Finite Mixture Modeling with Riemannian Gaussian Distributions on Hyperbolic Space

2026-04-27

AI Summary

This work addresses the lack of effective likelihood-based density modeling methods in hyperbolic space by introducing, for the first time, a finite mixture model of isotropic Riemannian Gaussian distributions on the hyperboloid. We derive the explicit density function and normalization constant, establishing a complete theoretical framework that includes the existence and uniqueness of weighted single-component estimators, an analysis of mixture likelihood singularities, and the existence of constrained parameter estimators. To facilitate practical inference, we propose both an exact EM algorithm and a generalized EM algorithm based on truncated hyperbolic majorization–minimization, both guaranteeing monotonic convergence. Experiments demonstrate that the proposed approach accurately recovers mixture components, effectively supports model selection, and achieves substantially improved computational efficiency with the generalized EM variant, enabling successful non-Euclidean exploratory clustering on real-world network embedding data.

Abstract

Hyperbolic space is increasingly used for hierarchical, tree-like, and network-structured data, but likelihood-based density modeling on hyperbolic space remains relatively limited. This paper develops finite mixture modeling with isotropic Riemannian Gaussian distributions on hyperbolic space under the hyperboloid model. We derive the density, radial normalizing constant, and a finite-sum representation involving the complementary error function. We then formulate weighted maximum likelihood estimation, which is the fundamental subproblem in mixture fitting: the location estimator is the weighted Fréchet mean, while the inverse-scale estimator is obtained from a one-dimensional strictly convex profile problem. For finite mixtures, we derive exact EM and generalized EM algorithms. The generalized version replaces exact barycenter solves with truncated hyperbolic majorization-minimization updates. We establish existence and uniqueness of the weighted single-component estimator, singularity of the unrestricted mixture likelihood, existence of a constrained mixture estimator, and monotonicity properties of the EM-type algorithms. Simulations show accurate weighted estimation, reliable mixture recovery, effective model selection, and substantial computational savings from generalized EM. Real network examples based on hyperbolic embeddings illustrate the method as an exploratory likelihood-based clustering tool for non-Euclidean data.

Problem

• hyperbolic space
• finite mixture modeling
• density estimation
• non-Euclidean data
• likelihood-based clustering

Innovation

• Riemannian Gaussian distribution
• hyperbolic space
• finite mixture model
• weighted Fréchet mean
• generalized EM algorithm