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1、TheoryandApplicationsofCompressedSensingGittaKutyniokAugust29,2012AbstractCompressedsensingisanovelresearcharea,whichwasintroducedin2006,andsincethenhasalreadybecomeakeyconceptinvariousareasofappliedmathematics,com-puterscience,andelectricalengineering.Itsurprisingly
2、predictsthathigh-dimensionalsignals,whichallowasparserepresentationbyasuitablebasisor,moregenerally,aframe,canberecoveredfromwhatwaspreviouslyconsideredhighlyincompletelinearmeasurementsbyusingecientalgorithms.Thisarticleshallserveasanintroductiontoandasurveyaboutco
3、mpressedsensing.KeyWords.Dimensionreduction.Frames.Greedyalgorithms.Ill-posedinverseproblems.`1minimization.Randommatrices.Sparseapproximation.Sparserecovery.Acknowledgements.Theauthorisgratefultothereviewersformanyhelpfulsugges-tionswhichimprovedthepresentationofthe
4、paper.ShewouldalsoliketothankEmmanuelCandes,DavidDonoho,MichaelElad,andYoninaEldarforvariousdiscussionsonrelatedtopics,andSadeghJokarforproducingFigure3.TheauthoracknowledgessupportbytheEinsteinFoundationBerlin,byDeutscheForschungsgemeinschaft(DFG)GrantsSPP-1324KU14
5、46/13andKU1446/14,andbytheDFGResearchCenterMatheonMathematicsforkeytechnologies"inBerlin.1IntroductionarXiv:1203.3815v2[cs.IT]28Aug2012Theareaofcompressedsensingwasinitiatedin2006bytwogroundbreakingpapers,namely[18]byDonohoand[11]byCandes,Romberg,andTao.Nowadays,af
6、teronly6years,anabundanceoftheoreticalaspectsofcompressedsensingareexploredinmorethan1000articles.Moreover,thismethodologyistodateextensivelyutilizedbyappliedmathemati-cians,computerscientists,andengineersforavarietyofapplicationsinastronomy,biology,medicine,radar,an
7、dseismology,tonameafew.Thekeyideaofcompressedsensingistorecoverasparsesignalfromveryfewnon-adaptive,linearmeasurementsbyconvexoptimization.Takingadierentviewpoint,itconcernstheexactrecoveryofahigh-dimensionalsparsevectorafteradimensionreductionstep.Fromayetanotherst
8、andpoint,wecanregardtheproblemascomputingasparsecoecientvectorforasignalwithrespecttoanovercompletesystem.Thetheoreticalfoun-datio