Table 2 MFB-CoSaMP algorithm

Initial: \({\overset \smile {\boldsymbol {\Psi }}^{\left (0 \right)}} \leftarrow {\mathbf {0}}\), v←y, k←0.

Repeat:

a). k=k+1.

b). Form the metric-vector proxy: u=Φ ^H v.

c). Identify the circle-cluster location according to u

W ₁={i:|u _i |= max{|u ₁|,|u ₂|,⋯,|u _P |}};

W ₁ ← the 2K indexes nearest to W ₁ in index set { 1,2, ⋯,P} including W ₁.

d). Merge the support set:

\({\mathrm {T}} \leftarrow {\text {supp}}\left ({{\overset \smile {\boldsymbol {\Psi }}^{\left ({k - 1} \right)}}} \right) \bigcup {{\mathbf {W}}_{1}}.\)

e). Least square estimation: b| _T←(Φ _T)^† y.

f). \(\phantom {\dot {i}\!}{\mathbf {b}}\left | {{~}_{{{{\mathrm {T}}}^{c}}}} \right. \leftarrow {\mathbf {0}}\).

g). Identify circle-cluster location according to b

W ₂={i:|b _i |= max{|b ₁|,|b ₂|,⋯,|b _P |}};

W ₂ ←the K indexes nearest to W ₂ in index set { 1,2, ⋯, P} including W ₂.

h). \({\mathbf {b}}\left | {{~}_{{\mathbf {W}}_{2}^{c}}} \right. \leftarrow {\mathbf {0}}.\)

i). Prune to obtain the next approximation:

\({\overset \smile {\boldsymbol {\Psi }}^{\left (k \right)}} \leftarrow {\mathbf {b}}.\)

j). Update current samples \({\mathbf {v }} \leftarrow {\mathbf {y}} - {\boldsymbol {\Phi }}{\overset \smile {\boldsymbol {\Psi }}^{\left (k \right)}}.\)

Until: k=K

\(\overset \smile {\boldsymbol {\Psi }} \leftarrow {\overset \smile {\boldsymbol {\Psi }}^{\left (K \right)}}\).