A finite sample estimator for large covariance matrices.

The present paper concerns large covariance matrix estimation via compositeminimization under the assumption of low rank plus sparse structure. In thisapproach, the low rank plus sparse decomposition of the covariance matrix isrecovered by least squares minimization under nuclear norm plus $l_1$ normpenalization. This paper proposes a new estimator of that family based on anadditional least-squares re-optimization step aimed at un-shrinking theeigenvalues of the low rank component estimated at the first step. We provethat such un-shrinkage causes the final estimate to approach the target asclosely as possible while recovering exactly the underlying low rank and sparsematrix varieties. In addition, consistency is guaranteed until $p\log(p)\gg n$,where $p$ is the dimension and $n$ is the sample size, and recovery is ensuredif the latent eigenvalues scale to $p^{\alpha}$, $\alpha \in[0,1]$. Theresulting estimator is called UNALCE (UNshrunk ALgebraic Covariance Estimator)and is shown to outperform both LOREC and POET estimators, especially for whatconcerns fitting properties and sparsity pattern detection.