Nonmonotone Adaptive Barzilai-Borwein Gradient Algorithm for Compressed Sensing

Yuanying Qiu,Jianlei Yan,Fanyong Xu

doi:10.1155/2014/410104

Abstract

We study a nonmonotone adaptive Barzilai-Borwein gradient algorithm forl1-norm minimization problems arising from compressed sensing. At each iteration, the generated search direction enjoys descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of thel1-norm. Under some suitable conditions, its global convergence result could be established. Numerical results illustrate that the proposed method is promising and competitive with the existing algorithms NBBL1 and TwIST.

Highlights

In recent years, algorithms for finding sparse solutions to underdetermined linear systems of equations have been intensively investigated in signal processing and compressed sensing
We describe some experiments to illustrate the good performance of the algorithm NABBL1 for reconstructing sparse signals
The relative error is used to measure the quality of the reconstructive signals which is defined as

Summary

Introduction

Algorithms for finding sparse solutions to underdetermined linear systems of equations have been intensively investigated in signal processing and compressed sensing. TwIST [12, 13] and FISTA [14] speed up the performance of IST and have virtually the same complexity but with better convergence properties Another closely related method is the sparse reconstruction algorithm SpaRSA [15], which is to minimize nonsmooth convex problem with separable structures. Xiao et al propose a Barzilai-Borwein gradient algorithm [21] for solving l1 regularized nonsmooth minimization problems (NBBL1) [22], in which they approximate f locally by a convex quadratic model at each iteration, where the Hessian is replaced by the multiples of a spectral coefficient with an identity matrix. We propose a nonmonotone adaptive Barzilai-Borwein gradient algorithm for l1-norm minimization in compressed sensing, which is based on a new quasi-Newton equation [23] and a new adaptive spectral coefficient.

Proposed Algorithm and Convergence Result

Experimental Results

Conclusion