Learning Mixtures of Gaussians Using Diffusion Models

TL;DR

This paper introduces a new analytic diffusion model-based algorithm for efficiently learning mixtures of Gaussians with theoretical guarantees, extending to complex distributions on low-dimensional manifolds.

Contribution

It provides the first end-to-end theoretical guarantees for diffusion models in learning Gaussian mixtures with quasi-polynomial time and sample complexity.

Findings

Achieves quasi-polynomial time and sample complexity for Gaussian mixture learning.

Extends to continuous mixtures supported on unions of balls or low-dimensional manifolds.

Provides theoretical guarantees for diffusion models in nontrivial distribution learning.

Abstract

We give a new algorithm for learning mixtures of $k$ Gaussians (with identity covariance in $\mathbb{R}^n$) to TV error $\varepsilon$, with quasi-polynomial ($O(n^{\text{poly\,log}\left(\frac{n+k}{\varepsilon}\right)})$) time and sample complexity, under a minimum weight assumption. Our results extend to continuous mixtures of Gaussians where the mixing distribution is supported on a union of $k$ balls of constant radius. In particular, this applies to the case of Gaussian convolutions of distributions on low-dimensional manifolds, or more generally sets with small covering number, for which no sub-exponential algorithm was previously known. Unlike previous approaches, most of which are algebraic in nature, our approach is analytic and relies on the framework of diffusion models. Diffusion models are a modern paradigm for generative modeling, which typically rely on learning the score function (gradient log-pdf) along a process transforming a pure noise distribution, in our case a Gaussian, to the data distribution. Despite their dazzling performance in tasks such as image generation, there are few end-to-end theoretical guarantees that they can efficiently learn nontrivial families of distributions; we give some of the first such guarantees. We proceed by deriving higher-order Gaussian noise sensitivity bounds for the score functions for a Gaussian mixture to show that that they can be inductively learned using piecewise polynomial regression (up to poly-logarithmic degree), and combine this with known convergence results for diffusion models.