  Convolutional Neural Networks (CNNs) have been applied to data with
underlying non-Euclidean structures and have achieved impressive successes.
This brings the stability analysis of CNNs on non-Euclidean domains into notice
because CNNs have been proved stable on Euclidean domains. This paper focuses
on the stability of CNNs on Riemannian manifolds. By taking the
Laplace-Beltrami operators into consideration, we construct an
$\alpha$-frequency difference threshold filter to help separate the spectrum of
the operator with an infinite dimensionality. We further construct a manifold
neural network architecture with these filters. We prove that both the manifold
filters and neural networks are stable under absolute perturbations to the
operators. The results also implicate a trade-off between the stability and
discriminability of manifold neural networks. Finally we verify our conclusions
with numerical experiments in a wireless adhoc network scenario.
