# Universal approximations of permutation invariant/equivariant functions by deep neural networks

arXiv: https://arxiv.org/abs/1903.01939 · HTML: https://mlsanity.com/paper/1903.01939/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Mar 2019.

## Built on by (most cited first, 29 of 29)

- [Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformers](https://mlsanity.com/paper/2107.13163/) (2021)
- [Equivariant Hypergraph Diffusion Neural Operators](https://mlsanity.com/paper/2207.06680/) (2022)
- [Low Dimensional Invariant Embeddings for Universal Geometric Learning](https://mlsanity.com/paper/2205.02956/) (2022)
- [The Exact Sample Complexity Gain from Invariances for Kernel Regression](https://mlsanity.com/paper/2303.14269/) (2023)
- [Explainable Equivariant Neural Networks for Particle Physics: PELICAN](https://mlsanity.com/paper/2307.16506/) (2023)
- [Permutation Invariant Representations with Applications to Graph Deep Learning](https://mlsanity.com/paper/2203.07546/) (2022)
- [Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation](https://mlsanity.com/paper/2112.03491/) (2021)
- [Neural Trees for Learning on Graphs](https://mlsanity.com/paper/2105.07264/) (2021)
- [Symmetry reduction for deep reinforcement learning active control of chaotic spatiotemporal dynamics](https://mlsanity.com/paper/2104.05437/) (2021)
- [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)
- [Postprocessing of Ensemble Weather Forecasts Using Permutation-invariant Neural Networks](https://mlsanity.com/paper/2309.04452/) (2023)
- [Polynomial Width is Sufficient for Set Representation with High-dimensional Features](https://mlsanity.com/paper/2307.04001/) (2023)
- [KASAM: Spline Additive Models for Function Approximation](https://mlsanity.com/paper/2205.06376/) (2022)
- [Addressing imperfect symmetry: A novel symmetry-learning actor-critic extension](https://mlsanity.com/paper/2309.02711/) (2023)
- [Learning multi-modal generative models with permutation-invariant encoders and tighter variational objectives](https://mlsanity.com/paper/2309.00380/) (2023)
- [Approximation of Permutation Invariant Polynomials by Transformers: Efficient Construction in Column-Size](https://mlsanity.com/paper/2502.11467/) (2025)
- [Rethinking Zero-Shot Time Series Classification: From Task-specific Classifiers to In-Context Inference](https://mlsanity.com/paper/2602.00620/) (2026)
- [Any-Dimensional Invariant Universality](https://mlsanity.com/paper/2605.23156/) (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)
- [P-Tensors: a General Formalism for Constructing Higher Order Message Passing Networks](https://mlsanity.com/paper/2306.10767/) (2023)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [Pointspectrum: Equivariance Meets Laplacian Filtering for Graph Representation Learning](https://mlsanity.com/paper/2109.02358/) (2021)
- [On the Approximation and Complexity of Deep Neural Networks to Invariant Functions](https://mlsanity.com/paper/2210.15279/) (2022)
- [A Structural Approach to the Design of Domain Specific Neural Network Architectures](https://mlsanity.com/paper/2301.09381/) (2023)
- [PolyGET: Accelerating Polymer Simulations by Accurate and Generalizable Forcefield with Equivariant Transformer](https://mlsanity.com/paper/2309.00585/) (2023)
- [Decomposition of Equivariant Maps via Invariant Maps: Application to Universal Approximation under Symmetry](https://mlsanity.com/paper/2409.16922/) (2024)
- [Transformers Are Universally Consistent](https://mlsanity.com/paper/2505.24531/) (2025)
- [On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions](https://mlsanity.com/paper/2511.12398/) (2025)

## Abstract (arXiv)

In this paper, we develop a theory about the relationship between G-invariant/equivariant functions and deep neural networks for finite group G. Especially, for a given G-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip G-actions and each affine transformations are G-equivariant/invariant. Due to representation theory, we can show that this approximator has exponentially fewer free parameters than usual models.