A New Bregman Iterative Regularization Algorithm for Sparse Least Squares Problems

The class of l1 norm regularization problems has received much attention recently because of the introduction of “compressed sensing” which allows images and signals to be reconstructed from small amounts of data. With an equivalent form of least squares problem and some techniques of Bregman iterative methods, we give out a derivation of A^+ linear Bregman iteration method that has already existed. Furthermore, combining with the continuous fixed-point iteration method and the new form of the non-surjective least square problem, a new method for solving the constrained l1 norm optimization problem is obtained. Simultaneously, the relationship between the A^+ linear Bregman iteration and the new iterative method is given. The solution obtained by the new method is proved to be the optimal solution of the constrained l1 norm optimization problem that we considered. Similar to A^+ linear Bregman iteration method that has exists, the new method needs only matrix-vector operation and shrinkage operator that is easy to implement in our considered problems. Finally, numerical results show that, for sparse signal recovery problem, the new method is faster, efficient and simple than A^+ Bregman iterative methods that have been existed. At the same time, the new method reduced the stagnation of iterative procedure.