doi: 10.3934/ipi.2022035
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Morozov's discrepancy principle for $ \alpha\ell_1-\beta\ell_2 $ sparsity regularization

1. 

Department of Mathematics, Northeast Forestry University, Harbin 150040, China

2. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA

* Corresponding author: Liang Ding

Received  January 2022 Revised  April 2022 Early access June 2022

In this paper, Morozov's discrepancy principle is considered for the non-convex $ \alpha\ell_1-\beta\ell_2 $ sparsity regularization ($ \alpha>\beta>0 $). It is shown that if $ \tau>1 $ satisfies some conditions, there exists a regularization parameter $ \alpha $ such that $ \delta\leq \|A(x_{\alpha,\beta}^{\delta})-y^{\delta}\|_Y\leq \tau \delta $ holds. Furthermore, it is shown that $ \alpha $ converges to 0 as $ \delta\rightarrow 0 $. In addition, well-posedness and convergence rate results are presented for the regularized solution under Morozov's discrepancy principle. Numerical simulation results are reported to illustrate the efficiency of the proposed approach.

Citation: Liang Ding, Weimin Han. Morozov's discrepancy principle for $ \alpha\ell_1-\beta\ell_2 $ sparsity regularization. Inverse Problems and Imaging, doi: 10.3934/ipi.2022035
References:
[1]

V. Albani and A. De Cezaro, A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles, Inverse Problems and Imaging, 13 (2019), 211-229.  doi: 10.3934/ipi.2019012.

[2]

S. W. Anzengruber and R. Ramlau, Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2010), 025001, 17 pp. doi: 10.1088/0266-5611/26/2/025001.

[3]

S. W. Anzengruber and R. Ramlau, Convergence rates for Morozov's discrepancy principle using variational inequalities, Inverse Problems, 27 (2011), 105007, 18 pp. doi: 10.1088/0266-5611/27/10/105007.

[4]

A. Beck and Y. C. Eldar, Sparsity constrained nonlinear optimization: Optimality conditions and algorithms, SIAM Journal on Optimization, 23 (2013), 1480-1509.  doi: 10.1137/120869778.

[5]

M. Benning and M. Burger, Modern regularization methods for inverse problems, Acta Numerica, 27 (2018), 1-111.  doi: 10.1017/S0962492918000016.

[6]

T. Blumensath, Compressed sensing with nonlinear observations and related nonlinear optimization problems, IEEE Transactions on Information Theory, 59 (2013), 3466-3474.  doi: 10.1109/TIT.2013.2245716.

[7]

T. Blumensath and M. E. Davies, Iterative thresholding for sparse approximations, J. Fourier Anal. Appl., 14 (2008), 629-654.  doi: 10.1007/s00041-008-9035-z.

[8]

T. Blumensath and M. E. Davies, Iterative hard thresholding for compressed sensing, Appl. Comput. Harmon. Anal., 27 (2009), 265-274.  doi: 10.1016/j.acha.2009.04.002.

[9]

T. Bonesky, Morozov's discrepancy principle and Tikhonov-type functionals, Inverse Problems, 25 (2009), 015015, 11 pp. doi: 10.1088/0266-5611/25/1/015015.

[10]

K. Bredies and D. A. Lorenz, Iterated hard shrinkage for minimization problems with sparsity constraints, SIAM J. Sci. Comput., 30 (2008), 657-683.  doi: 10.1137/060663556.

[11]

K. Bredies and D. A. Lorenz, Regularization with non-convex separable constraints, Inverse Problems, 25 (2009), 085011, 14 pp. doi: 10.1088/0266-5611/25/8/085011.

[12]

K. BrediesD. A. Lorenz and P. Maass, A generalized conditional gradient method and its connection to an iterative shrinkage method, Comput. Optim. Appl., 42 (2009), 173-193.  doi: 10.1007/s10589-007-9083-3.

