当前位置： 文档下载 > 所有分类 > Proximal support vector machine classifiersAbstract—A new approach to support vector machine (SVM) classification is proposed wherein each of two data sets are proximal to one of two distinct planes that are not parallel to each other. Each plane is generated such that it is closest to one of the
IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE,VOL.28,NO.1,JANUARY200669
MultisurfaceProximalSupportVectorMachineClassificationviaGeneralizedEigenvalues
OlviL.MangasarianandEdwardW.Wild
Abstract—Anewapproachtosupportvectormachine(SVM)classificationisproposedwhereineachoftwodatasetsareproximaltooneoftwodistinctplanesthatarenotparalleltoeachother.Eachplaneisgeneratedsuchthatitisclosesttooneofthetwodatasetsandasfaraspossiblefromtheotherdataset.EachofthetwononparallelproximalplanesisobtainedbyasingleMATLABcommandastheeigenvectorcorrespondingtoasmallesteigenvalueofageneralizedeigenvalueproblem.Classificationbyproximitytotwodistinctnonlinearsurfacesgeneratedbyanonlinearkernelalsoleadstotwosimplegeneralizedeigenvalueproblems.The
effectivenessoftheproposedmethodisdemonstratedbytestsonsimpleexamplesaswellasonanumberofpublicdatasets.Theseexamplesshowtheadvantagesoftheproposedapproachinbothcomputationtimeandtestsetcorrectness.IndexTerms—Supportvectormachines,proximalclassification,generalizedeigenvalues.
æ
1
INTRODUCTION
UPPORT
vectormachines(SVMs)[23],[4],[27]constitute
themethodofchoiceforclassificationproblemswhilethegeneralizedeigenvalueproblem[22],[5]isasimpleproblemofclassicallinearalgebrasolvablebyasinglecommandofMATLAB[17]orScilab[24]orbyusingstandardlinearalgebrasoftwaresuchLAPACK[1].Inproximalsupportvectorclassification[7],[25],[6],twoparallelplanesaregeneratedsuchthateachplaneisclosesttooneoftwodatasetstobeclassifiedandthetwoplanesareasfarapartaspossible.Inthepresentwork,wedroptheparallelismconditionontheproximalplanesandrequirethateachplanebeascloseaspossibletooneofthedatasetsandasfaraspossiblefromtheotherdataset.Thisformulationleadstotwogeneralizedeigenvalueproblems:Gz¼ HzandLz¼ Mz,whereG,H,L,andMaresymmetricpositivesemidefinitematrices.Eachofthenonparallelproximalplanesisgeneratedbyaneigenvectorcorrespondingtoasmallesteigenvalueofeachofthegeneralizedeigenvalueproblems.ApplicationofthismethodtotheclassicalXORproblemintwodimensionswherethetwosetsaref½00 ;½11 gandf½10 ;½01 gleadstoanexactclassificationbytwononparallelproximallineseachgoingthroughthetwopointsofeachset.
Relatedworkisthek-planeclusteringof[3],whereclustersaredeterminedbyproximitytovariousnonparallelplanesbasedonthesmallesteigenvectorofamatrixgeneratedbygivendatapoints.Wealsonotethat,in[11],thegeneralizedeigenvalueformulationwasusedforproteinfoldrecognitiontodetermineanoptimaltransformationofapermutationmatrixbasedonsimultaneouslyminimizingwithin-class
.O.L.MangasarianiswiththeComputerSciencesDepartment,UniversityofWisconsin,Madison,WI53706,andtheDepartmentofMathematics,UniversityofCaliforniaatSanDiego,LaJolla,CA92093.E-mail:olvi@cs.wisc.edu.
.E.W.WildiswiththeComputerSciencesDepartment,UniversityofWisconsin,Madison,WI53706.E-mail:wildt@cs.wisc.edu.Manuscriptreceived28Oct.2004;revised21Mar.2005;accepted6Apr.2005;publishedonline11Nov.2005.RecommendedforacceptancebyS.K.Pal.
Forinformationonobtainingreprintsofthisarticle,pleasesende-mailto:tpami@http://doc.xuehai.net,andreferenceIEEECSLogNumberTPAMI-0586-1004.
0162-8828/06/$20.00ß2006IEEE
S
variationandmaximizingbetween-classvariationofvariousproteinfolds.
Thisworkisorganizedasfollows:InSection2,webrieflydescribethegeneralclassificationproblemandourproximalmultiplanelinearkernelformulationasageneralizedeigenvalueproblem.InSection3,weextendourproximalresultstoaproximalmultisurfacenonlinearkernelformula-tion.InSection4,wetestournewapproachandcompareitwithstandardlinearandnonlinearkernelclassifiers.Section5concludesthepaper.
Awordaboutournotation.Allvectorswillbecolumnvectorsunlesstransposedtoarowvectorbyaprimesuperscript0.Thescalar(inner)productoftwovectorsxandyinthen-dimensionalrealspaceRnwillbedenotedbyx0y,the2-normofxwillbedenotedbykxk.ForamatrixA2RmÂn;AiistheithrowofAwhichisarowvectorinRn.AcolumnvectorofonesofarbitrarydimensionwillbedenotedbyeandtheidentitymatrixofarbitraryorderwillbedenotedbyI.ThegradientofadifferentiablefunctionfonRnisdefinedasthecolumnvectoroffirstpartialderivatives:
ðxÞ@fðxÞ0mÂn
andB2RnÂk;arfðxÞ:¼½@f1;...;n.ForA2R
kernelKðA;BÞmapsRmÂnÂRnÂkintoRmÂk.Inparticular,ifxandyarecolumnvectorsinRn,thenKðx0;yÞisarealnumber,Kðx0;A0ÞisarowvectorinRm,andKðA;A0ÞisanmÂmmatrix.Weshallmakenoassumptionsonourkernelsotherthansymmetry,thatis,Kðx0;yÞ0¼Kðy0;xÞand,inparticular,weshallnotassumeormakeuseofMercer’spositivedefinitenesscondition[27],[23].Thebaseofthenaturallogarithmwillbedenotedby".AfrequentlyusedkernelinnonlinearclassificationistheGaussiankernel[27],[15]whoseijthelement,i¼1...;m;j¼1...;k,isgivenby:
20
ðKðA;BÞÞij¼"À kAiÀBÁjk,whereA2RmÂn,B2RnÂk,and isapositiveconstant.
2THEMULTIPLANELINEARKERNELCLASSIFIER
Weconsidertheproblemofclassifyingmpointsinthen-dimensionalrealspaceRn,representedbythem1Ân
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