  This paper studies the relationship between a graph neural network (GNN) and
a manifold neural network (MNN) when the graph is constructed from a set of
points sampled from the manifold, thus encoding geometric information. We
consider convolutional MNNs and GNNs where the manifold and the graph
convolutions are respectively defined in terms of the Laplace-Beltrami operator
and the graph Laplacian. Using the appropriate kernels, we analyze both dense
and moderately sparse graphs. We prove non-asymptotic error bounds showing that
convolutional filters and neural networks on these graphs converge to
convolutional filters and neural networks on the continuous manifold. As a
byproduct of this analysis, we observe an important trade-off between the
discriminability of graph filters and their ability to approximate the desired
behavior of manifold filters. We then discuss how this trade-off is ameliorated
in neural networks due to the frequency mixing property of nonlinearities. We
further derive a transferability corollary for geometric graphs sampled from
the same manifold. We validate our results numerically on a navigation control
problem and a point cloud classification task.
