Stability to Deformations of Manifold Filters and Manifold Neural Networks

Stability is an important property of graph neural networks (GNNs) which explains their success in many problems of practical interest. Existing GNN stability results depend on the size of the graph, restricting applicability to graphs of moderate size. To understand the stability properties of GNNs on large graphs, we consider neural networks supported on manifolds. These are defined in terms of manifold diffusions mediated by the Laplace-Beltrami (LB) operator and are interpreted as limits of GNNs running on graphs of growing size. We define manifold deformations and show that they lead to perturbations of the manifold's LB operator that consist of an absolute and a relative perturbation term. We then define filters that split the infinite dimensional spectrum of the LB operator in finite partitions, and prove that manifold neural networks (MNNs) with these filters are stable to both, absolute and relative perturbations of the LB operator. Stability results are illustrated numerically in resource allocation problems in wireless networks.