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A theory of optimal convex regularization for low-dimensional recovery

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorTRAONMILIN, Yann
hal.structure.identifierRéseaux dynamiques : approche structurelle et temporelle [DANTE]
dc.contributor.authorGRIBONVAL, Rémi
hal.structure.identifierLaboratoire Jean Alexandre Dieudonné [LJAD]
dc.contributor.authorVAITER, Samuel
dc.date.accessioned2024-04-04T02:37:14Z
dc.date.available2024-04-04T02:37:14Z
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/190788
dc.description.abstractEnWe consider the problem of recovering elements of a low-dimensional model from under-determined linear measurements. To perform recovery, we consider the minimization of a convex regularizer subject to a data fit constraint. Given a model, we ask ourselves what is the ``best'' convex regularizer to perform its recovery. To answer this question, we define an optimal regularizer as a function that maximizes a compliance measure with respect to the model. We introduce and study several notions of compliance. We give analytical expressions for compliance measures based on the best-known recovery guarantees with the restricted isometry property. These expressions permit to show the optimality of the ℓ1-norm for sparse recovery and of the nuclear norm for low-rank matrix recovery for these compliance measures. We also investigate the construction of an optimal convex regularizer using the examples of sparsity in levels and of sparse plus low-rank models.
dc.description.sponsorshipRégularisation performante de problèmes inverses en grande dimension pour le traitement de données - ANR-20-CE40-0001
dc.description.sponsorshipMéthodes variationnelles pour les signaux sur graphe - ANR-18-CE40-0005
dc.description.sponsorshipAlgorithmes, Approximations, Parcimonie et Plongements pour l'IA - ANR-19-CHIA-0009
dc.language.isoen
dc.title.enA theory of optimal convex regularization for low-dimensional recovery
dc.typeDocument de travail - Pré-publication
dc.subject.halInformatique [cs]/Traitement du signal et de l'image
dc.identifier.arxiv2112.03540
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-03467123
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-03467123v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=TRAONMILIN,%20Yann&GRIBONVAL,%20R%C3%A9mi&VAITER,%20Samuel&rft.genre=preprint


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