# Computational and Statistical Guarantees of the textitc-Rectified flow

arXiv: https://arxiv.org/abs/2608.02487 · HTML: https://mlsanity.com/paper/2608.02487/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 16 of its 51 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 16 of 16)

- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [FLUX.1 Kontext: Flow Matching for In-Context Image Generation and Editing in Latent Space](https://mlsanity.com/paper/2506.15742/) (2025)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [From optimal score matching to optimal sampling](https://mlsanity.com/paper/2409.07032/) (2024)
- [On the Relation between Rectified Flows and Optimal Transport](https://mlsanity.com/paper/2505.19712/) (2025)
- [On the Convergence and Straightness of Rectified Flow](https://mlsanity.com/paper/2410.14949/) (2024)
- [Statistical Properties of Rectified Flow](https://mlsanity.com/paper/2511.03193/) (2025)
- [Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications](https://mlsanity.com/paper/2502.12326/) (2025)
- [On the minimax optimality of Flow Matching through the connection to kernel density estimation](https://mlsanity.com/paper/2504.13336/) (2025)
- [Diffusion & Adversarial Schr"odinger Bridges via Iterative Proportional Markovian Fitting](https://mlsanity.com/paper/2410.02601/) (2024)
- [Exponential Convergence Guarantees for Iterative Markovian Fitting](https://mlsanity.com/paper/2510.20871/) (2025)
- [Order-Optimal Sample Complexity of Rectified Flows](https://mlsanity.com/paper/2601.20250/) (2026)

## Abstract (arXiv)

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 textitc-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 textitc-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 H"older ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative textitc-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension (d ≥ 3) and a nearly parametric rate for (d=1,2).