Iterative Null-space Projection Method with Adaptive Thresholding in Sparse Signal Recovery and Matrix Completion

Adaptive thresholding methods have proved to yield high Signal-to-Noise Ratio (SNR) and fast convergence in finding the solution to the Compressed Sensing (CS) problems. Recently, It was observed that the robustness of a class of iterative sparse recovery algorithms such as Iterative Method with Adaptive Thresholding (IMAT) outperforms the well-known LASSO algorithm in terms of reconstruction quality, convergence speed, and the sensitivity to the noise. In this paper, we introduce a new method towards solving the CS problem. The logic of this method is based on iterative projections of the thresholded signal onto the null-space of the sensing matrix. The thresholding is carried out by recovering the support of the desired signal by projection onto thresholding subspaces. The simulations reveal that the proposed method has the capability of yielding noticeable output SNR values with about as many samples as twice the sparsity number, while other methods fail to recover the signals when approaching twice the sparsity number for recovery. The computational complexity of our method is also comparable to other methods in the simulations. We have also extended our algorithm to Matrix Completion (MC) scenarios and compared its efficiency to other well-known approaches for MC in the literature.