The Bregman chord divergence

TL;DR

This paper introduces Bregman chord divergences, a new class of distances that generalize Bregman divergences, are easy to tune with two parameters, and do not require gradient calculations, enhancing their practical applicability.

Contribution

The paper proposes Bregman chord divergences, extending Bregman divergences with a simpler, parameterized class that avoids gradient computations and generalizes asymptotically.

Findings

Bregman chord divergences do not require gradient calculations.

They are easily tunable with two scalar parameters.

They generalize asymptotically Bregman divergences.

Abstract

Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of testing one by one the entries of an ever-expanding dictionary of {\em ad hoc} distances, one rather prefers to consider parametric classes of distances that are exhaustively characterized by axioms derived from first principles. Bregman divergences are such a class. However fine-tuning a Bregman divergence is delicate since it requires to smoothly adjust a functional generator. In this work, we propose an extension of Bregman divergences called the Bregman chord divergences. This new class of distances does not require gradient calculations, uses two scalar parameters that can be easily tailored in applications, and generalizes asymptotically Bregman divergences.