Schur Nets: exploiting local structure for equivariance in higher order graph neural networks

Recent works have shown that extending the message passing paradigm to subgraphs communicating with other subgraphs, especially via higher order messages, can boost the expressivity of graph neural networks. In such architectures, to faithfully account for local structure such as cycles, the local operations must be equivariant to the automorphism group of the local environment. However, enumerating the automorphism groups of all subgraphs of interest and finding appropriate equivariant operations for each one of them separately is generally not feasible. In this paper we propose a solution to this problem based on spectral graph theory that bypasses having to determine the automorphism group entirely and constructs a basis for equivariant operations directly from the graph Laplacian. We show that this approach can boost the performance of GNNs on some standard benchmarks.