# Piecewise Linear Equivariant Maps for Compact Groups

arXiv: https://arxiv.org/abs/2608.21645 · HTML: https://mlsanity.com/paper/2608.21645/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 11 references that are in the map.
- First posted on arXiv 21 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Functional dimension of feedforward ReLU neural networks](https://mlsanity.com/paper/2209.04036/) (2022)
- [A Characterization Theorem for Equivariant Networks with Point-wise Activations](https://mlsanity.com/paper/2401.09235/) (2024)
- [Neural reproducing kernel Banach spaces and representer theorems for deep networks](https://mlsanity.com/paper/2403.08750/) (2024)
- [Equivariant neural networks and piecewise linear representation theory](https://mlsanity.com/paper/2408.00949/) (2024)

## Built on by (most cited first, 1 of 1)

- [Graded Representation Theory of Equivariant Neural Networks](https://mlsanity.com/paper/2609.25776/) (2026)

## Abstract (arXiv)

Motivated by equivariant neural networks, we study piecewise linear equivariant maps between finite-dimensional real representations of compact groups. We show that all genuinely non-linear piecewise linear behaviour is confined to the subspaces on which the identity component of the group acts trivially, while equivariance forces linearity on the corresponding orthogonal complements. As a consequence, we obtain a compact-group analogue of the finite-group existence criterion of Gibson--Tubbenhauer--Williamson for non-zero equivariant piecewise linear maps between irreducible representations, with the identity component giving rise to a rigidity phenomenon absent from the finite-group case.