The Methods For Sparse Least Squares Problems And Itsapplications

Because of the introduction of the new signal processing theory “compressed sensing”,the study on sparse representation theory has become hot topic in image processing, signalprocessing and other fields. The class of1regularized optimization problems has receivedmuch attention recently. In this dissertation, with an equivalent form of least squares problemand Bregman iterative methods, we study the algorithms of solving sparse least squaresproblems and the applications of related algorithms in spares signal recovery. The mainresearch results and innovations are made as follows:Firstly, we use some methods of the matrix to give out an equivalent form of leastsquares problem. Based on the research of the constraint least squares minimization problems,A linearized Bregman iterative formulas that have already existed are derived again.Secondly, combining with the alternating direction method, a new method calledpredictor-corrector method for solving the sparse least squares problems is proposed.Simultaneously, we prove that the solution sequence obtained by the new method is theoptimal solution of the sparse least squares problems. Finally, we use the new method to thesparse signal recovery problem. The numerical results show that the new method is faster,more efficient and simpler thanA linearized Bregman iterative method. At the same time,the new method can reduce the stagnation of the iterative procedure.At last, when recover the sparse signal of impluse noise, the sparse least squaresproblems of the2-norm constraint has some fault. Inspired by theTV1model of theimage processing, we change the2-norm constraint into1-norm constraint and proposethe sparse least squares problems of1-norm constraint. Simultaneously, we deduce the SplitBregman iterative algorithm of the new model and prove the convergence of the newalgorithm. Finally, numerical results show that for sparse signal recovery problem, themodified model can reconstruct signal from observational signal including noise-free signaland noise signal. Specially, for signal of impulse noise, the modified algorithm is moreexactly and quickly than the original one.