Coordinate descent optimization for <i>l</i><SUP>1</SUP> minimization with application to compressed sensing; a greedy algorithm

We propose a fast algorithm for solving the Basis Pursuit problem, min u $\{|u|_1\: \Au=f\}$, which has application to compressed sensing. We design an efficient method for solving the related unconstrained problem min u $E(u) = |u|_1 + \lambda \||Au-f\||^2_2$ based on a greedy coordinate descent method. We claim that in combination with a Bregman iterative method, our algorithm will achieve a solution with speed and accuracy competitive with some of the leading methods for the basis pursuit problem.