SPARSE RECOVERY BY NONCONVEX LIPSHITZIAN MAPPINGS

In recent years, the sparsity concept has attracted considerable attention in areas of applied mathematics and computer science, especially in signal and image processing fields. The general framework of sparse representation is now a mature concept with solid basis in relevant mathematical fields, such as probability, geometry of Banach spaces, harmonic analysis, theory of computability, and information-based complexity. Together with theoretical and practical advancements, also several numeric methods and algorithmic techniques have been developed in order to capture the complexity and the wide scope that the theory suggests. Sparse recovery relays over the fact that many signals can be represented in a sparse way, using only few nonzero coefficients in a suitable basis or overcomplete dictionary. Unfortunately, this problem, also called `0-norm minimization, is not only NP-hard, but also hard to approximate within an exponential factor of the optimal solution. Nevertheless, many heuristics for the problem has been obtained and proposed for many applications. This thesis provides new regularization methods for the sparse representation problem with application to face recognition and ECG signal compression. The proposed methods are based on fixed-point iteration scheme which combines nonconvex Lipschitzian-type mappings with canonical orthogonal projectors. The first are aimed at uniformly enhancing the sparseness level by shrinking effects, the latter to project back into the feasible space of solutions. In the second part of this thesis we study two applications in which sparseness has been successfully applied in recent areas of the signal and image processing: the face recognition problem and the ECG signal compression problem.