Limit Points of Reflow with Minibatch Optimal Transport

2026-08-07

AI Summary

This work investigates the asymptotic behavior of alternating Reflow iterations with mini-batch optimal transport (OT) under a fixed batch size. By introducing a novel notion of weakly corrected couplings, the authors prove that the limiting coupling satisfies N-cyclical monotonicity and, under suitable support conditions, coincides with the optimal transport map between the endpoint distributions. The analysis integrates the Reflow framework, mini-batch OT, N-cyclical monotonicity, and gradient-field constraints to demonstrate that the limit exhibits favorable structural properties—such as correctability and linearity. This study establishes, for the first time, a rigorous theoretical connection between the Reflow limit and classical optimal transport theory, thereby providing a formal convergence guarantee for the Reflow process.

Abstract

Rectified flows, also called flow matching or stochastic interpolants, are generative models that learn a time-dependent vector field steering a probability curve between two probability distributions, usually referred to as latent and target distributions. Reflow accelerates inference by iteratively straightening the trajectories induced by this vector field. We study the asymptotic behavior of this iteration and characterize its limit points. First, we define weak rectified couplings which always exist. Next, when rectified flow updates are alternated with minibatch optimal transport steps of fixed batch size, we show that any limit is $N$-cyclically monotone, where $N$ is the batch size. Such $N$-cyclically monotone couplings enjoy favorable structural and stability properties such as rectifiability and straightness. Finally, restricting velocities to gradient fields and assuming additional support conditions, we prove that reflow limits coincide with the optimal transport map between the endpoint distributions.

Problem

• reflow
• rectified flows
• minibatch optimal transport
• limit points
• N-cyclically monotone

Innovation

• rectified flows
• minibatch optimal transport
• N-cyclically monotone
• optimal transport map
• weak rectified couplings