A characterization of proximity operators

TL;DR

This paper provides a comprehensive characterization of proximity operators, including nonconvex penalties, by linking them to convex potentials and offering a test to identify such operators, with implications for various shrinkage methods.

Contribution

It extends the characterization of proximity operators to nonconvex penalties and introduces a test to verify if a function is a proximity operator.

Findings

Many known shrinkage operators are proximity operators.

Windowed Group-LASSO and Wiener shrinkage are generally not proximity operators.

Weighted non-overlapping group-sparse shrinkage can be proximity operators.

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 {\ell} 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 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.