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A characterization of proximity operators

Rémi Gribonval 1, 2, * Mila Nikolova 3
* Corresponding author
1 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio
Inria Rennes – Bretagne Atlantique , IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE
2 DANTE - Dynamic Networks : Temporal and Structural Capture Approach
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme, IXXI - Institut Rhône-Alpin des systèmes complexes
Abstract : We characterize proximity operators, that is to say functions that map a vector to a solution of a penalized least squares optimization problem. Proximity operators of convex penalties have been widely studied and fully characterized by Moreau. They are also widely used in practice with nonconvex penalties such as the ℓ0 pseudo-norm, yet the extension of Moreau’s characterization to this setting seemed to be a missing element of the literature. We characterize proximity operators of (convex or nonconvex) penalties as functions that are the subdifferential of some convex potential. This is proved as a consequence of a more general characterization of so-called Bregman proximity operators of possibly nonconvex penalties in terms of certain convex potentials. As a side effect of our analysis, we obtain a test to verify whether a given function is the proximity operator of some penalty, or not. Many well-known shrinkage operators are indeed confirmed to be proximity operators. However, we prove that windowed Group-LASSO and persistent empirical Wiener shrinkage – two forms of so-called social sparsity shrinkage– are generally not the proximity operator of any penalty; the exception is when they are simply weighted versions of group-sparse shrinkage with non-overlapping groups.
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Submitted on : Saturday, February 15, 2020 - 12:31:51 PM
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Rémi Gribonval, Mila Nikolova. A characterization of proximity operators. Journal of Mathematical Imaging and Vision, Springer Verlag, 2020, 62, pp.773-789. ⟨10.1007/s10851-020-00951-y⟩. ⟨hal-01835101v5⟩

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