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.

翻译：Föllmer过程是一个布朗运动，其条件是在时间1具有预先指定的分布。该过程可以解释为去噪扩散概率模型（DDPM）反向随机微分方程（SDE）的一种“增广”时间压缩版本。虽然这一事实已被间接用于通过反向SDE的离散化来分析DDPM采样误差，但Föllmer过程的直接离散化与DDPM采样器之间的联系尚未被充分探索。本文旨在澄清这一要点，同时综述现有工作中的相关结果。我们证明，离散化的Föllmer过程为DDPM采样器提供了自然的超参数设置。此外，这使我们能够系统地恢复关于DDPM采样误差界的最新结果，并略有改进。