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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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Contributor : Claire Boyer <>
Submitted on : Monday, November 26, 2018 - 3:46:21 PM
Last modification on : Thursday, January 28, 2021 - 10:28:02 AM
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 Journal on Optimization, Society for Industrial and Applied Mathematics, 2019, 29 (2), pp.1260-1281. ⟨10.1137/18M1200750⟩. ⟨hal-01823135v3⟩



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