Maps on positive definite matrices preserving Bregman and Jensen divergences

TL;DR

This paper characterizes bijective transformations on positive definite matrices that preserve specific Bregman and Jensen divergences, providing a detailed structure of these transformations for key divergence functions.

Contribution

It offers a complete description of structure-preserving maps for Bregman and Jensen divergences on positive definite matrices, extending understanding of their geometric and algebraic properties.

Findings

Identifies the structure of bijective maps preserving Bregman divergences.

Provides similar characterizations for Jensen divergence preservers.

Covers the most important Bregman divergences with explicit results.

Abstract

In this paper we determine those bijective maps of the set of all positive definite $n\times n$ complex matrices which preserve a given Bregman divergence corresponding to a differentiable convex function that satisfies certain conditions. We cover the cases of the most important Bregman divergences and present the precise structure of the mentioned transformations. Similar results concerning Jensen divergences and their preservers are also given.