分析扩散模型的收敛性，提出用海森矩阵加速采样。
Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Acceleration
- 对比不同离散化方法对收敛的影响。
- 利用海森矩阵信息实现更快的收敛速度。
- 适合研究生成模型理论与优化算法的学者。
基于得分的扩散模型在生成建模中表现强大，但其理论基础仍不完善。本文聚焦于得分扩散模型的Wasserstein收敛性分析，考察欧拉离散化、指数积分器及中点随机化等不同离散方案的影响。定量比较表明这些方法显著影响收敛行为。此外，当局部海森矩阵信息可用时，提出一种基于局部线性化的加速采样器。结果表明，该方法可达到$ ilde{ ext{O}}ig( rac{1}{\varepsilon}ig)$的收敛率，显著优于标准扩散模型的$ ilde{ ext{O}}ig( rac{1}{\varepsilon^2}ig)$，其中$\\varepsilon$为期望精度。
原文摘要 · Abstract (English)
Score-based diffusion models have emerged as powerful tools in generative modeling, yet their theoretical foundations remain underexplored. In this work, we focus on the Wasserstein convergence analysis of score-based diffusion models. Specifically, we investigate the impact of various discretization schemes, including Euler discretization, exponential integrators, and midpoint randomization methods. Our analysis provides a quantitative comparison of these discrete approximations, emphasizing their influence on convergence behavior. Furthermore, we explore scenarios where Hessian information is available and propose an accelerated sampler based on the local linearization method. We demonstrate that this Hessian-based approach achieves faster convergence rates of order $\widetilde{\mathcal{O}}\left(\frac{1}{\varepsilon}\right)$ significantly improving upon the standard rate $\widetilde{\mathcal{O}}\left(\frac{1}{\varepsilon^2}\right)$ of vanilla diffusion models, where $\varepsilon$ denotes the target accuracy.
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