Sparse decomposition by iterating Lipschitzian-type mappings

This paper provides the analysis of a fast iterative method for finding sparse solutions to underdetermined linear systems. It is based on a fixed-point iteration scheme which combines nonconvex Lipschitzian-type mappings with canonical orthogonal projectors. The former are aimed at uniformly enhancing the sparsity level by shrinkage effects, the latter are used to project back onto the space of feasible solutions. The iterative process is driven by an increasing sequence of a scalar parameter that mainly contributes to approach the sparsest solutions. It is shown that the minima are locally asymptotically stable for a specific smooth ℓ0-norm. Furthermore, it is shown that the points yielded by this iterative strategy are related to the optimal solutions measured in terms of a suitable smooth ℓ1-norm. Numerical simulations on phase transition show that the performances of the proposed technique overcome those yielded by well known methods for sparse recovery.