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Sharp Support Recovery from Noisy Random Measurements by L1 minimization

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorDOSSAL, Charles
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorCHABANOL, Marie-Line
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorPEYRÉ, Gabriel
hal.structure.identifierEquipe Image - Laboratoire GREYC - UMR6072
dc.contributor.authorFADILI, Jalal M.
dc.date.accessioned2024-04-04T02:26:44Z
dc.date.available2024-04-04T02:26:44Z
dc.date.created2010-01-01
dc.date.issued2012
dc.identifier.issn1063-5203
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/189976
dc.description.abstractEnIn this paper, we investigate the theoretical guarantees of penalized $\lun$ minimization (also called Basis Pursuit Denoising or Lasso) in terms of sparsity pattern recovery (support and sign consistency) from noisy measurements with non-necessarily random noise, when the sensing operator belongs to the Gaussian ensemble (i.e. random design matrix with i.i.d. Gaussian entries). More precisely, we derive sharp non-asymptotic bounds on the sparsity level and (minimal) signal-to-noise ratio that ensure support identification for most signals and most Gaussian sensing matrices by solving the Lasso problem with an appropriately chosen regularization parameter. Our first purpose is to establish conditions allowing exact sparsity pattern recovery when the signal is strictly sparse. Then, these conditions are extended to cover the compressible or nearly sparse case. In these two results, the role of the minimal signal-to-noise ratio is crucial. Our third main result gets rid of this assumption in the strictly sparse case, but this time, the Lasso allows only partial recovery of the support. We also provide in this case a sharp $\ell_2$-consistency result on the coefficient vector. The results of the present work have several distinctive features compared to previous ones. One of them is that the leading constants involved in all the bounds are sharp and explicit. This is illustrated by some numerical experiments where it is indeed shown that the sharp sparsity level threshold identified by our theoretical results below which sparsistency of the Lasso is guaranteed meets that empirically observed.
dc.description.sponsorshipAdaptivité pour la représentation des images naturelles et des textures - ANR-08-EMER-0009
dc.language.isoen
dc.publisherElsevier
dc.subject.enconsistency
dc.subject.enCompressed sensing
dc.subject.enL1 minimization
dc.subject.ensparsistency
dc.subject.enconsistency.
dc.title.enSharp Support Recovery from Noisy Random Measurements by L1 minimization
dc.typeArticle de revue
dc.identifier.doi10.1016/j.acha.2011.09.003
dc.subject.halInformatique [cs]/Traitement du signal et de l'image
dc.subject.halMathématiques [math]/Théorie de l'information et codage [math.IT]
dc.subject.halInformatique [cs]/Théorie de l'information [cs.IT]
dc.subject.halMathématiques [math]/Analyse numérique [math.NA]
dc.subject.halSciences de l'ingénieur [physics]/Traitement du signal et de l'image
dc.subject.halPhysique [physics]/Physique mathématique [math-ph]
dc.identifier.arxiv1101.1577
bordeaux.journalApplied and Computational Harmonic Analysis
bordeaux.page24-43
bordeaux.volume33
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue1
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-00553670
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00553670v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Applied%20and%20Computational%20Harmonic%20Analysis&rft.date=2012&rft.volume=33&rft.issue=1&rft.spage=24-43&rft.epage=24-43&rft.eissn=1063-5203&rft.issn=1063-5203&rft.au=DOSSAL,%20Charles&CHABANOL,%20Marie-Line&PEYR%C3%89,%20Gabriel&FADILI,%20Jalal%20M.&rft.genre=article


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