Algorithm 1 Accelerated Algorithm for Subproblem (6)

Input:$\tilde{\mu },{\alpha }_{-1}=0.5,{\mathbf{Y}}_{-1}^{\left(t\right)}={\mathbf{Y}}_0^{\left(t\right)}={\mathbf{Y}}_n^{\left(t-1\right)},L_{-1}=L_0,{\gamma }_0\ge \tilde{\mu },{\lambda }_0=1,\vartheta ,{\epsilon }_{\tilde{i}}>0$; Output:${\mathbf{Y}}_n^{\left(t\right)}$, $rank\left({\mathbf{Y}}_n^{\left(t\right)}\right)$; 1:for$l=0,1,2,\dots ,n$do 2:   while true do 3:      compute ${\alpha }_l\in \left(0,1\right)$ as the root of $L_l{\alpha }_l^2-\left(1-{\alpha }_l\right){\gamma }_l-{\alpha }_l\tilde{\mu }=0$, ${\gamma }_{l+1}=\left(1-{\alpha }_l\right){\gamma }_l+{\alpha }_l\tilde{\mu }$, ${\beta }_l=\frac{\left(1-{\alpha }_{l-1}\right){\gamma }_l}{\left({\gamma }_l+L_l{\alpha }_l\right){\alpha }_{l-1}}$; 4:      compute ${\mathbf{Z}}_l^{\left(t\right)}={\mathbf{Y}}_l^{\left(t\right)}+{\beta }_l\left({\mathbf{Y}}_l^{\left(t\right)}-{\mathbf{Y}}_{l-1}^{\left(t\right)}\right)$; ${\mathbf{Y}}_{l+1}^{\left(t\right)}={\mathbf{Z}}_l^{\left(t\right)}-\frac{1}{L_l}\nabla q\left({\mathbf{Z}}_l^{\left(t\right)}\right)$; $L_l=2L_l$; 5:   end while 6:   ${\lambda }_{l+1}=(1-{\alpha }_l){\lambda }_l$; 7:end for 8:return${\mathbf{Y}}_n^{\left(t\right)}$, $rank\left({\mathbf{Y}}_n^{\left(t\right)}\right)$;