揭示生成模型路径直度与采样效率的理论关系，实现一步生成。
On the Convergence and Straightness of Rectified Flow
- 提出分段直度参数γ₂,ₜ，量化路径弯曲程度。
- 证明最小化路径弯曲可实现单步高保真采样。
- 首次给出矩形流路径直度的理论保证，适合算法研究者。
Flow Matching已成为现代生成模型（如Stable Diffusion 3）的核心，其关键变体矩形流（Rectified Flow, RF）通过学习直线轨迹显著提升采样效率。然而，路径几何与采样效率之间的理论联系长期未被深入探讨。本文引入新的分段直度参数γ₂,ₜ，首次建立任意通用流模型的Wasserstein收敛误差上界，明确证明最小化路径曲率是实现高保真、单步采样的核心。基于此理论，我们构建首个分析矩形流直度的框架：先通过直观几何论证简单情形，再识别出单次修正（1-RF）达到完全直线或Monge最优耦合的充分条件。若满足这些条件，后续流（2-RF）将完全直线（γ₂,ₜ=0），使我们的上界中离散误差归零，从而实现完美单步采样。
原文摘要 · Abstract (English)
Flow Matching has become a cornerstone of modern generative models like Stable Diffusion 3, largely due to the efficiency of its Rectified Flow (RF) variant. The success of RF hinges on iteratively learning straight trajectories, pushing generation towards fewer sampling steps. However, the theoretical link between path geometry and sampling efficiency has been underexplored. This paper fills this gap by introducing a novel \textit{Piecewise Straightness} parameter, $γ_{2,T}$. We establish the first Wasserstein convergence bound that explicitly links the discretization error of \textit{any} general flow-model to $γ_{2,T}$, proving that minimizing curvature is the key to achieving high-fidelity, one-step sampling. Building on this theory, we establish the first theoretical framework to analyze the straightness of RF. We begin by offering intuitive geometric arguments for simple cases before identifying sufficient conditions under which a single rectification step (1-RF) yields a perfectly straight or even a Monge optimal coupling. While whether these sufficient conditions are met depends on the problem geometry, they enable the first concrete proofs in this area. Critically, fulfilling these conditions makes the subsequent flow (2-RF) perfectly straight ($γ_{2,T}=0$). This eliminates the discretization error in our bound and makes flawless, single-step sampling possible.
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