Information measures and geometry of the hyperbolic exponential families of Poincaré and hyperboloid distributions

TL;DR

This work develops a comprehensive information‑theoretic and geometric treatment of two hyperbolic distribution families: Poincaré distributions on the upper‑half plane and hyperboloid distributions on the Minkowski space. It shows that f‑divergences between distributions in each family can be expressed via a small set of canonical invariants, yielding symmetric behavior on appropriate foliations and enabling closed‑form KL, entropy, and Bhattacharyya quantities. A parallel hyperboloid theory is developed, including a universality result for hyperboloid mixtures as density estimators and a correspondence principle linking the two models in two dimensions. The paper also contributes practical tools: explicit Fisher information geometries, sampling schemes, and two Monte Carlo methods for numerically estimating divergences, facilitating clustering and inference on hyperbolic data. Overall, the results provide a principled, computable framework for statistical modeling and inference in hyperbolic spaces with potential impact on hierarchical data analysis and hyperbolic learning.

Abstract

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.

Figures (6)

• Figure 1: Statistical modeling in a hyperbolic model: A hierarchical structure (left) is embedded in a hyperbolic model with low-distortion (middle). The point data set is then modeled by a probability distribution in the hyperbolic model (right).
• Figure 2: Plotting the probability density function $p_\theta(x)$ of the Poincaré distribution on the Poincaré upper-half plane indexed by a $2\times 2$ positive-definite matrix $\theta$ (plotted for $x\in [-2,2]\times (0,2]\subset\mathbb{H}$).
• Figure 3: Correspondence between calculating the Kullback-Leibler divergence between parametric densities and the corresponding parameter divergence on the cone parameter space
• Figure 4: The Poincaré distributions $p_{\theta}$ and $p_{\theta^{\prime}}$ with their corresponding distributions $p_{g.\theta}$and $p_{g.\theta^{\prime}}$ obtained by the action of $g\in\mathrm{SL}(2,\mathbb{R})$. The Kullback-Leibler divergence is preserved: $D_\mathrm{KL}[p_{g.\theta}:p_{g.\theta^{\prime}}]=D_\mathrm{KL}[p_{\theta}:p_{\theta^{\prime}}]$
• Figure 5: The family of hyperboloid distributions has sample space the hyperboloid upper sheet and natural conic parameter space.

Theorems & Definitions (33)

• Definition 1: Maximal invariant of a group action Eaton-1989
• Proposition 1
• Proposition 2: Maximal invariant
• proof
• Theorem 1: Canonical terms of the $f$-divergences between Poincaré distributions
• Remark 1: Foliation
• Proposition 3
• proof
• Example 1
• Proposition 4