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

• URL: http://arxiv.org/abs/2304.14165v1
• Date: Thu, 27 Apr 2023 13:06:07 GMT
• Title: An Algorithm for Computing with Brauer's Group Equivariant Neural Network Layers
• Authors: Edward Pearce-Crump
• Abstract summary: 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 show that our approach extends to the symmetric group, $S_n$, recovering the algorithm of arXiv:2303.06208 in the process.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The learnable, linear neural network layers between tensor power spaces of $\mathbb{R}^{n}$ 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.

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