Abstract
We review three recovery algorithms used in Compressive Sensing for the reconstruction s-sparse vectors x∈ℂ N from the mere knowledge of linear measurements y=A x∈ℂ m, m<N. For each of the algorithms, we derive improved conditions on the restricted isometry constants of the measurement matrix A that guarantee the success of the reconstruction. These conditions are δ2s <0.4652 for basis pursuit, δ3s <0.5 and δ2s <0.25 for iterative hard thresholding, and δ4s <0.3843 for compressive sampling matching pursuit. The arguments also applies to almost sparse vectors and corrupted measurements. The analysis of iterative hard thresholding is surprisingly simple. The analysis of basis pursuit features a new inequality that encompasses several inequalities encountered in Compressive Sensing.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cai, T.T., Wang, L., Xu, G.: Shifting inequality and recovery of sparse signals. IEEE Transactions on Signal Processing 58, 1300–1308 (2010).
Candès, E.J.: The restricted isometry property and its implications for compressed sensing. Comptes Rendus de l’Académie des Sciences, Série I, 346, 589–592 (2008).
Candès, E., Tao. T.: Decoding by linear programing. IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
Davies, M.E., Blumensath, T.: Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal. 27, 265–274 (2009).
Foucart, S.: A note on guaranteed sparse recovery via ℓ 1-minimization. Applied and Comput. Harmonic Analysis, To appear. Appl. Comput. Harmon. Anal. 29, 97–103 (2010).
Garg, R., Khandekar, R.: Gradient descent with sparsification: An iterative algorithm for sparse recovery with restricted isometry property. In: Bottou, L., Littman, M. (eds.) Proceedings of the 26 th International Confer- ence on Machine Learning, pp. 337-344.
Needell, D., Tropp, J.A.: CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26 301–321 (2009).
Acknowledgements
The author thanks the meeting organizers, Mike Neamtu and Larry Schumaker, for welcoming a minisymposium on Compressive Sensing at the Approximation Theory conference. He also acknowledges support from the French National Research Agency (ANR) through project ECHANGE (ANR-08-EMER-006).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this paper
Cite this paper
Foucart, S. (2012). Sparse Recovery Algorithms: Sufficient Conditions in Terms of Restricted Isometry Constants. In: Neamtu, M., Schumaker, L. (eds) Approximation Theory XIII: San Antonio 2010. Springer Proceedings in Mathematics, vol 13. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0772-0_5
Download citation
DOI: https://doi.org/10.1007/978-1-4614-0772-0_5
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-0771-3
Online ISBN: 978-1-4614-0772-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)