A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels

TL;DR

This paper generalizes the Wigner-Eckart theorem to characterize G-steerable kernels in group equivariant convolutional networks, providing a comprehensive mathematical framework for understanding these kernels for any compact group.

Contribution

It offers a complete characterization of G-steerable kernels for compact groups, connecting steerability constraints to spherical tensor operators via a generalized Wigner-Eckart theorem.

Findings

Kernel spaces are fully characterized by reduced matrix elements, Clebsch-Gordan coefficients, and harmonic basis functions.

Provides a theoretical foundation for designing G-equivariant kernels in neural networks.

Bridges concepts from quantum mechanics and deep learning to advance symmetry-aware models.

Abstract

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.