Flow Matching from Viewpoint of Proximal Operators
This work provides theoretical insights into generative models for researchers in machine learning, though it appears incremental as it builds on existing OT-CFM frameworks.
The paper reformulates Optimal Transport Conditional Flow Matching (OT-CFM) as an exact proximal operator via an extended Brenier potential, enabling recovery of target points without density assumptions, and proves that for manifold-supported targets, the dynamics contracts exponentially in normal directions while remaining neutral tangentially after time rescaling.
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.