[13]

S. P. Chepuri and G. Leus, Sparsity-promoting sensor selection for non-linear measurement models, IEEE Trans. Signal Process., 63 (2015), 684-698.  doi: 10.1109/TSP.2014.2379662.

[14]

I. Daubechies, M. Defrise and C. De Mol, Sparsity-enforcing regularisation and ISTA revisited, Inverse Problems, 32 (2016), 104001, 15 pp. doi: 10.1088/0266-5611/32/10/104001.

[15]

L. Ding and W. Han, $\alpha\ell_{1}-\beta\ell_{2}$ regularization for sparse recovery, Inverse Problems, 35 (2019), 125009, 26 pp. doi: 10.1088/1361-6420/ab34b5.

[16]

L. Ding and W. Han, A projected gradient method for $\alpha\ell_{1}-\beta\ell_{2}$ sparsity regularization, Inverse Problems, 36 (2020), 125012, 30 pp. doi: 10.1088/1361-6420/abc857.

[17]

L. Ding and W. Han, $\alpha\ell_{1}-\beta\ell_{2}$ sparsity regularization for nonlinear ill-posed problems, preprint, 2020, arXiv: 2007.11377v1.

[18]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications vol 375: Dordrecht: Kluwer, 1996.

[19]

E. EsserY. Lou and J. Xin, A method for finding structured sparse solutions to non-negative least squares problems with applications, SIAM J. Imaging Sci., 6 (2013), 2010-2046.  doi: 10.1137/13090540X.

[20]

M. Fornasier, eds, Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyter, 2010. doi: 10.1515/9783110226157.93.

[21]

M. Fornasier and H. Rauhut, Iterative thresholding algorithms, Applied and Computational Harmonic Analysis, 25 (2008), 187-208.  doi: 10.1016/j.acha.2007.10.005.

[22]

A. Frommer and P. Maass, Fast CG-based methods for Tikhonov-Phillips regularization, SIAM J. Sci. Comp., 20 (1999), 1831-1850.  doi: 10.1137/S1064827596313310.

[23]

M. Grasmair, Well-posedness and convergence rates for sparse regularization with sublinear $\ell^q$ penalty term, Inverse Problems and Imaging, 3 (2009), 383-387.  doi: 10.3934/ipi.2009.3.383.

[24]

M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with $\ell^q$ penalty term, Inverse Problems, 24 (2008), 055020, 13 pp. doi: 10.1088/0266-5611/24/5/055020.

[25]

M. HankeA. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.  doi: 10.1007/s002110050158.

[26]

P. C. Hansen, Regularization Tools version 4.0 for Matlab 7.3, Numerical Algorithms, 46 (2007), 189-194.  doi: 10.1007/s11075-007-9136-9.

[27]

X.-L. HuangL. Shi and M. Yan, Nonconvex sorted $\ell_1$ minimization for sparse approximation, J. Oper. Res. Soc. China, 3 (2015), 207-229.  doi: 10.1007/s40305-014-0069-4.

[28]

B. Jin, D. A. Lorenz and S. Schiffer, Elastic-net regularization: Error estimates and active set methods, Inverse Problems, 25 (2009), 115022, 26 pp. doi: 10.1088/0266-5611/25/11/115022.

[29]

B. Jin and P. Maass, Sparsity regularization for parameter identification problems, Inverse Problems, 28 (2012), 123001, 70 pp. doi: 10.1088/0266-5611/28/12/123001.

[30]

B. Jin, P. Maass and O. Scherzer, Sparsity regularization in inverse problems, Inverse Problems, 33 (2017), 060301, 4 pp. doi: 10.1088/1361-6420/33/6/060301.

[31]

B. JinY. Zhao and J. Zou, Iterative parameter choice by discrepancy principle, IMA J. Numer. Anal., 32 (2012), 1714-1732.  doi: 10.1093/imanum/drr051.

[32]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-posed Problems, De Gruyter, 2012.

