The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing

IEEE Transactions on Information Theory

Volumn 60, Issue 2, 2014, Pages 1248-1259

  Tillmann, Andreas M ^a   Pfetsch, Marc E ^a  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Compressed sensing; computational complexity; sparse recovery conditions

Indexed keywords

ALGORITHMS; COMPRESSED SENSING; COMPUTATIONAL COMPLEXITY; LINEAR SYSTEMS; SIGNAL RECONSTRUCTION;

BEST CONSTANTS; MUTUAL COHERENCE; NULL SPACE; POSITIVE INTEGERS; RESTRICTED ISOMETRY PROPERTIES; RESTRICTED ISOMETRY PROPERTIES (RIP); SPARSE RECOVERY; UNDERDETERMINED LINEAR SYSTEMS;

MATRIX ALGEBRA;

EID: 84893320476     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2013.2290112     Document Type: Article

References (58)

1. D. L. Donoho, "Compressed sensing, " IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289-1306, Apr. 2006.
2. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. New York, NY, USA: Springer-Verlag, 2010.
3. M. Fornasier and H. Rauhut, "Compressive sensing, " in Handbook of mathematical Methods in Imaging, O. Scherzer, Ed. New York, NY, USA: Springer-Verlag, 2011, pp. 187-228.
4. M. R. Garey and D. S. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness. San Francisco, CA, USA: Freeman and Company, 1979.
5. B. K. Natarajan, "Sparse approximate solutions to linear systems, " SIAM J. Comput., vol. 24, no. 2, pp. 227-234, Apr. 1995.
6. S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit, " SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33-61, Aug. 1998. (Pubitemid 128689501)
7. E. Candès and T. Tao, "Decoding by linear programming, " IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4203-4215, Dec. 2005. (Pubitemid 41800353)
8. R. G. Baraniuk, M. A. Davenport, R. A. DeVore, and M. B. Wakin, "A simple proof of the restricted isometry property for random matrices, " Constructive Approx., vol. 28, no. 3, pp. 253-263, Dec. 2008.
9. J. D. Blanchard, C. Cartis, and J. Tanner, "Compressed sensing: How sharp is the RIP?" SIAM Rev., vol. 53, no. 1, pp. 105-125, Feb. 2011.
10. E. Candès, "The restricted isometry property and its implications for compressed sensing, " Comptes Rendus Math., vol. 346, nos. 9-10, pp. 589-592, 2008.
11. T. T. Cai, L. Wang, and G. Xu, "New bounds for restricted isometry constants, " IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4388-4394, Sep. 2010.
12. B. Bah and J. Tanner, "Improved bounds on restricted isometry constants for Gaussian matrices, " SIAM J. Matrix Anal. Appl., vol. 31, no. 5, pp. 2882-2892, 2010.
13. R. A. DeVore, "Deterministic constructions of compressed sensing matrices, " J. Complex., vol. 23, pp. 918-925, Aug. 2007.
14. M. A. Iwen, "Simple deterministically constructible RIP matrices with sublinear fourier sampling requirements, " in Proc. 43rd Ann. CISS, 2009, pp. 870-875.
15. J. Bourgain, S. J. Dilworth, K. Ford, S. Konyagin, and D. Kutzarova, "Explicit constructions of RIP matrices and related problems, " Duke Math. J., vol. 159, no. 1, pp. 145-185, 2011.
16. M. A. Davenport and M. B. Wakin, "Analysis of orthogonal matching pursuit using the restricted isometry property, " IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4395-4401, Sep. 2010.
17. J. A. Tropp, "Greed is good: Algorithmic results for sparse approximation, " IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2231-2242, Oct. 2004.
18. A. d'Aspremont, L. E. Ghaoui, M. I. Jordan, and G. R. G. Lanckriet, "A direct formulation for sparse PCA using semidefinite programming, " SIAM Rev., vol. 49, no. 3, pp. 434-448, 2007. (Pubitemid 47377693)
19. K. Lee and Y. Bressler, "Computing performance guarantees for compressed sensing, " in Proc. IEEE ICASSP, Apr. 2008, pp. 5129-5132.
20. P. Koiran and A. Zouzias, (2011). On the Certification of the Restricted Isometry Property [online] Available: http://arxiv.org/abs/1103.4984
21. P. Koiran and A. Zouzias, (2012). Hidden Cliques and the Certification of the Restricted Isometry Property [online] Available: http://arxiv.org/abs/1211. 0665
22. Q. Berthet and P. Rigollet, (2013). Computational Lower Bounds for Sparse PCA [online] Available: http://arxiv.org/abs/1304.0828
23. A. S. Bandeira, E. Dobriban, D. G. Mixon, and W. F. Sawin, "Certifying the restricted isometry property is hard, " IEEE Trans. Inf. Theory, vol. 59, no. 6, pp. 3448-3450, Jun. 2013.
24. D. L. Donoho and X. Huo, "Uncertainty principles and ideal atomic decomposition, " IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2845-2862, Nov. 2001. (Pubitemid 33053488)
25. 1- minimization: A non-RIP analysis and extensions, " Dept. Comput. Appl. Math., Rice Univ., Houston, TX, USA, Tech. Rep. TR08-11, 2008.
26. A. Cohen, W. Dahmen, and R. DeVore, "Compressed sensing and best k-term approximation, " J. Amer. Math. Soc, vol. 22, no. 1, pp. 211-231, 2009.
