A First-Order Augmented Lagrangian Method for Compressed Sensing

We propose a first-order augmented Lagrangian algorithm (FAL) for solving the basis pursuit problem. FAL computes a solution to this problem by inexactly solving a sequence of `1-regularized least squares sub-problems. These sub-problems are solved using an infinite memory proximal gradient algorithm wherein each update reduces to “shrinkage” or constrained “shrinkage”. We show that FAL converges to an optimal solution of the basis pursuit problem whenever the solution is unique, which is the case with very high probability for compressed sensing problems. We construct a parameter sequence such that the corresponding FAL iterates are -feasible and -optimal for all > 0 within O ( log ( −1 )) FAL iterations. Moreover, FAL requires at most O( −1) matrix-vector multiplications of the form Ax or AT y to compute an -feasible, -optimal solution. We show that FAL can be easily extended to solve the basis pursuit denoising problem when there is a non-trivial level of noise on the measurements. We report the results of numerical experiments comparing FAL with the state-of-the-art solvers for both noisy and noiseless compressed sensing problems. A striking property of FAL that we observed in the numerical experiments with randomly generated instances when there is no measurement noise was that FAL always correctly identifies the support of the target signal without any thresholding or post-processing, for moderately small error tolerance values.