[{"data":1,"prerenderedAt":11003},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-04-25-the-frechet-mean-prototype":3},{"id":4,"title":5,"body":6,"description":10989,"draft":10990,"enableComment":10991,"extension":10992,"image":10980,"meta":10993,"navigation":10991,"onday":10994,"path":10995,"seo":10996,"stem":10997,"summary":10998,"tags":10999,"__hash__":11002},"blog\u002Fblog\u002F2026\u002F2026-04-25-the-frechet-mean-prototype.md","The Fréchet Mean Prototype",{"type":7,"value":8,"toc":10979},"minimark",[9,62,65,70,77,141,373,686,728,1007,1010,1770,1773,1777,1786,2071,2462,2595,2598,2601,2608,3352,3427,3430,3433,3436,3512,3615,3881,4270,4426,4912,5363,5489,6105,6730,7072,7076,7085,7338,7634,7864,8257,8260,8263,8380,8383,8386,8442,8909,9001,9104,9108,9111,9184,9562,9634,10005,10077,10447,10450,10829,10976],[10,11,12,13,61],"p",{},"When the data being processed are not simple vectors (e.g., complex matrices, probability distributions, points on a curved surface) and cannot be directly summed and divided by ",[14,15,19,41],"span",{"className":16,"translate":18},[17],"katex","no",[14,20,23],{"className":21},[22],"katex-mathml",[24,25,27],"math",{"xmlns":26},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[28,29,30,37],"semantics",{},[31,32,33],"mrow",{},[34,35,36],"mi",{},"N",[38,39,36],"annotation",{"encoding":40},"application\u002Fx-tex",[14,42,46],{"className":43,"ariaHidden":45},[44],"katex-html","true",[14,47,50,55],{"className":48},[49],"base",[14,51],{"className":52,"style":54},[53],"strut","height:0.6833em;",[14,56,36],{"className":57,"style":60},[58,59],"mord","mathnormal","margin-right:0.10903em;",", the Fréchet mean becomes necessary.",[10,63,64],{},"The core value of the Fréchet mean prototype is that it generalizes the \"averaging\" operation from flat vector spaces to arbitrary curved spaces or complex objects, enabling many classical statistical methods and machine learning algorithms to handle non-vector data.",[66,67,69],"h2",{"id":68},"mean-prototype","Mean Prototype",[10,71,72,73,76],{},"The ",[74,75,69],"strong",{}," of a set of vectors refers to a new vector obtained by arithmetically averaging each corresponding component of the vectors.",[10,78,79,80,110,111,140],{},"More specifically: Suppose we have ",[14,81,83,97],{"className":82,"translate":18},[17],[14,84,86],{"className":85},[22],[24,87,88],{"xmlns":26},[28,89,90,95],{},[31,91,92],{},[34,93,94],{},"m",[38,96,94],{"encoding":40},[14,98,100],{"className":99,"ariaHidden":45},[44],[14,101,103,107],{"className":102},[49],[14,104],{"className":105,"style":106},[53],"height:0.4306em;",[14,108,94],{"className":109},[58,59]," 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vectors:",[14,142,145],{"className":143,"translate":18},[144],"katex-display",[14,146,148,196],{"className":147,"translate":18},[17],[14,149,151],{"className":150},[22],[24,152,154],{"xmlns":26,"display":153},"block",[28,155,156,193],{},[31,157,158,169,173,180,182,185,187],{},[159,160,161,165],"msub",{},[34,162,164],{"mathvariant":163},"bold","v",[166,167,168],"mn",{},"1",[170,171,172],"mo",{"separator":45},",",[159,174,175,177],{},[34,176,164],{"mathvariant":163},[166,178,179],{},"2",[170,181,172],{"separator":45},[170,183,184],{},"…",[170,186,172],{"separator":45},[159,188,189,191],{},[34,190,164],{"mathvariant":163},[34,192,94],{},[38,194,195],{"encoding":40},"  \\mathbf{v}_1, \\mathbf{v}_2, \\dots, \\mathbf{v}_m",[14,197,199],{"className":198,"ariaHidden":45},[44],[14,200,202,206,264,268,273,313,316,319,323,326,329,332],{"className":201},[49],[14,203],{"className":204,"style":205},[53],"height:0.6389em;vertical-align:-0.1944em;",[14,207,209,214],{"className":208},[58],[14,210,164],{"className":211,"style":213},[58,212],"mathbf","margin-right:0.01597em;",[14,215,218],{"className":216},[217],"msupsub",[14,219,223,255],{"className":220},[221,222],"vlist-t","vlist-t2",[14,224,227,250],{"className":225},[226],"vlist-r",[14,228,232],{"className":229,"style":231},[230],"vlist","height:0.3011em;",[14,233,235,240],{"style":234},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[14,236],{"className":237,"style":239},[238],"pstrut","height:2.7em;",[14,241,247],{"className":242},[243,244,245,246],"sizing","reset-size6","size3","mtight",[14,248,168],{"className":249},[58,246],[14,251,254],{"className":252},[253],"vlist-s","​",[14,256,258],{"className":257},[226],[14,259,262],{"className":260,"style":261},[230],"height:0.15em;",[14,263],{},[14,265,172],{"className":266},[267],"mpunct",[14,269],{"className":270,"style":272},[271],"mspace","margin-right:0.1667em;",[14,274,276,279],{"className":275},[58],[14,277,164],{"className":278,"style":213},[58,212],[14,280,282],{"className":281},[217],[14,283,285,305],{"className":284},[221,222],[14,286,288,302],{"className":287},[226],[14,289,291],{"className":290,"style":231},[230],[14,292,293,296],{"style":234},[14,294],{"className":295,"style":239},[238],[14,297,299],{"className":298},[243,244,245,246],[14,300,179],{"className":301},[58,246],[14,303,254],{"className":304},[253],[14,306,308],{"className":307},[226],[14,309,311],{"className":310,"style":261},[230],[14,312],{},[14,314,172],{"className":315},[267],[14,317],{"className":318,"style":272},[271],[14,320,184],{"className":321},[322],"minner",[14,324],{"className":325,"style":272},[271],[14,327,172],{"className":328},[267],[14,330],{"className":331,"style":272},[271],[14,333,335,338],{"className":334},[58],[14,336,164],{"className":337,"style":213},[58,212],[14,339,341],{"className":340},[217],[14,342,344,365],{"className":343},[221,222],[14,345,347,362],{"className":346},[226],[14,348,351],{"className":349,"style":350},[230],"height:0.1514em;",[14,352,353,356],{"style":234},[14,354],{"className":355,"style":239},[238],[14,357,359],{"className":358},[243,244,245,246],[14,360,94],{"className":361},[58,59,246],[14,363,254],{"className":364},[253],[14,366,368],{"className":367},[226],[14,369,371],{"className":370,"style":261},[230],[14,372],{},[10,374,375,376,685],{},"where each vector ",[14,377,379,446],{"className":378,"translate":18},[17],[14,380,382],{"className":381},[22],[24,383,384],{"xmlns":26},[28,385,386,443],{},[31,387,388,395,398,402,412,414,424,426,428,430,440],{},[159,389,390,392],{},[34,391,164],{"mathvariant":163},[34,393,394],{},"i",[170,396,397],{},"=",[170,399,401],{"stretchy":400},"false","(",[159,403,404,406],{},[34,405,164],{},[31,407,408,410],{},[34,409,394],{},[166,411,168],{},[170,413,172],{"separator":45},[159,415,416,418],{},[34,417,164],{},[31,419,420,422],{},[34,421,394],{},[166,423,179],{},[170,425,172],{"separator":45},[170,427,184],{},[170,429,172],{"separator":45},[159,431,432,434],{},[34,433,164],{},[31,435,436,438],{},[34,437,394],{},[34,439,125],{},[170,441,442],{"stretchy":400},")",[38,444,445],{"encoding":40},"\\mathbf{v}_i = (v_{i1}, v_{i2}, \\dots, v_{in})",[14,447,449,508],{"className":448,"ariaHidden":45},[44],[14,450,452,456,497,501,505],{"className":451},[49],[14,453],{"className":454,"style":455},[53],"height:0.5944em;vertical-align:-0.15em;",[14,457,459,462],{"className":458},[58],[14,460,164],{"className":461,"style":213},[58,212],[14,463,465],{"className":464},[217],[14,466,468,489],{"className":467},[221,222],[14,469,471,486],{"className":470},[226],[14,472,475],{"className":473,"style":474},[230],"height:0.3117em;",[14,476,477,480],{"style":234},[14,478],{"className":479,"style":239},[238],[14,481,483],{"className":482},[243,244,245,246],[14,484,394],{"className":485},[58,59,246],[14,487,254],{"className":488},[253],[14,490,492],{"className":491},[226],[14,493,495],{"className":494,"style":261},[230],[14,496],{},[14,498],{"className":499,"style":500},[271],"margin-right:0.2778em;",[14,502,397],{"className":503},[504],"mrel",[14,506],{"className":507,"style":500},[271],[14,509,511,515,519,567,570,573,619,622,625,628,631,634,637,681],{"className":510},[49],[14,512],{"className":513,"style":514},[53],"height:1em;vertical-align:-0.25em;",[14,516,401],{"className":517},[518],"mopen",[14,520,522,526],{"className":521},[58],[14,523,164],{"className":524,"style":525},[58,59],"margin-right:0.03588em;",[14,527,529],{"className":528},[217],[14,530,532,559],{"className":531},[221,222],[14,533,535,556],{"className":534},[226],[14,536,538],{"className":537,"style":474},[230],[14,539,541,544],{"style":540},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[14,542],{"className":543,"style":239},[238],[14,545,547],{"className":546},[243,244,245,246],[14,548,550,553],{"className":549},[58,246],[14,551,394],{"className":552},[58,59,246],[14,554,168],{"className":555},[58,246],[14,557,254],{"className":558},[253],[14,560,562],{"className":561},[226],[14,563,565],{"className":564,"style":261},[230],[14,566],{},[14,568,172],{"className":569},[267],[14,571],{"className":572,"style":272},[271],[14,574,576,579],{"className":575},[58],[14,577,164],{"className":578,"style":525},[58,59],[14,580,582],{"className":581},[217],[14,583,585,611],{"className":584},[221,222],[14,586,588,608],{"className":587},[226],[14,589,591],{"className":590,"style":474},[230],[14,592,593,596],{"style":540},[14,594],{"className":595,"style":239},[238],[14,597,599],{"className":598},[243,244,245,246],[14,600,602,605],{"className":601},[58,246],[14,603,394],{"className":604},[58,59,246],[14,606,179],{"className":607},[58,246],[14,609,254],{"className":610},[253],[14,612,614],{"className":613},[226],[14,615,617],{"className":616,"style":261},[230],[14,618],{},[14,620,172],{"className":621},[267],[14,623],{"className":624,"style":272},[271],[14,626,184],{"className":627},[322],[14,629],{"className":630,"style":272},[271],[14,632,172],{"className":633},[267],[14,635],{"className":636,"style":272},[271],[14,638,640,643],{"className":639},[58],[14,641,164],{"className":642,"style":525},[58,59],[14,644,646],{"className":645},[217],[14,647,649,673],{"className":648},[221,222],[14,650,652,670],{"className":651},[226],[14,653,655],{"className":654,"style":474},[230],[14,656,657,660],{"style":540},[14,658],{"className":659,"style":239},[238],[14,661,663],{"className":662},[243,244,245,246],[14,664,666],{"className":665},[58,246],[14,667,669],{"className":668},[58,59,246],"in",[14,671,254],{"className":672},[253],[14,674,676],{"className":675},[226],[14,677,679],{"className":678,"style":261},[230],[14,680],{},[14,682,442],{"className":683},[684],"mclose",".",[10,687,688,689,727],{},"The mean prototype ",[14,690,692,708],{"className":691,"translate":18},[17],[14,693,695],{"className":694},[22],[24,696,697],{"xmlns":26},[28,698,699,705],{},[31,700,701],{},[34,702,704],{"mathvariant":703},"bold-italic","μ",[38,706,707],{"encoding":40},"\\boldsymbol{\\mu}",[14,709,711],{"className":710,"ariaHidden":45},[44],[14,712,714,717],{"className":713},[49],[14,715],{"className":716,"style":205},[53],[14,718,720],{"className":719},[58],[14,721,723],{"className":722},[58],[14,724,704],{"className":725},[58,726],"boldsymbol"," is simply defined as:",[14,729,731],{"className":730,"translate":18},[144],[14,732,734,779],{"className":733,"translate":18},[17],[14,735,737],{"className":736},[22],[24,738,739],{"xmlns":26,"display":153},[28,740,741,776],{},[31,742,743,745,747,754,770],{},[34,744,704],{"mathvariant":703},[170,746,397],{},[748,749,750,752],"mfrac",{},[166,751,168],{},[34,753,94],{},[755,756,757,760,768],"munderover",{},[170,758,759],{},"∑",[31,761,762,764,766],{},[34,763,394],{},[170,765,397],{},[166,767,168],{},[34,769,94],{},[159,771,772,774],{},[34,773,164],{"mathvariant":163},[34,775,394],{},[38,777,778],{"encoding":40},"  \\boldsymbol{\\mu} = \\frac{1}{m} \\sum_{i=1}^{m} \\mathbf{v}_i",[14,780,782,806],{"className":781,"ariaHidden":45},[44],[14,783,785,788,797,800,803],{"className":784},[49],[14,786],{"className":787,"style":205},[53],[14,789,791],{"className":790},[58],[14,792,794],{"className":793},[58],[14,795,704],{"className":796},[58,726],[14,798],{"className":799,"style":500},[271],[14,801,397],{"className":802},[504],[14,804],{"className":805,"style":500},[271],[14,807,809,813,884,887,964,967],{"className":808},[49],[14,810],{"className":811,"style":812},[53],"height:2.9291em;vertical-align:-1.2777em;",[14,814,816,820,881],{"className":815},[58],[14,817],{"className":818},[518,819],"nulldelimiter",[14,821,823],{"className":822},[748],[14,824,826,872],{"className":825},[221,222],[14,827,829,869],{"className":828},[226],[14,830,833,846,857],{"className":831,"style":832},[230],"height:1.3214em;",[14,834,836,840],{"style":835},"top:-2.314em;",[14,837],{"className":838,"style":839},[238],"height:3em;",[14,841,843],{"className":842},[58],[14,844,94],{"className":845},[58,59],[14,847,849,852],{"style":848},"top:-3.23em;",[14,850],{"className":851,"style":839},[238],[14,853],{"className":854,"style":856},[855],"frac-line","border-bottom-width:0.04em;",[14,858,860,863],{"style":859},"top:-3.677em;",[14,861],{"className":862,"style":839},[238],[14,864,866],{"className":865},[58],[14,867,168],{"className":868},[58],[14,870,254],{"className":871},[253],[14,873,875],{"className":874},[226],[14,876,879],{"className":877,"style":878},[230],"height:0.686em;",[14,880],{},[14,882],{"className":883},[684,819],[14,885],{"className":886,"style":272},[271],[14,888,892],{"className":889},[890,891],"mop","op-limits",[14,893,895,955],{"className":894},[221,222],[14,896,898,952],{"className":897},[226],[14,899,902,924,937],{"className":900,"style":901},[230],"height:1.6514em;",[14,903,905,909],{"style":904},"top:-1.8723em;margin-left:0em;",[14,906],{"className":907,"style":908},[238],"height:3.05em;",[14,910,912],{"className":911},[243,244,245,246],[14,913,915,918,921],{"className":914},[58,246],[14,916,394],{"className":917},[58,59,246],[14,919,397],{"className":920},[504,246],[14,922,168],{"className":923},[58,246],[14,925,927,930],{"style":926},"top:-3.05em;",[14,928],{"className":929,"style":908},[238],[14,931,932],{},[14,933,759],{"className":934},[890,935,936],"op-symbol","large-op",[14,938,940,943],{"style":939},"top:-4.3em;margin-left:0em;",[14,941],{"className":942,"style":908},[238],[14,944,946],{"className":945},[243,244,245,246],[14,947,949],{"className":948},[58,246],[14,950,94],{"className":951},[58,59,246],[14,953,254],{"className":954},[253],[14,956,958],{"className":957},[226],[14,959,962],{"className":960,"style":961},[230],"height:1.2777em;",[14,963],{},[14,965],{"className":966,"style":272},[271],[14,968,970,973],{"className":969},[58],[14,971,164],{"className":972,"style":213},[58,212],[14,974,976],{"className":975},[217],[14,977,979,999],{"className":978},[221,222],[14,980,982,996],{"className":981},[226],[14,983,985],{"className":984,"style":474},[230],[14,986,987,990],{"style":234},[14,988],{"className":989,"style":239},[238],[14,991,993],{"className":992},[243,244,245,246],[14,994,394],{"className":995},[58,59,246],[14,997,254],{"className":998},[253],[14,1000,1002],{"className":1001},[226],[14,1003,1005],{"className":1004,"style":261},[230],[14,1006],{},[10,1008,1009],{},"That is:",[14,1011,1013],{"className":1012,"translate":18},[144],[14,1014,1016,1144],{"className":1015,"translate":18},[17],[14,1017,1019],{"className":1018},[22],[24,1020,1021],{"xmlns":26,"display":153},[28,1022,1023,1141],{},[31,1024,1025,1027,1029],{},[34,1026,704],{"mathvariant":703},[170,1028,397],{},[31,1030,1031,1033,1039,1053,1063,1065,1069,1075,1089,1099,1101,1103,1105,1107,1109,1115,1129,1139],{},[170,1032,401],{"fence":45},[748,1034,1035,1037],{},[166,1036,168],{},[34,1038,94],{},[755,1040,1041,1043,1051],{},[170,1042,759],{},[31,1044,1045,1047,1049],{},[34,1046,394],{},[170,1048,397],{},[166,1050,168],{},[34,1052,94],{},[159,1054,1055,1057],{},[34,1056,164],{},[31,1058,1059,1061],{},[34,1060,394],{},[166,1062,168],{},[170,1064,172],{"separator":45},[1066,1067,1068],"mtext",{}," ",[748,1070,1071,1073],{},[166,1072,168],{},[34,1074,94],{},[755,1076,1077,1079,1087],{},[170,1078,759],{},[31,1080,1081,1083,1085],{},[34,1082,394],{},[170,1084,397],{},[166,1086,168],{},[34,1088,94],{},[159,1090,1091,1093],{},[34,1092,164],{},[31,1094,1095,1097],{},[34,1096,394],{},[166,1098,179],{},[170,1100,172],{"separator":45},[1066,1102,1068],{},[170,1104,184],{},[170,1106,172],{"separator":45},[1066,1108,1068],{},[748,1110,1111,1113],{},[166,1112,168],{},[34,1114,94],{},[755,1116,1117,1119,1127],{},[170,1118,759],{},[31,1120,1121,1123,1125],{},[34,1122,394],{},[170,1124,397],{},[166,1126,168],{},[34,1128,94],{},[159,1130,1131,1133],{},[34,1132,164],{},[31,1134,1135,1137],{},[34,1136,394],{},[34,1138,125],{},[170,1140,442],{"fence":45},[38,1142,1143],{"encoding":40},"  \\boldsymbol{\\mu} = \\left( \\frac{1}{m}\\sum_{i=1}^{m} v_{i1},\\ \\frac{1}{m}\\sum_{i=1}^{m} v_{i2},\\ \\dots,\\ \\frac{1}{m}\\sum_{i=1}^{m} v_{in} \\right)",[14,1145,1147,1171],{"className":1146,"ariaHidden":45},[44],[14,1148,1150,1153,1162,1165,1168],{"className":1149},[49],[14,1151],{"className":1152,"style":205},[53],[14,1154,1156],{"className":1155},[58],[14,1157,1159],{"className":1158},[58],[14,1160,704],{"className":1161},[58,726],[14,1163],{"className":1164,"style":500},[271],[14,1166,397],{"className":1167},[504],[14,1169],{"className":1170,"style":500},[271],[14,1172,1174,1178],{"className":1173},[49],[14,1175],{"className":1176,"style":1177},[53],"height:3.0277em;vertical-align:-1.2777em;",[14,1179,1181,1191,1253,1256,1323,1326,1372,1375,1378,1381,1443,1446,1513,1516,1562,1565,1568,1571,1574,1577,1580,1583,1586,1648,1651,1718,1721,1764],{"className":1180},[322],[14,1182,1186],{"className":1183,"style":1185},[518,1184],"delimcenter","top:0em;",[14,1187,401],{"className":1188},[1189,1190],"delimsizing","size4",[14,1192,1194,1197,1250],{"className":1193},[58],[14,1195],{"className":1196},[518,819],[14,1198,1200],{"className":1199},[748],[14,1201,1203,1242],{"className":1202},[221,222],[14,1204,1206,1239],{"className":1205},[226],[14,1207,1209,1220,1228],{"className":1208,"style":832},[230],[14,1210,1211,1214],{"style":835},[14,1212],{"className":1213,"style":839},[238],[14,1215,1217],{"className":1216},[58],[14,1218,94],{"className":1219},[58,59],[14,1221,1222,1225],{"style":848},[14,1223],{"className":1224,"style":839},[238],[14,1226],{"className":1227,"style":856},[855],[14,1229,1230,1233],{"style":859},[14,1231],{"className":1232,"style":839},[238],[14,1234,1236],{"className":1235},[58],[14,1237,168],{"className":1238},[58],[14,1240,254],{"className":1241},[253],[14,1243,1245],{"className":1244},[226],[14,1246,1248],{"className":1247,"style":878},[230],[14,1249],{},[14,1251],{"className":1252},[684,819],[14,1254],{"className":1255,"style":272},[271],[14,1257,1259],{"className":1258},[890,891],[14,1260,1262,1315],{"className":1261},[221,222],[14,1263,1265,1312],{"className":1264},[226],[14,1266,1268,1288,1298],{"className":1267,"style":901},[230],[14,1269,1270,1273],{"style":904},[14,1271],{"className":1272,"style":908},[238],[14,1274,1276],{"className":1275},[243,244,245,246],[14,1277,1279,1282,1285],{"className":1278},[58,246],[14,1280,394],{"className":1281},[58,59,246],[14,1283,397],{"className":1284},[504,246],[14,1286,168],{"className":1287},[58,246],[14,1289,1290,1293],{"style":926},[14,1291],{"className":1292,"style":908},[238],[14,1294,1295],{},[14,1296,759],{"className":1297},[890,935,936],[14,1299,1300,1303],{"style":939},[14,1301],{"className":1302,"style":908},[238],[14,1304,1306],{"className":1305},[243,244,245,246],[14,1307,1309],{"className":1308},[58,246],[14,1310,94],{"className":1311},[58,59,246],[14,1313,254],{"className":1314},[253],[14,1316,1318],{"className":1317},[226],[14,1319,1321],{"className":1320,"style":961},[230],[14,1322],{},[14,1324],{"className":1325,"style":272},[271],[14,1327,1329,1332],{"className":1328},[58],[14,1330,164],{"className":1331,"style":525},[58,59],[14,1333,1335],{"className":1334},[217],[14,1336,1338,1364],{"className":1337},[221,222],[14,1339,1341,1361],{"className":1340},[226],[14,1342,1344],{"className":1343,"style":474},[230],[14,1345,1346,1349],{"style":540},[14,1347],{"className":1348,"style":239},[238],[14,1350,1352],{"className":1351},[243,244,245,246],[14,1353,1355,1358],{"className":1354},[58,246],[14,1356,394],{"className":1357},[58,59,246],[14,1359,168],{"className":1360},[58,246],[14,1362,254],{"className":1363},[253],[14,1365,1367],{"className":1366},[226],[14,1368,1370],{"className":1369,"style":261},[230],[14,1371],{},[14,1373,172],{"className":1374},[267],[14,1376,1068],{"className":1377},[271],[14,1379],{"className":1380,"style":272},[271],[14,1382,1384,1387,1440],{"className":1383},[58],[14,1385],{"className":1386},[518,819],[14,1388,1390],{"className":1389},[748],[14,1391,1393,1432],{"className":1392},[221,222],[14,1394,1396,1429],{"className":1395},[226],[14,1397,1399,1410,1418],{"className":1398,"style":832},[230],[14,1400,1401,1404],{"style":835},[14,1402],{"className":1403,"style":839},[238],[14,1405,1407],{"className":1406},[58],[14,1408,94],{"className":1409},[58,59],[14,1411,1412,1415],{"style":848},[14,1413],{"className":1414,"style":839},[238],[14,1416],{"className":1417,"style":856},[855],[14,1419,1420,1423],{"style":859},[14,1421],{"className":1422,"style":839},[238],[14,1424,1426],{"className":1425},[58],[14,1427,168],{"className":1428},[58],[14,1430,254],{"className":1431},[253],[14,1433,1435],{"className":1434},[226],[14,1436,1438],{"className":1437,"style":878},[230],[14,1439],{},[14,1441],{"className":1442},[684,819],[14,1444],{"className":1445,"style":272},[271],[14,1447,1449],{"className":1448},[890,891],[14,1450,1452,1505],{"className":1451},[221,222],[14,1453,1455,1502],{"className":1454},[226],[14,1456,1458,1478,1488],{"className":1457,"style":901},[230],[14,1459,1460,1463],{"style":904},[14,1461],{"className":1462,"style":908},[238],[14,1464,1466],{"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the mean vector is the \"center point\" or \"average position\" of the set of vectors. In data analysis and machine learning (e.g., clustering, PCA), it is commonly used to represent the overall trend of the dataset. The Mean Prototype can be seen as the \"optimal representative point under Euclidean geometry + Gaussian assumptions.\" If your samples are embedded on a unit sphere or follow a vMF distribution, the Euclidean mean is no longer optimal.",[66,1774,1776],{"id":1775},"geometric-median-prototype","Geometric Median Prototype",[10,1778,72,1779,1781,1782,1785],{},[74,1780,1776],{}," refers to: For a set of vectors (data points), find a point that minimizes the ",[74,1783,1784],{},"sum of Euclidean distances"," to all data points. This point is the geometric median of the set of vectors and serves as the \"prototype\" representing the entire dataset.",[10,1787,1788,1789,110,1817,1845,1846,2040,2041,2070],{},"Formal definition: Given ",[14,1790,1792,1805],{"className":1791,"translate":18},[17],[14,1793,1795],{"className":1794},[22],[24,1796,1797],{"xmlns":26},[28,1798,1799,1803],{},[31,1800,1801],{},[34,1802,94],{},[38,1804,94],{"encoding":40},[14,1806,1808],{"className":1807,"ariaHidden":45},[44],[14,1809,1811,1814],{"className":1810},[49],[14,1812],{"className":1813,"style":106},[53],[14,1815,94],{"className":1816},[58,59],[14,1818,1820,1833],{"className":1819,"translate":18},[17],[14,1821,1823],{"className":1822},[22],[24,1824,1825],{"xmlns":26},[28,1826,1827,1831],{},[31,1828,1829],{},[34,1830,125],{},[38,1832,125],{"encoding":40},[14,1834,1836],{"className":1835,"ariaHidden":45},[44],[14,1837,1839,1842],{"className":1838},[49],[14,1840],{"className":1841,"style":106},[53],[14,1843,125],{"className":1844},[58,59],"-dimensional vectors ",[14,1847,1849,1887],{"className":1848,"translate":18},[17],[14,1850,1852],{"className":1851},[22],[24,1853,1854],{"xmlns":26},[28,1855,1856,1884],{},[31,1857,1858,1864,1866,1872,1874,1876,1878],{},[159,1859,1860,1862],{},[34,1861,164],{"mathvariant":163},[166,1863,168],{},[170,1865,172],{"separator":45},[159,1867,1868,1870],{},[34,1869,164],{"mathvariant":163},[166,1871,179],{},[170,1873,172],{"separator":45},[170,1875,184],{},[170,1877,172],{"separator":45},[159,1879,1880,1882],{},[34,1881,164],{"mathvariant":163},[34,1883,94],{},[38,1885,1886],{"encoding":40},"\\mathbf{v}_1, \\mathbf{v}_2, \\dots, 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the geometric median ",[14,2042,2044,2058],{"className":2043,"translate":18},[17],[14,2045,2047],{"className":2046},[22],[24,2048,2049],{"xmlns":26},[28,2050,2051,2055],{},[31,2052,2053],{},[34,2054,10],{"mathvariant":163},[38,2056,2057],{"encoding":40},"\\mathbf{p}",[14,2059,2061],{"className":2060,"ariaHidden":45},[44],[14,2062,2064,2067],{"className":2063},[49],[14,2065],{"className":2066,"style":205},[53],[14,2068,10],{"className":2069},[58,212]," satisfies:",[14,2072,2074],{"className":2073,"translate":18},[144],[14,2075,2077,2161],{"className":2076,"translate":18},[17],[14,2078,2080],{"className":2079},[22],[24,2081,2082],{"xmlns":26,"display":153},[28,2083,2084,2158],{},[31,2085,2086,2088,2090,2093,2096,2123,2137,2141,2143,2146,2152],{},[34,2087,10],{"mathvariant":163},[170,2089,397],{},[34,2091,2092],{},"arg",[170,2094,2095],{},"⁡",[2097,2098,2099,2106],"munder",{},[31,2100,2101,2104],{},[34,2102,2103],{},"min",[170,2105,2095],{},[31,2107,2108,2111,2114],{},[34,2109,2110],{"mathvariant":163},"x",[170,2112,2113],{},"∈",[2115,2116,2117,2121],"msup",{},[34,2118,2120],{"mathvariant":2119},"double-struck","R",[34,2122,125],{},[755,2124,2125,2127,2135],{},[170,2126,759],{},[31,2128,2129,2131,2133],{},[34,2130,394],{},[170,2132,397],{},[166,2134,168],{},[34,2136,94],{},[34,2138,2140],{"mathvariant":2139},"normal","∥",[34,2142,2110],{"mathvariant":163},[170,2144,2145],{},"−",[159,2147,2148,2150],{},[34,2149,164],{"mathvariant":163},[34,2151,394],{},[159,2153,2154,2156],{},[34,2155,2140],{"mathvariant":2139},[166,2157,179],{},[38,2159,2160],{"encoding":40},"\\mathbf{p} = \\arg\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\sum_{i=1}^m \\|\\mathbf{x} - \\mathbf{v}_i\\|_2",[14,2162,2164,2182,2375],{"className":2163,"ariaHidden":45},[44],[14,2165,2167,2170,2173,2176,2179],{"className":2166},[49],[14,2168],{"className":2169,"style":205},[53],[14,2171,10],{"className":2172},[58,212],[14,2174],{"className":2175,"style":500},[271],[14,2177,397],{"className":2178},[504],[14,2180],{"className":2181,"style":500},[271],[14,2183,2185,2188,2196,2199,2288,2291,2355,2358,2361,2364,2368,2372],{"className":2184},[49],[14,2186],{"className":2187,"style":812},[53],[14,2189,2191,2192],{"className":2190},[890],"ar",[14,2193,2195],{"style":2194},"margin-right:0.01389em;","g",[14,2197],{"className":2198,"style":272},[271],[14,2200,2202],{"className":2201},[890,891],[14,2203,2205,2279],{"className":2204},[221,222],[14,2206,2208,2276],{"className":2207},[226],[14,2209,2212,2265],{"className":2210,"style":2211},[230],"height:0.6679em;",[14,2213,2215,2218],{"style":2214},"top:-2.3518em;margin-left:0em;",[14,2216],{"className":2217,"style":839},[238],[14,2219,2221],{"className":2220},[243,244,245,246],[14,2222,2224,2227,2230],{"className":2223},[58,246],[14,2225,2110],{"className":2226},[58,212,246],[14,2228,2113],{"className":2229},[504,246],[14,2231,2233,2237],{"className":2232},[58,246],[14,2234,2120],{"className":2235},[58,2236,246],"mathbb",[14,2238,2240],{"className":2239},[217],[14,2241,2243],{"className":2242},[221],[14,2244,2246],{"className":2245},[226],[14,2247,2250],{"className":2248,"style":2249},[230],"height:0.5935em;",[14,2251,2253,2257],{"style":2252},"top:-2.786em;margin-right:0.0714em;",[14,2254],{"className":2255,"style":2256},[238],"height:2.5em;",[14,2258,2262],{"className":2259},[243,2260,2261,246],"reset-size3","size1",[14,2263,125],{"className":2264},[58,59,246],[14,2266,2268,2271],{"style":2267},"top:-3em;",[14,2269],{"className":2270,"style":839},[238],[14,2272,2273],{},[14,2274,2103],{"className":2275},[890],[14,2277,254],{"className":2278},[253],[14,2280,2282],{"className":2281},[226],[14,2283,2286],{"className":2284,"style":2285},[230],"height:0.7756em;",[14,2287],{},[14,2289],{"className":2290,"style":272},[271],[14,2292,2294],{"className":2293},[890,891],[14,2295,2297,2347],{"className":2296},[221,222],[14,2298,2300,2344],{"className":2299},[226],[14,2301,2303,2323,2333],{"className":2302,"style":901},[230],[14,2304,2305,2308],{"style":904},[14,2306],{"className":2307,"style":908},[238],[14,2309,2311],{"className":2310},[243,244,245,246],[14,2312,2314,2317,2320],{"className":2313},[58,246],[14,2315,394],{"className":2316},[58,59,246],[14,2318,397],{"className":2319},[504,246],[14,2321,168],{"className":2322},[58,246],[14,2324,2325,2328],{"style":926},[14,2326],{"className":2327,"style":908},[238],[14,2329,2330],{},[14,2331,759],{"className":2332},[890,935,936],[14,2334,2335,2338],{"style":939},[14,2336],{"className":2337,"style":908},[238],[14,2339,2341],{"className":2340},[243,244,245,246],[14,2342,94],{"className":2343},[58,59,246],[14,2345,254],{"className":2346},[253],[14,2348,2350],{"className":2349},[226],[14,2351,2353],{"className":2352,"style":961},[230],[14,2354],{},[14,2356],{"className":2357,"style":272},[271],[14,2359,2140],{"className":2360},[58],[14,2362,2110],{"className":2363},[58,212],[14,2365],{"className":2366,"style":2367},[271],"margin-right:0.2222em;",[14,2369,2145],{"className":2370},[2371],"mbin",[14,2373],{"className":2374,"style":2367},[271],[14,2376,2378,2381,2421],{"className":2377},[49],[14,2379],{"className":2380,"style":514},[53],[14,2382,2384,2387],{"className":2383},[58],[14,2385,164],{"className":2386,"style":213},[58,212],[14,2388,2390],{"className":2389},[217],[14,2391,2393,2413],{"className":2392},[221,222],[14,2394,2396,2410],{"className":2395},[226],[14,2397,2399],{"className":2398,"style":474},[230],[14,2400,2401,2404],{"style":234},[14,2402],{"className":2403,"style":239},[238],[14,2405,2407],{"className":2406},[243,244,245,246],[14,2408,394],{"className":2409},[58,59,246],[14,2411,254],{"className":2412},[253],[14,2414,2416],{"className":2415},[226],[14,2417,2419],{"className":2418,"style":261},[230],[14,2420],{},[14,2422,2424,2427],{"className":2423},[58],[14,2425,2140],{"className":2426},[58],[14,2428,2430],{"className":2429},[217],[14,2431,2433,2454],{"className":2432},[221,222],[14,2434,2436,2451],{"className":2435},[226],[14,2437,2439],{"className":2438,"style":231},[230],[14,2440,2442,2445],{"style":2441},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[14,2443],{"className":2444,"style":239},[238],[14,2446,2448],{"className":2447},[243,244,245,246],[14,2449,179],{"className":2450},[58,246],[14,2452,254],{"className":2453},[253],[14,2455,2457],{"className":2456},[226],[14,2458,2460],{"className":2459,"style":261},[230],[14,2461],{},[10,2463,2464,2465,2558,2559,2594],{},"where ",[14,2466,2468,2491],{"className":2467,"translate":18},[17],[14,2469,2471],{"className":2470},[22],[24,2472,2473],{"xmlns":26},[28,2474,2475,2488],{},[31,2476,2477,2479,2482],{},[34,2478,2140],{"mathvariant":2139},[170,2480,2481],{},"⋅",[159,2483,2484,2486],{},[34,2485,2140],{"mathvariant":2139},[166,2487,179],{},[38,2489,2490],{"encoding":40},"\\|\\cdot\\|_2",[14,2492,2494,2512],{"className":2493,"ariaHidden":45},[44],[14,2495,2497,2500,2503,2506,2509],{"className":2496},[49],[14,2498],{"className":2499,"style":514},[53],[14,2501,2140],{"className":2502},[58],[14,2504],{"className":2505,"style":2367},[271],[14,2507,2481],{"className":2508},[2371],[14,2510],{"className":2511,"style":2367},[271],[14,2513,2515,2518],{"className":2514},[49],[14,2516],{"className":2517,"style":514},[53],[14,2519,2521,2524],{"className":2520},[58],[14,2522,2140],{"className":2523},[58],[14,2525,2527],{"className":2526},[217],[14,2528,2530,2550],{"className":2529},[221,222],[14,2531,2533,2547],{"className":2532},[226],[14,2534,2536],{"className":2535,"style":231},[230],[14,2537,2538,2541],{"style":2441},[14,2539],{"className":2540,"style":239},[238],[14,2542,2544],{"className":2543},[243,244,245,246],[14,2545,179],{"className":2546},[58,246],[14,2548,254],{"className":2549},[253],[14,2551,2553],{"className":2552},[226],[14,2554,2556],{"className":2555,"style":261},[230],[14,2557],{}," is the Euclidean distance under the ",[14,2560,2562,2579],{"className":2561,"translate":18},[17],[14,2563,2565],{"className":2564},[22],[24,2566,2567],{"xmlns":26},[28,2568,2569,2576],{},[31,2570,2571,2574],{},[34,2572,2573],{},"L",[166,2575,179],{},[38,2577,2578],{"encoding":40},"L2",[14,2580,2582],{"className":2581,"ariaHidden":45},[44],[14,2583,2585,2588,2591],{"className":2584},[49],[14,2586],{"className":2587,"style":54},[53],[14,2589,2573],{"className":2590},[58,59],[14,2592,179],{"className":2593},[58]," norm.",[10,2596,2597],{},"Intuitively, in 1D: the geometric median is the ordinary median (minimizing the sum of absolute deviations). Extended to multiple dimensions: it is a point in space that minimizes the \"total straight-line distance\" to all data points. Imagine finding a location on a plane that minimizes the total travel distance to all given points — similar to finding a gathering point that minimizes total travel distance among several cities.",[10,2599,2600],{},"The Geometric Median Prototype is highly robust, which is its most important characteristic — even if the data contains a few outliers, the geometric median still reflects the center of the \"majority\" of points. It is equivariant under rotation and translation: if the data is rotated or translated, the geometric median changes accordingly. It can lie anywhere between data points (unlike K-medoids, which can only select actual data points).",[10,2602,2603,2604,2607],{},"Since there is no closed-form solution, the ",[74,2605,2606],{},"Weiszfeld algorithm"," (iteratively reweighted average) is commonly used:",[14,2609,2611],{"className":2610,"translate":18},[144],[14,2612,2614,2728],{"className":2613,"translate":18},[17],[14,2615,2617],{"className":2616},[22],[24,2618,2619],{"xmlns":26,"display":153},[28,2620,2621,2725],{},[31,2622,2623,2637,2639],{},[159,2624,2625,2627],{},[34,2626,10],{"mathvariant":163},[31,2628,2629,2632,2635],{},[34,2630,2631],{},"t",[170,2633,2634],{},"+",[166,2636,168],{},[170,2638,397],{},[748,2640,2641,2685],{},[31,2642,2643,2657],{},[755,2644,2645,2647,2655],{},[170,2646,759],{},[31,2648,2649,2651,2653],{},[34,2650,394],{},[170,2652,397],{},[166,2654,168],{},[34,2656,94],{},[748,2658,2659,2665],{},[159,2660,2661,2663],{},[34,2662,164],{"mathvariant":163},[34,2664,394],{},[31,2666,2667,2669,2675,2677,2683],{},[34,2668,2140],{"mathvariant":2139},[159,2670,2671,2673],{},[34,2672,10],{"mathvariant":163},[34,2674,2631],{},[170,2676,2145],{},[159,2678,2679,2681],{},[34,2680,164],{"mathvariant":163},[34,2682,394],{},[34,2684,2140],{"mathvariant":2139},[31,2686,2687,2701],{},[755,2688,2689,2691,2699],{},[170,2690,759],{},[31,2692,2693,2695,2697],{},[34,2694,394],{},[170,2696,397],{},[166,2698,168],{},[34,2700,94],{},[748,2702,2703,2705],{},[166,2704,168],{},[31,2706,2707,2709,2715,2717,2723],{},[34,2708,2140],{"mathvariant":2139},[159,2710,2711,2713],{},[34,2712,10],{"mathvariant":163},[34,2714,2631],{},[170,2716,2145],{},[159,2718,2719,2721],{},[34,2720,164],{"mathvariant":163},[34,2722,394],{},[34,2724,2140],{"mathvariant":2139},[38,2726,2727],{"encoding":40},"\\mathbf{p}_{t+1} = \\frac{\\sum_{i=1}^m \\frac{\\mathbf{v}_i}{\\|\\mathbf{p}_t - \\mathbf{v}_i\\|}}{\\sum_{i=1}^m \\frac{1}{\\|\\mathbf{p}_t - \\mathbf{v}_i\\|}}",[14,2729,2731,2797],{"className":2730,"ariaHidden":45},[44],[14,2732,2734,2738,2788,2791,2794],{"className":2733},[49],[14,2735],{"className":2736,"style":2737},[53],"height:0.6528em;vertical-align:-0.2083em;",[14,2739,2741,2744],{"className":2740},[58],[14,2742,10],{"className":2743},[58,212],[14,2745,2747],{"className":2746},[217],[14,2748,2750,2779],{"className":2749},[221,222],[14,2751,2753,2776],{"className":2752},[226],[14,2754,2756],{"className":2755,"style":231},[230],[14,2757,2758,2761],{"style":2441},[14,2759],{"className":2760,"style":239},[238],[14,2762,2764],{"className":2763},[243,244,245,246],[14,2765,2767,2770,2773],{"className":2766},[58,246],[14,2768,2631],{"className":2769},[58,59,246],[14,2771,2634],{"className":2772},[2371,246],[14,2774,168],{"className":2775},[58,246],[14,2777,254],{"className":2778},[253],[14,2780,2782],{"className":2781},[226],[14,2783,2786],{"className":2784,"style":2785},[230],"height:0.2083em;",[14,2787],{},[14,2789],{"className":2790,"style":500},[271],[14,2792,397],{"className":2793},[504],[14,2795],{"className":2796,"style":500},[271],[14,2798,2800,2804],{"className":2799},[49],[14,2801],{"className":2802,"style":2803},[53],"height:2.9694em;vertical-align:-1.2551em;",[14,2805,2807,2810,3349],{"className":2806},[58],[14,2808],{"className":2809},[518,819],[14,2811,2813],{"className":2812},[748],[14,2814,2816,3340],{"className":2815},[221,222],[14,2817,2819,3337],{"className":2818},[226],[14,2820,2823,3064,3072],{"className":2821,"style":2822},[230],"height:1.7143em;",[14,2824,2826,2829],{"style":2825},"top:-2.2649em;",[14,2827],{"className":2828,"style":839},[238],[14,2830,2832,2898,2901],{"className":2831},[58],[14,2833,2835,2840],{"className":2834},[890],[14,2836,759],{"className":2837,"style":2839},[890,935,2838],"small-op","position:relative;top:0em;",[14,2841,2843],{"className":2842},[217],[14,2844,2846,2889],{"className":2845},[221,222],[14,2847,2849,2886],{"className":2848},[226],[14,2850,2853,2874],{"className":2851,"style":2852},[230],"height:0.8043em;",[14,2854,2856,2859],{"style":2855},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[14,2857],{"className":2858,"style":239},[238],[14,2860,2862],{"className":2861},[243,244,245,246],[14,2863,2865,2868,2871],{"className":2864},[58,246],[14,2866,394],{"className":2867},[58,59,246],[14,2869,397],{"className":2870},[504,246],[14,2872,168],{"className":2873},[58,246],[14,2875,2877,2880],{"style":2876},"top:-3.2029em;margin-right:0.05em;",[14,2878],{"className":2879,"style":239},[238],[14,2881,2883],{"className":2882},[243,244,245,246],[14,2884,94],{"className":2885},[58,59,246],[14,2887,254],{"className":2888},[253],[14,2890,2892],{"className":2891},[226],[14,2893,2896],{"className":2894,"style":2895},[230],"height:0.2997em;",[14,2897],{},[14,2899],{"className":2900,"style":272},[271],[14,2902,2904,2907,3061],{"className":2903},[58],[14,2905],{"className":2906},[518,819],[14,2908,2910],{"className":2909},[748],[14,2911,2913,3052],{"className":2912},[221,222],[14,2914,2916,3049],{"className":2915},[226],[14,2917,2920,3026,3034],{"className":2918,"style":2919},[230],"height:0.8451em;",[14,2921,2923,2926],{"style":2922},"top:-2.655em;",[14,2924],{"className":2925,"style":839},[238],[14,2927,2929],{"className":2928},[243,244,245,246],[14,2930,2932,2935,2978,2981,3023],{"className":2931},[58,246],[14,2933,2140],{"className":2934},[58,246],[14,2936,2938,2941],{"className":2937},[58,246],[14,2939,10],{"className":2940},[58,212,246],[14,2942,2944],{"className":2943},[217],[14,2945,2947,2969],{"className":2946},[221,222],[14,2948,2950,2966],{"className":2949},[226],[14,2951,2954],{"className":2952,"style":2953},[230],"height:0.2963em;",[14,2955,2957,2960],{"style":2956},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[14,2958],{"className":2959,"style":2256},[238],[14,2961,2963],{"className":2962},[243,2260,2261,246],[14,2964,2631],{"className":2965},[58,59,246],[14,2967,254],{"className":2968},[253],[14,2970,2972],{"className":2971},[226],[14,2973,2976],{"className":2974,"style":2975},[230],"height:0.143em;",[14,2977],{},[14,2979,2145],{"className":2980},[2371,246],[14,2982,2984,2987],{"className":2983},[58,246],[14,2985,164],{"className":2986,"style":213},[58,212,246],[14,2988,2990],{"className":2989},[217],[14,2991,2993,3015],{"className":2992},[221,222],[14,2994,2996,3012],{"className":2995},[226],[14,2997,3000],{"className":2998,"style":2999},[230],"height:0.3281em;",[14,3001,3003,3006],{"style":3002},"top:-2.357em;margin-left:-0.016em;margin-right:0.0714em;",[14,3004],{"className":3005,"style":2256},[238],[14,3007,3009],{"className":3008},[243,2260,2261,246],[14,3010,394],{"className":3011},[58,59,246],[14,3013,254],{"className":3014},[253],[14,3016,3018],{"className":3017},[226],[14,3019,3021],{"className":3020,"style":2975},[230],[14,3022],{},[14,3024,2140],{"className":3025},[58,246],[14,3027,3028,3031],{"style":848},[14,3029],{"className":3030,"style":839},[238],[14,3032],{"className":3033,"style":856},[855],[14,3035,3037,3040],{"style":3036},"top:-3.394em;",[14,3038],{"className":3039,"style":839},[238],[14,3041,3043],{"className":3042},[243,244,245,246],[14,3044,3046],{"className":3045},[58,246],[14,3047,168],{"className":3048},[58,246],[14,3050,254],{"className":3051},[253],[14,3053,3055],{"className":3054},[226],[14,3056,3059],{"className":3057,"style":3058},[230],"height:0.52em;",[14,3060],{},[14,3062],{"className":3063},[684,819],[14,3065,3066,3069],{"style":848},[14,3067],{"className":3068,"style":839},[238],[14,3070],{"className":3071,"style":856},[855],[14,3073,3075,3078],{"style":3074},"top:-3.91em;",[14,3076],{"className":3077,"style":839},[238],[14,3079,3081,3141,3144],{"className":3080},[58],[14,3082,3084,3087],{"className":3083},[890],[14,3085,759],{"className":3086,"style":2839},[890,935,2838],[14,3088,3090],{"className":3089},[217],[14,3091,3093,3133],{"className":3092},[221,222],[14,3094,3096,3130],{"className":3095},[226],[14,3097,3099,3119],{"className":3098,"style":2852},[230],[14,3100,3101,3104],{"style":2855},[14,3102],{"className":3103,"style":239},[238],[14,3105,3107],{"className":3106},[243,244,245,246],[14,3108,3110,3113,3116],{"className":3109},[58,246],[14,3111,394],{"className":3112},[58,59,246],[14,3114,397],{"className":3115},[504,246],[14,3117,168],{"className":3118},[58,246],[14,3120,3121,3124],{"style":2876},[14,3122],{"className":3123,"style":239},[238],[14,3125,3127],{"className":3126},[243,244,245,246],[14,3128,94],{"className":3129},[58,59,246],[14,3131,254],{"className":3132},[253],[14,3134,3136],{"className":3135},[226],[14,3137,3139],{"className":3138,"style":2895},[230],[14,3140],{},[14,3142],{"className":3143,"style":272},[271],[14,3145,3147,3150,3334],{"className":3146},[58],[14,3148],{"className":3149},[518,819],[14,3151,3153],{"className":3152},[748],[14,3154,3156,3326],{"className":3155},[221,222],[14,3157,3159,3323],{"className":3158},[226],[14,3160,3163,3263,3271],{"className":3161,"style":3162},[230],"height:0.7212em;",[14,3164,3165,3168],{"style":2922},[14,3166],{"className":3167,"style":839},[238],[14,3169,3171],{"className":3170},[243,244,245,246],[14,3172,3174,3177,3217,3220,3260],{"className":3173},[58,246],[14,3175,2140],{"className":3176},[58,246],[14,3178,3180,3183],{"className":3179},[58,246],[14,3181,10],{"className":3182},[58,212,246],[14,3184,3186],{"className":3185},[217],[14,3187,3189,3209],{"className":3188},[221,222],[14,3190,3192,3206],{"className":3191},[226],[14,3193,3195],{"className":3194,"style":2953},[230],[14,3196,3197,3200],{"style":2956},[14,3198],{"className":3199,"style":2256},[238],[14,3201,3203],{"className":3202},[243,2260,2261,246],[14,3204,2631],{"className":3205},[58,59,246],[14,3207,254],{"className":3208},[253],[14,3210,3212],{"className":3211},[226],[14,3213,3215],{"className":3214,"style":2975},[230],[14,3216],{},[14,3218,2145],{"className":3219},[2371,246],[14,3221,3223,3226],{"className":3222},[58,246],[14,3224,164],{"className":3225,"style":213},[58,212,246],[14,3227,3229],{"className":3228},[217],[14,3230,3232,3252],{"className":3231},[221,222],[14,3233,3235,3249],{"className":3234},[226],[14,3236,3238],{"className":3237,"style":2999},[230],[14,3239,3240,3243],{"style":3002},[14,3241],{"className":3242,"style":2256},[238],[14,3244,3246],{"className":3245},[243,2260,2261,246],[14,3247,394],{"className":3248},[58,59,246],[14,3250,254],{"className":3251},[253],[14,3253,3255],{"className":3254},[226],[14,3256,3258],{"className":3257,"style":2975},[230],[14,3259],{},[14,3261,2140],{"className":3262},[58,246],[14,3264,3265,3268],{"style":848},[14,3266],{"className":3267,"style":839},[238],[14,3269],{"className":3270,"style":856},[855],[14,3272,3274,3277],{"style":3273},"top:-3.4101em;",[14,3275],{"className":3276,"style":839},[238],[14,3278,3280],{"className":3279},[243,244,245,246],[14,3281,3283],{"className":3282},[58,246],[14,3284,3286,3289],{"className":3285},[58,246],[14,3287,164],{"className":3288,"style":213},[58,212,246],[14,3290,3292],{"className":3291},[217],[14,3293,3295,3315],{"className":3294},[221,222],[14,3296,3298,3312],{"className":3297},[226],[14,3299,3301],{"className":3300,"style":2999},[230],[14,3302,3303,3306],{"style":3002},[14,3304],{"className":3305,"style":2256},[238],[14,3307,3309],{"className":3308},[243,2260,2261,246],[14,3310,394],{"className":3311},[58,59,246],[14,3313,254],{"className":3314},[253],[14,3316,3318],{"className":3317},[226],[14,3319,3321],{"className":3320,"style":2975},[230],[14,3322],{},[14,3324,254],{"className":3325},[253],[14,3327,3329],{"className":3328},[226],[14,3330,3332],{"className":3331,"style":3058},[230],[14,3333],{},[14,3335],{"className":3336},[684,819],[14,3338,254],{"className":3339},[253],[14,3341,3343],{"className":3342},[226],[14,3344,3347],{"className":3345,"style":3346},[230],"height:1.2551em;",[14,3348],{},[14,3350],{"className":3351},[684,819],[10,3353,3354,3355,3426],{},"The initial value can be the mean vector, iterating until convergence. Note: Special handling is required when ",[14,3356,3358,3376],{"className":3357,"translate":18},[17],[14,3359,3361],{"className":3360},[22],[24,3362,3363],{"xmlns":26},[28,3364,3365,3373],{},[31,3366,3367],{},[159,3368,3369,3371],{},[34,3370,10],{"mathvariant":163},[34,3372,2631],{},[38,3374,3375],{"encoding":40},"\\mathbf{p}_t",[14,3377,3379],{"className":3378,"ariaHidden":45},[44],[14,3380,3382,3385],{"className":3381},[49],[14,3383],{"className":3384,"style":205},[53],[14,3386,3388,3391],{"className":3387},[58],[14,3389,10],{"className":3390},[58,212],[14,3392,3394],{"className":3393},[217],[14,3395,3397,3418],{"className":3396},[221,222],[14,3398,3400,3415],{"className":3399},[226],[14,3401,3404],{"className":3402,"style":3403},[230],"height:0.2806em;",[14,3405,3406,3409],{"style":2441},[14,3407],{"className":3408,"style":239},[238],[14,3410,3412],{"className":3411},[243,244,245,246],[14,3413,2631],{"className":3414},[58,59,246],[14,3416,254],{"className":3417},[253],[14,3419,3421],{"className":3420},[226],[14,3422,3424],{"className":3423,"style":261},[230],[14,3425],{}," coincides exactly with a data point (skip that point).",[10,3428,3429],{},"It is used to represent \"typical samples\" instead of the mean when data contains outliers or noise. Hence, it serves as cluster centers in some clustering algorithms (more robust than K-means).",[10,3431,3432],{},"In video background modeling, the geometric median of multiple frames can serve as the background (unaffected by foreground moving objects). It is also used to find the optimal service point among multiple demand points (e.g., hospital, warehouse location).",[10,3434,3435],{},"The Geometric Median Prototype is the \"spatial center\" of data points, minimizing the total Euclidean distance to all points, and is more robust to outliers than the mean vector.",[66,3437,3439,3511],{"id":3438},"lpl_plp-euclidean-norm",[14,3440,3442,3460],{"className":3441,"translate":18},[17],[14,3443,3445],{"className":3444},[22],[24,3446,3447],{"xmlns":26},[28,3448,3449,3457],{},[31,3450,3451],{},[159,3452,3453,3455],{},[34,3454,2573],{},[34,3456,10],{},[38,3458,3459],{"encoding":40},"L_p",[14,3461,3463],{"className":3462,"ariaHidden":45},[44],[14,3464,3466,3470],{"className":3465},[49],[14,3467],{"className":3468,"style":3469},[53],"height:0.9694em;vertical-align:-0.2861em;",[14,3471,3473,3476],{"className":3472},[58],[14,3474,2573],{"className":3475},[58,59],[14,3477,3479],{"className":3478},[217],[14,3480,3482,3502],{"className":3481},[221,222],[14,3483,3485,3499],{"className":3484},[226],[14,3486,3488],{"className":3487,"style":350},[230],[14,3489,3490,3493],{"style":2441},[14,3491],{"className":3492,"style":239},[238],[14,3494,3496],{"className":3495},[243,244,245,246],[14,3497,10],{"className":3498},[58,59,246],[14,3500,254],{"className":3501},[253],[14,3503,3505],{"className":3504},[226],[14,3506,3509],{"className":3507,"style":3508},[230],"height:0.2861em;",[14,3510],{}," Euclidean Norm",[10,3513,3514,3515,3584,3585,3614],{},"The Euclidean norm is the most familiar \"vector distance\". Different norms essentially define \"what distance\u002Fmagnitude means\", and different definitions directly change the geometric structure, optimization behavior, and even model preferences. Generally, we refer to the ",[14,3516,3518,3535],{"className":3517,"translate":18},[17],[14,3519,3521],{"className":3520},[22],[24,3522,3523],{"xmlns":26},[28,3524,3525,3533],{},[31,3526,3527],{},[159,3528,3529,3531],{},[34,3530,2573],{},[34,3532,10],{},[38,3534,3459],{"encoding":40},[14,3536,3538],{"className":3537,"ariaHidden":45},[44],[14,3539,3541,3544],{"className":3540},[49],[14,3542],{"className":3543,"style":3469},[53],[14,3545,3547,3550],{"className":3546},[58],[14,3548,2573],{"className":3549},[58,59],[14,3551,3553],{"className":3552},[217],[14,3554,3556,3576],{"className":3555},[221,222],[14,3557,3559,3573],{"className":3558},[226],[14,3560,3562],{"className":3561,"style":350},[230],[14,3563,3564,3567],{"style":2441},[14,3565],{"className":3566,"style":239},[238],[14,3568,3570],{"className":3569},[243,244,245,246],[14,3571,10],{"className":3572},[58,59,246],[14,3574,254],{"className":3575},[253],[14,3577,3579],{"className":3578},[226],[14,3580,3582],{"className":3581,"style":3508},[230],[14,3583],{}," norm, where ",[14,3586,3588,3601],{"className":3587,"translate":18},[17],[14,3589,3591],{"className":3590},[22],[24,3592,3593],{"xmlns":26},[28,3594,3595,3599],{},[31,3596,3597],{},[34,3598,10],{},[38,3600,10],{"encoding":40},[14,3602,3604],{"className":3603,"ariaHidden":45},[44],[14,3605,3607,3611],{"className":3606},[49],[14,3608],{"className":3609,"style":3610},[53],"height:0.625em;vertical-align:-0.1944em;",[14,3612,10],{"className":3613},[58,59]," is a parameter.",[10,3616,3617,3618,3646,3647,3876,3877,3880],{},"For an ",[14,3619,3621,3634],{"className":3620,"translate":18},[17],[14,3622,3624],{"className":3623},[22],[24,3625,3626],{"xmlns":26},[28,3627,3628,3632],{},[31,3629,3630],{},[34,3631,125],{},[38,3633,125],{"encoding":40},[14,3635,3637],{"className":3636,"ariaHidden":45},[44],[14,3638,3640,3643],{"className":3639},[49],[14,3641],{"className":3642,"style":106},[53],[14,3644,125],{"className":3645},[58,59],"-dimensional vector ",[14,3648,3650,3700],{"className":3649,"translate":18},[17],[14,3651,3653],{"className":3652},[22],[24,3654,3655],{"xmlns":26},[28,3656,3657,3697],{},[31,3658,3659,3661,3663,3665,3671,3673,3679,3681,3683,3685,3687,3689,3695],{},[34,3660,2110],{"mathvariant":163},[170,3662,397],{},[170,3664,401],{"stretchy":400},[159,3666,3667,3669],{},[34,3668,2110],{},[166,3670,168],{},[170,3672,172],{"separator":45},[159,3674,3675,3677],{},[34,3676,2110],{},[166,3678,179],{},[170,3680,172],{"separator":45},[34,3682,685],{"mathvariant":2139},[34,3684,685],{"mathvariant":2139},[34,3686,685],{"mathvariant":2139},[170,3688,172],{"separator":45},[159,3690,3691,3693],{},[34,3692,2110],{},[34,3694,125],{},[170,3696,442],{"stretchy":400},[38,3698,3699],{"encoding":40},"\\mathbf{x} = (x_1, x_2, ..., x_n)",[14,3701,3703,3722],{"className":3702,"ariaHidden":45},[44],[14,3704,3706,3710,3713,3716,3719],{"className":3705},[49],[14,3707],{"className":3708,"style":3709},[53],"height:0.4444em;",[14,3711,2110],{"className":3712},[58,212],[14,3714],{"className":3715,"style":500},[271],[14,3717,397],{"className":3718},[504],[14,3720],{"className":3721,"style":500},[271],[14,3723,3725,3728,3731,3771,3774,3777,3817,3820,3823,3827,3830,3833,3873],{"className":3724},[49],[14,3726],{"className":3727,"style":514},[53],[14,3729,401],{"className":3730},[518],[14,3732,3734,3737],{"className":3733},[58],[14,3735,2110],{"className":3736},[58,59],[14,3738,3740],{"className":3739},[217],[14,3741,3743,3763],{"className":3742},[221,222],[14,3744,3746,3760],{"className":3745},[226],[14,3747,3749],{"className":3748,"style":231},[230],[14,3750,3751,3754],{"style":2441},[14,3752],{"className":3753,"style":239},[238],[14,3755,3757],{"className":3756},[243,244,245,246],[14,3758,168],{"className":3759},[58,246],[14,3761,254],{"className":3762},[253],[14,3764,3766],{"className":3765},[226],[14,3767,3769],{"className":3768,"style":261},[230],[14,3770],{},[14,3772,172],{"className":3773},[267],[14,3775],{"className":3776,"style":272},[271],[14,3778,3780,3783],{"className":3779},[58],[14,3781,2110],{"className":3782},[58,59],[14,3784,3786],{"className":3785},[217],[14,3787,3789,3809],{"className":3788},[221,222],[14,3790,3792,3806],{"className":3791},[226],[14,3793,3795],{"className":3794,"style":231},[230],[14,3796,3797,3800],{"style":2441},[14,3798],{"className":3799,"style":239},[238],[14,3801,3803],{"className":3802},[243,244,245,246],[14,3804,179],{"className":3805},[58,246],[14,3807,254],{"className":3808},[253],[14,3810,3812],{"className":3811},[226],[14,3813,3815],{"className":3814,"style":261},[230],[14,3816],{},[14,3818,172],{"className":3819},[267],[14,3821],{"className":3822,"style":272},[271],[14,3824,3826],{"className":3825},[58],"...",[14,3828,172],{"className":3829},[267],[14,3831],{"className":3832,"style":272},[271],[14,3834,3836,3839],{"className":3835},[58],[14,3837,2110],{"className":3838},[58,59],[14,3840,3842],{"className":3841},[217],[14,3843,3845,3865],{"className":3844},[221,222],[14,3846,3848,3862],{"className":3847},[226],[14,3849,3851],{"className":3850,"style":350},[230],[14,3852,3853,3856],{"style":2441},[14,3854],{"className":3855,"style":239},[238],[14,3857,3859],{"className":3858},[243,244,245,246],[14,3860,125],{"className":3861},[58,59,246],[14,3863,254],{"className":3864},[253],[14,3866,3868],{"className":3867},[226],[14,3869,3871],{"className":3870,"style":261},[230],[14,3872],{},[14,3874,442],{"className":3875},[684],", the ",[74,3878,3879],{},"Lp norm"," is defined as:",[14,3882,3884],{"className":3883,"translate":18},[144],[14,3885,3887,3969],{"className":3886,"translate":18},[17],[14,3888,3890],{"className":3889},[22],[24,3891,3892],{"xmlns":26,"display":153},[28,3893,3894,3966],{},[31,3895,3896,3898,3900,3906,3908,3954,3956,3959,3961,3964],{},[34,3897,2140],{"mathvariant":2139},[34,3899,2110],{"mathvariant":163},[159,3901,3902,3904],{},[34,3903,2140],{"mathvariant":2139},[34,3905,10],{},[170,3907,397],{},[2115,3909,3910,3945],{},[31,3911,3912,3914,3928,3931,3937,3943],{},[170,3913,401],{"fence":45},[755,3915,3916,3918,3926],{},[170,3917,759],{},[31,3919,3920,3922,3924],{},[34,3921,394],{},[170,3923,397],{},[166,3925,168],{},[34,3927,125],{},[34,3929,3930],{"mathvariant":2139},"∣",[159,3932,3933,3935],{},[34,3934,2110],{},[34,3936,394],{},[2115,3938,3939,3941],{},[34,3940,3930],{"mathvariant":2139},[34,3942,10],{},[170,3944,442],{"fence":45},[31,3946,3947,3949,3952],{},[166,3948,168],{},[34,3950,3951],{"mathvariant":2139},"\u002F",[34,3953,10],{},[170,3955,172],{"separator":45},[271,3957],{"width":3958},"1em",[34,3960,10],{},[170,3962,3963],{},"≥",[166,3965,168],{},[38,3967,3968],{"encoding":40},"\\|\\mathbf{x}\\|_p = \\left( \\sum_{i=1}^{n} |x_i|^p \\right)^{1\u002Fp}, \\quad p \\ge 1",[14,3970,3972,4034,4260],{"className":3971,"ariaHidden":45},[44],[14,3973,3975,3979,3982,3985,4025,4028,4031],{"className":3974},[49],[14,3976],{"className":3977,"style":3978},[53],"height:1.0361em;vertical-align:-0.2861em;",[14,3980,2140],{"className":3981},[58],[14,3983,2110],{"className":3984},[58,212],[14,3986,3988,3991],{"className":3987},[58],[14,3989,2140],{"className":3990},[58],[14,3992,3994],{"className":3993},[217],[14,3995,3997,4017],{"className":3996},[221,222],[14,3998,4000,4014],{"className":3999},[226],[14,4001,4003],{"className":4002,"style":350},[230],[14,4004,4005,4008],{"style":2441},[14,4006],{"className":4007,"style":239},[238],[14,4009,4011],{"className":4010},[243,244,245,246],[14,4012,10],{"className":4013},[58,59,246],[14,4015,254],{"className":4016},[253],[14,4018,4020],{"className":4019},[226],[14,4021,4023],{"className":4022,"style":3508},[230],[14,4024],{},[14,4026],{"className":4027,"style":500},[271],[14,4029,397],{"className":4030},[504],[14,4032],{"className":4033,"style":500},[271],[14,4035,4037,4041,4235,4238,4241,4245,4248,4251,4254,4257],{"className":4036},[49],[14,4038],{"className":4039,"style":4040},[53],"height:3.3056em;vertical-align:-1.2777em;",[14,4042,4044,4203],{"className":4043},[322],[14,4045,4047,4053,4120,4123,4126,4166,4197],{"className":4046},[322],[14,4048,4050],{"className":4049,"style":1185},[518,1184],[14,4051,401],{"className":4052},[1189,1190],[14,4054,4056],{"className":4055},[890,891],[14,4057,4059,4112],{"className":4058},[221,222],[14,4060,4062,4109],{"className":4061},[226],[14,4063,4065,4085,4095],{"className":4064,"style":901},[230],[14,4066,4067,4070],{"style":904},[14,4068],{"className":4069,"style":908},[238],[14,4071,4073],{"className":4072},[243,244,245,246],[14,4074,4076,4079,4082],{"className":4075},[58,246],[14,4077,394],{"className":4078},[58,59,246],[14,4080,397],{"className":4081},[504,246],[14,4083,168],{"className":4084},[58,246],[14,4086,4087,4090],{"style":926},[14,4088],{"className":4089,"style":908},[238],[14,4091,4092],{},[14,4093,759],{"className":4094},[890,935,936],[14,4096,4097,4100],{"style":939},[14,4098],{"className":4099,"style":908},[238],[14,4101,4103],{"className":4102},[243,244,245,246],[14,4104,4106],{"className":4105},[58,246],[14,4107,125],{"className":4108},[58,59,246],[14,4110,254],{"className":4111},[253],[14,4113,4115],{"className":4114},[226],[14,4116,4118],{"className":4117,"style":961},[230],[14,4119],{},[14,4121],{"className":4122,"style":272},[271],[14,4124,3930],{"className":4125},[58],[14,4127,4129,4132],{"className":4128},[58],[14,4130,2110],{"className":4131},[58,59],[14,4133,4135],{"className":4134},[217],[14,4136,4138,4158],{"className":4137},[221,222],[14,4139,4141,4155],{"className":4140},[226],[14,4142,4144],{"className":4143,"style":474},[230],[14,4145,4146,4149],{"style":2441},[14,4147],{"className":4148,"style":239},[238],[14,4150,4152],{"className":4151},[243,244,245,246],[14,4153,394],{"className":4154},[58,59,246],[14,4156,254],{"className":4157},[253],[14,4159,4161],{"className":4160},[226],[14,4162,4164],{"className":4163,"style":261},[230],[14,4165],{},[14,4167,4169,4172],{"className":4168},[58],[14,4170,3930],{"className":4171},[58],[14,4173,4175],{"className":4174},[217],[14,4176,4178],{"className":4177},[221],[14,4179,4181],{"className":4180},[226],[14,4182,4185],{"className":4183,"style":4184},[230],"height:0.7144em;",[14,4186,4188,4191],{"style":4187},"top:-3.113em;margin-right:0.05em;",[14,4189],{"className":4190,"style":239},[238],[14,4192,4194],{"className":4193},[243,244,245,246],[14,4195,10],{"className":4196},[58,59,246],[14,4198,4200],{"className":4199,"style":1185},[684,1184],[14,4201,442],{"className":4202},[1189,1190],[14,4204,4206],{"className":4205},[217],[14,4207,4209],{"className":4208},[221],[14,4210,4212],{"className":4211},[226],[14,4213,4216],{"className":4214,"style":4215},[230],"height:2.0279em;",[14,4217,4219,4222],{"style":4218},"top:-4.2029em;margin-right:0.05em;",[14,4220],{"className":4221,"style":239},[238],[14,4223,4225],{"className":4224},[243,244,245,246],[14,4226,4228,4232],{"className":4227},[58,246],[14,4229,4231],{"className":4230},[58,246],"1\u002F",[14,4233,10],{"className":4234},[58,59,246],[14,4236],{"className":4237,"style":272},[271],[14,4239,172],{"className":4240},[267],[14,4242],{"className":4243,"style":4244},[271],"margin-right:1em;",[14,4246],{"className":4247,"style":272},[271],[14,4249,10],{"className":4250},[58,59],[14,4252],{"className":4253,"style":500},[271],[14,4255,3963],{"className":4256},[504],[14,4258],{"className":4259,"style":500},[271],[14,4261,4263,4267],{"className":4262},[49],[14,4264],{"className":4265,"style":4266},[53],"height:0.6444em;",[14,4268,168],{"className":4269},[58],[10,4271,4272,4273,4301,4302,4353,4354,4425],{},"Different values of ",[14,4274,4276,4289],{"className":4275,"translate":18},[17],[14,4277,4279],{"className":4278},[22],[24,4280,4281],{"xmlns":26},[28,4282,4283,4287],{},[31,4284,4285],{},[34,4286,10],{},[38,4288,10],{"encoding":40},[14,4290,4292],{"className":4291,"ariaHidden":45},[44],[14,4293,4295,4298],{"className":4294},[49],[14,4296],{"className":4297,"style":3610},[53],[14,4299,10],{"className":4300},[58,59]," yield distances under different norms. When ",[14,4303,4305,4323],{"className":4304,"translate":18},[17],[14,4306,4308],{"className":4307},[22],[24,4309,4310],{"xmlns":26},[28,4311,4312,4320],{},[31,4313,4314,4316,4318],{},[34,4315,10],{},[170,4317,397],{},[166,4319,168],{},[38,4321,4322],{"encoding":40},"p=1",[14,4324,4326,4344],{"className":4325,"ariaHidden":45},[44],[14,4327,4329,4332,4335,4338,4341],{"className":4328},[49],[14,4330],{"className":4331,"style":3610},[53],[14,4333,10],{"className":4334},[58,59],[14,4336],{"className":4337,"style":500},[271],[14,4339,397],{"className":4340},[504],[14,4342],{"className":4343,"style":500},[271],[14,4345,4347,4350],{"className":4346},[49],[14,4348],{"className":4349,"style":4266},[53],[14,4351,168],{"className":4352},[58],", we obtain the distance under the ",[14,4355,4357,4375],{"className":4356,"translate":18},[17],[14,4358,4360],{"className":4359},[22],[24,4361,4362],{"xmlns":26},[28,4363,4364,4372],{},[31,4365,4366],{},[159,4367,4368,4370],{},[34,4369,2573],{},[166,4371,168],{},[38,4373,4374],{"encoding":40},"L_1",[14,4376,4378],{"className":4377,"ariaHidden":45},[44],[14,4379,4381,4385],{"className":4380},[49],[14,4382],{"className":4383,"style":4384},[53],"height:0.8333em;vertical-align:-0.15em;",[14,4386,4388,4391],{"className":4387},[58],[14,4389,2573],{"className":4390},[58,59],[14,4392,4394],{"className":4393},[217],[14,4395,4397,4417],{"className":4396},[221,222],[14,4398,4400,4414],{"className":4399},[226],[14,4401,4403],{"className":4402,"style":231},[230],[14,4404,4405,4408],{"style":2441},[14,4406],{"className":4407,"style":239},[238],[14,4409,4411],{"className":4410},[243,244,245,246],[14,4412,168],{"className":4413},[58,246],[14,4415,254],{"className":4416},[253],[14,4418,4420],{"className":4419},[226],[14,4421,4423],{"className":4422,"style":261},[230],[14,4424],{}," norm:",[14,4427,4429],{"className":4428,"translate":18},[144],[14,4430,4432,4524],{"className":4431,"translate":18},[17],[14,4433,4435],{"className":4434},[22],[24,4436,4437],{"xmlns":26,"display":153},[28,4438,4439,4521],{},[31,4440,4441,4443,4445,4451,4453,4467,4469,4475,4477,4479,4481,4487,4489,4491,4493,4499,4501,4503,4505,4507,4509,4511,4513,4519],{},[34,4442,2140],{"mathvariant":2139},[34,4444,2110],{"mathvariant":163},[159,4446,4447,4449],{},[34,4448,2140],{"mathvariant":2139},[166,4450,168],{},[170,4452,397],{},[755,4454,4455,4457,4465],{},[170,4456,759],{},[31,4458,4459,4461,4463],{},[34,4460,394],{},[170,4462,397],{},[166,4464,168],{},[34,4466,125],{},[34,4468,3930],{"mathvariant":2139},[159,4470,4471,4473],{},[34,4472,2110],{},[34,4474,394],{},[34,4476,3930],{"mathvariant":2139},[170,4478,397],{},[34,4480,3930],{"mathvariant":2139},[159,4482,4483,4485],{},[34,4484,2110],{},[166,4486,168],{},[34,4488,3930],{"mathvariant":2139},[170,4490,2634],{},[34,4492,3930],{"mathvariant":2139},[159,4494,4495,4497],{},[34,4496,2110],{},[166,4498,179],{},[34,4500,3930],{"mathvariant":2139},[170,4502,2634],{},[34,4504,685],{"mathvariant":2139},[34,4506,685],{"mathvariant":2139},[34,4508,685],{"mathvariant":2139},[170,4510,2634],{},[34,4512,3930],{"mathvariant":2139},[159,4514,4515,4517],{},[34,4516,2110],{},[34,4518,125],{},[34,4520,3930],{"mathvariant":2139},[38,4522,4523],{"encoding":40},"\\|\\mathbf{x}\\|_1 = \\sum_{i=1}^{n} |x_i| = |x_1| + |x_2| + ... + |x_n|",[14,4525,4527,4588,4719,4780,4841,4860],{"className":4526,"ariaHidden":45},[44],[14,4528,4530,4533,4536,4539,4579,4582,4585],{"className":4529},[49],[14,4531],{"className":4532,"style":514},[53],[14,4534,2140],{"className":4535},[58],[14,4537,2110],{"className":4538},[58,212],[14,4540,4542,4545],{"className":4541},[58],[14,4543,2140],{"className":4544},[58],[14,4546,4548],{"className":4547},[217],[14,4549,4551,4571],{"className":4550},[221,222],[14,4552,4554,4568],{"className":4553},[226],[14,4555,4557],{"className":4556,"style":231},[230],[14,4558,4559,4562],{"style":2441},[14,4560],{"className":4561,"style":239},[238],[14,4563,4565],{"className":4564},[243,244,245,246],[14,4566,168],{"className":4567},[58,246],[14,4569,254],{"className":4570},[253],[14,4572,4574],{"className":4573},[226],[14,4575,4577],{"className":4576,"style":261},[230],[14,4578],{},[14,4580],{"className":4581,"style":500},[271],[14,4583,397],{"className":4584},[504],[14,4586],{"className":4587,"style":500},[271],[14,4589,4591,4594,4661,4664,4667,4707,4710,4713,4716],{"className":4590},[49],[14,4592],{"className":4593,"style":812},[53],[14,4595,4597],{"className":4596},[890,891],[14,4598,4600,4653],{"className":4599},[221,222],[14,4601,4603,4650],{"className":4602},[226],[14,4604,4606,4626,4636],{"className":4605,"style":901},[230],[14,4607,4608,4611],{"style":904},[14,4609],{"className":4610,"style":908},[238],[14,4612,4614],{"className":4613},[243,244,245,246],[14,4615,4617,4620,4623],{"className":4616},[58,246],[14,4618,394],{"className":4619},[58,59,246],[14,4621,397],{"className":4622},[504,246],[14,4624,168],{"className":4625},[58,246],[14,4627,4628,4631],{"style":926},[14,4629],{"className":4630,"style":908},[238],[14,4632,4633],{},[14,4634,759],{"className":4635},[890,935,936],[14,4637,4638,4641],{"style":939},[14,4639],{"className":4640,"style":908},[238],[14,4642,4644],{"className":4643},[243,244,245,246],[14,4645,4647],{"className":4646},[58,246],[14,4648,125],{"className":4649},[58,59,246],[14,4651,254],{"className":4652},[253],[14,4654,4656],{"className":4655},[226],[14,4657,4659],{"className":4658,"style":961},[230],[14,4660],{},[14,4662],{"className":4663,"style":272},[271],[14,4665,3930],{"className":4666},[58],[14,4668,4670,4673],{"className":4669},[58],[14,4671,2110],{"className":4672},[58,59],[14,4674,4676],{"className":4675},[217],[14,4677,4679,4699],{"className":4678},[221,222],[14,4680,4682,4696],{"className":4681},[226],[14,4683,4685],{"className":4684,"style":474},[230],[14,4686,4687,4690],{"style":2441},[14,4688],{"className":4689,"style":239},[238],[14,4691,4693],{"className":4692},[243,244,245,246],[14,4694,394],{"className":4695},[58,59,246],[14,4697,254],{"className":4698},[253],[14,4700,4702],{"className":4701},[226],[14,4703,4705],{"className":4704,"style":261},[230],[14,4706],{},[14,4708,3930],{"className":4709},[58],[14,4711],{"className":4712,"style":500},[271],[14,4714,397],{"className":4715},[504],[14,4717],{"className":4718,"style":500},[271],[14,4720,4722,4725,4728,4768,4771,4774,4777],{"className":4721},[49],[14,4723],{"className":4724,"style":514},[53],[14,4726,3930],{"className":4727},[58],[14,4729,4731,4734],{"className":4730},[58],[14,4732,2110],{"className":4733},[58,59],[14,4735,4737],{"className":4736},[217],[14,4738,4740,4760],{"className":4739},[221,222],[14,4741,4743,4757],{"className":4742},[226],[14,4744,4746],{"className":4745,"style":231},[230],[14,4747,4748,4751],{"style":2441},[14,4749],{"className":4750,"style":239},[238],[14,4752,4754],{"className":4753},[243,244,245,246],[14,4755,168],{"className":4756},[58,246],[14,4758,254],{"className":4759},[253],[14,4761,4763],{"className":4762},[226],[14,4764,4766],{"className":4765,"style":261},[230],[14,4767],{},[14,4769,3930],{"className":4770},[58],[14,4772],{"className":4773,"style":2367},[271],[14,4775,2634],{"className":4776},[2371],[14,4778],{"className":4779,"style":2367},[271],[14,4781,4783,4786,4789,4829,4832,4835,4838],{"className":4782},[49],[14,4784],{"className":4785,"style":514},[53],[14,4787,3930],{"className":4788},[58],[14,4790,4792,4795],{"className":4791},[58],[14,4793,2110],{"className":4794},[58,59],[14,4796,4798],{"className":4797},[217],[14,4799,4801,4821],{"className":4800},[221,222],[14,4802,4804,4818],{"className":4803},[226],[14,4805,4807],{"className":4806,"style":231},[230],[14,4808,4809,4812],{"style":2441},[14,4810],{"className":4811,"style":239},[238],[14,4813,4815],{"className":4814},[243,244,245,246],[14,4816,179],{"className":4817},[58,246],[14,4819,254],{"className":4820},[253],[14,4822,4824],{"className":4823},[226],[14,4825,4827],{"className":4826,"style":261},[230],[14,4828],{},[14,4830,3930],{"className":4831},[58],[14,4833],{"className":4834,"style":2367},[271],[14,4836,2634],{"className":4837},[2371],[14,4839],{"className":4840,"style":2367},[271],[14,4842,4844,4848,4851,4854,4857],{"className":4843},[49],[14,4845],{"className":4846,"style":4847},[53],"height:0.6667em;vertical-align:-0.0833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it is the sum of absolute values of each component. In a 2D plane, the ",[14,4916,4918,4935],{"className":4917,"translate":18},[17],[14,4919,4921],{"className":4920},[22],[24,4922,4923],{"xmlns":26},[28,4924,4925,4933],{},[31,4926,4927],{},[159,4928,4929,4931],{},[34,4930,2573],{},[166,4932,168],{},[38,4934,4374],{"encoding":40},[14,4936,4938],{"className":4937,"ariaHidden":45},[44],[14,4939,4941,4944],{"className":4940},[49],[14,4942],{"className":4943,"style":4384},[53],[14,4945,4947,4950],{"className":4946},[58],[14,4948,2573],{"className":4949},[58,59],[14,4951,4953],{"className":4952},[217],[14,4954,4956,4976],{"className":4955},[221,222],[14,4957,4959,4973],{"className":4958},[226],[14,4960,4962],{"className":4961,"style":231},[230],[14,4963,4964,4967],{"style":2441},[14,4965],{"className":4966,"style":239},[238],[14,4968,4970],{"className":4969},[243,244,245,246],[14,4971,168],{"className":4972},[58,246],[14,4974,254],{"className":4975},[253],[14,4977,4979],{"className":4978},[226],[14,4980,4982],{"className":4981,"style":261},[230],[14,4983],{}," distance is the ",[74,4986,4987],{},"Manhattan distance"," (moving only along coordinate axes). For