Computational Information Geometry Wonderland

In the paper The dual Voronoi diagrams with respect to representational Bregman divergences (ISVD 2009), International Symposium on Voronoi Diagrams, we show that Beta divergences is a (representational) Bregman divergence with

• Beta=1 -> Kullback-Leibler
• Beta=0 -> Itakura-Saito
• Beta=2 -> squared Euclidean distance

In the paper, we derive formula for the beta left and right-sided centroids. The program RepresentationalBetaBregman.java shows this equivalence (up to numerical errors).

Shows that beta divergences can be obtained from representational Bregman divergences
beta-div=0.0060607537896566754  equals Bregman rep. div=0.006060753789656717
beta-div=0.03596836005227946    equals Bregman rep. div=0.03596836005227946
beta-div=0.03786639961675385    equals Bregman rep. div=0.03786639961675385
beta-div=0.015356733711556145   equals Bregman rep. div=0.015356733711556173
beta-div=0.16665973512136045    equals Bregman rep. div=0.16665973512136045
beta-div=0.006143185064308276   equals Bregman rep. div=0.006143185064308207
beta-div=0.012777128199946086   equals Bregman rep. div=0.012777128199946072
beta-div=2.42453303134018E-4    equals Bregman rep. div=2.4245330313402494E-4
beta-div=0.07962156613977964    equals Bregman rep. div=0.07962156613977961
beta-div=2.6549301732092453E-4  equals Bregman rep. div=2.6549301732092974E-4
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Frank.