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 H/G can in a fairly general setting be understood as performing convolutions with G-steerable kernels (Cohen et al., 2019b) . Convolutional weight sharing hereby guarantees the equivariance under "translations" of the space while G-steerability is a constraint on the convolution kernel that ensures its equivariance under the action of the stabilizer subgroup G < H. Although the space of Gsteerable kernels has been characterized for specific choices of groups G and feature transformation laws, i.e., group representations ρ, see Section 5, no general solution was known so far. This work characterizes the solution space for arbitrary compact groups G. • We present a generalized Wigner-Eckart theorem 4.1 for G-steerable kernels. It describes the general structure of equivariant kernels in terms of 1) endomorphism bases, which generalize reduced matrix elements, 2) Clebsch-Gordan coefficients, and 3) harmonic basis functions on a suitable homogeneous space. In contrast to the usual formulation, we cover any compact group G and both real and complex representations. • Corollary 4.2 explains how to parameterize G-steerable kernels and thus GCNNs. • We apply the theorem exemplarily to solve for the kernel spaces for the symmetry groups SO(2) , Z/2 , SO(3) and O(3) , considering both real and complex representations. Thereby, we demonstrate that the endomorphism bases, Clebsch-Gordan coefficients, and harmonic basis functions can usually be determined for practically relevant symmetry groups.

2. SYMMETRY-CONSTRAINED OPERATORS AND THEIR MATRIX ELEMENTS

To motivate our generalized Wigner-Eckart theorem, we review quantum mechanical representation operators and G-steerable kernels with an emphasis on the similarity of their underlying symmetry constraints. Due to their symmetries, the matrix elements of such operators and kernels are fully specified by a comparatively small number of reduced matrix elements or learnable parameters, respectively. This reduction is for representation operators described by the Wigner-Eckart theorem. For clarity, we discuss this theorem in its most popular form, i.e., for spherical tensor operators (SO(3)-representation operators transforming under irreducible representations). The Representation Operator Constraint Consider a quantum mechanical system with symmetry under the action of some group G, for instance rotations. The action of this symmetry group on quantum states is modeled by some unitary G-representationfoot_0 U : G → U(H) on the Hilbert space H. More specifically, G acts on kets according to |ψ → |ψ := U (g) |ψ and on bras according to ψ| → ψ | := ψ| U (g) † , where U (g) † is the adjoint of U (g). Observables of the system correspond to self-adjoint operators A = A † . The expectation value of such an observable in some quantum state |ψ is given by ψ|A|ψ ∈ R. The transformation behaviors of states and observables need to be consistent with each other. As an example, consider a system consisting of a single, free particle in R 3 , which is (among other symmetries) symmetric under rotations G = SO(3). The momentum of the particle in the direction of the three frame axes is measured by the three momentum operators (P 1 , P 2 , P 3 ). Since the momentum of a classical particle transforms geometrically like a vector, one needs to demand the same for the momentum observable expectation values. If we denote by p i := ψ|P i |ψ the expected momentum in i-direction, this means that the expected momentum of a rotated system is given by p i = j R ij p j = j R ij ψ|P j |ψ , where R ∈ SO(3) is an element of the rotation group. This result should agree with the expectation values for rotated system states, that is, p i = ψ |P i |ψ = ψ|U (R) † P i U (R)|ψ . As this argument is independent from the particular choice of state |ψ , and making use of the linearity of the operations, this implies a consistency



Unitary representations are explained in Section 3. The notation U for the operator is distinct from the notation U of the unitary group U(H).



Our solution is motivated by the close resemblance of the G-steerability kernel constraint to the defining constraint of spherical tensor operators (or more general representation operators (Jeevanjee, 2011)) in quantum mechanics. The famous Wigner-Eckart theorem describes the general structure of these operators by Clebsch-Gordan coefficients, with the degrees of freedom given by reduced matrix elements. By generalizing this theorem, we find a general characterization and parameterization of G-steerable kernel spaces. For specific examples, like G = SO(3) or compact subgroups of G = O(2), our kernel space solution specializes to earlier work, e.g., Worrall et al. (2016); Thomas et al. (2018); Weiler & Cesa (2019). Our main contributions are the following:

