Algorithm 3 CoSaMP

Input: $A=\Phi \Psi ,y,k$  Initialize:  $\left\{\begin{array}{cc}{r}_0=y\hfill & //residual\hfill \\ t=0\hfill & \end{array}\right.$  ${\Lambda }_0=\underset{|\Lambda |=k}{argmax}{\parallel A_{\Lambda }^T{r}_0\parallel }_1$  // find k columns of $A^T$ that are most strongly correlated with residual $r_0$  Output: k-sparse coefficient vector x  while $t<N_{iter}$ do   $k_S=\gamma k,\gamma \in [0,1]$  // number of new columns to be selected   $W_t=\underset{\left|W\right|=k_S}{argmax}{\parallel A_W^T{r}_t\parallel }_1$  // find $k_S$ columns of $A^T$ that are most strongly correlated with residual $r_t$   $T_t={\Lambda }_t\cup W_t$  // merge the new columns such that $|T_t|=k+k_S$   $h=A_{T_t}^{†}y$  // find the best coefficients for residual approximation   ${\Lambda }_{t+1}=\underset{k,\Psi }{supp}\left(h\right)$  // find the set of sparsity ${\Lambda }_{t+1}$   $x=F(h,{\Lambda }_{t+1})$  // find sparse vector x   $r_{t+1}=y-A_{{\Lambda }_{t+1}}{x}_{{\Lambda }_{t+1}}$  // update the residual   $t=t+1$  end while