An Algorithm for Computing with Brauer's Group Equivariant Neural Network Layers

04/27/2023

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by   Edward Pearce-Crump, et al.

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The learnable, linear neural network layers between tensor power spaces of ℝ^n that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.