May  2015, 9(2): 601-624. doi: 10.3934/ipi.2015.9.601

Parallel matrix factorization for low-rank tensor completion

1. 

Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005

2. 

School of Mathematical Sciences, Dalian University of Technology, Dalian, China, China

3. 

Department of Mathematics, University of California, Los Angeles, CA 90095-1555

Received  December 2013 Revised  October 2014 Published  March 2015

Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data reconstruction, and so on. We propose a new model to recover a low-rank tensor by simultaneously performing low-rank matrix factorizations to the all-mode matricizations of the underlying tensor. An alternating minimization algorithm is applied to solve the model, along with two adaptive rank-adjusting strategies when the exact rank is not known.
    Phase transition plots reveal that our algorithm can recover a variety of synthetic low-rank tensors from significantly fewer samples than the compared methods, which include a matrix completion method applied to tensor recovery and two state-of-the-art tensor completion methods. Further tests on real-world data show similar advantages. Although our model is non-convex, our algorithm performs consistently throughout the tests and gives better results than the compared methods, some of which are based on convex models. In addition, subsequence convergence of our algorithm can be established in the sense that any limit point of the iterates satisfies the KKT condtions.
Citation: 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
References:
[1]

J. Cai, E. Candes and Z. Shen, A singular value thresholding algorithm for matrix completion,, SIAM J. Optim., 20 (2010), 1956. doi: 10.1137/080738970.

[2]

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

[3]

C. Chen, B. He and X. Yuan, Matrix completion via an alternating direction method,, IMA Journal of Numerical Analysis, 32 (2012), 227. doi: 10.1093/imanum/drq039.

[4]

S. Gandy, B. Recht and I. Yamada, Tensor completion and low-n-rank tensor recovery via convex optimization,, Inverse Problems, 27 (2011), 1. doi: 10.1088/0266-5611/27/2/025010.

[5]

L. Grippo and M. Sciandrone, On the convergence of the block nonlinear Gauss-Seidel method under convex constraints,, Oper. Res. Lett., 26 (2000), 127. doi: 10.1016/S0167-6377(99)00074-7.

[6]

B. Jiang, S. Ma and S. Zhang, Tensor principal component analysis via convex optimization,, Mathematical Programming, (2014), 1. doi: 10.1007/s10107-014-0774-0.

[7]

H. A. L. Kiers, Towards a standardized notation and terminology in multiway analysis,, Journal of Chemometrics, 14 (2000), 105. doi: 10.1002/1099-128X(200005/06)14:3<105::AID-CEM582>3.0.CO;2-I.

[8]

M. Kilmer, K. Braman, N. Hao and R. Hoover, Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging,, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 148. doi: 10.1137/110837711.

[9]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications,, SIAM review, 51 (2009), 455. doi: 10.1137/07070111X.

[10]

T. G. Kolda, B. W. Bader and J. P. Kenny, Higher-order web link analysis using multilinear algebra,, in Data Mining, (2005). doi: 10.1109/ICDM.2005.77.

[11]

N. Kreimer and M. D. Sacchi, A tensor higher-order singular value decomposition for prestack seismic data noise reduction and interpolation,, Geophysics, 77 (2012). doi: 10.1190/geo2011-0399.1.

[12]

D. Kressner, M. Steinlechner and B. Vandereycken, Low-rank tensor completion by riemannian optimization,, BIT, 54 (2014), 447. doi: 10.1007/s10543-013-0455-z.

[13]

M. Lai, Y. Xu and W. Yin, Improved iteratively reweighted least squares for unconstrained smoothed $l_q$ minimization,, SIAM Journal on Numerical Analysis, 51 (2013), 927. doi: 10.1137/110840364.

[14]

N. Li and B. Li, Tensor completion for on-board compression of hyperspectral images,, in 2010 17th IEEE International Conference on Image Processing (ICIP), (2010), 517. doi: 10.1109/ICIP.2010.5651225.

[15]

Q. Ling, Y. Xu, W. Yin and Z. Wen, Decentralized low-rank matrix completion,, in 2012 IEEE International Conference on Acoustics, (2012), 2925. doi: 10.1109/ICASSP.2012.6288528.

[16]

J. Liu, P. Musialski, P. Wonka and J. Ye, Tensor completion for estimating missing values in visual data,, IEEE Transactions on Pattern Analysis and Machine Intelligence, (2012), 208. doi: 10.1109/TPAMI.2012.39.

[17]

S. Ma, D. Goldfarb and L. Chen, Fixed point and bregman iterative methods for matrix rank minimization,, Mathematical Programming, 128 (2011), 321. doi: 10.1007/s10107-009-0306-5.

[18]

C. Mu, B. Huang, J. Wright and D. Goldfarb, Square deal: Lower bounds and improved relaxations for tensor recovery,, preprint, (2013).

