Recovery of High-Dimensional Sparse Signals via -Minimization

Shiqing Wang,Limin Su

doi:10.1155/2013/636094

Abstract

We consider the recovery of high-dimensional sparse signals via -minimization under mutual incoherence condition, which is shown to be sufficient for sparse signals recovery in the noiseless and noise cases. We study both -minimization under the constraint and the Dantzig selector. Using the two -minimization methods and a technical inequality, some results are obtained. They improve the results of the error bounds in the literature and are extended to the general case of reconstructing an arbitrary signal.

Highlights

The problem of recovering a high-dimensional sparse signal based on a small number of measurements, possibly corrupted by noise, has attracted much recent attention
We consider the recovery of high-dimensional sparse signals via l1-minimization under mutual incoherence condition, which is shown to be sufficient for sparse signals recovery in the noiseless and noise cases. We study both l1-minimization under the l2 constraint and the Dantzig selector
In the existing literature on sparse signals recovery and compressed sensing, the emphasis is on assessing sparse signal w ∈ Rn from an observationy ∈ Rm: y = Aw + z, (1)

Summary

Introduction

The problem of recovering a high-dimensional sparse signal based on a small number of measurements, possibly corrupted by noise, has attracted much recent attention. The Dantzig selector solves the sparse recovery problem through l1-minimization with a constraint on the correlation between the residuals and the column vectors of A:. We consider the problem of recovering a high-dimensional sparse signal via two well l1-minimization methods under the condition k < (1/2)(1/μ + 1). We study both l1-minimization under the l2 constraint (P1) and the Dantzig selector (P2). We begin the analysis of l1-minimization methods for sparse signals recovery by considering the exact recovery in the noise case in Section 3; our results are similar to those in [19] and to some extent, we provide tighter error bounds than the existing results in the literature.

Preliminaries

Recovery of k-Sparse Signals

Recovery of Approximately k-Sparse Signals

The Proofs of the Theorems