Table 1.
Comparison Results for Synthetic Data
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Online learning for supervised dimension reduction
May
2019, 2(2): 107-126.
doi: 10.3934/mfc.2019009
Nonconvex mixed matrix minimization
| 1. | Department of Mathematics, School of Science, Shijiazhuang University, Shijiazhuang 050035, China |
| 2. | Department of Applied Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, China |
| 3. | State Key Lab for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China |
Given the ultrahigh dimensionality and the complex structure, which contains matrices and vectors, the mixed matrix minimization becomes crucial for the analysis of those data. Recently, the nonconvex functions such as the smoothly clipped absolute deviation, the minimax concave penalty, the capped $ \ell_1 $-norm penalty and the $ \ell_p $ quasi-norm with $ 0<p<1 $ have been shown remarkable advantages in variable selection due to the fact that they can overcome the over-penalization. In this paper, we propose and study a novel nonconvex mixed matrix minimization, which combines the low-rank and sparse regularzations and nonconvex functions perfectly. The augmented Lagrangian method (ALM) is proposed to solve the noncovnex mixed matrix minimization problem. The resulting subproblems either have closed-form solutions or can be solved by fast solvers, which makes the ALM particularly efficient. In theory, we prove that the sequence generated by the ALM converges to a stationary point when the penalty parameter is above a computable threshold. Extensive numerical experiments illustrate that our proposed nonconvex mixed matrix minimization model outperforms the existing ones.
Keywords:
Complex structure,
mixed matrix minimization,
nonconvex optimization,
augmented Lagrangian method.
Mathematics Subject Classification:
Primary: 90C26, 90C46; Secondary: 90C90, 65K10.
Citation:
Meijuan Shang, Yanan Liu, Lingchen Kong, Xianchao Xiu, Ying Yang. Nonconvex mixed matrix minimization. Mathematical Foundations of Computing,
2019, 2
(2)
: 107-126.
doi: 10.3934/mfc.2019009
![]()
References:
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S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein,
Distributed optimization and statistical learning via alternating direction method of multipliers, Foundations and Trends® in Machine Learning, 3 (2011), 1-122.
doi: 10.1561/2200000016. |
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J. Cai, E. Cand$\grave{e}$s and Z. Shen,
A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), 1956-1982.
doi: 10.1137/080738970. |
| [3] |
X. Chen, M. Ng and C. Zhang,
Non-lipschitz $\ell_p$-regularization and box constrained model for image restoration, IEEE Transactions on Image Processing, 21 (2012), 4709-4721.
doi: 10.1109/TIP.2012.2214051. |
| [4] |
Y. Chen, N. Xiu and D. Peng,
Global solutions of non-Lipschitz $S_2-S_ p$ minimization over the positive semidefinite cone, Optimization Letters, 8 (2014), 2053-2064.
doi: 10.1007/s11590-013-0701-y. |
| [5] |
D. Donoho,
De-noising by soft-thresholding, IEEE Transactions on Information Theory, 41 (1995), 613-627.
doi: 10.1109/18.382009. |
| [6] |
M. Elad,
Why simple shrinkage is still relevant for redundant representations?, IEEE Transactions on Information Theory, 52 (2006), 5559-5569.
doi: 10.1109/TIT.2006.885522. |
| [7] |
J. Fan,
Comments on "wavelets in statistics: A review" by A. Antoniadis, Journal of the Italian Statistical Society, 6 (1997), 131-138.
doi: 10.1007/BF03178906. |
| [8] |
J. Fan and R. Li,
Variable selection via nonconcave penalized likelihood and its oracle properties, Journal of the American Statistical Association, 96 (2001), 1348-1360.
doi: 10.1198/016214501753382273. |
| [9] |
J. Fan, L. Xue and H. Zou,
Strong oracle optimality of folded concave penalized estimation, Annals of Statistics, 42 (2014), 819-849.
doi: 10.1214/13-AOS1198. |
| [10] |
M. Fazel, T. Pong, D. Sun and P. Tseng,
Hankel matrix rank minimization with applications to system identification and realization, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 946-977.
doi: 10.1137/110853996. |
| [11] |
D. Gabay,
Chapter ix applications of the method of multipliers to variational inequalities, Studies in Mathematics and Its Applications, 15 (1983), 299-331.
doi: 10.1016/S0168-2024(08)70034-1. |
| [12] |
D. Gabay and B. Mercier,
A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers and Mathematics with Applications, 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1. |
| [13] |
M. Hong, Z. Luo and M. Razaviyayn,
Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems, SIAM Journal on Optimization, 26 (2016), 337-364.
doi: 10.1137/140990309. |
| [14] |
G. Li and T. Peng.,
Douglass-Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems, Mathematical Programming, 159 (2016), 371-401.
doi: 10.1007/s10107-015-0963-5. |
| [15] |
S. Negahban and M. Wainwright,
Estimation of (near) low-rank matrices with noise and high-dimensional scaling, Annals of Statistics, 39 (2011), 1069-1097.
doi: 10.1214/10-AOS850. |
| [16] |
M. Nikolova, M. Ng, S. Zhang and W. Ching,
Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization, SIAM Journal on Imaging Sciences, 1 (2008), 2-25.
doi: 10.1137/070692285. |
| [17] |
G. Obozinski, M. Wainwright and M. Jordan,
Support union recovery in high-dimensional multivariate regression, Annals of Statistics, 39 (2011), 1-47.
doi: 10.1214/09-AOS776. |
| [18] |
L. Rudin and S. Osher,
Total variation based image restoration with free local constraints, In Proceedings of the IEEE International Conference on Image Processing, 1 (1994), 31-35.
doi: 10.1109/ICIP.1994.413269. |
| [19] |
R. Rockafellar and R. Wets, Variational Analysis, Springer Science and Business Media, 2009.
doi: 10.1007/978-3-642-02431-3. |
| [20] |
P. Shang and L. Kong, On the degrees of freedom of mixed matrix regression, Mathematical Problems in Engineering, 2017 (2017), Art. ID 6942865, 8 pp.
doi: 10.1155/2017/6942865. |
| [21] |
R. Tibshirani,
Regression shrinkage and selection via the Lasso, Journal of the Royal Statistical Society, Series B, 58 (1996), 267-288.
doi: 10.1111/j.2517-6161.1996.tb02080.x. |
| [22] |
R. Tibshirani, M. Saunders, S. Rosset, J. Zhu and K. Knight,
Sparsity and smoothness via the fused Lasso, Journal of the Royal Statistical Society, Series B, 67 (2005), 91-108.
doi: 10.1111/j.1467-9868.2005.00490.x. |
| [23] |
F. Wang, W. Cao and Z. Xu., Convergence of multi-block Bregman ADMM for nonconvex composite problems, Science China Information Sciences, 61 (2018), 122101, 12pp.
doi: 10.1007/s11432-017-9367-6. |
| [24] |
X. Xiu, L. Kong, Y. Li and H. Qi,
Iterative reweighted methods for $\ell_1-\ell_p$ minimization, Computational Optimization and Applications, 70 (2018), 201-219.
doi: 10.1007/s10589-017-9977-7. |
| [25] |
X. Xiu, W. Liu, L. Li and L. Kong,
Alternating direction method of multipliers for nonconvex fused regression problems, Computational Statistics and Data Analysis, 136 (2019), 59-71.
doi: 10.1016/j.csda.2019.01.002. |
| [26] |
L. Yang, T. Pong and X. Chen,
Alternating direction method of multipliers for a class of nonconvex and nonsmooth problems with applications to background/foreground extraction, SIAM Journal on Imaging Sciences, 10 (2017), 74-110.