[33]

D. Lazzaro, E. L. Piccolomini and F. Zama, A nonconvex penalization algorithm with automatic choice of the regularization parameter in sparse imaging, Inverse Problems, 35 (2019), 084002, 17 pp. doi: 10.1088/1361-6420/ab1c6b.

[34]

P. Li, W. Chen, H. Ge and M. K. Ng, $\ell_1$-$\alpha\ell_2$ minimization methods for signal and image reconstruction with impulsive noise removal, Inverse Problems, 36 (2020), 055009, 30 pp. doi: 10.1088/1361-6420/ab750c.

[35]

Y. Lou and M. Yan, Fast L1-L2 minimization via a proximal operator, J. Sci. Comput., 74 (2018), 767-785.  doi: 10.1007/s10915-017-0463-2.

[36]

A. K. Louis, Inverse und Schlecht Gestellte Probleme, Teubner: Stuttgart. 1989. doi: 10.1007/978-3-322-84808-6.

[37]

P. MaassS. V. PereverzevR. Ramlau and S. G. Solodky, An adaptive discretization scheme for Tikhonov-regularization with a posteriori parameter selection, Numerische Mathematik, 87 (2001), 485-502.  doi: 10.1007/PL00005421.

[38]

L. B. MontefuscoD. Lazzaro and S. Papi, A fast algorithm for nonconvex approaches to sparse recovery problems, Signal Proc., 93 (2013), 2636-2647. 

[39]

V. A. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., 7 (1966), 414-417. 

[40]

R. Ramlau, Morozov's discrepancy principle for Tikhonov regularization of nonlinear operators, Numer. Funct. Anal. Optim., 23 (2002), 147-172.  doi: 10.1081/NFA-120003676.

[41]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Applied Mathematical Sciences, Vol.167, Newyork: Springer, 2009.

[42]

T. Scherzer, B. Kaltenbacher, B. Hofmann and K. Kazimierski, Regularization Methods in Banach Spaces, De Gruyter, 2012. doi: 10.1515/9783110255720.

[43]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston Wiley: New York, 1977.

[44]

A. N. Tikhonov, A. S. Leonov and A. G. Yagola, Nonlinear Ill-Posed Problems, London: Chapman & Hall, 1998. doi: 10.1007/978-94-017-5167-4.

[45]

W. WangS. LuB. Hofmann and J. Cheng, Tikhonov regularization with $\ell_0$-term complementing a convex penalty: $\ell_1$-convergence under sparsity constraints, J. Inverse Ill-Posed Probl., 27 (2019), 575-590.  doi: 10.1515/jiip-2019-0008.

[46]

W. Wang, S. Lu, H. Mao and J. Cheng, Multi-parameter Tikhonov regularization with the $\ell_0$ sparsity constrain, Inverse Problems, 29 (2013), 065018, 18 pp. doi: 10.1088/0266-5611/29/6/065018.

[47]

L. Yan, Y. Shin and D. Xiu, Sparse approximation using $\ell_1$-$\ell_2$ minimization and its application to stochastic collocation, SIAM J. Sci. Comput., 39 (2017), A214–A239. doi: 10.1137/15M103947X.

[48]

S. YangM. WangP. LiL. JinB. Wu and L. Jiao, Compressive hyperspectral imaging via sparse tensor and nonlinear compressed sensing, IEEE Transactions on Geoscience and Remote Sensing, 53 (2015), 5943-5957. 

[49]

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.

show all references

References:
[1]

V. Albani and A. De Cezaro, A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles, Inverse Problems and Imaging, 13 (2019), 211-229.  doi: 10.3934/ipi.2019012.

[2]

S. W. Anzengruber and R. Ramlau, Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2010), 025001, 17 pp. doi: 10.1088/0266-5611/26/2/025001.

[3]

S. W. Anzengruber and R. Ramlau, Convergence rates for Morozov's discrepancy principle using variational inequalities, Inverse Problems, 27 (2011), 105007, 18 pp. doi: 10.1088/0266-5611/27/10/105007.

