Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs
This work addresses the need for more expressive models in graph-based signal processing for researchers and practitioners dealing with non-Euclidean data, though it is incremental in building on existing concepts.
The paper tackles the problem of representing complex relations in high-dimensional data by proposing combinatorial complexes as a framework that bridges hypergraphs and cell complexes, demonstrating through a numerical experiment that this flexibility can improve learning tasks.
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.