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1、1ConstructionofaLargeClassofDeterministicSensingMatricesthatSatisfyaStatisticalIsometryPropertyRobertCalderbank,Fellow,IEEE,StephenHoward,Member,IEEE,andSinaJafarpour,StudentMember,IEEEAbstract—CompressedSensingaimstocaptureattributesofThetwofundamentalquestionsincompressedsensing
2、are:k-sparsesignalsusingveryfewmeasurements.InthestandardhowtoconstructsuitablesensingmatricesΦ,andhowtoCompressedSensingparadigm,theN×Cmeasurementmatrixrecoverαfromfefficiently;itisalsoofpracticalimportanceΦisrequiredtoactasanearisometryonthesetofalltoberesilienttomeasurementnoise
3、andtobeabletorecon-k-sparsesignals(RestrictedIsometryPropertyorRIP).IfΦsatisfiestheRIP,thenBasisPursuitorMatchingPursuitrecoverystruct(approximationsto)k-compressiblesignals,i.e.signalsalgorithmscanbeusedtorecoveranyk-sparsevectorαfromthathavemorethanknonvanishingentries,butwhereon
4、lyktheNmeasurementsΦα.AlthoughitisknownthatcertainentriesaresignificantandtheremainingentriesareclosetoprobabilisticprocessesgenerateN×Cmatricesthatsatisfyzero.RIPwithhighprobability,thereisnopracticalalgorithmforTheworkofDonoho[9]andofCand`es,RombergandTaoverifyingwhetheragivensen
5、singmatrixΦhasthisproperty,[10],[2],[11]providesfundamentalinsightintothegeometrycrucialforthefeasibilityofthestandardrecoveryalgorithms.Incontrastthispaperprovidessimplecriteriathatguaranteethatofsensingmatrices.Thisgeometryisexpressedbye.g.theadeterministicsensingmatrixsatisfyin
6、gthesecriteriaactsasaRestrictedIsometryProperty(RIP),formulatedbyCand`esandnearisometryonanoverwhelmingmajorityofk-sparsesignals;Tao[10]:asensingmatrixsatisfiesthek-RestrictedIsometryinparticular,mostsuchsignalshaveauniquerepresentationinPropertyifitactsasanearisometryonallk-sparse
7、vectors;tothemeasurementdomain.Probabilitystillplaysacriticalrole,ensureuniqueandstablereconstructionofk-sparsevectors,itbutitentersthesignalmodelratherthantheconstructionofthesensingmatrix.AnessentialelementinourconstructionisissufficientthatΦsatisfy2k-RIP.WhenN/Cand/ork/Narethatw
8、erequirethecolumnsofthesensingmat
