A WIGNER-ECKART THEOREM FOR GROUP EQUIVARIANT CONVOLUTION KERNELS

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 Gsteerable 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.

1. INTRODUCTION

Undoubtedly, symmetries play a central role in the formulation of physical theories. Any imposed symmetry greatly reduces the set of admissible physical laws and dynamics. Specifically in quantum mechanics, the Hilbert space of a system is equipped with a group representation which specifies the transformation law of system states. Quantum mechanical operators, which map between different states, are required to respect these transformation laws. That is, any symmetry transformation of a state on which they act should lead to a corresponding transformation of the resulting state after their action. This requirement imposes a symmetry constraint on the operators themselves -only specific operators can map between a given pair of states. The situation in equivariant deep learning is remarkably similar to that in physics. Instead of a physical system, one considers in this case some learning task subject to symmetries. For instance, image segmentation is usually assumed to be translationally symmetric: a shift of the input image should lead to a corresponding shift of the predicted segmentation mask. Convolutional networks guarantee this property via their inherent translation equivariance. The role of the quantum states is in equivariant deep learning taken by the features in each layer, which are due to the enforced equivariance endowed with some transformation law. The analog of quantum mechanical operators, mapping between states, is the neural connectivity, mapping between features of consecutive layers. As in the case of operators, there is a symmetry (equivariance) constraint on the neural connectivity -only specific connectivity patterns guarantee a correct transformation law of the resulting features. In this work we are considering group equivariant convolutional networks (GCNNs), which are convolutional networks that are equivariant w.r.t. symmetries of the space on which the convolution is performed. Typical examples are isometry equivariant CNNs on Euclidean spaces (Weiler & Cesa, 2019) or spherical CNNs (Cohen et al., 2018) . Many different formulations of GCNNs have been proposed, however, it has recently been shown that H-equivariant GCNNs on homogeneous spaces

