  We study high-dimensional covariance/precision matrix estimation under the
assumption that the covariance/precision matrix can be decomposed into a
low-rank component L and a diagonal component D. The rank of L can either be
chosen to be small or controlled by a penalty function. Under moderate
conditions on the population covariance/precision matrix itself and on the
penalty function, we prove some consistency results for our estimators. A
blockwise coordinate descent algorithm, which iteratively updates L and D, is
then proposed to obtain the estimator in practice. Finally, various numerical
experiments are presented: using simulated data, we show that our estimator
performs quite well in terms of the Kullback-Leibler loss; using stock return
data, we show that our method can be applied to obtain enhanced solutions to
the Markowitz portfolio selection problem.
