Approximately Normalized Iterative Hard Thresholding for Nonlinear Compressive Sensing

Xunzhi Zhu

doi:10.1155/2016/2594752

Abstract

The nonlinear compressive sensing (NCS) is an extension of classical compressive sensing (CS) and the iterative hard thresholding (IHT) algorithm is a popular greedy-type method for solving CS. The normalized iterative hard thresholding (NIHT) is a modification of IHT and is more effective than IHT. In this paper, we propose an approximately normalized iterative hard thresholding (ANIHT) algorithm for NCS by using the approximate optimal stepsize combining with Armijo stepsize rule preiteration. Under the condition similar to restricted isometry property (RIP), we analyze the condition that can identify the iterative support sets in a finite number of iterations. Numerical experiments show the good performance of the new algorithm for the NCS.

Highlights

Compressed sensing (CS) [1, 2] deals with the problem of recovering sparse signals from underdetermined linear measurements
We propose an approximately normalized iterative hard thresholding (ANIHT) algorithm for nonlinear compressive sensing (NCS) by using the approximate optimal stepsize combining with Armijo stepsize rule preiteration
We have proposed an ANIHT algorithm for NCS and studied its convergence

Summary

Introduction

Compressed sensing (CS) [1, 2] deals with the problem of recovering sparse signals from underdetermined linear measurements. In [16], the authors showed that the numerical studies of IHT are not very promising and the algorithm often fails to converge when the conditions fail. They gave normalized IHT (NIHT) with an adaptive stepsize and line search and proved that it converges to a local minimum if A is row full-rank and s-regular, where s-regular means that any s columns of A are linear independent [17]. Blumensath [5] showed that IHT can recover signals from NCS under conditions similar to those required in CS Inspired by these works, we propose an approximately NIHT (ANIHT) algorithm to solve the NCS problem (2).

Algorithm

Numerical Experiments

Conclusion