Discrete diffusion models have gained increasing attention for their ability to model complex distributions with tractable sampling and inference. However, the error analysis for discrete diffusion models remains less well-understood. In this work, we propose a comprehensive framework for the error analysis of discrete diffusion models based on L\'evy-type stochastic integrals. By generalizing the Poisson random measure to that with a time-independent and state-dependent intensity, we rigorously establish a stochastic integral formulation of discrete diffusion models and provide the corresponding change of measure theorems that are intriguingly analogous to It\^o integrals and Girsanov's theorem for their continuous counterparts. Our framework unifies and strengthens the current theoretical results on discrete diffusion models and obtains the first error bound for the $\tau$-leaping scheme in KL divergence. With error sources clearly identified, our analysis gives new insight into the mathematical properties of discrete diffusion models and offers guidance for the design of efficient and accurate algorithms for real-world discrete diffusion model applications.

翻译：离散扩散模型因其能够对复杂分布进行建模并实现可处理的采样与推断而日益受到关注。然而，离散扩散模型的误差分析仍缺乏深入理解。本文提出了一种基于Lévy型随机积分的离散扩散模型误差分析综合框架。通过将泊松随机测度推广至具有时间无关且状态依赖强度的测度，我们严格建立了离散扩散模型的随机积分表述，并给出了相应的测度变换定理，这些定理与连续扩散模型中的Itô积分和Girsanov定理具有引人注目的相似性。我们的框架统一并强化了当前关于离散扩散模型的理论结果，并首次在KL散度下获得了$\tau$-leaping方案的误差界。通过清晰识别误差来源，我们的分析为理解离散扩散模型的数学特性提供了新的视角，并为现实世界离散扩散模型应用中高效、准确算法的设计提供了指导。