compressed-sensing1.pdf

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1、IntroductionCompressedsensingVariantsApplicationsCompressedsensingOr:theequationAx=b,revisitedTerenceTaoUniversityofCalifornia,LosAngelesMahlerLectureSeriesTerenceTaoCompressedsensingIntroductionCompressedsensingVariantsApplicationsAcquiringsignalsManytypes

2、ofreal-worldsignals(e.g.sound,images,video)canbeviewedasan01n-dimensionalvectorx1x=B..Cnofrealnumbers,wherenislarge(e.g.@.A2Rxnn106).Toacquirethissignal,weconsideralinearmeasurementmodel,inwhichwemeasureanm-dimensionalvectorb=Ax2Rmforsomemnmeasurementmatr

3、ixA(thuswemeasuretheinnerproductsofxwiththerowsofA).Forinstance,ifwearemeasuringatimeseriesinthefrequencydomain,AwouldbesomesortofFouriermatrix.TerenceTaoCompressedsensingIntroductionCompressedsensingVariantsApplicationsThisleadstothefollowingclassicalquest

4、ioninlinearalgebra:Question:Howmanymeasurementsmdoweneedtomakeinordertorecovertheoriginalsignalxexactlyfromb?Whataboutapproximately?Inotherwords:whencanwesolvetheequationAx=b?TerenceTaoCompressedsensingIntroductionCompressedsensingVariantsApplicationsThecla

5、ssicalanswerTheclassicaltheoryoflinearalgebra,whichwelearnasundergraduates,isasfollows:Ifthereareatleastasmanymeasurementsasunknowns(mn),andAhasfullrank,thentheproblemisdeterminedoroverdetermined,andonecaneasilysolveAx=buniquely(e.g.bygaussianelimination).

6、Iftherearefewermeasurementsthanunknowns(m

7、=argminx:Ax=bkxk`2=A(AA)1basthe“bestguess”forx.TerenceTaoCompressedsensingIntroductionCompressedsensingVariantsApplications#xAx=bAlow-dimensionalexampleofaleast-squaresguess.TerenceTaoCompressedsensingIntroductionCompressedsensingVariantsApplicationsInma

8、nysituationstheleastsquaressolutionisnotsatisfactory.Forinstance,considertheproblemofreconstructingaone-dimensionaldiscretesignalf:f1;:::;ng!Cfromapartialcollection^f(1);:::;^f(m)ofFouriercoefficients