Flow Matching from Viewpoint of Proximal Operators

TL;DR

This paper introduces an exact proximal formulation of OT-CFM, a dynamical generative model, enabling better understanding of its convergence and stability properties, especially for manifold-supported targets.

Contribution

It provides a novel proximal operator-based formulation of OT-CFM, extending its applicability without density assumptions and analyzing its convergence and hyperbolicity.

Findings

Exact proximal expression for the vector field in OT-CFM

Convergence of minibatch OT-CFM to the population version

Exponential contraction normal to data manifolds for manifold-supported targets

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.