Algorithm 2: Proposed method: Monte Carlo AMM with preprocessing

Require:  $\mathit{A}\in {\mathbb{R}}^{m\times k}$ and $\mathit{B}\in {\mathbb{R}}^{k\times n}$, and $c\in \mathbb{N}$

Ensure:  $\mathit{Z}\in {\mathbb{R}}^{m\times n}$, such that $\parallel \mathit{AB}-\mathit{Z}\parallel$ is small

1:  $\mathit{Z}\leftarrow \mathit{O}$

2:  ${\mathit{v}}_{\mathrm{c}}\leftarrow \mathit{A}\mathit{\beta }$, ${\mathit{v}}_{\mathrm{r}}\leftarrow {\mathbf{\alpha }}^{\top }\mathit{B}$, $w\leftarrow {\mathbf{\alpha }}^{\top }\mathit{\beta }$

3:  $Z_{ij}\leftarrow Z_{ij}-w+{\left[{\mathit{v}}_{\mathrm{c}}\right]}_i+{\left[{\mathit{v}}_{\mathrm{r}}\right]}_j$ for all $1\le i\le m$ and $1\le j\le n$

4:  Set $p_1,\dots ,p_k$ such that ${\sum }_{l=1}^k{p}_l=1$

5:  for  $t=1,\dots ,c$  do

6:        Pick $i_t$ with $\mathbb{P}[i_t=l]=p_l$

7:       $\mathit{Z}\leftarrow \mathit{Z}+{\displaystyle \frac{{\mathit{A}}^{\prime \left(i_t\right)}{\mathit{B}}_{\left(i_t\right)}^{\prime }}{c{p}_{i_t}}}$

8:  end for