The denoising diffusion model has recently emerged as a powerful generative technique that converts noise into data. While there are many studies providing theoretical guarantees for diffusion processes based on discretized stochastic differential equation (D-SDE), many generative samplers in real applications directly employ a discrete-time (DT) diffusion process. However, there are very few studies analyzing these DT processes, e.g., convergence for DT diffusion processes has been obtained only for distributions with bounded support. In this paper, we establish the convergence guarantee for substantially larger classes of distributions under DT diffusion processes and further improve the convergence rate for distributions with bounded support. In particular, we first establish the convergence rates for both smooth and general (possibly non-smooth) distributions having a finite second moment. We then specialize our results to a number of interesting classes of distributions with explicit parameter dependencies, including distributions with Lipschitz scores, Gaussian mixture distributions, and any distributions with early-stopping. We further propose a novel accelerated sampler and show that it improves the convergence rates of the corresponding regular sampler by orders of magnitude with respect to all system parameters. Our study features a novel analytical technique that constructs a tilting factor representation of the convergence error and exploits Tweedie's formula for handling Taylor expansion power terms.

翻译：去噪扩散模型作为一种强大的生成技术，已成功实现从噪声到数据的转换。尽管已有大量研究为基于离散化随机微分方程（D-SDE）的扩散过程提供了理论保证，但在实际应用中，许多生成采样器直接采用离散时间（DT）扩散过程。然而，目前对这类DT过程的分析研究仍非常有限，例如针对DT扩散过程的收敛性结果仅在有界支撑分布上得到证明。本文针对DT扩散过程，为更广泛的分布类建立了收敛性保证，并进一步改进了有界支撑分布的收敛速率。具体而言，我们首先建立了具有有限二阶矩的光滑分布及一般（可能非光滑）分布的收敛速率。随后，我们将结果具体应用于若干具有显式参数依赖性的重要分布类，包括具有Lipschitz评分的分布、高斯混合分布以及采用早停策略的任意分布。此外，我们提出了一种新颖的加速采样器，并证明其在所有系统参数维度上，相较于常规采样器能够实现数量级的收敛速率提升。本研究采用了一种创新的分析技术：通过构建收敛误差的倾斜因子表示，并利用Tweedie公式处理泰勒展开幂次项。