# Graded Representation Theory of Equivariant Neural Networks

arXiv: https://arxiv.org/abs/2609.25776 · HTML: https://mlsanity.com/paper/2609.25776/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 18 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 10 of 10)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity](https://mlsanity.com/paper/1602.05897/) (2016)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Universal Equivariant Multilayer Perceptrons](https://mlsanity.com/paper/2002.02912/) (2020)
- [Equivariant Polynomials for Graph Neural Networks](https://mlsanity.com/paper/2302.11556/) (2023)
- [Any-dimensional equivariant neural networks](https://mlsanity.com/paper/2306.06327/) (2023)
- [A Characterization Theorem for Equivariant Networks with Point-wise Activations](https://mlsanity.com/paper/2401.09235/) (2024)
- [Equivariant neural networks and piecewise linear representation theory](https://mlsanity.com/paper/2408.00949/) (2024)
- [Piecewise Linear Equivariant Maps for Compact Groups](https://mlsanity.com/paper/2608.21645/) (2026)

## Abstract (arXiv)

Nonlinear activations can create equivariant interactions between irreducible representations that linear maps cannot. We use the Gaussian degree decomposition to extend ordinary polynomial degree to such nonlinear maps, and prove that for a fixed coordinatewise equivariant layer each degree factors into a polynomial determined by the linear maps and a scalar determined by the activation. This separates three distinct obstructions, coming from symmetry, coordinates, and activation.