doi: 10.1137/15M1027528. |
| [27] |
M. Yuan, A. Ekici, Z. Lu and R. Monteiro,
Dimension reduction and coefficient estimation in multivariate linear regression, Journal of Royal Statistical Society, Series B, 69 (2007), 329-346.
doi: 10.1111/j.1467-9868.2007.00591.x. |
| [28] |
C. Zhang,
Nearly unbiased variable selection under minimax concave penalty, Annals of Statistics, 38 (2010), 894-942.
doi: 10.1214/09-AOS729. |
| [29] |
T. Zhang,
Analysis of multi-stage convex relaxation for sparse regularization, Journal of Machine Learning Research, 11 (2010), 1081-1107.
|
| [30] |
H. Zou and T. Hastie,
Regularization and variable selection via the elastic net, Journal of the Royal Statistical Society, Series B, 67 (2005), 301-320.
doi: 10.1111/j.1467-9868.2005.00503.x. |
| [31] |
H. Zhou and L. Li,
Regularized matrix regression, Journal of the Royal Statistical Society, Series B, 76 (2014), 463-483.
doi: 10.1111/rssb.12031. |
| [32] |
T. Zhou and D. Tao, Godec: Randomized low-rank and sparse matrix decomposition in noisy case, International Conference on Machine Learning, 2011. |
show all references
References:
| [1] |
S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein,
Distributed optimization and statistical learning via alternating direction method of multipliers, Foundations and Trends® in Machine Learning, 3 (2011), 1-122.
doi: 10.1561/2200000016. |
| [2] |
J. Cai, E. Cand$\grave{e}$s and Z. Shen,
A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), 1956-1982.
doi: 10.1137/080738970. |
| [3] |
X. Chen, M. Ng and C. Zhang,
Non-lipschitz $\ell_p$-regularization and box constrained model for image restoration, IEEE Transactions on Image Processing, 21 (2012), 4709-4721.
doi: 10.1109/TIP.2012.2214051. |
| [4] |
Y. Chen, N. Xiu and D. Peng,
Global solutions of non-Lipschitz $S_2-S_ p$ minimization over the positive semidefinite cone, Optimization Letters, 8 (2014), 2053-2064.
doi: 10.1007/s11590-013-0701-y. |
| [5] |
D. Donoho,
De-noising by soft-thresholding, IEEE Transactions on Information Theory, 41 (1995), 613-627.
doi: 10.1109/18.382009. |
| [6] |
M. Elad,
Why simple shrinkage is still relevant for redundant representations?, IEEE Transactions on Information Theory, 52 (2006), 5559-5569.
doi: 10.1109/TIT.2006.885522. |
| [7] |
J. Fan,
Comments on "wavelets in statistics: A review" by A. Antoniadis, Journal of the Italian Statistical Society, 6 (1997), 131-138.
doi: 10.1007/BF03178906. |
| [8] |
J. Fan and R. Li,
Variable selection via nonconcave penalized likelihood and its oracle properties, Journal of the American Statistical Association, 96 (2001), 1348-1360.
doi: 10.1198/016214501753382273. |
| [9] |
J. Fan, L. Xue and H. Zou,
Strong oracle optimality of folded concave penalized estimation, Annals of Statistics, 42 (2014), 819-849.
doi: 10.1214/13-AOS1198. |
| [10] |
M. Fazel, T. Pong, D. Sun and P. Tseng,
Hankel matrix rank minimization with applications to system identification and realization, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 946-977.
doi: 10.1137/110853996. |
| [11] |
D. Gabay,
Chapter ix applications of the method of multipliers to variational inequalities, Studies in Mathematics and Its Applications, 15 (1983), 299-331.
doi: 10.1016/S0168-2024(08)70034-1. |
| [12] |
D. Gabay and B. Mercier,
A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers and Mathematics with Applications, 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1. |
| [13] |
M. Hong, Z. Luo and M. Razaviyayn,
Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems, SIAM Journal on Optimization, 26 (2016), 337-364.
