Robust Face Recognition via Sparse Representation(8)

WRIGHTETAL.:ROBUSTFACERECOGNITIONVIASPARSEREPRESENTATION217

Fig.6.Examplesoffeatureextraction.(a)Originalfaceimage.(b)120Drepresentationsintermsoffourdifferentfeatures(fromlefttoright):Eigenfaces,Laplacianfaces,downsampled(12Â10pixel)image,andrandomprojection.Wewilldemonstratethatallthesefeaturescontainalmostthesameinformationabouttheidentityofthesubjectandgivesimilarlygoodrecognitionperformance.(c)Theeyeisapopularchoiceoffeaturefor

~¼Ry~.facerecognition.Inthiscase,thefeaturematrixRissimplyabinarymask.(a)Originaly.(b)120Dfeaturesyy.(c)Eyefeaturey

RA2IRdÂnofd-dimensionalfeatures;thetestimageyis

~.replacedbyitsfeaturesy

Forextantfacerecognitionmethods,empiricalstudieshaveshownthatincreasingthedimensiondofthefeaturespacegenerallyimprovestherecognitionrate,aslongasthedistributionoffeaturesRAidoesnotbecomedegenerate[42].Degeneracyisnotanissuefor‘1-minimization,sinceit

~beinorneartherangeofRAi—itmerelyrequiresthaty

T

doesnotdependonthecovarianceÆi¼ATiRRAibeingnonsingularasinclassicaldiscriminantanalysis.Thestableversionof‘1-minimization(10)or(17)isknowninstatisticalliteratureastheLasso[14].11Iteffectivelyregularizeshighlyunderdeterminedlinearregressionwhenthedesiredsolu-tionissparseandhasalsobeenprovenconsistentinsomenoisyoverdeterminedsettings[12].

Foroursparserepresentationapproachtorecognition,wewouldliketounderstandhowthechoiceofthefeatureextractionRaffectstheabilityofthe‘1-minimization(17)torecoverthecorrectsparsesolutionx0.Fromthegeometricinterpretationof‘1-minimizationgiveninSection2.2.1,theanswertothisdependsonwhethertheassociatednewpolytopeP¼RAðP1Þremainssufficientlyneighborly.ItiseasytoshowthattheneighborlinessofthepolytopeP¼RAðP1Þincreaseswithd[11],[15].InSection4,ourexperimentalresultswillverifytheabilityof‘1-minimiza-tion,inparticular,thestableversion(17),torecoversparserepresentationsforfacerecognitionusingavarietyoffeatures.Thissuggeststhatmostdata-dependentfeaturespopularinfacerecognition(e.g.,eigenfacesandLaplacian-faces)mayindeedgivehighlyneighborlypolytopesP.

Furtheranalysisofhigh-dimensionalpolytopegeometryhasrevealedasomewhatsurprisingphenomenon:ifthesolutionx0issparseenough,thenwithoverwhelmingprobability,itcanbecorrectlyrecoveredvia‘1-minimizationfromanysufficientlylargenumberdoflinearmeasurements~¼RAxyx0.Moreprecisely,ifx0hast(nnonzeros,thenwithoverwhelmingprobability

d!2tlogðn=dÞ

ð18Þ

randomlinearmeasurementsaresufficientfor‘1-minimiza-tion(17)torecoverthecorrectsparsesolutionx0[44].12Thissurprisingphenomenonhasbeendubbedthe“blessingofdimensionality”[15],[46].Randomfeaturescanbeviewedasaless-structuredcounterparttoclassicalfacefeaturessuchasEigenfacesorFisherfaces.Accordingly,wecallthelinearprojectiongeneratedbyaGaussianrandommatrixRandomfaces.13

Definition2(randomfaces).ConsideratransformmatrixR2IRdÂmwhoseentriesareindependentlysampledfromazero-meannormaldistribution,andeachrowisnormalizedtounitlength.TherowvectorsofRcanbeviewedasdrandomfacesinIRm.OnemajoradvantageofRandomfacesisthattheyareextremelyefficienttogenerate,asthetransformationRisindependentofthetrainingdataset.Thisadvantagecanbeimportantforafacerecognitionsystem,wherewemaynotbeabletoacquireacompletedatabaseofallsubjectsofinteresttoprecomputedata-dependenttransformationssuchasEigenfaces,orthesubjectsinthedatabasemaychangeovertime.Insuchcases,thereisnoneedforrecomputingtherandomtransformationR.

Aslongasthecorrectsparsesolutionx0canberecovered,Algorithm1willalwaysgivethesameclassifica-tionresult,regardlessofthefeatureactuallyused.Thus,whenthedimensionoffeaturedexceedstheabovebound(18),oneshouldexpectthattherecognitionperformanceofAlgorithm1withdifferentfeaturesquicklyconverges,andthechoiceofan“optimal”featuretransformationisnolongercritical:evenrandomprojectionsordownsampledimagesshouldperformaswellasanyothercarefullyengineeredfeatures.ThiswillbecorroboratedbytheexperimentalresultsinSection4.

3.2RobustnesstoOcclusionorCorruption

Inmanypracticalfacerecognitionscenarios,thetestimageycouldbepartiallycorruptedoroccluded.Inthiscase,theabovelinearmodel(3)shouldbemodifiedas

y¼y0þe0¼Ax0þe0;

ð19Þ

11.Classically,theLassosolutionisdefinedastheminimizerofkyÀAxxk22þ kxk1.Here, canbeviewedasinverseoftheLagrangemultiplierassociatedwithaconstraintkyÀAxxk22 ".Forevery ,thereisan"suchthatthetwoproblemshavethesamesolution.However,"canbeinterpretedasapixelnoiselevelandfixedacrossvariousinstancesoftheproblem,whereas cannot.OneshoulddistinguishtheLassooptimizationproblemfromtheLARSalgorithm,whichprovablysolvessomeinstancesofLassowithverysparseoptimizers[35].

12.Strictlyspeaking,thisthresholdholdswhenrandommeasurements

~¼Rxarecomputeddirectlyfromx0,i.e.,yx0.Nevertheless,ourexperiments

roughlyagreewiththeboundgivenby(18).Thecasewherex0isinsteadsparseinsomeovercompletebasisA,andweobservethatrandom

~¼RAxmeasurementsyx0hasalsobeenstudiedin[45].Whileconditionsfor

correctrecoveryhavebeengiven,theboundsarenotyetassharpas(18)above.

13.Randomprojectionhasbeenpreviouslystudiedasageneraldimensionality-reductionmethodfornumerousclusteringproblems[47],[48],[49],

aswellasforlearningnonlinearmanifolds[50],[51].

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