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.

翻译：在arXiv:2212.08630中，刻画了$\mathbb{R}^{n}$的张量幂空间之间可学习的线性神经网络层，这些层对正交群$O(n)$、特殊正交群$SO(n)$和辛群$Sp(n)$具有等变性。我们提出一种算法，通过范畴论构造实现上述各群的权重矩阵与向量的乘法。与朴素实现相比，利用Kronecker积矩阵执行乘法操作可显著降低计算成本。我们证明该方法可扩展至对称群$S_n$，并在此过程中恢复了arXiv:2303.06208中的算法。