Nonasymptotic support recovery for high-dimensional sparse covariance matrices

For high-dimensional data, the standard empirical estimator for the covariance matrix is very poor, and thus many methods have been proposed to more accurately estimate the covariance structure of high-dimensional data. In this article, we consider estimation under the assumption of sparsity but regularize with respect to the individual false-positive rate for incorrectly including a matrix entry in the support of the final estimator. The two benefits of this approach are (1) an interpretable regularization parameter removing the need for computationally expensive tuning and (2) extremely fast computation time arising from use of a binary search algorithm implemented to find the best estimator within a carefully constructed operator norm ball. We compare our approach to universal thresholding estimators and lasso-style penalized estimators on both simulated data and data from gene expression for cancerous tumours.