Large covariance matrix estimation via penalized log-det heuristics

Abstract: This paper provides a comprehensive estimation framework for large covariance matrices via a log-det heuristics augmented by a nuclear norm plus $\ell_{1}$-norm penalty. We develop the model framework, which includes high-dimensional approximate factor models with a sparse residual covariance. We prove that the aforementioned log-det heuristics is locally convex with a Lipschitz-continuous gradient, so that a proximal gradient algorithm may be stated to numerically solve the problem while controlling the threshold parameters. The proposed optimization strategy recovers in a single step both the covariance matrix components and the latent rank and the residual sparsity pattern with high probability, and performs systematically not worse than the corresponding estimators employing Frobenius loss in place of the log-det heuristics. The error bounds for the ensuing low rank and sparse covariance matrix estimators are established, and the identifiability conditions for the latent geometric manifolds are provided, improving existing literature. The validity of outlined results is highlighted by an exhaustive simulation study and a financial data example involving Euro Area banks.