[4]

A. Beck and Y. C. Eldar, Sparsity constrained nonlinear optimization: Optimality conditions and algorithms, SIAM Journal on Optimization, 23 (2013), 1480-1509.  doi: 10.1137/120869778.

[5]

M. Benning and M. Burger, Modern regularization methods for inverse problems, Acta Numerica, 27 (2018), 1-111.  doi: 10.1017/S0962492918000016.

[6]

T. Blumensath, Compressed sensing with nonlinear observations and related nonlinear optimization problems, IEEE Transactions on Information Theory, 59 (2013), 3466-3474.  doi: 10.1109/TIT.2013.2245716.

[7]

T. Blumensath and M. E. Davies, Iterative thresholding for sparse approximations, J. Fourier Anal. Appl., 14 (2008), 629-654.  doi: 10.1007/s00041-008-9035-z.

[8]

T. Blumensath and M. E. Davies, Iterative hard thresholding for compressed sensing, Appl. Comput. Harmon. Anal., 27 (2009), 265-274.  doi: 10.1016/j.acha.2009.04.002.

[9]

T. Bonesky, Morozov's discrepancy principle and Tikhonov-type functionals, Inverse Problems, 25 (2009), 015015, 11 pp. doi: 10.1088/0266-5611/25/1/015015.

[10]

K. Bredies and D. A. Lorenz, Iterated hard shrinkage for minimization problems with sparsity constraints, SIAM J. Sci. Comput., 30 (2008), 657-683.  doi: 10.1137/060663556.

[11]

K. Bredies and D. A. Lorenz, Regularization with non-convex separable constraints, Inverse Problems, 25 (2009), 085011, 14 pp. doi: 10.1088/0266-5611/25/8/085011.

[12]

K. BrediesD. A. Lorenz and P. Maass, A generalized conditional gradient method and its connection to an iterative shrinkage method, Comput. Optim. Appl., 42 (2009), 173-193.  doi: 10.1007/s10589-007-9083-3.

[13]

S. P. Chepuri and G. Leus, Sparsity-promoting sensor selection for non-linear measurement models, IEEE Trans. Signal Process., 63 (2015), 684-698.  doi: 10.1109/TSP.2014.2379662.

[14]

I. Daubechies, M. Defrise and C. De Mol, Sparsity-enforcing regularisation and ISTA revisited, Inverse Problems, 32 (2016), 104001, 15 pp. doi: 10.1088/0266-5611/32/10/104001.

[15]

L. Ding and W. Han, $\alpha\ell_{1}-\beta\ell_{2}$ regularization for sparse recovery, Inverse Problems, 35 (2019), 125009, 26 pp. doi: 10.1088/1361-6420/ab34b5.

[16]

L. Ding and W. Han, A projected gradient method for $\alpha\ell_{1}-\beta\ell_{2}$ sparsity regularization, Inverse Problems, 36 (2020), 125012, 30 pp. doi: 10.1088/1361-6420/abc857.

[17]

L. Ding and W. Han, $\alpha\ell_{1}-\beta\ell_{2}$ sparsity regularization for nonlinear ill-posed problems, preprint, 2020, arXiv: 2007.11377v1.

[18]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications vol 375: Dordrecht: Kluwer, 1996.

[19]

E. EsserY. Lou and J. Xin, A method for finding structured sparse solutions to non-negative least squares problems with applications, SIAM J. Imaging Sci., 6 (2013), 2010-2046.  doi: 10.1137/13090540X.

[20]

M. Fornasier, eds, Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyter, 2010. doi: 10.1515/9783110226157.93.

[21]

M. Fornasier and H. Rauhut, Iterative thresholding algorithms, Applied and Computational Harmonic Analysis, 25 (2008), 187-208.  doi: 10.1016/j.acha.2007.10.005.

