The Burbea-Rao and Bhattacharyya centroids

TL;DR

This paper introduces a new framework for computing centroids based on Burbea-Rao divergences, generalizing Jensen-Shannon divergence, and applies it to efficiently find Bhattacharyya centroids for exponential family distributions, enhancing clustering methods.

Contribution

It establishes the properties of Burbea-Rao centroids, links Bhattacharyya distances to these divergences, and develops an efficient algorithm for centroid computation in statistical exponential families.

Findings

Burbea-Rao centroids are unique and approximable via iterative algorithms.

Bhattacharyya distances for exponential families are equivalent to Burbea-Rao divergences.

The proposed centroid algorithm improves clustering of Gaussian mixtures.

Abstract

We study the centroid with respect to the class of information-theoretic Burbea-Rao divergences that generalize the celebrated Jensen-Shannon divergence by measuring the non-negative Jensen difference induced by a strictly convex and differentiable function. Although those Burbea-Rao divergences are symmetric by construction, they are not metric since they fail to satisfy the triangle inequality. We first explain how a particular symmetrization of Bregman divergences called Jensen-Bregman distances yields exactly those Burbea-Rao divergences. We then proceed by defining skew Burbea-Rao divergences, and show that skew Burbea-Rao divergences amount in limit cases to compute Bregman divergences. We then prove that Burbea-Rao centroids are unique, and can be arbitrarily finely approximated by a generic iterative concave-convex optimization algorithm with guaranteed convergence property. In the second part of the paper, we consider the Bhattacharyya distance that is commonly used to measure overlapping degree of probability distributions. We show that Bhattacharyya distances on members of the same statistical exponential family amount to calculate a Burbea-Rao divergence in disguise. Thus we get an efficient algorithm for computing the Bhattacharyya centroid of a set of parametric distributions belonging to the same exponential families, improving over former specialized methods found in the literature that were limited to univariate or "diagonal" multivariate Gaussians. To illustrate the performance of our Bhattacharyya/Burbea-Rao centroid algorithm, we present experimental performance results for $k$-means and hierarchical clustering methods of Gaussian mixture models.