On Representer Theorems and Convex Regularization Article - Mai 2019

Claire Boyer, Antonin Chambolle, Yohann de Castro, Vincent Duval, Frédéric de Gournay, Pierre Weiss

Claire Boyer, Antonin Chambolle, Yohann de Castro, Vincent Duval, Frédéric de Gournay, Pierre Weiss, « On Representer Theorems and Convex Regularization  », SIAM Journal on Optimization, mai 2019, pp. 1260–1281. ISSN 1052-6234

Abstract

We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which 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 still an unresolved problem.

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