Computational and Statistical Guarantees of the \textit{c}-Rectified flow
The paper says ordinary rectified flow can miss optimal transport, while its cost-aware variant converges under stated assumptions.
Wang, Xu, Liu, and Zhou study `c`-rectified flow, which projects velocity fields onto a gradient class while keeping endpoint marginals intact. In a Gaussian case, they show ordinary rectified flow reaches the optimal coupling only when source and target covariance matrices commute. They prove iterative `c`-rectified flow converges to the optimal transport coupling under compactness and uniform-integrability assumptions, with quantitative contraction and exponential convergence results under projection stability. They also give minimax-optimal score estimation rates under a Holder ball assumption and connect them to rate-optimal optimal-transport estimation in dimension `d >= 3`.
ArXiv · AI/CL/LG's note
Wang, Xu, Liu, and Zhou study `c`-rectified flow, which projects velocity fields onto a gradient class while keeping endpoint marginals intact. In a Gaussian case, they show ordinary rectified flow reaches the optimal coupling only when source and target covariance matrices commute. They prove iterative `c`-rectified flow converges to the optimal transport coupling under compactness and uniform-integrability assumptions, with quantitative contraction and exponential convergence results under projection stability. They also give minimax-optimal score estimation rates under a Holder ball assumption and connect them to rate-optimal optimal-transport estimation in dimension `d >= 3`.
ArXiv · AI/CL/LG's note
score 4