# Hypergraph Co-Optimal Transport: Metric and Categorical Properties

arXiv: https://arxiv.org/abs/2112.03904 · HTML: https://mlsanity.com/paper/2112.03904/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2021.

## Built on by (most cited first, 10 of 10)

- [The Z-Gromov-Wasserstein Distance](https://mlsanity.com/paper/2408.08233/) (2024)
- [Unbalanced CO-Optimal Transport](https://mlsanity.com/paper/2205.14923/) (2022)
- [G-Mapper: Learning a Cover in the Mapper Construction](https://mlsanity.com/paper/2309.06634/) (2023)
- [Conic Formulations of Transport Metrics for Unbalanced Measure Networks and Hypernetworks](https://mlsanity.com/paper/2508.10888/) (2025)
- [Hyperedge Modeling in Hypergraph Neural Networks by using Densest Overlapping Subgraphs](https://mlsanity.com/paper/2409.10340/) (2024)
- [Hypergraph clustering using Ricci curvature: an edge transport perspective](https://mlsanity.com/paper/2412.15695/) (2024)
- [Beyond Distance: Quantifying Point Cloud Dynamics with Persistent Homology and Dynamic Optimal Transport](https://mlsanity.com/paper/2603.15683/) (2026)
- [MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport](https://mlsanity.com/paper/2606.08258/) (2026)
- [Diffusion enabled Optimal Transport distances for graph matching](https://mlsanity.com/paper/2607.06646/) (2026)
- [A Unified Geometric Framework for Developmental Analysis of Spatial Transcriptomic Data](https://mlsanity.com/paper/2608.15306/) (2026)

## Abstract (arXiv)

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.