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

Tao Wu; Yu Lei; Jiao Shi; Maoguo Gong

doi:10.3934/bdia.2017006

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January 2017, 2(1): 23-37. doi: 10.3934/bdia.2017006

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

Tao Wu ^, , Yu Lei , Jiao Shi and Maoguo Gong

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.

Keywords: Matrix decomposition, evolutionary algorithm, multiobjective optimization, low-rank, sparse.

Mathematics Subject Classification: 90C29, 92D15.

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. Google Scholar
[2]	J.-F. Cai, E. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	V. Chandrasekaran, S. Sanghavi, P. A. Parrilo and A. S. Willsky, Rank-sparsity incoherence for matrix decomposition, SIAM Journal on Optimization, 21 (2011), 572-596. doi: 10.1137/090761793. Google Scholar
[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. Google Scholar
[8]	K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, Ltd. , Chichester, 2001. Google Scholar
[9]	K. Deb, A. Pratap, S. 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. Google Scholar
[10]	M. Fazel, H. 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. Google Scholar
[11]	M. Gong, L. Jiao, H. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, MA, 1999. Google Scholar
[15]	C. Qian, Y. Yu and Z.-H. Zhou, Pareto ensemble pruning, in AAAI, (2015), 2935-2941. Google Scholar
[16]	————, Subset selection by pareto optimization, in Advances in Neural Information Processing Systems, (2015), 1774-1782. Google Scholar
[17]	B. Recht, M. 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. Google Scholar
[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. Google Scholar
[19]	J. L. Starck, M. 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. Google Scholar
[20]	J. Yan, J. Liu, Y. Li, Z. 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. Google Scholar
[21]	X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods, Pacific Journal of Optimization, 9 (2013), 167-180. Google Scholar
[22]	C. Zhang, J. Liu, Q. Tian, C. Xu, H. 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. Google Scholar
[23]	Q. Zhang and H. Li, MOEA/D: A multiobjective evolutionary algorithm based on decomposition, IEEE Transactions on Evolutionary Computation, 11 (2007), 712-731. Google Scholar
[24]	M. Zibulevsky and B. A. Pearlmutter, Blind source separation by sparse decomposition in a signal dictionary, Neural Computation, 13 (2001), 863-882. Google Scholar
[25]	E. Zitzler, M. Laumanns and L. Thiele, SPEA2: Improving the strength pareto evolutionary algorithm, in Eurogen, 3242 (2001), 95-100. Google Scholar

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. Google Scholar
[2]	J.-F. Cai, E. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	V. Chandrasekaran, S. Sanghavi, P. A. Parrilo and A. S. Willsky, Rank-sparsity incoherence for matrix decomposition, SIAM Journal on Optimization, 21 (2011), 572-596. doi: 10.1137/090761793. Google Scholar
[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. Google Scholar
[8]	K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, Ltd. , Chichester, 2001. Google Scholar
[9]	K. Deb, A. Pratap, S. 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. Google Scholar
[10]	M. Fazel, H. 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. Google Scholar
[11]	M. Gong, L. Jiao, H. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, MA, 1999. Google Scholar
[15]	C. Qian, Y. Yu and Z.-H. Zhou, Pareto ensemble pruning, in AAAI, (2015), 2935-2941. Google Scholar
[16]	————, Subset selection by pareto optimization, in Advances in Neural Information Processing Systems, (2015), 1774-1782. Google Scholar
[17]	B. Recht, M. 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. Google Scholar
[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. Google Scholar
[19]	J. L. Starck, M. 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. Google Scholar
[20]	J. Yan, J. Liu, Y. Li, Z. 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. Google Scholar
[21]	X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods, Pacific Journal of Optimization, 9 (2013), 167-180. Google Scholar
[22]	C. Zhang, J. Liu, Q. Tian, C. Xu, H. 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. Google Scholar
[23]	Q. Zhang and H. Li, MOEA/D: A multiobjective evolutionary algorithm based on decomposition, IEEE Transactions on Evolutionary Computation, 11 (2007), 712-731. Google Scholar
[24]	M. Zibulevsky and B. A. Pearlmutter, Blind source separation by sparse decomposition in a signal dictionary, Neural Computation, 13 (2001), 863-882. Google Scholar
[25]	E. Zitzler, M. Laumanns and L. Thiele, SPEA2: Improving the strength pareto evolutionary algorithm, in Eurogen, 3242 (2001), 95-100. Google Scholar

Figure 1. The distributions of nondominated solutions, dominated solutions and Pareto front in the objective space of the two objectives problem.

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Figure 2. Error of sparse component recovering by the ADM method for LRSMD with different choices of the parameter $\lambda$

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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.

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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, $\sigma=15$.

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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 $\sigma =15$.

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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.

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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.

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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.

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Table 1. The Parameters in the MOLRSMD used in the Experiments

Parameter	Meaning	Value
$N$	The number of subproblems	100
$T$	The number of neighbors	20
$t_{max}$	The maximum of generations	300
$pc$	The probability of crossover	0.8
$pd$	The differential multiplier	0.5
$pm$	The probability of mutation	0.2
$ps$	The probability of mutation selection	0.85

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Parameter	Meaning	Value
$N$	The number of subproblems	100
$T$	The number of neighbors	20
$t_{max}$	The maximum of generations	300
$pc$	The probability of crossover	0.8
$pd$	The differential multiplier	0.5
$pm$	The probability of mutation	0.2
$ps$	The probability of mutation selection	0.85

Table 2. Errors of the proposed MOLRSMD compared with ADM and ALM.

Methods	$ErrLR$	$ErrSP$	$ErrT$
MOLRSMD	0.0100	0.1986	0
ADM	0.0299	0.1331	1.9135e-17
ALM	0.0148	0.0661	2.2412e-10

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Methods	$ErrLR$	$ErrSP$	$ErrT$
MOLRSMD	0.0100	0.1986	0
ADM	0.0299	0.1331	1.9135e-17
ALM	0.0148	0.0661	2.2412e-10

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