- Alternating Direction Algorithms for $\ell_1$-Problems in Compressive Sensing Junfeng Yang(jfyangnju.edu.cn) Yin Zhang(yin.zhangrice.edu) 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, 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 method to 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 each and every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical results in comparison with several state-of-the-art algorithms are given to demonstrate that the proposed algorithms are efficient, stable and robust. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points. One point is that algorithm speed should always be evaluated relative to appropriate solution accuracy; another is that whenever erroneous measurements possibly exist, the $\ell_1$-norm fidelity should be the fidelity of choice in compressive sensing. Keywords: Sparse solution recovery, compressive sensing, $\ell_1$-minimization, primal, dual, alternating direction method Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization ) Category 2: Convex and Nonsmooth Optimization (Convex Optimization ) Category 3: Convex and Nonsmooth Optimization Citation: TR09-37, CAAM, Rice University Download: [PDF]Entry Submitted: 12/07/2009Entry Accepted: 12/07/2009Entry Last Modified: 12/07/2009Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society.