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.

翻译：我们提出了一种新算法，用于在最小质量假设下学习具有单位协方差（在$\mathbb{R}^n$中）的$k$个高斯混合分布，使其达到TV误差$\varepsilon$，并且具有拟多项式（$O(n^{\text{poly log}\left(\frac{n+k}{\varepsilon}\right)})$）时间和样本复杂度。与以往大多基于代数的方法不同，我们的方法是解析的，并依赖于扩散模型框架。扩散模型是一种现代生成建模范式，通常通过学习沿一个将纯噪声分布（此处为高斯分布）转化为数据分布的过程中的得分函数（梯度对数概率密度）来实现。尽管扩散模型在图像生成等任务中表现出色，但其能否高效学习非平凡分布族仍缺乏端到端的理论保证；我们首次提供了此类保证。我们推导了高斯混合得分函数的高阶噪声敏感性界，表明可以通过分段多项式回归（达到多对数阶）对其进行归纳学习，并结合已知的扩散模型收敛结果。我们的结果可推广到连续高斯混合分布，其中混合分布支撑在$k个$恒定半径球体的并集上。特别地，这适用于低维流形上分布的高斯卷积，或更一般的具有小覆盖数的集合。