# On the Convergence and Straightness of Rectified Flow arXiv: https://arxiv.org/abs/2410.14949 · HTML: https://mlsanity.com/paper/2410.14949/ ## Key facts - Cited by 11 papers in the map (data through 14 Oct 2026). - Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. - Builds on 25 of its 37 references that are in the map. - First posted on arXiv 19 Oct 2024. ## Placed in - Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/) ## Builds on (most cited first, 25 of 25) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021) - [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020) - [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020) - [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022) - [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023) - [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021) - [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022) - [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023) - [eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoisers](https://mlsanity.com/paper/2211.01324/) (2022) - [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022) - [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023) - [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022) - [DPM-Solver-v3: Improved Diffusion ODE Solver with Empirical Model Statistics](https://mlsanity.com/paper/2310.13268/) (2023) - [Improving the Training of Rectified Flows](https://mlsanity.com/paper/2405.20320/) (2024) - [PeRFlow: Piecewise Rectified Flow as Universal Plug-and-Play Accelerator](https://mlsanity.com/paper/2405.07510/) (2024) - [Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models](https://mlsanity.com/paper/2306.09251/) (2023) - [Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance](https://mlsanity.com/paper/2212.06359/) (2022) - [On Kinetic Optimal Probability Paths for Generative Models](https://mlsanity.com/paper/2306.06626/) (2023) - [Towards a mathematical theory for consistency training in diffusion models](https://mlsanity.com/paper/2402.07802/) (2024) - [Improved Convergence of Score-Based Diffusion Models via Prediction-Correction](https://mlsanity.com/paper/2305.14164/) (2023) - [Improved Sample Complexity Bounds for Diffusion Model Training](https://mlsanity.com/paper/2311.13745/) (2023) ## Built on by (most cited first, 11 of 11) - [Curly Flow Matching for Learning Non-gradient Field Dynamics](https://mlsanity.com/paper/2510.26645/) (2025) - [On the Relation between Rectified Flows and Optimal Transport](https://mlsanity.com/paper/2505.19712/) (2025) - [Flow Matching is Adaptive to Manifold Structures](https://mlsanity.com/paper/2602.22486/) (2026) - [Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domains](https://mlsanity.com/paper/2510.08929/) (2025) - [Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective](https://mlsanity.com/paper/2601.15500/) (2026) - [Tessellations of Semi-Discrete Flow Matching](https://mlsanity.com/paper/2605.07513/) (2026) - [Total Variation Rates for Riemannian Flow Matching](https://mlsanity.com/paper/2602.05174/) (2026) - [One-Shot Generative Flows: Existence and Obstructions](https://mlsanity.com/paper/2604.15439/) (2026) - [An Eulerian Perspective on Straight-Line Sampling](https://mlsanity.com/paper/2510.11657/) (2025) - [Optimal Self-Distillation for Rectified Flow via Linear Probing](https://mlsanity.com/paper/2607.14947/) (2026) - [Computational and Statistical Guarantees of the textitc-Rectified flow](https://mlsanity.com/paper/2608.02487/) (2026) ## Abstract (arXiv) 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 textitPiecewise Straightness parameter, γ2,T. We establish the first Wasserstein convergence bound that explicitly links the discretization error of textitany general flow-model to γ2,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 (γ2,T=0). This eliminates the discretization error in our bound and makes flawless, single-step sampling possible.