Diffusion models have achieved great success in generating high-dimensional samples across various applications. While the theoretical guarantees for continuous-state diffusion models have been extensively studied, the convergence analysis of the discrete-state counterparts remains under-explored. In this paper, we study the theoretical aspects of score-based discrete diffusion models under the Continuous Time Markov Chain (CTMC) framework. We introduce a discrete-time sampling algorithm in the general state space $[S]^d$ that utilizes score estimators at predefined time points. We derive convergence bounds for the Kullback-Leibler (KL) divergence and total variation (TV) distance between the generated sample distribution and the data distribution, considering both scenarios with and without early stopping under specific assumptions. Notably, our KL divergence bounds are nearly linear in dimension $d$, aligning with state-of-the-art results for diffusion models. Our convergence analysis employs a Girsanov-based method and establishes key properties of the discrete score function, which are essential for characterizing the discrete-time sampling process.

翻译：扩散模型在生成高维样本方面取得了巨大成功，并在各种应用中表现出色。尽管连续状态扩散模型的理论保证已得到广泛研究，但其离散状态对应模型的收敛分析仍探索不足。本文在连续时间马尔可夫链框架下，研究了基于分数的离散扩散模型的理论特性。我们针对一般状态空间$[S]^d$，提出了一种离散时间采样算法，该算法利用预定义时间点的分数估计器。在特定假设下，我们分别推导了生成样本分布与数据分布之间库尔贝克-莱布勒散度和总变差距离的收敛界，涵盖了包含早停机制与不包含早停机制的两种情形。值得注意的是，我们的KL散度界在维度$d$上近乎线性，这与扩散模型的最先进成果保持一致。我们的收敛分析采用了基于Girsanov变换的方法，并建立了离散分数函数的关键性质，这些性质对于刻画离散时间采样过程至关重要。