Partial estimation of covariance matrices |
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Authors: | Elizaveta Levina Roman Vershynin |
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Institution: | 1. Department of Statistics, University of Michigan, 1085 S. University, Ann Arbor, MI, 48109, USA 2. Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI, 48109, USA
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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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