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  entropy related to the conformal flattening of the Fisher-Rao  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  flat divergences amount to dual Bregman Voronoi diagrams, and their dual complexes are regular triangulations. The primal Bregman Voronoi diagram is the Euclidean Voronoi diagram and the dual Bregman Voronoi diagram coincides with the Cauchy hyperbolic 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.
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.
A
' quadratic
A
curved
A
forward/reverse
R
Bregman Voronoi diagram is the Euclidean
Bregman-Tsallis Voronoi diagram corresponds to the hyperbolic
R
Bregman
Bregman-Tsallis
R
Cauchy hyperbolic
ordinary Euclidean