doi: 10.3934/jimo.2022045
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Low rank matrix minimization with a truncated difference of nuclear norm and Frobenius norm regularization

School of Mathematics, Tianjin University, Tianjin 300350, China

*Corresponding author: Xinzhen Zhang

Received  September 2021 Revised  January 2022 Early access March 2022

Fund Project: The third author is supported by NSFC grant 11871369

In this paper, we present a novel regularization with a truncated difference of nuclear norm and Frobenius norm of form $ L_{t,*-\alpha F} $ with an integer $ t $ and parameter $ \alpha $ for rank minimization problem. The forward-backward splitting (FBS) algorithm is proposed to solve such a regularization problem, whose subproblems are shown to have closed-form solutions. We show that any accumulation point of the sequence generated by the FBS algorithm is a first-order stationary point. In the end, the numerical results demonstrate that the proposed FBS algorithm outperforms the existing methods.

Citation: Huiyuan Guo, Quan Yu, Xinzhen Zhang, Lulu Cheng. Low rank matrix minimization with a truncated difference of nuclear norm and Frobenius norm regularization. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022045
References:
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A. ArgyriouT. Evgeniou and M. Pontil, Convex multi-task feature learning, Mach. Learn., 73 (2008), 243-272.  doi: 10.1007/s10994-007-5040-8.

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J. F. CaiE. J. Candes and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM J. Optim., 20 (2010), 1956-1928.  doi: 10.1137/080738970.

[3]

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

[4]

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

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F. H. Clarke, Optimization and Nonsmooth Analysis, 2$^{nd}$ edition, Society for Industrial and Applied Mathematics, 1990. doi: 10.1137/1.9781611971309.

[6]

M. FazelH. Hindi and S. P. Boyd, A rank minimization heuristic with application to minimum order system approximation, Proceedings of the 2001 American Control Conference, 6 (2001), 4734-4739.  doi: 10.1109/ACC.2001.945730.

[7]

M. FornasierH. Rauhut and R. Ward, Low-rank matrix recovery via iteratively reweighted least squares minimization, SIAM J. Optim., 21 (2011), 1614-1640.  doi: 10.1137/100811404.

[8]

M. LaiY. Xu and W. Yin, Improved iteratively reweighted least squares for unconstrained smoothed $l_{q}$ minimization, SIAM J. Numer. Anal, 51 (2013), 927-957.  doi: 10.1137/110840364.

[9]

A. S. Lewis and H. S. Sendov, Nonsmooth analysis of singular values. Part Ⅱ: Applications, Set-Valued Analysis, 13 (2005), 243-264.  doi: 10.1007/s11228-004-7198-6.

[10]

G. LiuZ. LinS. YanJ. SunY. Yu and Y. Ma, Robust recovery of subspace structures by low-rank representation, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2013), 171-184.  doi: 10.1109/TPAMI.2012.88.

[11]

Y. Lou and M. Yan, Fast $L_1-L_2$ minimization via a proximal operator, J. Sci. Comput., 74 (2018), 767-785.  doi: 10.1007/s10915-017-0463-2.

[12]

Z. LuY. Zhang and X. Li, Penalty decomposition methods for rank minimization, Optim. Methods Softw., 30 (2015), 531-558.  doi: 10.1080/10556788.2014.936438.

[13]

S. MaD. Goldfarb and L. Chen, Fixed point and Bregman iterative methods for matrix rank minimization, Math. Program., 128 (2011), 321-353.  doi: 10.1007/s10107-009-0306-5.

[14]

Z. MingL. ZhangY. Xu and M. Bakshi, An algorithm for matrix recovery of high-loss-rate network traffic data, Appl. Math. Model., 96 (2021), 645-656.  doi: 10.1016/j.apm.2021.03.036.

[15]

D. Phan and T. Nguyen, An accelerated IRNN-Iteratively reweighted nuclear norm algorithm for nonconvex nonsmooth low rank minimization problems, J. Comput. Appl. Math., 396 (2021), Paper No. 113602, 14 pp. doi: 10.1016/j.cam.2021.113602.

[16]

B. RechtM. Fazel and P. A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), 471-501.  doi: 10.1137/070697835.

[17]

Rockafellar. R. T, Convex Analysis, Princeton University Press, 1997.

[18]

K. Toh and S. Yun, An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems, Pac. J. Optim., 6 (2010), 615-640. 

[19]

Z. WenW. Yin and Y. Zhang, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Math. Program. Comput., 4 (2012), 333-361.  doi: 10.1007/s12532-012-0044-1.

[20]

P. YinY. LouQ. He and J. Xin, Minimization of $l_{1-2}$ for compressed sensing, SIAM J. Sci. Comput., 37 (2015), 536-563.  doi: 10.1137/140952363.

show all references

References:
[1]

A. ArgyriouT. Evgeniou and M. Pontil, Convex multi-task feature learning, Mach. Learn., 73 (2008), 243-272.  doi: 10.1007/s10994-007-5040-8.

[2]

J. F. CaiE. J. Candes and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM J. Optim., 20 (2010), 1956-1928.  doi: 10.1137/080738970.

[3]

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

[4]

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

[5]

F. H. Clarke, Optimization and Nonsmooth Analysis, 2$^{nd}$ edition, Society for Industrial and Applied Mathematics, 1990. doi: 10.1137/1.9781611971309.

[6]

M. FazelH. Hindi and S. P. Boyd, A rank minimization heuristic with application to minimum order system approximation, Proceedings of the 2001 American Control Conference, 6 (2001), 4734-4739.  doi: 10.1109/ACC.2001.945730.