doi: 10.1137/140990309. |
| [14] |
G. Li and T. Peng.,
Douglass-Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems, Mathematical Programming, 159 (2016), 371-401.
doi: 10.1007/s10107-015-0963-5. |
| [15] |
S. Negahban and M. Wainwright,
Estimation of (near) low-rank matrices with noise and high-dimensional scaling, Annals of Statistics, 39 (2011), 1069-1097.
doi: 10.1214/10-AOS850. |
| [16] |
M. Nikolova, M. Ng, S. Zhang and W. Ching,
Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization, SIAM Journal on Imaging Sciences, 1 (2008), 2-25.
doi: 10.1137/070692285. |
| [17] |
G. Obozinski, M. Wainwright and M. Jordan,
Support union recovery in high-dimensional multivariate regression, Annals of Statistics, 39 (2011), 1-47.
doi: 10.1214/09-AOS776. |
| [18] |
L. Rudin and S. Osher,
Total variation based image restoration with free local constraints, In Proceedings of the IEEE International Conference on Image Processing, 1 (1994), 31-35.
doi: 10.1109/ICIP.1994.413269. |
| [19] |
R. Rockafellar and R. Wets, Variational Analysis, Springer Science and Business Media, 2009.
doi: 10.1007/978-3-642-02431-3. |
| [20] |
P. Shang and L. Kong, On the degrees of freedom of mixed matrix regression, Mathematical Problems in Engineering, 2017 (2017), Art. ID 6942865, 8 pp.
doi: 10.1155/2017/6942865. |
| [21] |
R. Tibshirani,
Regression shrinkage and selection via the Lasso, Journal of the Royal Statistical Society, Series B, 58 (1996), 267-288.
doi: 10.1111/j.2517-6161.1996.tb02080.x. |
| [22] |
R. Tibshirani, M. Saunders, S. Rosset, J. Zhu and K. Knight,
Sparsity and smoothness via the fused Lasso, Journal of the Royal Statistical Society, Series B, 67 (2005), 91-108.
doi: 10.1111/j.1467-9868.2005.00490.x. |
| [23] |
F. Wang, W. Cao and Z. Xu., Convergence of multi-block Bregman ADMM for nonconvex composite problems, Science China Information Sciences, 61 (2018), 122101, 12pp.
doi: 10.1007/s11432-017-9367-6. |
| [24] |
X. Xiu, L. Kong, Y. Li and H. Qi,
Iterative reweighted methods for $\ell_1-\ell_p$ minimization, Computational Optimization and Applications, 70 (2018), 201-219.
doi: 10.1007/s10589-017-9977-7. |
| [25] |
X. Xiu, W. Liu, L. Li and L. Kong,
Alternating direction method of multipliers for nonconvex fused regression problems, Computational Statistics and Data Analysis, 136 (2019), 59-71.
doi: 10.1016/j.csda.2019.01.002. |
| [26] |
L. Yang, T. Pong and X. Chen,
Alternating direction method of multipliers for a class of nonconvex and nonsmooth problems with applications to background/foreground extraction, SIAM Journal on Imaging Sciences, 10 (2017), 74-110.
doi: 10.1137/15M1027528. |
| [27] |
M. Yuan, A. Ekici, Z. Lu and R. Monteiro,
Dimension reduction and coefficient estimation in multivariate linear regression, Journal of Royal Statistical Society, Series B, 69 (2007), 329-346.
doi: 10.1111/j.1467-9868.2007.00591.x. |
| [28] |
C. Zhang,
Nearly unbiased variable selection under minimax concave penalty, Annals of Statistics, 38 (2010), 894-942.
doi: 10.1214/09-AOS729. |
| [29] |
T. Zhang,
Analysis of multi-stage convex relaxation for sparse regularization, Journal of Machine Learning Research, 11 (2010), 1081-1107.
