# Convolutional Neural Networks on Manifolds: From Graphs and Back

arXiv: https://arxiv.org/abs/2210.00376 · HTML: https://mlsanity.com/paper/2210.00376/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 46 references that are in the map.
- First posted on arXiv 1 Oct 2022.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [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)
- [FFDNet: Toward a Fast and Flexible Solution for CNN based Image Denoising](https://mlsanity.com/paper/1710.04026/) (2017)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Generalization and Representational Limits of Graph Neural Networks](https://mlsanity.com/paper/2002.06157/) (2020)
- [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)
- [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)
- [Graph Convolutional Neural Networks via Scattering](https://mlsanity.com/paper/1804.00099/) (2018)
- [Graphon Signal Processing](https://mlsanity.com/paper/2003.05030/) (2020)
- [Stability of Graph Scattering Transforms](https://mlsanity.com/paper/1906.04784/) (2019)
- [Learning Decentralized Wireless Resource Allocations with Graph Neural Networks](https://mlsanity.com/paper/2107.01489/) (2021)
- [Spatiotemporal CNN for Video Object Segmentation](https://mlsanity.com/paper/1904.02363/) (2019)
- [Algebraic Neural Networks: Stability to Deformations](https://mlsanity.com/paper/2009.01433/) (2020)
- [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)
- [Graphons: A Nonparametric Method to Model, Estimate, and Design Algorithms for Massive Networks](https://mlsanity.com/paper/1706.01143/) (2017)
- [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)
- [Stability of Neural Networks on Riemannian Manifolds](https://mlsanity.com/paper/2103.02663/) (2021)

## Built on by (most cited first, 14 of 14)

- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [A Manifold Perspective on the Statistical Generalization of Graph Neural Networks](https://mlsanity.com/paper/2406.05225/) (2024)
- [Equivariant Neural Operator Learning with Graphon Convolution](https://mlsanity.com/paper/2311.10908/) (2023)
- [Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2210.15058/) (2022)
- [Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs](https://mlsanity.com/paper/2305.18467/) (2023)
- [A Convergence Rate for Manifold Neural Networks](https://mlsanity.com/paper/2212.12606/) (2022)
- [Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functions](https://mlsanity.com/paper/2409.05191/) (2024)
- [Convolutional Filtering on Sampled Manifolds](https://mlsanity.com/paper/2211.11058/) (2022)
- [Generalization of Graph Neural Networks is Robust to Model Mismatch](https://mlsanity.com/paper/2408.13878/) (2024)
- [Transferability of Graph Neural Networks using Graphon and Sampling Theories](https://mlsanity.com/paper/2307.13206/) (2023)
- [Manifold Filter-Combine Networks](https://mlsanity.com/paper/2307.04056/) (2023)
- [VDW-GNNs: Vector diffusion wavelets for geometric graph neural networks](https://mlsanity.com/paper/2510.01022/) (2025)
- [Manifold limit for the training of shallow graph convolutional neural networks](https://mlsanity.com/paper/2601.06025/) (2026)
- [Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves](https://mlsanity.com/paper/2605.06395/) (2026)

## Abstract (arXiv)

Geometric deep learning has gained much attention in recent years due to more available data acquired from non-Euclidean domains. Some examples include point clouds for 3D models and wireless sensor networks in communications. Graphs are common models to connect these discrete data points and capture the underlying geometric structure. With the large amount of these geometric data, graphs with arbitrarily large size tend to converge to a limit model -- the manifold. Deep neural network architectures have been proved as a powerful technique to solve problems based on these data residing on the manifold. In this paper, we propose a manifold neural network (MNN) composed of a bank of manifold convolutional filters and point-wise nonlinearities. We define a manifold convolution operation which is consistent with the discrete graph convolution by discretizing in both space and time domains. To sum up, we focus on the manifold model as the limit of large graphs and construct MNNs, while we can still bring back graph neural networks by the discretization of MNNs. We carry out experiments based on point-cloud dataset to showcase the performance of our proposed MNNs.