@article{2009-BregmanCentroids-TIT
, author={Frank Nielsen and Richard Nock}
, title={Sided and Symmetrized Bregman Centroids}
, journal={IEEE Transactions on Information Theory}
, month={June}
, year={2009}
, volume={55}
, number={6}
, pages={2048-2059}
, doi={10.1109/TIT.2009.2018176}
, abstract={
  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 form a rich and versatile family of distances that unifies quadratic
 Euclidean distances with various well-known statistical entropic measures. Since
 besides the squared Euclidean distance, Bregman divergences are asymmetric, we consider
 the left-sided and right-sided centroids and the symmetrized centroids as minimizers
 of average Bregman distortions. We prove that all three centroids are unique and
 give closed-form solutions for the sided centroids that are generalized means. Furthermore,
 we design a provably fast and efficient arbitrary close approximation algorithm
 for the symmetrized centroid based on its exact geometric characterization. The
 geometric approximation algorithm requires only to walk on a geodesic linking the
 two left/right-sided centroids. We report on our implementation for computing entropic
 centers of image histogram clusters and entropic centers of multivariate normal
 distributions that are useful operations for processing multimedia information and
 retrieval. These experiments illustrate that our generic methods compare favorably
 with former limited ad hoc methods.
  }
}