Empirical Bayes Estimators for Sparse Sequences

The problem of estimating a high-dimensional sparse vector theta is an element of R-n from an observation in i.i.d. Gaussian noise is considered. An empirical Bayes shrinkage estimator, derived using a Bernoulli-Gaussian prior, is analyzed and compared with the well-known soft-thresholding estimator using squared-error loss as a measure of performance. We obtain concentration inequalities for the Stein's unbiased risk estimate and the loss function of both estimators. Depending on the underlying theta, either the proposed empirical Bayes (eBayes) estimator or soft-thresholding may have smaller loss. We consider a hybrid estimator that attempts to pick the better of the soft-thresholding estimator and the eBayes estimator by comparing their risk estimates. It is shown that: i) the loss of the hybrid estimator concentrates on the minimum of the losses of the two competing estimators, and ii) the risk of the hybrid estimator is within order 1/root n of the minimum of the two risks. Simulation results are provided to support the theoretical results.