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arXiv · 2602.12683

Flow Matching from Viewpoint of Proximal Operators

Kenji Fukumizu·Wei Huang·Han Bao·Shuntuo Xu·Nisha Chandramoorthy
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Abstract

We reformulate Optimal Transport Conditional Flow Matching (OT-CFM), a class of dynamical generative models, showing that it admits an exact proximal formulation via an extended Brenier potential, without assuming that the target distribution has a density. In particular, the mapping to recover the target point is exactly given by a proximal operator, which yields an explicit proximal expression of the vector field. We also discuss the convergence of minibatch OT-CFM to the population formulation as the batch size increases. Finally, using second epi-derivatives of convex potentials, we prove that, for manifold-supported targets, OT-CFM is terminally normally hyperbolic: after time rescaling, the dynamics contracts exponentially in directions normal to the data manifold while remaining neutral along tangential directions.

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Kenji Fukumizu, Wei Huang, Han Bao, Shuntuo Xu, Nisha Chandramoorthy. 2026-02-13. Flow Matching from Viewpoint of Proximal Operators. https://arxiv.org/abs/2602.12683

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