Asymptotically minimax empirical Bayes estimation of a sparse normal mean

We consider the problem of estimating a sparse high-dimensional normal mean vector using an empirical Bayes approach. Our proposed model features a unique form of shrinkage, and has strong finite-sample performance. We also prove, under certain conditions, that our empirical Bayes posterior concentrates on balls, centered at the true mean, with squared radius proportional to the frequentist minimax rate for the given sparsity class. Asymptotic minimaxity of the corresponding empirical Bayes posterior mean is also shown.