On the Robustness of Distribution Support under Diffusion Guidance

TL;DR

This paper establishes a theoretical foundation for diffusion guidance, showing it nearly always generates samples close to the target support, explaining its empirical success in producing high-quality, structurally plausible samples.

Contribution

It introduces a robustness of support property for guided diffusion processes, providing a theoretical explanation for their effectiveness and high-quality sample generation.

Findings

Guided diffusion processes almost always generate samples near the target support.

The robustness property applies to both DDIM and DDPM models.

The analysis covers various discretization schemes induced by exponential integrators.

Abstract

Diffusion guidance is a powerful technique that enables controllable and high-fidelity sample generation with diffusion models. At a high level, it modifies the score function by incorporating a guidance term that steers the generative process toward a desired condition. Despite its empirical success, the theoretical properties of diffusion guidance remain largely unexplored, and it is not well understood why it consistently produces high-quality samples. In this work, we explain the effectiveness of diffusion guidance by establishing a \emph{robustness of support} property. Specifically, we show that, given exact access to the score functions, guided diffusion processes almost always generate samples that remain close to the target support. This property is particularly desirable, as samples that lie off the support are often structurally implausible and may adversely affect downstream tasks. Our analysis covers both Denoising Diffusion Implicit Models (DDIM) and Denoising Diffusion Probabilistic Models (DDPM), and applies to a wide range of discretization schemes induced by exponential integrators. Our results provide a rigorous foundation for understanding why diffusion guidance produces physically meaningful and structurally plausible samples.