Highly robust error correction by convex programming

IEEE Transactions on Information Theory

Volumn 54, Issue 7, 2008, Pages 2829-2840

  Candès, Emmanuel J ^a   Randall, Paige A ^a  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

1 minimization; Decoding of (random) linear codes; Gaussian random matrices and random projections; Linear programming; Lnear codes; Restricted orthonormality; Second order cone programming; Sparse solutions to underdetermined systems; The Dantzig selector

Indexed keywords

COMMUNICATION;

CONVEX PROGRAMMING;

ERROR CORRECTION;

EID: 46749134483     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2008.924688     Document Type: Article

References (26)

1. E. J. Candès and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4203-4215, Dec. 2005.
2. E. Candès, M. Rudelson, T. Tao, and R. Vershynin, "Error correction via linear programming," in Proc. 46th Annu. IEEE Symp. Foundations of Computer Science (FOCS), Pittsburgh, PA, Oct. 2005, pp. 295-308.
3. E. J. Candès, J. K. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Comm. Pure Appl. Math. vol. 59, no. 8, pp. 1207-1223, 2006
4. E. Candès and T. Tao, "The Dantzig selector: Statistical estimation when p is much larger than n," Ann. Statist., vol. 35, no. 6, pp. 2313-2351, 2007.
5. S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004.
6. D. Donoho, "Neighborly polytopes and sparse solutions of underdetermined linear equations," Stanford Univ., Stanford, CA, 2005, Tech. Rep.
7. C. Dwork, F. McSherry, and K. Talwar, "The price of privacy and the limits of LP decoding," in STOC '07: Proc. 39th Annu. ACM Symp. Theory of Computing, New York, NY., 2007, pp. 85-94.
8. E. Candès and J. Romberg, "Practical signal recovery from random projections," in Proc. SPIE Int. Symp. Electoric Imaging: Coniputational Imaging III, 2005.
9. B. Laurent and P. Massart, "Adaptive estimation of a quadratic functional by model selection," Ann. Statist., vol. 28, no. 5, pp. 1302-1338, 2000.
10. I. M. Johnstone, "Chi-square oracle inequalities," in State of the Art in Probability and Statistics (Leiden, 1999), ser. IMS Lecture Notes Monographs Ser.. Beachwood, OH: Inst. Math. Statist., 2001, vol. 36, pp. 399-418.
11. A. Barvinok, A Course in Convexity, ser. Graduate Studies in Mathematics. Providence, RI: AMS, 2002, vol. 54.
12. G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, ser. Cambridge, Tracts in Mathematics. Cambridge, U.K.: Cambridge, Univ. Press, 1989, vol. 94.
13. K. R. Davidson and S. J. Szarek, "Addenda and corrigenda to: Local operator theory, random matrices and banach spaces," in Handbook, of the Geometry of Banach Spaces. Amsterdam, The Netherlands: North-Holland, 2001, vol. I, pp. 317-366.
14. E. J. Candès and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?," IEEE Trans. Inf. Theory, vol. 52, no. 12, pp. 5406-5425, Dec. 2006.
15. 1 -norm solution is also the sparsest solution," Commun. Pure Appl. Math., vol. 59, no. 6, pp. 797-829, 2006.
16. M. Rudelson and R. Vershynin, "Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements," in Proc. 2006 40th. Annu. Conf. Information Sciences and Systems,, Mar. 2006, pp. 207-212.
17. J Feldman and D. R. Karger, "Decoding turbo-like codes via linear programming," J. Comput. Syst. Sci., vol. 688, no. 4, pp. 733--752, 2004.
18. J. Feldman and C. Stein, "LP decoding achieves capacity." in Proc. 16th Annu. ACM-SIAM Symp. Discrele Algorithms, New York, 2005, pp. 460-469.
19. J. Feldman, T. Malkin, R. A. Servedio, C. Stein, and M. J. Wainwright, "LP decoding corrects a constant fraction of errors," IEEE Trans. Inf. Theory, vol. 53, no. 1, pp. 82-89, Jan. 2007.
20. E. J. 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.
21. D. L. Donoho, "Compressed sensing," IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289-1306, Apr. 2006.
22. J. Tropp and A. Gilbert, "Signal Recovery From Partial Information via Orthogonal Matching Pursuit," 2005, unpublished manuscript.
23. A. Cohen, W. Dahmen, and R. DeVore, "Compressed Sensing and the Best κ-Term Approximation 2006, impublished manuscript.
24. M. Rudelson and R. Vershynin, "Geometric approach to error-correcting codes and reconstruction of signals," Int. Math. Res. Not., no. 64, pp. 4019-4041, 2005.
25. M. Vetterli, P. Marziliano, and T. Blu, "Sampling signals with finite rate of innovation," IEEE Trans. Signal Pnocess., vol. 50, no. 6, pp. 1417-1428, Jun. 2002.
26. P. Marziliano, M. Vetterli, and T. Blu, "Sampling and exact reconstruction of bandlimited signals with additive shot noise," IEEE Trans. Inf. Theory, vol. 52, no. 5, pp. 2230-2233, May 2006.