# Geometric Gaussian Approximations of Probability Distributions

arXiv: https://arxiv.org/abs/2507.00616 · HTML: https://mlsanity.com/paper/2507.00616/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 12 references that are in the map.
- First posted on arXiv 1 Jul 2025.

## Builds on (most cited first, 7 of 7)

- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Reparameterization invariance in approximate Bayesian inference](https://mlsanity.com/paper/2406.03334/) (2024)
- [Riemannian Laplace approximations for Bayesian neural networks](https://mlsanity.com/paper/2306.07158/) (2023)
- [Riemannian Laplace Approximation with the Fisher Metric](https://mlsanity.com/paper/2311.02766/) (2023)
- [Laplace Matching for fast Approximate Inference in Latent Gaussian Models](https://mlsanity.com/paper/2105.03109/) (2021)
- [Rethinking Approximate Gaussian Inference in Classification](https://mlsanity.com/paper/2502.03366/) (2025)
- [Likelihood approximations via Gaussian approximate inference](https://mlsanity.com/paper/2410.20754/) (2024)

## Built on by (most cited first, 2 of 2)

- [Staying on the Manifold: Geometry-Aware Noise Injection](https://mlsanity.com/paper/2509.20201/) (2025)
- [Tubular Riemannian Laplace Approximations for Bayesian Neural Networks](https://mlsanity.com/paper/2512.24381/) (2025)

## Abstract (arXiv)

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.