Hypergraphs capture multi-way relationships in data, and they have consequently seen a number of applications in higher-order network analysis, computer vision, geometry processing, and machine learning. In this paper, we develop theoretical foundations for studying the space of hypergraphs using ingredients from optimal transport. By enriching a hypergraph with probability measures on its nodes and hyperedges, as well as relational information capturing local and global structures, we obtain a general and robust framework for studying the collection of all hypergraphs. First, we introduce a hypergraph distance based on the co-optimal transport framework of Redko et al. and study its theoretical properties. Second, we formalize common methods for transforming a hypergraph into a graph as maps between the space of hypergraphs and the space of graphs, and study their functorial properties and Lipschitz bounds. Finally, we demonstrate the versatility of our Hypergraph Co-Optimal Transport (HyperCOT) framework through various examples.

翻译：测深仪捕捉了数据中的多路关系, 并因此在高阶网络分析、 计算机视觉、 几何处理和机器学习中看到了一些应用。 在本文中, 我们开发了理论基础, 用于使用来自最佳运输的成分来研究高音空间 。 通过丰富高射线及其节点和高端的概率测量, 以及收集地方和全球结构的关联信息, 我们获得了一个用于研究所有高射线集成的总和强大的框架 。 首先, 我们引入了一个基于Redko 等人共同最佳运输框架的高射距离, 并研究其理论特性 。 其次, 我们正式确定了将高射线转换为图的通用方法, 在高射线空间和图形空间之间绘制地图, 并研究其交替特性和利普施奇茨界限 。 最后, 我们通过各种实例展示了我们超光谱联合运输( HyperCOT) 框架的多功能性 。