Abstract
This chapter outlines several sparse reconstruction techniques analyzed throughout the book. More precisely, we present optimization methods, greedy methods, and thresholding-based methods. In each case, only intuition and basic facts about the algorithms are provided at this point.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)
E.J. Candès, T. Tao, The Dantzig selector: statistical estimation when p is much larger than n. Ann. Stat. 35(6), 2313–2351, (2007)
S. Chen, S. Billings, W. Luo, Orthogonal least squares methods and their application to nonlinear system identification. Intl. J. Contr. 50(5), 1873–1896 (1989)
S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by Basis Pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1999)
W. Dai, O. Milenkovic, Subspace Pursuit for Compressive Sensing Signal Reconstruction. IEEE Trans. Inform. Theor. 55(5), 2230–2249 (2009)
G. Davis, S. Mallat, Z. Zhang, Adaptive time-frequency decompositions. Opt. Eng. 33(7), 2183–2191 (1994)
D.L. Donoho, A. Maleki, A. Montanari, Message-passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. USA 106(45), 18914–18919 (2009)
J. Friedman, W. Stuetzle, Projection pursuit regressions. J. Am. Stat. Soc. 76, 817–823 (1981)
J. Högborn, Aperture synthesis with a non-regular distribution of interferometer baselines. Astronom. and Astrophys. 15, 417 (1974)
S. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)
D. Needell, J. Tropp, CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2008)
D. Needell, R. Vershynin, Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Found. Comput. Math. 9(3), 317–334 (2009)
D. Needell, R. Vershynin, Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit. IEEE J. Sel. Top. Signal Process. 4(2), 310–316 (April 2010)
J. Nocedal, S. Wright, Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering (Springer, New York, 2006)
Y.C. Pati, R. Rezaiifar, P.S. Krishnaprasad, Orthogonal Matching Pursuit: Recursive Function Approximation with Applications to Wavelet Decomposition. In 1993 Conference Record of The Twenty-Seventh Asilomar Conference on Signals, Systems and Computers, Nov. 1–3, 1993., pp. 40–44, 1993
S. Qian, D. Chen, Signal representation using adaptive normalized Gaussian functions. Signal Process. 36(1), 1–11 (1994)
V. Temlyakov, Nonlinear methods of approximation. Found. Comput. Math. 3(1), 33–107 (2003)
V. Temlyakov, Greedy approximation. Act. Num. 17, 235–409 (2008)
V. Temlyakov, Greedy Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 20 (Cambridge University Press, Cambridge, 2011)
R. Tibshirani, Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. B 58(1), 267–288 (1996)
J.A. Tropp, Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inform. Theor. 50(10), 2231–2242 (2004)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Foucart, S., Rauhut, H. (2013). Basic Algorithms. In: A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-4948-7_3
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4948-7_3
Published:
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-0-8176-4947-0
Online ISBN: 978-0-8176-4948-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)