July  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 ofdata imposes big challenge on how to process them effectively andefficiently. Fortunately, in practice data are not unstructured.Their samples usually lie around low-dimensional manifolds andhave high correlation among them. Such characteristics can beeffectively depicted by low rankness. As an extension to thesparsity of first order data, such as voices, low rankness is alsoan effective measure for the sparsity of second order data, suchas images. In this paper, I review the representative theories,algorithms and applications of the low rank subspace recoverymodels 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:
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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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[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. Google Scholar

[7]

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

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

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

[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. Google Scholar

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[45]

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