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1、606IEEEJOURNALOFSELECTEDTOPICSINSIGNALPROCESSING,VOL.1,NO.4,DECEMBER2007AnInterior-PointMethodforLarge-Scale`1-RegularizedLeastSquaresSeung-JeanKim,Member,IEEE,K.Koh,M.Lustig,StephenBoyd,Fellow,IEEE,andDimitryGorinevsky,Fellow,IEEEAbstract—Recently,alotofattention
2、hasbeenpaidto1A.-RegularizedLeastSquaresregularizationbasedmethodsforsparsesignalreconstruction(e.g.,basispursuitdenoisingandcompressedsensing)andfeatureselection(e.g.,theLassoalgorithm)insignalprocessing,statistics,Astandardtechniquetopreventover-fittingisorTikhon
3、ovandrelatedfields.Theseproblemscanbecastas1-regularizedleast-squaresprograms(LSPs),whichcanbereformulatedasregularization[38],whichcanbewrittenasconvexquadraticprograms,andthensolvedbyseveralstandardmethodssuchasinterior-pointmethods,atleastforsmallandmediumsizepr
4、oblems.Inthispaper,wedescribeaspecialized(1)interior-pointmethodforsolvinglarge-scale1-regularizedLSPsthatusesthepreconditionedconjugategradientsalgorithmtowhereistheregularizationparameter.TheTikhonovcomputethesearchdirection.Theinterior-pointmethodcansolveregula
5、rizationproblemor-regularizedleast-squaresprogramlargesparseproblems,withamillionvariablesandobservations,inafewtensofminutesonaPC.Itcanefficientlysolvelargedense(LSP)hastheanalyticsolutionproblems,thatariseinsparsesignalrecoverywithorthogonaltransforms,byexploitin
6、gfastalgorithmsforthesetransforms.Themethodisillustratedonamagneticresonanceimagingdata(2)set.WelistsomebasicpropertiesofTikhonovregularization,IndexTerms—Basispursuitdenoising,compressivesampling,compressedsensing,convexoptimization,interior-pointmethods,whichwer
7、efertolaterwhenwecompareitto-regularizedleastsquares,preconditionedconjugategradients,1regulariza-leastsquares.tion.•Linearity.From(2),weseethatthesolutiontotheTikhonovregularizationproblemisalinearfunctionof.•Limitingbehavioras.AsconvergesI.INTRODUCTIONtotheMoore
8、–Penrosesolution,whereistheMoore–Penrosepseudoinverseof.ThelimitpointEconsideralinearmodeloftheformWhastheminimum-normamongallpointsthatsatisfy:whereist
