Abstract
We present an algorithm for finding sparse solutions of the system of linear equations A x = b with the rectangular matrix A of size n ×N, where n < N. The algorithm basic constructive block is one iteration of the standard interior-point linear programming algorithm. To find the sparse representation we modify (reweight) each iteration in the spirit of Petukhov (Fast implementation of orthogonal greedy algorithm for tight wavelet frames. Signal Process. 86, 471–479 (2006)). However, the weights are selected according to the ℓ 1-greedy strategy developed in Kozlov and Petukhov (Sparse solutions for underdetermined systems of linear equations. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 1243–1259. Springer, Berlin (2010)). Our algorithm combines computational complexity close to plain ℓ 1-minimization with the significantly higher efficiency of the sparse representations recovery than the reweighted ℓ 1-minimization (Candes et al.: Enhancing sparsity by reweighted ℓ 1 minimization. J. Fourier Anal. Appl. 14, 877–905 (2008) (special issue on sparsity)), approaching the capacity of the ℓ 1-greedy algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Candès, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inform. Theor. 51, 4203–4215 (2005)
Candes, E.J., Wakin, M.B., Boyd, S.: Enhancing sparsity by reweighted ℓ 1 minimization. J. Fourier Anal. Appl. 14, 877–905 (2008) (special issue on sparsity)
Chartrand, R., Yin, W.: Iteratively reweighted algorithms for compressive sensing. In: Proceedings of International Conference on Acoustics, Speech, Signal Processing (ICASSP), pp. 3869–3872 (2008)
Daubechies, I., DeVore, R., Fornasier, M., Güntürk, C.S.: Iteratively re-weighted least squares minimization for sparse recovery. Commun. Pure Appl. Math. 63, 1–38 (2010)
Donoho, D.: Compressed sensing. IEEE Trans. Inform. Theor. 52, 1289–1306 (2006)
Donoho, D., Tsaig, Y., Drori, I., Starck, J.: Sparse solutions of underdetermined linear equations by stagewise orthogonal matching pursuit. Report, Stanford University (2006)
Foucart, S., Lai, M.J.: Sparsest solutions of underdetermined linear systems via ℓ q -minimization for 0 ≤ q ≤ 1. Appl. Comp. Harmonic Anal. 26, 395–407 (2009)
Kozlov, I., Petukhov, A.: Sparse solutions for underdetermined systems of linear equations. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 1243–1259. Springer, Berlin (2010)
Lai, M.-J., Wang, J.: An unconstrained l q minimization with 0 < q ≤ 1 for sparse solution of under-determined linear systems. SIAM J. Optim. 21, 82–101 (2010)
Natarajan, B.K.: Sparse approximate solution to linear systems. SIAM J. Comput. 24, 227–234 (1995)
Petukhov, A.: Fast implementation of orthogonal greedy algorithm for tight wavelet frames. Signal Process. 86, 471–479 (2006)
Rudelson, M., Vershynin, R.: Geometric approach to error correcting codes and reconstruction of signals. Int. Math. Res. Not. 64, 4019–4041 (2005)
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
Petukhov, A., Kozlov, I. (2012). Fast Implementation of ℓ 1-Greedy Algorithm. In: Bilyk, D., De Carli, L., Petukhov, A., Stokolos, A., Wick, B. (eds) Recent Advances in Harmonic Analysis and Applications. Springer Proceedings in Mathematics & Statistics, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4565-4_25
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4565-4_25
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4564-7
Online ISBN: 978-1-4614-4565-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)