# Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functions

arXiv: https://arxiv.org/abs/2409.05191 · HTML: https://mlsanity.com/paper/2409.05191/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 55 references that are in the map.
- First posted on arXiv 8 Sep 2024.

## Builds on (most cited first, 30 of 32)

- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Graph Neural Networks in Recommender Systems: A Survey](https://mlsanity.com/paper/2011.02260/) (2020)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Generalization and Representational Limits of Graph Neural Networks](https://mlsanity.com/paper/2002.06157/) (2020)
- [From Local Structures to Size Generalization in Graph Neural Networks](https://mlsanity.com/paper/2010.08853/) (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)
- [A PAC-Bayesian Approach to Generalization Bounds for Graph Neural Networks](https://mlsanity.com/paper/2012.07690/) (2020)
- [Stability and Generalization of Graph Convolutional Neural Networks](https://mlsanity.com/paper/1905.01004/) (2019)
- [Subgroup Generalization and Fairness of Graph Neural Networks](https://mlsanity.com/paper/2106.15535/) (2021)
- [Generalization Analysis of Message Passing Neural Networks on Large Random Graphs](https://mlsanity.com/paper/2202.00645/) (2022)
- [On Provable Benefits of Depth in Training Graph Convolutional Networks](https://mlsanity.com/paper/2110.15174/) (2021)
- [Transferability of Graph Neural Networks: an Extended Graphon Approach](https://mlsanity.com/paper/2109.10096/) (2021)
- [Learning Theory Can (Sometimes) Explain Generalisation in Graph Neural Networks](https://mlsanity.com/paper/2112.03968/) (2021)
- [Knowledge-aware Coupled Graph Neural Network for Social Recommendation](https://mlsanity.com/paper/2110.03987/) (2021)
- [CoCo: A Coupled Contrastive Framework for Unsupervised Domain Adaptive Graph Classification](https://mlsanity.com/paper/2306.04979/) (2023)
- [Transferability Properties of Graph Neural Networks](https://mlsanity.com/paper/2112.04629/) (2021)
- [Towards Understanding the Generalization of Graph Neural Networks](https://mlsanity.com/paper/2305.08048/) (2023)
- [A User Guide to Low-Pass Graph Signal Processing and its Applications](https://mlsanity.com/paper/2008.01305/) (2020)
- [Learning by Transference: Training Graph Neural Networks on Growing Graphs](https://mlsanity.com/paper/2106.03693/) (2021)
- [Generalization Bounds for Message Passing Networks on Mixture of Graphons](https://mlsanity.com/paper/2404.03473/) (2024)
- [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)
- [Coverage Control in Multi-Robot Systems via Graph Neural Networks](https://mlsanity.com/paper/2109.15278/) (2021)
- [Homophily modulates double descent generalization in graph convolution networks](https://mlsanity.com/paper/2212.13069/) (2022)
- [Stability to Deformations of Manifold Filters and Manifold Neural Networks](https://mlsanity.com/paper/2106.03725/) (2021)
- [Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs](https://mlsanity.com/paper/2305.18467/) (2023)
- [Training Graph Neural Networks on Growing Stochastic Graphs](https://mlsanity.com/paper/2210.15567/) (2022)

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

- [Improved Image Classification with Manifold Neural Networks](https://mlsanity.com/paper/2409.13063/) (2024)
- [Generalization Error Analysis for Attack-Free and Byzantine-Resilient Decentralized Learning with Data Heterogeneity](https://mlsanity.com/paper/2506.09438/) (2025)
- [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)
- [Graph Semi-Supervised Learning for Point Classification on Data Manifolds](https://mlsanity.com/paper/2506.12197/) (2025)
- [Graph Learning Should Move Beyond Restrictive Views of Spectral and Message-Passing GNNs](https://mlsanity.com/paper/2602.10031/) (2026)

## Abstract (arXiv)

In this paper, we study the generalization capabilities of geometric graph neural networks (GNNs). We consider GNNs over a geometric graph constructed from a finite set of randomly sampled points over an embedded manifold with topological information captured. We prove a generalization gap between the optimal empirical risk and the optimal statistical risk of this GNN, which decreases with the number of sampled points from the manifold and increases with the dimension of the underlying manifold. This generalization gap ensures that the GNN trained on a graph on a set of sampled points can be utilized to process other unseen graphs constructed from the same underlying manifold. The most important observation is that the generalization capability can be realized with one large graph instead of being limited to the size of the graph as in previous results. The generalization gap is derived based on the non-asymptotic convergence result of a GNN on the sampled graph to the underlying manifold neural networks (MNNs). We verify this theoretical result with experiments on multiple real-world datasets.