Statistical Properties of Rectified Flow

TL;DR

This paper investigates the statistical properties of rectified flow, a method for defining transport maps between distributions, providing theoretical guarantees and convergence rates that enhance its reliability in machine learning applications.

Contribution

It offers the first comprehensive theoretical analysis of rectified flow, including existence, uniqueness, regularity, and convergence rates, distinguishing between bounded and unbounded cases.

Findings

Faster convergence rates than standard nonparametric methods.

Established existence, uniqueness, and regularity of rectified flow.

Derived central limit theorems for estimators.

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, 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.