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We study high-dimensional covariance/precision matrix estimation under theassumption that the covariance/precision matrix can be decomposed into alow-rank component L and a diagonal component D. The rank of L can either bechosen to be small or controlled by a penalty function. Under moderateconditions on the population covariance/precision matrix itself and on thepenalty function, we prove some consistency results for our estimators. Ablockwise coordinate descent algorithm, which iteratively updates L and D, isthen proposed to obtain the estimator in practice. Finally, various numericalexperiments are presented: using simulated data, we show that our estimatorperforms quite well in terms of the Kullback-Leibler loss; using stock returndata, we show that our method can be applied to obtain enhanced solutions tothe Markowitz portfolio selection problem.

High-dimensional covariance matrix estimation using a low-rank and diagonal decomposition