Information Geometry: Duality, Convexity and Divergences
published: Dec. 5, 2008, recorded: November 2008, views: 1516
Report a problem or upload filesIf you have found a problem with this lecture or would like to send us extra material, articles, exercises, etc., please use our ticket system to describe your request and upload the data.
Enter your e-mail into the 'Cc' field, and we will keep you updated with your request's status.
In this talk, I explore the mathematical relationships between duality in information geometry, convex analysis, and divergence functions. First, from the fundamental inequality of a convex function, a family of divergence measures can be constructed, which specializes to the familiar Bregman divergence, Jenson difference, beta-divergence, and alpha-divergence, etc.
Second, the mixture parameter turns out to correspond to the alpha <-> -alpha duality in information geometry (which I call "referential duality", since it is related to the choice of a reference point for computing divergence).
Third, convex conjugate operation induces another kind of duality in information geometry, namely, that of biorthogonal coordinates and their transformation (which I call "representational duality", since it is related to the expression of geometric quantities, such as metric, affine connection, curvature, etc of the underlying manifold). Under this analysis, what is traditionally called "+1/-1 duality" and "e/m duality" in information geometry reflect two very different meanings of duality that are nevertheless intimately interwined for dually flat spaces.
Link this pageWould you like to put a link to this lecture on your homepage?
Go ahead! Copy the HTML snippet !