# Universal Equivariant Multilayer Perceptrons

arXiv: https://arxiv.org/abs/2002.02912 · HTML: https://mlsanity.com/paper/2002.02912/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Feb 2020.

## Built on by (most cited first, 30 of 49)

- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Sign and Basis Invariant Networks for Spectral Graph Representation Learning](https://mlsanity.com/paper/2202.13013/) (2022)
- [Weisfeiler and Leman go Machine Learning: The Story so far](https://mlsanity.com/paper/2112.09992/) (2021)
- [Understanding and Extending Subgraph GNNs by Rethinking Their Symmetries](https://mlsanity.com/paper/2206.11140/) (2022)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [The Exact Sample Complexity Gain from Invariances for Kernel Regression](https://mlsanity.com/paper/2303.14269/) (2023)
- [E(2)-Equivariant Vision Transformer](https://mlsanity.com/paper/2306.06722/) (2023)
- [A Canonicalization Perspective on Invariant and Equivariant Learning](https://mlsanity.com/paper/2405.18378/) (2024)
- [A PAC-Bayesian Generalization Bound for Equivariant Networks](https://mlsanity.com/paper/2210.13150/) (2022)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [A Simple and Universal Rotation Equivariant Point-cloud Network](https://mlsanity.com/paper/2203.01216/) (2022)
- [Universality of Group Convolutional Neural Networks Based on Ridgelet Analysis on Groups](https://mlsanity.com/paper/2205.14819/) (2022)
- [Deep Neural Network Approximation of Invariant Functions through Dynamical Systems](https://mlsanity.com/paper/2208.08707/) (2022)
- [Implicit Convolutional Kernels for Steerable CNNs](https://mlsanity.com/paper/2212.06096/) (2022)
- [A Characterization Theorem for Equivariant Networks with Point-wise Activations](https://mlsanity.com/paper/2401.09235/) (2024)
- [Universal approximation and model compression for radial neural networks](https://mlsanity.com/paper/2107.02550/) (2021)
- [A Unified Framework for Discovering Discrete Symmetries](https://mlsanity.com/paper/2309.02898/) (2023)
- [Identifiable Equivariant Networks are Layerwise Equivariant](https://mlsanity.com/paper/2601.21645/) (2026)
- [Geometrical aspects of lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2303.11448/) (2023)
- [A Classification of G-invariant Shallow Neural Networks](https://mlsanity.com/paper/2205.09219/) (2022)
- [How Jellyfish Characterise Alternating Group Equivariant Neural Networks](https://mlsanity.com/paper/2301.10152/) (2023)
- [Rotation Equivariant Arbitrary-scale Image Super-Resolution](https://mlsanity.com/paper/2508.05160/) (2025)
- [Graph Automorphism Group Equivariant Neural Networks](https://mlsanity.com/paper/2307.07810/) (2023)
- [Symmetry-Aware Generative Modeling through Learned Canonicalization](https://mlsanity.com/paper/2501.07773/) (2025)
- [On Universality Classes of Equivariant Networks](https://mlsanity.com/paper/2506.02293/) (2025)
- [On Universality of Deep Equivariant Networks](https://mlsanity.com/paper/2510.15814/) (2025)
- [Vanilla Group Equivariant Vision Transformer: Simple and Effective](https://mlsanity.com/paper/2602.08047/) (2026)
- [Capacity of Group-invariant Linear Readouts from Equivariant Representations: How Many Objects can be Linearly Classified Under All Possible Views?](https://mlsanity.com/paper/2110.07472/) (2021)
- [Densely Connected G-invariant Deep Neural Networks with Signed Permutation Representations](https://mlsanity.com/paper/2303.04614/) (2023)

## Abstract (arXiv)

Group invariant and equivariant Multilayer Perceptrons (MLP), also known as Equivariant Networks, have achieved remarkable success in learning on a variety of data structures, such as sequences, images, sets, and graphs. Using tools from group theory, this paper proves the universality of a broad class of equivariant MLPs with a single hidden layer. In particular, it is shown that having a hidden layer on which the group acts regularly is sufficient for universal equivariance (invariance). A corollary is unconditional universality of equivariant MLPs for Abelian groups, such as CNNs with a single hidden layer. A second corollary is the universality of equivariant MLPs with a high-order hidden layer, where we give both group-agnostic bounds and means for calculating group-specific bounds on the order of hidden layer that guarantees universal equivariance (invariance).