  Adaptive thresholding methods have proved to yield high SNRs 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)
has outperformed 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 on 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 the
algebraic bound for the number of samples required. The computational
complexity of our method is also comparable to other methods as observed in the
simulations. We have also extended our Algorithm to Matrix Completion (MC)
scenarios and compared its efficiency to other well-reputed approaches for MC
in the literature.
