Abstract
We review some recent approaches to robust approximations of low-rank data matrices. We consider the problem of estimating a low-rank mean matrix when the data matrix is subject to measurement errors as well as gross outliers in some of its entries. The purpose of the paper is to make various algorithms accessible with an understanding of their abilities and limitations to perform robust low-rank matrix approximations in both low and high dimensional problems.
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References
Agarwal A, Negahban S, Wainwright M J. Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions. Ann Statist, 2012, 40: 1171–1197
Candès E J, Li X, Ma Y, et al. Robust principal component analysis? J ACM, 2011, 58: 1–73
Chen C, He X, Wei Y. Lower rank approximation of matrices based on fast and robust alternating regression. J Comput Graph Statist, 2008, 17: 186–200
Eckart C, Young C. The approximation of one matrix by another of low rank. Psychometrika, 1936, 1: 211–218
Feng X, He X. Statistical inference based on robust low-rank data matrix approximation. Ann Statist, 2014, 42: 190–210
Gabriel K R, Zamir S. Lower rank approximation of matrices by least squares with any choice of weights. Technometrics, 1979, 21: 489–498
Huber P J. Robust estimation of a location parameters. Ann Math Statist, 1964, 35: 73–101
Johnstone I M. On the distribution of the lalarge eigenvalue in principal component analysis. Ann Statist, 2001, 29: 295–327
Johnstone I M, Lu A Y. On consistency and sparsity for principal components analysis in high dimension. J Amer Statist Assoc, 2009, 104: 682–693
Parikh N, Boyd S. Proximal algorithms. Found Trends Optim, 2013, 1: 123–231
Rousseeuw P J. Least median squares regression. J Amer Statist Assoc, 1984, 79: 871–880
Rousseeuw P J. Multivariate estimation with high breakdown point. In: Mathematical Statistics and Applications, vol. B. Dordrecht: Reidel, 1985, 283–297
She Y, Chen K. Robust reduced rank regression. ArXiv:1509.03938, 2015
She Y, Li S, Wu D. Robust orthogonal complement principal component analysis. J Amer Statist Assoc, 2016, 514: 41–64
Verboon P, Heiser W J. Resistent lower rank approximation of matrices by iterative majorization. Comput Statist Data Anal, 1994, 18: 457–467
Xu H, Caramanis C, Sanghavi S. Robust PCA via outlier pursuit. IEEE Trans Inform Theory, 2012, 58: 3047–3064
Zhang L, Shen H, Huang J. Robust regularized singular value decomposition with application to mortality data. Ann Appl Statist, 2013, 7: 1540–1561
Zhang T, Lerman G. A novel M-estimator for robust PCA. J Mach Learn Res, 2014, 15: 749–808
Zhou Z, Li X, Wright J, et al. Stable principal component pursuit. IEEE Internat Symp Inform Theory, 2010, 41: 1518–1522
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11571218), the State Key Program in the Major Research Plan of National Natural Science Foundation of China (Grant No. 91546202), Program for Changjiang Scholars and Innovative Research Team in Shanghai University of Finance and Economics (Grant No. IRT13077), and Program for Innovative Research Team of Shanghai University of Finance and Economics.
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Feng, X., He, X. Robust low-rank data matrix approximations. Sci. China Math. 60, 189–200 (2017). https://doi.org/10.1007/s11425-015-0484-1
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DOI: https://doi.org/10.1007/s11425-015-0484-1