  In this paper, we study the problem of high-dimensional approximately
low-rank covariance matrix estimation with missing observations. We propose a
simple procedure computationally tractable in high-dimension and that does not
require imputation of the missing data. We establish non-asymptotic sparsity
oracle inequalities for the estimation of the covariance matrix with the
Frobenius and spectral norms, valid for any setting of the sample size and the
dimension of the observations. We further establish minimax lower bounds
showing that our rates are minimax optimal up to a logarithmic factor.
