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

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, as well as others. We present and investigate two classes of algorithms derived from either the primal or the dual forms of the $\ell_1$-problems. The construction of the algorithms consists of two main steps: (1) to reformulate an $\ell_1$-problem into one having partially separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction…