Sparse Recovery by Semi-Iterative Hard Thresholding Algorithm

Xueqin Zhou,Mingli Jing,Xiangchu Feng

doi:10.1155/2013/498589

Abstract

We propose a computationally simple and efficient method for sparse recovery termed as the semi-iterative hard thresholding (SIHT). Unlike the existing iterative-shrinkage algorithms, which rely crucially on using negative gradient as the search direction, the proposed algorithm uses the linear combination of the current gradient and directions of few previous steps as the search direction. Compared to other iterative shrinkage algorithms, the performances of the proposed method show a clear improvement in iterations and error in noiseless, whilst the computational complexity does not increase.

Highlights

Compressed sensing (CS) [1,2,3] is a new framework for acquiring sparse signals based on the revelation that a small number of linear measurements of the signal contain enough information for its reconstruction
While it reveals that measurements of semi-iterative hard thresholding (SIHT) require less than those of iterative hard thresholding (IHT), GraDes, sparse reconstruction by separable approximation (SpaRSA), and fast iterative-shrinkage thresholding algorithm (FISTA) to recover the sparse vector for a given N and S
We show the number of iterations required by SIHT algorithm in comparison with four algorithms, namely, IHT, GraDes, SpaRSA, and FISTA for sparse ±1 spikes vector or Gaussian vector

Summary

Introduction

Compressed sensing (CS) [1,2,3] is a new framework for acquiring sparse signals based on the revelation that a small number of linear measurements of the signal contain enough information for its reconstruction. Approximation algorithms to find sparse solutions may be classified into greedy pursuits algorithms, convex relaxation algorithms, Bayesian framework, and nonconvex optimization. Iterative-shrinkage algorithms include iterative hard thresholding (IHT) [10] and gradient descent with sparsification (GraDeS) [11], parallel coordinate descent (PCD) [16], and fast iterative-shrinkage thresholding algorithm (FISTA) [17]. In these methods, each iteration consists of a multiplication by Φ and its transpose, along with a scalar shrinkage step on the obtained x. Inspired by the semi-iterative method [18] and hard thresholding, we present an algorithm for solving sparse recovery, which requires less time and fewer iterations

Background on Compressed Sensing

Semi-Iterative Hard Thresholding

Experimental Results

Conclusions