• Previous Article
    A moving block sequence-based evolutionary algorithm for resource investment project scheduling problems
  • BDIA Home
  • This Issue
  • Next Article
    Selective further learning of hybrid ensemble for class imbalanced increment learning
January  2017, 2(1): 23-37. doi: 10.3934/bdia.2017006

An evolutionary multiobjective method for low-rank and sparse matrix decomposition

School of Electronics and Information, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an Shaanxi, 710072, China

* Corresponding author: Tao Wu

Published  September 2017

This paper addresses the problem of finding the low-rank and sparse components of a given matrix. The problem involves two conflicting objective functions, reducing the rank and sparsity of each part simultaneously. Previous methods combine two objectives into a single objective penalty function to solve with traditional numerical optimization approaches. The main contribution of this paper is to put forward a multiobjective method to decompose the given matrix into low-rank component and sparse part. We optimize two objective functions with an evolutionary multiobjective algorithm MOEA/D. Another contribution of this paper, a modified low-rank and sparse matrix model is proposed, which simplifying the variable of objective functions and improving the efficiency of multiobjective optimization. The proposed method obtains a set of solutions with different trade-off between low-rank and sparse objectives, and decision makers can choose one or more satisfied decomposed results according to different requirements directly. Experiments conducted on artificial datasets and nature images, show that the proposed method always obtains satisfied results, and the convergence, stability and robustness of the proposed method is acceptable.

Citation: Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006
References:
[1]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Sciences, 2 (2009), 183-202.  doi: 10.1137/080716542.

[2]

J.-F. CaiE. J. Candès and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), 1956-1982.  doi: 10.1137/080738970.

[3]

Z. Cai and Y. Wang, A multiobjective optimization-based evolutionary algorithm for constrained optimization, IEEE Transactions on Evolutionary Computation, 10 (2006), 658-675.  doi: 10.1109/TEVC.2006.872344.

[4]

E. J. Candès, X. Li, Y. Ma and J. Wright, Robust principal component analysis? Journal of the ACM (JACM), 58 (2011), Art. 11, 37 pp. doi: 10.1145/1970392.1970395.

[5]

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

[6]

V. ChandrasekaranS. SanghaviP. A. Parrilo and A. S. Willsky, Rank-sparsity incoherence for matrix decomposition, SIAM Journal on Optimization, 21 (2011), 572-596.  doi: 10.1137/090761793.

[7]

C. A. C. Coello, D. A. Van~Veldhuizen and G. B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Genetic Algorithms and Evolutionary Computation, 5. Kluwer Academic/Plenum Publishers, New York, 2002. doi: 10.1007/978-1-4757-5184-0.

[8]

K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, Ltd. , Chichester, 2001.

[9]

K. DebA. PratapS. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.  doi: 10.1109/4235.996017.

[10]

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

[11]

M. GongL. JiaoH. Du and L. Bo, Multiobjective immune algorithm with nondominated neighbor-based selection, Evolutionary Computation, 16 (2008), 225-255.  doi: 10.1162/evco.2008.16.2.225.

[12]

Z. Lin, A. Ganesh, J. Wright, L. Wu, M. Chen and Y. Ma, Fast convex optimization algorithms for exact recovery of a corrupted low-rank matrix, Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), vol. 61,2009.

[13]

Z. Lin, M. Chen and Y. Ma, The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices, arXiv preprint, arXiv: 1009.5055, 2010.

[14]

K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, MA, 1999.

[15]

C. QianY. Yu and Z.-H. Zhou, Pareto ensemble pruning, in AAAI, (2015), 2935-2941. 

[16]

————, Subset selection by pareto optimization, in Advances in Neural Information Processing Systems, (2015), 1774-1782.

[17]

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

[18]

J. D. Schaffer, Multiple objective optimization with vector evaluated genetic algorithms, in Proceedings of the 1st international Conference on Genetic Algorithms. L. Erlbaum Associates Inc. , (1985), 93-100.

[19]

J. L. StarckM. Elad and D. L. Donoho, Image decomposition via the combination of sparse representations and a variational approach, IEEE Transactions on Image Processing, 14 (2005), 1570-1582.  doi: 10.1109/TIP.2005.852206.

[20]

J. YanJ. LiuY. LiZ. Niu and Y. Liu, Visual saliency detection via rank-sparsity decomposition, in IEEE International Conference on Image Processing, IEEE, (2010), 1089-1092.  doi: 10.1109/ICIP.2010.5652280.

[21]

X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods, Pacific Journal of Optimization, 9 (2013), 167-180. 

[22]

C. ZhangJ. LiuQ. TianC. XuH. Lu and S. Ma, Image classification by non-negative sparse coding, low-rank and sparse decomposition, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2011), 1673-1680.  doi: 10.1109/CVPR.2011.5995484.

[23]

Q. Zhang and H. Li, MOEA/D: A multiobjective evolutionary algorithm based on decomposition, IEEE Transactions on Evolutionary Computation, 11 (2007), 712-731. 

[24]

M. Zibulevsky and B. A. Pearlmutter, Blind source separation by sparse decomposition in a signal dictionary, Neural Computation, 13 (2001), 863-882. 

