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.

本研究解决了离散扩散模型在误差分析方面的不足，提出了基于Lévy型随机积分的综合框架。通过广义的泊松随机测度，严谨地建立了离散扩散模型的随机积分形式，并首次为$\tau$-跳跃方案提供了KL散度的误差界限，从而为离散扩散模型的数学性质提供了新见解，并支持高效准确算法的设计。

离散和连续扩散的结合：通过随机积分框架对离散扩散模型的全面分析