Computational and Statistical Guarantees of the \textit{c}-Rectified flow

2026-08-03

AI Summary

This work addresses the lack of theoretical guarantees for iterative rectified flow in recovering optimal transport couplings by introducing a cost-aware c-rectified flow, which projects the velocity field onto the gradient class to preserve endpoint marginal distributions. Under compactness and uniform integrability assumptions, the authors establish, for the first time, convergence of the iterative sequence to an optimal transport coupling and derive quantitative bounds demonstrating one-step contractivity and exponential convergence. Combining this with minimax-optimal score estimation rates under a Hölder ball assumption, the proposed method achieves minimax-optimal rates for optimal transport estimation when the dimension $d \geq 3$, and nearly parametric rates for $d = 1, 2$. Furthermore, in the Gaussian setting, the paper characterizes necessary and sufficient conditions for convergence of the classical rectified flow.

Abstract

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 \(d=1,2\).

Problem

• rectified flow
• optimal transport
• computational guarantees
• statistical guarantees
• c-rectified flow

Innovation

• c-rectified flow
• optimal transport
• computational guarantees
• statistical convergence
• score estimation