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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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This content will become publicly available on July 6, 2027
Geometry of the space of partitioned networks: a unified theoretical and computational framework
Abstract Network data are ubiquitous in the real world, capturing pairwise and high-order relations among objects. We introduce a class of measure-theoretic network objects, termed partitioned measure networks, that generalize many existing representations of network data, including graphs, hypergraphs and augmented graphs (i.e. graphs whose nodes are associated with categorical classes). We then propose a metric, the partitioned network distance, between partitioned measure networks, which extends both the Gromov-Wasserstein distance for graphs and the co-optimal transport distance for hypergraphs. By characterizing the geometry of the space of partitioned measure networks, we provide a unified theoretical treatment of generalized network structures encompassing both pairwise and higher-order relations. In particular, we show that this metric endows the space with an Alexandrov geometry of non-negative curvature and leverages this structure to define gradients for certain functionals commonly encountered in geometric data analysis. We further extend our framework to incorporate node-level label information and derive efficient computational schemes for utilizing the partitioned network distance in practice. Equipped with these theoretical and computational advances, we demonstrate the utility of our framework across diverse applications, including hypergraph alignment, clustering and dictionary learning from ensemble data, multi-omics alignment, as well as multiscale network alignment.
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- Award ID(s):
- 2324962
- PAR ID:
- 10707814
- Publisher / Repository:
- IMA
- Date Published:
- Journal Name:
- Information and Inference: A Journal of the IMA
- Volume:
- 15
- Issue:
- 3
- ISSN:
- 2049-8772
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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