  We introduce an architecture for processing signals supported on hypergraphs
via graph neural networks (GNNs), which we call a Hyper-graph Expansion Neural
Network (HENN), and provide the first bounds on the stability and
transferability error of a hypergraph signal processing model. To do so, we
provide a framework for bounding the stability and transferability error of
GNNs across arbitrary graphs via spectral similarity. By bounding the
difference between two graph shift operators (GSOs) in the positive
semi-definite sense via their eigenvalue spectrum, we show that this error
depends only on the properties of the GNN and the magnitude of spectral
similarity of the GSOs. Moreover, we show that existing transferability results
that assume the graphs are small perturbations of one another, or that the
graphs are random and drawn from the same distribution or sampled from the same
graphon can be recovered using our approach. Thus, both GNNs and our HENNs
(trained using normalized Laplacians as graph shift operators) will be
increasingly stable and transferable as the graphs become larger. Experimental
results illustrate the importance of considering multiple graph representations
in HENN, and show its superior performance when transferability is desired.
