AI 中文总结
该研究提出基于熵产生率的生成模型误差分析统一框架，将终端KL散度分解为三类误差，为Euler-Maruyama采样器得出$\boldsymbol{\textit{O}}(h^2)$收敛速率，统一三类生成模型分析并经实验验证。
AI 中文摘要
我们引入了一个基于前向-反向扩散过程对的熵产生率的生成模型误差分析的统一框架。对于一对连续性方程流，该速率具有闭合的速度形式恒等式，其时间积分将终端Kullback-Leibler（KL）散度分解为初始化误差、得分近似误差和时间离散化误差之和。通过在边际分布层面而非路径空间分析熵产生，我们的框架为Euler-Maruyama采样器得出了$\boldsymbol{\textit{O}}(h^2)$的尖锐收敛速率，其中$h$为步长，这优于通常从Girsanov路径空间分析得到的$\boldsymbol{\textit{O}}(h)$速率。此外，我们的框架通过在单个不等式内改变扩散系数，统一了基于得分的SDE、概率流ODE和随机插值的分析，揭示了确定性与随机采样之间的权衡。数值实验证实了预测的步长和终端时间的缩放关系。
英文摘要
We introduce a unified framework for the error analysis of generative models based on the entropy production rate of the forward-reverse diffusion process pair. For a pair of continuity equation flows, the rate admits a closed velocity form identity whose time integral decomposes the terminal Kullback--Leibler (KL) divergence into the sum of an initialization error, a score approximation error, and a time-discretization error. By analyzing the entropy production at the level of marginal distributions, rather than in path space, our framework yields a sharp convergence rate of $\mathcal{O}(h^2)$ for the Euler-Maruyama sampler, where $h$ is the step size. This improves upon the $\mathcal{O}(h)$ rates typically obtained from Girsanov's path-space analyses. Furthermore, our framework unifies the analysis of score-based SDEs, probability-flow ODEs, and stochastic interpolants by varying diffusion coefficients within a single inequality, revealing the trade-off between deterministic and stochastic sampling. Numerical experiments confirm the predicted scaling with step size and terminal time.
Comments37 pages, 15 figures
