A note on connections between the F\"ollmer process and the denoising diffusion probabilistic model

TL;DR

This paper explores the connection between the F"ollmer process and denoising diffusion probabilistic models (DDPM), revealing how discretized F"ollmer processes inform DDPM sampling and error analysis.

Contribution

It clarifies the link between F"ollmer processes and DDPM, providing new hyper-parameter insights and improved error bounds for DDPM sampling.

Findings

Discretized F"ollmer processes suggest optimal DDPM hyper-parameters.

Systematic recovery of state-of-the-art DDPM sampling error bounds.

Slight improvements over existing error bounds.

Abstract

The F\"ollmer 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 F\"ollmer 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 F\"ollmer 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.