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