Near optimal bound of orthogonal matching pursuit using restricted isometric constant

Jian Wang, Seokbeop Kwon, Byonghyo Shim

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

As a paradigm for reconstructing sparse signals using a set of under sampled measurements, compressed sensing has received much attention in recent years. In identifying the sufficient condition under which the perfect recovery of sparse signals is ensured, a property of the sensing matrix referred to as the restricted isometry property (RIP) is popularly employed. In this article, we propose the RIP based bound of the orthogonal matching pursuit (OMP) algorithm guaranteeing the exact reconstruction of sparse signals. Our proof is built on an observation that the general step of the OMP process is in essence the same as the initial step in the sense that the residual is considered as a new measurement preserving the sparsity level of an input vector. Our main conclusion is that if the restricted isometry constant δK of the sensing matrix satisfies δK < √ K-1/√K-1+K then the OMP algorithm can perfectly recover K(> 1)-sparse signals from measurements. We show that our bound is sharp and indeed close to the limit conjectured by Dai and Milenkovic.

Original languageEnglish
Article number8
JournalEurasip Journal on Advances in Signal Processing
Volume2012
Issue number1
DOIs
Publication statusPublished - 2012 Dec 1

Fingerprint

Compressed sensing
Recovery

Keywords

  • Compressed sensing
  • Orthogonal matching pursuit
  • Restricted isometric property
  • Sparse signal
  • Support

ASJC Scopus subject areas

  • Hardware and Architecture
  • Signal Processing
  • Electrical and Electronic Engineering

Cite this

Near optimal bound of orthogonal matching pursuit using restricted isometric constant. / Wang, Jian; Kwon, Seokbeop; Shim, Byonghyo.

In: Eurasip Journal on Advances in Signal Processing, Vol. 2012, No. 1, 8, 01.12.2012.

Research output: Contribution to journalArticle

@article{c7005a0794c344a0a82d2bb3697a97e7,
title = "Near optimal bound of orthogonal matching pursuit using restricted isometric constant",
abstract = "As a paradigm for reconstructing sparse signals using a set of under sampled measurements, compressed sensing has received much attention in recent years. In identifying the sufficient condition under which the perfect recovery of sparse signals is ensured, a property of the sensing matrix referred to as the restricted isometry property (RIP) is popularly employed. In this article, we propose the RIP based bound of the orthogonal matching pursuit (OMP) algorithm guaranteeing the exact reconstruction of sparse signals. Our proof is built on an observation that the general step of the OMP process is in essence the same as the initial step in the sense that the residual is considered as a new measurement preserving the sparsity level of an input vector. Our main conclusion is that if the restricted isometry constant δK of the sensing matrix satisfies δK < √ K-1/√K-1+K then the OMP algorithm can perfectly recover K(> 1)-sparse signals from measurements. We show that our bound is sharp and indeed close to the limit conjectured by Dai and Milenkovic.",
keywords = "Compressed sensing, Orthogonal matching pursuit, Restricted isometric property, Sparse signal, Support",
author = "Jian Wang and Seokbeop Kwon and Byonghyo Shim",
year = "2012",
month = "12",
day = "1",
doi = "10.1186/1687-6180-2012-8",
language = "English",
volume = "2012",
journal = "Eurasip Journal on Advances in Signal Processing",
issn = "1687-6172",
publisher = "Springer Publishing Company",
number = "1",

}

TY - JOUR

T1 - Near optimal bound of orthogonal matching pursuit using restricted isometric constant

AU - Wang, Jian

AU - Kwon, Seokbeop

AU - Shim, Byonghyo

PY - 2012/12/1

Y1 - 2012/12/1

N2 - As a paradigm for reconstructing sparse signals using a set of under sampled measurements, compressed sensing has received much attention in recent years. In identifying the sufficient condition under which the perfect recovery of sparse signals is ensured, a property of the sensing matrix referred to as the restricted isometry property (RIP) is popularly employed. In this article, we propose the RIP based bound of the orthogonal matching pursuit (OMP) algorithm guaranteeing the exact reconstruction of sparse signals. Our proof is built on an observation that the general step of the OMP process is in essence the same as the initial step in the sense that the residual is considered as a new measurement preserving the sparsity level of an input vector. Our main conclusion is that if the restricted isometry constant δK of the sensing matrix satisfies δK < √ K-1/√K-1+K then the OMP algorithm can perfectly recover K(> 1)-sparse signals from measurements. We show that our bound is sharp and indeed close to the limit conjectured by Dai and Milenkovic.

AB - As a paradigm for reconstructing sparse signals using a set of under sampled measurements, compressed sensing has received much attention in recent years. In identifying the sufficient condition under which the perfect recovery of sparse signals is ensured, a property of the sensing matrix referred to as the restricted isometry property (RIP) is popularly employed. In this article, we propose the RIP based bound of the orthogonal matching pursuit (OMP) algorithm guaranteeing the exact reconstruction of sparse signals. Our proof is built on an observation that the general step of the OMP process is in essence the same as the initial step in the sense that the residual is considered as a new measurement preserving the sparsity level of an input vector. Our main conclusion is that if the restricted isometry constant δK of the sensing matrix satisfies δK < √ K-1/√K-1+K then the OMP algorithm can perfectly recover K(> 1)-sparse signals from measurements. We show that our bound is sharp and indeed close to the limit conjectured by Dai and Milenkovic.

KW - Compressed sensing

KW - Orthogonal matching pursuit

KW - Restricted isometric property

KW - Sparse signal

KW - Support

UR - http://www.scopus.com/inward/record.url?scp=84865219384&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865219384&partnerID=8YFLogxK

U2 - 10.1186/1687-6180-2012-8

DO - 10.1186/1687-6180-2012-8

M3 - Article

AN - SCOPUS:84865219384

VL - 2012

JO - Eurasip Journal on Advances in Signal Processing

JF - Eurasip Journal on Advances in Signal Processing

SN - 1687-6172

IS - 1

M1 - 8

ER -