# Bases of Steerable Kernels for Equivariant CNNs: From 2D Rotations to the Lorentz Group

arXiv: https://arxiv.org/abs/2603.12459 · HTML: https://mlsanity.com/paper/2603.12459/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 25 references that are in the map.
- First posted on arXiv 12 Mar 2026.

## Builds on (most cited first, 12 of 12)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels](https://mlsanity.com/paper/2010.10952/) (2020)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Implicit Convolutional Kernels for Steerable CNNs](https://mlsanity.com/paper/2212.06096/) (2022)

## Abstract (arXiv)

We present an alternative way of solving the steerable kernel constraint that appears in the design of steerable equivariant convolutional neural networks. We find explicit real and complex bases which are ready to use, for different symmetry groups and for feature maps of arbitrary tensor type. A major advantage of this method is that it bypasses the need to numerically or analytically compute Clebsch-Gordan coefficients and works directly with the representations of the input and output feature maps. The strategy is to find a basis of kernels that respect a simpler invariance condition at some point x0, and then textitsteer it with the defining equation of steerability to move to some arbitrary point x=g· x0. This idea has already been mentioned in the literature before, but not advanced in depth and with some generality. Here we describe how it works with minimal technical tools to make it accessible for a general audience.