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2013 | OriginalPaper | Chapter

3. Basic Algorithms

Authors : Simon Foucart, Holger Rauhut

Published in: A Mathematical Introduction to Compressive Sensing

Publisher: Springer New York

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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.

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Literature
70.
go back to reference S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)MATHCrossRef S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)MATHCrossRef
98.
go back to reference E.J. Candès, T. Tao, The Dantzig selector: statistical estimation when p is much larger than n. Ann. Stat. 35(6), 2313–2351, (2007) E.J. Candès, T. Tao, The Dantzig selector: statistical estimation when p is much larger than n. Ann. Stat. 35(6), 2313–2351, (2007)
113.
go back to reference 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)MathSciNetMATHCrossRef 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)MathSciNetMATHCrossRef
114.
135.
go back to reference W. Dai, O. Milenkovic, Subspace Pursuit for Compressive Sensing Signal Reconstruction. IEEE Trans. Inform. Theor. 55(5), 2230–2249 (2009)MathSciNetCrossRef W. Dai, O. Milenkovic, Subspace Pursuit for Compressive Sensing Signal Reconstruction. IEEE Trans. Inform. Theor. 55(5), 2230–2249 (2009)MathSciNetCrossRef
145.
go back to reference G. Davis, S. Mallat, Z. Zhang, Adaptive time-frequency decompositions. Opt. Eng. 33(7), 2183–2191 (1994)CrossRef G. Davis, S. Mallat, Z. Zhang, Adaptive time-frequency decompositions. Opt. Eng. 33(7), 2183–2191 (1994)CrossRef
162.
go back to reference D.L. Donoho, A. Maleki, A. Montanari, Message-passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. USA 106(45), 18914–18919 (2009)CrossRef D.L. Donoho, A. Maleki, A. Montanari, Message-passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. USA 106(45), 18914–18919 (2009)CrossRef
277.
go back to reference J. Högborn, Aperture synthesis with a non-regular distribution of interferometer baselines. Astronom. and Astrophys. 15, 417 (1974) J. Högborn, Aperture synthesis with a non-regular distribution of interferometer baselines. Astronom. and Astrophys. 15, 417 (1974)
342.
go back to reference S. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)MATHCrossRef S. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)MATHCrossRef
361.
go back to reference D. Needell, J. Tropp, CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2008)MathSciNetCrossRef D. Needell, J. Tropp, CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2008)MathSciNetCrossRef
362.
go back to reference D. Needell, R. Vershynin, Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Found. Comput. Math. 9(3), 317–334 (2009)MathSciNetMATHCrossRef D. Needell, R. Vershynin, Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Found. Comput. Math. 9(3), 317–334 (2009)MathSciNetMATHCrossRef
363.
go back to reference 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)CrossRef 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)CrossRef
369.
go back to reference J. Nocedal, S. Wright, Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering (Springer, New York, 2006) J. Nocedal, S. Wright, Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering (Springer, New York, 2006)
378.
go back to reference 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 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
404.
go back to reference S. Qian, D. Chen, Signal representation using adaptive normalized Gaussian functions. Signal Process. 36(1), 1–11 (1994)MATHCrossRef S. Qian, D. Chen, Signal representation using adaptive normalized Gaussian functions. Signal Process. 36(1), 1–11 (1994)MATHCrossRef
472.
go back to reference V. Temlyakov, Greedy Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 20 (Cambridge University Press, Cambridge, 2011) V. Temlyakov, Greedy Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 20 (Cambridge University Press, Cambridge, 2011)
473.
go back to reference R. Tibshirani, Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. B 58(1), 267–288 (1996)MathSciNetMATH R. Tibshirani, Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. B 58(1), 267–288 (1996)MathSciNetMATH
476.
go back to reference J.A. Tropp, Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inform. Theor. 50(10), 2231–2242 (2004)MathSciNetCrossRef J.A. Tropp, Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inform. Theor. 50(10), 2231–2242 (2004)MathSciNetCrossRef
Metadata
Title
Basic Algorithms
Authors
Simon Foucart
Holger Rauhut
Copyright Year
2013
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-0-8176-4948-7_3

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