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Estimating covariance and precision matrices along subspaces

Kereta, Ž; Klock, T; (2021) Estimating covariance and precision matrices along subspaces. Electronic Journal of Statistics , 15 (1) pp. 554-588. 10.1214/20-EJS1782. Green open access

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Abstract

We study the accuracy of estimating the covariance and the precision matrix of a D-variate sub-Gaussian distribution along a prescribed subspace or direction using the finite sample covariance. Our results show that the estimation accuracy depends almost exclusively on the components of the distribution that correspond to desired subspaces or directions. This is relevant and important for problems where the behavior of data along a lower-dimensional space is of specific interest, such as dimension reduction or structured regression problems. We also show that estimation of precision matrices is almost independent of the condition number of the covariance matrix. The presented applications include direction-sensitive eigenspace perturbation bounds, relative bounds for the smallest eigenvalue, and the estimation of the single-index model. For the latter, a new estimator, derived from the analysis, with strong theoretical guarantees and superior numerical performance is proposed.

Type: Article
Title: Estimating covariance and precision matrices along subspaces
Open access status: An open access version is available from UCL Discovery
DOI: 10.1214/20-EJS1782
Publisher version: https://doi.org/10.1214/20-EJS1782
Language: English
Additional information: This is an Open Access article published under a Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/).
Keywords: Covariance matrix, finite sample bounds, dimension reduction, rate of convergence, ordinary least squares, single-index model, precision matrix
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/10119467
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