Convolutional Neural Networks on Manifolds: From Graphs and Back

In recent years, geometric deep learning has gained attraction due to both the need for machine learning on structured data (e.g., graphs) and the increasing availability of this type of data. Extensions of deep convolutional architectures to non-Euclidean domains in particular are a powerful technique in sensor network applications --- which can be seen as graphs --- and 3D model analysis --- which can be seen as manifolds. While recent works have provided a better theoretical understanding of why convolutional neural network architectures work well on graphs of moderate size, in the large-scale regime that is the setting of most problems of interest, their behavior is not as well understood. In this paper, we bridge this gap by modeling large graphs as samples from manifolds and studying manifold neural networks (MNNs). Our main contribution is to define a manifold convolution operation which, when ``discretized'' in both the space and time domains, is consistent with the practical implementation of a graph convolution. We then show that graph neural networks (GNNs) can be particularized from MNNs, which in turn are the limits of these GNNs. We conclude with numerical experiments showcasing an application of the MNN to point-cloud classification.