Computational Information Geometry Wonderland

In my paper at ISVD 2009 on

The dual Voronoi diagrams with respect to representational Bregman divergences

slides

I show that by using a representational function, we can obtain alpha-divergences as a representational Bregman divergences. Therefore, it is easy to extend algorithms tailored to Bregman divergences to alpha divergences.

Here is a code snippet in Java: RepresentationalAlphaBregman.java

Running it, you get something like

Shows that alpha divergences can be obtained from representational Bregman divergences
alpha-div=0.9008838766115398    equals Bregman rep. div=0.9008838766115399
alpha-div=0.14730849849416264   equals Bregman rep. div=0.14730849849416267
alpha-div=0.05455969651963932   equals Bregman rep. div=0.054559696519639225
alpha-div=1.2439567444853374    equals Bregman rep. div=1.2439567444853372
alpha-div=0.15345391125768915   equals Bregman rep. div=0.15345391125768917
alpha-div=0.12118392973570616   equals Bregman rep. div=0.12118392973570632
alpha-div=1.038494366079179 equals Bregman rep. div=1.0384943660791794
alpha-div=0.08541142546071197   equals Bregman rep. div=0.08541142546071195
alpha-div=0.06842729068201092   equals Bregman rep. div=0.06842729068201084
alpha-div=0.6941500904965174    equals Bregman rep. div=0.6941500904965173

In the ISVD 2009 paper, we give closed-form solutions for centroids of representational Bregman divergences, including alpha-means et beta-means.

Frank.