2016, 1(2/3 ): 139-161. doi: 10.3934/bdia.2016001

A review on low-rank models in data analysis

1. 

Key Lab. of Machine Perception (MOE), School of EECS, Peking University, Beijing, China

Received  May 2015 Revised  January 2016 Published  August 2016

Nowadays we are in the big data era. The high-dimensionality of data imposes big challenge on how to process them effectively and efficiently. Fortunately, in practice data are not unstructured. Their samples usually lie around low-dimensional manifolds and have high correlation among them. Such characteristics can be effectively depicted by low rankness. As an extension to the sparsity of first order data, such as voices, low rankness is also an effective measure for the sparsity of second order data, such as images. In this paper, I review the representative theories, algorithms and applications of the low rank subspace recovery models in data processing.
Citation: 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
References:
[1]

A. Adler, M. Elad and Y. Hel-Or, Probabilistic subspace clustering via sparse representations,, IEEE Signal Processing Letters, 20 (2013), 63. doi: 10.1109/LSP.2012.2229705.

[2]

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

[3]

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

[4]

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

[5]

E. Candès and Y. Plan, Matrix completion with noise,, Proceedings of the IEEE, 98 (2010), 925.

[6]

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.

[7]

V. Chandrasekaran, S. Sanghavi, P. Parrilo and A. Willsky, Sparse and low-rank matrix decompositions,, Annual Allerton Conference on Communication, (2009), 962.

[8]

C. Chen, B. He, Y. Ye and X. Yuan, The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent,, Mathematical Programming, 155 (2016), 57. doi: 10.1007/s10107-014-0826-5.

[9]

Y. Chen, H. Xu, C. Caramanis and S. Sanghavi, Robust matrix completion with corrupted columns,, International Conference on Machine Learning, (2011), 873.

[10]

B. Cheng, G. Liu, J. Wang, Z. Huang and S. Yan, Multi-task low-rank affinity pursuit for image segmentation,, International Conference on Computer Vision, (2011), 2439. doi: 10.1109/ICCV.2011.6126528.

[11]

A. Cichocki, R. Zdunek, A. H. Phan and S. Ichi Amari, Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation,, 1st edition, (2009). doi: 10.1002/9780470747278.

[12]

Y. Cui, C.-H. Zheng and J. Yang, Identifying subspace gene clusters from microarray data using low-rank representation,, PLoS One, 8 (2013). doi: 10.1371/journal.pone.0059377.

[13]

P. Drineas, R. Kannan and M. Mahoney, Fast Monte Carlo algorithms for matrices II: Computing a low rank approximation to a matrix,, SIAM Journal on Computing, 36 (2006), 158. doi: 10.1137/S0097539704442696.

[14]

E. Elhamifar and R. Vidal, Sparse subspace clustering,, in IEEE International Conference on Computer Vision and Pattern Recognition, (2009), 2790. doi: 10.1109/CVPR.2009.5206547.

[15]

E. Elhamifar and R. Vidal, Sparse subspace clustering: Algorithm, theory, and applications,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 2765. doi: 10.1109/TPAMI.2013.57.

[16]

P. Favaro, R. Vidal and A. Ravichandran, A closed form solution to robust subspace estimation and clustering,, IEEE Conference on Computer Vision and Pattern Recognition, (2011), 1801. doi: 10.1109/CVPR.2011.5995365.

[17]

J. Feng, Z. Lin, H. Xu and S. Yan, Robust subspace segmentation with block-diagonal prior,, IEEE Conference on Computer Vision and Pattern Recognition, (2014), 3818. doi: 10.1109/CVPR.2014.482.

[18]

M. Frank and P. Wolfe, An algorithm for quadratic programming,, Naval Research Logistics Quarterly, 3 (1956), 95. doi: 10.1002/nav.3800030109.

[19]

Y. Fu, J. Gao, D. Tien and Z. Lin, Tensor LRR based subspace clustering,, International Joint Conference on Neural Networks, (2014), 1877. doi: 10.1109/IJCNN.2014.6889472.

[20]

A. Ganesh, Z. Lin, J. Wright, L. Wu, M. Chen and Y. Ma, Fast algorithms for recovering a corrupted low-rank matrix,, International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, (2009), 213. doi: 10.1109/CAMSAP.2009.5413299.

[21]

H. Gao, J.-F. Cai, Z. Shen and H. Zhao, Robust principal component analysis-based four-dimensional computed tomography,, Physics in Medicine and Biology, 56 (2011), 3181. doi: 10.1088/0031-9155/56/11/002.

[22]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming (web page and software),, , (2009).

[23]

S. Gu, L. Zhang, W. Zuo and X. Feng, Weighted nuclear norm minimization with application to image denoising,, IEEE Conference on Computer Vision and Pattern Recognition, (2014), 2862. doi: 10.1109/CVPR.2014.366.

