---
title: "Computational and Statistical Guarantees of the c-Rectified Flow"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2608.02487v1"
permalink: "https://feed.astralor.com/items/itm_f286f4c695cd59e246a290ba"
published: "2026-08-03T16:53:29.000Z"
score: 83
channel: "ai"
tags: ["arXiv","FLUX.1","Stable Diffusion 3","Score Estimation","Rectified Flow","Optimal Transport"]
---

# Computational and Statistical Guarantees of the c-Rectified Flow

## AI commentary

> This paper moves rectified flow from 'works in practice' to 'theoretically controllable,' which is especially important for those working on the underlying algorithms of generative models. If you are tuning pipelines like FLUX or SD3, understanding that ordinary rectified flow can fail when covariances do not commute may help you avoid hidden training pitfalls.

## At a glance

This paper provides, for the first time, computational and statistical guarantees for iterative rectified flow, a framework powering large-scale image generation systems such as FLUX.1 and Stable Diffusion 3. The authors show that ordinary rectified flow can fail to recover the optimal transport coupling when the source and target covariance matrices do not commute. In contrast, under compactness and uniform integrability assumptions, iterative c-rectified flow always converges to the optimal transport coupling. They further establish one-step contraction and exponential convergence rates under projection-stability assumptions for quadratic and strongly convex displacement costs. Under a Hölder ball assumption, they derive minimax-optimal score estimation rates, yielding rate-optimal estimators of optimal transport for dimension d ≥ 3 and nearly parametric rates for d = 1, 2. This work bridges a critical gap between empirical success and theoretical understanding in generative modeling.

## Key points

- Ordinary iterated rectified flow may fail to recover the optimal transport coupling unless source and target covariance matrices commute; a Gaussian case study gives the exact condition.
- Iterative c-rectified flow always converges to the optimal transport coupling under compactness and uniform integrability assumptions.
- One-step contraction and exponential convergence guarantees are established under projection-stability assumptions for quadratic and strongly convex displacement costs.
- Minimax-optimal score estimation rates are obtained under a Hölder ball assumption, leading to rate-optimal estimators of optimal transport for d ≥ 3 and nearly parametric rates for d = 1, 2.

## Full text

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 \(d=1,2\).

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