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doi: 10.3934/jimo.2019035

Sparse signal reconstruction via the approximations of $ \ell_{0} $ quasinorm

Jun Wang and  Xing Tao Wang ,

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China

* Corresponding author: Xing Tao Wang

Received  June 2018 Revised  November 2018 Published  May 2019

In this paper, we propose two classes of the approximations to the cardinality function via the Moreau envelope of the $ \ell_{1} $ norm. We show that these two approximations are good choices of the merit function for sparsity and are essentially the truncated $ \ell_{1} $ norm and the truncated $ \ell_{2} $ norm. Moreover, we apply the approximations to solve sparse signal recovery problems and then provide new weights for reweighted $ \ell_{1} $ minimization and reweighted least squares to find sparse solutions of underdetermined linear systems of equations. Finally, we present some numerical experiments to illustrate our results.

Keywords: Reweighted $ \ell_{1} $-minimization, sparse approximation, merit function, sparse signal recovery.
Mathematics Subject Classification: Primary: 90C30, 65K05; Secondary: 49M37, 49M25.
Citation: Jun Wang, Xing Tao Wang. Sparse signal reconstruction via the approximations of $ \ell_{0} $ quasinorm. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019035
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Md. Z. A. Bhot, M. O. Ahmad and M. N. S. Swamy, An improved fast iterative shrinkage thresholding algorithm for image deblurring, SIAM J. Imaging Sci., 8 (2015), 1640-1657. doi: 10.1137/140970537.

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E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083.

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E. J. Candès, M. Wakin and S. Boyd, Enhancing sparsity by reweighted $\ell_{1}$ minimization, J. Fourier Anal. Appl., 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x.

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R. Chartrand and W. Yin, Iteratively reweighted algorithms for compressive sensing, in: Proc. Int. Conf. Accoust. Speech, Signal Process., 2008, 3869–3872. doi: 10.1109/ICASSP.2008.4518498.

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[13]

A. M. Christopher, M. Arian and G. B. Richard, From denoising to compressed sensing, IEEE Trans. Inf. Theory, 62 (2016), 5117-5144. doi: 10.1109/TIT.2016.2556683.

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[27]

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[28]

Z. S. Lu, Iterative reweighted minimization methods for $\ell_{p}$ regularized unconstrained nonlinear programming, Mathematical Program., 147 (2014), 277-307. doi: 10.1007/s10107-013-0722-4.

[29]

Z. S. Lu and X. Li, Sparse recovery via partial regularization: Models, theory, and algorithms, Math. Oper. Res., 43 (2018), 1290-1316. doi: 10.1287/moor.2017.0905.

[30]

J. Lv and Y. Fan, A unified approach to model selection and sparse recovery using regularized least squares, Ann. Statist., 37 (2009), 3498-3528. doi: 10.1214/09-AOS683.

[31]

H. Mansour and R. Saab, Recovery analysis for weighted $\ell_{1}$-minimization using the null space property, Appl. Comput. Harmon. Anal., 43 (2017), 23-38. doi: 10.1016/j.acha.2015.10.005.

[32]

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[33]

H. Mohimani, B. Z. Massoud and C. Jutten, A fast approach for overcomplete sparse decomposition based on smoothed $ \ell^{0} $ norm, IEEE Trans. Signal Process., 57 (2009), 289-301. doi: 10.1109/TSP.2008.2007606.

[34]

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[35]

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[36]

H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4$^{nd}$ edition, Macmillan Publishing Company, New York, 2010.
[37]

I. Selesnick, Sparse regularization via convex analysis, IEEE Trans. Signal Process., 65 (2017), 4481-4494. doi: 10.1109/TSP.2017.2711501.

[38]

I. Selesnick and M. Farshchian, Sparse signal approximation via nonseparable regularization, IEEE Trans. Signal Process., 65 (2017), 2561-2575. doi: 10.1109/TSP.2017.2669904.

[39]

G. W. Stewart, On scaled projections and pseudoinverses, Linear Algebra Appl., 112 (1989), 189-193. doi: 10.1016/0024-3795(89)90594-6.

[40]

B. Tian, X. Yang and K. Meng, An interior-point $\ell_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization, J. Ind. Manag. Optim., 12 (2016), 949-973. doi: 10.3934/jimo.2016.12.949.

[41]

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[42]

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[43]

J. Wang and X. T. Wang, Sparse approximate reconstruction decomposed by two optimization problems, Circ. Syst. Signal Process., 37 (2018), 2164-2178. doi: 10.1007/s00034-017-0667-6.

