# The Role of Fibration Symmetries in Geometric Deep Learning

arXiv: https://arxiv.org/abs/2408.15894 · HTML: https://mlsanity.com/paper/2408.15894/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 58 references that are in the map.
- First posted on arXiv 28 Aug 2024.

## Builds on (most cited first, 25 of 25)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [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)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [A Survey of Transformers](https://mlsanity.com/paper/2106.04554/) (2021)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [Neural Collapse: A Review on Modelling Principles and Generalization](https://mlsanity.com/paper/2206.04041/) (2022)
- [Reducing Overfitting in Deep Networks by Decorrelating Representations](https://mlsanity.com/paper/1511.06068/) (2015)
- [Theory of Graph Neural Networks: Representation and Learning](https://mlsanity.com/paper/2204.07697/) (2022)
- [GraphZoom: A multi-level spectral approach for accurate and scalable graph embedding](https://mlsanity.com/paper/1910.02370/) (2019)
- [Autobahn: Automorphism-based Graph Neural Nets](https://mlsanity.com/paper/2103.01710/) (2021)
- [Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks](https://mlsanity.com/paper/2306.04251/) (2023)
- [Natural Graph Networks](https://mlsanity.com/paper/2007.08349/) (2020)
- [MILE: A Multi-Level Framework for Scalable Graph Embedding](https://mlsanity.com/paper/1802.09612/) (2018)
- [Lov'asz Meets Weisfeiler and Leman](https://mlsanity.com/paper/1802.08876/) (2018)
- [The Expressive Power of Graph Neural Networks: A Survey](https://mlsanity.com/paper/2308.08235/) (2023)
- [Symmetry-Based Representations for Artificial and Biological General Intelligence](https://mlsanity.com/paper/2203.09250/) (2022)
- [Approximately Equivariant Graph Networks](https://mlsanity.com/paper/2308.10436/) (2023)
- [Low-Rank Learning by Design: the Role of Network Architecture and Activation Linearity in Gradient Rank Collapse](https://mlsanity.com/paper/2402.06751/) (2024)
- [Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks](https://mlsanity.com/paper/1306.0322/) (2013)
- [Learning Graph Neural Networks using Exact Compression](https://mlsanity.com/paper/2304.14793/) (2023)
- [Improving Expressivity of Graph Neural Networks](https://mlsanity.com/paper/2004.05994/) (2020)
- [Modular dynamical systems on networks](https://mlsanity.com/paper/1303.3907/) (2013)

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

- [Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks](https://mlsanity.com/paper/2606.02758/) (2026)
- [M-Fibration Theory with Applications to Weighted Graphs](https://mlsanity.com/paper/2608.25598/) (2026)

## Abstract (arXiv)

Geometric Deep Learning (GDL) unifies a broad class of machine learning techniques from the perspectives of symmetries, offering a framework for introducing problem-specific inductive biases like Graph Neural Networks (GNNs). However, the current formulation of GDL is limited to global symmetries that are not often found in real-world problems. We propose to relax GDL to allow for local symmetries, specifically fibration symmetries in graphs, to leverage regularities of realistic instances. We show that GNNs apply the inductive bias of fibration symmetries and derive a tighter upper bound for their expressive power. Additionally, by identifying symmetries in networks, we collapse network nodes, thereby increasing their computational efficiency during both inference and training of deep neural networks. The mathematical extension introduced here applies beyond graphs to manifolds, bundles, and grids for the development of models with inductive biases induced by local symmetries that can lead to better generalization.