2010/04/30 by Frank Nielsen, Sylvain Boltz · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Statistical Methods and Models #Algorithm #Applied mathematics #Artificial intelligence #Bayesian Methods and Mixture Models #Bhattacharyya distance #Bregman divergence #Centroid #Combinatorics #Computer science #Divergence (linguistics) #Exponential family #Mahalanobis distance #Mathematics #Multivariate statistics #Probability density function #Statistical Mechanics and Entropy #Statistics #Unimodality #Univariate #cs.CG #cs.IT #math.IT
paper · pdf · doi:10.1109/tit.2011.2159046
published as IEEE Transactions on Information Theory 57(8):5455-5466, 2011 · 13 pages
openalex publication_date 2011/08/01 · arxiv created 2012/04/19 · arxiv updated 2015/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 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 fork-means and hierarchical clustering methods of Gaussian mixture models.