## Electronic Journal of Statistics

### Group symmetry and covariance regularization

#### Abstract

Statistical models that possess symmetry arise in diverse settings such as random fields associated to geophysical phenomena, exchangeable processes in Bayesian statistics, and cyclostationary processes in engineering. We formalize the notion of a symmetric model via group invariance. We propose projection onto a group fixed point subspace as a fundamental way of regularizing covariance matrices in the high-dimensional regime. In terms of parameters associated to the group we derive precise rates of convergence of the regularized covariance matrix and demonstrate that significant statistical gains may be expected in terms of the sample complexity. We further explore the consequences of symmetry in related model-selection problems such as the learning of sparse covariance and inverse covariance matrices. We also verify our results with simulations.

#### Article information

Source
Electron. J. Statist., Volume 6 (2012), 1600-1640.

Dates
First available in Project Euclid: 26 September 2012

https://projecteuclid.org/euclid.ejs/1348665230

Digital Object Identifier
doi:10.1214/12-EJS723

Mathematical Reviews number (MathSciNet)
MR2988459

Zentralblatt MATH identifier
1295.62023

Subjects
Primary: 62F12: Asymptotic properties of estimators 62H12: Estimation

#### Citation

Shah, Parikshit; Chandrasekaran, Venkat. Group symmetry and covariance regularization. Electron. J. Statist. 6 (2012), 1600--1640. doi:10.1214/12-EJS723. https://projecteuclid.org/euclid.ejs/1348665230

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