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 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.

100 pages

References in corpus (10)

• Group Equivariant Convolutional Networks
• Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds
• On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups
• 3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data
• Spherical CNNs
• General $E(2)$-Equivariant Steerable CNNs
• Lorentz Group Equivariant Neural Network for Particle Physics
• N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials
• Deep Scale-spaces: Equivariance Over Scale
• Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs

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• Rotation Equivariant 3D Hand Mesh Generation from a Single RGB Image