Statistical Properties of Rectified Flow
This provides theoretical validation for a popular machine learning method, addressing a gap in understanding for researchers and practitioners in transport and generative modeling.
The paper tackles the lack of theoretical support for rectified flow, a method for defining transport maps between distributions, by analyzing its structural and statistical properties, establishing faster convergence rates than standard nonparametric methods for selected estimators.
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.