On a problem of Bauschke and Borwein

TL;DR

This paper investigates the convexity properties of Bregman distances induced by convex functions and their spectral counterparts, providing a negative answer to an open problem about joint convexity equivalence.

Contribution

It demonstrates that the joint convexity of Bregman distances for a convex function does not necessarily imply the same for its spectral function counterpart.

Findings

Counterexample to joint convexity equivalence

Clarification of spectral Bregman distance properties

Addresses an open problem in convex analysis

Abstract

Consider a differentiable convex function $f: \mathbb{R}^n \supset \mathrm{dom} f \rightarrow \mathbb{R}.$ The induced spectral function $F$ is given by $F=f \circ \lambda,$ where $\lambda: \mathbf{M}_n^{sa} \rightarrow \mathbb{R}^{n}$ is the eigenvalue map. Let us denote by $D_f$ and $D_F$ the Bregman distances associated with $f$ and $F,$ respectively. In the paper "Joint and separate convexity of the Bregman distance" written by H. Bauschke and J. Borwein the following open problem has been suggested. "Is $D_f$ jointly convex if and only if $D_F$ is?" In this short note we provide a negative answer to this question.