Diffusion models, which employ stochastic differential equations to sample images through integrals, have emerged as a dominant class of generative models. However, the rationality of the diffusion process itself receives limited attention, leaving the question of whether the problem is well-posed and well-conditioned. In this paper, we uncover a vexing propensity of diffusion models: they frequently exhibit the infinite Lipschitz near the zero point of timesteps. We provide theoretical proofs to illustrate the presence of infinite Lipschitz constants and empirical results to confirm it. The Lipschitz singularities pose a threat to the stability and accuracy during both the training and inference processes of diffusion models. Therefore, the mitigation of Lipschitz singularities holds great potential for enhancing the performance of diffusion models. To address this challenge, we propose a novel approach, dubbed E-TSDM, which alleviates the Lipschitz singularities of the diffusion model near the zero point. Remarkably, our technique yields a substantial improvement in performance. Moreover, as a byproduct of our method, we achieve a dramatic reduction in the Fréchet Inception Distance of acceleration methods relying on network Lipschitz, including DDIM and DPM-Solver, by over 33\%. Extensive experiments on diverse datasets validate our theory and method. Our work may advance the understanding of the general diffusion process, and also provide insights for the design of diffusion models.

Consistency models are a nascent family of generative models that can sample high quality data in one step without the need for adversarial training. Current consistency models achieve optimal sample quality by distilling from pre-trained diffusion models and employing learned metrics such as LPIPS. However, distillation limits the quality of consistency models to that of the pre-trained diffusion model, and LPIPS causes undesirable bias in evaluation. To tackle these challenges, we present improved techniques for consistency training, where consistency models learn directly from data without distillation. We delve into the theory behind consistency training and identify a previously overlooked flaw, which we address by eliminating Exponential Moving Average from the teacher consistency model. To replace learned metrics like LPIPS, we adopt Pseudo-Huber losses from robust statistics. Additionally, we introduce a lognormal noise schedule for the consistency training objective, and propose to double total discretization steps every set number of training iterations. Combined with better hyperparameter tuning, these modifications enable consistency models to achieve FID scores of 2.51 and 3.25 on CIFAR-10 and ImageNet $64\times 64$ respectively in a single sampling step. These scores mark a 3.5$\times$ and 4$\times$ improvement compared to prior consistency training approaches. Through two-step sampling, we further reduce FID scores to 2.24 and 2.77 on these two datasets, surpassing those obtained via distillation in both one-step and two-step settings, while narrowing the gap between consistency models and other state-of-the-art generative models.

We introduce Würstchen, a novel architecture for text-to-image synthesis that combines competitive performance with unprecedented cost-effectiveness for large-scale text-to-image diffusion models.A key contribution of our work is to develop a latent diffusion technique in which we learn a detailed but extremely compact semantic image representation used to guide the diffusion process. This highly compressed representation of an image provides much more detailed guidance compared to latent representations of language and this significantly reduces the computational requirements to achieve state-of-the-art results. Our approach also improves the quality of text-conditioned image generation based on our user preference study.The training requirements of our approach consists of 24,602 A100-GPU hours - compared to Stable Diffusion 2.1's 200,000 GPU hours. Our approach also requires less training data to achieve these results. Furthermore, our compact latent representations allows us to perform inference over twice as fast, slashing the usual costs and carbon footprint of a state-of-the-art (SOTA) diffusion model significantly, without compromising the end performance. In a broader comparison against SOTA models our approach is substantially more efficient and compares favourably in terms of image quality.We believe that this work motivates more emphasis on the prioritization of both performance and computational accessibility.