论文标题
在信息几何的cauchy歧管上
On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds
论文作者
论文摘要
我们通过考虑Fisher-Rao距离，Kullback-Leibler Divergence，Chi Square Divergence，Chi Square Divergence以及tsallis flat Divergent fistry fisher fisher fishers fishers fishers fishers fishers fishers fisher-raner-contrand-culane fistrand-culane vorne fistrane，我们从信息几何学的角度研究了一组有限的库奇分布及其双重复合物的Voronoi图及其双重复合物的Voronoi图。 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.相对于双向前/反向平坦差异的双伏罗尼亚图等于双Bregman Voronoi图，其双重复合物是常规的三角形。原始的Bregman-Tsallis voronoi图对应于双曲线伏罗尼亚图和双Bregman-Tsallis voronoi图与普通的欧几里得Voronoi图相吻合。此外，我们证明了库奇分布之间的kullback-leibler差异的平方根产生的度量距离，这是凯奇秤家庭的希尔伯特式。
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.
