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.

翻译：基于图的信号处理技术已成为处理非欧几里得空间数据的关键工具。然而，人们逐渐认识到，这些图模型可能需要扩展到“高阶”域，才能有效表示高维数据中存在的复杂关系。这类高阶域通常建模为超图，或单形复形、立方复形及其他细胞复形。在此背景下，细胞复形常被视为超图的一个子类，但具备可加以利用的额外代数结构（例如用于发展谱理论）。本文提出了一种替代视角。我们认为超图与细胞复形强调不同类型的关联，这些关联在不同应用场景中可能具有不同的效用。超图擅长建模实体间的集合型多体关系，而细胞复形则能有效刻画层次型的内-边界关系。我们讨论了这两种选择的相对优势，并详细阐述了此前提出的组合复形概念——该概念使得集合型关系与层次型关系能够共存。最后，我们通过简要数值实验证明，这种建模灵活性可在学习任务中发挥优势。