A finite sample estimator for large covariance matrices
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The present paper concerns large covariance matrix estimation via composite minimization under the assumption of low rank plus sparse structure. In this approach, the low rank plus sparse decomposition of the covariance matrix is recovered by least squares minimization under nuclear norm plus l_1 norm penalization. This paper proposes a new estimator of that family based on an additional least-squares re-optimization step aimed at un-shrinking the eigenvalues of the low rank component estimated at the first step. We prove that such un-shrinkage causes the final estimate to approach the target as closely as possible while recovering exactly the underlying low rank and sparse matrix varieties. In addition, consistency is guaranteed until p(p)≫ n, where p is the dimension and n is the sample size, and recovery is ensured if the latent eigenvalues scale to p^α, α∈[0,1]. The resulting estimator is called UNALCE (UNshrunk ALgebraic Covariance Estimator) and is shown to outperform both LOREC and POET estimators, especially for what concerns fitting properties and sparsity pattern detection.
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