Sparsity meets correlation in Gaussian sequence model
The Annals of Statistics · 2025 · Volume 53 · Issue 3
Abstract
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≍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.
Keywords
62C20, 62G05, 62G35, Correlation, High-dimensional statistics, Minimax rate, mode estimation
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