGhtY2pRQ0NDZ0NPNGlyd0dndWl1SGpDL1BDNWQ5dlJyOXZmZXc2WGM3bjFtL1UrYXpVQWJNTlZhQU1nQ25JcExoNjVlTFFEdmEzZTYwQ3Y3bEFQbytLdVB1ek1oTmIvbUFYQTRLdXRvYy9SS2hnMHBjenVEYU5SaFRKRUlSem5rMk9Lc3VHNm04TWxxVEJQZlVrOUs2U09wRnRaY3FoTG5QUklScm5iazRzNndyclptYkd2emxHL0s2bGF4NzE5dzZHKzM0dnV3RVJGVGpkOTdhMnRvV1cwQ3QyUG9XMHExM1QxaDVOUFMrVWpVV25JRkJSRVcvK0JXSHpRR1RPNXJXWGRzY3JJdVprK3Q2WGZiQnhaeW1YSHJIclo0TlZxMTdsZ0g3NERSdkE4VnAwcS81L01ZQmZUS1phclRUWjd2ZHZaK0p3TDc2NFBiS2VENjF6ZVE2dGdCZTlXUEczeXZJdmIyRkNHeGpTbmFRei9jdUJrc0RqWHIyVHN1aGIxRjIzT1hNQVdOOXR0MDkrcVlhYXJoMFRuZGFlT2pDN3RrMEpUY2pmdFNFOTVMWHBxZENEdnp0M3QzV0hIREdCQU5ROWFSYXZnZ05CV0p5b0t4NUd0akNuRGNjeGpiSytIRHhCUDk2Zy9WZWl3aVB2N2tsTzlFM05oTWFWbmhQUXg0VnljVEx0N1RlRnVrcUgydm85Rlc5VHNMZzJmRUtVRE03bG5uNVMxZVRJTHg1Q3JOL0FiMGQwemRxZy9rV2pNMHdHQVBzUjNtcnZMODdTK0NrWURYMS9nYUlTNE9aNXh2Rjd1NzJSNzlrbXBhR3VxTk1TeTQ5M2pvVkY4NTM2UzlHVUdIQjcxQXlRWmRyY1hBeUgxUjdXYWxubndCVjdHSWpnQmdBUFhlQU5McXEzQkYvZ09kR25Nb3hFdEpiUnRhOHI4cFF6M1F2a0NRbnFHNDlFT2t5NnR0K0JuMXIvQnR6RmRDQ2RZZGt1L2FCVjJOL3I5MyswZVNmTnN6NzdRWTkyaWxRbFBlSHV6Rzc1REg2RlYyQVcyclQ0d0dqcXdUOWFxUGpMM3ZjbDVxVEF2VHk1SXhmbEFjSnJHWk8yaU96RTdlbTRtMUxnT1o1ZEdzdm5jMGtoMlRLdGRlMUNqNko5b3RlRnBma3ZISFppNFJlMUVFK1FVYmtCb1o5R08xOUljNDUwSjFmZFVGb2NFM09YNGdTY1h0ZmYzdUtkSnlFRHY0cHZEam1EVDJJNFQzMEVvTUZHaHFXUkEvSnZYdGFuUmkyZ1ZPcmRvWk9iRlI5cnNaRzJsb01XNHA5UHFLVG1TYUtVaUpkdEdXdE1KOVM2ZzJzUGFHcnFBVnNHM1BTZTN4NlNXa28yUS9GNjRZdWszRDR1QlpNOUFLQzVOU0tGQ2QwaWdMakVjYzZZaS9mdDBFYi9vOEllSTFlZXdwbnA2TTE2UDNXK3dCaDEyc2NzOUpBVkh2RnJQUTdVMWRBaWYzc2lWbUNLUU5LYmFrY1pzeVpWM2hpTXNpdWQzbGp5WlBTL29Mc1NaYm5lSGRhZStzMCtYRHU2eXRxT3VlTXZseXluMTA2NHV1MzFIaThMdjBtb1AxTmJRWjJnRkxKdTFONTBuTktQbGVqTEdzcWx5a21YSGJYbHB0S1NGbzMrL0xxdm9IUUtLcWZjRWkyWVpkRWkxRmtqelRFa0FvRCtoRldnSG9Bd1ZUQ1l6aldrRjBjVjh5Y055TEt2M0NGOHZxK3RvUmYyRUZJbG53b01zcXJYK2J1M2Z0VFgwSy93YStMZnBUSlFOeC9HK2hyYzUzdGdRK21lWlJUczFGOElqQi9jdjZpWm5mVEsxNVd5K0p5cFRCNHVGYUNKNVVwY0ordDJSWXB4bUIzdVM5a0NzMjB6VFU5bmtuSkhWajh2Sko4T2FGbEJiUXgvaGw4QzVGWGNEVUhEeGhHOGFsT051WmVadHNpa3pFQmlYMGhHSDVIUk5vZDcydjBmMEFjdjRRdTZNTGgzc2NVVUNaWm9hNnRSYlVUQjdxMVUzODBUcHk5cEJnV28vclQyRnZvdC9DUXdBanlnYVI4c0Rhdmt0NXRqRE4rcjhvUitPb3c5d1FQdDJVYUlrU3FQVVNJM0UrSWNmM3hpNjhaUGo3OGR2MzQ2LzMvUXkzRVVyV2gzdDI4TnhwSHdVcmRSRlJKREhSVmpHeTFvTlFjdk9adC9uQlRzeEcyd0VGODBtaTJ1M3llTDhMd0FBQVAvL0F3QTYrQ1BGQUFFQUFBQUJHRkZ2TGRoVFh3ODg5UUFCQStnQUFBQUEyRjJnekFBQUFBRGRaaTgzL3IzKzNRZ2RBOGtBQWdBREFBSUFBQUFBQUFBQUFRQUFBOWorN3dBQUNFRCt2ZjI4Q0IwRDZBREMvOUVBQUFBQUFBQUFBQUFBQUNJQ2RBQWtBTWdBQUFKSEFDTUNKZ0E1QWZjQUl3SXZBQ01CemdBakFzRUFJd0ltQUNNQitnQU1BZjRBWFFJQS8vOENHUUFuQWJNQUpRSVhBQ2NCNFFBbEFoTUFBUUlMQUI4QTdRQWZBUGdBTEFJTkFCOENBd0FuQWhmLzlnRldBQjhCa3YvOEFVVUFQQUlRQURnQndBQTdBZUQvOXdIZ0FBQUJWUCs0QWVBQU1BRHRBQjhBQUFCSEFBQUFMZ0F1QUdZQW1BQ3dBTXdBM0FFS0FTNEJiZ0dDQVpvQjBnSUFBamdDY2dLNkF1UUM4QU1TQXp3RGFnT2tBOElEL2dRc0JGZ0VkZ1N5Qk9ZRTlnVUtCUmdGTGdBQUFBRUFBQUFpQUl3QURBQm1BQWNBQVFBQUFBQUFBQUFBQUFBQUFBQUVBQU40bkp5VTNXNGFWeFNGUDJLZ1RmOHVLaXR5YnF4em1Vck80RVp4bE1SWDR6cFdSa1dRTXFRL1VsVnBnREVnWUdiRUREak9FL1M2YjlHM3lGVWZvMDlSOWJyYW13MWhJcXRXVUJSckRXZi9yTFAyMmdmWTUxLzJxRlR2QW4vVmw0WXJITlovTm55SEwrcE53M3VjMVQ4elhPV285cmZoR29QYVc4TjFIdFE2aGovaFhmVVB3NS95dVBxYjRic2NWQzhNZjg2ajZyN2hML2NjL3hqK2lzZThXK0VLUE9WM3d4VU95QXpmWVo5ZkRlOXhENnRacVhLUFk4TTF2dWJRY0oxRG9NdVlnaWxqRW9ZNExoa3paTUdjbUp5UW1EbGpMb2taNEFqd21WTG9yd21SSXNmd3hsOGpRZ3JtUkZweFJJRmpTc2lVaUp5UlZYeXJXU212dEtQMG1TdVNiajRGSXlKNm1qRWhJc0dSTWlRbElXYWlkUW9LTXA3VG9FRk9YL2xtRk9SNDVJeVo0cEV5WjBpRE5oYzA2VEppVEk3alFpc0pzNUNVU3dxdWlMUy9zd2hScGs5TVFtNnNGaVFNbEtkVGZyRm1CelJ4dE1rMGRydHlzMVRoQ01lUG1pM2NKTWJ4TFo3K2Qxdk15bjNYYkNLV3FxZG8yR09xUEs5MGdpTlZYUHI2L0tDNDREbHVKMzNLVEc3WGFLRHNaTlpqZWl4MC9qSS95Uk1tQ2M0ZDRCT3FnMEozSDhlWmZnZUVxc2hQQkhSNVNadlhkUFhicDBNSG54WmRBbDVvYnBzT2p1OW8wK0pjTXdMRnE3TUxkVXlMWDNCOFQ2QXhVanMyZlVSeitYcERadTRXN3VJTXVjZU1UTGRBWml6M0YrZkhPMDFZZGlndHVTTlhWL1IxanlSUzlrQlVrVDJMR0pvck1uWEZUTFZjZXlNMy9RYm1DRGt0ZExidnoyVWJsclo3NnkyK0p0VWRkaXlNazB4dXhlaTJxWG9mZU9aSy8zbEVERWpwYVVTZmxCa05DcTcxYnErSVdEREZjYTQrZnFsYlV1QnZjcVRQdGJva1pxWU1BaEt0TXlmVHY1SDVKalR1L1pMdm5iMGxzbmt6M2RISjVsYTdkSkhLeWRhTVhxdDY2N1BRVkk2M0p0Q3dxYWFXSXhPV3lxdWJpdmNsMml2eEQ5WDl1YjV1S1E4SnRLNU1zbi9qSzNETU0wNDRaYVJkcEhyQ2NuTW1yNFJFbktvQ0UyS05ramZ6aldyMUNJOFRQSjV3d2duUGVQS0JsbXMyenZpNEVwL3Qvajdubk5IbTlOYmJyR0xiblBHaVZPMy9PLytwYkkvMVBvNkhtNituTzAvendVZTQ5aHVPU3U5bVIxOEQyYUNsVlh1L0hTMGlac0xGSFd4cjRlN2ovUWNBQVAvL0F3QnlvVkZBQUFBREFBRC85UUFBLzg0QU1nQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUEpfS5jb25uZWN0aW9uLC5zaGFwZXtzdHJva2UtbGluZWpvaW46cm91bmR9LnNoYXBle3NoYXBlLXJlbmRlcmluZzpnZW9tZXRyaWNQcmVjaXNpb259LmNvbm5lY3Rpb257c3Ryb2tlLWxpbmVjYXA6cm91bmR9LmQyLTM3NjI2NjgxNzMgLmZpbGwtTjF7ZmlsbDojMGEwZjI1fS5kMi0zNzYyNjY4MTczIC5maWxsLU4ye2ZpbGw6IzY3NmM3ZX0uZDItMzc2MjY2ODE3MyAuc3Ryb2tlLUIxe3N0cm9rZTojMGQzMmIyfTwvc3R5bGU+PGcgY2xhc3M9ImMzSmogYXJ0aWZhY3QiPjxwYXRoIGZpbGw9IiNmZmY4ZTYiIHN0cm9rZT0iI2M5YTIyNyIgZD0iTTAgMTQ5Vjg0aDE3N3Y2NWMtMjktMTQtNTktMTQtODggMC0zMCAxNS01OSAxNS04OSAwWiIgY2xhc3M9InNoYXBlIiBzdHlsZT0ic3Ryb2tlLXdpZHRoOjIiLz48dGV4dCB4PSI4OC41IiB5PSIxMTcuNjExIiBmaWxsPSIjMGEwZjI1IiBjbGFzcz0idGV4dC1ib2xkIGZpbGwtTjEiIHN0eWxlPSJ0ZXh0LWFuY2hvcjptaWRkbGU7Zm9udC1zaXplOjE2cHgiPua6kOS7o+eggSAvIOS6jOi\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\u002FlvoTkuI3lj6\u002Fovr7vvIzliarmnp08L3RzcGFuPjwvdGV4dD48L2c+PGcgY2xhc3M9ImNtVndjbTg9IHN0ZXAiPjxwYXRoIGZpbGw9IiNlZWY0ZmYiIHN0cm9rZT0iIzRhNmZhNSIgZD0iTTE2NDAgMTU0aDEyNHY4MmgtMTI0eiIgY2xhc3M9InNoYXBlIiBzdHlsZT0ic3Ryb2tlLXdpZHRoOjIiLz48dGV4dCB4PSIxNzAyIiB5PSIxOTIuNSIgZmlsbD0iIzBhMGYyNSIgY2xhc3M9InRleHQtYm9sZCBmaWxsLU4xIiBzdHlsZT0idGV4dC1hbmNob3I6bWlkZGxlO2ZvbnQtc2l6ZToxNnB4Ij48dHNwYW4geD0iMTcwMiIgZHk9IjAiPui+k+WFpeWbnuaUvjwvdHNwYW4+PHRzcGFuIHg9IjE3MDIiIGR5PSIxOC41Ij7noa7orqTltKnmuoM8L3RzcGFuPjwvdGV4dD48L2c+PGcgY2xhc3M9ImNtVndiM0owIHJlc3VsdCI+PGcgY2xhc3M9InNoYXBlIj48ZWxsaXBzZSBjeD0iMTk2NiIgY3k9IjE5NS41IiBmaWxsPSIjZWFmN2VlIiBzdHJva2U9IiM0YTlkNmIiIGNsYXNzPSJzaGFwZSIgcng9Ijg1IiByeT0iNDIuNSIgc3R5bGU9InN0cm9rZS13aWR0aDoyIi8+PC9nPjx0ZXh0IHg9IjE5NjYiIHk9IjE5MyIgZmlsbD0iIzBhMGYyNSIgY2xhc3M9InRleHQtYm9sZCBmaWxsLU4xIiBzdHlsZT0idGV4dC1hbmNob3I6bWlkZGxlO2ZvbnQtc2l6ZToxNnB4Ij48dHNwYW4geD0iMTk2NiIgZHk9IjAiPua8j+a0nuaKpeWRijwvdHNwYW4+PHRzcGFuIHg9IjE5NjYiIGR5PSIxOC41Ij4rIFBvQzwvdHNwYW4+PC90ZXh0PjwvZz48ZyBjbGFzcz0iS0hOeVl5QXRKbWQwT3lCemVXMHBXekJkIj48bWFya2VyIGlkPSJtay1kMi0zNzYyNjY4MTczLTM0ODgzNzgxMzQiIG1hcmtlckhlaWdodD0iMTIiIG1hcmtlclVuaXRzPSJ1c2VyU3BhY2VPblVzZSIgbWFya2VyV2lkdGg9IjEwIiBvcmllbnQ9ImF1dG8iIHJlZlg9IjciIHJlZlk9IjYiIHZpZXdCb3g9