Compressed sensing with nonlinear observations and related nonlinear optimization problems

IEEE Transactions on Information Theory

Volumn 59, Issue 6, 2013, Pages 3466-3474

  Blumensath, Thomas ^a  

^a UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

Author keywords

Compressed sensing (CS); inverse problems; nonconvex constraints; nonlinear optimization

Indexed keywords

COMPRESSIVE SENSING; ITERATIVE HARD THRESHOLDING; NON-CONVEX CONSTRAINTS; NON-LINEAR OPTIMIZATION; NON-LINEAR OPTIMIZATION PROBLEMS; OPTIMIZATION PROBLEMS; RESTRICTED ISOMETRY PROPERTIES; SPARSITY CONSTRAINTS;

INVERSE PROBLEMS; ITERATIVE METHODS; MATRIX ALGEBRA; OPTIMIZATION; SENSORS;

SIGNAL RECONSTRUCTION;

EID: 84877901629     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2013.2245716     Document Type: Article

References (23)

1. E. Candès, J. Romberg, and T. Tao, "Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information," IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489-509, Feb. 2006. (Pubitemid 43174093)
2. D. L. Donoho, "For most large underdetermined systems of linear equations the minimal 1-norm solution is also the sparsest solution," Commun. Pure Appl. Math., vol. 59, no. 6, pp. 797-829, 2006. (Pubitemid 43667226)
3. E. Candès and M. B. Wakin, "An introduction to compressive sampling," IEEE Signal Process. Mag., vol. 25, no. 2, pp. 21-30, Mar. 2008.
4. E. Candès, "The restricted isometry property and its implications for compressed sensing," Comptes Rendus de l'Academie des Sci., vol. 346, pp. 589-592, 2008.
5. D. Needell and J. Tropp, "COSAMP: Iterative signal recovery from incomplete and inaccurate samples," Appl. Comput. Harmonic Anal., vol. 26, pp. 301-321, 2008.
6. W. Dai and O.Milenkovic, "Subspace pursuit for compressed sensing: Closing the gap between performance and complexity," IEEE Trans. Inf. Theory, vol. 55, no. 5, pp. 2230-2249, May 2009.
7. T. Blumensath and M. E. Davies, "Iterative hard thresholding for compressed sensing," Appl. Comput. Harmon. Anal., vol. 27, no. 3, pp. 265-274, 2009.
8. S. Foucart, "Hard thresholding pursuit: An algorithm for compressive sensing," SIAM J. Numer. Anal., vol. 49, no. 6, pp. 2543-2563, 2011.
9. V. Cevher, "An ALPS view of sparse recovery," in Proc. IEEE. Conf. Acoust., Speech Signal Process., 2011, pp. 5808-5811.
10. 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.
11. D. Goldfarb and S. Ma, Convergence of fixed point continuation algorithms for matrix rank minimization 2010 [Online]. Available: arXiv:09063499v3
12. R. Baraniuk, V. Cevher, M. Duarte, and C. Hegde, "Model-based compressive sensing," IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 1982-2001, Apr. 2010.
13. R. T. Rockafellar, "Extended nonlinear programming," in Nonlinear Optimization and Related Topics, G. Di Pillo and F. Giannessi, Eds. Norwell, MA, USA: Kluwer, 1999.
14. A. S. Lewis, D. R. Luke, and J. Malick, "Local linear convergence for alternating and averaged nonconvex projections," Found. Comput. Math., vol. 9, no. 4, pp. 485-513, 2009.
15. D. R. Luke, "Local linear convergence of approximate projections onto regularized sets," Nonlinear Anal.: Theory, Methods Appl., vol. 75, no. 3, pp. 1531-1546, 2012.
16. T. Blumensath, Compressed sensingwith non-linear observations 2010 [Online]. Available: http://eprints.soton.ac.uk/164753/
17. W. Xu, M. Wang, and A. Tang, Sparse recovery from nonlinear measurements with applications in bad data detection for power networks 2011 [Online]. Available: arXiv:1112.6234v1
18. S. Bahmani, B. Raj, and P. Boufounos, Greedy sparsity-constrained optimization 2012 [Online]. Available: arXiv:1203.5483v1
19. H. Attouch, J. Bolte, and B. F. Svaiter, "Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, Forward-backward splitting, and regularized Gauss-Seidel methods," Math. Program., vol. 137, pp. 91-129, 2013.
20. T. Blumensath, "Sampling and reconstructing signals from a union of linear subspaces," IEEE Trans. Inf. Theory, vol. 57, no. 7, pp. 4660-4671, Jul. 2011.
21. A. Kyrillidis and V. Cevher, "Combinatorial selection and least absolute shrinkage via the CLASH algorithm," Preprint 2011.
22. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems. Norwell, MA, USA: Kluwer, 2000.
23. S. Negahban, P. Ravikumar, M. J. Wainwright, and B. Yu, "A unified framework for the analysis of regularized-estimators," presented at the Proc. Adv. Neural Inf. Process. Syst., Vancouver, BC, Canada, 2009.