标题： 使用奥恩斯坦-乌伦贝克过程理解去噪扩散概率模型及其噪声调度 显示英文标题

标题： Using Ornstein-Uhlenbeck Process to understand Denoising Diffusion Probabilistic Model and its Noise Schedules

摘要： 本文的目的是展示去噪扩散概率模型DDPM，一个非齐次离散时间马尔可夫过程，可以表示为在非均匀采样的离散时间点上观察到的齐次连续时间马尔可夫过程。令人惊讶的是，这个连续时间马尔可夫过程就是众所周知且广泛研究的奥恩斯坦-乌伦贝克（OU）过程，该过程是在1930年代为研究谐波势中的布朗粒子而提出的。我们使用其解析解建立了DDPM与OU过程之间的正式等价性。我们进一步证明了非齐次DDPM的噪声调度设计问题等价于为OU过程设计观测时间。我们基于自方差和费舍尔信息等原理性量提出了几种观测时间的启发式设计，并将它们与DDPM的临时噪声调度联系起来。有趣的是，我们展示了基于费舍尔信息的调度正好对应于没有理论基础但目前最先进的余弦调度。 显示英文摘要

摘要： The aim of this short note is to show that Denoising Diffusion Probabilistic Model DDPM, a non-homogeneous discrete-time Markov process, can be represented by a time-homogeneous continuous-time Markov process observed at non-uniformly sampled discrete times. Surprisingly, this continuous-time Markov process is the well-known and well-studied Ornstein-Ohlenbeck (OU) process, which was developed in 1930's for studying Brownian particles in Harmonic potentials. We establish the formal equivalence between DDPM and the OU process using its analytical solution. We further demonstrate that the design problem of the noise scheduler for non-homogeneous DDPM is equivalent to designing observation times for the OU process. We present several heuristic designs for observation times based on principled quantities such as auto-variance and Fisher Information and connect them to ad hoc noise schedules for DDPM. Interestingly, we show that the Fisher-Information-motivated schedule corresponds exactly the cosine schedule, which was developed without any theoretical foundation but is the current state-of-the-art noise schedule.