统一向量值最优传输的动态与静态理论，构建几何关联的距离度量。
Vector valued optimal transport: from dynamic to static formulations
- 将向量测度建模为产品空间上的概率测度，引入图结构定义传输几何。
- 证明四类传输距离互为双霍尔德等价，且存在紧致不等式关系。
- 静态形式可线性化，适合加速多物种PDE与数据分类中的计算应用。
受向量值测度分类与多物种偏微分方程应用的启发，本文建立了一个统一框架，融合了从动态（Benamou-Brenier型）到静态（Kantorovich型）的向量值最优传输理论。在该框架中，向量值测度被建模为乘积空间 $\mathbb{R}^d \times G$ 上的概率测度，其中 $G$ 是有限节点集上的加权图，其几何结构显著影响对应的动态与静态距离。我们得到了四类向量值最优传输度量之间的紧致不等式，并证明这些距离彼此为双霍尔德等价。论文讨论了各类度量的理论与实际优势，并指出其在多物种偏微分方程和数据分析中的潜在应用。特别地，文中讨论的一种静态形式具有可线性化特性，这一技术近年来被用于加速成对最优传输距离的计算。
原文摘要 · Abstract (English)
Motivated by applications in classification of vector valued measures and multispecies PDE, we develop a theory that unifies existing notions of vector valued optimal transport, from dynamic formulations (à la Benamou-Brenier) to static formulations (à la Kantorovich). In our framework, vector valued measures are modeled as probability measures on a product space $\mathbb{R}^d \times G$, where $G$ is a weighted graph over a finite set of nodes and the graph geometry strongly influences the associated dynamic and static distances. We obtain sharp inequalities relating four notions of vector valued optimal transport and prove that the distances are mutually bi-Hölder equivalent. We discuss the theoretical and practical advantages of each metric and indicate potential applications in multispecies PDE and data analysis. In particular, one of the static formulations discussed in the paper is amenable to linearization, a technique that has been explored in recent years to accelerate the computation of pairwise optimal transport distances.
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