On the Relation between Rectified Flows and Optimal Transport

TL;DR

This paper explores the relationship between rectified flows, flow matching, and optimal transport, providing theoretical insights, invariance properties, and counterexamples that challenge previous assumptions about their equivalence.

Contribution

It offers a detailed analysis of rectified flows, their invariance properties, explicit constructions in Gaussian settings, and critically examines the conditions under which they relate to optimal transport.

Findings

Rectified flows have specific invariance properties.

Gradient-constrained rectified flows do not generally solve optimal transport.

Counterexamples show the limitations of previous equivalence claims.

Abstract

This paper investigates the connections between rectified flows, flow matching, and optimal transport. Flow matching is a recent approach to learning generative models by estimating velocity fields that guide transformations from a source to a target distribution. Rectified flow matching aims to straighten the learned transport paths, yielding more direct flows between distributions. Our first contribution is a set of invariance properties of rectified flows and explicit velocity fields. In addition, we also provide explicit constructions and analysis in the Gaussian (not necessarily independent) and Gaussian mixture settings and study the relation to optimal transport. Our second contribution addresses recent claims suggesting that rectified flows, when constrained such that the learned velocity field is a gradient, can yield (asymptotically) solutions to optimal transport problems. We study the existence of solutions for this problem and demonstrate that they only relate to optimal transport under assumptions that are significantly stronger than those previously acknowledged. In particular, we present several counterexamples that invalidate earlier equivalence results in the literature, and we argue that enforcing a gradient constraint on rectified flows is, in general, not a reliable method for computing optimal transport maps.