Uncertainty principles and vector quantization

IEEE Transactions on Information Theory

Volumn 56, Issue 7, 2010, Pages 3491-3501

  Lyubarskii, Yurii ^a   Vershynin, Roman ^b  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

^b UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Frame representations; Kashin's representations; Restricted isometries; Uncertainty principles

Indexed keywords

A-FRAMES; COEFFICIENT LOSS; DYNAMIC RANGE; REDUCTION OF ERRORS; UNCERTAINTY PRINCIPLES;

VECTOR QUANTIZATION;

EID: 77953791245     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2010.2048458     Document Type: Article

References (39)

1. S. Artstein-Avidan, O. Friedland, V. Milman, and S. Sodin, "Polynomial bounds for large Bernoulli sections of the cross polytope," Israel J. Math, vol.156, pp. 141-151, 2006.
2. S. Artstein-Avidan and V. D. Milman, "Logarithmic reduction of the level of randomness in some probabilistic geometric constructions," J. Funct. Anal., vol.235, pp. 297-329, 2006.
3. E. J. Candés and J. Romberg, "Quantitative robust uncertainty principles and optimally sparse decompositions," Found. Comput. Math., vol.6, pp. 227-254, 2006.
4. E. J. Candés, J. Romberg, and T. Tao, "Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information," IEEE Trans. Inf. Theory, vol.52, pp. 489-509, 2006.
5. E. J. Candés, J. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math., vol.59, pp. 1207-1223, 2006.
6. E. J. Candés, M. Rudelson, T. Tao, and R. Vershynin, "Error correction via linear programming," in Proc. 46th Annu. IEEE Symp. Foundations of Computer Science, 2005, pp. 295-308.
7. E. J. Candés and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies," IEEE Trans. Inf. Theory, vol.52, pp. 5406-5425, 2006.
8. E. J. Candés and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory, vol.51, pp. 4203-4215, 2005.
9. Compressed Sensing [Online]. Available: http://www.dsp.ece.rice. edu/CS/
10. Z. Cvetkovic, "Resilience properties of redundant expansions under additive noise and quantization," IEEE Trans. Inf. Theory, vol.49, pp. 644-656, 2003.
11. I. Daubechies, "Ten lectures on wavelets," presented at the CBMS-NSF Regional Conf. Ser. Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
12. I. Daubechies and R. DeVore, "Approximating a bandlimited function using very coarsely quantized data: A family of stable sigmadelta modulators of arbitrary order," Ann. Math., vol.158, no.2, pp. 679-710, 2003.
13. I. Daubechies, R. DeVore, C. S. Güntürk, and V. A. Vaishampayan, "Beta expansions: A new approach to digitally correct A/D conversion," presented at the IEEE Int. Symp. Circuits and Systems ISCAS 2002, Phoenix, AZ, May 2002.
14. I. Daubechies, R. DeVore, C. S. Güntürk, and V. A. Vaishampayan, "A/D conversion with imperfect quantizers," IEEE Trans Inf. Theory, vol.53, pp. 874-885, 2006.
15. I. Daubechies and O. Ylmaz, "Robust and practical analog-to-digital conversion with exponential precision," IEEE Trans. Inf. Theory, vol.52, pp. 3533-3545, 2006.
16. D. Donoho and P. B. Stark, "Uncertainty principles and signal recovery," SIAM J. Appl. Math., vol.49, pp. 906-931, 1989.
17. A. Y. Garnaev and E. D. Gluskin, "The widths of a Euclidean ball," Soviet Math. Dokl., vol.30, pp. 200-204, 1984.
18. A. Gilbert, M. Strauss, J. Tropp, and R. Vershynin, "One sketch for all: Fast algorithms for compressed sensing," in Proc. 39th Annu. ACM Symp. Theory of Computing, New York, 2007, pp. 237-246.
19. E. D. Gluskin, Unpublished Communications.
20. V. K. Goyal, J. Kovacevic, and J. A. Kelner, "Quantized frame expansions with erasures," Appl. Comput. Harmon. Anal., vol.10, pp. 203-233, 2001.
21. V. K. Goyal, M. Vetterli, and N. T. Thao, "Quantized overcomplete expansions in : Analysis, synthesis, and algorithms," IEEE Trans. Inf. Theory, vol.44, pp. 16-31, 1998.
22. R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Inf. Theory, vol.44, pp. 2325-2383, 1998.
23. O. Guedon, S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann, "Subspaces and orthogonal decompositions generated by bounded orthogonal systems," Positivity, vol.11, pp. 269-283, 2007.
24. S. Güntúrk, "One-bit sigma-delta quantization with exponential accuracy," Commun. Pure Appl. Math., vol.56, pp. 1608-1630, 2003.
25. V. Havin and B. Jöricke, The Uncertainty Principle in Harmonic Analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics and Related Areas (3)). Berlin, Germany: Springer- Verlag, 1994, vol.28.
26. P. Indyk, "Stable distributions, pseudorandom generators, embeddings and data stream computation," presented at the 41st Symp. Foundations of Computer Science, 2000.
27. P. Indyk, Uncertainty Principles, Extractors, and Explicit Embeddings of L2 into L1, Electronic Colloquium on Computational Complexity, 2006, Rep. No.126.
28. B. Kashin, "Sections of some finite dimensional sets and classes of smooth functions," Izv. Acad. Nauk SSSR, vol.41, pp. 334-351, 1977.
29. M. Ledoux, "The concentration of measure phenomenon," AMS Math. Surv. Monogr., vol.89, 2001.
30. A. E. Litvak, A. Pajor, M. Rudelson, and N. Tomczak-Jaegermann, "Smallest singular value of random matrices and geometry of random polytopes," Adv. Math., vol.195, no.2, pp. 491-523, 2005.
31. A. E. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaegermann, and R. Vershynin, "Euclidean embeddings in spaces of finite volume ratio via random matrices," J. Reine Angew. Math., vol.589, pp. 1-19, 2005.
32. S. Mendelson and A. P. N. Tomczak-Jaegermann, "Reconstruction and subgaussian operators in asymptotic geometric analysis," Geom. Funct. Anal., vol.17, pp. 1248-1282, 2007.
33. S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann, "Uniform uncertainty principle for Bernoulli and subgaussian ensembles," Constr. Approx., vol.28, pp. 277-289, 2008.
34. G. Pisier, "The volume of convex bodies and Banach space geometry," in Cambridge Tracts in Mathematics 94. Cambridge, U.K.: Cambridge Univ. Press, 1989.
35. M. Rudelson and R. Vershynin, "Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements," presented at the CISS 2006 (40th Annu. Conf. Information Sciences and Systems).
36. M. Rudelson and R.Vershynin, On Sparse Reconstruction from Fourier and Gaussian Measurements, unpublished.
37. G. Schechtman, "Random embeddings of Euclidean spaces in sequence spaces," Israel J. Math., vol.40, pp. 187-192, 1981.
38. M. Talagrand, "Selecting a proportion of characters," Israel J. Math., vol.108, pp. 173-191, 1998.
39. A. W. van der Vaart and J. A.Wellner, "Weak convergence and empirical processes," in Springer Series in Statistics. New York: Springer- Verlag, 1996.