The Masked Sample Covariance Estimator: An Analysis via Matrix Concentration Inequalities

By Richard Y. Chen, Alex Gittens, Joel A. Tropp. 2011

73 citations recorded by the source corpus.

Covariance estimation becomes challenging in the regime where the number p of variables outstrips the number n of samples available to construct the estimate. One way to circumvent this problem is to assume that the covariance matrix is nearly sparse and to focus on estimating only the significant entries. To analyze this approach, Levina and Vershynin (2011) introduce a formalism called masked covariance estimation, where each entry of the sample covariance estimator is reweighted to reflect an a priori assessment of its importance. This paper provides a short analysis of the masked sample covariance estimator by means of a matrix concentration inequality. The main result applies to general distributions with at least four moments. Specialized to the case of a Gaussian distribution, the theory offers qualitative improvements over earlier work. For example, the new results show that n = O(B log^2 p) samples suffice to estimate a banded covariance matrix with bandwidth B up to a relative spectral-norm error, in contrast to the sample complexity n = O(B log^5 p) obtained by Levina and Vershynin.

Subfields: Computational Mechanics, Computer Networks and Communications, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/84a88232ca71ce0913dcc32b49b53c8cb61bd8f0

Citations in the corpus

• Communication-efficient algorithms for statistical optimization (2012, unknown): “For such matrices, we have: Lemma 16 (Chen et al. (2012), Theorem A.1(2)) Let Xi ∈ Rd×d be independent and symmetrically distributed Hermitian matrices.”