做扩散模型理论或采样加速的读者值得关注——这篇用 Föllmer 过程替代传统反向 OU 过程，给出了更紧的 Wasserstein 误差界，对理解 DDPM 的采样质量与步数关系有直接帮助。
该论文从 Föllmer 过程视角重新分析了去噪扩散概率模型（DDPM）的采样误差，在 2-Wasserstein 距离下给出了更紧的误差上界。主要贡献包括：在 Lipschitz 型分数函数假设下，对余弦调度等方差调度方案建立了维度与步数最优的误差界；证明了这些条件蕴含对数 Sobolev 不等式，从而将 KL 散度的最优误差界转化为 Wasserstein 界；对于一般对数凹目标分布，即使不满足二次运输成本不等式，也能达到最优 Wasserstein 误差界。该工作统一并改进了现有文献中的多个误差界结果。
Wasserstein bounds for denoising diffusion probabilistic models via the Föllmer process
This paper studies sampling error bounds for denoising diffusion probabilistic models (DDPMs) in the 2-Wasserstein distance. Our contributions are threefold. (i) Under general Lipschitz-type conditions on the score function and for a broad class of variance schedules, including the cosine schedule, we establish sharp upper bounds that are optimal in both the dimension and the number of steps, and recover several sharp error bounds previously obtained in the literature. (ii) We prove that the same Lipschitz-type conditions, which encompass those commonly imposed on the (learned) score, imply a logarithmic Sobolev inequality and hence a quadratic transportation cost inequality for the DDPM. As a consequence, in settings covered by existing work, an optimal Wasserstein bound, up to a logarithmic factor, follows from the recently obtained sharp error bound in the Kullback-Leibler divergence under geometric-type variance schedules. (iii) We show that for general log-concave target distributions, the optimal Wasserstein error bound remains attainable even without a quadratic transportation cost inequality for the target. Our analysis is based on viewing the DDPM sampler as a discretization of the Föllmer process rather than the conventional reverse Ornstein-Uhlenbeck process.