example, ",[14,4990,4992,5022],{"className":4991,"translate":18},[17],[14,4993,4995],{"className":4994},[22],[24,4996,4997],{"xmlns":26},[28,4998,4999,5019],{},[31,5000,5001,5003,5005,5007,5010,5012,5014,5017],{},[34,5002,2110],{"mathvariant":163},[170,5004,397],{},[170,5006,401],{"stretchy":400},[166,5008,5009],{},"3",[170,5011,172],{"separator":45},[170,5013,2145],{},[166,5015,5016],{},"4",[170,5018,442],{"stretchy":400},[38,5020,5021],{"encoding":40},"\\mathbf{x} = (3, 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norm ",[14,5142,5144,5197],{"className":5143,"translate":18},[17],[14,5145,5147],{"className":5146},[22],[24,5148,5149],{"xmlns":26},[28,5150,5151,5194],{},[31,5152,5153,5155,5157,5163,5165,5167,5169,5171,5173,5175,5177,5179,5181,5183,5185,5187,5189,5191],{},[34,5154,2140],{"mathvariant":2139},[34,5156,2110],{"mathvariant":163},[159,5158,5159,5161],{},[34,5160,2140],{"mathvariant":2139},[166,5162,168],{},[170,5164,397],{},[34,5166,3930],{"mathvariant":2139},[166,5168,5009],{},[34,5170,3930],{"mathvariant":2139},[170,5172,2634],{},[34,5174,3930],{"mathvariant":2139},[170,5176,2145],{},[166,5178,5016],{},[34,5180,3930],{"mathvariant":2139},[170,5182,397],{},[166,5184,5009],{},[170,5186,2634],{},[166,5188,5016],{},[170,5190,397],{},[166,5192,5193],{},"7",[38,5195,5196],{"encoding":40},"\\|\\mathbf{x}\\|_1 = |3| + |-4| = 3 + 4 = 7",[14,5198,5200,5261,5280,5298,5317,5336,5354],{"className":5199,"ariaHidden":45},[44],[14,5201,5203,5206,5209,5212,5252,5255,5258],{"className":5202},[49],[14,5204],{"className":5205,"style":514},[53],[14,5207,2140],{"className":5208},[58],[14,5210,2110],{"className":5211},[58,212],[14,5213,5215,5218],{"className":5214},[58],[14,5216,2140],{"className":5217},[58],[14,5219,5221],{"className":5220},[217],[14,5222,5224,5244],{"className":5223},[221,222],[14,5225,5227,5241],{"className":5226},[226],[14,5228,5230],{"className":5229,"style":231},[230],[14,5231,5232,5235],{"style":2441},[14,5233],{"className":5234,"style":239},[238],[14,5236,5238],{"className":5237},[243,244,245,246],[14,5239,168],{"className":5240},[58,246],[14,5242,254],{"className":5243},[253],[14,5245,5247],{"className":5246},[226],[14,5248,5250],{"className":5249,"style":261},[230],[14,5251],{},[14,5253],{"className":5254,"style":500},[271],[14,5256,397],{"className":5257},[504],[14,5259],{"className":5260,"style":500},[271],[14,5262,5264,5267,5271,5274,5277],{"className":5263},[49],[14,5265],{"className":5266,"style":514},[53],[14,5268,5270],{"className":5269},[58],"∣3∣",[14,5272],{"className":5273,"style":2367},[271],[14,5275,2634],{"className":5276},[2371],[14,5278],{"className":5279,"style":2367},[271],[14,5281,5283,5286,5289,5292,5295],{"className":5282},[49],[14,5284],{"className":5285,"style":514},[53],[14,5287,3930],{"className":5288},[58],[14,5290],{"className":5291,"style":2367},[271],[14,5293,2145],{"className":5294},[2371],[14,5296],{"className":5297,"style":2367},[271],[14,5299,5301,5304,5308,5311,5314],{"className":5300},[49],[14,5302],{"className":5303,"style":514},[53],[14,5305,5307],{"className":5306},[58],"4∣",[14,5309],{"className":5310,"style":500},[271],[14,5312,397],{"className":5313},[504],[14,5315],{"className":5316,"style":500},[271],[14,5318,5320,5324,5327,5330,5333],{"className":5319},[49],[14,5321],{"className":5322,"style":5323},[53],"height:0.7278em;vertical-align:-0.0833em;",[14,5325,5009],{"className":5326},[58],[14,5328],{"className":5329,"style":2367},[271],[14,5331,2634],{"className":5332},[2371],[14,5334],{"className":5335,"style":2367},[271],[14,5337,5339,5342,5345,5348,5351],{"className":5338},[49],[14,5340],{"className":5341,"style":4266},[53],[14,5343,5016],{"className":5344},[58],[14,5346],{"className":5347,"style":500},[271],[14,5349,397],{"className":5350},[504],[14,5352],{"className":5353,"style":500},[271],[14,5355,5357,5360],{"className":5356},[49],[14,5358],{"className":5359,"style":4266},[53],[14,5361,5193],{"className":5362},[58],[10,5364,5365,5366,5417,5418,5488],{},"When ",[14,5367,5369,5387],{"className":5368,"translate":18},[17],[14,5370,5372],{"className":5371},[22],[24,5373,5374],{"xmlns":26},[28,5375,5376,5384],{},[31,5377,5378,5380,5382],{},[34,5379,10],{},[170,5381,397],{},[166,5383,179],{},[38,5385,5386],{"encoding":40},"p=2",[14,5388,5390,5408],{"className":5389,"ariaHidden":45},[44],[14,5391,5393,5396,5399,5402,5405],{"className":5392},[49],[14,5394],{"className":5395,"style":3610},[53],[14,5397,10],{"className":5398},[58,59],[14,5400],{"className":5401,"style":500},[271],[14,5403,397],{"className":5404},[504],[14,5406],{"className":5407,"style":500},[271],[14,5409,5411,5414],{"className":5410},[49],[14,5412],{"className":5413,"style":4266},[53],[14,5415,179],{"className":5416},[58],", the distance under the ",[14,5419,5421,5439],{"className":5420,"translate":18},[17],[14,5422,5424],{"className":5423},[22],[24,5425,5426],{"xmlns":26},[28,5427,5428,5436],{},[31,5429,5430],{},[159,5431,5432,5434],{},[34,5433,2573],{},[166,5435,179],{},[38,5437,5438],{"encoding":40},"L_2",[14,5440,5442],{"className":5441,"ariaHidden":45},[44],[14,5443,5445,5448],{"className":5444},[49],[14,5446],{"className":5447,"style":4384},[53],[14,5449,5451,5454],{"className":5450},[58],[14,5452,2573],{"className":5453},[58,59],[14,5455,5457],{"className":5456},[217],[14,5458,5460,5480],{"className":5459},[221,222],[14,5461,5463,5477],{"className":5462},[226],[14,5464,5466],{"className":5465,"style":231},[230],[14,5467,5468,5471],{"style":2441},[14,5469],{"className":5470,"style":239},[238],[14,5472,5474],{"className":5473},[243,244,245,246],[14,5475,179],{"className":5476},[58,246],[14,5478,254],{"className":5479},[253],[14,5481,5483],{"className":5482},[226],[14,5484,5486],{"className":5485,"style":261},[230],[14,5487],{}," norm is defined 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0 400000 1080",[5832,6594],{"d":6595},"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",[14,6597,254],{"className":6598},[253],[14,6600,6602],{"className":6601},[226],[14,6603,6606],{"className":6604,"style":6605},[230],"height:0.1744em;",[14,6607],{},[14,6609],{"className":6610,"style":500},[271],[14,6612,397],{"className":6613},[504],[14,6615],{"className":6616,"style":500},[271],[14,6618,6620,6623],{"className":6619},[49],[14,6621],{"className":6622,"style":4266},[53],[14,6624,6321],{"className":6625},[58],". Geometrically, when ",[14,6628,6630,6656],{"className":6629,"translate":18},[17],[14,6631,6633],{"className":6632},[22],[24,6634,6635],{"xmlns":26},[28,6636,6637,6653],{},[31,6638,6639,6641,6643,6649,6651],{},[34,6640,2140],{"mathvariant":2139},[34,6642,2110],{"mathvariant":163},[159,6644,6645,6647],{},[34,6646,2140],{"mathvariant":2139},[166,6648,179],{},[170,6650,397],{},[166,6652,168],{},[38,6654,6655],{"encoding":40},"\\|\\mathbf{x}\\|_2 = 1",[14,6657,6659,6720],{"className":6658,"ariaHidden":45},[44],[14,6660,6662,6665,6668,6671,6711,6714,6717],{"className":6661},[49],[14,6663],{"className":6664,"style":514},[53],[14,6666,2140],{"className":6667},[58],[14,6669,2110],{"className":6670},[58,212],[14,6672,6674,6677],{"className":6673},[58],[14,6675,2140],{"className":6676},[58],[14,6678,6680],{"className":6679},[217],[14,6681,6683,6703],{"className":6682},[221,222],[14,6684,6686,6700],{"className":6685},[226],[14,6687,6689],{"className":6688,"style":231},[230],[14,6690,6691,6694],{"style":2441},[14,6692],{"className":6693,"style":239},[238],[14,6695,6697],{"className":6696},[243,244,245,246],[14,6698,179],{"className":6699},[58,246],[14,6701,254],{"className":6702},[253],[14,6704,6706],{"className":6705},[226],[14,6707,6709],{"className":6708,"style":261},[230],[14,6710],{},[14,6712],{"className":6713,"style":500},[271],[14,6715,397],{"className":6716},[504],[14,6718],{"className":6719,"style":500},[271],[14,6721,6723,6726],{"className":6722},[49],[14,6724],{"className":6725,"style":4266},[53],[14,6727,168],{"className":6728},[58],", the vectors lie on the unit circle.",[10,6731,6732,6733,6802,6803,6872,6873,6942,6943,6989,6990,7018,7019,685],{},"In summary, the distance under the ",[14,6734,6736,6753],{"className":6735,"translate":18},[17],[14,6737,6739],{"className":6738},[22],[24,6740,6741],{"xmlns":26},[28,6742,6743,6751],{},[31,6744,6745],{},[159,6746,6747,6749],{},[34,6748,2573],{},[166,6750,168],{},[38,6752,4374],{"encoding":40},[14,6754,6756],{"className":6755,"ariaHidden":45},[44],[14,6757,6759,6762],{"className":6758},[49],[14,6760],{"className":6761,"style":4384},[53],[14,6763,6765,6768],{"className":6764},[58],[14,6766,2573],{"className":6767},[58,59],[14,6769,6771],{"className":6770},[217],[14,6772,6774,6794],{"className":6773},[221,222],[14,6775,6777,6791],{"className":6776},[226],[14,6778,6780],{"className":6779,"style":231},[230],[14,6781,6782,6785],{"style":2441},[14,6783],{"className":6784,"style":239},[238],[14,6786,6788],{"className":6787},[243,244,245,246],[14,6789,168],{"className":6790},[58,246],[14,6792,254],{"className":6793},[253],[14,6795,6797],{"className":6796},[226],[14,6798,6800],{"className":6799,"style":261},[230],[14,6801],{}," norm is the sum of absolute values per coordinate (Manhattan distance), yielding sparse solutions. The ",[14,6804,6806,6823],{"className":6805,"translate":18},[17],[14,6807,6809],{"className":6808},[22],[24,6810,6811],{"xmlns":26},[28,6812,6813,6821],{},[31,6814,6815],{},[159,6816,6817,6819],{},[34,6818,2573],{},[166,6820,179],{},[38,6822,5438],{"encoding":40},[14,6824,6826],{"className":6825,"ariaHidden":45},[44],[14,6827,6829,6832],{"className":6828},[49],[14,6830],{"className":6831,"style":4384},[53],[14,6833,6835,6838],{"className":6834},[58],[14,6836,2573],{"className":6837},[58,59],[14,6839,6841],{"className":6840},[217],[14,6842,6844,6864],{"className":6843},[221,222],[14,6845,6847,6861],{"className":6846},[226],[14,6848,6850],{"className":6849,"style":231},[230],[14,6851,6852,6855],{"style":2441},[14,6853],{"className":6854,"style":239},[238],[14,6856,6858],{"className":6857},[243,244,245,246],[14,6859,179],{"className":6860},[58,246],[14,6862,254],{"className":6863},[253],[14,6865,6867],{"className":6866},[226],[14,6868,6870],{"className":6869,"style":261},[230],[14,6871],{}," norm is the square root of the sum of squares (Euclidean distance), smooth and easy to optimize. The general ",[14,6874,6876,6893],{"className":6875,"translate":18},[17],[14,6877,6879],{"className":6878},[22],[24,6880,6881],{"xmlns":26},[28,6882,6883,6891],{},[31,6884,6885],{},[159,6886,6887,6889],{},[34,6888,2573],{},[34,6890,10],{},[38,6892,3459],{"encoding":40},[14,6894,6896],{"className":6895,"ariaHidden":45},[44],[14,6897,6899,6902],{"className":6898},[49],[14,6900],{"className":6901,"style":3469},[53],[14,6903,6905,6908],{"className":6904},[58],[14,6906,2573],{"className":6907},[58,59],[14,6909,6911],{"className":6910},[217],[14,6912,6914,6934],{"className":6913},[221,222],[14,6915,6917,6931],{"className":6916},[226],[14,6918,6920],{"className":6919,"style":350},[230],[14,6921,6922,6925],{"style":2441},[14,6923],{"className":6924,"style":239},[238],[14,6926,6928],{"className":6927},[243,244,245,246],[14,6929,10],{"className":6930},[58,59,246],[14,6932,254],{"className":6933},[253],[14,6935,6937],{"className":6936},[226],[14,6938,6940],{"className":6939,"style":3508},[230],[14,6941],{}," formula is the ",[14,6944,6946,6968],{"className":6945,"translate":18},[17],[14,6947,6949],{"className":6948},[22],[24,6950,6951],{"xmlns":26},[28,6952,6953,6965],{},[31,6954,6955,6957,6959,6961,6963],{},[170,6956,401],{"stretchy":400},[166,6958,168],{},[34,6960,3951],{"mathvariant":2139},[34,6962,10],{},[170,6964,442],{"stretchy":400},[38,6966,6967],{"encoding":40},"(1\u002Fp)",[14,6969,6971],{"className":6970,"ariaHidden":45},[44],[14,6972,6974,6977,6980,6983,6986],{"className":6973},[49],[14,6975],{"className":6976,"style":514},[53],[14,6978,401],{"className":6979},[518],[14,6981,4231],{"className":6982},[58],[14,6984,10],{"className":6985},[58,59],[14,6987,442],{"className":6988},[684],"-th power of the sum of the ",[14,6991,6993,7006],{"className":6992,"translate":18},[17],[14,6994,6996],{"className":6995},[22],[24,6997,6998],{"xmlns":26},[28,6999,7000,7004],{},[31,7001,7002],{},[34,7003,10],{},[38,7005,10],{"encoding":40},[14,7007,7009],{"className":7008,"ariaHidden":45},[44],[14,7010,7012,7015],{"className":7011},[49],[14,7013],{"className":7014,"style":3610},[53],[14,7016,10],{"className":7017},[58,59],"-th powers of each component, approaching the maximum as ",[14,7020,7022,7042],{"className":7021,"translate":18},[17],[14,7023,7025],{"className":7024},[22],[24,7026,7027],{"xmlns":26},[28,7028,7029,7039],{},[31,7030,7031,7033,7036],{},[34,7032,10],{},[170,7034,7035],{},"→",[34,7037,7038],{"mathvariant":2139},"∞",[38,7040,7041],{"encoding":40},"p \\to \\infty",[14,7043,7045,7063],{"className":7044,"ariaHidden":45},[44],[14,7046,7048,7051,7054,7057,7060],{"className":7047},[49],[14,7049],{"className":7050,"style":3610},[53],[14,7052,10],{"className":7053},[58,59],[14,7055],{"className":7056,"style":500},[271],[14,7058,7035],{"className":7059},[504],[14,7061],{"className":7062,"style":500},[271],[14,7064,7066,7069],{"className":7065},[49],[14,7067],{"className":7068,"style":106},[53],[14,7070,7038],{"className":7071},[58],[66,7073,7075],{"id":7074},"coordinate-wise-median-prototype","Coordinate-wise Median Prototype",[10,7077,72,7078,7080,7081,7084],{},[74,7079,7075],{}," refers to: For a multi-dimensional dataset, compute the ",[74,7082,7083],{},"median"," on each dimension (coordinate) separately, and then combine these medians into a new vector that serves as the \"prototype\" or central representative of the entire dataset.",[10,7086,7087,7088,110,7116,1845,7144,7337],{},"Suppose we have ",[14,7089,7091,7104],{"className":7090,"translate":18},[17],[14,7092,7094],{"className":7093},[22],[24,7095,7096],{"xmlns":26},[28,7097,7098,7102],{},[31,7099,7100],{},[34,7101,94],{},[38,7103,94],{"encoding":40},[14,7105,7107],{"className":7106,"ariaHidden":45},[44],[14,7108,7110,7113],{"className":7109},[49],[14,7111],{"className":7112,"style":106},[53],[14,7114,94],{"className":7115},[58,59],[14,7117,7119,7132],{"className":7118,"translate":18},[17],[14,7120,7122],{"className":7121},[22],[24,7123,7124],{"xmlns":26},[28,7125,7126,7130],{},[31,7127,7128],{},[34,7129,125],{},[38,7131,125],{"encoding":40},[14,7133,7135],{"className":7134,"ariaHidden":45},[44],[14,7136,7138,7141],{"className":7137},[49],[14,7139],{"className":7140,"style":106},[53],[14,7142,125],{"className":7143},[58,59],[14,7145,7147,7184],{"className":7146,"translate":18},[17],[14,7148,7150],{"className":7149},[22],[24,7151,7152],{"xmlns":26},[28,7153,7154,7182],{},[31,7155,7156,7162,7164,7170,7172,7174,7176],{},[159,7157,7158,7160],{},[34,7159,164],{"mathvariant":163},[166,7161,168],{},[170,7163,172],{"separator":45},[159,7165,7166,7168],{},[34,7167,164],{"mathvariant":163},[166,7169,179],{},[170,7171,172],{"separator":45},[170,7173,184],{},[170,7175,172],{"separator":45},[159,7177,7178,7180],{},[34,7179,164],{"mathvariant":163},[34,7181,94],{},[38,7183,1886],{"encoding":40},[14,7185,7187],{"className":7186,"ariaHidden":45},[44],[14,7188,7190,7193,7233,7236,7239,7279,7282,7285,7288,7291,7294,7297],{"className":7189},[49],[14,7191],{"className":7192,"style":205},[53],[14,7194,7196,7199],{"className":7195},[58],[14,7197,164],{"className":7198,"style":213},[58,212],[14,7200,7202],{"className":7201},[217],[14,7203,7205,7225],{"className":7204},[221,222],[14,7206,7208,7222],{"className":7207},[226],[14,7209,7211],{"className":7210,"style":231},[230],[14,7212,7213,7216],{"style":234},[14,7214],{"className":7215,"style":239},[238],[14,7217,7219],{"className":7218},[243,244,245,246],[14,7220,168],{"className":7221},[58,246],[14,7223,254],{"className":7224},[253],[14,7226,7228],{"className":7227},[226],[14,7229,7231],{"className":7230,"style":261},[230],[14,7232],{},[14,7234,172],{"className":7235},[267],[14,7237],{"className":7238,"style":272},[271],[14,7240,7242,7245],{"className":7241},[58],[14,7243,164],{"className":7244,"style":213},[58,212],[14,7246,7248],{"className":7247},[217],[14,7249,7251,7271],{"className":7250},[221,222],[14,7252,7254,7268],{"className":7253},[226],[14,7255,7257],{"className":7256,"style":231},[230],[14,7258,7259,7262],{"style":234},[14,7260],{"className":7261,"style":239},[238],[14,7263,7265],{"className":7264},[243,244,245,246],[14,7266,179],{"className":7267},[58,246],[14,7269,254],{"className":7270},[253],[14,7272,7274],{"className":7273},[226],[14,7275,7277],{"className":7276,"style":261},[230],[14,7278],{},[14,7280,172],{"className":7281},[267],[14,7283],{"className":7284,"style":272},[271],[14,7286,184],{"className":7287},[322],[14,7289],{"className":7290,"style":272},[271],[14,7292,172],{"className":7293},[267],[14,7295],{"className":7296,"style":272},[271],[14,7298,7300,7303],{"className":7299},[58],[14,7301,164],{"className":7302,"style":213},[58,212],[14,7304,7306],{"className":7305},[217],[14,7307,7309,7329],{"className":7308},[221,222],[14,7310,7312,7326],{"className":7311},[226],[14,7313,7315],{"className":7314,"style":350},[230],[14,7316,7317,7320],{"style":234},[14,7318],{"className":7319,"style":239},[238],[14,7321,7323],{"className":7322},[243,244,245,246],[14,7324,94],{"className":7325},[58,59,246],[14,7327,254],{"className":7328},[253],[14,7330,7332],{"className":7331},[226],[14,7333,7335],{"className":7334,"style":261},[230],[14,7336],{},", where",[14,7339,7341],{"className":7340,"translate":18},[144],[14,7342,7344,7405],{"className":7343,"translate":18},[17],[14,7