-
Previous Article
Indirect methods for fuel-minimal rendezvous with a large population of temporarily captured orbiters
- NACO Home
- This Issue
-
Next Article
Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm
Homotopy method for matrix rank minimization based on the matrix hard thresholding method
1. | Ruijie Networks Co., Ltd, Fuzhou 350108, China |
2. | Center for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou 350108, China |
Based on the matrix hard thresholding method, a homotopy method is proposed for solving the matrix rank minimization problem. This method iteratively solves a series of regularization subproblems, whose solutions are given in closed form by the matrix hard thresholding operator. Under some mild assumptions, convergence of the proposed method is proved. The proposed method does not depend on a prior knowledge of exact rank value. Numerical experiments demonstrate that the proposed homotopy method weakens the affection of the choice of the regularization parameter, and is more efficient and effective than the existing sate-of-the-art methods.
References:
[1] |
J. D. Blanchard, J. Tanner and K. Wei,
CGIHT: Conjugate gradient iterative hard thresholding for compressed sensing and matrix completion, Information and Inference: A Journal of the IMA, 4 (2014), 289-327.
doi: 10.1093/imaiai/iav011. |
[2] |
T. Blumensath and M. E. Davies,
Iterative thresholding for sparse approximations, Journal of Constructive Approximation, 14 (2008), 629-654.
doi: 10.1007/s00041-008-9035-z. |
[3] |
T. Blumensath and M. E. Davies, Normalised itertive hard thresholding: guaranteed stability and performance, IEEE Journal of Selected Topics in Signal Processing, 4 (2010), 298-309. Google Scholar |
[4] |
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. |
[5] |
E. J. Candès and T. Tao,
The power of convex relaxation: Near-optimal matrix completion, IEEE Transactions on Information Theory, 56 (2009), 2053-1080.
doi: 10.1109/TIT.2010.2044061. |
[6] |
M. Fazel, H. Hindi and S. Boyd, Rank minimization and applications in system theory, Proceedings of the American Control Conference, 4 (2004), 3273-3278. Google Scholar |
[7] |
M. Fazel, T. K. 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. |
[8] |
D. Goldfarb and S. Ma,
Convergence of fixed-point continuation algorithms for matrix rank minimization, Foundations of Computational Mathematics, 11 (2011), 183-210.
doi: 10.1007/s10208-011-9084-6. |
[9] |
J. P. Haldar and D. Hernando, Rank-constrained solutions to linear matrix equations using powerfactorization, IEEE Signal Processing Letters, 16 (2009), 584-587. Google Scholar |
[10] |
N. J. A. Harvey, D. R. Karger and S. Yekhanin, The complexity of matrix completion, Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithm, (2006), 1103–1111.
doi: 10.1145/1109557.1109679. |
[11] |
A. Kyrillidis and V. Cevher, Martix ALPS: Accelerated low rank and sparse matrix reconstruction, Technical Report, 2012. Google Scholar |
[12] |
A. Kyrillidis and V. Cevher,
Martix recips for hard thresholding methods, Journal of Mathematical Imaging and Vision, 48 (2014), 235-265.
doi: 10.1007/s10851-013-0434-7. |
[13] |
Y. Liu, J. Tao, H. Zhang, X. Xiu and L. Kong, Fused LASSO penalized least absolute deviation estimator for high dimensional linear regression, Numerical Algebra, Control & Optimization, 8 (2018), 97–117.
doi: 10.3934/naco.2018006. |
[14] |
Z. Liu and L. Vandenberghe,
Interior-point method for nuclear norm approximation with application to system identification, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 1235-1256.
doi: 10.1137/090755436. |
[15] |
C. Lu, J. Tang, S. Yan and Z. Lin, Generalized nonconvex nonsmooth low-rank minimization, IEEE Conference on Computer Vision and Pattern Recognition, 2014. Google Scholar |
[16] |
Z. Lu,
Iterative hard thresholding methods for $l_0$ regularized convex cone programming, Mathematical Programming, 147 (2014), 125-154.
doi: 10.1007/s10107-013-0714-4. |
[17] |
Z. Lu and Y. Zhang,
Penalty decomposition methods for rank minimization, Optimization Methods and Software, 30 (2015), 531-558.
doi: 10.1080/10556788.2014.936438. |
[18] |
S. Ma, D. Goldfarb and L. Chen,
Fixed point and bregman iterative methods for matrix rank minimization, Mathematical Programming, 128 (2011), 321-353.
doi: 10.1007/s10107-009-0306-5. |
[19] |
K. Mohan and M. Fazel, Reweighted nuclear norm minimization with application to system identification, Proceedings of the American Control Conference, 2010. Google Scholar |
[20] |
K. Mohan and M. Fazel,
Iterative reweighted algorithms for matrix rank minimization, Journal of Machine Learning Research, 13 (2012), 3441-3473.
|
[21] |
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. |
[22] |
J. Tanner and K. Wei, Normalized iterative hard thresholding for matrix completion, SIAM Journal on Scientific Computing, 35 (2013), S104–S125.
doi: 10.1137/120876459. |
[23] |
Z. Wen, W. Yin and Y. Zhang,
Solving a low-rank factorization model for matrix completion by a non-linear successive over-relaxation algorithm, Mathematical Programming Computation, 4 (2012), 333-361.
doi: 10.1007/s12532-012-0044-1. |
[24] |
Z. Weng and X. Wang, Low-rank matrix completion for array signal processing, IEEE International Conference on Speech and Signal Processing, (2012), 2697–2700. Google Scholar |
show all references
References:
[1] |
J. D. Blanchard, J. Tanner and K. Wei,
CGIHT: Conjugate gradient iterative hard thresholding for compressed sensing and matrix completion, Information and Inference: A Journal of the IMA, 4 (2014), 289-327.
doi: 10.1093/imaiai/iav011. |
[2] |
T. Blumensath and M. E. Davies,
Iterative thresholding for sparse approximations, Journal of Constructive Approximation, 14 (2008), 629-654.
doi: 10.1007/s00041-008-9035-z. |
[3] |
T. Blumensath and M. E. Davies, Normalised itertive hard thresholding: guaranteed stability and performance, IEEE Journal of Selected Topics in Signal Processing, 4 (2010), 298-309. Google Scholar |
[4] |
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. |
[5] |
E. J. Candès and T. Tao,
The power of convex relaxation: Near-optimal matrix completion, IEEE Transactions on Information Theory, 56 (2009), 2053-1080.
doi: 10.1109/TIT.2010.2044061. |
[6] |
M. Fazel, H. Hindi and S. Boyd, Rank minimization and applications in system theory, Proceedings of the American Control Conference, 4 (2004), 3273-3278. Google Scholar |
[7] |
M. Fazel, T. K. 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. |
[8] |
D. Goldfarb and S. Ma,
Convergence of fixed-point continuation algorithms for matrix rank minimization, Foundations of Computational Mathematics, 11 (2011), 183-210.
doi: 10.1007/s10208-011-9084-6. |
[9] |
J. P. Haldar and D. Hernando, Rank-constrained solutions to linear matrix equations using powerfactorization, IEEE Signal Processing Letters, 16 (2009), 584-587. Google Scholar |
[10] |
N. J. A. Harvey, D. R. Karger and S. Yekhanin, The complexity of matrix completion, Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithm, (2006), 1103–1111.
doi: 10.1145/1109557.1109679. |
[11] |
A. Kyrillidis and V. Cevher, Martix ALPS: Accelerated low rank and sparse matrix reconstruction, Technical Report, 2012. Google Scholar |
[12] |
A. Kyrillidis and V. Cevher,
Martix recips for hard thresholding methods, Journal of Mathematical Imaging and Vision, 48 (2014), 235-265.
doi: 10.1007/s10851-013-0434-7. |
[13] |
Y. Liu, J. Tao, H. Zhang, X. Xiu and L. Kong, Fused LASSO penalized least absolute deviation estimator for high dimensional linear regression, Numerical Algebra, Control & Optimization, 8 (2018), 97–117.
doi: 10.3934/naco.2018006. |
[14] |
Z. Liu and L. Vandenberghe,
Interior-point method for nuclear norm approximation with application to system identification, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 1235-1256.
doi: 10.1137/090755436. |
[15] |
C. Lu, J. Tang, S. Yan and Z. Lin, Generalized nonconvex nonsmooth low-rank minimization, IEEE Conference on Computer Vision and Pattern Recognition, 2014. Google Scholar |
[16] |
Z. Lu,
Iterative hard thresholding methods for $l_0$ regularized convex cone programming, Mathematical Programming, 147 (2014), 125-154.
doi: 10.1007/s10107-013-0714-4. |
[17] |
Z. Lu and Y. Zhang,
Penalty decomposition methods for rank minimization, Optimization Methods and Software, 30 (2015), 531-558.
doi: 10.1080/10556788.2014.936438. |
[18] |
S. Ma, D. Goldfarb and L. Chen,
Fixed point and bregman iterative methods for matrix rank minimization, Mathematical Programming, 128 (2011), 321-353.
doi: 10.1007/s10107-009-0306-5. |
[19] |
K. Mohan and M. Fazel, Reweighted nuclear norm minimization with application to system identification, Proceedings of the American Control Conference, 2010. Google Scholar |
[20] |
K. Mohan and M. Fazel,
Iterative reweighted algorithms for matrix rank minimization, Journal of Machine Learning Research, 13 (2012), 3441-3473.
|
[21] |
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. |
[22] |
J. Tanner and K. Wei, Normalized iterative hard thresholding for matrix completion, SIAM Journal on Scientific Computing, 35 (2013), S104–S125.
doi: 10.1137/120876459. |
[23] |
Z. Wen, W. Yin and Y. Zhang,
Solving a low-rank factorization model for matrix completion by a non-linear successive over-relaxation algorithm, Mathematical Programming Computation, 4 (2012), 333-361.
doi: 10.1007/s12532-012-0044-1. |
[24] |
Z. Weng and X. Wang, Low-rank matrix completion for array signal processing, IEEE International Conference on Speech and Signal Processing, (2012), 2697–2700. Google Scholar |
r=5 | r=10 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 50 | 6.71 | 5 | 9.22E-06 | 50 | 13.61 | 10 | 2.66E-05 |
PD | 50 | 24.08 | 5 | 7.70E-05 | 50 | 33.65 | 10 | 8.96E-05 |
HIHT | 50 | 19.37 | 5 | 5.33E-05 | 50 | 16.12 | 10 | 4.78E-05 |
r=30 | r=50 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 0 | 43.74 | 16.6 | 3.80E-02 | - | - | - | - |
PD | 50 | 39.55 | 30 | 7.42E-05 | 50 | 53.83 | 50 | 1.00E-04 |
HIHT | 50 | 36.34 | 30 | 3.52E-05 | 50 | 47.36 | 50 | 9.93E-06 |
r=70 | r=90 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | - | - | - | - | - | - | - | - |
PD | 50 | 87.17 | 70 | 1.51E-04 | 50 | 294.05 | 90 | 2.38E-04 |
HIHT | 50 | 70.3 | 70 | 6.31E-05 | 50 | 82.79 | 90 | 2.02E-04 |
r=5 | r=10 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 50 | 6.71 | 5 | 9.22E-06 | 50 | 13.61 | 10 | 2.66E-05 |
PD | 50 | 24.08 | 5 | 7.70E-05 | 50 | 33.65 | 10 | 8.96E-05 |
HIHT | 50 | 19.37 | 5 | 5.33E-05 | 50 | 16.12 | 10 | 4.78E-05 |
r=30 | r=50 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 0 | 43.74 | 16.6 | 3.80E-02 | - | - | - | - |
PD | 50 | 39.55 | 30 | 7.42E-05 | 50 | 53.83 | 50 | 1.00E-04 |
HIHT | 50 | 36.34 | 30 | 3.52E-05 | 50 | 47.36 | 50 | 9.93E-06 |
r=70 | r=90 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | - | - | - | - | - | - | - | - |
PD | 50 | 87.17 | 70 | 1.51E-04 | 50 | 294.05 | 90 | 2.38E-04 |