[7]

M. FornasierH. Rauhut and R. Ward, Low-rank matrix recovery via iteratively reweighted least squares minimization, SIAM J. Optim., 21 (2011), 1614-1640.  doi: 10.1137/100811404.

[8]

M. LaiY. Xu and W. Yin, Improved iteratively reweighted least squares for unconstrained smoothed $l_{q}$ minimization, SIAM J. Numer. Anal, 51 (2013), 927-957.  doi: 10.1137/110840364.

[9]

A. S. Lewis and H. S. Sendov, Nonsmooth analysis of singular values. Part Ⅱ: Applications, Set-Valued Analysis, 13 (2005), 243-264.  doi: 10.1007/s11228-004-7198-6.

[10]

G. LiuZ. LinS. YanJ. SunY. Yu and Y. Ma, Robust recovery of subspace structures by low-rank representation, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2013), 171-184.  doi: 10.1109/TPAMI.2012.88.

[11]

Y. Lou and M. Yan, Fast $L_1-L_2$ minimization via a proximal operator, J. Sci. Comput., 74 (2018), 767-785.  doi: 10.1007/s10915-017-0463-2.

[12]

Z. LuY. Zhang and X. Li, Penalty decomposition methods for rank minimization, Optim. Methods Softw., 30 (2015), 531-558.  doi: 10.1080/10556788.2014.936438.

[13]

S. MaD. Goldfarb and L. Chen, Fixed point and Bregman iterative methods for matrix rank minimization, Math. Program., 128 (2011), 321-353.  doi: 10.1007/s10107-009-0306-5.

[14]

Z. MingL. ZhangY. Xu and M. Bakshi, An algorithm for matrix recovery of high-loss-rate network traffic data, Appl. Math. Model., 96 (2021), 645-656.  doi: 10.1016/j.apm.2021.03.036.

[15]

D. Phan and T. Nguyen, An accelerated IRNN-Iteratively reweighted nuclear norm algorithm for nonconvex nonsmooth low rank minimization problems, J. Comput. Appl. Math., 396 (2021), Paper No. 113602, 14 pp. doi: 10.1016/j.cam.2021.113602.

[16]

B. RechtM. Fazel and P. A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), 471-501.  doi: 10.1137/070697835.

[17]

Rockafellar. R. T, Convex Analysis, Princeton University Press, 1997.

[18]

K. Toh and S. Yun, An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems, Pac. J. Optim., 6 (2010), 615-640. 

[19]

Z. WenW. Yin and Y. Zhang, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Math. Program. Comput., 4 (2012), 333-361.  doi: 10.1007/s12532-012-0044-1.

[20]

P. YinY. LouQ. He and J. Xin, Minimization of $l_{1-2}$ for compressed sensing, SIAM J. Sci. Comput., 37 (2015), 536-563.  doi: 10.1137/140952363.

Figure 1.  Comparison of random data with $ p = 0.5 $
Figure 2.  As can be seen, the distribution of the singular values has a fast decaying distribution, and the information is dominated by the top 50 singular values
Figure 3.  An empirical study of Male images data
Figure 4.  Comparison of the four algorithms with different sampling ratio $ p $
Figure 5.  Testing images. "Clock": grayscale image of $ 256 \times 256 $ pixels. "Male": grayscale image of $ 512 \times 512 $ pixels. "Pixel ruler": grayscale image of $ 512 \times 512 $ pixels. "Walter Cronkite": grayscale image of $ 256 \times 256 $ pixels
Figure 6.  Recovered images of the masked images
Figure 7.  Comparison of four algorithms for the first $ 100 $ frames
Figure 8.  Comparison results of the MRI volume dataset
Table 1.  Numerical results of different images
Method Clock Male Pixel ruler Walter Cronkite
rel.err time rel.err time rel.err time rel.err time
FBS-TM 2.96e-02 5.47 4.63e-02 40.28 4.31e-02 9.66 1.77e-02 7.62
LMaFit 7.95e-02 0.05 1.74e-01 0.16 8.74e-02 1.43 1.27e-01 0.05
SVT 4.93e-02 19.05 1.02e-01 39.51 3.09e-01 45.64 7.23e-02 19.39
FPCA 7.60e-02 2.82 1.36e-01 14.69 1.30e-01 14.69 8.29e-02 2.94
Method Clock Male Pixel ruler Walter Cronkite
rel.err time rel.err time rel.err time rel.err time
FBS-TM 2.96e-02 5.47 4.63e-02 40.28 4.31e-02 9.66 1.77e-02 7.62
LMaFit 7.95e-02 0.05 1.74e-01 0.16 8.74e-02 1.43 1.27e-01 0.05
SVT 4.93e-02 19.05 1.02e-01 39.51 3.09e-01 45.64 7.23e-02 19.39
FPCA 7.60e-02 2.82 1.36e-01 14.69 1.30e-01 14.69 8.29e-02 2.94
Table 2.  Numerical results of masked images
FBS-TM LMaFit SVT FPCA
time rel.err time rel.err time rel.err time
2.95e-02 25.45 1.43e-01 0.49 1.70e-01 11.26 7.26e-01 16.51
1.50e-02 25.10 1.12e-01 0.26 3.04e-02 52.21 5.35e-01 15.43
FBS-TM LMaFit SVT FPCA
time rel.err time rel.err time rel.err time
2.95e-02 25.45 1.43e-01 0.49 1.70e-01 11.26 7.26e-01 16.51
1.50e-02 25.10 1.12e-01 0.26 3.04e-02 52.21 5.35e-01 15.43
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