Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911837590650880 |
|---|---|
| author | Hajij, Mustafa Zamzmi, Ghada Papamarkou, Theodore Guzmán-Sáenz, Aldo Birdal, Tolga Schaub, Michael T. |
| author_facet | Hajij, Mustafa Zamzmi, Ghada Papamarkou, Theodore Guzmán-Sáenz, Aldo Birdal, Tolga Schaub, Michael T. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09504 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs Hajij, Mustafa Zamzmi, Ghada Papamarkou, Theodore Guzmán-Sáenz, Aldo Birdal, Tolga Schaub, Michael T. Machine Learning Social and Information Networks Algebraic Topology Combinatorics 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. |
| title | Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs |
| topic | Machine Learning Social and Information Networks Algebraic Topology Combinatorics |
| url | https://arxiv.org/abs/2312.09504 |