论文标题
庞加莱和倍曲面分布的双曲线指数家族的信息度量和几何形状
Information measures and geometry of the hyperbolic exponential families of Poincaré and hyperboloid distributions
论文作者
论文摘要
我们研究了各种信息理论措施以及庞加莱分布的信息几何形状和相关的倍增分布，并证明它们的统计混合模型是双曲线空间中光滑密度的通用密度估计器。 Poincaré和倍增分布是使用不同模型的双曲几何形状定义的两种类型的双曲线概率分布。也就是说，庞加莱分布形成了一个三位化双变量指数家族，其样品空间是poincaré上半平面，自然参数空间是开放的3D凸锥，该凸锥具有两乘两个正定原始矩阵。双曲线分布的家族形成了另一个指数族家族，该家族具有样品空间的两片单元倍增双曲线几何形状的前向纸。在第一部分中，我们证明，庞加莱分布之间的所有$ f $ diverences都可以使用伊顿的最大群体不变性框架使用三个规范术语来表达。我们还表明，任何两个庞加莱分布之间的$ f $ diverence是不对称的，除非这些分布属于参数空间的特定叶子的同一叶子。我们报告了Fisher信息矩阵，香农的差分熵和Kullback-Leibler Divergence的封闭式公式。使用指数家庭的框架，这种分布之间的Bhattacharyya距离。在第二部分中，我们通过突出庞加莱和倍虫分布之间的参数对应关系来指出双粘性分布的指数家族的相应结果。最后，我们描述了一个随机生成器来绘制变体并提出两种蒙特卡洛方法，以随机估计双曲线分布之间的数值$ f $ diverences。
We study various information-theoretic measures and the information geometry of the Poincaré distributions and the related hyperboloid distributions, and prove that their statistical mixture models are universal density estimators of smooth densities in hyperbolic spaces. The Poincaré and the hyperboloid distributions are two types of hyperbolic probability distributions defined using different models of hyperbolic geometry. Namely, the Poincaré distributions form a triparametric bivariate exponential family whose sample space is the hyperbolic Poincaré upper-half plane and natural parameter space is the open 3D convex cone of two-by-two positive-definite matrices. The family of hyperboloid distributions form another exponential family which has sample space the forward sheet of the two-sheeted unit hyperboloid modeling hyperbolic geometry. In the first part, we prove that all $f$-divergences between Poincaré distributions can be expressed using three canonical terms using Eaton's framework of maximal group invariance. We also show that the $f$-divergences between any two Poincaré distributions are asymmetric except when those distributions belong to a same leaf of a particular foliation of the parameter space. We report closed-form formula for the Fisher information matrix, the Shannon's differential entropy and the Kullback-Leibler divergence. and Bhattacharyya distances between such distributions using the framework of exponential families. In the second part, we state the corresponding results for the exponential family of hyperboloid distributions by highlighting a parameter correspondence between the Poincaré and the hyperboloid distributions. Finally, we describe a random generator to draw variates and present two Monte Carlo methods to stochastically estimate numerically $f$-divergences between hyperbolic distributions.
