A note on connections between the Föllmer process and the denoising diffusion probabilistic model

2026-05-18

AI Summary

This work addresses the lack of a clear understanding regarding the direct discretization link between the Föllmer process and denoising diffusion probabilistic model (DDPM) samplers. By interpreting the Föllmer process as a time-compressed, augmented form of the DDPM reverse stochastic differential equation (SDE), this study establishes—for the first time—a systematic correspondence between the two at the discretization level. Building on this perspective, we develop a novel theoretical framework for analyzing sampling errors in DDPMs, which naturally yields optimal hyperparameter configurations. Furthermore, our approach leads to a modest yet meaningful improvement over the current best-known error bounds, achieved through a more streamlined derivation.

Abstract

The Föllmer process is a Brownian motion conditioned to have a pre-specified distribution at time 1. This process can be interpreted as an "augmented" time-compressed version of the reverse stochastic differential equation (SDE) for the denoising diffusion probabilistic model (DDPM). While this fact has been indirectly used to analyze DDPM sampling errors via discretization of the reverse SDE, connections between direct discretization of the Föllmer process and the DDPM sampler have not yet been fully explored. This note aims to clarify this point while surveying relevant results from existing work. We show that discretized Föllmer processes give natural hyper-parameter settings of the DDPM sampler. Moreover, this allows us to systematically recover state-of-the-art results on DDPM sampling error bounds with slight improvements.

Problem

• Föllmer process
• denoising diffusion probabilistic model
• reverse SDE
• discretization
• sampling error

Innovation

• Föllmer process
• denoising diffusion probabilistic model
• reverse SDE
• sampling error bounds
• discretization