Re-examination of Bregman functions and new properties of their divergences

TL;DR

This paper re-examines Bregman functions and divergences, introduces new properties and functions, and broadens the understanding of their theoretical foundations and applications across various fields.

Contribution

It provides new sufficient conditions for constructing Bregman functions, introduces novel Bregman functions like negative iterated log entropy, and reinterprets known functions within an expanded theoretical framework.

Findings

Introduction of the concept of relative uniform convexity

New properties of uniformly and strongly convex functions

Reclassification of the negative Burg entropy as a Bregman function

Abstract

The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the "Bregman function"). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, operations research, information theory, nonlinear analysis, machine learning and more. This paper re-examines various aspects related to the theory of Bregman functions and divergences. In particular, it presents many sufficient conditions which allow the construction of Bregman functions in a general setting and introduces new Bregman functions (such as a negative iterated log entropy). Moreover, it sheds new light on several known Bregman functions such as quadratic entropies, the negative Havrda-Charv\'at-Tsallis entropy, and the negative Boltzmann-Gibbs-Shannon entropy, and it shows that the negative Burg entropy, which is not a Bregman function according to the classical theory but nevertheless is known to have "Bregmanian properties", can, by our re-examination of the theory, be considered as a Bregman function. Our analysis yields several by-products of independent interest such as the introduction of the concept of relative uniform convexity (a certain generalization of uniform convexity), new properties of uniformly and strongly convex functions, and results in Banach space theory.