HAL : hal-00627452, version 4

arXiv : 1109.6222

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v1 (28-09-2011)

v2 (22-02-2012)

v3 (02-03-2012)

v4 (14-04-2012)

Robust Sparse Analysis Regularization

Samuel Vaiter 1, Gabriel Peyré 1, Charles Dossal 2, Jalal Fadili 3

(09/2011)

This paper studies the properties of L1-analysis regularization for the resolution of linear inverse problems. Most previous works consider sparse synthesis priors where the sparsity is measured as the L1 norm of the coefficients that synthesize the signal in a given dictionary. In contrast, the more general analysis regularization minimizes the L1 norm of the correlations between the signal and the atoms in the dictionary. The corresponding variational problem includes several well-known regularizations such as the discrete total variation and the fused lasso. We give a sufficient condition to ensure that a signal is the unique solution of the analysis regularization when there is no noise in the observations. The same criterion ensures the robustness of the sparse analysis solution to a small noise in the observations. We also define a stronger sufficient condition that ensures robustness to an arbitrary bounded noise. In the special case of synthesis regularization, our contributions recover already known results, that are hence generalized to the analysis setting. We illustrate these theoritical results on practical examples to study the robustness of the total variation and the fused lasso regularizations.

1 :	CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
	CNRS : UMR7534 – Université Paris IX - Paris Dauphine
2 :	Institut de Mathématiques de Bordeaux (IMB)
	CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II
3 :	Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen (GREYC)
	CNRS : UMR6072 – Université de Caen – Ecole Nationale Supérieure d'Ingénieurs de Caen

Domaine

:

Mathématiques/Théorie de l'information et codage Informatique/Théorie de l'information et codage Sciences de l'ingénieur/Traitement du signal et de l'image Informatique/Traitement du signal et de l'image

Mots Clés : sparsity – analysis regularization – synthesis regularization – inverse problems – L1 minimization – union of subspaces – noise robustness – total variation – wavelets – Fused Lasso

hal-00627452, version 4

http://hal.archives-ouvertes.fr/hal-00627452

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Contributeur : Gabriel Peyré <>

Soumis le : Samedi 14 Avril 2012, 14:28:23

Dernière modification le : Samedi 14 Avril 2012, 22:31:57