IjAgMCAxMCAxMiI+PHBhdGggc3Ryb2tlLXdpZHRoPSIyIiBkPSJtMCAwIDEwIDYtMTAgNnoiIGNsYXNzPSJjb25uZWN0aW9uIiBzdHlsZT0iZmlsbDojMGQzMmIyIi8+PC9tYXJrZXI+PHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSIjMGQzMmIyIiBtYXJrZXItZW5kPSJ1cmwoI21rLWQyLTM3NjI2NjgxNzMtMzQ4ODM3ODEzNCkiIGQ9Ik0xNzkgMTIxLjk4M2M0NC44LS4zODMgNjguMy0uNDgzIDExMS41LS40ODMiIGNsYXNzPSJjb25uZWN0aW9uIHN0cm9rZS1CMSIgbWFzaz0idXJsKCNkMi0zNzYyNjY4MTczKSIgc3R5bGU9InN0cm9rZS13aWR0aDoyIi8+PC9nPjxnIGNsYXNzPSJLSE41YlNBdEptZDBPeUJ3WVhSb0tWc3dYUT09Ij48cGF0aCBmaWxsPSJub25lIiBzdHJva2U9IiMwZDMyYjIiIG1hcmtlci1lbmQ9InVybCgjbWstZDItMzc2MjY2ODE3My0zNDg4Mzc4MTM0KSIgZD0iTTQ3MS41IDEyMS41YzQ1LjIgMCA2OC43LS4xIDExMS41LS40NjYiIGNsYXNzPSJjb25uZWN0aW9uIHN0cm9rZS1CMSIgbWFzaz0idXJsKCNkMi0zNzYyNjY4MTczKSIgc3R5bGU9InN0cm9rZS13aWR0aDoyIi8+PC9nPjxnIGNsYXNzPSJLSEJoZEdnZ0xTWm5kRHNnYzIxMEtWc3dYUT09Ij48cGF0aCBmaWxsPSJub25lIiBzdHJva2U9IiMwZDMyYjIiIG1hcmtlci1lbmQ9InVybCgjbWstZDItMzc2MjY2ODE3My0zNDg4Mzc4MTM0KSIgZD0iTTczNyAxMjEuMDE3YzQ0LjguMzgzIDY4LjMuNDgzIDExMS41LjQ4MyIgY2xhc3M9ImNvbm5lY3Rpb24gc3Ryb2tlLUIxIiBtYXNrPSJ1cmwoI2QyLTM3NjI2NjgxNzMpIiBzdHlsZT0ic3Ryb2tlLXdpZHRoOjIiLz48L2c+PGcgY2xhc3M9IktITnRkQ0F0Sm1kME95QnpZWFFwV3pCZCI+PHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSIjMGQzMmIyIiBtYXJrZXItZW5kPSJ1cmwoI21rLWQyLTM3NjI2NjgxNzMtMzQ4ODM3ODEzNCkiIGQ9Ik0xMDQxLjg2OSAxNTguNzEzQzExMTcuNiAxODcuNiAxMTYyLjIgMTk1IDEyNTkgMTk1IiBjbGFzcz0iY29ubmVjdGlvbiBzdHJva2UtQjEiIG1hc2s9InVybCgjZDItMzc2MjY2ODE3MykiIHN0eWxlPSJzdHJva2Utd2lkdGg6MiIvPjx0ZXh0IHg9IjExNDkiIHk9IjE5OCIgZmlsbD0iIzY3NmM3ZSIgY2xhc3M9InRleHQtaXRhbGljIGZpbGwtTjIiIHN0eWxlPSJ0ZXh0LWFuY2hvcjptaWRkbGU7Zm9udC1zaXplOjE2cHgiPuWPr+a7oei2szwvdGV4dD48L2c+PGcgY2xhc3M9IktITnRkQ0F0Sm1kME95QjFibk5oZENsYk1GMD0iPjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iIzBkM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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。",[7246,72376,72377],{},"公式建得对不对，直接决定 SMT 的结论有没有意义。",[11,72379,72380,72381,72384],{},"所以，SMT 在漏洞检测里的正确定位是：",[7246,72382,72383],{},"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。",[18,72386,72388],{"id":72387},"六小结","六、小结",[11,72390,72391,72392,72487,72488,72491],{},"回到开头那道小学题。