将流匹配推广到概率分布的分布空间，实现高效生成复杂结构数据
Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once
- 基于嵌套沃瑟斯坦几何，用内外层传输计划构造元分布流
- 引入切片与线性沃瑟斯坦距离，显著降低计算成本并保持轨迹稳定
- 适用于点云、集合等复杂数据生成，理论扎实且可实用
流匹配最近作为生成建模的灵活高效框架出现，通过学习概率测度间的确定性传输动态。本文将其扩展至概率测度上的概率测度空间，提出沃瑟斯坦-沃瑟斯坦（WoW）公式。利用嵌套沃瑟斯坦几何，我们证明测度上的传输计划自然诱导出实现元测度流的速度场。这通过耦合的外层与内层传输计划，为沃瑟斯坦流匹配提供了严谨的推广。为应对WoW传输的高昂计算成本，我们提出基于切片和线性沃瑟斯坦距离的可扩展近似方法，在保证数值稳定性和近似直线轨迹的同时实现高效训练。该框架统一并扩展了现有点云与集合生成方法，提供了一种在WoW空间中实用且理论严谨的生成建模方式。
原文摘要 · Abstract (English)
Flow matching has recently emerged as a flexible and efficient framework for generative modelling by learning deterministic transport dynamics between probability measures. In this work, we extend flow matching to the space of probability measures over probability measures, introducing a Wasserstein-on-Wasserstein (WoW) formulation. Leveraging the nested Wasserstein geometry, we show that measures over transport plans naturally induce velocity fields that realize metameasure flows. This yields a principled generalization of Wasserstein flow matching via coupled outer and inner transport plans. To address the substantial computational cost of WoW transport, we propose scalable approximations based on sliced and linear Wasserstein distances, enabling efficient training while promoting numerically stable, near-straight trajectories. Our framework unifies and extends existing approaches to point cloud and set generation, providing a practical and theoretically grounded method for generative modelling in WoW spaces.
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