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Uniform uncertainty principle and signal recovery via Regularized orthogonal matching pursuit. (English) Zbl 1183.68739
Summary: This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements—${\text{L}}_{1}$-minimization methods and iterative methods (Matching Pursuits). We find a simple Regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of ${\text{L}}_{1}$-minimization. Our algorithm, ROMP, reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the uniform uncertainty principle.
##### MSC:
 68W20 Randomized algorithms 65T50 Discrete and fast Fourier transforms (numerical methods) 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
##### Keywords:
sparse signal recovery
##### References:
 [1] E. Candès, Compressive sampling, in Proceedings of International Congress of Mathematics, vol. 3, Madrid, Spain, 2006, pp. 1433–1452. [2] E. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory 52, 489–509 (2006). · Zbl 1231.94017 · doi:10.1109/TIT.2005.862083 [3] E. Candès, T. Tao, Near-optimal signal recovery from random projections: universal encoding strategies, IEEE Trans. Inf. Theory 52, 5406–5425 (2004). · Zbl 05455295 · doi:10.1109/TIT.2006.885507 [4] E.J. Candès, T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory 51, 4203–4215 (2005). · Zbl 1264.94121 · doi:10.1109/TIT.2005.858979 [5] A. Cohen, W. Dahmen, R. DeVore, Compressed sensing and k-term approximation, Manuscript (2007). [6] Compressed sensing, webpage, http://www.dsp.ece.rice.edu/cs/ . [7] D. Donoho, Compressed sensing, IEEE Trans. Inf. Theory 52, 1289–1306 (2006). · Zbl 05454299 · doi:10.1109/TIT.2006.871582 [8] D. Donoho, M. Elad, V. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise, IEEE Trans. Inf. Theory 52, 6–18 (2006). · Zbl 05454300 · doi:10.1109/TIT.2005.860430 [9] D. Donoho, M. Elad, V. Temlyakov, On the Lebesgue type inequalities for greedy approximation, J. Approx. Theory 147, 185–195 (2007). · Zbl 1120.41007 · doi:10.1016/j.jat.2007.01.004 [10] D. Donoho, P. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49, 906–931 (1989). · Zbl 0689.42001 · doi:10.1137/0149053 [11] A. Gilbert, S. Muthukrishnan, M. Strauss, Approximation of functions over redundant dictionaries using coherence, in The 14th Annual ACM–SIAM Symposium on Discrete Algorithms (2003). [12] A. Gilbert, M. Strauss, J. Tropp, R. Vershynin, Algorithmic linear dimension reduction in the L1 norm for sparse vectors, submitted. Conference version, in Algorithmic Linear Dimension Reduction in the L 1 Norm for Sparse Vectors, Allerton, 2006. 44th Annual Allerton Conference on Communication, Control, and Computing. [13] A. Gilbert, M. Strauss, J. Tropp, R. Vershynin, One sketch for all: fast algorithms for compressed sensing, in STOC 2007. 39th ACM Symposium on Theory of Computing, San Diego, 2007, to appear. [14] Y. Lyubarskii, R. Vershynin, Uncertainty principles and vector quantization, submitted. [15] S. Mendelson, A. Pajor, N. Tomczak-Jaegermann, Uniform uncertainty principle for Bernoulli and subgaussian ensembles, Constr. Approx., submitted. [16] H. Rauhut, On the impossibility of uniform recovery using greedy methods, in Sample Theory Signal Image Process., to appear. [17] M. Rudelson, R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements, Commun. Pure Appl. Math., to appear. Conference version in CISS 2006. 40th Annual Conference on Information Sciences and Systems, Princeton. [18] D. Spielman, S.-H. Teng, Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time, J. ACM 51, 385–463 (2004). · Zbl 1192.90120 · doi:10.1145/990308.990310 [19] V. Temlyakov, Nonlinear methods of approximation, Found. Comput. Math. 3, 33–107 (2003). · Zbl 1039.41012 · doi:10.1007/s102080010029 [20] J.A. Tropp, A.C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007). · Zbl 05455712 · doi:10.1109/TIT.2007.909108 [21] R. Vershynin, Beyond Hirsch Conjecture: walks on random polytopes and smoothed complexity of the simplex method, submitted. Conference version in FOCS 2006. 47th Annual Symposium on Foundations of Computer Science, Berkeley, pp. 133–142.