Tackling the Singularities at the Endpoints of Time Intervals in Diffusion Models

Most diffusion models assume that the reverse process adheres to a Gaussian distribution. However, this approximation has not been rigorously validated, especially at singularities, where $t=0$ and $t=1$. Improperly dealing with such singularities leads to an average brightness issue in applications, and limits the generation of images with extreme brightness or darkness. We primarily focus on tackling singularities from both theoretical and practical perspectives. Initially, we establish the error bounds for the reverse process approximation, and showcase its Gaussian characteristics at singularity time steps. Based on this theoretical insight, we confirm the singularity at $t=1$ is conditionally removable while it at $t=0$ is an inherent property. Upon these significant conclusions, we propose a novel plug-and-play module to address the initial singular time step sampling, which not only effectively resolves the average brightness issue for a wide range of diffusion models without extra training efforts, but also enhances their generation capability in achieving notable lower FID scores.