Large Covariance Matrix Estimation by Composite Minimization

The present thesis concerns large covariance matrix estimation via composite minimization under the assumption of low rank plus sparse structure. Existing methods like POET (Principal Orthogonal complEment Thresholding) perform estimation by extracting principal components and then applying a soft-thresholding algorithm. In contrast, the numerical approach recovers the low rank plus sparse decomposition of the covariance matrix by least squares minimization under nuclear norm plus $l_1$ norm penalization. This non-smooth convex minimization procedure is based on semidefinite programming and subdifferential methods, resulting in two separable problems solved by a singular value thresholding plus a soft-thresholding algorithm. The most recent estimator in the literature is called LOREC (LOw Rank and sparsE Covariance estimator) and provides non-asymptotic error rates as well as identifiability conditions in the context of algebraic geometry. We propose 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 and recovery are guaranteed until $p^\alpha\log(p)\gg n$, where $p$ is the dimension and $n$ is the sample size, if the latent eigenvalues scale to $p^{\alpha}$, $\alpha \in[0,1]$. In the same context, an ad-hoc model selection criterion which detects the optimal point in terms of composite penalty is proposed. Empirical results, coming from a wide original simulation study where various low rank plus sparse settings are simulated according to different parameter values, are described outlining in detail the improvements over existing methods. Two real data-sets, regarding ECB banking supervisory data and UK market data respectively, are finally explored highlighting the usefulness of our method in practical applications.