New and improved johnson-lindenstrauss embeddings via the restricted isometry property

SIAM Journal on Mathematical Analysis

Volumn 43, Issue 3, 2011, Pages 1269-1281

  Krahmer, Felix ^a   Ward, Rachel ^b  

^a UNIVERSITY OF BONN   (Germany)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

Johnson Lindenstrauss lemma; Optimal asymptotics; Rademacher chaos; Restricted isometry property

Indexed keywords

COMPRESSED SENSING; EMBEDDING DIMENSIONS; EMBEDDINGS; HADAMARD MATRICES; HIGH PROBABILITY; JOHNSON-LINDENSTRAUSS LEMMATA; MATRIX; MULTIPLICATIVE FACTORS; N-MATRIX; OPTIMAL ASYMPTOTICS; PARTIAL-FOURIER; RADEMACHER CHAOS; RANDOM MATRICES; REDUNDANT DICTIONARIES; RESTRICTED ISOMETRY PROPERTY; SPARSE VECTORS;

OPTIMIZATION;

MATRIX ALGEBRA;

EID: 79960436599     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/100810447     Document Type: Article

References (40)

1. N. Ailon and B. Chazelle, Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform, in Proceedings of the Thirty-Eighth Annual ACMSymposium on Theory of Computing, 2006, pp. 557-563. (Pubitemid 44306590)
2. N. Ailon and E. Liberty, An almost optimal unrestricted fast Johnson-Lindenstrauss transform, in Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '11), SIAM, Philadelphia, 2011, pp. 185-191.
3. N. Ailon and E. Liberty, Fast dimension reduction using Rademacher series on dual BCH codes, in Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '08), SIAM, Philadelphia, 2008, pp. 1-9.
4. N. Ailon, E. Liberty, and A. Singer, Dense fast random projections and Lean Walsh transforms, in Proceedings of the Twelfth International Workshop on Randomization and Computation (RANDOM), 2008, pp. 512-522.
5. N. Alon, Problems and results in extremal combinatorics, Discrete Math., 273 (2003), pp. 31-53.
6. R. G. Baraniuk, M. Davenport, R. A. DeVore, and M. Wakin, A simple proof of the restricted isometry property for random matrices, Constr. Approx., 28 (2008), pp. 253-263.
7. R. Baraniuk and M. Wakin, Random projections of smooth manifolds, Found. Comput. Math., 9 (2009), pp. 51-77.
8. S. Boucheron, G. Lugosi, and P. Massart, Concentration inequalities using the entropy method, Ann. Probab., 31 (2003), pp. 1583-1614. (Pubitemid 36881408)
9. J. Bourgain, S. Dilworth, K. Ford, S. Konyagin, and D. Kutzarova, Explicit constructions of RIP matrices and related problems, Duke Math. J., to appear.
10. E. Candès, Y. Eldar, and D. Needell, Compressed sensing with coherent and redundant dictionaries, Appl. Comput. Harmon. Anal., 31 (2011), pp. 59-73.
11. E. J. Candès, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), pp. 489-509. (Pubitemid 43174093)
12. E. J. Candès, J. Romberg, and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., 59 (2006), pp. 1207-1223. (Pubitemid 43988295)
13. 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)
14. S. Dasgupta and A. Gupta, An elementary proof of a theorem of Johnson and Lindenstrauss, Random Structures Algorithms, 22 (2003), pp. 60-65.
15. A. Dasgupta, R. Kumar, and T. Sarlos, A sparse Johnson-Lindenstrauss transform, in Proceedings of the Forty-Second Annual ACM Symposium on Theory of Computing (STOC), 2010, pp. 341-350.
16. R. DeVore, Deterministic constructions of compressed sensing matrices, J. Complexity, 23 (2007), pp. 918-925. (Pubitemid 350122965)
