Bregman Iteration Applied In Dual Problem Of L1-norm Minimization And Its Relation Of Equivalence

Due to the emerge of new signal processing theory “compressed sensing”,the study on sparse representation theory has become research hot topic in image processing,signal processing and other fields.The class of regularized optimization problems has attracted much attention recently.Based on regularized optimization theory and Bregman iteration,this paper mainly discusses the algorithms to solve the basis pursuit problem and their application to the signal processing.The main subjects of this paper include in the following three areas.Firstly,to overcome the default of stagnation in linearized Bregman algorithm,we propose a novel Bregman iterative algorithm,based on the residual linearized Bregman iterative algorithm.The nonlinear problem is linearized by using fixed point iteration and a new threshold operator is defined to denoise by threshold operator.Furthermore,the convergence of the iterative algorithm is proved and its high convergence speed and the efficiency of the proposed method are illustrated by numerical experiments for signal recovery.Secondly,inspired by the equivalence of the linearized Bregman and the gradient descent algorithm of the dual problem,we put forward a new algorithm by introducing Nesterov accelerating technique into the predictor-corrector method for solving the sparse least squares problems.Simultaneously,we prove that the solution sequence obtained by the new method is the optimal solution of the sparse least squares problems.Finally,we use the new method to the sparse signal recovery problem.The numerical results show that the new method is faster and more efficient than the old ones.At last,combining with the linearized Bregman iterative algorithm and the splitting Bregman iterative algorithm for solving the sparse least squares problems,we propose a new iterative algorithm.And when A is full rank,the new algorithm is equivalent to the splitting Bregman iterative algorithm.We prove the convergence of the new algorithm and apply it into the signal recovery,then get a better result,compared with the A+ linearized Bregman iterative procedure.