27. R. Gribonval and M. Nielsen, "Sparse representations in unions of bases, " IEEE Trans. Inf. Theory, vol. 49, no. 12, pp. 3320-3325, Dec. 2003.
28. p sparse recovery can fail for 0 < p < 1, " IEEE Trans. Inf. Theory vol. 55, no. 5, pp. 2203-2214 May 2009.
29. A. d'Aspremont, F. Bach, and L. E. Ghaoui, "Optimal solutions for sparse principal component analysis, " J. Mach. Learn. Res., vol. 9, pp. 1269-1294, Jul. 2008.
30. A. d'Aspremont and L. E. Ghaoui, "Testing the nullspace property using semidefinite programming, " Math. Program., vol. 127, no. 1, pp. 123-144, 2011.
31. D. L. Donoho and M. Elad, "Optimally sparse representation in general (non-orthogonal) dictionaries via i minimization, " Pattern. Nat. Acad. Sci. USA, vol. 100, no. 5, pp. 2197-2202, Mar. 2003. (Pubitemid 36297476)
32. L. Khachiyan, "On the complexity of approximating extremat determinants in matrices, " J. Complex., vol. 11, pp. 138-153, Jan. 1995.
33. A. Chistov, H. Fournier, L. Gurvits, and P. Koiran, "Vandermonde matrices, NP-completeness, and transversal subspaces, " Found. Comput. Math., vol. 3, no. 4, pp. 421-427, Oct. 2003. (Pubitemid 37503686)
34. E. Amaldi and V. Kann, "The complexity and approximability of finding maximum feasible subsystems of linear relations, " Theoretical Comput. Sci., vol. 147, nos. 1-2, pp. 181-210, Aug. 1995.
35. T. F. Coleman and A. Pothen, "The null space problem I. Complexity, " SIAM J. Algebra Discribing, vol. 7, no. 4, pp. 527-537, 1986.
36. S. T. McCormick, "A combinatorial approach to some sparse matrix problems, " Ph.D. dissertation, Dept. Oper. Res., Stanford Univ., Stanford, CA, USA, 1983.
37. B. Alexeev, J. Cahill, and D. G. Mixon, "Full spark frames, " J. Fourier Anal. Appl, vol. 3, pp. 2-5, Oct. 2011.
38. M. J. Piff, and D. J. A. Welsh, "On the vector representation of matroids, " J. London Math. Soc, vol. 2, no. 2, pp. 284-288, 1970.
39. R. Luss and M. Teboulle, "Conditional gradient algorithms for rank-one matrix approximations with a sparsity constraint, " SIAM Rev., vol. 55, no. 1, pp. 65-98, 2013.
40. B. Korte and J. Vygen, Combinatorial optimization. Theory and Algorithms, vol. 21. 4th ed., New York, NY, USA: Springer-Verlag, 2008.
41. J. G. Oxley, Matroid Theory, vol. 3. 1st ed., London, U.K.: Oxford Univ. Press, 1992.
42. C. van Nuffelen, "On the incidence matrix of a graph, " IEEE Trans. Circuits Syst, vol. 23, no. 9, pp. 572-574, Sep. 1976.
43. S. De Marchi, "Generalized Vandermonde determinants, Toeplitz matrices and Schur functions, " Ergebnisberichte Angewandte Mathematik, Universität Dortmund, Tech. Rep. 545, Dortmund, Germany, 1999.
44. H. P. Hirst and W. T. Macey, "Bounding the roots of polynomials, " College Math. J., vol. 28, no. 4, pp. 292-295, Sep. 1997.
45. M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, vol. 2. 2nd ed. New York, NY, USA: Springer-Verlag, 1993.
46. R. M. Karp, "Reducibility among combinatorial problems, " in Complexity of Computer Computations, R. Miller and J. W. Thatcher, Eds. New York, NY, USA: Plenum Press, 1972, pp. 85-103.
47. A. Itai and M. Rodeh, "Finding a minimum circuit in a graph, " SIAM J. Comput, vol. 7, no. 4, pp. 413-423, 1978.
48. A. Vardy, "The intractability of computing the minimum distance of a code, " IEEE Trans. Inf. Theory, vol. 43, no. 6, pp. 1757-1766, Nov. 1997. (Pubitemid 127826468)
49. E. R. Berlekamp, R. J. McEliece, and H. C. A. van Tilborg, "On the inherent intractability of certain coding problems, " IEEE Trans. Inf. Theory, vol. 24, no. 3, pp. 384-386, May 1978.
50. J. F. Queiró, "On the interlacing property for singular values and eigenvalues, " Linear Algebra Appl., vol. 97, pp. 23-28, Dec. 1987.
51. S. Foucart and M.-J. Lai, "Sparsest solutions of underdetermined linear systems via tq-minimization for 0 < q < 1, " Appl. Comput. Harmon. Anal. vol. 26, no. 3, pp. 395-407, 2009.
52. S. Friedland, "Bounds on the spectral radius of graphs with e edges, " Linear Algorithms Appl., vol. 101, pp. 81-86, Sep. 1988.
53. Q. Berthet and P. Rigollet, (2012). "Optimal detection of sparse principal components in high dimensions, " [online] Available: http://arxiv.org/abs/1202.5070
54. P. E. Gill, W. Murray, and M. H. Wright, Numerical Linear Algebra and Optimization, vol. 1. Redwood City, CA, USA: Addison-Wesley Publishing Company, 1991.
55. G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. Baltimore, MD, USA: Johns Hopkins Univ. Press, 1996.
56. A. M. Bruckstein, D. L. Donoho, and M. Elad, "From sparse solutions of systems of equations to sparse modeling of signals and images, " SIAM Rev., vol. 51, no. 1, pp. 34-81, 2009.
57. E. Amaldi and V. Kann, "On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems, " Theory Comput. Sci., vol. 209, nos. 1-2, pp. 237-260, 1998. (Pubitemid 128450440)
58. M. Cho and W. Xu, (2013). Precisely Verifying the Null Space Conditions in Compressed Sensing: A Sandwiching Algorithm [online] Available: http://arxiv.org/abs/1306.2665