On the natural domain of Bregman operators

TL;DR

This paper redefines Bregman operators on their natural domain, improving theoretical understanding and applicability, especially for nonconvex functions, by adopting a domain-aware perspective that generalizes and simplifies existing results.

Contribution

It introduces a domain-aware framework for Bregman analysis, extending the theory to nonconvex functions and simplifying proofs and assumptions.

Findings

Generalizes Bregman operators to natural domains

Enables analysis of nonconvex functions like weakly convex ones

Simplifies theoretical results and proofs

Abstract

The Bregman proximal mapping and Bregman-Moreau envelope are traditionally studied for functions defined on the entire space $\mathbb{R}^n$, even though these constructions depend only on the values of the function within (the interior of) the domain of the distance-generating function (dgf). While this convention is largely harmless in the convex setting, it leads to substantial limitations in the nonconvex case, as it fails to embrace important classes of functions such as relatively weakly convex ones. In this work, we revisit foundational aspects of Bregman analysis by adopting a domain-aware perspective: we define functions on the natural domain induced by the dgf and impose properties only relative to this set. This framework not only generalizes existing results but also rectifies and simplifies their statements and proofs. Several examples illustrate both the necessity of our assumptions and the advantages of this refined approach.