Cell Complex Neural Networks

Mustafa Hajij, Kyle Istvan, Ghada Zamzmi

Keywords: Graph Neural Networks, Cell Complex, Simplicial Complex

Abstract: Cell complexes are topological spaces constructed from simple blocks called cells. They generalize graphs, simplicial complexes, and polyhedral complexes that form important domains for practical applications. They also provide a combinatorial formalism that allows the inclusion of complicated relationships of restrictive structures such as graphs and meshes. In this paper, we propose \textbf{cell complexes neural networks (CXNs)} a general, combinatorial, and unifying construction for performing neural network-type computations on cell complexes. We introduce an inter-cellular message passing scheme on cell complexes that takes the topology of the underlying space into account and generalizes message passing scheme to graphs. Finally, we introduce a unified cell complex encoder-decoder framework that enables learning representation of cells for a given complex inside the Euclidean spaces. In particular, we show how our cell complex autoencoder construction can give in the special case \textbf{cell2vec}, a generalization for node2vec.

Previous Submission: No

TL;DR: Cell complex networks are massively general computational schemes that allow performing neural network-type computations on cell complexes, domains that generalize graphs and simplicial complexes and have geometric and combinatorial characteristics.