Group Symmetry and Covariance Regularization

Details Group Symmetry and Covariance Regularization Group Symmetry and Covariance Regularization

2011-11-30

arxiv.org/abs/1111.7061v1

Mathematics

Statistics Theory

38 pages, 3 figures, submitted to Electronic Journal of Statistics

Scientific paper

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 on related model-selection problems such as the learning of sparse covariance and inverse covariance matrices. We also verify our results with simulations.

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