  In this tutorial, we provide a didactic treatment of the emerging topic of
signal processing on higher-order networks. Drawing analogies from discrete and
graph signal processing, we introduce the building blocks for processing data
on simplicial complexes and hypergraphs, two common higher-order network
abstractions that can incorporate polyadic relationships. We provide brief
introductions to simplicial complexes and hypergraphs, with a special emphasis
on the concepts needed for the processing of signals supported on these
structures. Specifically, we discuss Fourier analysis, signal denoising, signal
interpolation, node embeddings, and nonlinear processing through neural
networks, using these two higher-order network models. In the context of
simplicial complexes, we specifically focus on signal processing using the
Hodge Laplacian matrix, a multi-relational operator that leverages the special
structure of simplicial complexes and generalizes desirable properties of the
Laplacian matrix in graph signal processing. For hypergraphs, we present both
matrix and tensor representations, and discuss the trade-offs in adopting one
or the other. We also highlight limitations and potential research avenues,
both to inform practitioners and to motivate the contribution of new
researchers to the area.
