  This paper provides a comprehensive estimation framework via nuclear norm
plus $l_1$ norm penalization for 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.
In that context, existing approaches based on principal components may lead to
misestimate the latent rank, due to the numerical instability of sample
eigenvalues. On the contrary, the proposed optimization problem retrieves the
latent covariance structure and exactly recovers the latent rank and the
residual sparsity pattern. Conditioning on them, the asymptotic rates of the
subsequent ordinary least squares estimates of loadings and factor scores are
provided, the recovered latent eigenvalues are shown to be maximally
concentrated and the estimates of factor scores via Bartlett's and Thompson's
methods are proved to be the most precise given the data. The validity of
outlined results is highlighted in an exhaustive simulation study and in a real
financial data example.
