---
title: "c-Rectified Flow 的计算与统计保证"
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","分数估计","整流流","最优传输"]
---

# c-Rectified Flow 的计算与统计保证

## AI 点评

> 这篇论文真正把整流流从“经验好用”推向了“理论可控”，对做生成模型底层算法的人尤其重要。如果你在调 FLUX 或 SD3 这类管线，理解普通整流流在协方差不可交换时可能失效，能帮你避开一些隐蔽的训练陷阱。

## 速览

最新论文首次为迭代整流流（Rectified Flow）建立计算与统计保证。理论表明，普通整流流在源和目标协方差矩阵不可交换时可能无法恢复最优传输耦合；而引入成本感知的 c-Rectified Flow 后，在紧性与一致可积条件下总能收敛到最优传输。同时给出二次与强凸位移成本下的一步收缩与指数收敛速率，并在 Hölder 球假设下获得最优分数估计率，为图像生成模型的训练与推理提供理论支撑。

## 关键点

- 普通整流流迭代可能无法收敛到最优传输耦合，除非源与目标协方差矩阵可交换；高斯案例给出了充要条件。
- c-Rectified Flow 在紧性与一致可积条件下，总能保证迭代收敛到最优传输耦合。
- 对于二次和强凸位移成本，建立了投影稳定假设下的一步收缩与指数收敛速率。
- 在 Hölder 球假设下取得 minimax 最优分数估计率，并用于构造最优传输的最优速率估计器。

## 全文

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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