Masked Toeplitz covariance estimation.

The problem of estimating the covariance matrix $Sigma$ of a $p$-variate distribution based on its $n$ observations arises in many data analysis contexts. While for $nu003ep$, the classical sample covariance matrix $hat{Sigma}_n$ is a good estimator for $Sigma$, it fails in the high-dimensional setting when $nll p$. In this scenario one requires prior knowledge about the structure of the covariance matrix in order to construct reasonable estimators. Under the common assumption that $Sigma$ is sparse, a refined estimator is given by $Mcdothat{Sigma}_n$, where $M$ is a suitable symmetric mask matrix indicating the nonzero entries of $Sigma$ and $cdot$ denotes the entrywise product of matrices. In the present work we assume that $Sigma$ has Toeplitz structure corresponding to stationary signals. This suggests to average the sample covariance $hat{Sigma}_n$ over the diagonals in order to obtain an estimator $tilde{Sigma}_n$ of Toeplitz structure. Assuming in addition that $Sigma$ is sparse suggests to study estimators of the form $Mcdottilde{Sigma}_n$. For Gaussian random vectors and, more generally, random vectors satisfying the convex concentration property, our main result bounds the estimation error in terms of $n$ and $p$ and shows that accurate estimation is indeed possible when $n ll p$. The new bound significantly generalizes previous results by Cai, Ren and Zhou and provides an alternative proof. Our analysis exploits the connection between the spectral norm of a Toeplitz matrix and the supremum norm of the corresponding spectral density function.