TRAONMILIN, Yann; GRIBONVAL, Rémi; VAITER, Samuel

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TRAONMILIN, Yann
Institut de Mathématiques de Bordeaux [IMB]

GRIBONVAL, Rémi
Réseaux dynamiques : approche structurelle et temporelle [DANTE]

VAITER, Samuel
Laboratoire Jean Alexandre Dieudonné [LJAD]

Langue

Document de travail - Pré-publication

Résumé en anglais

We 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.< Réduire

URI

https://oskar-bordeaux.fr/handle/20.500.12278/190788

Project ANR

Régularisation performante de problèmes inverses en grande dimension pour le traitement de données - ANR-20-CE40-0001
Méthodes variationnelles pour les signaux sur graphe - ANR-18-CE40-0005
Algorithmes, Approximations, Parcimonie et Plongements pour l'IA - ANR-19-CHIA-0009

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Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251