HIHT | 50 | 70.3 | 70 | 6.31E-05 | 50 | 82.79 | 90 | 2.02E-04 |
IHT | HIHT | |||||||
NS | time | rank | rel.err. | NS | time | rank | rel.err. | |
50 | 4 | 122.43 | 38.56 | 1.29E-01 | 50 | 35.33 | 40 | 1.17E-05 |
20 | 50 | 18.83 | 40.00 | 5.82E-05 | 50 | 19.86 | 40 | 1.17E-05 |
10 | 47 | 23.69 | 40.06 | 2.52E-03 | 50 | 23.25 | 40 | 1.38E-05 |
5 | 0 | 52.92 | 227.10 | 6.33E-01 | 0 | 45.41 | 227.72 | 6.34E-01 |
IHT | HIHT | |||||||
NS | time | rank | rel.err. | NS | time | rank | rel.err. | |
50 | 4 | 122.43 | 38.56 | 1.29E-01 | 50 | 35.33 | 40 | 1.17E-05 |
20 | 50 | 18.83 | 40.00 | 5.82E-05 | 50 | 19.86 | 40 | 1.17E-05 |
10 | 47 | 23.69 | 40.06 | 2.52E-03 | 50 | 23.25 | 40 | 1.38E-05 |
5 | 0 | 52.92 | 227.10 | 6.33E-01 | 0 | 45.41 | 227.72 | 6.34E-01 |
HIHT | ||||
NS | time | rank | rel.err. | |
5 | 50 | 16.72 | 40 | 4.35E-05 |
10 | 50 | 18.13 | 40 | 2.09E-05 |
20 | 50 | 18.97 | 40 | 1.35E-05 |
30 | 50 | 19.29 | 40 | 1.16E-05 |
40 | 50 | 19.27 | 40 | 1.16E-05 |
50 | 50 | 19.24 | 40 | 1.17E-05 |
60 | 50 | 19.72 | 40 | 1.01E-05 |
70 | 50 | 19.71 | 40 | 1.02E-05 |
HIHT | ||||
NS | time | rank | rel.err. | |
5 | 50 | 16.72 | 40 | 4.35E-05 |
10 | 50 | 18.13 | 40 | 2.09E-05 |
20 | 50 | 18.97 | 40 | 1.35E-05 |
30 | 50 | 19.29 | 40 | 1.16E-05 |
40 | 50 | 19.27 | 40 | 1.16E-05 |
50 | 50 | 19.24 | 40 | 1.17E-05 |
60 | 50 | 19.72 | 40 | 1.01E-05 |
70 | 50 | 19.71 | 40 | 1.02E-05 |
r=5 | r=10 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 50 | 4.69 | 5 | 1.24E-06 | 50 | 5.19 | 10 | 2.90E-06 |
PD | 50 | 95.92 | 5 | 5.92E-05 | 50 | 102.83 | 10 | 8.54E-05 |
HIHT | 50 | 10.91 | 5 | 7.49E-05 | 50 | 12.54 | 10 | 7.02E-05 |
r=30 | r=50 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 50 | 5.78 | 30 | 1.44E-05 | 50 | 255.32 | 50 | 1.61E-04 |
PD | 50 | 102.38 | 30 | 9.40E-05 | 50 | 106.19 | 50 | 1.50E-04 |
HIHT | 50 | 18.57 | 30 | 6.63E-05 | 50 | 24.13 | 50 | 2.67E-05 |
r=70 | r=90 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 50 | 279.55 | 70 | 2.23E-04 | 50 | 332.90 | 90 | 2.71E-04 |
PD | 50 | 164.06 | 70 | 1.78E-04 | 50 | 471.48 | 90 | 2.51E-04 |
HIHT | 50 | 38.23 | 70 | 5.01E-05 | 50 | 84.32 | 90 | 2.08E-04 |
r=5 | r=10 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 50 | 4.69 | 5 | 1.24E-06 | 50 | 5.19 | 10 | 2.90E-06 |
PD | 50 | 95.92 | 5 | 5.92E-05 | 50 | 102.83 | 10 | 8.54E-05 |
HIHT | 50 | 10.91 | 5 | 7.49E-05 | 50 | 12.54 | 10 | 7.02E-05 |
r=30 | r=50 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 50 | 5.78 | 30 | 1.44E-05 | 50 | 255.32 | 50 | 1.61E-04 |
PD | 50 | 102.38 | 30 | 9.40E-05 | 50 | 106.19 | 50 | 1.50E-04 |
HIHT | 50 | 18.57 | 30 | 6.63E-05 | 50 | 24.13 | 50 | 2.67E-05 |
r=70 | r=90 | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 50 | 279.55 | 70 | 2.23E-04 | 50 | 332.90 | 90 | 2.71E-04 |
PD | 50 | 164.06 | 70 | 1.78E-04 | 50 | 471.48 | 90 | 2.51E-04 |
HIHT | 50 | 38.23 | 70 | 5.01E-05 | 50 | 84.32 | 90 | 2.08E-04 |
Gaussian matrix | uniform matrix | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 50 | 284.36 | 200 | 8.28E-05 | 0 | 1887.44 | 1.06 | 2.35E-02 |
PD | 50 | 13232.85 | 200 | 1.02E-04 | 50 | 5976.50 | 200 | 9.89E-05 |
