Introduction to Sparse Optimization

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1、WhatareSparseOptimizationandCompressiveSensing?WotaoYinDefinition•Sparseoptimizationistheclassofoptimizationmodels,methods,andalgorithmsforseekingsparsesolutions•“Sparse”ismorebroaderthan“fewnonzero.”Itroughlymeans“havingasimplestructure”•Sparseoptim

2、izationisnotoptimizationwithsparsedatabutwithunknownsparsesolutions•Sparseoptimizationhasbeenwidelyusedsince1960sbuthasbecomepopularduetocompressedsensingCompressiveSensingBasics•a.k.a.CompressedSensing~=CompressiveSampling•Goal:digitallyacquireanunk

3、nownsparseobjectx•Approach–Linearencoding:b=Ax,dimensionreduced–Recordsb–RecoverxfrombandACompressiveSensingBasics•TherowsofAarechoseninadvance•Amusthaveenoughrows,thoughstillfewerthandim(x)•Allmeasurementshaveroughlyequalinfoaboutx,nocriticalmeasure

4、ments•xcanbetheoreticallyrecoverbyL0minimization,practicallybyL1minimization•CSdoesnotrecordsupportinfo;ifonedoes,CShasnoadvantagesTraditionalsensingvsCompressiveSensing•TS:compressaftersensing•CS:compressduringsensing•TC:compressdigitally•CS:compres

5、sphysically(orinothermeans)•TSsensesmore;compressefficiently•CSsensesless;compresslessefficiently•TScompressionisadaptive;CScompressionisnon-adaptive(atleastintheoriginalCS)•TSandCSarecomplementary;neitheronewillreplacetheotherBottomline:CSsensesless

6、,computesmore,integratescompressionwithsensingExample1:RiceSinglePixelCameraExample1:RiceSinglePixelCameraExample2:SpectrumSensing•High-speedwirelesscommunicationrequires–Widebands–Hightransmissionpower–Advancecoding(GSM,CDMA,LTE…)•Spectraarescarcena

7、turalresources;theyaredividedintobandsandlicensedtovarioususers;butmostofthemareavailableatatime•Spectrumsharing,secondaryusersuseprimaryuses’bandswhentheyareavailable•SpectrumsharingrequiresspectrumsensingExample2:SpectrumSensing•Occupiedbandhashigh

8、power•Powervectorissparse•Swipingfrequenciesistooslow•FrequencyselectivefiltersallowCSsensing•Solution:–sensesb=Ax–recoversasparsex.•Similartoimaging,remotesensing,surveillanceHowtoRecognizeCSApplications•Recognizesparsity•LookforsensingmeansoftheCS-