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1、SPARSEAPPROXIMATIONVIAITERATIVETHRESHOLDINGKyleK.Herrity,AnnaC.Gilbert,andJoelA.TroppUniversityofMichiganDepartmentofMathematicsAnnArbor,MI48109ABSTRACTthresholdingateachiteration.GENERALITisaLandweberiterationwithnonlinearshrinkageateachstepandismoti-Thewell-knownshrink
2、agetechniqueisstillrelevantforcon-vatedbytheanalysisof[1].BLOCKITisusedwhenourtemporarysignalprocessingproblemsoverredundantdictio-redundantdictionaryisaunionoforthonormalbases(e.g.,naries.Wepresenttheoreticalandempiricalanalysesfortwomorphologicalcomponents[3]).TheBLOCK
3、ITalgorithmiterativealgorithmsforsparseapproximationthatuseshrink-thresholdsineachbasissequentiallyand,assuch,isthemoreage.TheGENERALITalgorithmamountstoaLandweberpracticalalgorithm.Eachsubstepofafulliterationinvolvesaiterationwithnonlinearshrinkageateachiterationstep.Th
4、esingleunitarytransform(possiblywithafastimplementation)BLOCKITalgorithmarisesinmorphologicalcomponentsanal-andweneedonlyworkwithresidualandcoefficientvectors,ysis.AsufficientconditionforwhichGeneralITexactlyre-whichareequalinlengthtotheoriginalsignal.GENERALcoversasparses
5、ignalispresented,inwhichthecumulativeITrequiresthatwemanipulatealargercoefficientvectorandcoherencefunctionnaturallyarises.Thisanalysisextendsperformtwomatrix–vectormultiplicationsmuchlargerthanpreviousresultsconcerningtheOrthogonalMatchingPursuitthesingleunitarytransform
6、s.Thisalgorithmdoes,however,(OMP)andBasisPursuit(BP)algorithmstoITalgorithms.takeintoaccounttheinteractionsamongthevectorsinthedictionary.1.INTRODUCTIONWeprovideasufficientconditionforwhichguaranteesthatGENERALITrecoversexactlysparsesignals.Thissuf-Sparseapproximationprob
7、lemshavebeenstudiedfornearlyficientconditionmatchesthesufficientgeometricconditionsacentury,andtheyariseinmanyarenas,fromcompressionfortheOrthogonalMatchingPursuit(OMP)andBasisPur-andanalysisofaudio,image,andvideosignals,tomachinesuit(BP)algorithms.Wealsoprovideanalysisoft
8、hefixedlearning,denoising,andregularization.Ineachoftheseap-pointsoftheBLOCKITalgorithm.Inthefollowingse
