[NIPS 2011] Learning Anchor Planes for Classification

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1、LearningAnchorPlanesforClassificationyzyyxZimingZhangL’uborLadickýPhilipH.S.TorrAmirSaffariyDepartmentofComputing,OxfordBrookesUniversity,Wheatley,Oxford,OX331HX,U.K.zDepartmentofEngineeringScience,UniversityofOxford,ParksRoad,Oxford,OX13PJ,U.K.xSonyComputerEntert

2、ainmentEurope,London,UK{ziming.zhang,philiptorr}@brookes.ac.uklubor@robots.ox.ac.ukamir@ymer.orgAbstractLocalCoordinateCoding(LCC)[18]isamethodformodelingfunctionsofdatalyingonnon-linearmanifolds.Itprovidesasetofanchorpointswhichformalocalcoordinatesystem,suchtha

3、teachdatapointonthemanifoldcanbeapproximatedbyalinearcombinationofitsanchorpoints,andthelinearweightsbecomethelocalcoordinatecoding.Inthispaperweproposeencodingdatausingorthogonalanchorplanes,ratherthananchorpoints.Ourmethodneedsonlyafeworthogonalanchorplanesforc

4、oding,anditcanlinearizeany(;;p)-Lipschitzsmoothnon-linearfunctionwithafixedexpectedvalueoftheupper-boundapproximationerroronanyhighdimensionaldata.Inpractice,theorthogonalcoordinatesystemcanbeeasilylearnedbyminimizingthisupperboundusingsingularvaluedecomposition

5、(SVD).WeapplyourmethodtomodelthecoordinateslocallyinlinearSVMsforclassificationtasks,andourexperimentonMNISTshowsthatusingonly50anchorplanesourmethodachieves1.72%errorrate,whileLCCachieves1.90%errorrateusing4096anchorpoints.1IntroductionLocalCoordinateCoding(LCC)[

6、18]isacodingschemethatencodesthedatalocallysothatanynon-linear(;;p)-Lipschitzsmoothfunction(seeDefinition1inSection2fordetails)overthedatamanifoldcanbeapproximatedusinglinearfunctions.Therearetwocomponentsinthismethod:(1)asetofanchorpointswhichdecidethelocalcoor

7、dinates,and(2)thecodingforeachdatabasedonthelocalcoordinatesgiventheanchorpoints.Theoretically[18]suggeststhatundercertainassumptions,localityismoreessentialthansparsityfornon-linearfunctionapproximation.LCChasbeensuccessfullyappliedtomanyapplicationssuchlikeobje

8、ctrecognition(e.g.locality-constraintlinearcoding(LLC)[16])inVOC2009challenge[7].OnebigissueinLCCisthatitsclassificationperformanceishighlydependentonthenumbero