On the Null Space Property oflq-Minimization for0&lt;q≤1in Compressed Sensing

Yi Gao,Jigen Peng,Yuan Zhao,Shigang Yue

doi:10.1155/2015/579853

Abstract

The paper discusses the relationship between the null space property (NSP) and thelq-minimization in compressed sensing. Several versions of the null space property, that is, thelqstable NSP, thelqrobust NSP, and thelq,probust NSP for0<p≤q<1based on the standardlqNSP, are proposed, and their equivalent forms are derived. Consequently, reconstruction results for thelq-minimization can be derived easily under the NSP condition and its equivalent form. Finally, thelqNSP is extended to thelq-synthesis modeling and the mixedl2/lq-minimization, which deals with the dictionary-based sparse signals and the block sparse signals, respectively.

Highlights

Compressed sensing has been drawing extensive and hot attention as soon as it was proposed since 2006 [1,2,3,4]
The lq null space property (NSP) is extended to the lq-synthesis modeling and the mixed l2/lq-minimization, which deals with the dictionary-based sparse signals and the block sparse signals, respectively
We introduce the definition of the null space property [6, 8, 22]

Summary

Introduction

Compressed sensing has been drawing extensive and hot attention as soon as it was proposed since 2006 [1,2,3,4]. Given a matrix A ∈ Rn×N, every s-sparse vector x ∈ RN is the unique solution of the lq-minimization with y = Ax if and only if A satisfies the lq null space property of order s.

Results

Conclusion