  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 $p-2s \asymp \sqrt{p}$. 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.
