Information measures and geometry of the hyperbolic exponential families of Poincar\'e and hyperboloid distributions

We study various information-theoretic measures and the information geometry of the Poincar\'e distributions and the related hyperboloid distributions, and prove that their statistical mixture models are universal density estimators of smooth densities in hyperbolic spaces. The Poincar\'e and the hyperboloid distributions are two types of hyperbolic probability distributions defined using different models of hyperbolic geometry. Namely, the Poincar\'e distributions form a triparametric bivariate exponential family whose sample space is the hyperbolic Poincar\'e 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 Poincar\'e 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 Poincar\'e 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 Poincar\'e 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.