1、譜聚類演算法步驟公式
（1）整理資料集，使資料集中資料在0-1之間。假設資料集m行n列。
（2）求鄰接矩陣W。元素值為每一點到其他點之間距離，即權重。
（3）求相似度矩陣S，相似度矩陣的定義。相似度矩陣由權值矩陣得到，實踐中一般用高斯核函式（也稱徑向基函式核）計算相似度，距離越大，代表其相似度越小。（這裡我認為相似度矩陣與鄰接矩陣一個概念，鄰接矩陣即在圖中表示頂點之間相鄰關係的矩陣，因此這裡用相似度矩陣表示該圖頂點之間的相互關係）

（4）求度矩陣D。把的每一列元素加起來得到個數，然後把它們放在對角線上（其它地方都是零），組成一個的對角矩陣，記為度矩陣。
（5）求拉普拉斯矩陣L。使用Normalized相似變換 L=I-D-(1/2)SD-(1/2) (I單位矩陣)
（6）求出L的最小的k個特徵值和對應的特徵向量V

（7）將特徵向量V進行kmeans聚類(少量的特徵向量)，得到m行1列的矩陣，即分類結果。

2、譜聚類演算法matlab實現
function C = spectral(W,sigma, num_clusters)
% 譜聚類演算法
% 使用Normalized相似變換
% 輸入  : W              : N-by-N 矩陣, 即連線矩陣
%        sigma          : 高斯核函式,sigma值不能為0
%        num_clusters   : 分類數
%
% 輸出  : C : N-by-1矩陣 聚類結果，標籤值
%
    format long
    m = size
 
(W, 1);
    %計算相似度矩陣  相似度矩陣由權值矩陣得到，實踐中一般用高斯核函式
    W = W.*W;   %平方
    W = -W/(2*sigma*sigma);
    S = full(spfun(@exp, W)); % 在這裡S即為相似度矩陣，也就是這不在以鄰接矩陣計算，而是採用相似度矩陣

    %獲得度矩陣D
    D = full(sparse(1:m, 1:m, sum(S))); %所以此處D為相似度矩陣S中一列元素加起來放到對角線上，得到度矩陣D

    % 獲得拉普拉斯矩陣 Do laplacian, L = D^(-1/2) * S * D^(-1/2)
 

    L = eye(m)-(D^(-1/2) * S * D^(-1/2)); %拉普拉斯矩陣

    % 求特徵向量 V
    %  eigs 'SM';絕對值最小特徵值
    [V, ~] = eigs(L, num_clusters, 'SM');
    % 對特徵向量求k-means
    C=kmeans(V,num_clusters);
end

呼叫

clc
clf
clear
%twoCircles資料集
load('twoCircles.mat');
dataSet=twoCircles;
dataSet=dataSet/max(max(abs(dataSet)));
num_clusters=2;
sigma=0.1;
%XOR資料集
% load('XOR.mat');
% dataSet=XOR;
% dataSet=dataSet/max(max(abs(dataSet)));
% num_clusters=4;
% sigma=0.1;
Z=pdist(dataSet);
W=squareform(Z);
C = spectral(W,sigma, num_clusters);
plot(dataSet(C==1,1),dataSet(C==1,2),'r.', dataSet(C==2,1),dataSet(C==2,2),'b.', dataSet(C==3,1),dataSet(C==3,2),'g.', dataSet(C==4,1),dataSet(C==4,2),'m.');

