普通Rectified flow可能不收敛到最优传输，c-Rectified flow能保证收敛，d=1,2还有近参数速率。做生成模型可以读。
该论文研究c-Rectified flow，这是对纠正流引入成本项的一类变体，FLUX.1和Stable Diffusion 3背后的生成框架正是纠正流。高斯案例显示，普通纠正流只有在源和目标协方差矩阵可交换时才收敛到最优传输耦合；而迭代c-Rectified flow在紧性和一致可积条件下总能收敛到最优传输耦合。作者还针对二次和强凸位移成本建立了投影稳定假设下的单步收缩与指数收敛保证。在Hölder球假设下，他们得到极小极大最优的分数估计速率，用于迭代c-Rectified flow后可在d≥3获得速率最优的最优传输估计，d=1,2时接近参数速率。
Computational and Statistical Guarantees of the \textit{c}-Rectified flow
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 \(d=1,2\).