Algorithm 1 Gibbs sampler with auxiliary variables in order to eliminate the coupling induced by $\mathsf{\Lambda }$.

Initialize: ${\mathbf{x}}^{(0)}\in {\mathbb{R}}^Q$, ${\mathbf{v}}^{(0)}\in {\mathbb{R}}^N$, $\mu >0$ such that ${\mu \parallel \mathsf{\Lambda }\parallel }_{\mathrm{S}}<1$ 1:for $t=0,1,\dots$ do 2: Generate ${\mathbf{v}}^{(t+1)}\sim \mathcal{N}\left(\mathsf{\Gamma }\mathbf{H}{\mathbf{x}}^{(t)},\mathsf{\Gamma }\right)$ where  $\mathsf{\Gamma }={\displaystyle \frac{1}{\mu }}{\mathbf{I}}_N-\mathsf{\Lambda }$ 3: Generate ${\mathbf{x}}^{(t+1)}\sim {\mathcal{P}}_{\mathbf{x}|{\mathbf{v}}^{(t+1)},\mathbf{z}}$ 4:end for