  The paper defines and studies manifold (M) convolutional filters and neural
networks (NNs). \emph{Manifold} filters and MNNs are defined in terms of the
Laplace-Beltrami operator exponential and are such that \emph{graph} (G)
filters and neural networks (NNs) are recovered as discrete approximations when
the manifold is sampled. These filters admit a spectral representation which is
a generalization of both the spectral representation of graph filters and the
frequency response of standard convolutional filters in continuous time. The
main technical contribution of the paper is to analyze the stability of
manifold filters and MNNs to smooth deformations of the manifold. This analysis
generalizes known stability properties of graph filters and GNNs and it is also
a generalization of known stability properties of standard convolutional
filters and neural networks in continuous time. The most important observation
that follows from this analysis is that manifold filters, same as graph filters
and standard continuous time filters, have difficulty discriminating high
frequency components in the presence of deformations. This is a challenge that
can be ameliorated with the use of manifold, graph, or continuous time neural
networks. The most important practical consequence of this analysis is to shed
light on the behavior of graph filters and GNNs in large-scale graphs.
