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1、IEEETRANSACTIONSONSIGNALPROCESSING,VOL.47,NO.1,JANUARY1999187AnAffineScalingMethodologyforBestBasisSelectionBhaskarD.Rao,SeniorMember,IEEE,andKennethKreutz-Delgado,SeniorMember,IEEEAbstractÐAmethodologyisdevelopedtoderivealgorithmsalgorithmdoesnotresultineffective
2、sparserepresentationsforoptimalbasisselectionbyminimizingdiversitymeasures[7],[13],[14].AnothereffectiveapproachtobasisselectionproposedbyWickerhauserandDonoho.Thesemeasuresincludewasdevelopedin[8]and[9]inthecontextofspecialdictio-thep-norm-like(`(p1))diversitym
3、easuresandtheGaussianandShannonentropies.Thealgorithmdevelopmentmethodologynaries,waveletpackets,andcosinepackets.Anentropy-basedusesafactoredrepresentationforthegradientandinvolvesmeasureofsparsitywasusedtochoosetheoptimalbasis,andsuccessiverelaxationoftheLagran
4、giannecessarycondition.Thisanefficientalgorithmwasdeveloped,exploitingthespecialyieldsalgorithmsthatareintimatelyrelatedtotheAffineScalingstructureinthedictionaryvectors.ThegeneralproblemofTransformation(AST)basedmethodscommonlyemployedbythebasisselectionwasaddress
5、edin[10]and[11],whereinaninteriorpointapproachtononlinearoptimization.Thealgorithmsminimizingthe`(p1)diversitymeasuresareequivalenttoanormmeasurewasusedasameasureofsparsity.BasisvectorsrecentlydevelopedclassofalgorithmscalledFOCalUnderde-werechosenthatresultedin
6、arepresentationwiththesmallestterminedSystemSolver(FOCUSS).Thegeneralnatureofthenorm,andthemethodwasshowntobequiteeffective.methodologyprovidesasystematicapproachforderivingthisInterestingly,theproblemofbasisselectionarisesinclassofalgorithmsandanaturalmechanismf
7、orextendingthem.Italsofacilitatesabetterunderstandingoftheconvergencemanyotherapplications,andresearchersinotherareashavebehaviorandastrengtheningoftheconvergenceresults.ThealsoattemptedtodefinediversitymeasuresandtocomputeGaussianentropyminimizationalgorithmissho
8、wntobeequiv-sparse/concentratedsolutionsbasedonminimizingthemalenttoawell-behavedp=0norm-likeoptimizationalgorithm.[15]±[19].Theuseofthetermªdiversityºinthispa
