Stability to Deformations in 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 define manifold convolutions and 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 two frequency dependent manifold filters that split the infinite dimensional spectrum of the LB operator in finite partitions, and prove that these filters are stable to absolute and relative perturbations of the LB operator respectively. We also observe a trade-off between the stability and the discriminability from the stability bounds. Moreover, manifold neural networks (MNNs) composed of these filters inherit the stability properties while the nonlinear activation function helps to improve the discriminability. Therefore, the MNNs can be both stable and discriminative. We verify our results numerically in shape classification with point cloud datasets.