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On Representer Theorems and Convex Regularization

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BCDDGW_18.pdf (338.1Kb)
Date
2019
Dewey
Analyse
Sujet
Total variation; Vector space; Convex regularization; Representer theorem; Inverse problems
Journal issue
SIAM Journal on Optimization
Volume
29
Number
2
Publication date
2019
Article pages
1260–1281
Publisher
SIAM - Society for Industrial and Applied Mathematics
DOI
http://dx.doi.org/10.1137/18M1200750
URI
https://basepub.dauphine.fr/handle/123456789/19905
Collections
  • CEREMADE : Publications
Metadata
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Author
Boyer, Claire
66 Département de Mathématiques et Applications - ENS Paris [DMA]
542130 Laboratoire de Probabilités, Statistiques et Modélisations [LPSM (UMR_8001)]
Chambolle, Antonin
89626 Centre de Mathématiques Appliquées - Ecole Polytechnique [CMAP]
De Castro, Yohann
245281 Laboratoire de Mathématiques d'Orsay [LMO]
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Duval, Vincent
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
de Gournay, Frédéric
1954 Institut de Mathématiques de Toulouse UMR5219 [IMT]
Weiss, Pierre
Type
Article accepté pour publication ou publié
Abstract (EN)
We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions that are convex combinations of a small number of atoms. These atoms are identified with the extreme points and elements of the extreme rays of the regularizer level sets. An extension to a broader class of quasi-convex regularizers is also discussed. As a side result, we characterize the minimizers of the total gradient variation, which was previously an unresolved problem.

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