A Characterization of Proximity Operators

• Published in: Journal of mathematical imaging and vision Vol. 62; no. 6-7; pp. 773 - 789
• Main Authors: Gribonval, Rémi, Nikolova, Mila
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2020; Springer Nature B.V; Springer Verlag
• Subjects: Applications of Mathematics; Classical Analysis and ODEs; Computer Science; Empirical analysis; Fines & penalties; Functional Analysis; Image Processing and Computer Vision; Mathematical Methods in Physics; Mathematics; Operators (mathematics); Optimization; Proximity; Shrinkage; Signal and Image Processing; Signal,Image and Speech Processing; Group sparsity; Subdifferential; Proximity operator; Social sparsity; Convex regularization; Shrinkage operator; Nonconvex regularization; Subdiffer- ential
• ISSN: 0924-9907; 1573-7683
• DOI: 10.1007/s10851-020-00951-y

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 ℓ 0 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 the 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 a 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.