Statistical Properties of Rectified Flow

2025-11-05

AI Summary

This paper establishes the statistical foundations of Rectified Flow (RF) transport maps. Addressing existence, uniqueness, and regularity, it provides the first rigorous characterization under both bounded and unbounded distribution settings. Methodologically, it proposes novel construction schemes based on regression and density estimation—bypassing classical nonparametric rate limitations and achieving faster statistical convergence. It further derives the first central limit theorem for the corresponding estimators. Integrating tools from optimal transport theory, empirical process theory, and nonparametric statistics, the work delivers the first comprehensive asymptotic framework for data-driven transport maps. The results substantially enhance the interpretability and reliability of RF methods in statistical learning and generative modeling.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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. Because of this, one can leverage standard data analysis tools for regression and density estimation 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 separately the bounded and unbounded cases as each presents unique challenges. In both cases, we are able to establish convergence at faster rates than the ones for the usual nonparametric regression and density estimation.

Problem

• Rectified flow defines transport maps between distributions
• It approximates optimal transport without complex optimization
• The paper analyzes statistical properties and convergence rates

Innovation

• Rectified flow approximates optimal transport without optimization
• It uses standard regression and density estimation tools
• Convergence rates are faster than nonparametric methods