research-article

New bounds for restricted isometry constants

Authors:

Guangwu XuAuthors Info & Claims

IEEE Transactions on Information Theory, Volume 56, Issue 9

Pages 4388 - 4394

https://doi.org/10.1109/TIT.2010.2054730

Published: 01 September 2010 Publication History

Abstract

This paper discusses new bounds for restricted isometry constants in compressed sensing. Let φ be an n×p real matrix and k be a positive integer with k ≤ n. One of the main results of this paper shows that if the restricted isometry constant δ_k of φ satisfies δ_k < 0.307 then k-sparse signals are guaranteed to be recovered exactly via l₁ minimization when no noise is present and k-sparse signals can be estimated stably in the noisy case. It is also shown that the bound cannot be substantially improved. An explicit example is constructed in which δ_k = k-1/2k-1 < 0.5, but it is impossible to recover certain k-sparse signals.

References

[1]

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, "A simple proof of the restricted isometry property for random matrices," Constr. Approx., vol. 28, 2008.

[2]

T. Cai, L. Wang, and G. Xu, IEEE Trans. Inf. Theory, vol. 56, pp. 3516-3522, 2010.

Digital Library

[3]

T. Cai, L. Wang, and G. Xu, "Shifting inequality and recovery of sparse signals," IEEE Trans. Signal Process., vol. 58, pp. 1300-1308, 2010.

Digital Library

[4]

T. Cai, G. Xu, and J. Zhang, "On recovery of sparse signals via l ₁ minimization," IEEE Trans. Inf. Theory, vol. 55, pp. 3388-3397, 2009.

Digital Library

[5]

E. J. Candès, "The restricted isometry property and its Implications for compressed sensing," Compte Rendus de l'Academie des Sciences. Paris, France, ser. I, vol. 346, pp. 589-592.

[6]

E. J. Candès, J. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Comm. Pure Appl. Math., vol. 59, pp. 1207-1223, 2006.

[7]

E. J. Candès and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory, vol. 51, pp. 4203-4215, 2005.

Digital Library

[8]

E. J. Candès and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?," IEEE Trans. Inf. Theory, vol. 52, pp. 5406-5425, 2006.

Digital Library

[9]

E. J. Candès and T. Tao, "The Dantzig selector: Statistical estimation when p is much larger than n (with discussion)," Ann. Statist., vol. 35, pp. 2313-2351, 2007.

[10]

M. E. Davies and R. Gribonval, "Restricted isometry constants where l_p sparse recove:ry can fail for 0 < p ≤ 1," IEEE Trans. Inf. Theory, to be published.

Digital Library

[11]

D. L. Donoho, "Compressed sensing," IEEE Trans. Inf. Theory, vol. 52, pp. 1289-1306, 2006.

Digital Library

[12]

D. L. Donoho, M. Elad, and V. N. Temlyakov, "Stable recovery of sparse overcomplete representations in the presence of noise," IEEE Trans. Inf. Theory, vol. 52, pp. 6-18, 2006.

Digital Library

[13]

D. L. Donoho and X. Huo, "Uncertainty principles and ideal atomic decomposition," IEEE Trans. Inf. Theory, vol. 47, pp. 2845-2862, 2001.

Digital Library

[14]

S. Foucart and M. Lai, "Sparsest solutions of underdetermined linear systems via l_q -minimization for 0 < q ≤ 1," Appl. Comput. Harmon. Anal., vol. 26/3, pp. 395-407, 2009.

[15]

M. Rudelson and R. Vershynin, "Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements," presented at the 40th Annu. Conf. Information Sciences and Systems, 2006.

Cited By

Vidyasagar M(2021)A tutorial introduction to compressed sensing2016 IEEE 55th Conference on Decision and Control (CDC)10.1109/CDC.2016.7799048(5091-5104)Online publication date: 10-Mar-2021
https://dl.acm.org/doi/10.1109/CDC.2016.7799048
Axiotis KSviridenko MDaumé HSingh A(2020)Sparse convex optimization via adaptively regularized hard thresholdingProceedings of the 37th International Conference on Machine Learning10.5555/3524938.3524981(452-462)Online publication date: 13-Jul-2020
https://dl.acm.org/doi/10.5555/3524938.3524981
Zhao YJiang HLuo Z(2019)Weak Stability of ℓ1-Minimization Methods in Sparse Data ReconstructionMathematics of Operations Research10.1287/moor.2017.091944:1(173-195)Online publication date: 1-Feb-2019
https://dl.acm.org/doi/10.1287/moor.2017.0919
Show More Cited By

