Flow Matching from Viewpoint of Proximal Operators

We reformulate Optimal Transport Conditional Flow Matching (OT-CFM), showing that it admits an exact proximal form via an extended Brenier potential, without assuming that the target distribution has a density. In particular, the mapping to recover the target point is expressed 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 OT formulation as the sample and batch sizes increase. Using second epi-derivatives of convex potentials, we prove that, for manifold-supported targets, the manifold structure is stable by perturbation to the dynamics: after time rescaling, the dynamics contracts exponentially in directions normal to the manifold while remaining neutral along tangential directions.