  We present simplicial neural networks (SNNs), a generalization of graph
neural networks to data that live on a class of topological spaces called
simplicial complexes. These are natural multi-dimensional extensions of graphs
that encode not only pairwise relationships but also higher-order interactions
between vertices - allowing us to consider richer data, including vector fields
and $n$-fold collaboration networks. We define an appropriate notion of
convolution that we leverage to construct the desired convolutional neural
networks. We test the SNNs on the task of imputing missing data on coauthorship
complexes.
