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

TL;DR

This paper explores the geometric properties of Voronoi diagrams and Delaunay complexes for Cauchy distributions within the framework of information geometry, revealing their hyperbolic and Euclidean structures and their relationships with divergence measures.

Contribution

It establishes the coincidence of certain Voronoi diagrams with hyperbolic diagrams and characterizes dual complexes as regular triangulations, linking divergence measures to geometric structures.

Findings

Voronoi diagrams of Fisher-Rao, chi square, and Kullback-Leibler divergences coincide with hyperbolic Voronoi diagrams.

Dual Cauchy hyperbolic Delaunay complexes are Fisher orthogonal to the Voronoi diagrams.

Square root of Kullback-Leibler divergence is a Hilbertian metric for Cauchy scale families.

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.