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.
@article{arxiv.2602.12683,
title = {Flow Matching from Viewpoint of Proximal Operators},
author = {Kenji Fukumizu and Wei Huang and Han Bao and Shuntuo Xu and Nisha Chandramoorthy},
journal= {arXiv preprint arXiv:2602.12683},
year = {2026}
}