Diffusion models have shown remarkable empirical success in sampling from rich multi-modal distributions. Their inference relies on numerically solving a certain differential equation. This differential equation cannot be solved in closed form, and its resolution via discretization typically requires many small iterations to produce \emph{high-quality} samples. More precisely, prior works have shown that the iteration complexity of discretization methods for diffusion models scales polynomially in the ambient dimension and the inverse accuracy $1/\varepsilon$. In this work, we propose a new solver for diffusion models relying on a subtle interplay between low-degree approximation and the collocation method (Lee, Song, Vempala 2018), and we prove that its iteration complexity scales \emph{polylogarithmically} in $1/\varepsilon$, yielding the first ``high-accuracy'' guarantee for a diffusion-based sampler that only uses (approximate) access to the scores of the data distribution. In addition, our bound does not depend explicitly on the ambient dimension; more precisely, the dimension affects the complexity of our solver through the \emph{effective radius} of the support of the target distribution only.

翻译：扩散模型在从丰富的多模态分布中采样方面展现出卓越的实证效果。其推断依赖于对特定微分方程的数值求解。该微分方程无法以闭式解形式求解，通过离散化方法求解通常需要大量微步迭代才能生成*高质量*样本。更准确地说，已有研究表明扩散模型离散化方法的迭代复杂度随环境维度和逆精度 $1/\varepsilon$ 呈多项式增长。本工作提出一种基于低阶逼近与配置法（Lee, Song, Vempala 2018）精妙相互作用的新型扩散模型求解器，并证明其迭代复杂度在 $1/\varepsilon$ 上呈*多对数*增长，这为仅使用（近似）数据分布分数访问的扩散采样器提供了首个“高精度”理论保证。此外，所得界限不显式依赖于环境维度；更准确地说，维度仅通过目标分布支撑集的*有效半径*影响求解器的复杂度。