[24]

H. Hu, Z. Lin, J. Feng and J. Zhou, Smooth representation clustering,, IEEE Conference on Computer Vision and Pattern Recognition, (2014), 3834. doi: 10.1109/CVPR.2014.484.

[25]

Y. Hu, D. Zhang, J. Ye, X. Li and X. He, Fast and accurate matrix completion via truncated nuclear norm regularization,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 2117. doi: 10.1109/TPAMI.2012.271.

[26]

M. Jaggi, Revisiting Frank-Wolfe: Projection-free sparse convex optimization,, in International Conference on Machine Learning, (2013), 427.

[27]

M. Jaggi and M. Sulovský, A simple algorithm for nuclear norm regularized problems,, in International Conference on Machine Learning, (2010), 471.

[28]

I. Jhuo, D. Liu, D. Lee and S. Chang, Robust visual domain adaptation with low-rank reconstruction,, IEEE Conference on Computer Vision and Pattern Recognition, (2012), 2168.

[29]

H. Ji, C. Liu, Z. Shen and Y. Xu, Robust video denoising using low rank matrix completion,, IEEE Conference on Computer Vision and Pattern Recognition, (2010), 1791. doi: 10.1109/CVPR.2010.5539849.

[30]

Y. Jin, Q. Wu and L. Liu, Unsupervised upright orientation of man-made models,, Graphical Models, 74 (2012), 99. doi: 10.1016/j.gmod.2012.03.007.

[31]

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

[32]

C. Lang, G. Liu, J. Yu and S. Yan, Saliency detection by multitask sparsity pursuit,, IEEE Transactions on Image Processing, 21 (2012), 1327. doi: 10.1109/TIP.2011.2169274.

[33]

R. M. Larsen, http://sun.stanford.edu/~rmunk/PROPACK/,, 2004., ().

[34]

D. Lee and H. Seung, Learning the parts of objects by non-negative matrix factorization,, Nature, 401 (1999).

[35]

X. Liang, X. Ren, Z. Zhang and Y. Ma, Repairing sparse low-rank texture,, in European Conference on Computer Vision, 7576 (2012), 482. doi: 10.1007/978-3-642-33715-4_35.

[36]

Z. Lin, R. Liu and H. Li, Linearized alternating direction method with parallel splitting and adaptive penality for separable convex programs in machine learning,, Machine Learning, 99 (2015), 287. doi: 10.1007/s10994-014-5469-5.

[37]

Z. Lin, R. Liu and Z. Su, Linearized alternating direction method with adaptive penalty for low-rank representation,, Advances in Neural Information Processing Systems, (2011), 612.

[38]

G. Liu, Z. Lin, S. Yan, J. Sun and Y. Ma, Robust recovery of subspace structures by low-rank representation,, IEEE Transactions Pattern Analysis and Machine Intelligence, 35 (2013), 171. doi: 10.1109/TPAMI.2012.88.

[39]

G. Liu, Z. Lin and Y. Yu, Robust subspace segmentation by low-rank representation,, in International Conference on Machine Learning, (2010), 663.

[40]

G. Liu, H. Xu and S. Yan, Exact subspace segmentation and outlier detection by low-rank representation,, International Conference on Artificial Intelligence and Statistics, (2012), 703.

[41]

G. Liu and S. Yan, Latent low-rank representation for subspace segmentation and feature extraction,, in IEEE International Conference on Computer Vision, (2011), 1615. doi: 10.1109/ICCV.2011.6126422.

[42]

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, 35 (2013), 208. doi: 10.1109/TPAMI.2012.39.

[43]

R. Liu, Z. Lin, Z. Su and J. Gao, Linear time principal component pursuit and its extensions using $l_1$ filtering,, Neurocomputing, 142 (2014), 529.

[44]

R. Liu, Z. Lin, F. Torre and Z. Su, Fixed-rank representation for unsupervised visual learning,, IEEE Conference on Computer Vision and Pattern Recognition, (2012), 598.

[45]

C. Lu, J. Feng, Z. Lin and S. Yan, Correlation adaptive subspace segmentation by trace lasso,, International Conference on Computer Vision, (2013), 1345. doi: 10.1109/ICCV.2013.170.

[46]

C. Lu, Z. Lin and S. Yan, Smoothed low rank and sparse matrix recovery by iteratively reweighted least squared minimization,, IEEE Transactions on Image Processing, 24 (2015), 646. doi: 10.1109/TIP.2014.2380155.

[47]

C. Lu, H. Min, Z. Zhao, L. Zhu, D. Huang and S. Yan, Robust and efficient subspace segmentation via least squares regression,, European Conference on Computer Vision, 7578 (2012), 347. doi: 10.1007/978-3-642-33786-4_26.

