Adaptive Thresholding for Sparse Covariance Matrix Estimation

Abstract In this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries. The estimators are fully data-driven and demonstrate excellent performance both theoretically and numerically. It is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm. In contrast, the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces. Support recovery is discussed as well. The adaptive thresholding estimators are easy to implement. The numerical performance of the estimators is studied using both simulated and real data. Simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators. The method is also illustrated in an analysis on a dataset from a small round blue-cell tumor microarray experiment. A supplement to this article presenting additional technical proofs is available online. Keywords: : Frobenius normOptimal rate of convergenceSpectral normSupport recoveryUniversal thresholding