[44]

D. Wipf and S. Nagarajan, Iterative reweighted $\ell_1 $ and $\ell_2 $ methods for finding sparse solutions, IEEE J. Select. Topics Signal Process., 4 (2010), 317-329. doi: 10.1109/JSTSP.2010.2042413.

[45]

J. Wright, A. Y. Yang and G. Arvind et al., Robust face recognition via sparse representation, IEEE Trans. Pattern Analysis and Machine Intelligence, 31 (2008), 210-227. doi: 10.1109/AFGR.2008.4813404.

[46]

P. Yin, Y. Lou, Q. He and J. Xin, Minimization of $\ell_{1-2}$ for compressed sensing, SIAM J. Sci. Comput., 37 (2015), A536–A563. doi: 10.1137/140952363.

[47]

C.-H. Zhang, Nearly unbiased variable selection under minimax concave penalty, Statist. Sci., 2 (2010), 894-942. doi: 10.1214/09-AOS729.

[48]

C.-H. Zhang and T. Zhang, A general theory of concave regularization for high-dimensional sparse estimation problems, Statist. Sci., 27 (2012), 576-593. doi: 10.1214/12-STS399.

[49]

T. Zhang, Analysis of multi-stage convex relaxation for sparse regularization, J. Mach. Learn. Res., 11 (2010), 1081-1107.

[50]

Y.-B. Zhao, RSP-based analysis for sparsest and least $\ell_{1}$-norm solutions to underdetermined linear systems, IEEE Trans. Signal Process., 61 (2013), 5777-5788. doi: 10.1109/TSP.2013.2281030.

[51]

Y.-B. Zhao and M. Kocvara, A new computational method for the sparsest solutions to systems of linear equations, SIAM J. Optim., 25 (2015), 1110-1134. doi: 10.1137/140968240.

[52]

Y. Zhao and D. Li, Reweighted $\ell_{1}$-minimization for sparse solutions to underdetermined linear systems, SIAM J. Optim., 22 (2012), 1065-1088. doi: 10.1137/110847445.

show all references

References:
[1]

Q. Berthet and P. Rigollet, Optimal detection of sparse principal components in high dimension, Annal. Statistics, 41 (2013), 1780-1815. doi: 10.1214/13-AOS1127.

[2]

Md. Z. A. Bhot, M. O. Ahmad and M. N. S. Swamy, An improved fast iterative shrinkage thresholding algorithm for image deblurring, SIAM J. Imaging Sci., 8 (2015), 1640-1657. doi: 10.1137/140970537.

[3]

A. M. Bruckstein, D. L. Donoho and M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Review, 51 (2009), 34-81. doi: 10.1137/060657704.

[4]

E.J. Candès and B. Recht, Exact matrix completion via convex optimization, Found. Comput. Math., 9 (2009), 717-772. doi: 10.1007/s10208-009-9045-5.

[5]

E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083.

[6]

E. J. Candés, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., 59 (2006), 1207-1223. doi: 10.1002/cpa.20124.

[7]

E. J. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979.

[8]

E. J. Candès and T. Tao, Near-optimal signal recovery from random projections: Universal encoding strategies?, IEEE Trans. Inf. Theory, 52 (2006), 5406-5425. doi: 10.1109/TIT.2006.885507.

[9]

E. J. Candès, M. Wakin and S. Boyd, Enhancing sparsity by reweighted $\ell_{1}$ minimization, J. Fourier Anal. Appl., 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x.

[10]

R. Chartrand, Exact reconstruction of sparse signals via nonconvex minimization, IEEE Signal Process. Lett., 14 (2007), 707-710. doi: 10.1109/LSP.2007.898300.

[11]

R. Chartrand and W. Yin, Iteratively reweighted algorithms for compressive sensing, in: Proc. Int. Conf. Accoust. Speech, Signal Process., 2008, 3869–3872. doi: 10.1109/ICASSP.2008.4518498.

[12]

S. S. Chen, D. L. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 43 (2001), 129-159. doi: 10.1137/S003614450037906X.

[13]

A. M. Christopher, M. Arian and G. B. Richard, From denoising to compressed sensing, IEEE Trans. Inf. Theory, 62 (2016), 5117-5144. doi: 10.1109/TIT.2016.2556683.

[14]

A. Cohen, W. Dahmen and R. DeVore, Compressed sensing and best k-term approximation, J. American Math. Soc., 22 (2009), 211-231. doi: 10.1090/S0894-0347-08-00610-3.

[15]

I. Daubechies, M. Defrise and M. C. De, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pur. Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042.

[16]

D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.

