Estimating Structured High-Dimensional Covariance and Precision Matrices: Optimal Rates and Adaptive Estimation
نویسندگان
چکیده
This is an expository paper that reviews recent developments on optimal estimation of structured high-dimensional covariance and precision matrices. Minimax rates of convergence for estimating several classes of structured covariance and precision matrices, including bandable, Toeplitz, and sparse covariance matrices as well as sparse precision matrices, are given under the spectral norm loss. Data-driven adaptive procedures for estimating various classes of matrices are presented. Some key technical tools including large deviation results and minimax lower bound arguments that are used in the theoretical analyses are discussed. In addition, estimation under other losses and a few related problems such as Gaussian graphical models, sparse principal component analysis, and hypothesis testing on the covariance structure are considered. Some open problems on estimating high-dimensional covariance and precision matrices and their functionals are also discussed.
منابع مشابه
Rejoinder of “Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation”∗
We are deeply grateful to the discussants for providing constructive and stimulating comments and suggestions. Our paper gives a survey of recent optimality and adaptivity results on estimating various families of structured covariance and precision matrices in the high-dimensional setting, with a focus on understanding the intrinsic difficulty of the problems. To achieve this goal, we present ...
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