Bhattacharyya Clustering with Applications to Mixture Simplifications

Bhattacharrya distance (BD) is a widely used distance in statistics to compare probability density functions (PDFs). It has shown strong statistical properties (in terms of Bayes error) and it relates to Fisher information. It has also practical advantages, since it strongly relates on measuring the overlap of the supports of the PDFs. Unfortunately, even with common parametric models on PDFs, few closed-form formulas are known. Moreover, the BD centroid estimation was limited to univariate gaussian PDFs in the literature and no convergence guarantees were provided. In this paper, we propose a closed-form formula for BD on a general class of parametric distributions named exponential families. We show that the BD is a Burbea-Rao divergence for the log normalizer of the exponential family. We propose an efficient iterative scheme to compute a BD centroid on exponential families. Finally, these results allow us to define a Bhattacharrya hierarchical clustering algorithms (BHC). It can be viewed as a generalization of k-means on BD. Results on image segmentation shows the stability of the method.