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

《Compressed sensing, sparse approximation, and low-rank matrix estimation》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

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,anoshootofCSandSAhasbeenafocusofresearchonotherlow-dimensionalsignalstructuressuchasmatricesoflowrank.Low-rankmatrixrecovery(LRMR)isdemonstratingarapidlygrowingarrayofimportantappli-cationssuchasquantumstatetomography,triangulationfr

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

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

7、inparticular`1andnuclear-normminimizationbasedprograms

8、inCS,SA,andLR

9、MR.Wederiveupperboundsforthenumberofmeasurementsrequiredandtheerrorderivedbyconvexoptimization,whichinmanycasesmatchthelowerboundsuptoconstantorlogarithmicfactors.Themajorityoftheseresultsdonotrequiretherestrictedisometryproperty(