# Statistical Properties of Rectified Flow

arXiv: https://arxiv.org/abs/2511.03193 · HTML: https://mlsanity.com/paper/2511.03193/

## 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 16 of its 101 references that are in the map.
- First posted on arXiv 5 Nov 2025.

## 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, 16 of 16)

- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Let us Build Bridges: Understanding and Extending Diffusion Generative Models](https://mlsanity.com/paper/2208.14699/) (2022)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Iterative α-(de)Blending: a Minimalist Deterministic Diffusion Model](https://mlsanity.com/paper/2305.03486/) (2023)
- [On the Relation between Rectified Flows and Optimal Transport](https://mlsanity.com/paper/2505.19712/) (2025)
- [An Optimal Transport Approach to Estimating Causal Effects via Nonlinear Difference-in-Differences](https://mlsanity.com/paper/2108.05858/) (2021)
- [Dimension-independent rates for structured neural density estimation](https://mlsanity.com/paper/2411.15095/) (2024)
- [Language Rectified Flow: Advancing Diffusion Language Generation with Probabilistic Flows](https://mlsanity.com/paper/2403.16995/) (2024)
- [Optimal Transport for Deep Joint Transfer Learning](https://mlsanity.com/paper/1709.02995/) (2017)
- [Log-concave Density Estimation with Independent Components](https://mlsanity.com/paper/2401.01500/) (2024)

## Built on by (most cited first, 11 of 11)

- [Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective](https://mlsanity.com/paper/2601.15500/) (2026)
- [A Kinetic Energy Perspective of Flow Matching](https://mlsanity.com/paper/2602.07928/) (2026)
- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [Tessellations of Semi-Discrete Flow Matching](https://mlsanity.com/paper/2605.07513/) (2026)
- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)
- [Total Variation Rates for Riemannian Flow Matching](https://mlsanity.com/paper/2602.05174/) (2026)
- [Limit Points of Reflow with Minibatch Optimal Transport](https://mlsanity.com/paper/2608.07042/) (2026)
- [The Fundamental Limits of Valid Transport Map Estimation](https://mlsanity.com/paper/2606.30574/) (2026)
- [Computational and Statistical Guarantees of the textitc-Rectified flow](https://mlsanity.com/paper/2608.02487/) (2026)
- [Sample complexity bounds for categorical Markov random fields via Discrete Diffusions](https://mlsanity.com/paper/2610.02128/) (2026)
- [When do explicit penalties improve flow matching?](https://mlsanity.com/paper/2610.15689/) (2026)

## Abstract (arXiv)

Rectified flow (Liu et al., 2022; Liu, 2022; Wu et al., 2023) is a method for defining a transport map between two distributions, and enjoys popularity in machine learning, although theoretical results supporting the validity of these methods are scant. The rectified flow can be regarded as an approximation to optimal transport, but in contrast to other transport methods that require optimization over a function space, computing the rectified flow only requires standard statistical tools such as regression or density estimation, which we leverage to develop empirical versions of transport maps. We study some structural properties of the rectified flow, including existence, uniqueness, and regularity, as well as the related statistical properties, such as rates of convergence and central limit theorems, for some selected estimators. To do so, we analyze the bounded and unbounded cases separately as each presents unique challenges. In both cases, we are able to establish convergence at faster rates than those for the usual nonparametric regression and density estimation.