Doklady Mathematics

, Volume 98, Issue 2, pp 511–514 | Cite as

Confidence Sets for Spectral Projectors of Covariance Matrices

  • A. A. Naumov
  • V. G. Spokoiny
  • V. V. Ulyanov


A sample X1,...,Xn consisting of independent identically distributed vectors in ℝp with zero mean and a covariance matrix Σ is considered. The recovery of spectral projectors of high-dimensional covariance matrices from a sample of observations is a key problem in statistics arising in numerous applications. In their 2015 work, V. Koltchinskii and K. Lounici obtained nonasymptotic bounds for the Frobenius norm \(\parallel {P_r} - {\hat P_r}{\parallel _2}\) of the distance between sample and true projectors and studied asymptotic behavior for large samples. More specifically, asymptotic confidence sets for the true projector Pr were constructed assuming that the moment characteristics of the observations are known. This paper describes a bootstrap procedure for constructing confidence sets for the spectral projector Pr of the covariance matrix Σ from given data. This approach does not use the asymptotical distribution of \(\parallel {P_r} - {\hat P_r}{\parallel _2}\) and does not require the computation of its moment characteristics. The performance of the bootstrap approximation procedure is analyzed.


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  1. 1.
    J. Tropp, Found. Comput. Math. 12 (4), 389–434 (2012).MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Vershynin, “Introduction to the non-asymptotic analysis of random matrices,” in Compressed Sensing: Theory and Applications (Cambridge Univ. Press, Cambridge, 2012), pp. 210–268.CrossRefGoogle Scholar
  3. 3.
    R. van Handel, Trans. Am. Math. Soc. 369 (11), 8161–8178 (2017).CrossRefGoogle Scholar
  4. 4.
    V. Koltchinskii and K. Lounici, “Concentration inequalities and moment bounds for sample covariance operators” (2015). ArXiv:1405.2468.zbMATHGoogle Scholar
  5. 5.
    V. Koltchinskii and K. Lounici, Ann. Stat. 45 (1), 121–157 (2017).CrossRefGoogle Scholar
  6. 6.
    V. Spokoiny, and M. Zhilova, Ann. Stat. 43 (6), 2653–2675 (2015).CrossRefGoogle Scholar
  7. 7.
    V. Chernozhukov, D. Chetverikov, and K. Kato, Ann. Stat. 41 (6), 2786–2819 (2013).CrossRefGoogle Scholar
  8. 8.
    V. Chernozhukov, D. Chetverikov, and K. Kato, Ann. Probab. 45 (4), 2309–2352 (2017).MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Naumov, V. Spokoiny, and V. Ulyanov, “Bootstrap confidence sets for spectral projectors of sample covariance” (2017). arXiv:1703.00871.Google Scholar
  10. 10.
    F. Götze, A. Naumov, V. Spokoiny, and V. Ulyanov, “Large ball probabilities, Gaussian comparison and anti-concentration,” Bernoulli 25 (2019). arXiv:1708.08663v2.Google Scholar
  11. 11.
    A. Naumov, V. Spokoiny, Yu. Tavyrikov and V. Ulyanov, “Nonasymptotic estimates for the closeness of Gaussian measures on balls”, Dokl. Math. 98 (2018).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • A. A. Naumov
    • 1
    • 2
  • V. G. Spokoiny
    • 2
    • 3
    • 4
    • 5
  • V. V. Ulyanov
    • 1
    • 6
  1. 1.National Research University Higher School of EconomicsMoscowRussia
  2. 2.Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia
  3. 3.Skolkovo Institute of Science and Technology, SkolkovoMoscowRussia
  4. 4.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  5. 5.Humboldt University of BerlinBerlinGermany
  6. 6.Faculty of Computational Mathematics and CyberneticsMoscow State UniversityMoscowRussia

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