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The singular values and vectors of low rank perturbations of large rectangular random matrices

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  • Benaych-Georges, Florent
  • Nadakuditi, Raj Rao

Abstract

In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate projections of the corresponding singular vectors of the perturbed matrix.

Suggested Citation

  • Benaych-Georges, Florent & Nadakuditi, Raj Rao, 2012. "The singular values and vectors of low rank perturbations of large rectangular random matrices," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 120-135.
    • Handle: RePEc:eee:jmvana:v:111:y:2012:i:c:p:120-135
    DOI: 10.1016/j.jmva.2012.04.019
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    References listed on IDEAS

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    Cited by:

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    3. Couillet, Romain, 2015. "Robust spiked random matrices and a robust G-MUSIC estimator," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 139-161.
    4. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    5. Barigozzi, Matteo & Trapani, Lorenzo, 2020. "Sequential testing for structural stability in approximate factor models," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5149-5187.
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    7. Monika Bhattacharjee & Arup Bose, 2017. "Matrix polynomial generalizations of the sample variance-covariance matrix when pn−1 → y ∈ (0, ∞)," Indian Journal of Pure and Applied Mathematics, Springer, vol. 48(4), pages 575-607, December.
    8. Kargin, Vladislav, 2015. "On estimation in the reduced-rank regression with a large number of responses and predictors," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 377-394.
    9. Philippe Loubaton, 2016. "On the Almost Sure Location of the Singular Values of Certain Gaussian Block-Hankel Large Random Matrices," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1339-1443, December.
    10. Leeb, William, 2022. "Optimal singular value shrinkage for operator norm loss: Extending to non-square matrices," Statistics & Probability Letters, Elsevier, vol. 186(C).
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    12. Ding, Xiucai & Yang, Fan, 2022. "Edge statistics of large dimensional deformed rectangular matrices," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
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    14. Xinyi Zhong & Chang Su & Zhou Fan, 2022. "Empirical Bayes PCA in high dimensions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(3), pages 853-878, July.
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