将流匹配重新表述为近端算子，揭示其数学本质并证明收敛性与稳定性。
Flow Matching from Viewpoint of Proximal Operators
- 用扩展的Brenier势函数构建精确近端形式，无需假设目标分布有密度。
- 证明了小批量流匹配随批次增大收敛到总体形式。
- 首次证明流匹配在流形支持下具有指数收缩的法向动态，适合数据流形建模。
我们将最优传输条件流匹配（OT-CFM）重新表述为一种精确的近端算子形式，通过扩展的Brenier势函数实现，无需假设目标分布具有密度。具体而言，恢复目标点的映射恰好由一个近端算子给出，从而导出向量场的显式表达式。我们还讨论了小批量OT-CFM随批次大小增加向总体形式收敛的性质。最后，利用凸势函数的二阶上导数，证明对于流形支持的目标，OT-CFM在时间重标度后呈终端正常双曲性：沿数据流形法向方向动态指数收缩，而切向方向保持中性。
原文摘要 · Abstract (English)
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.
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