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This work provides a unified analysis of the properties of the sample covariance matrix $\Sigma_n$ over the class of $p\times p$ population covariance matrices $\Sigma$ of reduced effective rank $r_e(\Sigma)$.
- This class includes scaled factor models and covariance matrices with decaying spectrum.
- We consider $r_e(\Sigma)$ as a measure of matrix complexity, and obtain sharp minimax rates on the operator and Frobenius norm of $\Sigma_n-\Sigma$, as a function of $r_e(\Sigma)$ and $\|\Sigma\|_2$, the operator norm of $\Sigma$.
- With guidelines offered by the optimal rates, we define classes of matrices of reduced effective rank over which $\Sigma_n$ is an accurate estimator.
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