  This paper provides a comprehensive estimation framework for large covariance
matrices via a log-det heuristics augmented by a nuclear norm plus $l_{1}$ norm
penalty. %We develop the model framework, which includes high-dimensional
approximate factor models with a sparse residual covariance. The underlying
assumptions allow for non-pervasive latent eigenvalues and a prominent residual
covariance pattern. 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 with high probability both the covariance matrix components and the
latent rank and the residual sparsity pattern, 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 condition
for the latent geometric manifolds is provided. The validity of outlined
results is highlighted by means of an exhaustive simulation study and a real
financial data example involving euro zone banks.
