Beckmann Transport Models: From Autonomous Flows to One-Step Maps

Abstract:We propose an instantiation of flow matching that relies on a time-independent velocity field (an \emph{autonomous flow}) to exactly map between two distributions, so long as the target is singular, i.e.\ supported on a lower-dimensional data manifold. We also show that the one-step generative map associated with this flow is the unique solution of a simple conservation equation, which can be used to learn the map directly from samples. These autonomous flows and maps give a dynamical meaning to the flux constraint of Beckmann's transportation problem. Their construction provides a unifying framework that recovers, for instance, the closed-form Poisson-flow generative model and equilibrium matching with a quadratic flow-matching regression loss. We illustrate how this theory corrects inconsistencies in existing methods and demonstrate the effectiveness of the autonomous flow and the one-step map on ImageNet 256x256.

Machine Learning (cs.LG)

arXiv:2608.01692 [cs.LG]

(or arXiv:2608.01692v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.01692 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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