  In this paper, we generalize the notions of centroids and barycenters to the
broad class of information-theoretic distortion measures called Bregman
divergences. Bregman divergences are versatile, and unify quadratic geometric
distances with various statistical entropic measures. Because Bregman
divergences are typically asymmetric, we consider both the left-sided and
right-sided centroids and the symmetrized centroids, and prove that all three
are unique. We give closed-form solutions for the sided centroids that are
generalized means, and design a provably fast and efficient approximation
algorithm for the symmetrized centroid based on its exact geometric
characterization that requires solely to walk on the geodesic linking the two
sided centroids. We report on our generic implementation for computing entropic
centers of image clusters and entropic centers of multivariate normals, and
compare our results with former ad-hoc methods.
