Algorithm 1:

Algorithm $A{\star }_{\mathrm{\text{M}}}B$ for invertible $\mathbf{M}$ from ref. 11

INPUT: $A\in {\mathbb{C}}^{m\times p\times n}$, $B\in {\mathbb{C}}^{p\times \ell \times n}$, invertible $\mathbf{M}\in {\mathbb{C}}^{n\times n}$

1: Define $\hat{A}≔A{\times }_3\mathbf{M}$, $\hat{B}≔B{\times }_3\mathbf{M}$

2: for $i=1,\dots ,n$ do

3: ${\hat{C}}_{:,:,i}={\hat{A}}_{:,:,i}{\hat{B}}_{:,:,i}$

4: end for

5: Define-$C=\hat{C}{\times }_3{\mathbf{M}}^{-1}.\text{Now}C=A{\star }_{\mathrm{\text{M}}}B\in {\mathbb{C}}^{m\times \ell \times n}$