Solution Of Sparsity Problems By (?)_q Regularization Technique

The sparsity problems emerging in many areas of scientific research and engineer-ing practice have attracted considerable attention in recent years. After the presentation of compressed sensing theory, the sparsity problems get a wide range of applications in signal recovery, image processing and statistical inference. Therefore, it has great important theoretical and practical significance for studying the sparsity problems.In this dissertation, we mainly focus on the solution of sparsity problems by (?)_q regu-larization. When 0< q<1,(?)_q regularization leads to a nonconvex, nonsmooth and even non-Lipschitz optimization problem. In general, it is difficult to solve the problem fast and efficiently. Instead of solving the (?)_q regularization directly, we find the minimizer of (?)_q regularization by solving a fixed point equation. On the one hand, through presenting a general thresholding representation form and making detailed analysis of the properties of thresholding function, we propose an iterative (?)_q thresholding algorithm for solutions of the fixed point equation effectively. Some preliminary convergence results are also reported. On the other hand, since the low convergence speed of the (?)_q algorithm, we propose an accelerated version of (?)_q thresholding algorithm. Extensive numerical ex-periments performed demonstrate the effective performance of the proposed algorithm. Inspired by the fact that Elastic Net regularization is very effective for variable selection, we consider an elastic l2-(?)_q minimization for sparse vector recovery. We provide two different algorithms, iterative algorithm and iterative reweighted l1 algorithm, for solv-ing this problem. By using an algebraic method, we prove the convergence results of the two algorithms. A series of numerical simulations on sparse vector recovery are re-ported. Compared with some existing algorithms, our proposed algorithm shows higher recovery accuracy, stable and effective performance.The nonconvex (?)_q (0< q< 1) regularization on sparse signal recovery has attracted a great deal of attention all over the world. The nonconvex (?)_q (0< q< 1) regulariza-tion generalized the convex optimization to many areas of engineering practice, such as signal reconstruction, image processing, machine learning, statistical inference and vari-able selection. Therefore, studying sparse signal recovery problem via (?)_q (0< q< 1) regularization is of great value and significance.