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.
The symmetrized Kullback-Leibler of multivariate normal distributions. We describe the geodesic-walk for this particular mixed-type exponential family of multivariate normals, and explain the Legendre mixed-type vector matrix dual convex conjugates defining the corresponding Bregman divergences. This yields a simple, fast and elegant geometric method compared to the former overly complex method of Myrvoll and Soong @cite_9 that relies on solving Riccati matrix equations.
@cite_9
143794185
A locking cone chassis and several use methods to eliminate double and triple handling of shipping freight containers. The chassis device comprises two parallel I-beam main rails, a plurality of transverse ribs, a forward cone receiver formed from steel square tubing affixed to a forward end of the main rails, and a similar rear cone receiver affixed to the rear end of the main rails, the only essential difference between the receivers being that the forward cone receiver has an upper flange to guide and to stop a freight container. Each receiver has two box-shaped end portions, which operate like the corner pockets of a freight container-that is, each box-shaped end portion receives a locking cone through an upper surface cone receiving aperture. Each end portion also has access apertures for manually unlocking a cone.