Sparsity meets correlation in Gaussian sequence model

We study estimation of an s-sparse signal in the p-dimensional Gaussian sequence model with equicorrelated observations and derive the minimax rate. A new phenomenon emerges from correlation, namely the rate scales with respect to p-2s and exhibits a phase transition at p-2s asymp √p. Correlation is shown to be a blessing provided it is sufficiently strong, and the critical correlation level exhibits a delicate dependence on the sparsity level. Due to correlation, the minimax rate is driven by two subproblems: estimation of a linear functional (the average of the signal) and estimation of the signal's (p-1)-dimensional projection onto the orthogonal subspace. The high-dimensional projection is estimated via sparse regression and the linear functional is cast as a robust location estimation problem. Existing robust estimators turn out to be suboptimal, and we show a kernel mode estimator with a widening bandwidth exploits the Gaussian character of the data to achieve the optimal estimation rate.