3、結果
使用twoCircles資料集結果如下

使用XOR資料集結果如下

4、資料集
twoCircles資料集
0.100267601303493 -0.0525014663431810
0.135362505728604 -0.0459654893365564
0.126933443606530 -0.0164012350510703
0.115293882173694 -0.0473205001143649
0.139683311317968 -0.0501011177695915
0.143195450367134 0.0122605442686863
0.106009774791485 -0.0168037897565408
0.118163138199935 -0.00666648250514589
0.110780653575040 0.0265235804271568
0.100737917234786 0.0156260544496707
0.126536424113384 0.0560221149798287
0.130698280785180 0.0684661165422064
0.104542664983812 0.0809424119400617
0.0726902879896738 0.0562442697844981
0.0637149247388668 0.0892398222314587
0.0853220620494712 0.0587256576192461
0.0641053278934059 0.0947081798114114
0.0540426146452954 0.106785289158448
0.0642233489137368 0.0773824972648639
0.0350076911071655 0.111209797092187
0.0575752186252200 0.139474771321149
0.0126290954650543 0.108504680216120
0.00349847022973863 0.112987536527351
0.0105711038041949 0.155177325738704
-0.0102810207109895 0.140100147985530
0.0305894017235263 0.124743038795609
0.0186284478675877 0.134335541069807
-0.0173374800758943 0.164502347969208
-0.0400122017295506 0.162977680067346
-0.0335834225808840 0.150519495750737
-0.0438278406773373 0.173784298945292
-0.0345953885966573 0.182458289569849
-0.0635677163520193 0.159022637074378
-0.0405818861039921 0.169202540121774
-0.0527891130200498 0.168094775301963
-0.0603092138248180 0.194748108368073
-0.106572021716591 0.157099440212529
-0.119891726239592 0.203801706866876
-0.0940989249650586 0.167363595210180
-0.144698032772901 0.179866876326964
-0.159434519297908 0.196417018620475
-0.146015734820454 0.189134524311375
-0.132834910083993 0.156211145302309
-0.172867395587399 0.187335748061479
-0.193854848542921 0.151202690306914
-0.211766599907447 0.197752526283055
-0.170091451059050 0.191328759522471
-0.199365119129387 0.165811645197576
-0.222213410804137 0.172067404952940
-0.214140822664785 0.185931958176438
-0.215512624668578 0.130509784136747
-0.251860997790970 0.163417827951663
-0.238942511928649 0.162858234955589
-0.276695845932721 0.154731684809285
-0.239602463560443 0.113577060960391
-0.262380455618860 0.121797461707344
-0.302793019604424 0.0905875818722080
-0.293697339739456 0.102574990330745
-0.289468524988789 0.0841848131601263
-0.278550398589779 0.0958968428628290
-0.309193942355570 0.0956952453943073
-0.309665221370501 0.101114961370419
-0.312388873911295 0.0707668589797037
-0.303505342893366 0.0692910517672390
-0.355052139807089 0.0844176064770470
-0.344861224975699 0.0741505640116930
-0.355087516529306 0.0440336650435923
-0.336475652845790 0.0563709794282264
-0.328576905148420 -0.00379063760949573
-0.356115327189969 0.0322631886597826
-0.340366743529895 -0.00564538741063073
-0.371908493540811 -0.0216241755498018
-0.360894413187727 0.00429001151502345
-0.372562268165067 -0.0339568582868362
-0.343688383714249 -0.0547917622629392
-0.358143770979545 -0.0729574893403541
-0.349822126123939 -0.0272893847258684
-0.393336391064242 -0.0537848133661400
-0.383204007322266 -0.0529678498497836
-0.353515524006840 -0.0582479249357490
-0.385827080066973 -0.0837505105693941
-0.395565518334062 -0.0794460419109608
-0.369838780380090 -0.128964105956146
-0.337849121293584 -0.101031682031104
-0.375974210376338 -0.108682708099617
-0.354766754663879 -0.155495771701691
-0.364247967067762 -0.178592840253556
-0.380703233598296 -0.146316315538511
-0.348409149249923 -0.196479292584203
-0.324568709628736 -0.166904266048153
-0.354047306608441 -0.192633432918573
-0.361878508037289 -0.189117407045915
-0.344545303997592 -0.226502649530425
-0.314056774268735 -0.213047594950611
-0.327977346511586 -0.214936421244042
-0.301897033254626 -0.259903925262516
-0.334105134733781 -0.240055452972030
-0.284924107698938 -0.222357013912615
-0.297379072167748 -0.275672884069141
-0.311400589636153 -0.262648154727971
-0.281353262796720 -0.256011103516970
-0.254789900987687 -0.283968058929749
-0.234305948705366 -0.250380189311949
-0.222139655356567 -0.271161175379960
-0.213559952965771 -0.305485551482142
-0.209975366956662 -0.289335212588093
-0.202818085453815 -0.274235106083386
-0.197541571095125 -0.297528078237609
-0.202986511362828 -0.289497670612192
