Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs

• URL: http://arxiv.org/abs/2312.09504v1
• Date: Fri, 15 Dec 2023 03:04:28 GMT
• Title: Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs
• Authors: Mustafa Hajij, Ghada Zamzmi, Theodore Papamarkou, Aldo Guzm\'an-S\'aenz, Tolga Birdal, Michael T. Schaub
• Abstract summary: We argue that hypergraphs and cell complexes emphasize emphdifferent types of relations, which may have different utility depending on the application context. We discuss the relative advantages of these two choices and elaborate on the previously introduced concept of a complex that enables co-existing set-type and hierarchical relations.
• Score: 18.793940779717627
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively represent the complex relations found in high-dimensional data. Such higher-order domains are typically modeled either as hypergraphs, or as simplicial, cubical or other cell complexes. In this context, cell complexes are often seen as a subclass of hypergraphs with additional algebraic structure that can be exploited, e.g., to develop a spectral theory. In this article, we promote an alternative perspective. We argue that hypergraphs and cell complexes emphasize \emph{different} types of relations, which may have different utility depending on the application context. Whereas hypergraphs are effective in modeling set-type, multi-body relations between entities, cell complexes provide an effective means to model hierarchical, interior-to-boundary type relations. We discuss the relative advantages of these two choices and elaborate on the previously introduced concept of a combinatorial complex that enables co-existing set-type and hierarchical relations. Finally, we provide a brief numerical experiment to demonstrate that this modelling flexibility can be advantageous in learning tasks.

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