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：超图能够捕捉数据中的多元关系，因此在高阶网络分析、计算机视觉、几何处理及机器学习等领域得到了广泛应用。本文利用最优传输理论的相关工具，为研究超图空间奠定了理论基础。通过在超图节点和超边上定义概率测度，并引入刻画局部与全局结构的关系信息，我们构建了一个通用且稳健的框架，用于研究所有超图的集合。首先，基于Redko等人提出的协同最优传输框架，我们定义了一种超图距离，并探讨其理论性质。其次，我们将超图转化为图的常见方法形式化为超图空间与图空间之间的映射，研究了这些映射的函子性质与Lipschitz界。最后，通过多个实例展示了超图协同最优传输（HyperCOT）框架的广泛适用性。