-0.201700888643611 -0.284206470938609
-0.209630925383829 -0.290272082259436
-0.168298490585481 -0.335428344847537
-0.145963190943968 -0.334038327241760
-0.171474058532450 -0.337884423131229
-0.127628295490400 -0.346688626968460
-0.163836129157153 -0.348800171515015
-0.135368304936212 -0.314081776188499
-0.132007097436545 -0.313449799619478
-0.128926470738468 -0.294581296451970
-0.0902652793447816 -0.305026685606573
-0.0992012727673828 -0.329898372201677
-0.0507464573856505 -0.316762063098592
-0.0685488580119897 -0.329141751075382
-0.0444113533236343 -0.298403913030859
-0.0783191135006571 -0.332725423603294
-0.0332333013765817 -0.291940950102696
-0.0255661481044777 -0.284709797688673
-0.00245296851181924 -0.284803597794617
-0.00249863019152190 -0.305142315144786
0.0198228715332239 -0.285804165103858
0.0300100433770582 -0.287865952823912
-0.00646122214190621 -0.305561033396500
0.00428173125450494 -0.255530516108207
0.0482792519180897 -0.258225143471595
0.0123155252390966 -0.277151618583252
0.0396615132170281 -0.228522339341406
0.0416024960684403 -0.268866466045177
0.0747102721122029 -0.254810237522588
0.0436779492222656 -0.240772920306928
0.0914459988808117 -0.243627308241155
0.0970110968333145 -0.238233512837646
0.0878270794000305 -0.199494381203487
0.0765908466258044 -0.222637671377960
0.0708152910911313 -0.193570495703914
0.108928079105592 -0.198380235456279
0.0773284701063797 -0.179437889536288
0.0836765261597315 -0.142573610997107
0.0914692129480362 -0.175860041459249
0.114093997612083 -0.179336608807588
0.130382722297946 -0.115457499962721
0.144010623150724 -0.119396339545842
0.124763921481018 -0.119550424368215
0.0996057698713907 -0.109291370936644
0.116935511527647 -0.124543384023350
0.137633394319036 -0.0609595214130716
0.146950113500472 -0.0892548421397118
0.117347761318916 -0.0632817695214719
0.379647097515717 -0.0614002586252345
0.397881577085257 -0.0645688296012094
0.382685746530102 -0.0123937535394865
0.376571015503113 0.00526568870359465
0.360854505995832 0.00923618270753422
0.361534611077772 0.0167124782472926
0.379525822624179 0.0838695030300762
0.346759278173418 0.0554243132780597
0.347442760581728 0.0904281335775545
0.334523949996636 0.137099373387937
0.345257112013698 0.166779404797340
0.329850886009906 0.161630987533876
0.325478864590708 0.190771254398351
0.305729085928220 0.205005461086872
0.289377833644790 0.217239719283975
0.282662806695955 0.234449280180130
0.245906090317521 0.266819781549496
0.253842647226271 0.258021761453835
0.273989951944179 0.264725787116273
0.248765919679540 0.295404095896161
0.231178953356613 0.295397753861568
0.218352310580521 0.290136905199436
0.202267722976175 0.321104094721482
0.191437763754652 0.339566316931083
0.170554341287904 0.341721171850985
0.166129078208872 0.352545814265809
0.0967436810674985 0.383681171204894
0.0960953748434208 0.379190363603298
0.0837673594924505 0.385807952149335
0.0513156132273566 0.369505219697132
0.0193368368407719 0.386295498951418
0.0566033158446816 0.404403969159148
0.0225105750076566 0.421370129571270
-0.0316187666303336 0.430428396403566
-0.0512198377212044 0.451407595962523
-0.0375447511302303 0.450983972374145
-0.0768275219623830 0.402995944672121
-0.115035872740353 0.456288634618542
-0.114952165516496 0.409659621623729
-0.151021813205498 0.458135112803455
-0.131649339625237 0.434944387104889
-0.143705628511081 0.411955302052513
-0.219094098803580 0.399381111766744
-0.200120592939383 0.409995446343507
-0.218014262677533 0.436848150425081
-0.262938083606404 0.387899788368698
-0.246788821726377 0.399054144152815
-0.298730596387592 0.379430618109544
-0.333478792626487 0.373970635322010
-0.347918173555825 0.360435503767655
-0.323744109495305 0.390313479309319
-0.346679114506107 0.344170933324307
-0.359721999121444 0.366066149191139
-0.386243474098894 0.328875690392725
-0.390668503973189 0.346254510097211
-0.454142427723858 0.325161895529989
-0.447475711019174 0.299748117578652
-0.471869692480270 0.319336223353854
-0.472502727888249 0.304838269344423
-0.462450815819334 0.297691775190530
-0.511820295556556 0.226539941566540
-0.505467277760517 0.249540131137678
-0.551970521879217 0.195656800623545
-0.555714847641494 0.185227125097265
