  In this paper, we propose two new algorithms for maximum-likelihood
estimation (MLE) of high dimensional sparse covariance matrices. Unlike most of
the state of-the-art methods, which either use regularization techniques or
penalize the likelihood to impose sparsity, we solve the MLE problem based on
an estimated covariance graph. More specifically, we propose a two-stage
procedure: in the first stage, we determine the sparsity pattern of the target
covariance matrix (in other words the marginal independence in the covariance
graph under a Gaussian graphical model) using the multiple hypothesis testing
method of false discovery rate (FDR), and in the second stage we use either a
block coordinate descent approach to estimate the non-zero values or a proximal
distance approach that penalizes the distance between the estimated covariance
graph and the target covariance matrix. Doing so gives rise to two different
methods, each with its own advantage: the coordinate descent approach does not
require tuning of any hyper-parameters, whereas the proximal distance approach
is computationally fast but requires a careful tuning of the penalty parameter.
Both methods are effective even in cases where the number of observed samples
is less than the dimension of the data. For performance evaluation, we test the
proposed methods on both simulated and real-world data and show that they
provide more accurate estimates of the sparse covariance matrix than two
state-of-the-art methods.
