# Stability of Neural Networks on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2103.02663 · HTML: https://mlsanity.com/paper/2103.02663/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Mar 2021.

## Built on by (most cited first, 7 of 7)

- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [Convolutional Neural Networks on Manifolds: From Graphs and Back](https://mlsanity.com/paper/2210.00376/) (2022)
- [Stability to Deformations of Manifold Filters and Manifold Neural Networks](https://mlsanity.com/paper/2106.03725/) (2021)
- [Stability of Neural Networks on Manifolds to Relative Perturbations](https://mlsanity.com/paper/2110.04702/) (2021)
- [Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2210.15058/) (2022)
- [Convolutional Filtering on Sampled Manifolds](https://mlsanity.com/paper/2211.11058/) (2022)
- [Graph Semi-Supervised Learning for Point Classification on Data Manifolds](https://mlsanity.com/paper/2506.12197/) (2025)

## Abstract (arXiv)

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 α-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.