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.

翻译：我们重新表述了最优传输条件流匹配（OT-CFM）——一类动态生成模型，证明其可通过扩展的Brenier势函数获得精确的邻近算子表述，且无需假设目标分布具有密度。特别地，恢复目标点的映射恰好由邻近算子给出，这导出了向量场的显式邻近表达式。我们还讨论了小批量OT-CFM随着批量增大向总体表述的收敛性。最后，利用凸势函数的二阶上图导数，我们证明了对于流形支撑的目标分布，OT-CFM具有终端正规双曲性：经过时间重标度后，动力学在垂直于数据流形的方向上呈指数收缩，而在切向方向上保持中性。