Using Ornstein-Uhlenbeck Process to understand Denoising Diffusion Probabilistic Model and its Noise Schedules
This provides a theoretical foundation for noise schedules in diffusion models, benefiting researchers in generative AI.
The paper shows that Denoising Diffusion Probabilistic Models (DDPM) can be represented as an Ornstein-Uhlenbeck process, linking noise schedule design to observation times, and demonstrates that a Fisher-Information-motivated schedule corresponds to the state-of-the-art cosine schedule.
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.