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Sparse Representations Are Most Likely to Be the Sparsest Possible

Abstract

Given a signal and a full-rank matrix with, we define the signal's overcomplete representations as all satisfying. Among all the possible solutions, we have special interest in the sparsest one—the one minimizing. Previous work has established that a representation is unique if it is sparse enough, requiring. The measure stands for the minimal number of columns from that are linearly dependent. This bound is tight—examples can be constructed to show that with or more nonzero entries, uniqueness is violated. In this paper we study the behavior of overcomplete representations beyond the above bound. While tight from a worst-case standpoint, a probabilistic point of view leads to uniqueness of representations satisfying. Furthermore, we show that even beyond this point, uniqueness can still be claimed with high confidence. This new result is important for the study of the average performance of pursuit algorithms—when trying to show an equivalence between the pursuit result and the ideal solution, one must also guarantee that the ideal result is indeed the sparsest.

References

  1. Chen SS, Donoho DL, Saunders MA: Atomic decomposition by basis pursuit. SIAM Review 2001, 43(1):129–159. 10.1137/S003614450037906X

    Article  MathSciNet  Google Scholar 

  2. Natarajan BK: Sparse approximate solutions to linear systems. SIAM Journal on Computing 1995, 24(2):227–234. 10.1137/S0097539792240406

    Article  MathSciNet  Google Scholar 

  3. Davis G, Mallat SG, Avellaneda M: Adaptive greedy approximations. Constructive Approximations 1997, 13(1):57–98.

    Article  MathSciNet  Google Scholar 

  4. Mallat SG, Zhang Z: Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing 1993, 41(12):3397–3415. 10.1109/78.258082

    Article  Google Scholar 

  5. Pati YC, Rezaiifar R, Krishnaprasad PS: Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition. Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, November 1993, Pacific Grove, Calif, USA 1: 40–44.

    Article  Google Scholar 

  6. Couvreur C, Bresler Y: On the optimality of the backward greedy algorithm for the subset selection problem. SIAM Journal on Matrix Analysis and Applications 2000, 21(3):797–808. 10.1137/S0895479898332928

    Article  MathSciNet  Google Scholar 

  7. Gilbert AC, Muthukrishnan S, Strauss MJ: Approximation of functions over redundant dictionaries using coherence. Proceedings of 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '03), January 2003, Baltimore, Md, USA 243–252.

    Google Scholar 

  8. Tropp JA, Gilbert AC, Muthukrishnan S, Strauss MJ: Improved sparse approximation over quasi-incoherent dictionaries. Proceedings of IEEE International Conference on Image Processing (ICIP '03), September 2003, Barcelona, Spain 1: 37–40.

    Google Scholar 

  9. Tropp JA: Greed is good: algorithmic results for sparse approximation. IEEE Transactions on Information Theory 2004, 50(10):2231–2242. 10.1109/TIT.2004.834793

    Article  MathSciNet  Google Scholar 

  10. Donoho DL, Elad M, Temlyakov VN: Stable recovery of sparse overcomplete representations in the presence of noise. to appear in IEEE Transactions on Information Theory, February 2004

    MATH  Google Scholar 

  11. Tropp JA: Just relax: Convex programming methods for subset selection and sparse approximation. In ICES Report 04-04. University of Texas at Austin, Austin, Tex, USA; February 2004.

    Google Scholar 

  12. Temlyakov VN:Greedy algorithms and-term approximation with regard to redundant dictionaries. Journal of Approximation Theory 1999, 98(1):117–145. 10.1006/jath.1998.3265

    Article  MathSciNet  Google Scholar 

  13. Temlyakov VN: Weak greedy algorithms. Advances in Computational Mathematics 2000, 12(2–3):213–227.

    Article  MathSciNet  Google Scholar 

  14. Gorodnitsky IF, Rao BD: Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm. IEEE Transactions on Signal Processing 1997, 45(3):600–616. 10.1109/78.558475

    Article  Google Scholar 

  15. Donoho DL, Huo X: Uncertainty principles and ideal atomic decomposition. IEEE Transactions on Information Theory 2001, 47(7):2845–2862. 10.1109/18.959265

    Article  MathSciNet  Google Scholar 

  16. Huo X: Sparse image representation via combined transforms, M.S. thesis. Stanford University, Stanford, Calif, USA; 1999.

    Google Scholar 

  17. Elad M, Bruckstein AM: A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Transactions on Information Theory 2002, 48(9):2558–2567. 10.1109/TIT.2002.801410

    Article  MathSciNet  Google Scholar 

  18. Donoho DL, Elad M:Optimally sparse representation in general (non-orthogonal) dictionaries via minimization. Proceedings of the National Academy of Sciences of the United States of America 2003, 100(5):2197–2202. 10.1073/pnas.0437847100

    Article  MathSciNet  Google Scholar 

  19. Gribonval R, Nielsen M: Sparse representations in unions of bases. IEEE Transactions on Information Theory 2003, 49(12):3320–3325. 10.1109/TIT.2003.820031

    Article  MathSciNet  Google Scholar 

  20. Fuchs J-J: On sparse representations in arbitrary redundant bases. IEEE Transactions on Information Theory 2004, 50(6):1341–1344. 10.1109/TIT.2004.828141

    Article  MathSciNet  Google Scholar 

  21. Candès EJ, Romberg J: Quantitative robust uncertainty principles and optimally sparse decompositions. to appear in Foundations of Computational Mathematics, 2004

    MATH  Google Scholar 

  22. Donoho DL:For most large underdetermined systems of linear equations the minimal-norm solution is also the sparsest solution. In Tech. Rep. 2004-10. Department of Statistics, Stanford University, Stanford, Calif, USA; September 2004.

    Google Scholar 

  23. Edelman A: Eigenvalues and condition numbers of random matrices. SIAM Journal on Matrix Analysis and Applications 1988, 9(4):543–560. 10.1137/0609045

    Article  MathSciNet  Google Scholar 

  24. Shen J: On the singular values of Gaussian random matrices. Linear Algebra and its Applications 2001, 326(1–3):1–14.

    Article  MathSciNet  Google Scholar 

  25. Strohmer T, Heath RW Jr.: Grassmannian frames with applications to coding and communication. Applied and Computational Harmonic Analysis 2003, 14(3):257–275. 10.1016/S1063-5203(03)00023-X

    Article  MathSciNet  Google Scholar 

  26. Tropp JA, Dhillon IS, Heath RW Jr., Strohmer T: Designing structured tight frames via an alternating projection method. IEEE Transactions on Information Theory 2005, 51(1):188–209.

    Article  MathSciNet  Google Scholar 

  27. Candès EJ, Donoho DL:New tight frames of curvelets and the problem of approximating piecewise images with piecewise edges. 2002.

    Google Scholar 

  28. Starck J-L, Candès EJ, Donoho DL: The curvelet transform for image denoising. IEEE Transactions on Image Processing 2002, 11(6):670–684. 10.1109/TIP.2002.1014998

    Article  MathSciNet  Google Scholar 

  29. Björner A: Some matroid inequalities. Discrete Mathematics 1980, 31(1):101–103. 10.1016/0012-365X(80)90179-X

    Article  MathSciNet  Google Scholar 

  30. Goldberg F: Overcomplete Bases, Spark, and Matroids. 2004.

    Google Scholar 

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Elad, M. Sparse Representations Are Most Likely to Be the Sparsest Possible. EURASIP J. Adv. Signal Process. 2006, 096247 (2006). https://doi.org/10.1155/ASP/2006/96247

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