345,7347],{"className":7346},[22],[24,7348,7349],{"xmlns":26,"display":153},[28,7350,7351,7403],{},[31,7352,7353,7359,7361,7363,7373,7375,7385,7387,7389,7391,7401],{},[159,7354,7355,7357],{},[34,7356,164],{"mathvariant":163},[34,7358,394],{},[170,7360,397],{},[170,7362,401],{"stretchy":400},[159,7364,7365,7367],{},[34,7366,164],{},[31,7368,7369,7371],{},[34,7370,394],{},[166,7372,168],{},[170,7374,172],{"separator":45},[159,7376,7377,7379],{},[34,7378,164],{},[31,7380,7381,7383],{},[34,7382,394],{},[166,7384,179],{},[170,7386,172],{"separator":45},[170,7388,184],{},[170,7390,172],{"separator":45},[159,7392,7393,7395],{},[34,7394,164],{},[31,7396,7397,7399],{},[34,7398,394],{},[34,7400,125],{},[170,7402,442],{"stretchy":400},[38,7404,445],{"encoding":40},[14,7406,7408,7463],{"className":7407,"ariaHidden":45},[44],[14,7409,7411,7414,7454,7457,7460],{"className":7410},[49],[14,7412],{"className":7413,"style":455},[53],[14,7415,7417,7420],{"className":7416},[58],[14,7418,164],{"className":7419,"style":213},[58,212],[14,7421,7423],{"className":7422},[217],[14,7424,7426,7446],{"className":7425},[221,222],[14,7427,7429,7443],{"className":7428},[226],[14,7430,7432],{"className":7431,"style":474},[230],[14,7433,7434,7437],{"style":234},[14,7435],{"className":7436,"style":239},[238],[14,7438,7440],{"className":7439},[243,244,245,246],[14,7441,394],{"className":7442},[58,59,246],[14,7444,254],{"className":7445},[253],[14,7447,7449],{"className":7448},[226],[14,7450,7452],{"className":7451,"style":261},[230],[14,7453],{},[14,7455],{"className":7456,"style":500},[271],[14,7458,397],{"className":7459},[504],[14,7461],{"className":7462,"style":500},[271],[14,7464,7466,7469,7472,7518,7521,7524,7570,7573,7576,7579,7582,7585,7588,7631],{"className":7465},[49],[14,7467],{"className":7468,"style":514},[53],[14,7470,401],{"className":7471},[518],[14,7473,7475,7478],{"className":7474},[58],[14,7476,164],{"className":7477,"style":525},[58,59],[14,7479,7481],{"className":7480},[217],[14,7482,7484,7510],{"className":7483},[221,222],[14,7485,7487,7507],{"className":7486},[226],[14,7488,7490],{"className":7489,"style":474},[230],[14,7491,7492,7495],{"style":540},[14,7493],{"className":7494,"style":239},[238],[14,7496,7498],{"className":7497},[243,244,245,246],[14,7499,7501,7504],{"className":7500},[58,246],[14,7502,394],{"className":7503},[58,59,246],[14,7505,168],{"className":7506},[58,246],[14,7508,254],{"className":7509},[253],[14,7511,7513],{"className":7512},[226],[14,7514,7516],{"className":7515,"style":261},[230],[14,7517],{},[14,7519,172],{"className":7520},[267],[14,7522],{"className":7523,"style":272},[271],[14,7525,7527,7530],{"className":7526},[58],[14,7528,164],{"className":7529,"style":525},[58,59],[14,7531,7533],{"className":7532},[217],[14,7534,7536,7562],{"className":7535},[221,222],[14,7537,7539,7559],{"className":7538},[226],[14,7540,7542],{"className":7541,"style":474},[230],[14,7543,7544,7547],{"style":540},[14,7545],{"className":7546,"style":239},[238],[14,7548,7550],{"className":7549},[243,244,245,246],[14,7551,7553,7556],{"className":7552},[58,246],[14,7554,394],{"className":7555},[58,59,246],[14,7557,179],{"className":7558},[58,246],[14,7560,254],{"className":7561},[253],[14,7563,7565],{"className":7564},[226],[14,7566,7568],{"className":7567,"style":261},[230],[14,7569],{},[14,7571,172],{"className":7572},[267],[14,7574],{"className":7575,"style":272},[271],[14,7577,184],{"className":7578},[322],[14,7580],{"className":7581,"style":272},[271],[14,7583,172],{"className":7584},[267],[14,7586],{"className":7587,"style":272},[271],[14,7589,7591,7594],{"className":7590},[58],[14,7592,164],{"className":7593,"style":525},[58,59],[14,7595,7597],{"className":7596},[217],[14,7598,7600,7623],{"className":7599},[221,222],[14,7601,7603,7620],{"className":7602},[226],[14,7604,7606],{"className":7605,"style":474},[230],[14,7607,7608,7611],{"style":540},[14,7609],{"className":7610,"style":239},[238],[14,7612,7614],{"className":7613},[243,244,245,246],[14,7615,7617],{"className":7616},[58,246],[14,7618,669],{"className":7619},[58,59,246],[14,7621,254],{"className":7622},[253],[14,7624,7626],{"className":7625},[226],[14,7627,7629],{"className":7628,"style":261},[230],[14,7630],{},[14,7632,442],{"className":7633},[684],[10,7635,72,7636,110,7638,3880],{},[74,7637,7075],{},[14,7639,7641,7687],{"className":7640,"translate":18},[17],[14,7642,7644],{"className":7643},[22],[24,7645,7646],{"xmlns":26},[28,7647,7648,7684],{},[31,7649,7650,7652,7654,7656,7662,7664,7670,7672,7674,7676,7682],{},[34,7651,10],{"mathvariant":163},[170,7653,397],{},[170,7655,401],{"stretchy":400},[159,7657,7658,7660],{},[34,7659,10],{},[166,7661,168],{},[170,7663,172],{"separator":45},[159,7665,7666,7668],{},[34,7667,10],{},[166,7669,179],{},[170,7671,172],{"separator":45},[170,7673,184],{},[170,7675,172],{"separator":45},[159,7677,767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= (p_1, p_2, \\dots, p_n)",[14,7688,7690,7708],{"className":7689,"ariaHidden":45},[44],[14,7691,7693,7696,7699,7702,7705],{"className":7692},[49],[14,7694],{"className":7695,"style":205},[53],[14,7697,10],{"className":7698},[58,212],[14,7700],{"className":7701,"style":500},[271],[14,7703,397],{"className":7704},[504],[14,7706],{"className":7707,"style":500},[271],[14,7709,7711,7714,7717,7757,7760,7763,7803,7806,7809,7812,7815,7818,7821,7861],{"className":7710},[49],[14,7712],{"className":7713,"style":514},[53],[14,7715,401],{"className":7716},[518],[14,7718,7720,7723],{"className":7719},[58],[14,7721,10],{"className":7722},[58,59],[14,7724,7726],{"className":7725},[217],[14,7727,7729,7749],{"className":7728},[221,222],[14,7730,7732,7746],{"className":7731},[226],[14,7733,7735],{"className":7734,"style":231},[230],[14,7736,7737,7740],{"style":2441},[14,7738],{"className":7739,"style":239},[238],[14,7741,7743],{"className":7742},[243,244,245,246],[14,7744,168],{"className":7745},[58,246],[14,7747,254],{"className":7748},[253],[14,7750,7752],{"className":7751},[226],[14,7753,7755],{"className":7754,"style":261},[230],[14,7756],{},[14,7758,172],{"className":7759},[267],[14,7761],{"className":7762,"style":272},[271],[14,7764,7766,7769],{"className":7765},[58],[14,7767,10],{"className":7768},[58,59],[14,7770,7772],{"className":7771},[217],[14,7773,7775,7795],{"className":7774},[221,222],[14,7776,7778,7792],{"className":7777},[226],[14,7779,7781],{"className":7780,"style":231},[230],[14,7782,7783,7786],{"style":2441},[14,7784],{"className":7785,"style":239},[238],[14,7787,7789],{"className":7788},[243,244,245,246],[14,7790,179],{"className":7791},[58,246],[14,7793,254],{"className":7794},[253],[14,7796,7798],{"className":7797},[226],[14,7799,7801],{"className":7800,"style":261},[230],[14,7802],{},[14,7804,172],{"className":7805},[267],[14,7807],{"className":7808,"style":272},[271],[14,7810,184],{"className":7811},[322],[14,7813],{"className":7814,"style":272},[271],[14,7816,172],{"className":7817},[267],[14,7819],{"className":7820,"style":272},[271],[14,7822,7824,7827],{"className":7823},[58],[14,7825,10],{"className":7826},[58,59],[14,7828,7830],{"className":7829},[217],[14,7831,7833,7853],{"className":7832},[221,222],[14,7834,7836,7850],{"className":7835},[226],[14,7837,7839],{"className":7838,"style":350},[230],[14,7840,7841,7844],{"style":2441},[14,7842],{"className":7843,"style":239},[238],[14,7845,7847],{"className":7846},[243,244,245,246],[14,7848,125],{"className":7849},[58,59,246],[14,7851,254],{"className":7852},[253],[14,7854,7856],{"className":7855},[226],[14,7857,7859],{"className":7858,"style":261},[230],[14,7860],{},[14,7862,442],{"className":7863},[684],[14,7865,7867],{"className":7866,"translate":18},[144],[14,7868,7870,7957],{"className":7869,"translate":18},[17],[14,7871,7873],{"className":7872},[22],[24,7874,7875],{"xmlns":26,"display":153},[28,7876,7877,7954],{},[31,7878,7879,7886,7888,7890,7892,7902,7904,7914,7916,7918,7920,7930,7932,7934,7936,7938,7940,7942,7944,7946,7948,7950,7952],{},[159,7880,7881,7883],{},[34,7882,10],{},[34,7884,7885],{},"j",[170,7887,397],{},[1066,7889,7083],{},[170,7891,401],{"stretchy":400},[159,7893,7894,7896],{},[34,7895,164],{},[31,7897,7898,7900],{},[166,7899,168],{},[34,7901,7885],{},[170,7903,172],{"separator":45},[159,7905,7906,7908],{},[34,7907,164],{},[31,7909,7910,7912],{},[166,7911,179],{},[34,7913,7885],{},[170,7915,172],{"separator":45},[170,7917,184],{},[170,7919,172],{"separator":45},[159,7921,7922,7924],{},[34,7923,164],{},[31,7925,7926,7928],{},[34,7927,94],{},[34,7929,7885],{},[170,7931,442],{"stretchy":400},[170,7933,172],{"separator":45},[271,7935],{"width":3958},[34,7937,7885],{},[170,7939,397],{},[166,7941,168],{},[170,7943,172],{"separator":45},[166,7945,179],{},[170,7947,172],{"separator":45},[170,7949,184],{},[170,7951,172],{"separator":45},[34,7953,125],{},[38,7955,7956],{"encoding":40},"p_j = \\text{median}(v_{1j}, v_{2j}, \\dots, v_{mj}), \\quad j = 1, 2, \\dots, n",[14,7958,7960,8017,8217],{"className":7959,"ariaHidden":45},[44],[14,7961,7963,7967,8008,8011,8014],{"className":7962},[49],[14,7964],{"className":7965,"style":7966},[53],"height:0.7167em;vertical-align:-0.2861em;",[14,7968,7970,7973],{"className":7969},[58],[14,7971,10],{"className":7972},[58,59],[14,7974,7976],{"className":7975},[217],[14,7977,7979,8000],{"className":7978},[221,222],[14,7980,7982,7997],{"className":7981},[226],[14,7983,7985],{"className":7984,"style":474},[230],[14,7986,7987,7990],{"style":2441},[14,7988],{"className":7989,"style":239},[238],[14,7991,7993],{"className":7992},[243,244,245,246],[14,7994,7885],{"className":7995,"style":7996},[58,59,246],"margin-right:0.05724em;",[14,7998,254],{"className":7999},[253],[14,8001,8003],{"className":8002},[226],[14,8004,8006],{"className":8005,"style":3508},[230],[14,8007],{},[14,8009],{"className":8010,"style":500},[271],[14,8012,397],{"className":8013},[504],[14,8015],{"className":8016,"style":500},[271],[14,8018,8020,8023,8030,8033,8079,8082,8085,8131,8134,8137,8140,8143,8146,8149,8193,8196,8199,8202,8205,8208,8211,8214],{"className":8019},[49],[14,8021],{"className":8022,"style":3978},[53],[14,8024,8027],{"className":8025},[58,8026],"text",[14,8028,7083],{"className":8029},[58],[14,8031,401],{"className":8032},[518],[14,8034,8036,8039],{"className":8035},[58],[14,8037,164],{"className":8038,"style":525},[58,59],[14,8040,8042],{"className":8041},[217],[14,8043,8045,8071],{"className":8044},[221,222],[14,8046,8048,8068],{"className":8047},[226],[14,8049,8051],{"className":8050,"style":474},[230],[14,8052,8053,8056],{"style":540},[14,8054],{"className":8055,"style":239},[238],[14,8057,8059],{"className":8058},[243,244,245,246],[14,8060,8062,8065],{"className":8061},[58,246],[14,8063,168],{"className":8064},[58,246],[14,8066,7885],{"className":8067,"style":7996},[58,59,246],[14,8069,254],{"className":8070},[253],[14,8072,8074],{"className":8073},[226],[14,8075,8077],{"className":8076,"style":3508},[230],[14,8078],{},[14,8080,172],{"className":8081},[267],[14,8083],{"className":8084,"style":272},[271],[14,8086,8088,8091],{"className":8087},[58],[14,8089,164],{"className":8090,"style":525},[58,59],[14,8092,8094],{"className":8093},[217],[14,8095,8097,8123],{"className":8096},[221,222],[14,8098,8100,8120],{"className":8099},[226],[14,8101,8103],{"className":8102,"style":474},[230],[14,8104,8105,8108],{"style":540},[14,8106],{"className":8107,"style":239},[238],[14,8109,8111],{"className":8110},[243,244,245,246],[14,8112,8114,8117],{"className":8113},[58,246],[14,8115,179],{"className":8116},[58,246],[14,8118,7885],{"className":8119,"style":7996},[58,59,246],[14,8121,254],{"className":8122},[253],[14,8124,8126],{"className":8125},[226],[14,8127,8129],{"className":8128,"style":3508},[230],[14,8130],{},[14,8132,172],{"className":8133},[267],[14,8135],{"className":8136,"style":272},[271],[14,8138,184],{"className":8139},[322],[14,8141],{"className":8142,"style":272},[271],[14,8144,172],{"className":8145},[267],[14,8147],{"className":8148,"style":272},[271],[14,8150,8152,8155],{"className":8151},[58],[14,8153,164],{"className":8154,"style":525},[58,59],[14,8156,8158],{"className":8157},[217],[14,8159,8161,8185],{"className":8160},[221,222],[14,8162,8164,8182],{"className":8163},[226],[14,8165,8167],{"className":8166,"style":474},[230],[14,8168,8169,8172],{"style":540},[14,8170],{"className":8171,"style":239},[238],[14,8173,8175],{"className":8174},[243,244,245,246],[14,8176,8178],{"className":8177},[58,246],[14,8179,8181],{"className":8180,"style":7996},[58,59,246],"mj",[14,8183,254],{"className":8184},[253],[14,8186,8188],{"className":8187},[226],[14,8189,8191],{"className":8190,"style":3508},[230],[14,8192],{},[14,8194,442],{"className":8195},[684],[14,8197,172],{"className":8198},[267],[14,8200],{"className":8201,"style":4244},[271],[14,8203],{"className":8204,"style":272},[271],[14,8206,7885],{"className":8207,"style":7996},[58,59],[14,8209],{"className":8210,"style":500},[271],[14,8212,397],{"className":8213},[504],[14,8215],{"className":8216,"style":500},[271],[14,8218,8220,8224,8227,8230,8233,8236,8239,8242,8245,8248,8251,8254],{"className":8219},[49],[14,8221],{"className":8222,"style":8223},[53],"height:0.8389em;vertical-align:-0.1944em;",[14,8225,168],{"className":8226},[58],[14,8228,172],{"className":8229},[267],[14,8231],{"className":8232,"style":272},[271],[14,8234,179],{"className":8235},[58],[14,8237,172],{"className":8238},[267],[14,8240],{"className":8241,"style":272},[271],[14,8243,184],{"className":8244},[322],[14,8246],{"className":8247,"style":272},[271],[14,8249,172],{"className":8250},[267],[14,8252],{"className":8253,"style":272},[271],[14,8255,125],{"className":8256},[58,59],[10,8258,8259],{},"That is, each component independently takes the median of all data points on that dimension.",[10,8261,8262],{},"If rotated by 45 degrees, the coordinate-wise median result changes (not rotation equivariant), whereas the geometric median rotates accordingly.",[10,8264,8265,8266,8334,8335,8379],{},"In computation, it only requires sorting per dimension ",[14,8267,8269,8297],{"className":8268,"translate":18},[17],[14,8270,8272],{"className":8271},[22],[24,8273,8274],{"xmlns":26},[28,8275,8276,8294],{},[31,8277,8278,8281,8283,8285,8288,8290,8292],{},[34,8279,8280],{},"O",[170,8282,401],{"stretchy":400},[34,8284,94],{},[34,8286,8287],{},"log",[170,8289,2095],{},[34,8291,94],{},[170,8293,442],{"stretchy":400},[38,8295,8296],{"encoding":40},"O(m \\log m)",[14,8298,8300],{"className":8299,"ariaHidden":45},[44],[14,8301,8303,8306,8310,8313,8316,8319,8325,8328,8331],{"className":8302},[49],[14,8304],{"className":8305,"style":514},[53],[14,8307,8280],{"className":8308,"style":8309},[58,59],"margin-right:0.02778em;",[14,8311,401],{"className":8312},[518],[14,8314,94],{"className":8315},[58,59],[14,8317],{"className":8318,"style":272},[271],[14,8320,8322,8323],{"className":8321},[890],"lo",[14,8324,2195],{"style":2194},[14,8326],{"className":8327,"style":272},[271],[14,8329,94],{"className":8330},[58,59],[14,8332,442],{"className":8333},[684]," or using a selection algorithm ",[14,8336,8338,8358],{"className":8337,"translate":18},[17],[14,8339,8341],{"className":8340},[22],[24,8342,8343],{"xmlns":26},[28,8344,8345,8355],{},[31,8346,8347,8349,8351,8353],{},[34,8348,8280],{},[170,8350,401],{"stretchy":400},[34,8352,94],{},[170,8354,442],{"stretchy":400},[38,8356,8357],{"encoding":40},"O(m)",[14,8359,8361],{"className":8360,"ariaHidden":45},[44],[14,8362,8364,8367,8370,8373,8376],{"className":8363},[49],[14,8365],{"className":8366,"style":514},[53],[14,8368,8280],{"className":8369,"style":8309},[58,59],[14,8371,401],{"className":8372},[518],[14,8374,94],{"className":8375},[58,59],[14,8377,442],{"className":8378},[684],", and it can resist up to 50% of arbitrary extreme outliers.",[10,8381,8382],{},"However, the choice of coordinate axes affects the result. If the data has correlations (e.g., a tilted ellipse), the coordinate-wise median can deviate completely from the true center. It necessarily lies within the value range of each dimension, so it remains within the axis-aligned bounding box of the data, but not necessarily within the convex hull of the data points (because each dimension is chosen independently). In machine learning, it lacks a global optimization objective, unlike the median or geometric median which minimize some sum of distances. It is not suitable for objective functions.",[10,8384,8385],{},"It is robust to outliers when the data has outliers and the dimensions are relatively independent.",[10,8387,8388,8389,8441],{},"For example, in 3D space, we have a ",[14,8390,8392,8411],{"className":8391,"translate":18},[17],[14,8393,8395],{"className":8394},[22],[24,8396,8397],{"xmlns":26},[28,8398,8399,8408],{},[31,8400,8401,8403,8406],{},[166,8402,5016],{},[170,8404,8405],{},"×",[166,8407,5009],{},[38,8409,8410],{"encoding":40},"4 \\times 3",[14,8412,8414,8432],{"className":8413,"ariaHidden":45},[44],[14,8415,8417,8420,8423,8426,8429],{"className":8416},[49],[14,8418],{"className":8419,"style":5323},[53],[14,8421,5016],{"className":8422},[58],[14,8424],{"className":8425,"style":2367},[271],[14,8427,8405],{"className":8428},[2371],[14,8430],{"className":8431,"style":2367},[271],[14,8433,8435,8438],{"className":8434},[49],[14,8436],{"className":8437,"style":4266},[53],[14,8439,5009],{"className":8440},[58]," vector