A note on the approximate symmetry of Bregman distances

TL;DR

This paper investigates conditions under which Bregman distances exhibit approximate symmetry, providing bounds and demonstrating this property for p-powers of lp and Lp norms when 1 < p < ancy{infinity}.

Contribution

It establishes sufficient conditions for the approximate symmetry of Bregman distances and proves this symmetry for specific norm powers.

Findings

Bregman distances can be bounded against each other when arguments are switched.

The approximate symmetry holds for p-powers of lp and Lp norms for 1 < p < ancy{infinity}.

Abstract

The Bregman distance $B_{\xi_x}(y,x)$, $\xi_x \in \partial J(y),$ associated to a convex sub-differentiable functional $J$ is known to be in general non-symmetric in its arguments $x$, $y$. In this note we address the question when Bregman distances can be bounded against each other when the arguments are switched, i.e., if some constant $C>0$ exists such that for all $x,y$ on a convex set $M$ it holds that $\frac{1}{C} B_{\xi_x}(y,x) \leq B_{\xi_y}(x,y) \leq C B_{\xi_x}(y,x).$ We state sufficient conditions for such an inequality and prove in particular that it holds for the $p$-powers of the $\ell_p$ and $L^p$-norms when $1 < p <\infty$.