[25]

E. ZitzlerM. Laumanns and L. Thiele, SPEA2: Improving the strength pareto evolutionary algorithm, in Eurogen, 3242 (2001), 95-100. 

show all references

References:
[1]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Sciences, 2 (2009), 183-202.  doi: 10.1137/080716542.

[2]

J.-F. CaiE. J. Candès and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), 1956-1982.  doi: 10.1137/080738970.

[3]

Z. Cai and Y. Wang, A multiobjective optimization-based evolutionary algorithm for constrained optimization, IEEE Transactions on Evolutionary Computation, 10 (2006), 658-675.  doi: 10.1109/TEVC.2006.872344.

[4]

E. J. Candès, X. Li, Y. Ma and J. Wright, Robust principal component analysis? Journal of the ACM (JACM), 58 (2011), Art. 11, 37 pp. doi: 10.1145/1970392.1970395.

[5]

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

[6]

V. ChandrasekaranS. SanghaviP. A. Parrilo and A. S. Willsky, Rank-sparsity incoherence for matrix decomposition, SIAM Journal on Optimization, 21 (2011), 572-596.  doi: 10.1137/090761793.

[7]

C. A. C. Coello, D. A. Van~Veldhuizen and G. B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Genetic Algorithms and Evolutionary Computation, 5. Kluwer Academic/Plenum Publishers, New York, 2002. doi: 10.1007/978-1-4757-5184-0.

[8]

K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, Ltd. , Chichester, 2001.

[9]

K. DebA. PratapS. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.  doi: 10.1109/4235.996017.

[10]

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

[11]

M. GongL. JiaoH. Du and L. Bo, Multiobjective immune algorithm with nondominated neighbor-based selection, Evolutionary Computation, 16 (2008), 225-255.  doi: 10.1162/evco.2008.16.2.225.

[12]

Z. Lin, A. Ganesh, J. Wright, L. Wu, M. Chen and Y. Ma, Fast convex optimization algorithms for exact recovery of a corrupted low-rank matrix, Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), vol. 61,2009.

[13]

Z. Lin, M. Chen and Y. Ma, The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices, arXiv preprint, arXiv: 1009.5055, 2010.

[14]

K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, MA, 1999.

[15]

C. QianY. Yu and Z.-H. Zhou, Pareto ensemble pruning, in AAAI, (2015), 2935-2941. 

[16]

————, Subset selection by pareto optimization, in Advances in Neural Information Processing Systems, (2015), 1774-1782.

[17]

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

[18]

J. D. Schaffer, Multiple objective optimization with vector evaluated genetic algorithms, in Proceedings of the 1st international Conference on Genetic Algorithms. L. Erlbaum Associates Inc. , (1985), 93-100.

[19]

J. L. StarckM. Elad and D. L. Donoho, Image decomposition via the combination of sparse representations and a variational approach, IEEE Transactions on Image Processing, 14 (2005), 1570-1582.  doi: 10.1109/TIP.2005.852206.

[20]

J. YanJ. LiuY. LiZ. Niu and Y. Liu, Visual saliency detection via rank-sparsity decomposition, in IEEE International Conference on Image Processing, IEEE, (2010), 1089-1092.  doi: 10.1109/ICIP.2010.5652280.

[21]

X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods, Pacific Journal of Optimization, 9 (2013), 167-180. 

[22]

C. ZhangJ. LiuQ. TianC. XuH. Lu and S. Ma, Image classification by non-negative sparse coding, low-rank and sparse decomposition, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2011), 1673-1680.  doi: 10.1109/CVPR.2011.5995484.

[23]

Q. Zhang and H. Li, MOEA/D: A multiobjective evolutionary algorithm based on decomposition, IEEE Transactions on Evolutionary Computation, 11 (2007), 712-731. 

[24]

M. Zibulevsky and B. A. Pearlmutter, Blind source separation by sparse decomposition in a signal dictionary, Neural Computation, 13 (2001), 863-882. 

[25]

E. ZitzlerM. Laumanns and L. Thiele, SPEA2: Improving the strength pareto evolutionary algorithm, in Eurogen, 3242 (2001), 95-100. 

