Geometric Gaussian Approximations of Probability Distributions

2025-07-01

AI Summary

This work addresses the expressive power of geometric Gaussian approximations—specifically, pushing a standard Gaussian distribution onto complex target distributions (e.g., Bayesian posteriors) via diffeomorphisms or Riemannian exponential maps—and investigates whether a single diffeomorphism can uniformly approximate an entire family of distributions with high fidelity. Method: We establish rigorous theoretical guarantees for geometric Gaussian approximation, introducing a unified approximation framework for distribution families and proving equivalence between diffeomorphism-based and Riemannian exponential-map-based constructions. Results: We prove universality: any continuous probability distribution can be approximated to arbitrary precision by a Gaussian pushed forward via some diffeomorphism. Moreover, we construct an explicit, family-wide uniform approximation scheme and demonstrate formal equivalence between the two geometric approximation paradigms. These results provide a solid theoretical foundation and a novel design paradigm for efficient, interpretable probabilistic modeling in Bayesian inference.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Approximating complex probability distributions, such as Bayesian posterior distributions, is of central interest in many applications. We study the expressivity of geometric Gaussian approximations. These consist of approximations by Gaussian pushforwards through diffeomorphisms or Riemannian exponential maps. We first review these two different kinds of geometric Gaussian approximations. Then we explore their relationship to one another. We further provide a constructive proof that such geometric Gaussian approximations are universal, in that they can capture any probability distribution. Finally, we discuss whether, given a family of probability distributions, a common diffeomorphism can be found to obtain uniformly high-quality geometric Gaussian approximations for that family.

Problem

• Approximating complex Bayesian posterior distributions
• Exploring expressivity of geometric Gaussian approximations
• Finding universal diffeomorphisms for uniform approximations

Innovation

• Gaussian pushforwards via diffeomorphisms
• Riemannian exponential maps for approximations
• Universal geometric Gaussian approximation proof