A Sharp RIP Condition for Orthogonal Matching Pursuit

Algorithm 1

Orthogonal matching pursuit—OMP (Φ,𝑦).

Input: Sampling matrix Φ, observation 𝑦

Output: Reconstructed sparse vector 𝑥^∗ and index set

INITIALIZATION: Let the index set Ω₀=Ø and the residual 𝑟₀=𝑦. Let the iteration counter 𝑡=1.

IDENTIFICATION: Choose the index 𝑖 subject to |Φ^𝑇_𝑖𝑟_𝑡−1|>max_𝑗≠𝑖|Φ^𝑇_𝑗𝑟_𝑡−1|.

UPDATE: Add the new index 𝑖 to the index set: Ω_𝑡=Ω_𝑡−1∪𝑖, and update the signal and the residual

             𝑥_𝑡|_Ω𝑡=argmin_𝑧||𝑦−Φ_Ω𝑡𝑧||₂,𝑥_𝑡|_Ω𝑡=0;

               𝑟_𝑡=𝑦−Φ𝑥_𝑡.

If 𝑟_𝑡=0, stop the algorithm. Otherwise, update the iteration counter 𝑡=𝑡+1 and return to Step IDENTIFICATION.