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.

翻译：我们研究了在$p$维等相关高斯序列模型中估计$s$-稀疏信号的问题，并推导了极小化极大速率。相关性催生了一个新现象：估计速率与$p-2s$成比例，并在$p-2s \asymp \sqrt{p}$处发生相变。研究表明，当相关性足够强时，它反而成为一种有利因素，而临界相关水平对稀疏程度表现出微妙的依赖性。由于相关性的存在，极小化极大速率由两个子问题驱动：线性泛函的估计（信号均值）以及信号在正交补空间上的$(p-1)$维投影估计。高维投影通过稀疏回归进行估计，而线性泛函则被转化为稳健位置估计问题。现有稳健估计量被证明是次优的，我们提出了一种带宽递增的核众数估计器，该估计器利用数据的高斯特性实现了最优估计速率。