Low-Complexity Iterative Algorithms for (Discrete) Compressed Sensing

We consider iterative (`turbo') algorithms for compressed sensing. First, a unified exposition of the different approaches available in the literature is given, thereby enlightening the general principles and main differences. In particular we discuss i) the estimation step (matched filter vs. optimum MMSE estimator), ii) the unbiasing operation (implicitly or explicitly done and equivalent to the calculation of extrinsic information), and iii) thresholding vs. the calculation of soft values. Based on these insights we propose a low-complexity but well-performing variant utilizing a Krylov space approximation of the optimum linear MMSE estimator. The derivations are valid for any probability density of the signal vector. However, numerical results are shown for the discrete case. The novel algorithms shows very good performance and even slightly faster convergence compared to approximative message passing.