[19]

K. A. Patwardhan, G. Sapiro and M. Bertalmío, Video inpainting under constrained camera motion,, IEEE Transactions on Image Processing, 16 (2007), 545. doi: 10.1109/TIP.2006.888343.

[20]

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.

[21]

B. Romera-Paredes and M. Pontil, A new convex relaxation for tensor completion,, preprint, (2013).

[22]

A. C. Sauve, A. O. Hero III, W. Leslie Rogers, S. J. Wilderman and N. H. Clinthorne, 3d image reconstruction for a compton spect camera model,, Nuclear Science, 46 (1999), 2075. doi: 10.1109/23.819285.

[23]

J. Sun, H. Zeng, H. Liu, Y. Lu and Z. Chen, Cubesvd: a novel approach to personalized web search,, in Proceedings of the 14th international conference on World Wide Web, (2005), 382. doi: 10.1145/1060745.1060803.

[24]

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.

[25]

P. Tseng, Convergence of a block coordinate descent method for nondifferentiable minimization,, Journal of Optimization Theory and Applications, 109 (2001), 475. doi: 10.1023/A:1017501703105.

[26]

L. R. Tucker, Some mathematical notes on three-mode factor analysis,, Psychometrika, 31 (1966), 279. doi: 10.1007/BF02289464.

[27]

Z. Wen, W. Yin and Y. Zhang, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm,, Mathematical Programming Computation, 4 (2012), 333. doi: 10.1007/s12532-012-0044-1.

[28]

Z. Xing, M. Zhou, A. Castrodad, G. Sapiro and L. Carin, Dictionary learning for noisy and incomplete hyperspectral images,, SIAM Journal on Imaging Sciences, 5 (2012), 33. doi: 10.1137/110837486.

[29]

Y. Xu and W. Yin, A block coordinate descent method for regularized multi-convex optimization with applications to nonnegative tensor factorization and completion,, SIAM Journal on Imaging Sciences, 6 (2013), 1758. doi: 10.1137/120887795.

[30]

Y. Xu and W. Yin, A globally convergent algorithm for nonconvex optimization based on block coordinate update,, , (2014).

[31]

Y. Xu, W. Yin, Z. Wen and Y. Zhang, An alternating direction algorithm for matrix completion with nonnegative factors,, Journal of Frontiers of Mathematics in China, 7 (2011), 365. doi: 10.1007/s11464-012-0194-5.

[32]

Z. Zhang, G. Ely, S. Aeron, N. Hao and M. Kilmer, Novel factorization strategies for higher order tensors: Implications for compression and recovery of multi-linear data,, preprint, (2013).

show all references

References:
[1]

J. Cai, E. Candes and Z. Shen, A singular value thresholding algorithm for matrix completion,, SIAM J. Optim., 20 (2010), 1956. doi: 10.1137/080738970.

[2]

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

[3]

C. Chen, B. He and X. Yuan, Matrix completion via an alternating direction method,, IMA Journal of Numerical Analysis, 32 (2012), 227. doi: 10.1093/imanum/drq039.

[4]

S. Gandy, B. Recht and I. Yamada, Tensor completion and low-n-rank tensor recovery via convex optimization,, Inverse Problems, 27 (2011), 1. doi: 10.1088/0266-5611/27/2/025010.

[5]

L. Grippo and M. Sciandrone, On the convergence of the block nonlinear Gauss-Seidel method under convex constraints,, Oper. Res. Lett., 26 (2000), 127. doi: 10.1016/S0167-6377(99)00074-7.

[6]

B. Jiang, S. Ma and S. Zhang, Tensor principal component analysis via convex optimization,, Mathematical Programming, (2014), 1. doi: 10.1007/s10107-014-0774-0.

[7]

H. A. L. Kiers, Towards a standardized notation and terminology in multiway analysis,, Journal of Chemometrics, 14 (2000), 105. doi: 10.1002/1099-128X(200005/06)14:3<105::AID-CEM582>3.0.CO;2-I.

[8]

M. Kilmer, K. Braman, N. Hao and R. Hoover, Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging,, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 148. doi: 10.1137/110837711.

[9]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications,, SIAM review, 51 (2009), 455. doi: 10.1137/07070111X.

[10]

T. G. Kolda, B. W. Bader and J. P. Kenny, Higher-order web link analysis using multilinear algebra,, in Data Mining, (2005). doi: 10.1109/ICDM.2005.77.

[11]

N. Kreimer and M. D. Sacchi, A tensor higher-order singular value decomposition for prestack seismic data noise reduction and interpolation,, Geophysics, 77 (2012). doi: 10.1190/geo2011-0399.1.

[12]

D. Kressner, M. Steinlechner and B. Vandereycken, Low-rank tensor completion by riemannian optimization,, BIT, 54 (2014), 447. doi: 10.1007/s10543-013-0455-z.

[13]

M. Lai, Y. Xu and W. Yin, Improved iteratively reweighted least squares for unconstrained smoothed $l_q$ minimization,, SIAM Journal on Numerical Analysis, 51 (2013), 927. doi: 10.1137/110840364.

