Algorithm 1 BCD on dual of Problem (2.1).

Inputs: S, λ, and weights matrices, W(ℓ). Initialize Â(ℓ) = 0 for all ℓ.

For ℓ = 1, …, p − 1:

• Compute ${\widehat{\mathbf{R}}}^{(\ell )}\leftarrow \mathbf{S}-\lambda {\sum }_{{\ell }^{\prime }=1}^{p-1}{W}^{({\ell }^{\prime })}\ast {\widehat{\mathbf{A}}}^{({\ell }^{\prime })}$  • For m ≤ ℓ, set ${\widehat{\mathbf{A}}}_{s_m}^{(\ell )}\leftarrow \frac{w_{\ell m}}{\lambda (w_{\ell m}^2+max\{{\widehat{\nu }}_{\ell },0\})}{\widehat{\mathbf{R}}}_{s_m}^{(\ell )}$ where ${\widehat{\nu }}_{\ell }$ satisfies ${\lambda }^2=h_{\ell }({\widehat{\nu }}_{\ell })$, as in (3.2).

$\{{\widehat{\mathbf{A}}}^{(\ell )}\}$ is a solution to (3.1) and ${\widehat{\mathbf{R}}}^{(p)}=\mathbf{S}-\lambda {\sum }_{\ell =1}^{p-1}{\mathbf{W}}^{(\ell )}\ast {\widehat{\mathbf{A}}}^{(\ell )}$ is the solution to (2.1).