An unconstrained lq minimization with 0 < q ≤ 1 for sparse solution of underdetermined linear systems

SIAM Journal on Optimization

Volumn 21, Issue 1, 2011, Pages 82-101

  Lap, Ming Jun ^a   Wang, Jingyue ^a  

^a UNIVERSITY OF GEORGIA   (United States)

Author keywords

Compressed sensing; lq minimization; Sparse solution

Indexed keywords

COMPRESSED SENSING; ITERATIVE ALGORITHM; ITERATIVE SOLUTIONS; LQ MINIMIZATION; NONCONVEX; NUMERICAL EXPERIMENTS; SPARSE SOLUTION; SPARSE SOLUTIONS;

ALGORITHMS; LINEAR SYSTEMS;

ITERATIVE METHODS;

EID: 79957452717     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/090775397     Document Type: Article

References (37)

1. R. BARANIUK, Compressive sensing, IEEE Signal Process. Mag., 24 (2007), pp. 118-121.
2. A. M. BRUCKSTEIN, D. L. DONOHO, AND M. ELAD, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Review, 51 (2009), pp. 34-81.
3. E. J. Candès, http://www-stat.stanford.edu/-candes/.
4. E. J. CANDÈS, Compressive sampling, in Proceedings of the International Congress of Mathematicians, Eur. Math. Soc., Madrid, Spain, 2006, pp. 1433-1452.
5. E. J. CANDÈS, The restricted isometry property and its implications for compressed sensing, Comptes Rendus de l'Académie des Sciences, Série I, 346 (2008), pp. 589-592.
6. E. J. CANDÈS, J. K. ROMBERG, AND T. TAO, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., 59 (2006), pp. 1207-1223. (Pubitemid 43988295)
7. E. J. CANDÈS AND T. TAO, Decoding by linear programming, IEEE Trans. Inform. Theory, 51 (2005), pp. 4203-4215. (Pubitemid 41800353)
8. E. J. CANDÈS AND T. TAO, Near-optimal signal recovery from random projections: Universal encoding strategies, IEEE Trans. Inform. Theory, 52 (2006), pp. 5406-5425. (Pubitemid 46445381)
9. 1 minimization, J. Fourier Anal. Appl., 14 (2008), pp. 877-905.
10. R. CHARTRAND, Exact reconstruction of sparse signals via nonconvex minimization, IEEE Signal Process. Lett., 14 (2007), pp. 707-710. (Pubitemid 47434090)
11. R. CHARTRAND AND V. STANEVA, Restricted isometry properties and nonconvex compressive sensing, Inverse Problems, 24 (2008), pp. 1-14.
12. p Minimization, http://www.stanford.edu/- yyye/
13. I. DAUBECHIES, M. DEFREISE, AND C. DE MOL, An iterative thresholding algorithm for linear inverse problem with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), pp. 1413-1457. (Pubitemid 39427442)
14. I. DAUBECHIES, R. DEVORE, M. FORNASIER, AND C. S. GÜNTUK, Iteratively re-weighted least squares minimization for sparse recovery, Comm. Pure Appl. Math., 63 (2010), pp. 1-38.
15. D. L. DONOHO, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), pp. 1289-1306.
16. 1 minimization, Proc. Natl. Acad. Sci. USA, 100 (2003), pp. 2197-2202. (Pubitemid 36297476)
17. D. L. DONOHO, M. ELAD, AND V. N. TEMLYAKOV, Stable recovery of sparse overcomplete representations in the presence of noise, IEEE Trans. Inform. Theory, 52 (2006), pp. 6-18. (Pubitemid 43263116)
18. D. L. DONOHO AND B. F. LOGAN, Signal recovery and the large sieve, SIAM J. Appl. Math., 52 (1992), pp. 577-591.
19. i-minimization, Appl. Comput. Harmon. Anal., 29 (2010), pp. 97-103.
20. q minimization for 0 < q < 1, Appl. Comput. Harmon. Anal., 26 (2009), pp. 395-407.
21. S. FOUCART AND M. J. LAI, Sparse recovery with pre-Gaussian random matrices, Studia Math., 200 (2010), pp. 91-102.
22. E. DE GIORGI AND T. FRANZONI, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. (8), 58 (1975), pp. 842-850.
23. p minimization, Math. Program., submitted, 2010.
24. R. GRIBNOVAL AND M. NIELSEN, Sparse decompositions in unions of bases, IEEE Trans. Inform. Theory, 49 (2003), pp. 3320-3325.
25. 1 Minimization, manuscript, 2008.
26. S. KARLIN AND W. J. STUDDEN, Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience, New York, 1966.
27. I. KOZLOV AND A. PETUKHOV, Sparse solutions of underdetermined linear systems, in Handbook of Geomathematics, W. Freeden, M. Z. Nashed, and T. Sonar, eds., Springer, Berlin, 2010, pp. 1243-1259.
28. M. J. LAI, On sparse solution of underdetermined linear systems, J. Concr. Appl. Math., 8 (2010), pp. 296-327.
29. G. DAL MASO, An Introduction to G-Convergence, Birkhauser, Basel, Switzerland, 1993.
30. B. K. NATARAJAN, Sparse approximate solutions to linear systems, SIAM J. Comput., 24 (1995), pp. 227-234.
31. D. NEEDELL AND R. VERSHYNIN, Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit, Found. Comput. Math., 9 (2009), pp. 317-334.
32. M. NIKOLOVA, M. NG, S. ZHANG, AND W.-K. CHING, Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization, SIAM J. Imaging Sci., 1 (2008), pp. 2-25.
33. A. PETUKHOV, Fast implementation of orthogonal greedy algorithm for tight wavelet frames, Signal Process., 86 (2006), pp. 471-479. (Pubitemid 43106566)
34. Q. SUN, Recovery of sparsest signals via lq minimization, Appl. Comput. Harmon. Anal., submitted, 2010.
35. V. N. TEMLYAKOV, Nonlinear methods of approximation, Found. Comput. Math., 3 (2003), pp. 33-107.
36. J. A. TROPP, Greed is good: Algorithmic results for sparse approximation, IEEE Trans. Inform. Theory, 50 (2004), pp. 2231-2242.
37. 1minimization with applications to compressed sensing, SIAM J. Imaging Sci., 1 (2008), pp. 143-168.