[14]

N. Li and B. Li, Tensor completion for on-board compression of hyperspectral images,, in 2010 17th IEEE International Conference on Image Processing (ICIP), (2010), 517. doi: 10.1109/ICIP.2010.5651225.

[15]

Q. Ling, Y. Xu, W. Yin and Z. Wen, Decentralized low-rank matrix completion,, in 2012 IEEE International Conference on Acoustics, (2012), 2925. doi: 10.1109/ICASSP.2012.6288528.

[16]

J. Liu, P. Musialski, P. Wonka and J. Ye, Tensor completion for estimating missing values in visual data,, IEEE Transactions on Pattern Analysis and Machine Intelligence, (2012), 208. doi: 10.1109/TPAMI.2012.39.

[17]

S. Ma, D. Goldfarb and L. Chen, Fixed point and bregman iterative methods for matrix rank minimization,, Mathematical Programming, 128 (2011), 321. doi: 10.1007/s10107-009-0306-5.

[18]

C. Mu, B. Huang, J. Wright and D. Goldfarb, Square deal: Lower bounds and improved relaxations for tensor recovery,, preprint, (2013).

[19]

K. A. Patwardhan, G. Sapiro and M. Bertalmío, Video inpainting under constrained camera motion,, IEEE Transactions on Image Processing, 16 (2007), 545. doi: 10.1109/TIP.2006.888343.

[20]

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.

[21]

B. Romera-Paredes and M. Pontil, A new convex relaxation for tensor completion,, preprint, (2013).

[22]

A. C. Sauve, A. O. Hero III, W. Leslie Rogers, S. J. Wilderman and N. H. Clinthorne, 3d image reconstruction for a compton spect camera model,, Nuclear Science, 46 (1999), 2075. doi: 10.1109/23.819285.

[23]

J. Sun, H. Zeng, H. Liu, Y. Lu and Z. Chen, Cubesvd: a novel approach to personalized web search,, in Proceedings of the 14th international conference on World Wide Web, (2005), 382. doi: 10.1145/1060745.1060803.

[24]

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.

[25]

P. Tseng, Convergence of a block coordinate descent method for nondifferentiable minimization,, Journal of Optimization Theory and Applications, 109 (2001), 475. doi: 10.1023/A:1017501703105.

[26]

L. R. Tucker, Some mathematical notes on three-mode factor analysis,, Psychometrika, 31 (1966), 279. doi: 10.1007/BF02289464.

[27]

Z. Wen, W. Yin and Y. Zhang, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm,, Mathematical Programming Computation, 4 (2012), 333. doi: 10.1007/s12532-012-0044-1.

[28]

Z. Xing, M. Zhou, A. Castrodad, G. Sapiro and L. Carin, Dictionary learning for noisy and incomplete hyperspectral images,, SIAM Journal on Imaging Sciences, 5 (2012), 33. doi: 10.1137/110837486.

[29]

Y. Xu and W. Yin, A block coordinate descent method for regularized multi-convex optimization with applications to nonnegative tensor factorization and completion,, SIAM Journal on Imaging Sciences, 6 (2013), 1758. doi: 10.1137/120887795.

[30]

Y. Xu and W. Yin, A globally convergent algorithm for nonconvex optimization based on block coordinate update,, , (2014).

[31]

Y. Xu, W. Yin, Z. Wen and Y. Zhang, An alternating direction algorithm for matrix completion with nonnegative factors,, Journal of Frontiers of Mathematics in China, 7 (2011), 365. doi: 10.1007/s11464-012-0194-5.

[32]

Z. Zhang, G. Ely, S. Aeron, N. Hao and M. Kilmer, Novel factorization strategies for higher order tensors: Implications for compression and recovery of multi-linear data,, preprint, (2013).

[1]

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

[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]

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

[4]

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

[5]

Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051

[6]

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

[7]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2018134

[8]

H. D. Scolnik, N. E. Echebest, M. T. Guardarucci. Extensions of incomplete oblique projections method for solving rank-deficient least-squares problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 175-191. doi: 10.3934/jimo.2009.5.175

[9]

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

[10]

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

[11]

Sihem Guerarra. Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 15-22. doi: 10.3934/naco.2019002

[12]

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

[13]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

[14]

Yanfei Lu, Qingfei Yin, Hongyi Li, Hongli Sun, Yunlei Yang, Muzhou Hou. Solving higher order nonlinear ordinary differential equations with least squares support vector machines. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2019012

[15]

Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381

[16]

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

[17]

Meixia Li, Changyu Wang, Biao Qu. Non-convex semi-infinite min-max optimization with noncompact sets. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1859-1881. doi: 10.3934/jimo.2017022

[18]

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

[19]

Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019

[20]

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

2017 Impact Factor: 1.465

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (23)

Other articles
by authors

[Back to Top]