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1、IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.50,NO.6,JUNE20041341OnSparseRepresentationsinArbitraryRedundantBasesthenitistheuniquesparsestsolutionandthatunderthestrongercon-ditionJean-JacquesFuchs,Member,IEEE11kxk0<1+;withMasabove(2)2MAbstract—Thepurposeofthiscontributionistog
2、eneralizesomerecentresultsonsparserepresentationsofsignalsinredundantbases.Theques-theuniquesparsestsolutionisalsotheuniqueminimumpointof(LP).tionthatisconsideredisthefollowing:givenamatrixofdimensionStrongerresultsareestablishedin[2]forthiskindofAmatrices.()withandavecto
3、r=,findasufficientconditionfortohaveauniquesparsestrepresentationasalinearcombinationofInthefollowing,weprovethatifcondition(2)issatisfiedbyasolu-columnsof.tionxoofAx=b,withAamatrixbuiltuponanarbitrarynumberofAnswerstothisquestionareknownwhenistheconcatenationoftwovectors
4、withunitEuclideannorm,thenitistheuniqueminimumpointunitarymatricesandeitheranextensivecombinatorialsearchisperformedofbothaparametrizedquadraticprogram(QP)inasensetobedefinedoralinearprogramissolved.Weconsiderarbitrarymatricesandgiveasufficientconditionfortheuniquesparses
5、tsolutiontobetheuniqueso-later(Theorems2and3),andthelinearprogram(LP)(Theorems3andlutiontobothalinearprogramoraparametrizedquadraticprogram.The4).ButthisimpliesthatxoisalsothetheuniquesparsestsolutionofproofiselementaryandthepossibilityofusingaquadraticprogramopensAx=bsin
6、ceifasparserrepresentationexisted,itwouldsatisfy(2),perspectivestothecasewhere=+withavectorofnoiseorthesamereasoningwouldholdandonewouldarriveatacontradic-modelingerrors.tionsinceboth(QP)and(LP)areconvexprogramsthathaveauniqueIndexTerms—Basispursuit,globalmatchedfilter,li
7、nearprogram,optimumattainedatasinglepointunder(2).Asimilarresulthasin-quadraticprogram,redundantdictionaries,sparserepresentations.dependentlybeenobtainedin[3]for(LP)usingacompletelydifferentapproach.Toestablishthisresult,wemerelyapplymoregeneralbutnonex-I.INTRODUCTIONpli
8、citresultspresentedin[4],[5]tothisveryspecificproblemwhereLetusconsiderasetofmn-dimensionalvecto