[22]

A. Frommer and P. Maass, Fast CG-based methods for Tikhonov-Phillips regularization, SIAM J. Sci. Comp., 20 (1999), 1831-1850.  doi: 10.1137/S1064827596313310.

[23]

M. Grasmair, Well-posedness and convergence rates for sparse regularization with sublinear $\ell^q$ penalty term, Inverse Problems and Imaging, 3 (2009), 383-387.  doi: 10.3934/ipi.2009.3.383.

[24]

M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with $\ell^q$ penalty term, Inverse Problems, 24 (2008), 055020, 13 pp. doi: 10.1088/0266-5611/24/5/055020.

[25]

M. HankeA. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.  doi: 10.1007/s002110050158.

[26]

P. C. Hansen, Regularization Tools version 4.0 for Matlab 7.3, Numerical Algorithms, 46 (2007), 189-194.  doi: 10.1007/s11075-007-9136-9.

[27]

X.-L. HuangL. Shi and M. Yan, Nonconvex sorted $\ell_1$ minimization for sparse approximation, J. Oper. Res. Soc. China, 3 (2015), 207-229.  doi: 10.1007/s40305-014-0069-4.

[28]

B. Jin, D. A. Lorenz and S. Schiffer, Elastic-net regularization: Error estimates and active set methods, Inverse Problems, 25 (2009), 115022, 26 pp. doi: 10.1088/0266-5611/25/11/115022.

[29]

B. Jin and P. Maass, Sparsity regularization for parameter identification problems, Inverse Problems, 28 (2012), 123001, 70 pp. doi: 10.1088/0266-5611/28/12/123001.

[30]

B. Jin, P. Maass and O. Scherzer, Sparsity regularization in inverse problems, Inverse Problems, 33 (2017), 060301, 4 pp. doi: 10.1088/1361-6420/33/6/060301.

[31]

B. JinY. Zhao and J. Zou, Iterative parameter choice by discrepancy principle, IMA J. Numer. Anal., 32 (2012), 1714-1732.  doi: 10.1093/imanum/drr051.

[32]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-posed Problems, De Gruyter, 2012.

[33]

D. Lazzaro, E. L. Piccolomini and F. Zama, A nonconvex penalization algorithm with automatic choice of the regularization parameter in sparse imaging, Inverse Problems, 35 (2019), 084002, 17 pp. doi: 10.1088/1361-6420/ab1c6b.

[34]

P. Li, W. Chen, H. Ge and M. K. Ng, $\ell_1$-$\alpha\ell_2$ minimization methods for signal and image reconstruction with impulsive noise removal, Inverse Problems, 36 (2020), 055009, 30 pp. doi: 10.1088/1361-6420/ab750c.

[35]

Y. Lou and M. Yan, Fast L1-L2 minimization via a proximal operator, J. Sci. Comput., 74 (2018), 767-785.  doi: 10.1007/s10915-017-0463-2.

[36]

A. K. Louis, Inverse und Schlecht Gestellte Probleme, Teubner: Stuttgart. 1989. doi: 10.1007/978-3-322-84808-6.

[37]

P. MaassS. V. PereverzevR. Ramlau and S. G. Solodky, An adaptive discretization scheme for Tikhonov-regularization with a posteriori parameter selection, Numerische Mathematik, 87 (2001), 485-502.  doi: 10.1007/PL00005421.

[38]

L. B. MontefuscoD. Lazzaro and S. Papi, A fast algorithm for nonconvex approaches to sparse recovery problems, Signal Proc., 93 (2013), 2636-2647. 

[39]

V. A. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., 7 (1966), 414-417. 

[40]

R. Ramlau, Morozov's discrepancy principle for Tikhonov regularization of nonlinear operators, Numer. Funct. Anal. Optim., 23 (2002), 147-172.  doi: 10.1081/NFA-120003676.

[41]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Applied Mathematical Sciences, Vol.167, Newyork: Springer, 2009.

[42]

T. Scherzer, B. Kaltenbacher, B. Hofmann and K. Kazimierski, Regularization Methods in Banach Spaces, De Gruyter, 2012. doi: 10.1515/9783110255720.

