Stability of Neural Networks on Riemannian Manifolds

Zhiyang Wang; Luana Ruiz; Alejandro Ribeiro

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arXiv · 2103.02663

Stability of Neural Networks on Riemannian Manifolds

Zhiyang Wang·Luana Ruiz·Alejandro Ribeiro

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Abstract

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.

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BibTeX RIS

Zhiyang Wang, Luana Ruiz, Alejandro Ribeiro. 2021-03-03. Stability of Neural Networks on Riemannian Manifolds. https://arxiv.org/abs/2103.02663

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