# On Universality Classes of Equivariant Networks

arXiv: https://arxiv.org/abs/2506.02293 · HTML: https://mlsanity.com/paper/2506.02293/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 60 references that are in the map.
- First posted on arXiv 2 Jun 2025.

## Builds on (most cited first, 30 of 42)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [On the equivalence between graph isomorphism testing and function approximation with GNNs](https://mlsanity.com/paper/1905.12560/) (2019)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Equivariant Subgraph Aggregation Networks](https://mlsanity.com/paper/2110.02910/) (2021)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Understanding and Extending Subgraph GNNs by Rethinking Their Symmetries](https://mlsanity.com/paper/2206.11140/) (2022)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Expressive Power of Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/2006.15646/) (2020)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [On the Expressive Power of Geometric Graph Neural Networks](https://mlsanity.com/paper/2301.09308/) (2023)
- [Subgraph Neural Networks](https://mlsanity.com/paper/2006.10538/) (2020)
- [Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces](https://mlsanity.com/paper/2211.14400/) (2022)
- [Universal Equivariant Multilayer Perceptrons](https://mlsanity.com/paper/2002.02912/) (2020)
- [On Universal Equivariant Set Networks](https://mlsanity.com/paper/1910.02421/) (2019)

## Built on by (most cited first, 4 of 4)

- [On Universality of Deep Equivariant Networks](https://mlsanity.com/paper/2510.15814/) (2025)
- [If you can distinguish, you can express: Galois theory, Stone--Weierstrass, machine learning, and linguistics](https://mlsanity.com/paper/2510.09902/) (2025)
- [A Unified Framework for Vision Transformers Equivariant to Discrete Subgroups of O(2)](https://mlsanity.com/paper/2606.27864/) (2026)
- [Transversal Pooling Neural Networks](https://mlsanity.com/paper/2609.36237/) (2026)

## Abstract (arXiv)

Equivariant neural networks provide a principled framework for incorporating symmetry into learning architectures and have been extensively analyzed through the lens of their separation power, that is, the ability to distinguish inputs modulo symmetry. This notion plays a central role in settings such as graph learning, where it is often formalized via the Weisfeiler-Leman hierarchy. In contrast, the universality of equivariant models-their capacity to approximate target functions-remains comparatively underexplored. In this work, we investigate the approximation power of equivariant neural networks beyond separation constraints. We show that separation power does not fully capture expressivity: models with identical separation power may differ in their approximation ability. To demonstrate this, we characterize the universality classes of shallow invariant networks, providing a general framework for understanding which functions these architectures can approximate. Since equivariant models reduce to invariant ones under projection, this analysis yields sufficient conditions under which shallow equivariant networks fail to be universal. Conversely, we identify settings where shallow models do achieve separation-constrained universality. These positive results, however, depend critically on structural properties of the symmetry group, such as the existence of adequate normal subgroups, which may not hold in important cases like permutation symmetry.