Effect of errors in partially known support on the recoverability of modified compressive sensing

Jun Zhang,Yuanqing Li,Zhuliang Yu,Zhenghui Gu

doi:10.1186/1687-6180-2012-199

Abstract

The recently proposed modified compressive sensing (modified-CS), which utilizes the partially known support as prior knowledge, significantly improves the performance of compressive sensing. In practice, the known part will inevitably involve some errors, which may degrade the gain of modified-CS. Within the stochastic framework, this article discuss the effect of errors in known part on the recoverability of modified-CS. First, based on the probabilistic measure of recoverability, two probability inequalities on recoverability are established, which reflect the changing tendencies of the recoverability of modified-CS with respect to the addition of errors in the known support and sparsity of original sparse vector. A direct corollary reveals further the effect degree of recoverability as adding an error in known support. Second, the maximum number of errors that the modified-CS can bear is also analyzed. We prove a quantitative-bound of errors in known part that relates with the number of samples and the sparsity of original vector. This bound mirrors the fault-tolerance capability of modified-CS. Simulation experiments have been carried out to validate our theoretical results.

Highlights

The problem of finding sparse solutions to underdetermined linear systems from limited data arises in many applications, including biomedical imaging [1], sensor network [2], wireless communication [3], pattern recognition [4]
The goal is to recover a high-dimensional vector x∗ from its lowerdimensional observable vector y. It is one of the central problems in compressed sensing (CS) and the major breakthrough in this area has been the demonstration that 1 minimization can efficiently recover the sparse vector x∗ via far smaller number of measurements y than its ambient dimension [5,6]
Vaswani and Lu [7] analyzed when is the solution of modified-CS equal to the original vector x∗, i.e., the recoverability problem

Summary

Introduction

The problem of finding sparse solutions to underdetermined linear systems from limited data arises in many applications, including biomedical imaging [1], sensor network [2], wireless communication [3], pattern recognition [4]. The goal is to recover a high-dimensional vector x∗ from its lowerdimensional observable vector y It is one of the central problems in compressed sensing (CS) and the major breakthrough in this area has been the demonstration that 1 minimization can efficiently recover the sparse vector x∗ via far smaller number of measurements y than its ambient dimension [5,6]. Vaswani and Lu [7] analyzed when is the solution of modified-CS equal to the original vector x∗, i.e., the recoverability problem They demonstrated when the sizes of the unknown part of the support and of errors in the known part are small compared to the support size, the sufficient conditions on recoverability problem are much weaker than those needed for classical 1 minimization method. We derived a sufficient and necessary condition on recoverability of modified-CS and investigated the recoverability in a probability way [10]

Objectives

Conclusion