matrix:",[14,8443,8445],{"className":8444,"translate":18},[144],[14,8446,8448,8557],{"className":8447,"translate":18},[17],[14,8449,8451],{"className":8450},[22],[24,8452,8453],{"xmlns":26,"display":153},[28,8454,8455,8554],{},[31,8456,8457,8459,8552],{},[170,8458,401],{"fence":45},[8460,8461,8464,8494,8522],"mtable",{"rowspacing":8462,"columnalign":8463,"columnspacing":3958},"0.16em","center center center 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3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664\nc-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11\nc0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17\nc242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558\nl0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z",[14,8899,254],{"className":8900},[253],[14,8902,8904],{"className":8903},[226],[14,8905,8907],{"className":8906,"style":8614},[230],[14,8908],{},[10,8910,8911,8912,8941,8942,8971,8972,685],{},"The median of the 1st dimension is ",[14,8913,8915,8929],{"className":8914,"translate":18},[17],[14,8916,8918],{"className":8917},[22],[24,8919,8920],{"xmlns":26},[28,8921,8922,8927],{},[31,8923,8924],{},[166,8925,8926],{},"4.5",[38,8928,8926],{"encoding":40},[14,8930,8932],{"className":8931,"ariaHidden":45},[44],[14,8933,8935,8938],{"className":8934},[49],[14,8936],{"className":8937,"style":4266},[53],[14,8939,8926],{"className":8940},[58],", the 2nd dimension is ",[14,8943,8945,8959],{"className":8944,"translate":18},[17],[14,8946,8948],{"className":8947},[22],[24,8949,8950],{"xmlns":26},[28,8951,8952,8957],{},[31,8953,8954],{},[166,8955,8956],{},"6.5",[38,8958,8956],{"encoding":40},[14,8960,8962],{"className":8961,"ariaHidden":45},[44],[14,8963,8965,8968],{"className":8964},[49],[14,8966],{"className":8967,"style":4266},[53],[14,8969,8956],{"className":8970},[58],", and the 3rd dimension is ",[14,8973,8975,8989],{"className":8974,"translate":18},[17],[14,8976,8978],{"className":8977},[22],[24,8979,8980],{"xmlns":26},[28,8981,8982,8987],{},[31,8983,8984],{},[166,8985,8986],{},"12.5",[38,8988,8986],{"encoding":40},[14,8990,8992],{"className":8991,"ariaHidden":45},[44],[14,8993,8995,8998],{"className":8994},[49],[14,8996],{"className":8997,"style":4266},[53],[14,8999,8986],{"className":9000},[58],[10,9002,9003,9004,685],{},"Its Coordinate-wise Median Prototype is ",[14,9005,9007,9038],{"className":9006,"translate":18},[17],[14,9008,9010],{"className":9009},[22],[24,9011,9012],{"xmlns":26},[28,9013,9014,9035],{},[31,9015,9016,9018,9020,9022,9024,9026,9028],{},[170,9017,401],{"stretchy":400},[166,9019,8926],{},[170,9021,172],{"separator":45},[166,9023,8956],{},[170,9025,172],{"separator":45},[166,9027,8986],{},[2115,9029,9030,9032],{},[170,9031,442],{"stretchy":400},[34,9033,9034],{},"T",[38,9036,9037],{"encoding":40},"(4.5, 6.5, 12.5)^T",[14,9039,9041],{"className":9040,"ariaHidden":45},[44],[14,9042,9044,9048,9051,9054,9057,9060,9063,9066,9069,9072],{"className":9043},[49],[14,9045],{"className":9046,"style":9047},[53],"height:1.0913em;vertical-align:-0.25em;",[14,9049,401],{"className":9050},[518],[14,9052,8926],{"className":9053},[58],[14,9055,172],{"className":9056},[267],[14,9058],{"className":9059,"style":272},[271],[14,9061,8956],{"className":9062},[58],[14,9064,172],{"className":9065},[267],[14,9067],{"className":9068,"style":272},[271],[14,9070,8986],{"className":9071},[58],[14,9073,9075,9078],{"className":9074},[684],[14,9076,442],{"className":9077},[684],[14,9079,9081],{"className":9080},[217],[14,9082,9084],{"className":9083},[221],[14,9085,9087],{"className":9086},[226],[14,9088,9091],{"className":9089,"style":9090},[230],"height:0.8413em;",[14,9092,9094,9097],{"style":9093},"top:-3.063em;margin-right:0.05em;",[14,9095],{"className":9096,"style":239},[238],[14,9098,9100],{"className":9099},[243,244,245,246],[14,9101,9034],{"className":9102,"style":9103},[58,59,246],"margin-right:0.13889em;",[66,9105,9107],{"id":9106},"extend-to-high-norm","Extend to High Norm",[10,9109,9110],{},"Having defined the Euclidean norm, let's revisit the definitions of the mean prototype, geometric median prototype, and coordinate-wise median prototype. Clearly, we can define them using the Euclidean norm.",[10,9112,9113,9114,9183],{},"The mean prototype is the point that minimizes the sum of squared Euclidean distances (",[14,9115,9117,9134],{"className":9116,"translate":18},[17],[14,9118,9120],{"className":9119},[22],[24,9121,9122],{"xmlns":26},[28,9123,9124,9132],{},[31,9125,9126],{},[159,9127,9128,9130],{},[34,9129,2573],{},[166,9131,179],{},[38,9133,5438],{"encoding":40},[14,9135,9137],{"className":9136,"ariaHidden":45},[44],[14,9138,9140,9143],{"className":9139},[49],[14,9141],{"className":9142,"style":4384},[53],[14,9144,9146,9149],{"className":9145},[58],[14,9147,2573],{"className":9148},[58,59],[14,9150,9152],{"className":9151},[217],[14,9153,9155,9175],{"className":9154},[221,222],[14,9156,9158,9172],{"className":9157},[226],[14,9159,9161],{"className":9160,"style":231},[230],[14,9162,9163,9166],{"style":2441},[14,9164],{"className":9165,"style":239},[238],[14,9167,9169],{"className":9168},[243,244,245,246],[14,9170,179],{"className":9171},[58,246],[14,9173,254],{"className":9174},[253],[14,9176,9178],{"className":9177},[226],[14,9179,9181],{"className":9180,"style":261},[230],[14,9182],{}," distance norm) to all points:",[14,9185,9187],{"className":9186,"translate":18},[144],[14,9188,9190,9264],{"className":9189,"translate":18},[17],[14,9191,9193],{"className":9192},[22],[24,9194,9195],{"xmlns":26,"display":153},[28,9196,9197,9261],{},[31,9198,9199,9201,9203,9205,9207,9227,9241,9243,9245,9247,9253],{},[34,9200,704],{"mathvariant":163},[170,9202,397],{},[34,9204,2092],{},[170,9206,2095],{},[2097,9208,9209,9215],{},[31,9210,9211,9213],{},[34,9212,2103],{},[170,9214,2095],{},[31,9216,9217,9219,9221],{},[34,9218,2110],{"mathvariant":163},[170,9220,2113],{},[2115,9222,9223,9225],{},[34,9224,2120],{"mathvariant":2119},[34,9226,125],{},[755,9228,9229,9231,9239],{},[170,9230,759],{},[31,9232,9233,9235,9237],{},[34,9234,394],{},[170,9236,397],{},[166,9238,168],{},[34,9240,94],{},[34,9242,2140],{"mathvariant":2139},[34,9244,2110],{"mathvariant":163},[170,9246,2145],{},[159,9248,9249,9251],{},[34,9250,164],{"mathvariant":163},[34,9252,394],{},[5536,9254,9255,9257,9259],{},[34,9256,2140],{"mathvariant":2139},[166,9258,179],{},[166,9260,179],{},[38,9262,9263],{"encoding":40},"\\mathbf{\\mu} = \\arg\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\sum_{i=1}^m \\|\\mathbf{x} - \\mathbf{v}_i\\|_2^2",[14,9265,9267,9285,9463],{"className":9266,"ariaHidden":45},[44],[14,9268,9270,9273,9276,9279,9282],{"className":9269},[49],[14,9271],{"className":9272,"style":3610},[53],[14,9274,704],{"className":9275},[58,59],[14,9277],{"className":9278,"style":500},[271],[14,9280,397],{"className":9281},[504],[14,9283],{"className":9284,"style":500},[271],[14,9286,9288,9291,9296,9299,9378,9381,9445,9448,9451,9454,9457,9460],{"className":9287},[49],[14,9289],{"className":9290,"style":812},[53],[14,9292,2191,9294],{"className":9293},[890],[14,9295,2195],{"style":2194},[14,9297],{"className":9298,"style":272},[271],[14,9300,9302],{"className":9301},[890,891],[14,9303,9305,9370],{"className":9304},[221,222],[14,9306,9308,9367],{"className":9307},[226],[14,9309,9311,9357],{"className":9310,"style":2211},[230],[14,9312,9313,9316],{"style":2214},[14,9314],{"className":9315,"style":839},[238],[14,9317,9319],{"className":9318},[243,244,245,246],[14,9320,9322,9325,9328],{"className":9321},[58,246],[14,9323,2110],{"className":9324},[58,212,246],[14,9326,2113],{"className":9327},[504,246],[14,9329,9331,9334],{"className":9330},[58,246],[14,9332,2120],{"className":9333},[58,2236,246],[14,9335,9337],{"className":9336},[217],[14,9338,9340],{"className":9339},[221],[14,9341,9343],{"className":9342},[226],[14,9344,9346],{"className":9345,"style":2249},[230],[14,9347,9348,9351],{"style":2252},[14,9349],{"className":9350,"style":2256},[238],[14,9352,9354],{"className":9353},[243,2260,2261,246],[14,9355,125],{"className":9356},[58,59,246],[14,9358,9359,9362],{"style":2267},[14,9360],{"className":9361,"style":839},[238],[14,9363,9364],{},[14,9365,2103],{"className":9366},[890],[14,9368,254],{"className":9369},[253],[14,9371,9373],{"className":9372},[226],[14,9374,9376],{"className":9375,"style":2285},[230],[14,9377],{},[14,9379],{"className":9380,"style":272},[271],[14,9382,9384],{"className":9383},[890,891],[14,9385,9387,9437],{"className":9386},[221,222],[14,9388,9390,9434],{"className":9389},[226],[14,9391,9393,9413,9423],{"className":9392,"style":901},[230],[14,9394,9395,9398],{"style":904},[14,9396],{"className":9397,"style":908},[238],[14,9399,9401],{"className":9400},[243,244,245,246],[14,9402,9404,9407,9410],{"className":9403},[58,246],[14,9405,394],{"className":9406},[58,59,246],[14,9408,397],{"className":9409},[504,246],[14,9411,168],{"className":9412},[58,246],[14,9414,9415,9418],{"style":926},[14,9416],{"className":9417,"style":908},[238],[14,9419,9420],{},[14,9421,759],{"className":9422},[890,935,936],[14,9424,9425,9428],{"style":939},[14,9426],{"className":9427,"style":908},[238],[14,9429,9431],{"className":9430},[243,244,245,246],[14,9432,94],{"className":9433},[58,59,246],[14,9435,254],{"className":9436},[253],[14,9438,9440],{"className":9439},[226],[14,9441,9443],{"className":9442,"style":961},[230],[14,9444],{},[14,9446],{"className":9447,"style":272},[271],[14,9449,2140],{"className":9450},[58],[14,9452,2110],{"className":9453},[58,212],[14,9455],{"className":9456,"style":2367},[271],[14,9458,2145],{"className":9459},[2371],[14,9461],{"className":9462,"style":2367},[271],[14,9464,9466,9470,9510],{"className":9465},[49],[14,9467],{"className":9468,"style":9469},[53],"height:1.1141em;vertical-align:-0.25em;",[14,9471,9473,9476],{"className":9472},[58],[14,9474,164],{"className":9475,"style":213},[58,212],[14,9477,9479],{"className":9478},[217],[14,9480,9482,9502],{"className":9481},[221,222],[14,9483,9485,9499],{"className":9484},[226],[14,9486,9488],{"className":9487,"style":474},[230],[14,9489,9490,9493],{"style":234},[14,9491],{"className":9492,"style":239},[238],[14,9494,9496],{"className":9495},[243,244,245,246],[14,9497,394],{"className":9498},[58,59,246],[14,9500,254],{"className":9501},[253],[14,9503,9505],{"className":9504},[226],[14,9506,9508],{"className":9507,"style":261},[230],[14,9509],{},[14,9511,9513,9516],{"className":9512},[58],[14,9514,2140],{"className":9515},[58],[14,9517,9519],{"className":9518},[217],[14,9520,9522,9554],{"className":9521},[221,222],[14,9523,9525,9551],{"className":9524},[226],[14,9526,9529,9540],{"className":9527,"style":9528},[230],"height:0.8641em;",[14,9530,9531,9534],{"style":6042},[14,9532],{"className":9533,"style":239},[238],[14,9535,9537],{"className":9536},[243,244,245,246],[14,9538,179],{"className":9539},[58,246],[14,9541,9542,9545],{"style":4187},[14,9543],{"className":9544,"style":239},[238],[14,9546,9548],{"className":9547},[243,244,245,246],[14,9549,179],{"className":9550},[58,246],[14,9552,254],{"className":9553},[253],[14,9555,9557],{"className":9556},[226],[14,9558,9560],{"className":9559,"style":6073},[230],[14,9561],{},[10,9563,9564,9565,9183],{},"The geometric median prototype is the point that minimizes the sum of Euclidean distances (",[14,9566,9568,9585],{"className":9567,"translate":18},[17],[14,9569,9571],{"className":9570},[22],[24,9572,9573],{"xmlns":26},[28,9574,9575,9583],{},[31,9576,9577],{},[159,9578,9579,9581],{},[34,9580,2573],{},[166,9582,179],{},[38,9584,5438],{"encoding":40},[14,9586,9588],{"className":9587,"ariaHidden":45},[44],[14,9589,9591,9594],{"className":9590},[49],[14,9592],{"className":9593,"style":4384},[53],[14,9595,9597,9600],{"className":9596},[58],[14,9598,2573],{"className":9599},[58,59],[14,9601,9603],{"className":9602},[217],[14,9604,9606,9626],{"className":9605},[221,222],[14,9607,9609,9623],{"className":9608},[226],[14,9610,9612],{"className":9611,"style":231},[230],[14,9613,9614,9617],{"style":2441},[14,9615],{"className":9616,"style":239},[238],[14,9618,9620],{"className":9619},[243,244,245,246],[14,9621,179],{"className":9622},[58,246],[14,9624,254],{"className":9625},[253],[14,9627,9629],{"className":9628},[226],[14,9630,9632],{"className":9631,"style":261},[230],[14,9633],{},[14,9635,9637],{"className":9636,"translate":18},[144],[14,9638,9640,9716],{"className":9639,"translate":18},[17],[14,9641,9643],{"className":9642},[22],[24,9644,9645],{"xmlns":26,"display":153},[28,9646,9647,9713],{},[31,9648,9649,9651,9653,9655,9657,9677,9691,9693,9695,9697,9699,9705,9707],{},[34,9650,10],{"mathvariant":163},[170,9652,397],{},[34,9654,2092],{},[170,9656,2095],{},[2097,9658,9659,9665],{},[31,9660,9661,9663],{},[34,9662,2103],{},[170,9664,2095],{},[31,9666,9667,9669,9671],{},[34,9668,2110],{"mathvariant":163},[170,9670,2113],{},[2115,9672,9673,9675],{},[34,9674,2120],{"mathvariant":2119},[34,9676,125],{},[755,9678,9679,9681,9689],{},[170,9680,759],{},[31,9682,9683,9685,9687],{},[34,9684,394],{},[170,9686,397],{},[166,9688,168],{},[34,9690,94],{},[34,9692,3930],{"mathvariant":2139},[34,9694,3930],{"mathvariant":2139},[34,9696,2110],{"mathvariant":163},[170,9698,2145],{},[159,9700,9701,9703],{},[34,9702,164],{"mathvariant":163},[34,9704,394],{},[34,9706,3930],{"mathvariant":2139},[159,9708,9709,9711],{},[34,9710,3930],{"mathvariant":2139},[166,9712,179],{},[38,9714,9715],{"encoding":40},"\\mathbf{p} = \\arg\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\sum_{i=1}^m ||\\mathbf{x} - \\mathbf{v}_i||_2",[14,9717,9719,9737,9916],{"className":9718,"ariaHidden":45},[44],[14,9720,9722,9725,9728,9731,9734],{"className":9721},[49],[14,9723],{"className":9724,"style":205},[53],[14,9726,10],{"className":9727},[58,212],[14,9729],{"className":9730,"style":500},[271],[14,9732,397],{"className":9733},[504],[14,9735],{"className":9736,"style":500},[271],[14,9738,9740,9743,9748,9751,9830,9833,9897,9900,9904,9907,9910,9913],{"className":9739},[49],[14,9741],{"className":9742,"style":812},[53],[14,9744,2191,9746],{"className":9745},[890],[14,9747,2195],{"style":2194},[14,9749],{"className":9750,"style":272},[271],[14,9752,9754],{"className":9753},[890,891],[14,9755,9757,9822],{"className":9756},[221,222],[14,9758,9760,9819],{"className":9759},[226],[14,9761,9763,9809],{"className":9762,"style":2211},[230],[14,9764,9765,9768],{"style":2214},[14,9766],{"className":9767,"style":839},[238],[14,9769,9771],{"className":9770},[243,244,245,246],[14,9772,9774,9777,9780],{"className":9773},[58,246],[14,9775,2110],{"className":9776},[58,212,246],[14,9778,2113],{"className":9779},[504,246],[14,9781,9783,9786],{"className":9782},[58,246],[14,9784,2120],{"className":9785},[58,2236,246],[14,9787,9789],{"className":9788},[217],[14,9790,9792],{"className":9791},[221],[14,9793,9795],{"className":9794},[226],[14,9796,9798],{"className":9797,"style":2249},[230],[14,9799,9800,9803],{"style":2252},[14,9801],{"className":9802,"style":2256},[238],[14,9804,9806],{"className":9805},[243,2260,2261,246],[14,9807,125],{"className":9808},[58,59,246],[14,9810,9811,9814],{"style":2267},[14,9812],{"className":9813,"style":839},[238],[14,9815,9816],{},[14,9817,2103],{"className":9818},[890],[14,9820,254],{"className":9821},[253],[14,9823,9825],{"className":9824},[226],[14,9826,9828],{"className":9827,"style":2285},[230],[14,9829],{},[14,9831],{"className":9832,"style":272},[271],[14,9834,9836],{"className":9835},[890,891],[14,9837,9839,9889],{"className":9838},[221,222],[14,9840,9842,9886],{"className":9841},[226],[14,9843,9845,9865,9875],{"className":9844,"style":901},[230],[14,9846,9847,9850],{"style":904},[14,9848],{"className":9849,"style":908},[238],[14,9851,9853],{"className":9852},[243,244,245,246],[14,9854,9856,9859,9862],{"className":9855},[58,246],[14,9857,394],{"className":9858},[58,59,246],[14,9860,397],{"className":9861},[504,246],[14,9863,168],{"className":9864},[58,246],[14,9866,9867,9870],{"style":926},[14,9868],{"className":9869,"style":908},[238],[14,9871,9872],{},[14,9873,759],{"className":9874},[890,935,936],[14,9876,9877,9880],{"style":939},[14,9878],{"className":9879,"style":908},[238],[14,9881,9883],{"className":9882},[243,244,245,246],[14,9884,94],{"className":9885},[58,59,246],[14,9887,254],{"className":9888},[253],[14,9890,9892],{"className":9891},[226],[14,9893,9895],{"className":9894,"style":961},[230],[14,9896],{},[14,9898],{"className":9899,"style":272},[271],[14,9901,9903],{"className":9902},[58],"∣∣",[14,9905,2110],{"className":9906},[58,212],[14,9908],{"className":9909,"style":2367},[271],[14,9911,2145],{"className":9912},[2371],[14,9914],{"className":9915,"style":2367},[271],[14,9917,9919,9922,9962,9965],{"className":9918},[49],[14,9920],{"className":9921,"style":514},[53],[14,9923,9925,9928],{"className":9924},[58],[14,9926,164],{"className":9927,"style":213},[58,212],[14,9929,9931],{"className":9930},[217],[14,9932,9934,9954],{"className":9933},[221,222],[14,9935,9937,9951],{"className":9936},[226],[14,9938,9940],{"className":9939,"style":474},[230],[14,9941,9942,9945],{"style":234},[14,9943],{"className":9944,"style":239},[238],[14,9946,9948],{"className":9947},[243,244,245,246],[14,9949,394],{"className":9950},[58,59,246],[14,9952,254],{"className":9953},[253],[14,9955,9957],{"className":9956},[226],[14,9958,9960],{"className":9959,"style":261},[230],[14,9961],{},[14,9963,3930],{"className":9964},[58],[14,9966,9968,9971],{"className":9967},[58],[14,9969,3930],{"className":9970},[58],[14,9972,9974],{"className":9973},[217],[14,9975,9977,9997],{"className":9976},[221,222],[14,9978,9980,9994],{"className":9979},[226],[14,9981,9983],{"className":9982,"style":231},[230],[14,9984,9985,9988],{"style":2441},[14,9986],{"className":9987,"style":239},[238],[14,9989,9991],{"className":9990},[243,244,245,246],[14,9992,179],{"className":9993},[58,246],[14,9995,254],{"className":9996},[253],[14,9998,10000],{"className":9999},[226],[14,10001,10003],{"className":10002,"style":261},[230],[14,10004],{},[10,10006,10007,10008,9183],{},"The