2020/06/12 by Frank Nielsen, Nielsen, Frank
Computer Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Computational Geometry (cs.CG) #FOS: Computer and information sciences #Information Theory (cs.IT) #Machine Learning (cs.LG) #Statistical Mechanics and Entropy #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2006.07020
openalex publication_date 2020/06/12 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We study the Voronoi diagrams of a finite set of Cauchy distributions and\ntheir dual complexes from the viewpoint of information geometry by considering\nthe Fisher-Rao distance, the Kullback-Leibler divergence, the chi square\ndivergence, and a flat divergence derived from Tsallis' quadratic entropy\nrelated to the conformal flattening of the Fisher-Rao curved geometry. We prove\nthat the Voronoi diagrams of the Fisher-Rao distance, the chi square\ndivergence, and the Kullback-Leibler divergences all coincide with a hyperbolic\nVoronoi diagram on the corresponding Cauchy location-scale parameters, and that\nthe dual Cauchy hyperbolic Delaunay complexes are Fisher orthogonal to the\nCauchy hyperbolic Voronoi diagrams. The dual Voronoi diagrams with respect to\nthe dual forward/reverse flat divergences amount to dual Bregman Voronoi\ndiagrams, and their dual complexes are regular triangulations. The primal\nBregman-Tsallis Voronoi diagram corresponds to the hyperbolic Voronoi diagram\nand the dual Bregman-Tsallis Voronoi diagram coincides with the ordinary\nEuclidean Voronoi diagram. Besides, we prove that the square root of the\nKullback-Leibler divergence between Cauchy distributions yields a metric\ndistance which is Hilbertian for the Cauchy scale families.\n