Abstract
We propose a novel sparse signal reconstruction method aiming to directly minimize \(\ell _{0}\)-quasinorm. Based on the smoothed \(\ell _{0}\)-quasinorm, we show that there exists an unconstrained optimization problem such that both this problem and the basis pursuit problem are subproblems of the \(\ell _{0}\) minimization problem. Moreover, we can obtain a sparse solution to the \(\ell _{0}\) minimization by solving these two subproblems. In addition, we establish the relation between solutions to the \(\ell _{0}\) minimization and the least square solutions of a linear system. Finally, we present some numerical experiments to illustrate our results.
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Md.Z. Ali Bhotto, M.O. Ahmad, M.N.S. Swamy, An improved fast iterative shrinkage thresholding algorithm for image deblurring. SIAM J. Imaging Sci. 8(3), 1640–1657 (2015)
A.S. Bandeira, D.G. Mixon, Certifying the restricted isometry property is hard. IEEE Trans. Inf. Theory 59(6), 3448–3450 (2013). doi:10.1109/TIT.2013.2248414
P. Bloomfield, S. William, Least Absolute Deviations: Theory, Applications and Algorithms (Springer, New York, 2012)
T. Blumensath, M.E. Davies, Iterative thresholding for sparse approximations. J. Fourier Anal. Appl. 14(5–6), 629–654 (2008). doi:10.1007/s00041-008-9035-z
T. Blumensath, M.E. Davies, Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal. 27(3), 265–274 (2009)
E.J. Candès, M.B. Wakin, An introduction to compressive sampling. IEEE Signal Proc. Mag. 25(2), 21–30 (2008). doi:10.1109/MSP.2007.914731
E.J. Candès, K.R. Justin, Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 6(2), 227–254 (2006). doi:10.1007/s10208-004-0162-x
E.J. Candès, K.J. Justin, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006). doi:10.1109/TIT.2005.862083
E.J. Candès, K.R. Justin, T. Tao, Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006). doi:10.1002/cpa.20124
E.J. Candès, T. Tao, Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005). doi:10.1109/TIT.2005.858979
E.J. Candès, T. Tao, Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006). doi:10.1109/TIT.2006.885507
S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998)
D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). doi:10.1109/TIT.2006.871582
D.L. Donoho, De-noising by soft-thresholding. IEEE Trans. Inf. Theory 41(3), 613–627 (1995). doi:10.1109/18.382009
D.L. Donoho, M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via \(l_{1}\) minimization. Proc. Natl. Acad. Sci. 100(5), 2197–2202 (2003). doi:10.1073/pnas.0437847100
D.L. Donoho, X.M. Huo, Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory 47(7), 2845–2862 (2001)
M.F. Duarte, R.G. Baraniuk, Spectral compressive sensing. Appl. Comput. Harmon. Anal. 35(1), 111–129 (2013)
S. Foucart, H. Rauhut, A Mathematical Introduction to Compressive Sensing (Birkhäuser, Basel, 2013)
I.F. Gorodnitsky, D.R. Bhaskar, Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans. Signal Process. 45(3), 600–616 (1997). doi:10.1109/78.558475
X.L. Huang, Y.P. Liu, L. Shi, Two-level \(\ell _{1}\) minimization for compressed sensing. Signal Process. 108, 459–475 (2015)
S. Jalali, M. Arian, From compression to compressed sensing. Appl. Comput. Harmon. Anal. 40(2), 352–385 (2016)
F. Kittaneh, Singular values of companion matrices and bounds on zeros of polynomials. SIAM J. Matrix Anal. Appl. 16(1), 333–340 (1995)
S.G. Mallat, Z.F. Zhang, Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993). doi:10.1109/78.258082
H. Mansour, S. Rayan, Recovery analysis for weighted \(\ell _{1}\)-minimization using the null space property. Appl. Comput. Harmon. Anal. 43(1), 23–38 (2017)
H. Mohimani, B.Z. Massoud, C. Jutten, A fast approach for overcomplete sparse decomposition based on smoothed \(\ell _{0}\) norm. IEEE Trans. Signal Process. 57(1), 289–301 (2009). doi:10.1109/TSP.2008.2007606
B.K. Natarajan, Sparse approximate solutions to linear systems. SIAM J. Comput. 24(2), 227–234 (1995)
D. Needell, J.A. Tropp, CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2009)
S.Z. Tao, B. Daniel, S.Z. Zhang, Local linear convergence of ISTA and FISTA on the LASSO problem. SIAM J. Optim. 26(1), 313–336 (2016)
R. Tibshirani, Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.) 58(1), 267–288 (1996)
J.A. Tropp, A.C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007). doi:10.1109/TIT.2007.909108
Y. Tsaig, D.L. Donoho, Extensions of compressed sensing. Signal Process. 86(3), 549–571 (2006)
B. Wang, L.L. Hu, J.Y. An et al., Recovery error analysis of noisy measurement in compressed sensing. Circ. Syst. Signal Process. 36(1), 137–155 (2017). doi:10.1007/s00034-016-0296-5
Y. Wang, X.Q. Xiang et al., Compressed sensing based on trust region method. Circ. Syst. Signal Process. 36(1), 202–218 (2017). doi:10.1007/s00034-016-0299-2
D. Wei, Y. Li, Generalized sampling expansions with multiple sampling rates for lowpass and bandpass signals in the fractional fourier transform domain. IEEE Trans. Signal Process. 64(18), 4861–4874 (2016). doi:10.1109/TSP.2016.2560148
D.L. Wu, W.P. Zhu, N.S. Swamy, The theory of compressive sensing matching pursuit considering time-domain noise with application to speech enhancement. IEEE/ACM Trans. Audio Speech Lang. Process. 22(3), 682–696 (2014). doi:10.1109/TASLP.2014.2300336
Acknowledgements
We thank the anonymous referees, associate editors, and editor-in-chief for their thoughtful and insightful comments, which improved the paper greatly. The research was supported partially by National Natural Science Foundation of China (Grant Nos. 10871056 and 10971150).
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Wang, J., Wang, X.T. Sparse Approximate Reconstruction Decomposed by Two Optimization Problems. Circuits Syst Signal Process 37, 2164–2178 (2018). https://doi.org/10.1007/s00034-017-0667-6
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DOI: https://doi.org/10.1007/s00034-017-0667-6