  High-dimensional data arises in numerous applications, and the rapidly
developing field of geometric deep learning seeks to develop neural network
architectures to analyze such data in non-Euclidean domains, such as graphs and
manifolds. Recent work by Z. Wang, L. Ruiz, and A. Ribeiro has introduced a
method for constructing manifold neural networks using the spectral
decomposition of the Laplace Beltrami operator. Moreover, in this work, the
authors provide a numerical scheme for implementing such neural networks when
the manifold is unknown and one only has access to finitely many sample points.
The authors show that this scheme, which relies upon building a data-driven
graph, converges to the continuum limit as the number of sample points tends to
infinity. Here, we build upon this result by establishing a rate of convergence
that depends on the intrinsic dimension of the manifold but is independent of
the ambient dimension. We also discuss how the rate of convergence depends on
the depth of the network and the number of filters used in each layer.
