Computable performance guarantees for compressed sensing matrices

Myung Cho, Kumar Vijay Mishra, Weiyu Xu

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

The null space condition for ℓ1 minimization in compressed sensing is a necessary and sufficient condition on the sensing matrices under which a sparse signal can be uniquely recovered from the observation data via ℓ1 minimization. However, verifying the null space condition is known to be computationally challenging. Most of the existing methods can provide only upper and lower bounds on the proportion parameter that characterizes the null space condition. In this paper, we propose new polynomial-time algorithms to establish upper bounds of the proportion parameter. We leverage on these techniques to find upper bounds and further develop a new procedure—tree search algorithm—that is able to precisely and quickly verify the null space condition. Numerical experiments show that the execution speed and accuracy of the results obtained from our methods far exceed those of the previous methods which rely on linear programming (LP) relaxation and semidefinite programming (SDP).

Original languageEnglish (US)
Article number16
JournalEurasip Journal on Advances in Signal Processing
Volume2018
Issue number1
DOIs
StatePublished - Dec 1 2018

Fingerprint

Compressed sensing
Linear programming
Polynomials
Experiments

All Science Journal Classification (ASJC) codes

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

Cite this

@article{aa54cb6bc2bd4db88239a88a258ea3ef,
title = "Computable performance guarantees for compressed sensing matrices",
abstract = "The null space condition for ℓ1 minimization in compressed sensing is a necessary and sufficient condition on the sensing matrices under which a sparse signal can be uniquely recovered from the observation data via ℓ1 minimization. However, verifying the null space condition is known to be computationally challenging. Most of the existing methods can provide only upper and lower bounds on the proportion parameter that characterizes the null space condition. In this paper, we propose new polynomial-time algorithms to establish upper bounds of the proportion parameter. We leverage on these techniques to find upper bounds and further develop a new procedure—tree search algorithm—that is able to precisely and quickly verify the null space condition. Numerical experiments show that the execution speed and accuracy of the results obtained from our methods far exceed those of the previous methods which rely on linear programming (LP) relaxation and semidefinite programming (SDP).",
author = "Myung Cho and {Vijay Mishra}, Kumar and Weiyu Xu",
year = "2018",
month = "12",
day = "1",
doi = "10.1186/s13634-018-0535-y",
language = "English (US)",
volume = "2018",
journal = "Eurasip Journal on Advances in Signal Processing",
issn = "1687-6172",
publisher = "Springer Publishing Company",
number = "1",

}

Computable performance guarantees for compressed sensing matrices. / Cho, Myung; Vijay Mishra, Kumar; Xu, Weiyu.

In: Eurasip Journal on Advances in Signal Processing, Vol. 2018, No. 1, 16, 01.12.2018.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Computable performance guarantees for compressed sensing matrices

AU - Cho, Myung

AU - Vijay Mishra, Kumar

AU - Xu, Weiyu

PY - 2018/12/1

Y1 - 2018/12/1

N2 - The null space condition for ℓ1 minimization in compressed sensing is a necessary and sufficient condition on the sensing matrices under which a sparse signal can be uniquely recovered from the observation data via ℓ1 minimization. However, verifying the null space condition is known to be computationally challenging. Most of the existing methods can provide only upper and lower bounds on the proportion parameter that characterizes the null space condition. In this paper, we propose new polynomial-time algorithms to establish upper bounds of the proportion parameter. We leverage on these techniques to find upper bounds and further develop a new procedure—tree search algorithm—that is able to precisely and quickly verify the null space condition. Numerical experiments show that the execution speed and accuracy of the results obtained from our methods far exceed those of the previous methods which rely on linear programming (LP) relaxation and semidefinite programming (SDP).

AB - The null space condition for ℓ1 minimization in compressed sensing is a necessary and sufficient condition on the sensing matrices under which a sparse signal can be uniquely recovered from the observation data via ℓ1 minimization. However, verifying the null space condition is known to be computationally challenging. Most of the existing methods can provide only upper and lower bounds on the proportion parameter that characterizes the null space condition. In this paper, we propose new polynomial-time algorithms to establish upper bounds of the proportion parameter. We leverage on these techniques to find upper bounds and further develop a new procedure—tree search algorithm—that is able to precisely and quickly verify the null space condition. Numerical experiments show that the execution speed and accuracy of the results obtained from our methods far exceed those of the previous methods which rely on linear programming (LP) relaxation and semidefinite programming (SDP).

UR - http://www.scopus.com/inward/record.url?scp=85042718747&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042718747&partnerID=8YFLogxK

U2 - 10.1186/s13634-018-0535-y

DO - 10.1186/s13634-018-0535-y

M3 - Article

C2 - 29503664

AN - SCOPUS:85042718747

VL - 2018

JO - Eurasip Journal on Advances in Signal Processing

JF - Eurasip Journal on Advances in Signal Processing

SN - 1687-6172

IS - 1

M1 - 16

ER -