The Group Lasso for stable recovery of block-sparse signal representations

IEEE Transactions on Signal Processing

Volumn 59, Issue 4, 2011, Pages 1371-1382

  Lv, Xiaolei ^a   Bi, Guoan ^a   Wan, Chunru ^a  

^a NANYANG TECHNOLOGICAL UNIVERSITY   (Singapore)

Author keywords

Block coherence; group Lasso; orthogonal greedy algorithm; sparse representation

Indexed keywords

BLOCK STRUCTURES; BLOCK-COHERENCE; ERROR BOUND; GREEDY ALGORITHMS; GROUP LASSO; MACHINE-LEARNING; NOISY DATA; NUMERICAL EXPERIMENTS; ORIGINAL SIGNAL; ORTHOGONAL GREEDY ALGORITHM; OVERCOMPLETE DICTIONARIES; QUADRATIC PROGRAMS; REGULARIZATION METHODS; SOLUTION UNIQUENESS; SPARSE REPRESENTATION; SPARSE SIGNAL REPRESENTATION; SPARSE SIGNALS; STABILITY RESULTS; THEORETICAL RESULT;

RECOVERY;

ALGORITHMS;

EID: 79952633750     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2011.2105478     Document Type: Article

References (33)

1. S. Mallat and Z. Zhang, "Matching pursuits with time-frequency dictionaries IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3397-3415, Dec. 1993.
2. J. A. Tropp and A. C. Gilbert, "Signal recovery from random measurements via orthogonal matching pursuit IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4655-4666, Dec. 2007.
3. D. Needell and R. Vershynin, "Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit Found. Comput. Math., vol. 9, pp. 317-334, 2009.
4. D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, "Sparse Solution of Underdetermined Linear Equations by Stagewise Orthogonal Matching Pursuit (StOMP) [Online]. Available: http://sparselab.stanford.edu/SparseLab-files/ local-files/StOMP.pdf [Online], Available:
5. I. F. Gorodnitsky and B. D. Rao, "Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm IEEE Trans. Signal Process., vol. 45, no. 3, pp. 600-616, Mar. 1997. (Pubitemid 127765947)
6. S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit SIAM J. Sci. Comp., vol. 20, no. 1, pp. 33-61, 1998. (Pubitemid 128689501)
7. J. J. Fuchs, "On sparse representations in arbitrary redundant bases IEEE Trans. Inf. Theory, vol. 50, no. 6, pp. 1341-1344, Jun. 2004.
8. R. Tibshirani, "Regression shrinkage and selection via the lasso J. Roy. Statist. Soc. Ser. B, vol. 58, no. 1, pp. 267-288, 1996.
9. E. J. Candes and Y. Plan, "Near-ideal model selection by l1 minimization Ann. Statist., vol. 37, no. 5A, pp. 2145-2177, 2009.
10. P. Zhao and B.Yu, "On model selection consistency of Lasso J. Mach. Learn. Res., vol. 7, pp. 2541-2563, 2006. (Pubitemid 44866738)
11. X. Lv, C. Wan, and G. Bi, "Block orthogonal greedy algorithm for stable recovery of block-sparse signal representations Signal Process., vol. 90, no. 12, pp. 3265-3277, Dec. 2010.
12. P. Tseng, "Further results on stable recovery of sparse overcomplete representations in the presence of noise IEEE Trans. Inf. Theory, vol. 55, no. 2, pp. 888-899, Feb. 2009.
13. D. L. Donoho, M. Elad, and V. Temlyakov, "Stable recovery of sparse overcomplete representations in the presence of noise IEEE Trans. Inf. Theory, vol. 52, no. 1, pp. 6-18, Jan. 2006. (Pubitemid 43263116)
14. M. Mishali and Y. C. Eldar, "Blind multi-band signal reconstruction: Compressed sensing for analog signals IEEE Trans. Signal Process., vol. 57, no. 3, pp. 993-1009, Mar. 2009.
15. R. Gribonval and E. Bacry, "Harmonic decomposition of audio signals with matching pursuit IEEE Trans. Signal Process., vol. 51, no. 1, pp. 101-110, Jan. 2003.
16. L. Peotta and P. Vandergheynst, "Matching pursuit with block incoherent dictionaries IEEE Trans. Signal Process., vol. 55, no. 9, pp. 4549-4557, Sep. 2007. (Pubitemid 47323629)
17. Y. C. Eldar, P. Kuppinger, and H. Bolcskei, "Block-sparse signals: Uncertainty relations and efficient recovery IEEE Trans. Signal Process., vol. 58, no. 6, pp. 3042-3054, Jun. 2010.
18. Y. C. Eldar and M. Mishali, "Robust recovery of signals from a structured union of subspaces IEEE Trans. Inf. Theory, vol. 55, no. 11, pp. 5302-5316, Nov. 2009.
19. T. Shimamura, H. Minami, and M. Mizuta, "Regularization Parameter Selection in the Group Lasso [Online]. Available:
20. M.Yuan and Y. Lin, "Model selection and estimation in regression with grouped variables J. Royal Statist. Soc. B, vol. 68, no. 1, pp. 49-67, 2006. (Pubitemid 43415335)
21. F. Bach, "Consistency of the group lasso and multiple kernel learning J. Mach. Learn. Res., vol. 9, pp. 1179-1225, 2008.
22. Y. Nardi and A. Rinaldo, "On the asymptotic properties of the group lasso estimator for linear models Electron. J. Statist. vol. 2, pp. 605-633, 2008 [Online]. Available: http://www.projecteuclid.org/DPubS?service= UI&version=1.0&verb=Display&handle=euclid.ejs/1217450797
23. J. Huang and T. Zhang, The Benefit of Group Sparsity [Online]. Available:
24. J. J. Fuchs, "Recovery of exact sparse representations in the presence of bounded noise IEEE Trans. Inf. Theory, vol. 51, no. 10, pp. 3601-3608, Oct. 2005. (Pubitemid 41448600)
25. F. Santosa and W. W. Symes, "Linear inversion of band-limited reflection seismograms SIAM J. Sci. Stat. Comput., vol. 7, no. 4, pp. 1307-1330, 1986.
26. R. Gribonval and M. Nielsen, "Sparse representations in unions of bases IEEE Trans. Inf. Theory, vol. 49, no. 12, pp. 3320-3325, Dec. 2003.
27. H. Liu and J. Zhang, "Estimation consistency of the group Lasso and its applications in Proc. 12th Int. Conf. AISTATS, FL, 2009, pp. 376-383.
28. N. Meinshausen and B. Yu, "Lasso-type recovery of sparse representations for high-dimensional data Ann. Statist., vol. 37, no. 1, pp. 246-270, 2009.
29. J. A. Tropp, I. S. Dhillon, R. W. Heath, and T. Strohmer, "Designing structured tight frames via alternating projection IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 188-209, Jan. 2005. (Pubitemid 40145995)
30. I. S. Dhillon, R. W. Heath, T. Strohmer, and J. A. Tropp, "Constructing packings in Grassmannian manifolds via alternating projection, Exper. Math., vol. 17, no. 1, pp. 9-35, Jan. 2008.
31. H. Wang and C. Leng, "A note on adaptive group Lasso Computat. Statist. Data Anal., vol. 52, pp. 5277-5286, 2008.
32. H. Zou, "The adaptive Lasso and its oracle properties J. Amer. Statist. Assoc., vol. 101, no. 476, pp. 1418-1429, Dec. 2006.
33. Z. Ben-Haim and Y. C. Eldar, "Near-Oracle performance of greedy block-sparse estimation techniques from noisy measurements Sep. 2010 [Online]. Available: