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 formalizethe notion of asymmetric modelvia groupinvariance. 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 interms of the sample complexity. We further explore the consequences of symmetry inrelated model-selection problems such as the learning of sparse covariance and inverse covarianc
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