Table 4 Greedy algorithms

1. Let $\widehat{\mathbf{s}}={\mathbf{0}}_{n\times 1},{\mathbf{r}}^{\left(0\right)}=\mathbf{x}$, ${\mathcal{S}}^{\left(0\right)}=\varnothing$ and i=1.

2. Evaluate c _j =〈r⁽ⁱ⁻¹⁾,a _j 〉for $j=1,\dots ,n$ where a _j ’s are the columns of the mixing matrix A(atoms) and sort c _j ’s as $\left|c_{j_1}\right|\ge \cdots \ge \left|c_{j_n}\right|$.

3. ● MP: Set ${\mathcal{S}}^{\left(i\right)}={\mathcal{S}}^{(i-1)}\cup j_1$.

 ● OMP: Set ${\mathcal{S}}^{\left(i\right)}={\mathcal{S}}^{(i-1)}\cup j_1$ and ${\mathbf{A}}_{m\times \left|{\mathcal{S}}^{\left(i\right)}\right|}={\left[{\mathbf{a}}_j\right]}_{j\in {\mathcal{S}}^{\left(i\right)}}$.

 ● CoSaMP: Set ${\mathcal{S}}^{\left(i\right)}={\mathcal{S}}^{(i-1)}\cup \{j_1,\dots ,j_{2k}\}$ and ${\mathbf{A}}_{m\times \left|{\mathcal{S}}^{\left(i\right)}\right|}={\left[{\mathbf{a}}_j\right]}_{j\in {\mathcal{S}}^{\left(i\right)}}$.

4. ● MP: −−−

 ● OMP & CoSaMP: Find $\tilde{\mathbf{s}}$ that ${\mathbf{A}}^{\left(i\right)}·\tilde{\mathbf{s}}=\mathbf{x}$.

5. ● MP & OMP: −−−

 ● CoSaMP: Sort the values of $\tilde{\mathbf{s}}$ as $\left|{\tilde{s}}_{t_1}\right|\ge \left|{\tilde{s}}_{t_2}\right|\ge \dots$ and redefine $j_1,\dots ,j_k$ as the indices of the columns in A that correspond to the columns $t_1,\dots ,t_k$ in A⁽ⁱ⁾. Also set ${\mathcal{S}}^{\left(i\right)}=\{j_1,\dots ,j_k\}$.

6. ● MP: Set ${ŝ}_{j_1}=c_{j_1}$.

 ● OMP & CoSaMP: Set ${ŝ}_{j_l}={\tilde{s}}_l$ for $l=1,\dots ,k$ and ${ŝ}_l=0$ where $l\notin {\mathcal{S}}^{\left(i\right)}$.

7. Set ${\mathbf{r}}^{\left(i\right)}=\mathbf{x}-\mathbf{A}·\widehat{\mathbf{s}}$.

8. Stop if $\parallel {\mathbf{r}}^{\left(i\right)}{\parallel }_{{\ell }_2}$ is smaller than a desired threshold or when a maximum number of iterations is reached; otherwise, increase i and go to step 2.