提出新型连续流模型，精确实现任意p-距离的最优传输
Potential Matching Optimal Transport: Continuous Normalizing Flows for Exact $p$-Wasserstein Dynamics
- 用标量势函数参数化速度场，基于自洽路径匹配训练
- 零损失解能精确恢复p-最优传输映射与动态过程
- 适用于高维数据建模和灵活的样本分布变换
我们提出潜在匹配最优传输（PMOT），一种用于一般p-代价最优传输（$c_p(x,y)=\|x-y\|^p$）的势流框架。PMOT在选定指数p下，以广义Benamou--Brenier形式参数化连续归一化流（CNF）的速度场。它通过模型自身端点决定的直线桥路，利用自诱导匹配损失训练势函数梯度，同时允许灵活的终态分布匹配。主要结果表明：在给定正则性、精确终态匹配及唯一性假设下，任何零损失解均满足广义Benamou--Brenier最优系统，并恢复对应的p-最优传输映射与动力学。在合成基准上，PMOT学习到与对应p-匹配最优传输参考一致的映射；在高维表格数据上仍保持竞争力作为似然密度模型；基于MMD的色彩转换实验展示了灵活的基于样本的终态匹配能力。
原文摘要 · Abstract (English)
We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general $p$-cost optimal transport with $c_p(x,y)=\|x-y\|^p$. PMOT parameterizes the CNF velocity field with a scalar potential in the generalized Benamou--Brenier form for the chosen exponent $p$. It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model's own endpoints, while allowing flexible terminal distribution matching. Our main result establishes zero-loss exactness: under the stated regularity, exact terminal matching, and uniqueness assumptions, any zero-loss solution satisfies the generalized Benamou--Brenier optimality system and recovers the corresponding $p$-optimal transport map and dynamics. On synthetic benchmarks, PMOT learns $p$-specific maps that agree with the corresponding $p$-matched OT references. It also remains competitive as a likelihood-based density model on high-dimensional tabular data, and an MMD-based color transformation experiment demonstrates flexible sample-based terminal matching.
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