# Universal Approximation Theorem for Equivariant Maps by Group CNNs

arXiv: https://arxiv.org/abs/2012.13882 · HTML: https://mlsanity.com/paper/2012.13882/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Dec 2020.

## Built on by (most cited first, 12 of 12)

- [fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions](https://mlsanity.com/paper/2406.07456/) (2024)
- [Brauer's Group Equivariant Neural Networks](https://mlsanity.com/paper/2212.08630/) (2022)
- [Approximation Properties of Deep ReLU CNNs](https://mlsanity.com/paper/2109.00190/) (2021)
- [Universality of Group Convolutional Neural Networks Based on Ridgelet Analysis on Groups](https://mlsanity.com/paper/2205.14819/) (2022)
- [Approximately Equivariant Neural Processes](https://mlsanity.com/paper/2406.13488/) (2024)
- [Ghosts in Neural Networks: Existence, Structure and Role of Infinite-Dimensional Null Space](https://mlsanity.com/paper/2106.04770/) (2021)
- [3D Scalable Quantum Convolutional Neural Networks for Point Cloud Data Processing in Classification Applications](https://mlsanity.com/paper/2210.09728/) (2022)
- [On Non-Linear operators for Geometric Deep Learning](https://mlsanity.com/paper/2207.03485/) (2022)
- [Equivariant geometric convolutions for emulation of dynamical systems](https://mlsanity.com/paper/2305.12585/) (2023)
- [Equivariant and Steerable Neural Networks: A review with special emphasis on the symmetric group](https://mlsanity.com/paper/2301.03019/) (2023)
- [A unified Fourier slice method to derive ridgelet transform for a variety of depth-2 neural networks](https://mlsanity.com/paper/2402.15984/) (2024)
- [Design equivariant neural networks for 3D point cloud](https://mlsanity.com/paper/2205.00630/) (2022)

## Abstract (arXiv)

Group symmetry is inherent in a wide variety of data distributions. Data processing that preserves symmetry is described as an equivariant map and often effective in achieving high performance. Convolutional neural networks (CNNs) have been known as models with equivariance and shown to approximate equivariant maps for some specific groups. However, universal approximation theorems for CNNs have been separately derived with individual techniques according to each group and setting. This paper provides a unified method to obtain universal approximation theorems for equivariant maps by CNNs in various settings. As its significant advantage, we can handle non-linear equivariant maps between infinite-dimensional spaces for non-compact groups.