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Journal Articles SIAM Journal on Optimization Year : 2019

On Representer Theorems and Convex Regularization

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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Dates and versions

hal-01823135 , version 1 (25-06-2018)
hal-01823135 , version 2 (16-07-2018)
hal-01823135 , version 3 (26-11-2018)

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Cite

Claire Boyer, Antonin Chambolle, Yohann de Castro, Vincent Duval, Frédéric de Gournay, et al.. On Representer Theorems and Convex Regularization. SIAM Journal on Optimization, 2019, 29 (2), pp.1260-1281. ⟨10.1137/18M1200750⟩. ⟨hal-01823135v3⟩
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