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On Representer Theorems and Convex Regularization

Boyer, Claire; Chambolle, Antonin; De Castro, Yohann; Duval, Vincent; de Gournay, Frédéric; Weiss, Pierre (2019), On Representer Theorems and Convex Regularization, SIAM Journal on Optimization, 29, 2, p. 1260–1281. 10.1137/18M1200750

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Type
Article accepté pour publication ou publié
Date
2019
Journal name
SIAM Journal on Optimization
Volume
29
Number
2
Publisher
SIAM - Society for Industrial and Applied Mathematics
Pages
1260–1281
Publication identifier
Metadata
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Author(s)
Boyer, Claire
Département de Mathématiques et Applications - ENS Paris [DMA]
Laboratoire de Probabilités, Statistiques et Modélisations [LPSM (UMR_8001)]
Chambolle, Antonin cc
Centre de Mathématiques Appliquées - Ecole Polytechnique [CMAP]
De Castro, Yohann
Laboratoire de Mathématiques d'Orsay [LMO]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Duval, Vincent cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
de Gournay, Frédéric cc
Institut de Mathématiques de Toulouse UMR5219 [IMT]
Weiss, Pierre cc
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.
Subjects / Keywords
Total variation; Vector space; Convex regularization; Representer theorem; Inverse problems