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

The learnable, linear neural network layers between tensor power spaces of ℝn 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, Sn, recovering the algorithm of arXiv:2303.06208 in the process.