# Manifold Filter-Combine Networks

arXiv: https://arxiv.org/abs/2307.04056 · HTML: https://mlsanity.com/paper/2307.04056/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 73 references that are in the map.
- First posted on arXiv 8 Jul 2023.

## Builds on (most cited first, 30 of 37)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [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)
- [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017)
- [ETA Prediction with Graph Neural Networks in Google Maps](https://mlsanity.com/paper/2108.11482/) (2021)
- [Graphon Neural Networks and the Transferability of Graph Neural Networks](https://mlsanity.com/paper/2006.03548/) (2020)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Transferability of Graph Neural Networks: an Extended Graphon Approach](https://mlsanity.com/paper/2109.10096/) (2021)
- [Geometric Scattering for Graph Data Analysis](https://mlsanity.com/paper/1810.03068/) (2018)
- [A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction](https://mlsanity.com/paper/1512.06293/) (2015)
- [Graph Convolutional Neural Networks via Scattering](https://mlsanity.com/paper/1804.00099/) (2018)
- [Diffusion Scattering Transforms on Graphs](https://mlsanity.com/paper/1806.08829/) (2018)
- [Scaling the Scattering Transform: Deep Hybrid Networks](https://mlsanity.com/paper/1703.08961/) (2017)
- [Stability of Graph Scattering Transforms](https://mlsanity.com/paper/1906.04784/) (2019)
- [Separation and Concentration in Deep Networks](https://mlsanity.com/paper/2012.10424/) (2020)
- [A variational approach to the consistency of spectral clustering](https://mlsanity.com/paper/1508.01928/) (2015)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms](https://mlsanity.com/paper/1805.09450/) (2018)
- [Overcoming Oversmoothness in Graph Convolutional Networks via Hybrid Scattering Networks](https://mlsanity.com/paper/2201.08932/) (2022)
- [Geometric Scattering on Measure Spaces](https://mlsanity.com/paper/2208.08561/) (2022)
- [Graph Neural Tangent Kernel: Convergence on Large Graphs](https://mlsanity.com/paper/2301.10808/) (2023)
- [Convolutional Neural Networks on Manifolds: From Graphs and Back](https://mlsanity.com/paper/2210.00376/) (2022)
- [Stability of Neural Networks on Manifolds to Relative Perturbations](https://mlsanity.com/paper/2110.04702/) (2021)
- [The Manifold Scattering Transform for High-Dimensional Point Cloud Data](https://mlsanity.com/paper/2206.10078/) (2022)
- [Learnable Filters for Geometric Scattering Modules](https://mlsanity.com/paper/2208.07458/) (2022)
- [Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs](https://mlsanity.com/paper/2305.18467/) (2023)

## Built on by (most cited first, 2 of 2)

- [VDW-GNNs: Vector diffusion wavelets for geometric graph neural networks](https://mlsanity.com/paper/2510.01022/) (2025)
- [InfoGain Wavelets: Furthering the Design of Graph Diffusion Wavelets](https://mlsanity.com/paper/2504.08802/) (2025)

## Abstract (arXiv)

In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). Our filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as manifold analogues of various popular GNNs. We propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating an underlying manifold by a sparse graph. We then prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity, and we numerically demonstrate its effectiveness on real-world and synthetic data sets.