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Phase transition in limiting distributions of coherence of high-dimensional random matrices

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  • Tony Cai, T.
  • Jiang, Tiefeng

Abstract

The coherence of a random matrix, which is defined to be the largest magnitude of the Pearson correlation coefficients between the columns of the random matrix, is an important quantity for a wide range of applications including high-dimensional statistics and signal processing. Inspired by these applications, this paper studies the limiting laws of the coherence of n×p random matrices for a full range of the dimension p with a special focus on the ultra high-dimensional setting. Assuming the columns of the random matrix are independent random vectors with a common spherical distribution, we give a complete characterization of the behavior of the limiting distributions of the coherence. More specifically, the limiting distributions of the coherence are derived separately for three regimes: 1nlogp→0, 1nlogp→β∈(0,∞), and 1nlogp→∞. The results show that the limiting behavior of the coherence differs significantly in different regimes and exhibits interesting phase transition phenomena as the dimension p grows as a function of n. Applications to statistics and compressed sensing in the ultra high-dimensional setting are also discussed.

Suggested Citation

  • Tony Cai, T. & Jiang, Tiefeng, 2012. "Phase transition in limiting distributions of coherence of high-dimensional random matrices," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 24-39.
    • Handle: RePEc:eee:jmvana:v:107:y:2012:i:c:p:24-39
    DOI: 10.1016/j.jmva.2011.11.008
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    References listed on IDEAS

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    1. Jianqing Fan & Jinchi Lv, 2008. "Sure independence screening for ultrahigh dimensional feature space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(5), pages 849-911, November.
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    Cited by:

    1. Long Feng & Tiefeng Jiang & Binghui Liu & Wei Xiong, 2020. "Max-sum tests for cross-sectional dependence of high-demensional panel data," Papers 2007.03911, arXiv.org.
    2. Ian W. McKeague & Min Qian, 2015. "An Adaptive Resampling Test for Detecting the Presence of Significant Predictors," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(512), pages 1422-1433, December.
    3. Tiefeng Jiang & Danning Li, 2015. "Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations," Journal of Theoretical Probability, Springer, vol. 28(3), pages 804-847, September.
    4. Ley, Christophe & Paindaveine, Davy & Verdebout, Thomas, 2015. "High-dimensional tests for spherical location and spiked covariance," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 79-91.
    5. Christine Cutting & Davy Paindaveine & Thomas Verdebout, 2015. "Tests of Concentration for Low-Dimensional and High-Dimensional Directional Data," Working Papers ECARES ECARES 2015-05, ULB -- Universite Libre de Bruxelles.
    6. Christine Cutting & Davy Paindaveine & Thomas Verdebout, 2015. "Testing Uniformity on High-Dimensional Spheres against Contiguous Rotationally Symmetric Alternatives," Working Papers ECARES ECARES 2015-04, ULB -- Universite Libre de Bruxelles.
    7. Bar, Haim & Wells, Martin T., 2023. "On graphical models and convex geometry," Computational Statistics & Data Analysis, Elsevier, vol. 187(C).
    8. Davy Paindaveine & Thomas Verdebout, 2013. "Universal Asymptotics for High-Dimensional Sign Tests," Working Papers ECARES ECARES 2013-40, ULB -- Universite Libre de Bruxelles.
    9. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.

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