Diffusion models have achieved huge empirical success in data generation tasks. Recently, some efforts have been made to adapt the framework of diffusion models to discrete state space, providing a more natural approach for modeling intrinsically discrete data, such as language and graphs. This is achieved by formulating both the forward noising process and the corresponding reversed process as Continuous Time Markov Chains (CTMCs). In this paper, we investigate the theoretical properties of the discrete diffusion model. Specifically, we introduce an algorithm leveraging the uniformization of continuous Markov chains, implementing transitions on random time points. Under reasonable assumptions on the learning of the discrete score function, we derive Total Variation distance and KL divergence guarantees for sampling from any distribution on a hypercube. Our results align with state-of-the-art achievements for diffusion models in $\mathbb{R}^d$ and further underscore the advantages of discrete diffusion models in comparison to the $\mathbb{R}^d$ setting.

翻译：扩散模型在数据生成任务中取得了巨大的实证成功。近年来，学界致力于将扩散模型框架调整至离散状态空间，为建模语言和图形等本质离散数据提供了更自然的方法。这通过将前向加噪过程和相应的反向过程均表述为连续时间马尔可夫链（CTMC）来实现。本文探究离散扩散模型的理论性质。具体而言，我们引入一种利用连续马尔可夫链均匀化的算法，在随机时间点上执行转移。在关于离散评分函数学习的合理假设下，我们推导出从超立方体上任意分布进行采样的总变差距离和KL散度保证。我们的结果与$\mathbb{R}^d$中扩散模型的最新成果一致，并进一步凸显离散扩散模型相较于$\mathbb{R}^d$设置的优势。