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Partial estimation of covariance matrices
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  • Published: 01 March 2011

Partial estimation of covariance matrices

  • Elizaveta Levina1 &
  • Roman Vershynin2 

Probability Theory and Related Fields volume 153, pages 405–419 (2012)Cite this article

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  • 13 Citations

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Abstract

A classical approach to accurately estimating the covariance matrix Σ of a p-variate normal distribution is to draw a sample of size n > p and form a sample covariance matrix. However, many modern applications operate with much smaller sample sizes, thus calling for estimation guarantees in the regime \({n \ll p}\). We show that a sample of size n = O(m log6 p) is sufficient to accurately estimate in operator norm an arbitrary symmetric part of Σ consisting of m ≤ n nonzero entries per row. This follows from a general result on estimating Hadamard products M · Σ, where M is an arbitrary symmetric matrix.

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Author information

Authors and Affiliations

  1. Department of Statistics, University of Michigan, 1085 S. University, Ann Arbor, MI, 48109, USA

    Elizaveta Levina

  2. Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI, 48109, USA

    Roman Vershynin

Authors
  1. Elizaveta Levina
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  2. Roman Vershynin
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Corresponding author

Correspondence to Roman Vershynin.

Additional information

Partially supported by NSF grants DMS 0805798 (E.L.) and FRG DMS 0918623, DMS 1001829 (R.V.).

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Cite this article

Levina, E., Vershynin, R. Partial estimation of covariance matrices. Probab. Theory Relat. Fields 153, 405–419 (2012). https://doi.org/10.1007/s00440-011-0349-4

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  • Received: 07 October 2010

  • Revised: 11 February 2011

  • Published: 01 March 2011

  • Issue Date: August 2012

  • DOI: https://doi.org/10.1007/s00440-011-0349-4

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Mathematics Subject Classification (2000)

  • 62H12 (primary)
  • 60B20 (secondary)
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