A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels

• URL: http://arxiv.org/abs/2010.10952v4
• Date: Thu, 21 Jan 2021 10:00:28 GMT
• Title: A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels
• Authors: Leon Lang, Maurice Weiler
• Abstract summary: Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with G-steerable kernels.
• Score: 16.143012623830792
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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