# Stability to Deformations of Manifold Filters and Manifold Neural Networks

arXiv: https://arxiv.org/abs/2106.03725 · HTML: https://mlsanity.com/paper/2106.03725/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 53 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Graph Neural Networks for Social Recommendation](https://mlsanity.com/paper/1902.07243/) (2019)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Session-based Recommendation with Graph Neural Networks](https://mlsanity.com/paper/1811.00855/) (2018)
- [Stability Properties of Graph Neural Networks](https://mlsanity.com/paper/1905.04497/) (2019)
- [Graphon Neural Networks and the Transferability of Graph Neural Networks](https://mlsanity.com/paper/2006.03548/) (2020)
- [Convolutional Neural Network Architectures for Signals Supported on Graphs](https://mlsanity.com/paper/1805.00165/) (2018)
- [Transferability of Spectral Graph Convolutional Neural Networks](https://mlsanity.com/paper/1907.12972/) (2019)
- [Stability and Generalization of Graph Convolutional Neural Networks](https://mlsanity.com/paper/1905.01004/) (2019)
- [Graph Neural Networks: Architectures, Stability and Transferability](https://mlsanity.com/paper/2008.01767/) (2020)
- [Convergence and Stability of Graph Convolutional Networks on Large Random Graphs](https://mlsanity.com/paper/2006.01868/) (2020)
- [Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks](https://mlsanity.com/paper/2003.03777/) (2020)
- [Unfolding WMMSE using Graph Neural Networks for Efficient Power Allocation](https://mlsanity.com/paper/2009.10812/) (2020)
- [Graph Convolutional Neural Networks via Scattering](https://mlsanity.com/paper/1804.00099/) (2018)
- [Autoregressive Moving Average Graph Filtering](https://mlsanity.com/paper/1602.04436/) (2016)
- [Stability of Graph Scattering Transforms](https://mlsanity.com/paper/1906.04784/) (2019)
- [Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs](https://mlsanity.com/paper/1910.13476/) (2019)
- [Graph and graphon neural network stability](https://mlsanity.com/paper/2010.12529/) (2020)
- [Unsupervised Learning for Asynchronous Resource Allocation in Ad-hoc Wireless Networks](https://mlsanity.com/paper/2011.02644/) (2020)
- [Stability of Neural Networks on Riemannian Manifolds](https://mlsanity.com/paper/2103.02663/) (2021)

## Built on by (most cited first, 9 of 9)

- [A Manifold Perspective on the Statistical Generalization of Graph Neural Networks](https://mlsanity.com/paper/2406.05225/) (2024)
- [Convolutional Neural Networks on Manifolds: From Graphs and Back](https://mlsanity.com/paper/2210.00376/) (2022)
- [Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs](https://mlsanity.com/paper/2305.18467/) (2023)
- [Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functions](https://mlsanity.com/paper/2409.05191/) (2024)
- [Generalization of Graph Neural Networks is Robust to Model Mismatch](https://mlsanity.com/paper/2408.13878/) (2024)
- [Improved Image Classification with Manifold Neural Networks](https://mlsanity.com/paper/2409.13063/) (2024)
- [VDW-GNNs: Vector diffusion wavelets for geometric graph neural networks](https://mlsanity.com/paper/2510.01022/) (2025)
- [Size-Transferability of Graph Transformers with Convolutional Positional Encodings](https://mlsanity.com/paper/2602.15239/) (2026)
- [Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves](https://mlsanity.com/paper/2605.06395/) (2026)

## Abstract (arXiv)

The paper defines and studies manifold (M) convolutional filters and neural networks (NNs). Manifold filters and MNNs are defined in terms of the Laplace-Beltrami operator exponential and are such that 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.