Cell Complex Neural Networks

TL;DR

This paper introduces Cell Complex Neural Networks (CXNs), a novel framework for neural computations on topological cell complexes, generalizing message passing and embedding methods from graphs to complex topological structures.

Contribution

It proposes a unifying, combinatorial neural network framework for cell complexes, extending message passing and embedding techniques to complex topological spaces.

Findings

Developed a message passing scheme that incorporates topology.

Created a cell complex autoencoder for learning representations.

Generalized node2vec to cell2vec in complex spaces.

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.