SPARSITY MEETS CORRELATION IN GAUSSIAN SEQUENCE MODEL

• 成果类型: Article
• 署名作者: Kotekal, Subhodh; Gao, Chao
• 署名单位: University of Chicago
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/25-AOS2494
• 发表日期: 2025
• 页码: 1095-1122
• 关键词: linear functionals; Nonparametric Regression; Adaptive estimation; Minimax Estimation; wavelet shrinkage; Robust Estimation; density; estimators; rates; CONVERGENCE

摘要

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 asymptotic to root root p root 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.