[48]

C. Lu, C. Zhu, C. Xu, S. Yan and Z. Lin, Generalized singular value thresholding,, AAAI Conference on Artificial Intelligence, (2015), 1805.

[49]

X. Lu, Y. Wang and Y. Yuan, Graph-regularized low-rank representation for destriping of hyperspectral images,, IEEE Transactions on Geoscience and Remote Sensing, 51 (2013), 4009. doi: 10.1109/TGRS.2012.2226730.

[50]

Y. Ma, S. Soatto, J. Kosecka and S. Sastry, An Invitation to 3-D Vision: From Images to Geometric Models,, 1st edition, (2004). doi: 10.1007/978-0-387-21779-6.

[51]

K. Min, Z. Zhang, J. Wright and Y. Ma, Decomposing background topics from keywords by principal component pursuit,, in ACM International Conference on Information and Knowledge Management, (2010), 269. doi: 10.1145/1871437.1871475.

[52]

Y. Ming and Q. Ruan, Robust sparse bounding sphere for 3D face recognition,, Image and Vision Computing, 30 (2012), 524. doi: 10.1016/j.imavis.2012.05.001.

[53]

L. Mukherjee, V. Singh, J. Xu and M. Collins, Analyzing the subspace structure of related images: Concurrent segmentation of image sets,, European Conference on Computer Vision, 7575 (2012), 128. doi: 10.1007/978-3-642-33765-9_10.

[54]

Y. Nesterov, A method of solving a convex programming problem with convergence rate $O(1/k^2)$,, (Russian) Dokl. Akad. Nauk SSSR, 269 (1983), 543.

[55]

Y. Panagakis and C. Kotropoulos, Automatic music tagging by low-rank representation,, International Conference on Acoustics, (2012), 497. doi: 10.1109/ICASSP.2012.6287925.

[56]

Y. Peng, A. Ganesh, J. Wright, W. Xu and Y. Ma, RASL: Robust alignment by sparse and low-rank decomposition for linearly correlated images,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 34 (2012), 2233.

[57]

J. Qian, J. Yang, F. Zhang and Z. Lin, Robust low-rank regularized regression for face recognition with occlusion,, The Workshop of IEEE Conference on Computer Vision and Pattern Recognition, (2014), 21. doi: 10.1109/CVPRW.2014.9.

[58]

X. Ren and Z. Lin, Linearized alternating direction method with adaptive penalty and warm starts for fast solving transform invariant low-rank textures,, International Journal of Computer Vision, 104 (2013), 1. doi: 10.1007/s11263-013-0611-6.

[59]

A. P. Singh and G. J. Gordon, A unified view of matrix factorization models,, in Proceedings of Machine Learning and Knowledge Discovery in Databases, 5212 (2008), 358. doi: 10.1007/978-3-540-87481-2_24.

[60]

H. Tan, J. Feng, G. Feng, W. Wang and Y. Zhang, Traffic volume data outlier recovery via tensor model,, Mathematical Problems in Engineering, 2013 (2013). doi: 10.1155/2013/164810.

[61]

M. Tso, Reduced-rank regression and canonical analysis,, Journal of the Royal Statistical Society, 43 (1981), 183.

[62]

R. Vidal, Y. Ma and S. Sastry, Generalized principal component analysis,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 27 (2005), 1945.

[63]

R. Vidal, Subspace clustering,, IEEE Signal Processing Magazine, 28 (2011), 52. doi: 10.1109/MSP.2010.939739.

[64]

J. Wang, V. Saligrama and D. Castanon, Structural similarity and distance in learning,, Annual Allerton Conf. Communication, (2011), 744. doi: 10.1109/Allerton.2011.6120242.

[65]

Y.-X. Wang and Y.-J. Zhang, Nonnegative matrix factorization: A comprehensive review,, IEEE Transactions on Knowledge and Data Engineering, 25 (2013), 1336. doi: 10.1109/TKDE.2012.51.

[66]

S. Wei and Z. Lin, Analysis and improvement of low rank representation for subspace segmentation,, , ().

[67]

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.

[68]

J. Wright, A. Ganesh, S. Rao, Y. Peng and Y. Ma, Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization,, Advances in Neural Information Processing Systems, (2009), 2080.

[69]

L. Wu, A. Ganesh, B. Shi, Y. Matsushita, Y. Wang and Y. Ma, Robust photometric stereo via low-rank matrix completion and recovery,, Asian Conference on Computer Vision, (2010), 703. doi: 10.1007/978-3-642-19318-7_55.

[70]

L. Yang, Y. Lin, Z. Lin and H. Zha, Low rank global geometric consistency for partial-duplicate image search,, International Conference on Pattern Recognition, (2014), 3939. doi: 10.1109/ICPR.2014.675.

