The Bregman proximal average

TL;DR

This paper introduces a Bregman proximal average based on a 1-coercive Legendre function, showing how Bregman envelopes and proximal mappings of the average relate to those of individual functions, using variational analysis techniques.

Contribution

It defines a novel Bregman proximal average and characterizes its envelopes and mappings in terms of individual functions, advancing the theoretical framework of Bregman proximal methods.

Findings

Bregman envelope of the average is a convex combination of individual envelopes.

Bregman proximal mapping of the average is a convex combination of convexified mappings.

Uses variational analysis techniques to establish properties of the Bregman proximal average.

Abstract

We provide a proximal average with repect to a $1$-coercive Legendre function. In the sense of Bregman distance, the Bregman envelope of the proximal average is a convex combination of Bregman envelopes of individual functions. The Bregman proximal mapping of the average is a convex combination of convexified proximal mappings of individual functions. Techniques from variational analysis provide the keys for the Bregman proximal average.