Analysis of phase transition using deterministic matrix in Compressed Sensing

Phase transition, an important property in Compressed Sensing (CS), is the sparsity undersampling tradeoff. It serves as a performance criterion for a particular recovery algorithm. The phase transition of well-known algorithms including Basis Pursuit (BP), Orthogonal Matching Pursuit (OMP) and Approximate Message Passing (AMP) have been considered in this paper, by employing sensing matrix that is deterministic in nature. The phase boundary separating the successful and unsuccessful recovery of sparse signal is already defined in different papers by sensing matrix having random Gaussian entries. The random matrices satisfy the RIP condition for sensing matrices. While the deterministic sensing matrix used in this work satisfies the condition of coherence. The phase transition obtained using the deterministic setting shows an improvement for OMP, but is almost the same for the other two algorithms. Generally, AMP and BP are considered superior to OMP for the parameter of phase transition. While for the random case, the curve for OMP is lower than that of BP and AMP. However, the new curve obtained by deterministic sensing matrix for OMP is even higher than that of the other two algorithms. Hence, sensing matrix with lower coherence improves phase transition for OMP. Thus OMP algorithm when using deterministic matrix is superior to AMP and BP as opposed to previous research.