When the data are high dimensional, widely used multivariate statistical methods such as principal component analysis can behave in unexpected ways. In settings where the dimension of the observations is comparable to the sample size, upward bias in sample eigenvalues and inconsistency of sample eigenvectors are among the most notable phenomena that appear. These phenomena, and the limiting behavior of the rescaled extreme sample eigenvalues, have recently been investigated in detail under the spiked covariance model. The behavior of the bulk of the sample eigenvalues under weak distributional assumptions on the observations has been described. These results have been exploited to develop new estimation and hypothesis testing methods for the population covariance matrix. Furthermore, partly in response to these phenomena, alternative classes of estimation procedures have been developed by exploiting sparsity of the eigenvectors or the covariance matrix. This paper gives an orientation to these areas.

We study the problem of estimating the leading eigenvectors of a high-dimensional population covariance matrix based on independent Gaussian observations. We establish lower bounds on the rates of convergence of the estimators of the leading eigenvectors under $l^q$-sparsity constraints when an $l^2$ loss function is used. We also propose an estimator of the leading eigenvectors based on a coordinate selection scheme combined with PCA and show that the proposed estimator achieves the optimal rate of convergence under a sparsity regime. Moreover, we establish that under certain scenarios, the usual PCA achieves the minimax convergence rate.

We consider the linear inverse problem of estimating an unknown signal $f$ from noisy measurements on $Kf$ where the linear operator $K$ admits a wavelet-vaguelette decomposition (WVD). We formulate the problem in the Gaussian sequence model and propose estimation based on complexity penalized regression on a level-by-level basis. We adopt squared error loss and show that the estimator achieves exact rate-adaptive optimality as $f$ varies over a wide range of Besov function classes.

We study the problem of estimating the leading eigenvectors of a high-dimensional population covariance matrix based on independent Gaussian observations. We establish a lower bound on the minimax risk of estimators under the $l_2$ loss, in the joint limit as dimension and sample size increase to infinity, under various models of sparsity for the population eigenvectors. The lower bound on the risk points to the existence of different regimes of sparsity of the eigenvectors. We also propose a new method for estimating the eigenvectors by a two-stage coordinate selection scheme.