Computational and Statistical Guarantees of the \textit{c}-Rectified flow
This work provides theoretical foundations for a widely used generative modeling framework, benefiting researchers and practitioners in optimal transport and generative models by clarifying when rectified flow converges to optimal transport and offering a theoretically justified variant.
The paper provides the first computational and statistical guarantees for iterative rectified flow, showing that ordinary rectified flow may fail to recover optimal transport couplings, while a cost-aware variant (c-rectified flow) always converges to the optimal coupling under mild assumptions. They also establish convergence rates and minimax-optimal score estimation rates, leading to rate-optimal transport estimators for d≥3 and nearly parametric rates for d=1,2.
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 \(d=1,2\).