# Decomposition of Equivariant Maps via Invariant Maps: Application to Universal Approximation under Symmetry

arXiv: https://arxiv.org/abs/2409.16922 · HTML: https://mlsanity.com/paper/2409.16922/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 53 references that are in the map.
- First posted on arXiv 25 Sep 2024.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [Sign and Basis Invariant Networks for Spectral Graph Representation Learning](https://mlsanity.com/paper/2202.13013/) (2022)
- [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)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations](https://mlsanity.com/paper/1708.02691/) (2017)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [On the Universality of Rotation Equivariant Point Cloud Networks](https://mlsanity.com/paper/2010.02449/) (2020)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [Universal Approximation of Functions on Sets](https://mlsanity.com/paper/2107.01959/) (2021)
- [Universal Equivariant Multilayer Perceptrons](https://mlsanity.com/paper/2002.02912/) (2020)
- [On Universal Equivariant Set Networks](https://mlsanity.com/paper/1910.02421/) (2019)
- [Transformers Generalize DeepSets and Can be Extended to Graphs and Hypergraphs](https://mlsanity.com/paper/2110.14416/) (2021)
- [Improved Generalization Bounds of Group Invariant / Equivariant Deep Networks via Quotient Feature Spaces](https://mlsanity.com/paper/1910.06552/) (2019)
- [Universal approximations of permutation invariant/equivariant functions by deep neural networks](https://mlsanity.com/paper/1903.01939/) (2019)
- [Towards fully covariant machine learning](https://mlsanity.com/paper/2301.13724/) (2023)

## Abstract (arXiv)

In this paper, we develop a theory about the relationship between invariant and equivariant maps with regard to a group G. We then leverage this theory in the context of deep neural networks with group symmetries in order to obtain novel insight into their mechanisms. More precisely, we establish a one-to-one relationship between equivariant maps and certain invariant maps. This allows us to reduce arguments for equivariant maps to those for invariant maps and vice versa. As an application, we propose a construction of universal equivariant architectures built from universal invariant networks. We, in turn, explain how the universal architectures arising from our construction differ from standard equivariant architectures known to be universal. Furthermore, we explore the complexity, in terms of the number of free parameters, of our models, and discuss the relation between invariant and equivariant networks' complexity. Finally, we also give an approximation rate for G-equivariant deep neural networks with ReLU activation functions for finite group G.