发表机构
Yale University; University of Texas at Austin（耶鲁大学; 德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对迭代整流流的计算与统计保证问题，提出c-整流流方法，证明其在合适假设下可收敛到最优传输耦合，并建立相关收敛保证与最优估计速率，为整流流提供理论支撑。
AI 中文摘要
近期，整流流（rectified flow）已成为大规模图像生成的基础框架，支撑着FLUX.1和Stable Diffusion 3等达到顶尖水平的系统。尽管其在经验上取得显著成功，迭代整流流的计算与统计保证仍在很大程度上未被探索。我们通过研究c-整流流（一种感知成本的整流流类别，其将速度场投影到梯度类中同时保留端点边际分布）解决该问题。普通整流流可能无法恢复最优传输耦合：在高斯案例研究中，当且仅当源协方差矩阵与目标协方差矩阵可交换时，迭代才会收敛到最优耦合。相比之下，在合适的紧性和一致可积性假设下，迭代c-整流流始终收敛到最优传输耦合。我们进一步在投影稳定性假设下，针对二次和强凸位移成本建立了定量的一步收缩与指数收敛保证。最后，在Hölder球假设下，我们提出了新的极小极大最优评分估计率，并表明当与迭代c-整流流结合时，它们会产生维度d≥3时的最优速率最优估计量，以及d=1、2时的近参数速率的最优传输估计量。
英文摘要
Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying \textit{c}-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. In contrast, under suitable compactness and uniform-integrability assumptions, iterative \textit{c}-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a Hölder ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative \textit{c}-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension \(d \ge 3\) and a nearly parametric rate for \(d=1,2\).
