Hi everyone, You are all cordially invited to the AMLab Seminar on Thursday 12nd Nov. at 16:00 CET on Zoom, where Leon Lang will give a talk titled “A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels“.
Title: 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) ClebschGordan coefficients, and 3) harmonic basis functions on homogeneous spaces.
Link to paper: https://arxiv.org/pdf/2010.10952.pdf
To gain more deep insights into Group Equivariance, feel free to join and discuss it 🙂 !