Alternating Direction Algorithms for $\ell_1$-Problems in Compressive Sensing

Abstract

In this paper, we propose and study the use of alternating direction algorithms for several $\ell_1$-norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basis pursuit denoising problems of both unconstrained and constrained forms, and others. We present and investigate two classes of algorithms derived from either the primal or the dual form of $\ell_1$-problems. The construction of the algorithms consists of two main steps: (1) to reformulate an $\ell_1$-problem into one having blockwise separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction method to the augmented Lagrangian function of the resulting problem. The derived alternating direction algorithms can be regarded as first-order primal-dual algorithms because both primal and dual variables are updated at every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical experiments are performed, using randomized partial Walsh–Hadamard sensing matrices, to demonstrate the versatility and effectiveness of the proposed approach. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points: (i) that algorithm speed should be evaluated relative to appropriate solution accuracy; and (ii) that when erroneous measurements possibly exist, the $\ell_1$-fidelity should generally be preferable to the $\ell_2$-fidelity.