-0.525810838500301 0.168429835638935
-0.574289349890964 0.166809846044552
-0.545635651284044 0.141529296950356
-0.583865841637917 0.129927382439677
-0.571884196344373 0.105807686150180
-0.616270722993104 0.0954205924778298
-0.581864710787012 0.0785497954177747
-0.601703525051394 0.0725376628135345
-0.580053039284052 0.0620171518880286
-0.623464756517173 0.0114109890977535
-0.621672284794125 0.0261452976867265
-0.640064677610222 0.000108697156637055
-0.590828127767378 -0.0146681274947393
-0.623799802780419 -0.0732747659392776
-0.644908537176083 -0.100277452718236
-0.606413711833031 -0.0875732059574307
-0.617640064830947 -0.102052945534801
-0.626198284694807 -0.157330319367997
-0.597715034374953 -0.178056796073903
-0.589020489814766 -0.167580959497453
-0.619429629614866 -0.174196295521016
-0.597283540414415 -0.193338165826052
-0.597168701903739 -0.255901732252636
-0.606988796774917 -0.236188327416440
-0.572848380762827 -0.244342689069538
-0.544322309131500 -0.289631534143429
-0.585075126906007 -0.330368760062847
-0.527806671611551 -0.337602584622756
-0.538351777507918 -0.325233350555587
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-0.521276649672564 -0.385241118629563
-0.496275646942683 -0.375611322249414
-0.477542224199524 -0.434961142088870
-0.474046480222339 -0.454614136269792
-0.434202530151097 -0.436340960041392
-0.412423548399363 -0.481529498157065
-0.405509437595275 -0.441323575634914
-0.408920353040217 -0.497091528299050
-0.380353653282453 -0.486610507329795
-0.361009657383465 -0.484313117375382
-0.374300424468568 -0.493141352891889
-0.334278320943441 -0.522523400728082
-0.327180651476797 -0.514420331510727
-0.292540064690979 -0.512619812658298
-0.251800146578828 -0.520182110976352
-0.263709023622946 -0.567341993513358
-0.268023520168569 -0.552560001970452
-0.194739028009272 -0.542712013354447
-0.175339173096868 -0.581755463596484
-0.189543416715458 -0.551446557340132
-0.175548554711783 -0.569770480214661
-0.123976564892378 -0.564555345969930
-0.120499554453321 -0.584365688282749
-0.0974472943526174 -0.594557046909839
-0.0701458099188990 -0.580606917064844
-0.0857861704911607 -0.541518134770355
-0.0191656181726216 -0.565571445395348
0.00473997594657107 -0.574101145702962
-0.0265136243266595 -0.540173306084224
0.0377824100859994 -0.534526125402419
0.0607127170553193 -0.573714123533801
0.0256991639305313 -0.515568692662187
0.0750293804771959 -0.513104127725868
0.100909125259251 -0.533433406421375
0.102938144215622 -0.503905120993026
0.136360882424846 -0.516724237714893
0.160351512887989 -0.468463720117496
0.165875735585508 -0.496423463021149
0.185553922917037 -0.491627279645423
0.180159052973811 -0.444826340073898
0.229183556823824 -0.448061358573410
0.219204028289945 -0.424033207830737
0.245742156083010 -0.391214678624873
0.271531239374451 -0.415838279344467
0.250818397502540 -0.414908934695387
0.271124073472142 -0.356374642544174
0.292164827520917 -0.365814651340193
0.296720828006570 -0.336049546556310
0.286019255902477 -0.343139336378628
0.335077527086029 -0.332101862006207
0.301485445863558 -0.309695995639731
0.333707792193999 -0.251151746792849
0.317821117540833 -0.264234266154152
0.381050549849280 -0.240494338034531
0.360004479542351 -0.205456598355826
0.388637523745346 -0.176202259871168
0.361675847978310 -0.159811109806494
0.395587569270693 -0.163709904402880
0.360873026344104 -0.155004205758110
0.402373685593358 -0.107735146318242
0.402966624140516 -0.126363800088237
0.373867567207756 -0.0896191882457839

XOR資料集
11.7019498550828 11.9864146917909
11.2854197541034 9.61106025805058
12.9293997424851 7.77139077294215
11.0678743631211 10.4674361646635
10.2184969161492 10.0020184568202
13.0269044940811 9.51316600308935
11.8515238069851 8.48249559804150
13.1609598294686 10.3732294253150
10.9263007068675 9.96449067911523
12.9089787789130 10.9301847688819
12.9750456160589 11.9504775516646
10.9553894933476 9.91018786085430
12.1339163659016 9.87242404255941
11.2520226218202 11.0674309284256
13.5473713727711 11.9613951483563
11.9684417926901 11.5074099651376
12.6895038389352 11.1186184934507
10.2832003244487 11.5415772866841
12.4229965264968 11.1877890494122
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