Learning Mixtures of Gaussians Using Diffusion Models

• URL: http://arxiv.org/abs/2404.18869v1
• Date: Mon, 29 Apr 2024 17:00:20 GMT
• Title: Learning Mixtures of Gaussians Using Diffusion Models
• Authors: Khashayar Gatmiry, Jonathan Kelner, Holden Lee,
• Abstract summary: We give a new algorithm for learning mixtures of $k$ Gaussians to TV error. Our approach is analytic and relies on the framework of diffusion models.
• Score: 9.118706387430883
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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. 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. 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.

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