Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choices

2026-06-22

AI Summary

This work investigates whether the adaptability of diffusion models to low-dimensional data structures during sampling depends on specific update coefficients. By analyzing convergence in total variation (TV) distance and leveraging tools from high-dimensional probability and diffusion process theory, the study establishes for the first time that such low-dimensional adaptability is an intrinsic and robust property of diffusion models, independent of the choice of sampler or ambient dimensionality. Theoretically, the analysis shows that only $\tilde{O}(k/\varepsilon)$ iterations are required to generate samples within $\varepsilon$ TV distance of the target distribution, substantially broadening the class of samplers with dimension-free acceleration guarantees. These results provide a unified theoretical foundation for the widely adopted diffusion-based sampling methods in practice.

Abstract

Diffusion models are known to exploit unknown low-dimensional structure to accelerate sampling. However, existing convergence theory under low-dimensional data structure has largely focused on update rules with narrowly prescribed coefficient choices. This raises a fundamental question: is adaptation to low-dimensional structure sensitive to the precise choice of update coefficients? In this paper, we show that such adaptation is a robust property of diffusion models. For a broad class of update coefficients, we prove that $\widetilde{O}(k/\varepsilon)$ iterations suffice to generate an $\varepsilon$-accurate sample in total variation (TV) distance, independently of the ambient dimension. Our framework substantially broadens the class of diffusion samplers known to enjoy low dimensional adaptation and applies to several commonly used methods in practice. These results provide a theoretical justification for the empirical effectiveness of diffusion samplers across different coefficient choices when applied to structured, high-dimensional data.

Problem

• diffusion models
• low-dimensional structure
• update coefficients
• convergence theory
• sampling

Innovation

• diffusion models
• low-dimensional structure
• coefficient robustness
• sampling complexity
• total variation distance