Limit Points of Reflow with Minibatch Optimal Transport

• URL: http://arxiv.org/abs/2608.07042v1
• Date: Fri, 07 Aug 2026 09:51:33 GMT
• Title: Limit Points of Reflow with Minibatch Optimal Transport
• Abstract summary: Rectified flows are generative models that learn a time-dependent vector field steering a probability curve between two probability distributions.<n>We study the behavior of this iteration and characterize its limit points.<n>Such $N$-cyclically monotone couplings enjoy favorable structural and stability properties such as rectifiability and straightness.
• Score: 5.12067518301035
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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