Nielsen Frank
Sony Computer Science Laboratories, Tokyo 141-0022, Japan.
Entropy (Basel). 2020 Jun 28;22(7):713. doi: 10.3390/e22070713.
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. In addition, 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.
我们从信息几何的角度，通过考虑费希尔 - 拉奥距离、库尔贝克 - 莱布勒散度、卡方散度以及与费希尔 - 拉奥几何的共形平坦化相关的源自Tsallis熵的平坦散度，研究了一组有限柯西分布的Voronoi图及其对偶复形。我们证明，费希尔 - 拉奥距离、卡方散度和库尔贝克 - 莱布勒散度的Voronoi图在相应的柯西位置 - 尺度参数上都与双曲Voronoi图一致，并且对偶柯西双曲德劳内复形与柯西双曲Voronoi图是费希尔正交的。关于对偶平坦散度的对偶Voronoi图相当于对偶布雷格曼Voronoi图，并且它们的对偶复形是正则三角剖分。原始布雷格曼Voronoi图是欧几里得Voronoi图，对偶布雷格曼Voronoi图与柯西双曲Voronoi图一致。此外，我们证明柯西分布之间库尔贝克 - 莱布勒散度的值的平方根产生了一种度量距离，对于柯西尺度族来说它是希尔伯特型的。
