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Spiked singular values and vectors under extreme aspect ratios

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  • Feldman, Michael J.

Abstract

The behavior of the leading singular values and vectors of noisy low-rank matrices is fundamental to many statistical and scientific problems. Theoretical understanding currently derives from asymptotic analysis under one of two regimes: classical, with a fixed number of rows, large number of columns or vice versa; and proportional, with large numbers of rows and columns, proportional to one another. This paper is concerned with the disproportional regime, where the matrix is either “tall and narrow” or “short and wide”: we study sequences of matrices of size n×mn with aspect ratio n/mn→0 or n/mn→∞ as n→∞. This regime has important “big data” applications.

Suggested Citation

  • Feldman, Michael J., 2023. "Spiked singular values and vectors under extreme aspect ratios," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    • Handle: RePEc:eee:jmvana:v:196:y:2023:i:c:s0047259x23000337
    DOI: 10.1016/j.jmva.2023.105187
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    1. John Novembre & Toby Johnson & Katarzyna Bryc & Zoltán Kutalik & Adam R. Boyko & Adam Auton & Amit Indap & Karen S. King & Sven Bergmann & Matthew R. Nelson & Matthew Stephens & Carlos D. Bustamante, 2008. "Genes mirror geography within Europe," Nature, Nature, vol. 456(7219), pages 274-274, November.
    2. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
    3. 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.
    4. John Novembre & Toby Johnson & Katarzyna Bryc & Zoltán Kutalik & Adam R. Boyko & Adam Auton & Amit Indap & Karen S. King & Sven Bergmann & Matthew R. Nelson & Matthew Stephens & Carlos D. Bustamante, 2008. "Genes mirror geography within Europe," Nature, Nature, vol. 456(7218), pages 98-101, November.
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