Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many domains encountered in scientific computations. In this paper, we present a unifying deep learning framework built upon a richer data structure that includes widely adopted topological domains. Specifically, we first introduce combinatorial complexes, a novel type of topological domain. Combinatorial complexes can be seen as generalizations of graphs that maintain certain desirable properties. Similar to hypergraphs, combinatorial complexes impose no constraints on the set of relations. In addition, combinatorial complexes permit the construction of hierarchical higher-order relations, analogous to those found in simplicial and cell complexes. Thus, combinatorial complexes generalize and combine useful traits of both hypergraphs and cell complexes, which have emerged as two promising abstractions that facilitate the generalization of graph neural networks to topological spaces. Second, building upon combinatorial complexes and their rich combinatorial and algebraic structure, we develop a general class of message-passing combinatorial complex neural networks (CCNNs), focusing primarily on attention-based CCNNs. We characterize permutation and orientation equivariances of CCNNs, and discuss pooling and unpooling operations within CCNNs in detail. Third, we evaluate the performance of CCNNs on tasks related to mesh shape analysis and graph learning. Our experiments demonstrate that CCNNs have competitive performance as compared to state-of-the-art deep learning models specifically tailored to the same tasks. Our findings demonstrate the advantages of incorporating higher-order relations into deep learning models in different applications.

翻译：拓扑深度学习是一个快速发展的领域，涉及为支持在拓扑域（如单纯复形、胞腔复形和超图）上的数据开发深度学习模型，这些拓扑域概括了科学计算中遇到的许多域。在本文中，我们提出了一个统一的深度学习框架，该框架构建于一个包含广泛采用的拓扑域的更丰富的数据结构之上。具体来说，我们首先引入组合复形，这是一种新型拓扑域。组合复形可被视为保留了某些理想性质的图的推广。与超图类似，组合复形对关系集合没有施加任何约束。此外，组合复形允许构建分层的高阶关系，类似于单纯复形和胞腔复形中发现的那些关系。因此，组合复形概括并结合了超图和胞腔复形的有用特性，后两者已成为促进图神经网络推广到拓扑空间的两种有前景的抽象。其次，基于组合复形及其丰富的组合与代数结构，我们开发了一类通用的消息传递组合复形神经网络（CCNN），主要关注基于注意力的CCNN。我们刻画了CCNN的置换等变性和定向等变性，并详细讨论了CCNN中的池化和反池化操作。第三，我们在网格形状分析和图学习相关任务上评估了CCNN的性能。我们的实验表明，与专门针对相同任务的最先进深度学习模型相比，CCNN具有竞争力的性能。我们的研究结果证明了在不同应用中向深度学习模型引入高阶关系的优势。