Recommendations

New Restricted Isometry Property Analysis for $\ell_1-\ell_2$ Minimization Methods

The $\ell_1-\ell_2$ regularization is a popular nonconvex yet Lipschitz continuous metric, which has been widely used in signal and image processing. The theory for the $\ell_1-\ell_2$ minimization method shows that it has superior sparse recovery ...
Restricted isometry constants where l^psparse recovery can fail for 0 < p ≤ 1

This paper investigates conditions under which the solution of an underdetermined linear system with minimal l^P norm, 0 < p ≤ 1, is guaranteed to be also the sparsest one. Matrices are constructed with restricted isometry constants (RIC) δ_2m arbitrarily ...
Compressed Sensing: How Sharp Is the Restricted Isometry Property?

Compressed sensing (CS) seeks to recover an unknown vector with $N$ entries by making far fewer than $N$ measurements; it posits that the number of CS measurements should be comparable to the information content of the vector, not simply $N$. CS ...

Comments

Information & Contributors

Information

Published In

cover image IEEE Transactions on Information Theory

IEEE Transactions on Information Theory Volume 56, Issue 9

September 2010

584 pages

ISSN:0018-9448

Issue’s Table of Contents

Copyright © 2010.

Publisher

IEEE Press

Publication History

Published: 01 September 2010

Revised: 11 March 2010

Received: 08 November 2009

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

19
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 11 Aug 2024

Other Metrics

View Author Metrics

Citations

Cited By

Vidyasagar M(2021)A tutorial introduction to compressed sensing2016 IEEE 55th Conference on Decision and Control (CDC)10.1109/CDC.2016.7799048(5091-5104)Online publication date: 10-Mar-2021
https://dl.acm.org/doi/10.1109/CDC.2016.7799048
Axiotis KSviridenko MDaumé HSingh A(2020)Sparse convex optimization via adaptively regularized hard thresholdingProceedings of the 37th International Conference on Machine Learning10.5555/3524938.3524981(452-462)Online publication date: 13-Jul-2020
https://dl.acm.org/doi/10.5555/3524938.3524981
Zhao YJiang HLuo Z(2019)Weak Stability of ℓ1-Minimization Methods in Sparse Data ReconstructionMathematics of Operations Research10.1287/moor.2017.091944:1(173-195)Online publication date: 1-Feb-2019
https://dl.acm.org/doi/10.1287/moor.2017.0919
Qi HZhang XGao Y(2018)Channel Energy Statistics Learning in Compressive Spectrum SensingIEEE Transactions on Wireless Communications10.1109/TWC.2018.287271217:12(7910-7921)Online publication date: 1-Dec-2018
https://dl.acm.org/doi/10.1109/TWC.2018.2872712
Ji STan CYang PSun YFu DWang J(2018)Compressive sampling and data fusion-based structural damage monitoring in wireless sensor networkThe Journal of Supercomputing10.1007/s11227-016-1938-x74:3(1108-1131)Online publication date: 1-Mar-2018
https://dl.acm.org/doi/10.1007/s11227-016-1938-x
Huo HSun WXiao L(2018)New Conditions on Stable Recovery of Weighted Sparse Signals via Weighted $$l_1$$l1 MinimizationCircuits, Systems, and Signal Processing10.1007/s00034-017-0691-637:7(2866-2883)Online publication date: 1-Jul-2018
https://dl.acm.org/doi/10.1007/s00034-017-0691-6
Shen JLi P(2017)On the iteration complexity of support recovery via hard thresholding pursuitProceedings of the 34th International Conference on Machine Learning - Volume 7010.5555/3305890.3306003(3115-3124)Online publication date: 6-Aug-2017
https://dl.acm.org/doi/10.5555/3305890.3306003
Shen JLi P(2017)A tight bound of hard thresholdingThe Journal of Machine Learning Research10.5555/3122009.324206518:1(7650-7691)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.5555/3122009.3242065
Liang SLu WSong RWang L(2017)Sparse concordance-assisted learning for optimal treatment decisionThe Journal of Machine Learning Research10.5555/3122009.324205918:1(7375-7400)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.5555/3122009.3242059
Chiani M(2017)On the Probability That All Eigenvalues of Gaussian, Wishart, and Double Wishart Random Matrices Lie Within an IntervalIEEE Transactions on Information Theory10.1109/TIT.2017.269484663:7(4521-4531)Online publication date: 14-Jun-2017
https://dl.acm.org/doi/10.1109/TIT.2017.2694846
Show More Cited By

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents