Non-convex Lasso-kind approach to compressed sensing for finite-valued signals
In this paper, we bring together two trends that have recently emerged in sparse signal recovery: the problem of sparse signals that stem from finite alphabets and the techniques that introduce concave penalties. Specifically, we show that using a minimax concave penalty (MCP) the recovery of finite-valued sparse signals is enhanced with respect to classical Lasso, in terms of estimation accuracy, number of necessary measurements, and run time. We focus on problems where sparse signals can be recovered from few linear measurements, as stated in compressed sensing theory. We start by proposing a Lasso-kind functional with MCP, whose minimum is the desired signal in the noise-free case, under null space conditions. We analyze its robustness to noise as well. We then propose an efficient ADMM-based algorithm to search the minimum. The algorithm is proved to converge to the set of stationary points, and its performance is evaluated through numerical experiments, both on randomly generated data and on a real localization problem. Furthermore, in the noise-free case, it is possible to check the exactness of the solution, and we test a version of the algorithm that exploits this fact to look for the right signal.