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arXiv · 2112.03904

Hypergraph Co-Optimal Transport: Metric and Categorical Properties

Samir Chowdhury·Tom Needham·Ethan Semrad·Bei Wang·Youjia Zhou
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Abstract

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.

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Samir Chowdhury, Tom Needham, Ethan Semrad, Bei Wang, Youjia Zhou. 2023-02-02. Hypergraph Co-Optimal Transport: Metric and Categorical Properties. https://arxiv.org/abs/2112.03904

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