Further results on stable recovery of sparse overcomplete representations in the presence of noise

IEEE Transactions on Information Theory

Volumn 55, Issue 2, 2009, Pages 888-899

  Tseng, Paul ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

1 norm minimization; Basis pursuit; Greedy algorithm; Matching pursuit; Mutual coherence; Overcomplete representation; Sparse representation

Indexed keywords

OPTIMIZATION; SIGNAL PROCESSING;

BASIS PURSUIT; GREEDY ALGORITHM; MATCHING PURSUIT; MUTUAL COHERENCE; OVERCOMPLETE REPRESENTATION; SPARSE REPRESENTATION;

ALGORITHMS;

EID: 61349087401     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2008.2009812     Document Type: Article

References (33)

1. A. R. Barron, A. Cohen, W. Dahmen, and R. A. DeVore, "Approximation and learning by greedy algorithms," Ann. Statist., vol. 36, no. 1, pp. 64-94, 2008.
2. A. Antoniadis and J. Fan, "Regularization of wavelet approximations," J. Amer. Statist. Assoc., vol. 96, no. 455, pp. 939-967, 2001.
3. E. J. Candès, J. Romberg, and T. Tao, "Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information," IEEE Inf. Theory, vol. 52, no. 2, pp. 489-509, Feb. 2006.
4. E. Candès, J. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. vol. 59, pp. 1207-1223, 2006.
5. E. Candes and T. Tao, "Decoding by linear programming," IEEE Inf. Theory, vol. 51, no. 12, pp. 4203-4215, Dec. 2005.
6. S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33-61, 1998.
7. G. Davis, S. Mallat, and Z. Zhang, "Adaptive time-frequency decompositions," Opt. Eng., vol. 33, no. 7, pp. 2183-2191, 1994.
8. R. A. DeVore and V. N. Temlyakov, "Some remarks on greedy algorithms," Adv. Comput. Math., vol. 5, pp. 173-187, 1996.
9. 1-norm near-solution approximates the sparsest near-soltuilon," Dept. Statist., Stanford Univ., Stanford, CA, Tech. Rep., 2006.
10. 1-norm solution is also the sparsest solution," Commun. Pure Appl. Math., vol. 59, no. 6, pp. 797-929, 2006.
11. 1 minimization," Proc. Nat. Acad. Sci., vol. 100, pp. 2197-2202, 2003.
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, no. 1, pp. 6-18, Jan. 2006.
13. D. L. Donoho and X. Huo, "Uncertainty principles and ideal atomic decomposition," IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2845-2862, Nov. 2001.
14. D. L. Donoho, Y. Tsaig, I. Drori, and J.-L. Starck, "Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit," Dept. Statistics, Stanford Univ., Stanford, CA, Tech. Rep., 2006.
15. M. Elad and A. M. Bruckstein, "A generalized uncertainty principle and sparse representation in pairs of bases," IEEE Trans. Inf. Theory vol. 48, no. 9, pp. 2558-2567, Sep. 2002.
16. M. P. Freidlander and P. Tseng, "Exact regularization of convex programs," SIAM J. Optim., vol. 18, no. 4, pp. 1326-1350, Oct. 2007.
17. J.-J. Fuchs. "On sparse representations in arbitrary redundant bases," IEEE Trans. Inf. Theory, vol. 50, no. 6, pp. 1341-1344, Jun. 2004.
18. 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.
19. A. C. Gilbert, S. S. Muthukrishnan, and M. J. Strauss, "Approximation of functions over redundant dictionaries using coherence," in Proc. 14th Annu. ACM-SIAM Symp. Discrete Algorithms, 2003, pp. 243-252.
20. G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: The Johns Hopkins Univ. Press, 1985.
21. 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.
22. R. Gribonval and M. Nielsen, "Sparse representations in unions of bases," IEEE Trans. Inf. Theory, vol. 49, no. 12, pp. 3320-3325, Dec. 2003.
23. S. Mallat and Z. Zhang, "Matching pursuit in a time-frequency dictionary," IEEE Trans. Signal Process. vol. 41, no 12, pp. 3397-3415, Dec. 1993.
24. Y. C. Pati, R. Rezaiifar. and P. S. Krishnaprasad, "Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition," in Proc. 27th Annu. Asilomar Conf. Signals Syst., Comput., Nov. 1993, pp. 40-44.
25. R. T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton Univ. Press, 1970.
26. S. Sardy, A. Bruce, and P. Tseng, "Block coordinate relaxation methods for nonparametric wavelet denoising," J. Comput. Graph. Statist., vol. 9, no. 2, pp. 361-379, 2000.
27. T. Strohmer and R. Heath, Jr., "Grassmannian frames with applications to coding and communications," Appl. Comp. Harm. Anal., vol. 14, no. 3, pp. 257-275, May 2003.
28. V. N. Temlyakov, "Weak greedy algorithms," Adv. Comput. Math., vol. 12, pp. 213-227, 2000.
29. R. Tibshirani, "Regression shrinkage and selection via the lasso," J. Roy. Statist. Soc. Ser. B, vol. 58, no. 1, pp. 267-288, 1996.
30. J. A. Tropp, "Greed is good: Algorithmic results for sparse approximation," IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2231-2242, Oct. 2004.
31. J. A. Tropp, "Just relax: Convex programming methods tor identifying sparse signals in noise," IEEE Trans. Inf. Theory, vol. 52, no. 3, pp. 1030-1051, Mar. 2006.
32. 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.
33. P. Tseng and Z.-Q. Luo, "On the convergence of the affine-scaling algorithm," Math, Program., vol. 56, no. 3, pp. 301-319, 1992.