[71]

M. Yin, J. Gao and Z. Lin, Laplacian regularized low-rank representation and its applications,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38 (2016), 504. doi: 10.1109/TPAMI.2015.2462360.

[72]

Y. Yu and D. Schuurmans, Rank/norm regularization with closed-form solutions: Application to subspace clustering,, Uncertainty in Artificial Intelligence, (2011), 778.

[73]

H. Zhang, Z. Lin and C. Zhang, A counterexample for the validity of using nuclear norm as a convex surrogate of rank,, European Conference on Machine Learning, 8189 (2013), 226. doi: 10.1007/978-3-642-40991-2_15.

[74]

H. Zhang, Z. Lin, C. Zhang and E. Chang, Exact recoverability of robust PCA via outlier pursuit with tight recovery bounds,, AAAI Conference on Artificial Intelligence, (2015), 3143.

[75]

H. Zhang, Z. Lin, C. Zhang and J. Gao, Robust latent low rank representation for subspace clustering,, Neurocomputing, 145 (2014), 369. doi: 10.1016/j.neucom.2014.05.022.

[76]

H. Zhang, Z. Lin, C. Zhang and J. Gao, Relation among some low rank subspace recovery models,, Neural Computation, 27 (2015), 1915. doi: 10.1162/NECO_a_00762.

[77]

T. Zhang, B. Ghanem, S. Liu and N. Ahuja, Low-rank sparse learning for robust visual tracking,, European Conference on Computer Vision, 7577 (2012), 470. doi: 10.1007/978-3-642-33783-3_34.

[78]

Z. Zhang, A. Ganesh, X. Liang and Y. Ma, TILT: Transform invariant low-rank textures,, International Journal of Computer Vision, 99 (2012), 1. doi: 10.1007/s11263-012-0515-x.

[79]

Z. Zhang, X. Liang and Y. Ma, Unwrapping low-rank textures on generalized cylindrical surfaces,, International Conference on Computer Vision, (2011), 1347. doi: 10.1109/ICCV.2011.6126388.

[80]

Z. Zhang, Y. Matsushita and Y. Ma, Camera calibration with lens distortion from low-rank textures,, IEEE Conference on Computer Vision and Pattern Recognition, (2011), 2321. doi: 10.1109/CVPR.2011.5995548.

[81]

Y. Zheng, X. Zhang, S. Yang and L. Jiao, Low-rank representation with local constraint for graph construction,, Neurocomputing, 122 (2013), 398. doi: 10.1016/j.neucom.2013.06.013.

[82]

X. Zhou, C. Yang, H. Zhao and W. Yu, Low-rank modeling and its applications in image analysis,, ACM Computing Surveys, 47 (2014). doi: 10.1145/2674559.

[83]

G. Zhu, S. Yan and Y. Ma, Image tag refinement towards low-rank, content-tag prior and error sparsity,, in International conference on Multimedia, (2010), 461. doi: 10.1145/1873951.1874028.

[84]

L. Zhuang, H. Gao, Z. Lin, Y. Ma, X. Zhang and N. Yu, Non-negative low rank and sparse graph for semi-supervised learning,, IEEE International Conference on Computer Vision and Pattern Recognition, (2012), 2328.

[85]

W. Zuo and Z. Lin, A generalized accelerated proximal gradient approach for total-variation-based image restoration,, IEEE Transactions on Image Processing, 20 (2011), 2748. doi: 10.1109/TIP.2011.2131665.

show all references

References:
[1]

A. Adler, M. Elad and Y. Hel-Or, Probabilistic subspace clustering via sparse representations,, IEEE Signal Processing Letters, 20 (2013), 63. doi: 10.1109/LSP.2012.2229705.

[2]

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

[3]

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

[4]

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

[5]

E. Candès and Y. Plan, Matrix completion with noise,, Proceedings of the IEEE, 98 (2010), 925.

[6]

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.

[7]

V. Chandrasekaran, S. Sanghavi, P. Parrilo and A. Willsky, Sparse and low-rank matrix decompositions,, Annual Allerton Conference on Communication, (2009), 962.

[8]

C. Chen, B. He, Y. Ye and X. Yuan, The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent,, Mathematical Programming, 155 (2016), 57. doi: 10.1007/s10107-014-0826-5.

[9]

Y. Chen, H. Xu, C. Caramanis and S. Sanghavi, Robust matrix completion with corrupted columns,, International Conference on Machine Learning, (2011), 873.

[10]

B. Cheng, G. Liu, J. Wang, Z. Huang and S. Yan, Multi-task low-rank affinity pursuit for image segmentation,, International Conference on Computer Vision, (2011), 2439. doi: 10.1109/ICCV.2011.6126528.

