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

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 block‐wise 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. The Canadian Journal of Statistics 48: 308–337; 2020 © 2019 Statistical Society of Canada