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1、ManifoldAlignmentusingProcrustesAnalysisChangWangchwang@cs.umass.eduSridharMahadevanmahadeva@cs.umass.eduComputerScienceDepartment,UniversityofMassachusetts,Amherst,MA01003USAAbstractonlyhavealimitednumberofdegreesoffreedom,im-plyingthedatasethasalow
2、intrinsicdimensionality.InthispaperweintroduceanovelapproachSimilartocurrentworkinthe¯eld,weassumekernelstomanifoldalignment,basedonProcrustesforcomputingthesimilaritybetweendatapointsinanalysis.Ourapproachdi®ersfromsemi-theoriginalspacearealreadygi
3、ven.Inthe¯rststep,supervisedalignment"inthatitresultsinawemapthedatasetstolowdimensionalspacesre°ect-mappingthatisde¯nedeverywhere{wheningtheirintrinsicgeometriesusingastandard(nonlin-usedwithasuitabledimensionalityreductionearorlinear)dimensionality
4、reductionapproach.Formethod{ratherthanjustonthetrainingexample,usingagraph-basednonlineardimensional-datapoints.Wedescribeandevaluateourityreductionmethodprovidesadiscretizedapproxi-approachboththeoreticallyandexperimen-mationtothemanifolds,sothenewr
5、epresentationstally,providingresultsshowingusefulknowl-characterizetherelationshipsbetweenpointsbutnotedgetransferfromonedomaintoanother.theoriginalfeatures.Bydoingthis,wecancompareNovelapplicationsofourmethodincludingtheembeddingsofthetwosetsinstead
6、oftheiroriginalcross-lingualinformationretrievalandtrans-representations.Generallyspeaking,iftwodatasetsferlearninginMarkovdecisionprocessesareS1andS2havesimilarintrinsicgeometrystructures,presented.theyhavesimilarembeddings.Inoursecondstep,weapplyPr
7、ocrustesanalysistoalignthetwolowdimen-sionalembeddingsofthedatasetsbasedonanumber1.Introductionoflandmarkpoints.Procrustesanalysis,whichhasbeenusedforstatisticalshapeanalysisandimagereg-Manifoldalignmentisveryusefulinavarietyofappli-istrationof2D/3Dd
8、ata(Luoetal.,1999),removesthecationssinceitprovidesknowledgetransferbetweentranslational,rotationalandscalingcomponentsfromtwoseeminglydisparatedatasets.Sampleapplica-onesetsothattheoptimalalignmentbetweenthetwotionsincludeautomaticmachinetranslation