[11]

A. Cichocki, R. Zdunek, A. H. Phan and S. Ichi Amari, Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation,, 1st edition, (2009). doi: 10.1002/9780470747278.

[12]

Y. Cui, C.-H. Zheng and J. Yang, Identifying subspace gene clusters from microarray data using low-rank representation,, PLoS One, 8 (2013). doi: 10.1371/journal.pone.0059377.

[13]

P. Drineas, R. Kannan and M. Mahoney, Fast Monte Carlo algorithms for matrices II: Computing a low rank approximation to a matrix,, SIAM Journal on Computing, 36 (2006), 158. doi: 10.1137/S0097539704442696.

[14]

E. Elhamifar and R. Vidal, Sparse subspace clustering,, in IEEE International Conference on Computer Vision and Pattern Recognition, (2009), 2790. doi: 10.1109/CVPR.2009.5206547.

[15]

E. Elhamifar and R. Vidal, Sparse subspace clustering: Algorithm, theory, and applications,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 2765. doi: 10.1109/TPAMI.2013.57.

[16]

P. Favaro, R. Vidal and A. Ravichandran, A closed form solution to robust subspace estimation and clustering,, IEEE Conference on Computer Vision and Pattern Recognition, (2011), 1801. doi: 10.1109/CVPR.2011.5995365.

[17]

J. Feng, Z. Lin, H. Xu and S. Yan, Robust subspace segmentation with block-diagonal prior,, IEEE Conference on Computer Vision and Pattern Recognition, (2014), 3818. doi: 10.1109/CVPR.2014.482.

[18]

M. Frank and P. Wolfe, An algorithm for quadratic programming,, Naval Research Logistics Quarterly, 3 (1956), 95. doi: 10.1002/nav.3800030109.

[19]

Y. Fu, J. Gao, D. Tien and Z. Lin, Tensor LRR based subspace clustering,, International Joint Conference on Neural Networks, (2014), 1877. doi: 10.1109/IJCNN.2014.6889472.

[20]

A. Ganesh, Z. Lin, J. Wright, L. Wu, M. Chen and Y. Ma, Fast algorithms for recovering a corrupted low-rank matrix,, International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, (2009), 213. doi: 10.1109/CAMSAP.2009.5413299.

[21]

H. Gao, J.-F. Cai, Z. Shen and H. Zhao, Robust principal component analysis-based four-dimensional computed tomography,, Physics in Medicine and Biology, 56 (2011), 3181. doi: 10.1088/0031-9155/56/11/002.

[22]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming (web page and software),, , (2009).

[23]

S. Gu, L. Zhang, W. Zuo and X. Feng, Weighted nuclear norm minimization with application to image denoising,, IEEE Conference on Computer Vision and Pattern Recognition, (2014), 2862. doi: 10.1109/CVPR.2014.366.

[24]

H. Hu, Z. Lin, J. Feng and J. Zhou, Smooth representation clustering,, IEEE Conference on Computer Vision and Pattern Recognition, (2014), 3834. doi: 10.1109/CVPR.2014.484.

[25]

Y. Hu, D. Zhang, J. Ye, X. Li and X. He, Fast and accurate matrix completion via truncated nuclear norm regularization,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 2117. doi: 10.1109/TPAMI.2012.271.

[26]

M. Jaggi, Revisiting Frank-Wolfe: Projection-free sparse convex optimization,, in International Conference on Machine Learning, (2013), 427.

[27]

M. Jaggi and M. Sulovský, A simple algorithm for nuclear norm regularized problems,, in International Conference on Machine Learning, (2010), 471.

[28]

I. Jhuo, D. Liu, D. Lee and S. Chang, Robust visual domain adaptation with low-rank reconstruction,, IEEE Conference on Computer Vision and Pattern Recognition, (2012), 2168.

[29]

H. Ji, C. Liu, Z. Shen and Y. Xu, Robust video denoising using low rank matrix completion,, IEEE Conference on Computer Vision and Pattern Recognition, (2010), 1791. doi: 10.1109/CVPR.2010.5539849.

[30]

Y. Jin, Q. Wu and L. Liu, Unsupervised upright orientation of man-made models,, Graphical Models, 74 (2012), 99. doi: 10.1016/j.gmod.2012.03.007.

[31]

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

[32]

C. Lang, G. Liu, J. Yu and S. Yan, Saliency detection by multitask sparsity pursuit,, IEEE Transactions on Image Processing, 21 (2012), 1327. doi: 10.1109/TIP.2011.2169274.

