October  2016, 12(4): 1507-1519. doi: 10.3934/jimo.2016.12.1507

On linear convergence of projected gradient method for a class of affine rank minimization problems

1. 

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

2. 

Business School and China Academy of Corporate Governance, Nankai University, Tianjin 300071, China

Received  June 2015 Revised  October 2015 Published  January 2016

The affine rank minimization problem is to find a low-rank matrix satisfying a set of linear equations, which includes the well-known matrix completion problem as a special case and draws much attention in recent years. In this paper, a new model for affine rank minimization problem is proposed. The new model not only enhances the robustness of affine rank minimization problem, but also leads to high nonconvexity. We show that if the classical projected gradient method is applied to solve our new model, the linear convergence rate can be established under some conditions. Some preliminary experiments have been conducted to show the efficiency and effectiveness of our method.
Citation: 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
References:
[1]

A. Beck and M. Teboulle, A linearly convergent algorithm for solving a class of nonconvex/affine feasibility problems,, In Fixed-Point Algorithms for Inverse Problems in Science and Engineering, 49 (2011), 33.  doi: 10.1007/978-1-4419-9569-8_3.  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.  doi: 10.1137/080738970.  Google Scholar

[3]

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

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

[5]

M. Fazel, H. Hindi and S. P. Boyd, A rank minimization heuristic with application to minimum order system approximation,, In American Control Conference, 6 (2001), 4734.  doi: 10.1109/ACC.2001.945730.  Google Scholar

[6]

D. Goldfarb and S. Ma, Convergence of fixed-point continuation algorithms for matrix rank minimization,, Foundations of Computational Mathematics, 11 (2011), 183.  doi: 10.1007/s10208-011-9084-6.  Google Scholar

[7]

P. J. Huber, Robust statistics,, Springer, (2011).   Google Scholar

[8]

P. Jain, R. Meka and I. S. Dhillon, Guaranteed rank minimization via singular value projection,, In NIPS, 23 (2010), 937.   Google Scholar

[9]

L. Kong, J. Sun and N. Xiu, S-semigoodness for low-rank semidefinite matrix recovery,, Pacific Journal of Optimization, 10 (2014), 73.   Google Scholar

[10]

L. Kong, J. Sun, J. Tao and N. Xiu, Sparse recovery on Euclidean Jordan algebras,, Linear Algebra and Applications, 465 (2015), 65.  doi: 10.1016/j.laa.2014.09.018.  Google Scholar

[11]

Z. Lin, M. Chen and Y. Ma, The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices,, , (2010).   Google Scholar

[12]

N. Linial, E. London and Y. Rabinovich, The geometry of graphs and some of its algorithmic applications,, Combinatorica, 15 (1995), 215.  doi: 10.1007/BF01200757.  Google Scholar

[13]

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.  doi: 10.1137/090755436.  Google Scholar

[14]

N. Natarajan and I. S. Dhillon, Inductive matrix completion for predicting gene-disease associations,, Bioinformatics, 30 (2014).  doi: 10.1093/bioinformatics/btu269.  Google Scholar

[15]

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.  doi: 10.1137/070697835.  Google Scholar

[16]

K.-C. Toh and S. Yun, An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems,, Pacific Journal of Optimization, 6 (2010), 615.   Google Scholar

[17]

X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods,, Pac. J. Optim., 9 (2013), 167.   Google Scholar

[18]

S. Zhang, J. Ang and J. Sun, An alternating direction method for solving convex nonlinear semidefinite programming problems,, Optimization, 62 (2013), 527.  doi: 10.1080/02331934.2011.611883.  Google Scholar

show all references

References:
[1]

A. Beck and M. Teboulle, A linearly convergent algorithm for solving a class of nonconvex/affine feasibility problems,, In Fixed-Point Algorithms for Inverse Problems in Science and Engineering, 49 (2011), 33.  doi: 10.1007/978-1-4419-9569-8_3.  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.  doi: 10.1137/080738970.  Google Scholar

[3]

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

[4]

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

[5]

M. Fazel, H. Hindi and S. P. Boyd, A rank minimization heuristic with application to minimum order system approximation,, In American Control Conference, 6 (2001), 4734.  doi: 10.1109/ACC.2001.945730.  Google Scholar

[6]

D. Goldfarb and S. Ma, Convergence of fixed-point continuation algorithms for matrix rank minimization,, Foundations of Computational Mathematics, 11 (2011), 183.  doi: 10.1007/s10208-011-9084-6.  Google Scholar

[7]

P. J. Huber, Robust statistics,, Springer, (2011).   Google Scholar

[8]

P. Jain, R. Meka and I. S. Dhillon, Guaranteed rank minimization via singular value projection,, In NIPS, 23 (2010), 937.   Google Scholar

[9]

L. Kong, J. Sun and N. Xiu, S-semigoodness for low-rank semidefinite matrix recovery,, Pacific Journal of Optimization, 10 (2014), 73.   Google Scholar

[10]

L. Kong, J. Sun, J. Tao and N. Xiu, Sparse recovery on Euclidean Jordan algebras,, Linear Algebra and Applications, 465 (2015), 65.  doi: 10.1016/j.laa.2014.09.018.  Google Scholar

[11]

Z. Lin, M. Chen and Y. Ma, The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices,, , (2010).   Google Scholar

[12]

N. Linial, E. London and Y. Rabinovich, The geometry of graphs and some of its algorithmic applications,, Combinatorica, 15 (1995), 215.  doi: 10.1007/BF01200757.  Google Scholar

[13]

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.  doi: 10.1137/090755436.  Google Scholar

[14]

N. Natarajan and I. S. Dhillon, Inductive matrix completion for predicting gene-disease associations,, Bioinformatics, 30 (2014).  doi: 10.1093/bioinformatics/btu269.  Google Scholar

[15]

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.  doi: 10.1137/070697835.  Google Scholar

[16]

K.-C. Toh and S. Yun, An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems,, Pacific Journal of Optimization, 6 (2010), 615.   Google Scholar

[17]

X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods,, Pac. J. Optim., 9 (2013), 167.   Google Scholar

[18]

S. Zhang, J. Ang and J. Sun, An alternating direction method for solving convex nonlinear semidefinite programming problems,, Optimization, 62 (2013), 527.  doi: 10.1080/02331934.2011.611883.  Google Scholar

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