Diffusion models have demonstrated remarkable performance in generating high-dimensional samples across domains such as vision, language, and the sciences. Although continuous-state diffusion models have been extensively studied both empirically and theoretically, discrete-state diffusion models, essential for applications involving text, sequences, and combinatorial structures, they remain significantly less understood from a theoretical standpoint. In particular, all existing analyses of discrete-state models assume access to an empirical risk minimizer. In this work, we present a principled theoretical framework analyzing diffusion models, providing a state-of-the-art sample complexity bound of $\widetilde{\mathcal{O}}(\epsilon^{-4})$. Our structured decomposition of the score estimation error into statistical and optimization components offers critical insights into how diffusion models can be trained efficiently. This analysis addresses a fundamental gap in the literature and establishes the theoretical tractability and practical relevance of diffusion models.

翻译：扩散模型在生成高维样本方面展现出卓越性能，其应用涵盖视觉、语言及科学计算等多个领域。尽管连续状态扩散模型已在实证与理论层面得到广泛研究，但针对文本、序列及组合结构等应用至关重要的离散状态扩散模型，其理论理解仍存在显著不足。特别值得注意的是，现有所有离散状态模型分析均假设可访问经验风险最小化器。本研究提出一个理论严谨的分析框架用于研究扩散模型，实现了$\widetilde{\mathcal{O}}(\epsilon^{-4})$的先进样本复杂度边界。通过将分数估计误差结构化分解为统计误差与优化误差，本工作揭示了扩散模型高效训练的内在机制。该分析弥补了现有文献的理论空白，确立了扩散模型的理论可处理性与实践相关性。