Algorithm 1: solve (25) by ALM

Input: attractor matrix $X\in {ℝ}^{n_1\times n_2}$  Parameter: number of anchor points: $q$; local weight coefficient matrix: $K\in {ℝ}^{n_1\times n_2}$; regularization parameter: $\lambda =1/\sqrt{\max (n_1,n_2)}$  for all i=1:$q$, parallel do   1. select $s_i(a_i,b_i)$ uniformly in $X$    for all a=1: $n_1$, do     $K(a_i,a)=1-d^2(a_i,a)$    end for all b=1:$n_2$, do     $K(b_i,b)=1-d^2(b_i,b)$    end for all $a\in [1,n_1],b\in [1,n_2]$     $K(a,b)=K(a_i,a)K(b_i,b)$    end   2. $\mathcal{T}\left(s_i\right)=K\odot X$   Initialize: $L_{s_i}^0=E_{s_i}^0=Y^0=0$, ${\tau }_0=1e^3$, ${\tau }_{\min }=1e^{-10}$, $\beta =0.9$, $\epsilon =1e-8$   while not converged do    3. fix the others and update $L_{s_i}^{k+1}$ by     ${\mathbf{L}}_{s_i}^{k+1}=\underset{{\mathbf{L}}_{s_i}}{argmin}:{∥{\mathbf{L}}_{s_i}∥}_{*}+\frac{1}{2{\tau }_k}{∥{\mathbf{L}}_{s_i}+{\mathbf{E}}_{s_i}^k-\mathcal{T}\left(s_i\right)+{\mathbf{Y}}^k∥}_F^2$    4. fix the others and update $S_{s_i}^{k+1}$ by     ${\mathbf{E}}_{s_i}^{k+1}=\underset{{\mathbf{E}}_{s_i}}{argmin}:\lambda {∥{\mathbf{E}}_{s_i}∥}_1+\frac{1}{2{\tau }_k}{∥{\mathbf{L}}_{s_i}^{k+1}+{\mathbf{E}}_{s_i}-\mathcal{T}\left(s_i\right)+{\mathbf{Y}}^k∥}_F^2$    5. update the Lagrange multiplier $Y$: ${\mathbf{Y}}^{k+1}={\mathbf{Y}}^k+\mathcal{T}\left(s_i\right)-{\mathbf{L}}_{s_i}^{k+1}-{\mathbf{E}}_{s_i}^{k+1}$    6. update $\tau$: ${\tau }_{k+1}=\max (\rho {\tau }_k,{\tau }_{\min })$    7. check the convergence conditions:     ${\Vert L_{s_i}^{k+1}-L_{s_i}^k\Vert }_{\infty }\le \epsilon$, ${\Vert E_{s_i}^{k+1}-E_{s_i}^k\Vert }_{\infty }\le \epsilon$, ${∥{\mathbf{L}}_{s_i}^{k+1}+{\mathbf{E}}_{s_i}^{k+1}-\mathcal{T}\left(s_i\right)∥}_{\infty }\le \epsilon$   end  end  output: $L_{s_1},\cdots ,L_{s_q}$