A First-Order Smoothed Penalty Method for Compressed Sensing

We propose a first-order smoothed penalty algorithm (SPA) to solve the sparse recovery problem min{‖x‖1 : Ax = b}. SPA is efficient as long as the matrix-vector product Ax and AT y can be computed efficiently; in particular, A need not have orthogonal rows. SPA converges to the target signal by solving a sequence of penalized optimization sub-problems, and each sub-problem is solved using Nesterov’s optimal algorithm for simple sets [18, 19]. We show that the SPA iterates xk are -feasible, i.e. ‖Axk − b‖2 ≤ and -optimal, i.e. | ‖xk‖1 − ‖x∗‖1| ≤ after Õ( − 3 2 ) iterations. SPA is able to work with `1, `2 or `∞ penalty on the infeasibility, and SPA can be easily extended to solve the relaxed recovery problem min{‖x‖1 : ‖Ax− b‖2 ≤ }.