[33]

R. M. Larsen, http://sun.stanford.edu/~rmunk/PROPACK/,, 2004., ().

[34]

D. Lee and H. Seung, Learning the parts of objects by non-negative matrix factorization,, Nature, 401 (1999).

[35]

X. Liang, X. Ren, Z. Zhang and Y. Ma, Repairing sparse low-rank texture,, in European Conference on Computer Vision, 7576 (2012), 482. doi: 10.1007/978-3-642-33715-4_35.

[36]

Z. Lin, R. Liu and H. Li, Linearized alternating direction method with parallel splitting and adaptive penality for separable convex programs in machine learning,, Machine Learning, 99 (2015), 287. doi: 10.1007/s10994-014-5469-5.

[37]

Z. Lin, R. Liu and Z. Su, Linearized alternating direction method with adaptive penalty for low-rank representation,, Advances in Neural Information Processing Systems, (2011), 612.

[38]

G. Liu, Z. Lin, S. Yan, J. Sun and Y. Ma, Robust recovery of subspace structures by low-rank representation,, IEEE Transactions Pattern Analysis and Machine Intelligence, 35 (2013), 171. doi: 10.1109/TPAMI.2012.88.

[39]

G. Liu, Z. Lin and Y. Yu, Robust subspace segmentation by low-rank representation,, in International Conference on Machine Learning, (2010), 663.

[40]

G. Liu, H. Xu and S. Yan, Exact subspace segmentation and outlier detection by low-rank representation,, International Conference on Artificial Intelligence and Statistics, (2012), 703.

[41]

G. Liu and S. Yan, Latent low-rank representation for subspace segmentation and feature extraction,, in IEEE International Conference on Computer Vision, (2011), 1615. doi: 10.1109/ICCV.2011.6126422.

[42]

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, 35 (2013), 208. doi: 10.1109/TPAMI.2012.39.

[43]

R. Liu, Z. Lin, Z. Su and J. Gao, Linear time principal component pursuit and its extensions using $l_1$ filtering,, Neurocomputing, 142 (2014), 529.

[44]

R. Liu, Z. Lin, F. Torre and Z. Su, Fixed-rank representation for unsupervised visual learning,, IEEE Conference on Computer Vision and Pattern Recognition, (2012), 598.

[45]

C. Lu, J. Feng, Z. Lin and S. Yan, Correlation adaptive subspace segmentation by trace lasso,, International Conference on Computer Vision, (2013), 1345. doi: 10.1109/ICCV.2013.170.

[46]

C. Lu, Z. Lin and S. Yan, Smoothed low rank and sparse matrix recovery by iteratively reweighted least squared minimization,, IEEE Transactions on Image Processing, 24 (2015), 646. doi: 10.1109/TIP.2014.2380155.

[47]

C. Lu, H. Min, Z. Zhao, L. Zhu, D. Huang and S. Yan, Robust and efficient subspace segmentation via least squares regression,, European Conference on Computer Vision, 7578 (2012), 347. doi: 10.1007/978-3-642-33786-4_26.

[48]

C. Lu, C. Zhu, C. Xu, S. Yan and Z. Lin, Generalized singular value thresholding,, AAAI Conference on Artificial Intelligence, (2015), 1805.

[49]

X. Lu, Y. Wang and Y. Yuan, Graph-regularized low-rank representation for destriping of hyperspectral images,, IEEE Transactions on Geoscience and Remote Sensing, 51 (2013), 4009. doi: 10.1109/TGRS.2012.2226730.

[50]

Y. Ma, S. Soatto, J. Kosecka and S. Sastry, An Invitation to 3-D Vision: From Images to Geometric Models,, 1st edition, (2004). doi: 10.1007/978-0-387-21779-6.

[51]

K. Min, Z. Zhang, J. Wright and Y. Ma, Decomposing background topics from keywords by principal component pursuit,, in ACM International Conference on Information and Knowledge Management, (2010), 269. doi: 10.1145/1871437.1871475.

[52]

Y. Ming and Q. Ruan, Robust sparse bounding sphere for 3D face recognition,, Image and Vision Computing, 30 (2012), 524. doi: 10.1016/j.imavis.2012.05.001.

[53]

L. Mukherjee, V. Singh, J. Xu and M. Collins, Analyzing the subspace structure of related images: Concurrent segmentation of image sets,, European Conference on Computer Vision, 7575 (2012), 128. doi: 10.1007/978-3-642-33765-9_10.

[54]

Y. Nesterov, A method of solving a convex programming problem with convergence rate $O(1/k^2)$,, (Russian) Dokl. Akad. Nauk SSSR, 269 (1983), 543.

[55]

Y. Panagakis and C. Kotropoulos, Automatic music tagging by low-rank representation,, International Conference on Acoustics, (2012), 497. doi: 10.1109/ICASSP.2012.6287925.

