Algorithm 4 NIHT

Input: $A=\Phi \Psi ,y,k$  Initialize:  $\left\{\begin{array}{cc}{\Lambda }_0=\underset{|\Lambda |=k}{argmax}{\parallel A_{\Lambda }^{T}y\parallel }_1\hfill & \hfill //find the columns of{A}^T that are the\\ & \hfill // most strongly correlated with residual\\ x_0=A_{{\Lambda }_0}^{†}y\hfill & \hfill // find the best coefficients for residual approximation\\ t=0\hfill & \end{array}\right.$  Output: k-sparse coefficient vector x  while$t<N_{iter}$do   $r_t=A{x}_t-y$ // update residual   ${\displaystyle \rho =\underset{j}{min}{\parallel a_j\parallel }_2\left(\frac{1}{{\parallel a_1\parallel }_2},\dots ,\frac{1}{{\parallel a_N\parallel }_2}\right)}$ // step size vector   $q_t=\rho \odot \left(A^T{r}_t\right)$ // normalized gradient vector   ${\Lambda }_{t+1}={\Lambda }_{t+1}^{★}$ // initialize the set of sparsity ${\Lambda }_{t+1}$   if$t>0$then    while (stop criterion on ${\Lambda }_{t+1}$) do    ${\tilde{x}}_{t+1}=x_t-{\mu }_t\left({\Lambda }_{t+1}\right)q_t$  // update $x_t$ with step size ${\mu }_t$ given by (46)    ${\Lambda }_{t+1}=\underset{k,\Psi }{supp}\left({\tilde{x}}_{t+1}\right)$  // update set of sparsity ${\Lambda }_{t+1}$    $x_{t+1}=F({\tilde{x}}_{t+1},{\Lambda }_{t+1})$  // find sparse vector $x_{t+1}$    end while   else    ${\tilde{x}}_{t+1}=x_t-{\mu }_t\left(I_N\right)q_t$    ${\left[x_{t+1}\right]}_k=F({\tilde{x}}_{t+1},\underset{k,\Psi }{supp}\left({\tilde{x}}_{t+1}\right))$  // find sparse vector $x_{t+1}$   end if   $t=t+1$  end while