In recent years, the knowledge surrounding diffusion models(DMs) has grown significantly, though several theoretical gaps remain. Particularly noteworthy is prior error, defined as the discrepancy between the termination distribution of the forward process and the initial distribution of the reverse process. To address these deficiencies, this paper explores the deeper relationship between optimal transport(OT) theory and DMs with discrete initial distribution. Specifically, we demonstrate that the two stages of DMs fundamentally involve computing time-dependent OT. However, unavoidable prior error result in deviation during the reverse process under quadratic transport cost. By proving that as the diffusion termination time increases, the probability flow exponentially converges to the gradient of the solution to the classical Monge-Amp\`ere equation, we establish a vital link between these fields. Therefore, static OT emerges as the most intrinsic single-step method for bridging this theoretical potential gap. Additionally, we apply these insights to accelerate sampling in both unconditional and conditional generation scenarios. Experimental results across multiple image datasets validate the effectiveness of our approach.

翻译：近年来，关于扩散模型的知识体系已显著扩展，但仍存在若干理论空白。尤其值得关注的是先验误差，其定义为前向过程终止分布与反向过程初始分布之间的差异。为弥补这些不足，本文深入探讨了最优传输理论与离散初始分布扩散模型之间的内在联系。具体而言，我们证明扩散模型的两个阶段本质上都涉及计算时间依赖的最优传输。然而，在二次传输代价下，不可避免的先验误差会导致反向过程产生偏差。通过证明当扩散终止时间增大时，概率流会指数收敛于经典蒙日-安培方程解的梯度，我们在两个领域间建立了关键联系。因此，静态最优传输成为弥合这一理论潜在鸿沟的最本质单步方法。此外，我们将这些见解应用于无条件生成和条件生成场景中的采样加速。在多个图像数据集上的实验结果验证了我们方法的有效性。