[43]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston Wiley: New York, 1977.

[44]

A. N. Tikhonov, A. S. Leonov and A. G. Yagola, Nonlinear Ill-Posed Problems, London: Chapman & Hall, 1998. doi: 10.1007/978-94-017-5167-4.

[45]

W. WangS. LuB. Hofmann and J. Cheng, Tikhonov regularization with $\ell_0$-term complementing a convex penalty: $\ell_1$-convergence under sparsity constraints, J. Inverse Ill-Posed Probl., 27 (2019), 575-590.  doi: 10.1515/jiip-2019-0008.

[46]

W. Wang, S. Lu, H. Mao and J. Cheng, Multi-parameter Tikhonov regularization with the $\ell_0$ sparsity constrain, Inverse Problems, 29 (2013), 065018, 18 pp. doi: 10.1088/0266-5611/29/6/065018.

[47]

L. Yan, Y. Shin and D. Xiu, Sparse approximation using $\ell_1$-$\ell_2$ minimization and its application to stochastic collocation, SIAM J. Sci. Comput., 39 (2017), A214–A239. doi: 10.1137/15M103947X.

[48]

S. YangM. WangP. LiL. JinB. Wu and L. Jiao, Compressive hyperspectral imaging via sparse tensor and nonlinear compressed sensing, IEEE Transactions on Geoscience and Remote Sensing, 53 (2015), 5943-5957. 

[49]

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.

Figure 1.  (a) $ \alpha $ vs. $ \delta $ where $ \alpha: \|A(x_{\alpha, \beta}^{\delta})-y^{\delta}\|_Y = c(\delta) $; (b) Rerror vs. $ \delta $
Figure 2.  (a) $ \alpha $ vs. $ \delta $ where $ \alpha $ is determined by Algorithm 1; (b) Rerror vs. $ \delta $
Figure 3.  (a) True image; (b) Observed image; (c) Recovered image with $ \alpha: \|A(x_{\alpha_j, \beta_j}^{\delta})-y^{\delta}\|_Y = c(\delta) $; (d) Recovered image with $ \alpha $ is determined by Algorithm 1
Figure 4.  (a) $ \alpha $ vs. $ \delta $ where $ \alpha: \|A(x_{\alpha, \beta}^{\delta})-y^{\delta}\|_Y = c(\delta) $; (b) Rerror vs. $ \delta $
Figure 5.  (a) $ \alpha $ vs. $ \delta $ where $ \alpha $ is determined by Algorithm 1; (b) Rerror vs. $ \delta $
Figure 6.  (a) True signal; (b) Observed data; (c) Recovered signal with $ \alpha: \|A(x_{\alpha_j, \beta_j}^{\delta})-y^{\delta}\|_Y = c(\delta) $; Rerror = 0.0828; (d) Recovered signal with $ \alpha $ is determined by Algorithm 1; Rerror = 0.0465
Figure 7.  $ \delta $, the upper bound $ c(\delta) $ and $ \|A(x_{\alpha,\beta}^{\delta})-y^{\delta}\| $ vs. $ \alpha $
Figure 8.  (a) $ \alpha $ vs. $ \delta $ where $ \alpha: \|A(x_{\alpha, \beta}^{\delta})-y^{\delta}\|_Y = c(\delta) $; (b) Rerror vs. $ \delta $
Figure 9.  (a) $ \alpha $ vs. $ \delta $ where $ \alpha $ is determined by Algorithm 1; (b) Rerror vs. $ \delta $
Figure 10.  (a) True signal; (b) Observed data; (c) Recovered signal with $ \alpha: \|A(x_{\alpha, \beta}^{\delta})-y^{\delta}\|_Y = c(\delta) $ Rerror = 0.0734; (d) Recovered signal with $ \alpha $ is determined by Algorithm 1 Rerror = 0.0137
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