On the Convergence and Straightness of Rectified Flow

Vansh Bansal, Saptarshi Roy, Alessandro Rinaldo, Purnamrita Sarkar
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:4636-4644, 2026.

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 under-explored. 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-bansal26a, title = { On the Convergence and Straightness of Rectified Flow }, author = {Bansal, Vansh and Roy, Saptarshi and Rinaldo, Alessandro and Sarkar, Purnamrita}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {4636--4644}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/bansal26a/bansal26a.pdf}, url = {https://proceedings.mlr.press/v300/bansal26a.html}, 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 under-explored. 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. } }
Endnote
%0 Conference Paper %T On the Convergence and Straightness of Rectified Flow %A Vansh Bansal %A Saptarshi Roy %A Alessandro Rinaldo %A Purnamrita Sarkar %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-bansal26a %I PMLR %P 4636--4644 %U https://proceedings.mlr.press/v300/bansal26a.html %V 300 %X 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 under-explored. 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.
APA
Bansal, V., Roy, S., Rinaldo, A. & Sarkar, P.. (2026). On the Convergence and Straightness of Rectified Flow . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:4636-4644 Available from https://proceedings.mlr.press/v300/bansal26a.html.

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