[17]

E. Esser, Y. F. Lou and J. Xin, A method for finding structured sparse solutions to nonnegative least squares problems with applications, SIAM J. maging Sci., 6 (2013), 2010-2046. doi: 10.1137/13090540X.

[18]

J. Fan and R. Li, Variable selection via nonconcave penalized likelihood and its oracle properties, J. Amer. Statist. Assoc., 96 (2001), 1348-1360. doi: 10.1198/016214501753382273.

[19]

M. Fornasier, S. Peter, H. Rauhut and S. Worm, Conjugate gradient acceleration of iteratively re-weighted least squares methods, Comput. Optim. Appl., 65 (2016), 205-259. doi: 10.1007/s10589-016-9839-8.

[20]

S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Birkhaäuser, Basel, 2013. doi: 10.1007/978-0-8176-4948-7.

[21]

S. Foucart and M. Lai, Sparsest solutions of underdetermined linear systems via $\ell_{q}$-minimization for $0 <q\leq1$, ppl. Comput. Harmon. Anal., 26 (2009), 395-407. doi: 10.1016/j.acha.2008.09.001.

[22]

I. F. Gorodnitsky and B. D. Rao, Sparse signal reconstruction from limited data using FOCUSS, A re-weighted minimum norm algorithm, IEEE Trans. Signal Process., 45 (1997), 600-616. doi: 10.1109/78.558475.

[23]

J. Huang, Y. Jiao, B. Jin, J. Liu, X. Lu and C. Yang., A unified primal dual active set algorithm for nonconvex sparse recovery, arXiv: 1310.1147.
[24]

J. Huang, Y. Jiao, Y. Liu and X. Lu, A constructive approach to $L_{0}$-penalized regression, J. Mach. Learn. Res., 19 (2018), Paper No. 10, 37 pp.

[25]

Y. Jiao, B. Jin and X. Lu, A primal dual active set with continuation algorithm for the $\ell^{0}$-regularized optimization problem, Appl. Comput. Harmon. Anal., 39 (2015), 400-426. doi: 10.1016/j.acha.2014.10.001.

[26]

R. Kueng, H. Rauhut and U. Terstiege., Low rank matrix recovery from rank one measurements, Appl. Comput. Harmon. Anal., 42 (2017), 88-116. doi: 10.1016/j.acha.2015.07.007.

[27]

M. Lai and J. Wang, An unconstrained $\ell_{q}$ minimization with $0 < q\leq 1$ for sparse solution of underdetermined linear systems, SIAM J. Optim., 21 (2011), 82-101. doi: 10.1137/090775397.

[28]

Z. S. Lu, Iterative reweighted minimization methods for $\ell_{p}$ regularized unconstrained nonlinear programming, Mathematical Program., 147 (2014), 277-307. doi: 10.1007/s10107-013-0722-4.

[29]

Z. S. Lu and X. Li, Sparse recovery via partial regularization: Models, theory, and algorithms, Math. Oper. Res., 43 (2018), 1290-1316. doi: 10.1287/moor.2017.0905.

[30]

J. Lv and Y. Fan, A unified approach to model selection and sparse recovery using regularized least squares, Ann. Statist., 37 (2009), 3498-3528. doi: 10.1214/09-AOS683.

[31]

H. Mansour and R. Saab, Recovery analysis for weighted $\ell_{1}$-minimization using the null space property, Appl. Comput. Harmon. Anal., 43 (2017), 23-38. doi: 10.1016/j.acha.2015.10.005.

[32]

F. D. Marco, A. D. Mark and T. Dharmpal et al., Single-pixel imaging via compressive sampling, IEEE Trans. Signal Magazine, 25 (2008), 83-91. doi: 10.1109/MSP.2007.914730.

[33]

H. Mohimani, B. Z. Massoud and C. Jutten, A fast approach for overcomplete sparse decomposition based on smoothed $ \ell^{0} $ norm, IEEE Trans. Signal Process., 57 (2009), 289-301. doi: 10.1109/TSP.2008.2007606.

[34]

B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM J. Comput., 24 (1995), 227-234. doi: 10.1137/S0097539792240406.

[35]

D. Needell and J. A. Tropp, CoSaMP: Iterative signal recovery from incomplete and inaccurate samples, Appl. Comput. Harmon. Anal., 26 (2009), 301-321. doi: 10.1016/j.acha.2008.07.002.

[36]

H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4$^{nd}$ edition, Macmillan Publishing Company, New York, 2010.
[37]

I. Selesnick, Sparse regularization via convex analysis, IEEE Trans. Signal Process., 65 (2017), 4481-4494. doi: 10.1109/TSP.2017.2711501.

