On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds

Frank Nielsen

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arXiv · 2006.07020

On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds

Frank Nielsen

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Abstract

We study the Voronoi diagrams of a finite set of Cauchy distributions and their dual complexes from the viewpoint of information geometry by considering the Fisher-Rao distance, the Kullback-Leibler divergence, the chi square divergence, and a flat divergence derived from Tsallis' quadratic entropy related to the conformal flattening of the Fisher-Rao curved geometry. We prove that the Voronoi diagrams of the Fisher-Rao distance, the chi square divergence, and the Kullback-Leibler divergences all coincide with a hyperbolic Voronoi diagram on the corresponding Cauchy location-scale parameters, and that the dual Cauchy hyperbolic Delaunay complexes are Fisher orthogonal to the Cauchy hyperbolic Voronoi diagrams. The dual Voronoi diagrams with respect to the dual forward/reverse flat divergences amount to dual Bregman Voronoi diagrams, and their dual complexes are regular triangulations. The primal Bregman-Tsallis Voronoi diagram corresponds to the hyperbolic Voronoi diagram and the dual Bregman-Tsallis Voronoi diagram coincides with the ordinary Euclidean Voronoi diagram. Besides, we prove that the square root of the Kullback-Leibler divergence between Cauchy distributions yields a metric distance which is Hilbertian for the Cauchy scale families.

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Frank Nielsen. 2020-06-12. On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds. https://arxiv.org/abs/2006.07020

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