HIHT | 50 | 4427.63 | 200 | 1.63E-05 | 50 | 7637.74 | 200 | 1.70E-05 |
Gaussian matrix | uniform matrix | |||||||
Alg. | NS | time | rank | rel.err. | NS | time | rank | rel.err. |
FPCA | 50 | 284.36 | 200 | 8.28E-05 | 0 | 1887.44 | 1.06 | 2.35E-02 |
PD | 50 | 13232.85 | 200 | 1.02E-04 | 50 | 5976.50 | 200 | 9.89E-05 |
HIHT | 50 | 4427.63 | 200 | 1.63E-05 | 50 | 7637.74 | 200 | 1.70E-05 |
[1] |
Yuan Shen, Xin Liu. An alternating minimization method for matrix completion problems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020103 |
[2] |
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 |
[3] |
Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems & Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601 |
[4] |
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 |
[5] |
Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507 |
[6] |
Yi Yang, Jianwei Ma, Stanley Osher. Seismic data reconstruction via matrix completion. Inverse Problems & Imaging, 2013, 7 (4) : 1379-1392. doi: 10.3934/ipi.2013.7.1379 |
[7] |
El-Sayed M.E. Mostafa. A nonlinear conjugate gradient method for a special class of matrix optimization problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 883-903. doi: 10.3934/jimo.2014.10.883 |
[8] |
Lingling Lv, Zhe Zhang, Lei Zhang, Weishu Wang. An iterative algorithm for periodic sylvester matrix equations. Journal of Industrial & Management Optimization, 2018, 14 (1) : 413-425. doi: 10.3934/jimo.2017053 |
[9] |
Jie Huang, Marco Donatelli, Raymond H. Chan. Nonstationary iterated thresholding algorithms for image deblurring. Inverse Problems & Imaging, 2013, 7 (3) : 717-736. doi: 10.3934/ipi.2013.7.717 |
[10] |
Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357 |
[11] |
Victor Kozyakin. Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 143-158. doi: 10.3934/dcdsb.2010.14.143 |
[12] |
Ai-Guo Wu, Ying Zhang, Hui-Jie Sun. Parametric Smith iterative algorithms for discrete Lyapunov matrix equations. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019093 |
[13] |
Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 289-326. doi: 10.3934/naco.2015.5.289 |
[14] |
Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127 |
[15] |
Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201 |
[16] |
Soumya Kundu, Soumitro Banerjee, Damian Giaouris. Vanishing singularity in hard impacting systems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 319-332. doi: 10.3934/dcdsb.2011.16.319 |
[17] |
Frédéric Lebon, Raffaella Rizzoni. Modeling a hard, thin curvilinear interface. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1569-1586. doi: 10.3934/dcdss.2013.6.1569 |
[18] |
Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic & Related Models, 2012, 5 (3) : 459-484. doi: 10.3934/krm.2012.5.459 |
[19] |
Yuhong Dai, Nobuo Yamashita. Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 61-69. doi: 10.3934/naco.2011.1.61 |
[20] |
Qingshan You, Qun Wan, Yipeng Liu. A short note on strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2013, 7 (1) : 305-306. doi: 10.3934/ipi.2013.7.305 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]