17. T. Do, L. Gan, Y. Chen, N. Nguyen, and T. Tran, Fast and efficient dimensionality reduction using structurally random matrices, in Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2009, pp. 1821-1824.
18. 1 solution is also the sparsest solution, Commun. Pure Appl. Anal., 59 (2006), pp. 797-829. (Pubitemid 43667226)
19. 1- minimization, Appl. Comput. Harmon. Anal., 29 (2010), pp. 97-103.
20. S. Foucart, A. Pajor, H. Rauhut, and T. Ullrich, The Gelfand widths of lp-balls for 0 0 < p ≤1, J. Complexity, 26 (2010), pp. 629-640.
21. P. Frankl and H. Maehara, The Johnson-Lindenstrauss Lemma and the sphericity of some graphs, J. Combin. Theory Ser. B, 44 (1988), pp. 355-362.
22. A. Garnaev and E. Gluskin, The widths of a Euclidean ball, Dokl. Akad. Nauk SSSR, 277 (1984), pp. 1048-1052 (in Russian).
23. N. Halko, P. G. Martinsson, and J. A. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), pp. 217-288.
24. D. L. Hanson and F. T. Wright, A bound on tail probabilities for quadratic forms in independent random variables, Ann. Math. Statist., 42 (1971), pp. 1079-1083.
25. W. Hoeffding, Probability inequalities for sums of bounded random variables, J.Amer. Statist. Assoc., 58 (1963), pp. 13-30.
26. P. Indyk, Algorithmic applications of low-distortion embeddings, in Proceedings of the Forty-Second IEEE Symposium on Foundations of Computer Science (FOCS), 2001, p. 10.
27. M. Iwen, Simple deterministically constructible RIP matrices with sublinear Fourier sampling requirements, in Proceedings of the Forty-Third Annual Conference on Information Sciences and Systems (CISS 2009), 2009, pp. 870-875.
28. W. B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Contemp. Math, 26 (1984), pp. 189-206.
29. D. Kane and J. Nelson, A Derandomized Sparse Johnson-Lindenstrauss Transform, Preprint, 2010.
30. E. Liberty, F. Woolfe, P.-G. Martinsson, V. Rokhlin, and M. Tygert, Randomized algorithms for the low-rank approximation of matrices, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 20167-20172.
31. H. Rauhut, Circulant and Toeplitz matrices in compressed sensing, in Proceedings of the Workshop on Signal Processing with Adaptive Sparse Structured Representations (SPARS'09), Saint-Malo, France, 2009.
32. H. Rauhut, Compressive sensing and structured random matrices, in Theoretical Foundations and Numerical Methods for Sparse Recovery, M. Fornasier, ed., Radon Series Comp. Appl. Math. 9, de Gruyter, 2010, pp. 1-92.
33. H. Rauhut, J. Romberg, and J. Tropp, Restricted isometries for partial random circulant matrices, Appl. Comput. Harmon. Anal., to appear.
34. 1 Minimization, Preprint, 2010.
35. J. Romberg, Compressive sensing by random convolution, SIAM J. Imaging Sci., 2 (2009), pp. 1098-1128.
36. M. Rudelson and R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements, Comm. Pure Appl. Math., 61 (2008), pp. 1025-1045.
37. T. Sarlos, Improved approximation algorithms for large matrices via random projections, in Proceedings of the Forty-Seventh IEEE Symposium on Foundations of Computer Science (FOCS), 2006, pp. 143-152. (Pubitemid 351175504)
38. J. Tropp, J. Laska, M. Duarte, J. Romberg, and R. G. Baraniuk, Beyond Nyquist: Efficient sampling of sparse, bandlimited signals, IEEE Trans. Inform. Theory, 56 (2010), pp. 520-544.
39. J. Vybíral, A variant of the Johnson-Lindenstrauss lemma for circulant matrices, J. Funct. Anal., 260 (2011), pp. 1096-1105.
40. R. Ward, Compressed sensing with cross validation, IEEE Trans. Inform. Theory, 55 (2009), pp. 5773-5782.