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1、ShiftingInequalityandRecoveryofSparseSignalsLieWangIntroductionòTheproblemofrecoveringahighdimensionalsparsesignalbasedonasmallnumberofmeasurementshasattractedmuchattentionrecently.òModelselection.òConstructionapproximation.òCompressivesensing.IntroductionòMainmodel:òFisannbypmatrix,wherencou
2、ldbemuchlessthanp.òZisthevectorofmeasurementerror.òβistheunknownvectorofcoefficients,ourgoalistoreconstructβ.IntroductionòTheerrorvectorzcaneitherbezero(noiselesscase),bounded,orGaussian(i.i.d.standardnormal).òβisassumedtobesparse,usuallyintermsofLnorm(numberofnonzero0coefficients).òLminimiza
3、tioniscomputationally0undoable.MethodsòInmanycasesthesparsesolutioncanbefoundthroughLminimization.1òThisLminimizationproblemhasbeen1studied,forexample,inFuchs(2004),CandesandTao(2005)andDonoho(2006).MethodsòNoisycase,twoLminimizationmethods.1òUnderLconstraintofresiduals.2òDantzigselector,byCa
4、ndesandTaoConditionsòItisclearthatregularityconditionsareneededinorderforthesemethodstobewellbehaved.Nearorthogonalcondition.òRestrictedIsometryProperty(RIP).òCandesandTaoconsideredsparserecoveryproblemsintheRIPframework.Conditionsòk-restrictedisometryconstantδofFkforanyksparsevectorc.òkk’-re
5、strictedorthogonalityconstantθk,k’foranykandk’sparsevectorsc,c’withdisjointsupport.ConditionsòDifferentconditionsonδandθhavebeenusedintheliterature.Forexample,CandesandTao(2007)imposesòCandes(2008)usesòActually,thesecondconditionisstronger.NoiselessCaseòUnderstandingthenoiselesscaseisnotonlyo
6、finterestonitsownright,italsoprovidesdeepinsightintotheproblemofreconstructingsparsesignalsinthenoisycase.òInthiscase,weneedtorecoverthesparsesignalexactly.NoiselessCaseò(CandesandTao)LetFbeann*pmatrix.Supposek>1satisfiesòLetβbeak-sparsevectorandY=Fβ.ThenβistheuniqueminimizertoUnifiedArgument
7、òWefoundthatallthoseresultscanbederivedfromthefollowingelementaryinequality(calledshiftinginequality):òSupposer≤q≤3r,andthenNoiselessCaseòOurresult:òLetFbeann*pmatrix.Supposek>1satisfiesandY=Fβ.Then,theminimizertosatisfiesNoiselessCaseòSuppos
