Reconsidering Noise for Denoising Diffusion Probabilistic Models

Denoising Diffusion Probabilistic Models (DDPMs) typically utilize white Gaussian noise in their processes. In this paper, we explore several theoretical aspects of noise in DDPMs. We derive a necessary condition for the input of the forward diffusion process to match the denoised output, as well as a sufficient condition for when they differ. Our findings show that minimizing the Mean Square Error (MSE) between the actual and predicted noise in a DDPM is more effective with colored Gaussian noise than with white Gaussian noise, and that non-Gaussian noise offers further improvements in MSE minimization. Additionally, we demonstrate that the probability of error between the input and denoised output in a DDPM is reduced when using colored Gaussian noise compared to white Gaussian noise. Furthermore, we show that a DDPM trained with white Gaussian noise can effectively denoise processes involving any zero-mean symmetric distribution noise. Theoretical results are validated through experiments using the Hugging Face Hub 1000 butterfly pictures dataset and the LSUN Church-256 dataset, with experimental outcomes confirming our theoretical findings.