Conformal Flattening on the Probability Simplex and Its Applications to Voronoi Partitions and Centroids
Conformal Flattening on the Probability Simplex and Its Applications to Voronoi Partitions and Centroids
复制标题
概率单纯形的共形展平及其在 Voronoi 分区和质心中的应用
DOI：
10.1007/978-3-030-02520-5_4
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Atsumi Ohara
中科院分区：
文献类型：
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作者：
佐藤巌;瀬川悦生;松江要;Atsumi Ohara
一类信息几何结构可以共形变换为对偶平坦结构。本文从微分几何的角度研究了概率单纯形的变换，并给出了它的应用。通过在一定条件下限制仿射浸入，实现了概率单形为浸入的1-共形平坦统计流形。利用这一事实，我们为这类流形引入了共形平坦的概念，以得到相应的具有共形发散的对偶平坦统计(Hessian)流形，并给出了势函数和仿射坐标的显式形式。最后，我们展示了扁平化在非广延统计物理、Voronoi划分和加权质心关于共度散度的概率单纯形上的应用，这些散度不一定是Bregman型的。
A certain class of information geometric structure can be conformally transformed to dually flat one. This paper studies the transformation on the probability simplex from a viewpoint ofaffine differential geometryand provides its applications. By restricting affine immersions with certain conditions, the probability simplex is realized to be 1-conformally flat statistical manifolds immersed in. Using this fact, we introduce a concept ofconformal flatteningfor such manifolds in order to obtain the corresponding dually flat statistical (Hessian) ones with conformal divergences, and show explicit forms of potential functions and affine coordinates. Finally, we demonstrate applications of the flattening to nonextensive statistical physics, Voronoi partitions and weighted centroids on the probability simplex with respect togeometric divergences, which are not necessarily of Bregman type.