  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 in Frobenius norm while recovering exactly the underlying
low rank and sparsity pattern. Consistency is guaranteed when $n$ is at least
$O(p^{\frac{3}{2}\delta})$, provided that the maximum number of non-zeros per
row in the sparse component is $O(p^{\delta})$ with $\delta \leq \frac{1}{2}$.
Consistent recovery is ensured if the latent eigenvalues scale to $p^{\alpha}$,
$\alpha \in[0,1]$, while rank consistency is ensured if $\delta \leq \alpha$.
The resulting estimator is called UNALCE (UNshrunk ALgebraic Covariance
Estimator) and is shown to outperform state of the art estimators, especially
for what concerns fitting properties and sparsity pattern detection. The
effectiveness of UNALCE is highlighted on a real example regarding ECB banking
supervisory data.
