Compressed sensing, sparse approximation, and low-rank matrix estimation

Compressed sensing, sparse approximation, and low-rank matrix estimation

ID:40600723

大小:1.13 MB

页数:171页

时间:2019-08-04

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1、Compressedsensing,sparseapproximation,andlow-rankmatrixestimationThesisbyYanivPlanInPartialFul llmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCaliforniaInstituteofTechnologyPasadena,California2011(DefendedFebruary1,2011)ii©2011YanivPlanAllRightsReservediiiDedicatedtomywife,JasminivAbstr

2、actTheimportanceofsparsesignalstructureshasbeenrecognizedinaplethoraofapplicationsrangingfrommedicalimagingtogroupdiseasetestingtoradartechnology.Ithasbeenshowninpracticethatvarioussignalsofinterestmaybe(approximately)sparselymodeled,andthatsparsemodelingisoftenbene cial,orevenindispensabletosi

3、gnalrecovery.Alongsideanincreaseinapplications,arichtheoryofsparseandcompressiblesignalrecoveryhasrecentlybeendevelopedunderthenamescompressedsensing(CS)andsparseapproximation(SA).Thisrevolutionaryresearchhasdemon-stratedthatmanysignalscanberecoveredfromseverelyundersampledmeasurementsbytakinga

4、dvantageoftheirinherentlow-dimensionalstructure.Morerecently,ano shootofCSandSAhasbeenafocusofresearchonotherlow-dimensionalsignalstructuressuchasmatricesoflowrank.Low-rankmatrixrecovery(LRMR)isdemonstratingarapidlygrowingarrayofimportantappli-cationssuchasquantumstatetomography,triangulationfr

5、omincompletedistancemeasurements,recommendersystems(e.g.,theNet ixproblem),andsystemidenti cationandcontrol.Inthisdissertation,weexamineCS,SA,andLRMRfromatheoreticalperspective.Weconsideravarietyofdi erentmeasurementandsignalmodels,bothrandomanddeterministic,andmainlyasktwoquestions.Howmanymeas

6、urementsarenecessary?Howlargeistherecoveryerror?Wegivetheoreticallowerboundsforbothofthesequestions,includingoracleandminimaxlowerboundsfortheerror.However,themainemphasisofthethesisistodemonstratetheecacyofconvexoptimization

7、inparticular`1andnuclear-normminimizationbasedprograms

8、inCS,SA,andLR

9、MR.Wederiveupperboundsforthenumberofmeasurementsrequiredandtheerrorderivedbyconvexoptimization,whichinmanycasesmatchthelowerboundsuptoconstantorlogarithmicfactors.Themajorityoftheseresultsdonotrequiretherestrictedisometryproperty(

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