Bregman sided and symmetrized centroids

We generalize the notions of centroids and barycenters to the broad class of information-theoretic distortion measures called Bregman divergences. 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.