The Dual Voronoi Diagrams with Respect to Representational Bregman Divergences

We present a generalization of Bregman Voronoi diagrams induced by a Bregman divergence acting on a representation function.Bregman divergences are canonical distortion measures of flat spaces induced by strictly convex and differentiable functions, called Bregman generators.Considering a representation function further allows us to conveniently embed the not necessarily flat source space into a dually flat space for which the dual Voronoi diagrams can be derived from an equivalent power affine diagram. We explain the fundamental dualities induced by the pair of Legendre convex conjugates coupled with a pair of conjugate representations.In particular, we show that Amari´s celebrated family of ¿-divergences and Eguchi and Copas´s ß-divergences are two cases of representational Bregman divergences that are often considered in information geometry. We report closed-form formula for their centroids and describe their dual Voronoi diagrams on the induced statistical manifolds.