Algorithm 3:

Return t-SVDMII under ${\star }_{\mathrm{\text{M}}}$, $\mathit{\rho }$ to meet energy constraint

INPUT: $A$, $\mathbf{M}$ a multiple of unitary matrix; desired energy $\gamma \in \left(\mathrm{0,1}\right]$.

1: Compute t-SVDM of $A$.

2: Concatenate $\left(\left({\hat{S}}_{j,j,i}\right){.}^2\right)$ for all $i,j$ into a vector $\mathbf{v}$.

3: $\mathbf{v}\leftarrow \text{sort}\left(\mathbf{v}{,}^{\prime }{\text{descend}}^{\prime }\right)$.

4: Let $\mathbf{w}$ be the vector of cumulative sums: i.e., ${\mathbf{w}}_k={\sum }_{i=1}^k{\mathbf{v}}_i$

5: Find the first index $J$ such that ${\mathbf{w}}_J/{\Vert \widehat{S}\Vert }_F^2>\gamma$.

6: Define $\tau ≔{\mathbf{v}}_J$.

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

8: Set ${\rho }_i$ as number of singular values for ${\hat{A}}_{:,:,i}$ greater or equal to $\tau$.

9: Keep only the $m\times {\rho }_i\hspace{0.17em}{\hat{U}}_{:,1:{\rho }_i,i}$ and ${\hat{G}}_{\mathit{\rho }}≔{\hat{S}}_{1:{\rho }_i,1:{\rho }_i,i}{\hat{V}}_{:,1:{\rho }_i,i}^{\mathrm{H}}$.

End for