[56]

Y. Peng, A. Ganesh, J. Wright, W. Xu and Y. Ma, RASL: Robust alignment by sparse and low-rank decomposition for linearly correlated images,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 34 (2012), 2233.

[57]

J. Qian, J. Yang, F. Zhang and Z. Lin, Robust low-rank regularized regression for face recognition with occlusion,, The Workshop of IEEE Conference on Computer Vision and Pattern Recognition, (2014), 21. doi: 10.1109/CVPRW.2014.9.

[58]

X. Ren and Z. Lin, Linearized alternating direction method with adaptive penalty and warm starts for fast solving transform invariant low-rank textures,, International Journal of Computer Vision, 104 (2013), 1. doi: 10.1007/s11263-013-0611-6.

[59]

A. P. Singh and G. J. Gordon, A unified view of matrix factorization models,, in Proceedings of Machine Learning and Knowledge Discovery in Databases, 5212 (2008), 358. doi: 10.1007/978-3-540-87481-2_24.

[60]

H. Tan, J. Feng, G. Feng, W. Wang and Y. Zhang, Traffic volume data outlier recovery via tensor model,, Mathematical Problems in Engineering, 2013 (2013). doi: 10.1155/2013/164810.

[61]

M. Tso, Reduced-rank regression and canonical analysis,, Journal of the Royal Statistical Society, 43 (1981), 183.

[62]

R. Vidal, Y. Ma and S. Sastry, Generalized principal component analysis,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 27 (2005), 1945.

[63]

R. Vidal, Subspace clustering,, IEEE Signal Processing Magazine, 28 (2011), 52. doi: 10.1109/MSP.2010.939739.

[64]

J. Wang, V. Saligrama and D. Castanon, Structural similarity and distance in learning,, Annual Allerton Conf. Communication, (2011), 744. doi: 10.1109/Allerton.2011.6120242.

[65]

Y.-X. Wang and Y.-J. Zhang, Nonnegative matrix factorization: A comprehensive review,, IEEE Transactions on Knowledge and Data Engineering, 25 (2013), 1336. doi: 10.1109/TKDE.2012.51.

[66]

S. Wei and Z. Lin, Analysis and improvement of low rank representation for subspace segmentation,, , ().

[67]

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.

[68]

J. Wright, A. Ganesh, S. Rao, Y. Peng and Y. Ma, Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization,, Advances in Neural Information Processing Systems, (2009), 2080.

[69]

L. Wu, A. Ganesh, B. Shi, Y. Matsushita, Y. Wang and Y. Ma, Robust photometric stereo via low-rank matrix completion and recovery,, Asian Conference on Computer Vision, (2010), 703. doi: 10.1007/978-3-642-19318-7_55.

[70]

L. Yang, Y. Lin, Z. Lin and H. Zha, Low rank global geometric consistency for partial-duplicate image search,, International Conference on Pattern Recognition, (2014), 3939. doi: 10.1109/ICPR.2014.675.

[71]

M. Yin, J. Gao and Z. Lin, Laplacian regularized low-rank representation and its applications,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38 (2016), 504. doi: 10.1109/TPAMI.2015.2462360.

[72]

Y. Yu and D. Schuurmans, Rank/norm regularization with closed-form solutions: Application to subspace clustering,, Uncertainty in Artificial Intelligence, (2011), 778.

[73]

H. Zhang, Z. Lin and C. Zhang, A counterexample for the validity of using nuclear norm as a convex surrogate of rank,, European Conference on Machine Learning, 8189 (2013), 226. doi: 10.1007/978-3-642-40991-2_15.

[74]

H. Zhang, Z. Lin, C. Zhang and E. Chang, Exact recoverability of robust PCA via outlier pursuit with tight recovery bounds,, AAAI Conference on Artificial Intelligence, (2015), 3143.

[75]

H. Zhang, Z. Lin, C. Zhang and J. Gao, Robust latent low rank representation for subspace clustering,, Neurocomputing, 145 (2014), 369. doi: 10.1016/j.neucom.2014.05.022.

[76]

H. Zhang, Z. Lin, C. Zhang and J. Gao, Relation among some low rank subspace recovery models,, Neural Computation, 27 (2015), 1915. doi: 10.1162/NECO_a_00762.

[77]

T. Zhang, B. Ghanem, S. Liu and N. Ahuja, Low-rank sparse learning for robust visual tracking,, European Conference on Computer Vision, 7577 (2012), 470. doi: 10.1007/978-3-642-33783-3_34.

[78]

Z. Zhang, A. Ganesh, X. Liang and Y. Ma, TILT: Transform invariant low-rank textures,, International Journal of Computer Vision, 99 (2012), 1. doi: 10.1007/s11263-012-0515-x.

[79]

Z. Zhang, X. Liang and Y. Ma, Unwrapping low-rank textures on generalized cylindrical surfaces,, International Conference on Computer Vision, (2011), 1347. doi: 10.1109/ICCV.2011.6126388.

[80]

Z. Zhang, Y. Matsushita and Y. Ma, Camera calibration with lens distortion from low-rank textures,, IEEE Conference on Computer Vision and Pattern Recognition, (2011), 2321. doi: 10.1109/CVPR.2011.5995548.

[81]

Y. Zheng, X. Zhang, S. Yang and L. Jiao, Low-rank representation with local constraint for graph construction,, Neurocomputing, 122 (2013), 398. doi: 10.1016/j.neucom.2013.06.013.

[82]

X. Zhou, C. Yang, H. Zhao and W. Yu, Low-rank modeling and its applications in image analysis,, ACM Computing Surveys, 47 (2014). doi: 10.1145/2674559.

[83]

G. Zhu, S. Yan and Y. Ma, Image tag refinement towards low-rank, content-tag prior and error sparsity,, in International conference on Multimedia, (2010), 461. doi: 10.1145/1873951.1874028.

[84]

L. Zhuang, H. Gao, Z. Lin, Y. Ma, X. Zhang and N. Yu, Non-negative low rank and sparse graph for semi-supervised learning,, IEEE International Conference on Computer Vision and Pattern Recognition, (2012), 2328.

[85]

W. Zuo and Z. Lin, A generalized accelerated proximal gradient approach for total-variation-based image restoration,, IEEE Transactions on Image Processing, 20 (2011), 2748. doi: 10.1109/TIP.2011.2131665.

[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]

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

[3]

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

[4]

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

[5]

Guojun Gan, Kun Chen. A soft subspace clustering algorithm with log-transformed distances. Big Data & Information Analytics, 2016, 1 (1) : 93-109. doi: 10.3934/bdia.2016.1.93

[6]

Xin Zhang, Jie Wen, Qin Ni. Subspace trust-region algorithm with conic model for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 223-234. doi: 10.3934/naco.2013.3.223

[7]

Michael Kiermaier, Reinhard Laue. Derived and residual subspace designs. Advances in Mathematics of Communications, 2015, 9 (1) : 105-115. doi: 10.3934/amc.2015.9.105

[8]

Heide Gluesing-Luerssen, Carolyn Troha. Construction of subspace codes through linkage. Advances in Mathematics of Communications, 2016, 10 (3) : 525-540. doi: 10.3934/amc.2016023

[9]

Ernst M. Gabidulin, Pierre Loidreau. Properties of subspace subcodes of Gabidulin codes. Advances in Mathematics of Communications, 2008, 2 (2) : 147-157. doi: 10.3934/amc.2008.2.147

[10]

Afaf Bouharguane, Pascal Azerad, Frédéric Bouchette, Fabien Marche, Bijan Mohammadi. Low complexity shape optimization & a posteriori high fidelity validation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 759-772. doi: 10.3934/dcdsb.2010.13.759

[11]

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

[12]

Qiao-Fang Lian, Yun-Zhang Li. Reducing subspace frame multiresolution analysis and frame wavelets. Communications on Pure & Applied Analysis, 2007, 6 (3) : 741-756. doi: 10.3934/cpaa.2007.6.741

[13]

Xin Zhao, Jinyan Fan. On subspace properties of the quadratically constrained quadratic program. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1625-1640. doi: 10.3934/jimo.2017010

[14]

Thomas Honold, Michael Kiermaier, Sascha Kurz. Constructions and bounds for mixed-dimension subspace codes. Advances in Mathematics of Communications, 2016, 10 (3) : 649-682. doi: 10.3934/amc.2016033

[15]

Qiao Liang, Qiang Ye. Deflation by restriction for the inverse-free preconditioned Krylov subspace method. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 55-71. doi: 10.3934/naco.2016.6.55

[16]

Lori Badea, Marius Cocou. Approximation results and subspace correction algorithms for implicit variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1507-1524. doi: 10.3934/dcdss.2013.6.1507

[17]

Suqi Ma. Low viral persistence of an immunological model. Mathematical Biosciences & Engineering, 2012, 9 (4) : 809-817. doi: 10.3934/mbe.2012.9.809

[18]

Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365

[19]

Nian Li, Xiaohu Tang, Tor Helleseth. A class of quaternary sequences with low correlation. Advances in Mathematics of Communications, 2015, 9 (2) : 199-210. doi: 10.3934/amc.2015.9.199

[20]

Jinyuan Zhang, Aimin Zhou, Guixu Zhang, Hu Zhang. A clustering based mate selection for evolutionary optimization. Big Data & Information Analytics, 2017, 2 (1) : 77-85. doi: 10.3934/bdia.2017010

 Impact Factor: 

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]