On the Convergence and Straightness of Rectified Flow

TL;DR

This paper introduces a theoretical framework linking flow path straightness to sampling efficiency in generative models, demonstrating how minimizing curvature can enable high-fidelity, single-step sampling.

Contribution

It establishes the first Wasserstein convergence bound related to flow straightness and provides conditions under which rectified flow achieves perfect straightness and optimal coupling.

Findings

Introducing the Piecewise Straightness parameter $oldsymbol{_{2,T}}$

Proving bounds that connect flow curvature to sampling error

Identifying conditions for perfect straightness in rectified flow

Abstract

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, $\gamma_{2,T}$. We establish the first Wasserstein convergence bound that explicitly links the discretization error of \textit{any} general flow-model to $\gamma_{2,T}$, 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 ($\gamma_{2,T}=0$). This eliminates the discretization error in our bound and makes flawless, single-step sampling possible.