# A Convergence Rate for Manifold Neural Networks

arXiv: https://arxiv.org/abs/2212.12606 · HTML: https://mlsanity.com/paper/2212.12606/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 25 references that are in the map.
- First posted on arXiv 23 Dec 2022.

## Builds on (most cited first, 12 of 12)

- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [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)
- [ETA Prediction with Graph Neural Networks in Google Maps](https://mlsanity.com/paper/2108.11482/) (2021)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [Geometric Scattering on Measure Spaces](https://mlsanity.com/paper/2208.08561/) (2022)
- [Balancing Geometry and Density: Path Distances on High-Dimensional Data](https://mlsanity.com/paper/2012.09385/) (2020)
- [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)
- [Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scaling](https://mlsanity.com/paper/2209.08004/) (2022)
- [Convolutional Filtering on Sampled Manifolds](https://mlsanity.com/paper/2211.11058/) (2022)

## Built on by (most cited first, 6 of 6)

- [A Manifold Perspective on the Statistical Generalization of Graph Neural Networks](https://mlsanity.com/paper/2406.05225/) (2024)
- [Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs](https://mlsanity.com/paper/2305.18467/) (2023)
- [Generalization of Graph Neural Networks is Robust to Model Mismatch](https://mlsanity.com/paper/2408.13878/) (2024)
- [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)
- [Convergence of Manifold Filter-Combine Networks](https://mlsanity.com/paper/2410.14639/) (2024)

## Abstract (arXiv)

High-dimensional data arises in numerous applications, and the rapidly developing field of geometric deep learning seeks to develop neural network architectures to analyze such data in non-Euclidean domains, such as graphs and manifolds. Recent work by Z. Wang, L. Ruiz, and A. Ribeiro has introduced a method for constructing manifold neural networks using the spectral decomposition of the Laplace Beltrami operator. Moreover, in this work, the authors provide a numerical scheme for implementing such neural networks when the manifold is unknown and one only has access to finitely many sample points. The authors show that this scheme, which relies upon building a data-driven graph, converges to the continuum limit as the number of sample points tends to infinity. Here, we build upon this result by establishing a rate of convergence that depends on the intrinsic dimension of the manifold but is independent of the ambient dimension. We also discuss how the rate of convergence depends on the depth of the network and the number of filters used in each layer.