[38]

I. Selesnick and M. Farshchian, Sparse signal approximation via nonseparable regularization, IEEE Trans. Signal Process., 65 (2017), 2561-2575. doi: 10.1109/TSP.2017.2669904.

[39]

G. W. Stewart, On scaled projections and pseudoinverses, Linear Algebra Appl., 112 (1989), 189-193. doi: 10.1016/0024-3795(89)90594-6.

[40]

B. Tian, X. Yang and K. Meng, An interior-point $\ell_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization, J. Ind. Manag. Optim., 12 (2016), 949-973. doi: 10.3934/jimo.2016.12.949.

[41]

R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288. doi: 10.1111/j.2517-6161.1996.tb02080.x.

[42]

J. A. Tropp and A. C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inf. Theory, 53 (2007), 4655-4666. doi: 10.1109/TIT.2007.909108.

[43]

J. Wang and X. T. Wang, Sparse approximate reconstruction decomposed by two optimization problems, Circ. Syst. Signal Process., 37 (2018), 2164-2178. doi: 10.1007/s00034-017-0667-6.

[44]

D. Wipf and S. Nagarajan, Iterative reweighted $\ell_1 $ and $\ell_2 $ methods for finding sparse solutions, IEEE J. Select. Topics Signal Process., 4 (2010), 317-329. doi: 10.1109/JSTSP.2010.2042413.

[45]

J. Wright, A. Y. Yang and G. Arvind et al., Robust face recognition via sparse representation, IEEE Trans. Pattern Analysis and Machine Intelligence, 31 (2008), 210-227. doi: 10.1109/AFGR.2008.4813404.

[46]

P. Yin, Y. Lou, Q. He and J. Xin, Minimization of $\ell_{1-2}$ for compressed sensing, SIAM J. Sci. Comput., 37 (2015), A536–A563. doi: 10.1137/140952363.

[47]

C.-H. Zhang, Nearly unbiased variable selection under minimax concave penalty, Statist. Sci., 2 (2010), 894-942. doi: 10.1214/09-AOS729.

[48]

C.-H. Zhang and T. Zhang, A general theory of concave regularization for high-dimensional sparse estimation problems, Statist. Sci., 27 (2012), 576-593. doi: 10.1214/12-STS399.

[49]

T. Zhang, Analysis of multi-stage convex relaxation for sparse regularization, J. Mach. Learn. Res., 11 (2010), 1081-1107.

[50]

Y.-B. Zhao, RSP-based analysis for sparsest and least $\ell_{1}$-norm solutions to underdetermined linear systems, IEEE Trans. Signal Process., 61 (2013), 5777-5788. doi: 10.1109/TSP.2013.2281030.

[51]

Y.-B. Zhao and M. Kocvara, A new computational method for the sparsest solutions to systems of linear equations, SIAM J. Optim., 25 (2015), 1110-1134. doi: 10.1137/140968240.

[52]

Y. Zhao and D. Li, Reweighted $\ell_{1}$-minimization for sparse solutions to underdetermined linear systems, SIAM J. Optim., 22 (2012), 1065-1088. doi: 10.1137/110847445.

Figure 1.  Comparison of success rates of finding the $ s $-sparse solution $ \mathbf{y} $ of $ \mathbf{b = Ax} $, where $ \mathbf{A} \in \mathbb{R}^{50\times 250} $ and $ \mathbf{y} \in \mathbb{R}^{250} $. For each $ s $-sparsity, $ 500 $ attempts were made
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Figure 2.  Comparison of $ \Vert \mathbf{y}^{k}\Vert_{1} $ and weighted $ \Vert \mathbf{y}^{k}\Vert_{1} $ at each iteration $ k $ with $ \mathbf{A} \in \mathbb{R}^{50\times 250} $, $ \mathbf{y} \in \mathbb{R}^{250} $ and $ p = 0.618, s = 10 $
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Figure 3.  Comparison of success rates of finding the $ s $-sparse solution $ \mathbf{y} $ of $ \mathbf{b = Ay} $, where $ \mathbf{A} \in \mathbb{R}^{50\times 250} $ and $ \mathbf{y} \in \mathbb{R}^{250} $. For each $ s $-sparsity, $ 500 $ attempts were made
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Figure 4.  Comparison of $ \Vert \mathbf{y}^{k}\Vert_{2}^{2} $ and weighted $ \Vert \mathbf{y}^{k}\Vert_{2}^{2} $ at each iteration $ k $ with $ \mathbf{A} \in \mathbb{R}^{50\times 250} $, $ \mathbf{y} \in \mathbb{R}^{250} $ and $ p = 0.618, s = 10 $
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