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1、SIAMJ.MATRIXANAL.APPL.c2010SocietyforIndustrialandAppliedMathematicsVol.31,No.5,pp.2623–2649STRUCTUREDTOTALMAXIMUMLIKELIHOOD:ANALTERNATIVETOSTRUCTUREDTOTALLEASTSQUARES∗AMIRBECK†ANDYONINAC.ELDAR‡Abstract.Linearinverseproblemswithuncertainmeasurementma
2、tricesappearinmanydiffer-entapplications.Oneofthestandardtechniquesforsolvingsuchproblemsisthetotalleastsquares(TLS)method.Recently,analternativeapproachhasbeensuggested,basedonmaximizinganap-propriatelikelihoodfunctionassumingthatthemeasurementmatrixc
3、onsistsofrandomGaussianvariables.Werefertothistechniqueasthetotalmaximumlikelihood(TML)method.Hereweextendthisstrategytothecaseinwhichthemeasurementmatrixisstructuredsothattheperturbationsarenotarbitrarybutratherfollowafixedpattern.Theresultingestimate
4、isreferredtoasthestructuredTML(STML).Asweshow,theSTMLcanbeviewedasaregularizedversionofthestructuredTLS(STLS)approachinwhichtheregularizationconsistsofalogarithmicpenalty.IncontrasttotheSTLSsolution,theSTMLalwaysexists.Furthermore,itsperformanceinprac
5、ticetendstobesuperiortothatoftheSTLSandcompetitivetootherregularizedsolvers,asweillustrateviaseveralexamples.WealsoconsiderafewinterestingspecialcasesinwhichtheSTMLcanbecomputedefficientlyeitherbyreducingitintoaone-dimensionalproblemregardlessoftheprobl
6、emsizeorbyadecompositionviaadiscreteFouriertransform.Keywords.totalleastsquares,maximumlikelihood,nonconvexprogramming,linearinverseproblem,circulantstructuresAMSsubjectclassifications.90C26,90C06,65F22,62J99DOI.10.1137/0907563381.Introduction.Thetotal
7、leastsquares(TLS)method,introducedfirstbyGolubandvanLoanin[23],isapopularapproachtodealwithapproximatelinearsystemsAx≈binwhichboththemodelmatrixAandtheright-handsidevectorbaresubjecttouncertainties[23,24,40].OneoftheappealingfeaturesoftheTLSalgorithmis
8、thatitcoincideswiththemaximumlikelihood(ML)solutionwhenAandbareknownuptoanadditiveGaussiandistortion.ThederivationofTLSasanMLestimateassumesthatnoisymeasurementsofAandbaregivenandjointlyestimatesxandA.Despiteitspopularity,inpractice,theperform
