# A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels

arXiv: https://arxiv.org/abs/2010.10952 · HTML: https://mlsanity.com/paper/2010.10952/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2020.

## Built on by (most cited first, 30 of 46)

- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [DISCO: accurate Discrete Scale Convolutions](https://mlsanity.com/paper/2106.02733/) (2021)
- [Fourier Transporter: Bi-Equivariant Robotic Manipulation in 3D](https://mlsanity.com/paper/2401.12046/) (2024)
- [ConDor: Self-Supervised Canonicalization of 3D Pose for Partial Shapes](https://mlsanity.com/paper/2201.07788/) (2022)
- [Informing Geometric Deep Learning with Electronic Interactions to Accelerate Quantum Chemistry](https://mlsanity.com/paper/2105.14655/) (2021)
- [INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservation](https://mlsanity.com/paper/2212.14365/) (2022)
- [A case for using rotation invariant features in state of the art feature matchers](https://mlsanity.com/paper/2204.10144/) (2022)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Steerable Partial Differential Operators for Equivariant Neural Networks](https://mlsanity.com/paper/2106.10163/) (2021)
- [Homogeneous vector bundles and G-equivariant convolutional neural networks](https://mlsanity.com/paper/2105.05400/) (2021)
- [Leveraging Symmetries in Pick and Place](https://mlsanity.com/paper/2308.07948/) (2023)
- [E2PN: Efficient SE(3)-Equivariant Point Network](https://mlsanity.com/paper/2206.05398/) (2022)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Clouds](https://mlsanity.com/paper/2111.15341/) (2021)
- [M"obius Convolutions for Spherical CNNs](https://mlsanity.com/paper/2201.12212/) (2022)
- [Lie Group Decompositions for Equivariant Neural Networks](https://mlsanity.com/paper/2310.11366/) (2023)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Implicit Convolutional Kernels for Steerable CNNs](https://mlsanity.com/paper/2212.06096/) (2022)
- [A Characterization Theorem for Equivariant Networks with Point-wise Activations](https://mlsanity.com/paper/2401.09235/) (2024)
- [SCARP: 3D Shape Completion in ARbitrary Poses for Improved Grasping](https://mlsanity.com/paper/2301.07213/) (2023)
- [ShapeMatcher: Self-Supervised Joint Shape Canonicalization, Segmentation, Retrieval and Deformation](https://mlsanity.com/paper/2311.11106/) (2023)
- [Universal approximation and model compression for radial neural networks](https://mlsanity.com/paper/2107.02550/) (2021)
- [Equivariant Single View Pose Prediction Via Induced and Restricted Representations](https://mlsanity.com/paper/2307.03704/) (2023)
- [E(2)-Equivariant Graph Planning for Navigation](https://mlsanity.com/paper/2309.13043/) (2023)
- [Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras](https://mlsanity.com/paper/2310.04521/) (2023)
- [Quiver neural networks](https://mlsanity.com/paper/2207.12773/) (2022)
- [PDO-s3DCNNs: Partial Differential Operator Based Steerable 3D CNNs](https://mlsanity.com/paper/2208.03720/) (2022)

## Abstract (arXiv)

Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with G-steerable kernels, that is, kernels that satisfy an equivariance constraint themselves. While the G-steerability constraint has been derived, it has to date only been solved for specific use cases - a general characterization of G-steerable kernel spaces is still missing. This work provides such a characterization for the practically relevant case of G being any compact group. Our investigation is motivated by a striking analogy between the constraints underlying steerable kernels on the one hand and spherical tensor operators from quantum mechanics on the other hand. By generalizing the famous Wigner-Eckart theorem for spherical tensor operators, we prove that steerable kernel spaces are fully understood and parameterized in terms of 1) generalized reduced matrix elements, 2) Clebsch-Gordan coefficients, and 3) harmonic basis functions on homogeneous spaces.