SMT 的价值，不是把 ",[31,72393,72395,72421],{"className":72394,"translate":35},[34],[31,72396,72398],{"className":72397},[39],[41,72399,72400],{"xmlns":43},[45,72401,72402,72418],{},[48,72403,72404,72406,72408,72410,72412,72414,72416],{},[51,72405,53],{},[95,72407,16040],{},[91,72409,93],{},[95,72411,53038],{},[51,72413,53],{},[95,72415,9758],{},[91,72417,7853],{},[55,72419,72420],{"encoding":57},"x > 3 \\wedge x \u003C 5",[31,72422,72424,72442,72460,72478],{"className":72423,"ariaHidden":62},[61],[31,72425,72427,72430,72433,72436,72439],{"className":72426},[66],[31,72428],{"className":72429,"style":21113},[70],[31,72431,53],{"className":72432},[75,76],[31,72434],{"className":72435,"style":293},[119],[31,72437,16040],{"className":72438},[297],[31,72440],{"className":72441,"style":293},[119],[31,72443,72445,72448,72451,72454,72457],{"className":72444},[66],[31,72446],{"className":72447,"style":326},[70],[31,72449,93],{"className":72450},[75],[31,72452],{"className":72453,"style":120},[119],[31,72455,53038],{"className":72456},[124],[31,72458],{"className":72459,"style":120},[119],[31,72461,72463,72466,72469,72472,72475],{"className":72462},[66],[31,72464],{"className":72465,"style":21113},[70],[31,72467,53],{"className":72468},[75,76],[31,72470],{"className":72471,"style":293},[119],[31,72473,9758],{"className":72474},[297],[31,72476],{"className":72477,"style":293},[119],[31,72479,72481,72484],{"className":72480},[66],[31,72482],{"className":72483,"style":326},[70],[31,72485,7853],{"className":72486},[75]," 算得更快，而是让机器",[7246,72489,72490],{},"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」。",[11,72493,72494,72497,72498,72501],{},[7246,72495,72496],{},"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。 ",[7246,72499,72500],{},"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。",[72503,72504,72505],"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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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。",[72551],"logic","2026-10-29T09:03:03+08:00","1C-2B31uaFHKLW1GJ3_3j1Cbaf3yP10u6PKSP7bBHZ8",1793450670763]