Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models

TL;DR

This paper provides a theoretical analysis of score-based diffusion models, showing how they can adapt to unknown low-dimensional structures in data, with improved error bounds and coefficient design.

Contribution

It introduces a new theoretical framework demonstrating that DDPM can adapt to unknown low-dimensional structures, emphasizing the importance of coefficient design.

Findings

Error dependency on ambient dimension is unavoidable in general.

A specific coefficient design achieves near-optimal convergence rate.

First theoretical demonstration of adaptation to low-dimensional structures in DDPM.

Abstract

This paper investigates score-based diffusion models when the underlying target distribution is concentrated on or near low-dimensional manifolds within the higher-dimensional space in which they formally reside, a common characteristic of natural image distributions. Despite previous efforts to understand the data generation process of diffusion models, existing theoretical support remains highly suboptimal in the presence of low-dimensional structure, which we strengthen in this paper. For the popular Denoising Diffusion Probabilistic Model (DDPM), we find that the dependency of the error incurred within each denoising step on the ambient dimension $d$ is in general unavoidable. We further identify a unique design of coefficients that yields a converges rate at the order of $O(k^{2}/\sqrt{T})$ (up to log factors), where $k$ is the intrinsic dimension of the target distribution and $T$ is the number of steps. This represents the first theoretical demonstration that the DDPM sampler can adapt to unknown low-dimensional structures in the target distribution, highlighting the critical importance of coefficient design. All of this is achieved by a novel set of analysis tools that characterize the algorithmic dynamics in a more deterministic manner.