|
| [30] |
H. Zou and T. Hastie,
Regularization and variable selection via the elastic net, Journal of the Royal Statistical Society, Series B, 67 (2005), 301-320.
doi: 10.1111/j.1467-9868.2005.00503.x. |
| [31] |
H. Zhou and L. Li,
Regularized matrix regression, Journal of the Royal Statistical Society, Series B, 76 (2014), 463-483.
doi: 10.1111/rssb.12031. |
| [32] |
T. Zhou and D. Tao, Godec: Randomized low-rank and sparse matrix decomposition in noisy case, International Conference on Machine Learning, 2011. |
| Cases | RMSE(B) | RMSE( |
|||||
| |
|
LMM | LSMM | NonMM | LMM | LSMM | NonMM |
| 5 | 0.01 | 0.0000 | 0.0000 | 0.0000 | 0.1806 | 0.1421 | 0.1101 |
| 0.05 | 0.0456 | 0.0437 | 0.0274 | 0.6453 | 0.2742 | 0.2336 | |
| 0.1 | 0.0905 | 0.0688 | 0.0496 | 0.9162 | 0.3788 | 0.3176 | |
| 0.2 | 0.1314 | 0.1082 | 0.0911 | 1.0822 | 0.5063 | 0.4410 | |
| 0.5 | 0.2744 | 0.2152 | 0.1013 | 1.5868 | 0.7371 | 0.7092 | |
| 10 | 0.01 | 0.0000 | 0.0000 | 0.0000 | 0.1927 | 0.1513 | 0.1217 |
| 0.05 | 0.0781 | 0.05115 | 0.0382 | 0.7663 | 0.3312 | 0.2961 | |
| 0.1 | 0.1072 | 0.0749 | 0.0536 | 1.0452 | 0.3879 | 0.3401 | |
| 0.2 | 0.1339 | 0.1217 | 0.1161 | 1.2696 | 0.5305 | 0.4918 | |
| 0.5 | 0.2947 | 0.2440 | 0.1278 | 1.9561 | 0.7971 | 0.7330 | |
| 15 | 0.01 | 0.0000 | 0.0000 | 0.0000 | 0.2197 | 0.1920 | 0.1405 |
| 0.05 | 0.0982 | 0.0749 | 0.0654 | 0.8408 | 0.3223 | 0.2893 | |
| 0.1 | 0.1323 | 0.1130 | 0.1108 | 1.2501 | 0.3534 | 0.3293 | |
| 0.2 | 0.1846 | 0.1560 | 0.1272 | 1.6730 | 0.4988 | 0.4494 | |
| 0.5 | 0.3176 | 0.2812 | 0.2176 | 2.3608 | 0.7817 | 0.7502 | |
| 20 | 0.01 | 0.0000 | 0.0000 | 0.0000 | 0.1936 | 0.1532 | 0.1509 |
| 0.05 | 0.0987 | 0.0791 | 0.0732 | 0.9081 | 0.3168 | 0.2719 | |
| 0.1 | 0.1528 | 0.1454 | 0.1402 | 1.416 | 0.4583 | 0.3961 | |
| 0.2 | 0.1904 | 0.1821 | 0.1723 | 1.8304 | 0.4722 | 0.4148 | |
| 0.5 | 0.3876 | 0.2922 | 0.2134 | 2.4961 | 0.8049 | 0.7891 | |
| 30 | 0.01 | 0.0000 | 0.0000 | 0.0000 | 0.1768 | 0.1620 | 0.1400 |
| 0.05 | 0.1071 | 0.0932 | 0.0858 | 1.0089 | 0.2360 | 0.2232 | |
| 0.1 | 0.1608 | 0.1572 | 0.1469 | 1.5286 | 0.3241 | 0.3062 | |
