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 method SingDiffusion 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. Code and models are released at https://github.com/PangzeCheung/SingDiffusion.

翻译：大多数扩散模型假设反向过程服从高斯分布。然而，这一近似尚未得到严格验证，尤其是在t=0和t=1的奇异点处。对这些奇异点的不当处理会导致应用中的平均亮度问题，并限制极端亮度或暗度图像的生成。本文主要从理论和实践两个角度解决奇异点问题。首先，我们建立了反向过程近似的误差界，并展示了其在奇异时间步长上的高斯特性。基于这一理论洞见，我们确认t=1处的奇异性是条件可移除的，而t=0处的奇异性则是固有问题。基于这些重要结论，我们提出了一种新颖的即插即用方法SingDiffusion，用于处理初始奇异时间步长的采样。该方法不仅无需额外训练即可有效解决多种扩散模型的平均亮度问题，还能提升其生成能力，获得显著更低的FID分数。代码和模型已发布在https://github.com/PangzeCheung/SingDiffusion。