Figure 1.  The distributions of nondominated solutions, dominated solutions and Pareto front in the objective space of the two objectives problem.
Figure 2.  Error of sparse component recovering by the ADM method for LRSMD with different choices of the parameter $\lambda$
Figure 3.  The Pareto front, two objective functions corresponding to the solutions and box-plot for $ErrLR$ and $ErrSP$ by running 30 times independent experiments. Three rows represent experiment on the data with size:$20\times 20$, rank: 5, sparsity: 0.2, size: $50\times 50$, rank: 5, sparsity: 0.2 and size: $100\times 100$, rank: 5, sparsity: 0.5 respectively.
Figure 4.  Box-plot about $ErrLR$ and $ErrSP$ for data with different noise. (a): number of noise points is 20, $\sigma=1$. (b): number of noise points is 20, $\sigma=5$. (c): number of noise points is 20, $\sigma=15$. (d): number of noise points is 50, $\sigma=5$. (e): number of noise points is 50, $\sigma=15$.
Figure 5.  The Pareto front and two objective functions corresponding to the solutions for noise data. Noise type: number of noise points is 50 and $\sigma =15$.
Figure 6.  The Pareto front and decomposed results with image lena. (a) is Pareto front of MOLRSMD and (b) is three different decomposed results selected form Pareto front, and they locate at the top, middle and bottom of Pareto front, respectively.
Figure 7.  The Pareto front and decomposed results with image face. (a) is Pareto front of MOLRSMD and (b) is three different decomposed results selected form Pareto front, and they locate at the top, middle and bottom of Pareto front, respectively.
Figure 8.  Results of the proposed MOLRSMD compared with ADM and ALM. The test dataset size is $50\times 50$, rank equals 5 and sparsity is 0.2.
Table 1.  The Parameters in the MOLRSMD used in the Experiments
ParameterMeaningValue
$N$The number of subproblems100
$T$The number of neighbors20
$t_{max}$The maximum of generations300
$pc$The probability of crossover0.8
$pd$The differential multiplier0.5
$pm$The probability of mutation0.2
$ps$The probability of mutation selection0.85
ParameterMeaningValue
$N$The number of subproblems100
$T$The number of neighbors20
$t_{max}$The maximum of generations300
$pc$The probability of crossover0.8
$pd$The differential multiplier0.5
$pm$The probability of mutation0.2
$ps$The probability of mutation selection0.85
Table 2.  Errors of the proposed MOLRSMD compared with ADM and ALM.
Methods$ErrLR$ $ErrSP$ $ErrT$
MOLRSMD0.01000.19860
ADM0.02990.13311.9135e-17
ALM0.01480.06612.2412e-10
Methods$ErrLR$ $ErrSP$ $ErrT$
MOLRSMD0.01000.19860
ADM0.02990.13311.9135e-17
ALM0.01480.06612.2412e-10
[1]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems and Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

[2]

Simon Arridge, Pascal Fernsel, Andreas Hauptmann. Joint reconstruction and low-rank decomposition for dynamic inverse problems. Inverse Problems and Imaging, 2022, 16 (3) : 483-523. doi: 10.3934/ipi.2021059

[3]

Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems and Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601

[4]

Ke Wei, Jian-Feng Cai, Tony F. Chan, Shingyu Leung. Guarantees of riemannian optimization for low rank matrix completion. Inverse Problems and Imaging, 2020, 14 (2) : 233-265. doi: 10.3934/ipi.2020011

[5]

Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127

[6]

Yun Cai, Song Li. Convergence and stability of iteratively reweighted least squares for low-rank matrix recovery. Inverse Problems and Imaging, 2017, 11 (4) : 643-661. doi: 10.3934/ipi.2017030

[7]

Wei Wan, Weihong Guo, Jun Liu, Haiyang Huang. Non-local blind hyperspectral image super-resolution via 4d sparse tensor factorization and low-rank. Inverse Problems and Imaging, 2020, 14 (2) : 339-361. doi: 10.3934/ipi.2020015

[8]

Zhouchen Lin. A review on low-rank models in data analysis. Big Data & Information Analytics, 2016, 1 (2&3) : 139-161. doi: 10.3934/bdia.2016001

[9]

Simon Foucart, Richard G. Lynch. Recovering low-rank matrices from binary measurements. Inverse Problems and Imaging, 2019, 13 (4) : 703-720. doi: 10.3934/ipi.2019032

[10]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial and Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134

[11]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407

[12]

Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 289-326. doi: 10.3934/naco.2015.5.289

[13]

Changming Song, Yun Wang. Nonlocal latent low rank sparse representation for single image super resolution via self-similarity learning. Inverse Problems and Imaging, 2021, 15 (6) : 1347-1362. doi: 10.3934/ipi.2021017

[14]

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, 2022  doi: 10.3934/jimo.2022045

[15]

Nguyen Van Thoai. Decomposition branch and bound algorithm for optimization problems over efficient sets. Journal of Industrial and Management Optimization, 2008, 4 (4) : 647-660. doi: 10.3934/jimo.2008.4.647

[16]

Haixia Liu, Jian-Feng Cai, Yang Wang. Subspace clustering by (k,k)-sparse matrix factorization. Inverse Problems and Imaging, 2017, 11 (3) : 539-551. doi: 10.3934/ipi.2017025

[17]

Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems and Imaging, 2021, 15 (3) : 519-537. doi: 10.3934/ipi.2021003

[18]

Yitong Guo, Bingo Wing-Kuen Ling. Principal component analysis with drop rank covariance matrix. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2345-2366. doi: 10.3934/jimo.2020072

[19]

Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015

[20]

Chunrong Chen, T. C. Edwin Cheng, Shengji Li, Xiaoqi Yang. Nonlinear augmented Lagrangian for nonconvex multiobjective optimization. Journal of Industrial and Management Optimization, 2011, 7 (1) : 157-174. doi: 10.3934/jimo.2011.7.157

 Impact Factor: 

Metrics

  • PDF downloads (181)
  • HTML views (136)
  • Cited by (0)

Other articles
by authors

[Back to Top]