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.

中文翻译：

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关于信息几何柯西流形上的 Voronoi 图和对偶 Delaunay 复形

我们通过考虑 Fisher-Rao 距离、Kullback-Leibler 散度、卡方散度和从 Tsallis 二次导出的平坦散度，从信息几何学的角度研究了一组有限 Cauchy 分布及其对偶复形的 Voronoi 图与 Fisher-Rao 弯曲几何的共形展平相关的熵。我们证明了 Fisher-Rao 距离、卡方散度和 Kullback-Leibler 散度的 Voronoi 图都与对应的 Cauchy 位置尺度参数上的双曲 Voronoi 图重合，并且对偶 Cauchy 双曲 Delaunay 复形是 Fisher正交于柯西双曲 Voronoi 图。关于双正向/反向平坦发散的双 Voronoi 图相当于双 Bregman Voronoi 图，它们的对偶复合体是规则三角剖分。原始 Bregman-Tsallis Voronoi 图对应于双曲 Voronoi 图，对偶 Bregman-Tsallis Voronoi 图与普通欧几里得 Voronoi 图重合。此外，我们证明了柯西分布之间的 Kullback-Leibler 散度的平方根产生了一个度量距离，它是柯西尺度家族的希尔伯特距离。