coordinate-wise median prototype is the point that minimizes the sum of Manhattan distances (",[14,10009,10011,10028],{"className":10010,"translate":18},[17],[14,10012,10014],{"className":10013},[22],[24,10015,10016],{"xmlns":26},[28,10017,10018,10026],{},[31,10019,10020],{},[159,10021,10022,10024],{},[34,10023,2573],{},[166,10025,168],{},[38,10027,4374],{"encoding":40},[14,10029,10031],{"className":10030,"ariaHidden":45},[44],[14,10032,10034,10037],{"className":10033},[49],[14,10035],{"className":10036,"style":4384},[53],[14,10038,10040,10043],{"className":10039},[58],[14,10041,2573],{"className":10042},[58,59],[14,10044,10046],{"className":10045},[217],[14,10047,10049,10069],{"className":10048},[221,222],[14,10050,10052,10066],{"className":10051},[226],[14,10053,10055],{"className":10054,"style":231},[230],[14,10056,10057,10060],{"style":2441},[14,10058],{"className":10059,"style":239},[238],[14,10061,10063],{"className":10062},[243,244,245,246],[14,10064,168],{"className":10065},[58,246],[14,10067,254],{"className":10068},[253],[14,10070,10072],{"className":10071},[226],[14,10073,10075],{"className":10074,"style":261},[230],[14,10076],{},[14,10078,10080],{"className":10079,"translate":18},[144],[14,10081,10083,10159],{"className":10082,"translate":18},[17],[14,10084,10086],{"className":10085},[22],[24,10087,10088],{"xmlns":26,"display":153},[28,10089,10090,10156],{},[31,10091,10092,10094,10096,10098,10100,10120,10134,10136,10138,10140,10142,10148,10150],{},[34,10093,10],{"mathvariant":163},[170,10095,397],{},[34,10097,2092],{},[170,10099,2095],{},[2097,10101,10102,10108],{},[31,10103,10104,10106],{},[34,10105,2103],{},[170,10107,2095],{},[31,10109,10110,10112,10114],{},[34,10111,2110],{"mathvariant":163},[170,10113,2113],{},[2115,10115,10116,10118],{},[34,10117,2120],{"mathvariant":2119},[34,10119,125],{},[755,10121,10122,10124,10132],{},[170,10123,759],{},[31,10125,10126,10128,10130],{},[34,10127,394],{},[170,10129,397],{},[166,10131,168],{},[34,10133,94],{},[34,10135,3930],{"mathvariant":2139},[34,10137,3930],{"mathvariant":2139},[34,10139,2110],{"mathvariant":163},[170,10141,2145],{},[159,10143,10144,10146],{},[34,10145,164],{"mathvariant":163},[34,10147,394],{},[34,10149,3930],{"mathvariant":2139},[159,10151,10152,10154],{},[34,10153,3930],{"mathvariant":2139},[166,10155,168],{},[38,10157,10158],{"encoding":40},"\\mathbf{p} = \\arg\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\sum_{i=1}^m ||\\mathbf{x} - \\mathbf{v}_i||_1",[14,10160,10162,10180,10358],{"className":10161,"ariaHidden":45},[44],[14,10163,10165,10168,10171,10174,10177],{"className":10164},[49],[14,10166],{"className":10167,"style":205},[53],[14,10169,10],{"className":10170},[58,212],[14,10172],{"className":10173,"style":500},[271],[14,10175,397],{"className":10176},[504],[14,10178],{"className":10179,"style":500},[271],[14,10181,10183,10186,10191,10194,10273,10276,10340,10343,10346,10349,10352,10355],{"className":10182},[49],[14,10184],{"className":10185,"style":812},[53],[14,10187,2191,10189],{"className":10188},[890],[14,10190,2195],{"style":2194},[14,10192],{"className":10193,"style":272},[271],[14,10195,10197],{"className":10196},[890,891],[14,10198,10200,10265],{"className":10199},[221,222],[14,10201,10203,10262],{"className":10202},[226],[14,10204,10206,10252],{"className":10205,"style":2211},[230],[14,10207,10208,10211],{"style":2214},[14,10209],{"className":10210,"style":839},[238],[14,10212,10214],{"className":10213},[243,244,245,246],[14,10215,10217,10220,10223],{"className":10216},[58,246],[14,10218,2110],{"className":10219},[58,212,246],[14,10221,2113],{"className":10222},[504,246],[14,10224,10226,10229],{"className":10225},[58,246],[14,10227,2120],{"className":10228},[58,2236,246],[14,10230,10232],{"className":10231},[217],[14,10233,10235],{"className":10234},[221],[14,10236,10238],{"className":10237},[226],[14,10239,10241],{"className":10240,"style":2249},[230],[14,10242,10243,10246],{"style":2252},[14,10244],{"className":10245,"style":2256},[238],[14,10247,10249],{"className":10248},[243,2260,2261,246],[14,10250,125],{"className":10251},[58,59,246],[14,10253,10254,10257],{"style":2267},[14,10255],{"className":10256,"style":839},[238],[14,10258,10259],{},[14,10260,2103],{"className":10261},[890],[14,10263,254],{"className":10264},[253],[14,10266,10268],{"className":10267},[226],[14,10269,10271],{"className":10270,"style":2285},[230],[14,10272],{},[14,10274],{"className":10275,"style":272},[271],[14,10277,10279],{"className":10278},[890,891],[14,10280,10282,10332],{"className":10281},[221,222],[14,10283,10285,10329],{"className":10284},[226],[14,10286,10288,10308,10318],{"className":10287,"style":901},[230],[14,10289,10290,10293],{"style":904},[14,10291],{"className":10292,"style":908},[238],[14,10294,10296],{"className":10295},[243,244,245,246],[14,10297,10299,10302,10305],{"className":10298},[58,246],[14,10300,394],{"className":10301},[58,59,246],[14,10303,397],{"className":10304},[504,246],[14,10306,168],{"className":10307},[58,246],[14,10309,10310,10313],{"style":926},[14,10311],{"className":10312,"style":908},[238],[14,10314,10315],{},[14,10316,759],{"className":10317},[890,935,936],[14,10319,10320,10323],{"style":939},[14,10321],{"className":10322,"style":908},[238],[14,10324,10326],{"className":10325},[243,244,245,246],[14,10327,94],{"className":10328},[58,59,246],[14,10330,254],{"className":10331},[253],[14,10333,10335],{"className":10334},[226],[14,10336,10338],{"className":10337,"style":961},[230],[14,10339],{},[14,10341],{"className":10342,"style":272},[271],[14,10344,9903],{"className":10345},[58],[14,10347,2110],{"className":10348},[58,212],[14,10350],{"className":10351,"style":2367},[271],[14,10353,2145],{"className":10354},[2371],[14,10356],{"className":10357,"style":2367},[271],[14,10359,10361,10364,10404,10407],{"className":10360},[49],[14,10362],{"className":10363,"style":514},[53],[14,10365,10367,10370],{"className":10366},[58],[14,10368,164],{"className":10369,"style":213},[58,212],[14,10371,10373],{"className":10372},[217],[14,10374,10376,10396],{"className":10375},[221,222],[14,10377,10379,10393],{"className":10378},[226],[14,10380,10382],{"className":10381,"style":474},[230],[14,10383,10384,10387],{"style":234},[14,10385],{"className":10386,"style":239},[238],[14,10388,10390],{"className":10389},[243,244,245,246],[14,10391,394],{"className":10392},[58,59,246],[14,10394,254],{"className":10395},[253],[14,10397,10399],{"className":10398},[226],[14,10400,10402],{"className":10401,"style":261},[230],[14,10403],{},[14,10405,3930],{"className":10406},[58],[14,10408,10410,10413],{"className":10409},[58],[14,10411,3930],{"className":10412},[58],[14,10414,10416],{"className":10415},[217],[14,10417,10419,10439],{"className":10418},[221,222],[14,10420,10422,10436],{"className":10421},[226],[14,10423,10425],{"className":10424,"style":231},[230],[14,10426,10427,10430],{"style":2441},[14,10428],{"className":10429,"style":239},[238],[14,10431,10433],{"className":10432},[243,244,245,246],[14,10434,168],{"className":10435},[58,246],[14,10437,254],{"className":10438},[253],[14,10440,10442],{"className":10441},[226],[14,10443,10445],{"className":10444,"style":261},[230],[14,10446],{},[10,10448,10449],{},"At this point, we can unify all the above concepts:",[14,10451,10453],{"className":10452,"translate":18},[144],[14,10454,10456,10531],{"className":10455,"translate":18},[17],[14,10457,10459],{"className":10458},[22],[24,10460,10461],{"xmlns":26,"display":153},[28,10462,10463,10528],{},[31,10464,10465,10467,10469,10471,10473,10493,10507,10509,10511,10513,10519],{},[34,10466,10],{"mathvariant":163},[170,10468,397],{},[34,10470,2092],{},[170,10472,2095],{},[2097,10474,10475,10481],{},[31,10476,10477,10479],{},[34,10478,2103],{},[170,10480,2095],{},[31,10482,10483,10485,10487],{},[34,10484,2110],{"mathvariant":163},[170,10486,2113],{},[2115,10488,10489,10491],{},[34,10490,2120],{"mathvariant":2119},[34,10492,125],{},[755,10494,10495,10497,10505],{},[170,10496,759],{},[31,10498,10499,10501,10503],{},[34,10500,394],{},[170,10502,397],{},[166,10504,168],{},[34,10506,94],{},[34,10508,2140],{"mathvariant":2139},[34,10510,2110],{"mathvariant":163},[170,10512,2145],{},[159,10514,10515,10517],{},[34,10516,164],{"mathvariant":163},[34,10518,394],{},[5536,10520,10521,10523,10525],{},[34,10522,2140],{"mathvariant":2139},[34,10524,10],{},[34,10526,10527],{},"q",[38,10529,10530],{"encoding":40},"\\mathbf{p} = \\arg\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\sum_{i=1}^m \\|\\mathbf{x} - \\mathbf{v}_i\\|_p^q",[14,10532,10534,10552,10730],{"className":10533,"ariaHidden":45},[44],[14,10535,10537,10540,10543,10546,10549],{"className":10536},[49],[14,10538],{"className":10539,"style":205},[53],[14,10541,10],{"className":10542},[58,212],[14,10544],{"className":10545,"style":500},[271],[14,10547,397],{"className":10548},[504],[14,10550],{"className":10551,"style":500},[271],[14,10553,10555,10558,10563,10566,10645,10648,10712,10715,10718,10721,10724,10727],{"className":10554},[49],[14,10556],{"className":10557,"style":812},[53],[14,10559,2191,10561],{"className":10560},[890],[14,10562,2195],{"style":2194},[14,10564],{"className":10565,"style":272},[271],[14,10567,10569],{"className":10568},[890,891],[14,10570,10572,10637],{"className":10571},[221,222],[14,10573,10575,10634],{"className":10574},[226],[14,10576,10578,10624],{"className":10577,"style":2211},[230],[14,10579,10580,10583],{"style":2214},[14,10581],{"className":10582,"style":839},[238],[14,10584,10586],{"className":10585},[243,244,245,246],[14,10587,10589,10592,10595],{"className":10588},[58,246],[14,10590,2110],{"className":10591},[58,212,246],[14,10593,2113],{"className":10594},[504,246],[14,10596,10598,10601],{"className":10597},[58,246],[14,10599,2120],{"className":10600},[58,2236,246],[14,10602,10604],{"className":10603},[217],[14,10605,10607],{"className":10606},[221],[14,10608,10610],{"className":10609},[226],[14,10611,10613],{"className":10612,"style":2249},[230],[14,10614,10615,10618],{"style":2252},[14,10616],{"className":10617,"style":2256},[238],[14,10619,10621],{"className":10620},[243,2260,2261,246],[14,10622,125],{"className":10623},[58,59,246],[14,10625,10626,10629],{"style":2267},[14,10627],{"className":10628,"style":839},[238],[14,10630,10631],{},[14,10632,2103],{"className":10633},[890],[14,10635,254],{"className":10636},[253],[14,10638,10640],{"className":10639},[226],[14,10641,10643],{"className":10642,"style":2285},[230],[14,10644],{},[14,10646],{"className":10647,"style":272},[271],[14,10649,10651],{"className":10650},[890,891],[14,10652,10654,10704],{"className":10653},[221,222],[14,10655,10657,10701],{"className":10656},[226],[14,10658,10660,10680,10690],{"className":10659,"style":901},[230],[14,10661,10662,10665],{"style":904},[14,10663],{"className":10664,"style":908},[238],[14,10666,10668],{"className":10667},[243,244,245,246],[14,10669,10671,10674,10677],{"className":10670},[58,246],[14,10672,394],{"className":10673},[58,59,246],[14,10675,397],{"className":10676},[504,246],[14,10678,168],{"className":10679},[58,246],[14,10681,10682,10685],{"style":926},[14,10683],{"className":10684,"style":908},[238],[14,10686,10687],{},[14,10688,759],{"className":10689},[890,935,936],[14,10691,10692,10695],{"style":939},[14,10693],{"className":10694,"style":908},[238],[14,10696,10698],{"className":10697},[243,244,245,246],[14,10699,94],{"className":10700},[58,59,246],[14,10702,254],{"className":10703},[253],[14,10705,10707],{"className":10706},[226],[14,10708,10710],{"className":10709,"style":961},[230],[14,10711],{},[14,10713],{"className":10714,"style":272},[271],[14,10716,2140],{"className":10717},[58],[14,10719,2110],{"className":10720},[58,212],[14,10722],{"className":10723,"style":2367},[271],[14,10725,2145],{"className":10726},[2371],[14,10728],{"className":10729,"style":2367},[271],[14,10731,10733,10737,10777],{"className":10732},[49],[14,10734],{"className":10735,"style":10736},[53],"height:1.1331em;vertical-align:-0.3831em;",[14,10738,10740,10743],{"className":10739},[58],[14,10741,164],{"className":10742,"style":213},[58,212],[14,10744,10746],{"className":10745},[217],[14,10747,10749,10769],{"className":10748},[221,222],[14,10750,10752,10766],{"className":10751},[226],[14,10753,10755],{"className":10754,"style":474},[230],[14,10756,10757,10760],{"style":234},[14,10758],{"className":10759,"style":239},[238],[14,10761,10763],{"className":10762},[243,244,245,246],[14,10764,394],{"className":10765},[58,59,246],[14,10767,254],{"className":10768},[253],[14,10770,10772],{"className":10771},[226],[14,10773,10775],{"className":10774,"style":261},[230],[14,10776],{},[14,10778,10780,10783],{"className":10779},[58],[14,10781,2140],{"className":10782},[58],[14,10784,10786],{"className":10785},[217],[14,10787,10789,10820],{"className":10788},[221,222],[14,10790,10792,10817],{"className":10791},[226],[14,10793,10795,10806],{"className":10794,"style":4184},[230],[14,10796,10797,10800],{"style":6042},[14,10798],{"className":10799,"style":239},[238],[14,10801,10803],{"className":10802},[243,244,245,246],[14,10804,10],{"className":10805},[58,59,246],[14,10807,10808,10811],{"style":4187},[14,10809],{"className":10810,"style":239},[238],[14,10812,10814],{"className":10813},[243,244,245,246],[14,10815,10527],{"className":10816,"style":525},[58,59,246],[14,10818,254],{"className":10819},[253],[14,10821,10823],{"className":10822},[226],[14,10824,10827],{"className":10825,"style":10826},[230],"height:0.3831em;",[14,10828],{},[10,10830,2464,10831,10859,10860,10888,10889,10917,10918,10946,10947,10975],{},[14,10832,10834,10847],{"className":10833,"translate":18},[17],[14,10835,10837],{"className":10836},[22],[24,10838,10839],{"xmlns":26},[28,10840,10841,10845],{},[31,10842,10843],{},[34,10844,10],{},[38,10846,10],{"encoding":40},[14,10848,10850],{"className":10849,"ariaHidden":45},[44],[14,10851,10853,10856],{"className":10852},[49],[14,10854],{"className":10855,"style":3610},[53],[14,10857,10],{"className":10858},[58,59]," is the distance norm, and ",[14,10861,10863,10876],{"className":10862,"translate":18},[17],[14,10864,10866],{"className":10865},[22],[24,10867,10868],{"xmlns":26},[28,10869,10870,10874],{},[31,10871,10872],{},[34,10873,10527],{},[38,10875,10527],{"encoding":40},[14,10877,10879],{"className":10878,"ariaHidden":45},[44],[14,10880,10882,10885],{"className":10881},[49],[14,10883],{"className":10884,"style":3610},[53],[14,10886,10527],{"className":10887,"style":525},[58,59]," is the power to which the distance under that norm is raised. ",[14,10890,10892,10905],{"className":10891,"translate":18},[17],[14,10893,10895],{"className":10894},[22],[24,10896,10897],{"xmlns":26},[28,10898,10899,10903],{},[31,10900,10901],{},[34,10902,10],{},[38,10904,10],{"encoding":40},[14,10906,10908],{"className":10907,"ariaHidden":45},[44],[14,10909,10911,10914],{"className":10910},[49],[14,10912],{"className":10913,"style":3610},[53],[14,10915,10],{"className":10916},[58,59]," determines how distance is measured, while ",[14,10919,10921,10934],{"className":10920,"translate":18},[17],[14,10922,10924],{"className":10923},[22],[24,10925,10926],{"xmlns":26},[28,10927,10928,10932],{},[31,10929,10930],{},[34,10931,10527],{},[38,10933,10527],{"encoding":40},[14,10935,10937],{"className":10936,"ariaHidden":45},[44],[14,10938,10940,10943],{"className":10939},[49],[14,10941],{"className":10942,"style":3610},[53],[14,10944,10527],{"className":10945,"style":525},[58,59]," determines the penalty on outliers: the larger ",[14,10948,10950,10963],{"className":10949,"translate":18},[17],[14,10951,10953],{"className":10952},[22],[24,10954,10955],{"xmlns":26},[28,10956,10957,10961],{},[31,10958,10959],{},[34,10960,10527],{},[38,10962,10527],{"encoding":40},[14,10964,10966],{"className":10965,"ariaHidden":45},[44],[14,10967,10969,10972],{"className":10968},[49],[14,10970],{"className":10971,"style":3610},[53],[14,10973,10527],{"className":10974,"style":525},[58,59],", the more sensitive to outliers.",[10,10977,10978],{},"This formula also describes the core objective function of the Generalized Weber Location Problem.",{"title":10980,"searchDepth":10981,"depth":10981,"links":10982},"",2,[10983,10984,10985,10987,10988],{"id":68,"depth":10981,"text":69},{"id":1775,"depth":10981,"text":1776},{"id":3438,"depth":10981,"text":10986},"LpL_pLp​ Euclidean Norm",{"id":7074,"depth":10981,"text":7075},{"id":9106,"depth":10981,"text":9107},"When the data being processed are not simple vectors (e.g., complex matrices, probability distributions, points on a curved surface) and cannot be directly summed and divided by NNN, the Fréchet mean becomes necessary.",false,true,"md",{},"2026-04-25","\u002Fblog\u002F2026\u002F2026-04-25-the-frechet-mean-prototype",{"title":5,"description":10989},"blog\u002F2026\u002F2026-04-25-the-frechet-mean-prototype","The Fréchet Mean Prototype is the center point in a metric space that minimizes the sum of the squared distances to all data points. It extends the averaging operation from flat vector spaces to arbitrary curved spaces or complex objects.",[11000,11001],"mathematics","machine-learning","CasS5dZECil4WWXrmNYpfAtanFx5bar__Q68oIjy57I",1777090294521]