# Equivariant non-linear maps for neural networks on homogeneous spaces

arXiv: https://arxiv.org/abs/2504.20974 · HTML: https://mlsanity.com/paper/2504.20974/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2025.

## Built on by (most cited first, 4 of 4)

- [Platonic Transformers: A Solid Choice For Equivariance](https://mlsanity.com/paper/2510.03511/) (2025)
- [The Geometry of Polynomial Group Convolutional Neural Networks](https://mlsanity.com/paper/2603.29566/) (2026)
- [Conditional Clifford-Steerable CNNs for PDE Modeling](https://mlsanity.com/paper/2510.14007/) (2025)
- [Steerable Neural ODEs on Homogeneous Spaces](https://mlsanity.com/paper/2605.11133/) (2026)

## Abstract (arXiv)

This paper presents a novel framework for non-linear equivariant neural network layers on homogeneous spaces. The seminal work of Cohen et al. on equivariant G-CNNs on homogeneous spaces characterized the representation theory of such layers in the linear setting, finding that they are given by convolutions with kernels satisfying so-called steerability constraints. Motivated by the empirical success of non-linear layers, such as self-attention or input dependent kernels, we set out to generalize these insights to the non-linear setting. We derive generalized steerability constraints that any such layer needs to satisfy and prove the universality of our construction. The insights gained into the symmetry-constrained functional dependence of equivariant operators on feature maps and group elements informs the design of future equivariant neural network layers. We demonstrate how several common equivariant network architectures - G-CNNs, implicit steerable kernel networks, conventional and relative position embedded attention based transformers, and LieTransformers - may be derived from our framework.