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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https://hal.archives-ouvertes.fr/hal-01823135
Contributor : Claire Boyer <>
Submitted on : Monday, November 26, 2018 - 3:46:21 PM
Last modification on : Friday, October 11, 2019 - 8:22:43 PM
Long-term archiving on : Wednesday, February 27, 2019 - 12:31:16 PM

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

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