  In this paper, we propose and analyze an accelerated linearized Bregman (ALB)
method for solving the basis pursuit and related sparse optimization problems.
This accelerated algorithm is based on the fact that the linearized Bregman
(LB) algorithm is equivalent to a gradient descent method applied to a certain
dual formulation. We show that the LB method requires $O(1/\epsilon)$
iterations to obtain an $\epsilon$-optimal solution and the ALB algorithm
reduces this iteration complexity to $O(1/\sqrt{\epsilon})$ while requiring
almost the same computational effort on each iteration. Numerical results on
compressed sensing and matrix completion problems are presented that
demonstrate that the ALB method can be significantly faster than the LB method.