| 0.2 | 0.2487 | 0.2371 | 0.2206 | 2.082 | 0.4920 | 0.4599 | |
| 0.5 | 0.4170 | 0.3943 | 0.3725 | 2.6203 | 0.7856 | 0.8041 | |
| Cases | RMSE(B) | RMSE( |
|||||
| |
|
LMM | LSMM | NonMM | LMM | LSMM | NonMM |
| 5 | 0.01 | 0.0000 | 0.0000 | 0.0000 | 0.1806 | 0.1421 | 0.1101 |
| 0.05 | 0.0456 | 0.0437 | 0.0274 | 0.6453 | 0.2742 | 0.2336 | |
| 0.1 | 0.0905 | 0.0688 | 0.0496 | 0.9162 | 0.3788 | 0.3176 | |
| 0.2 | 0.1314 | 0.1082 | 0.0911 | 1.0822 | 0.5063 | 0.4410 | |
| 0.5 | 0.2744 | 0.2152 | 0.1013 | 1.5868 | 0.7371 | 0.7092 | |
| 10 | 0.01 | 0.0000 | 0.0000 | 0.0000 | 0.1927 | 0.1513 | 0.1217 |
| 0.05 | 0.0781 | 0.05115 | 0.0382 | 0.7663 | 0.3312 | 0.2961 | |
| 0.1 | 0.1072 | 0.0749 | 0.0536 | 1.0452 | 0.3879 | 0.3401 | |
| 0.2 | 0.1339 | 0.1217 | 0.1161 | 1.2696 | 0.5305 | 0.4918 | |
| 0.5 | 0.2947 | 0.2440 | 0.1278 | 1.9561 | 0.7971 | 0.7330 | |
| 15 | 0.01 | 0.0000 | 0.0000 | 0.0000 | 0.2197 | 0.1920 | 0.1405 |
| 0.05 | 0.0982 | 0.0749 | 0.0654 | 0.8408 | 0.3223 | 0.2893 | |
| 0.1 | 0.1323 | 0.1130 | 0.1108 | 1.2501 | 0.3534 | 0.3293 | |
| 0.2 | 0.1846 | 0.1560 | 0.1272 | 1.6730 | 0.4988 | 0.4494 | |
| 0.5 | 0.3176 | 0.2812 | 0.2176 | 2.3608 | 0.7817 | 0.7502 | |
| 20 | 0.01 | 0.0000 | 0.0000 | 0.0000 | 0.1936 | 0.1532 | 0.1509 |
| 0.05 | 0.0987 | 0.0791 | 0.0732 | 0.9081 | 0.3168 | 0.2719 | |
| 0.1 | 0.1528 | 0.1454 | 0.1402 | 1.416 | 0.4583 | 0.3961 | |
| 0.2 | 0.1904 | 0.1821 | 0.1723 | 1.8304 | 0.4722 | 0.4148 | |
| 0.5 | 0.3876 | 0.2922 | 0.2134 | 2.4961 | 0.8049 | 0.7891 | |
| 30 | 0.01 | 0.0000 | 0.0000 | 0.0000 | 0.1768 | 0.1620 | 0.1400 |
| 0.05 | 0.1071 | 0.0932 | 0.0858 | 1.0089 | 0.2360 | 0.2232 | |
| 0.1 | 0.1608 | 0.1572 | 0.1469 | 1.5286 | 0.3241 | 0.3062 | |
| 0.2 | 0.2487 | 0.2371 | 0.2206 | 2.082 | 0.4920 | 0.4599 | |
| 0.5 | 0.4170 | 0.3943 | 0.3725 | 2.6203 | 0.7856 | 0.8041 | |
Table 2.
Comparison Results for Real-world Data
| Rate | LMM | LSMM | NonMM |
| 5-fold | 0.1321 | 0.1178 | 0.1025 |
| 10-fold | 0.1972 | 0.1630 | 0.1589 |
| Rate | LMM | LSMM | NonMM |
| 5-fold | 0.1321 | 0.1178 | 0.1025 |
| 10-fold | 0.1972 | 0.1630 | 0.1589 |
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