MINIMAX-OPTIMAL NONPARAMETRIC REGRESSION IN HIGH DIMENSIONS

• 成果类型: Article
• 署名作者: Yang, Yun; Tokdar, Surya T.
• 署名单位: University of California System; University of California Berkeley; Duke University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/14-AOS1289
• 发表日期: 2015
• 页码: 652-674
• 关键词: posterior distributions; sparsity recovery; Gaussian process; rates; CONVERGENCE; selection; Lasso; Consistency

摘要

Minimax L-2 risks for high-dimensional nonparametric regression are derived under two sparsity assumptions: (1) the true regression surface is a sparse function that depends only on d = O(log n) important predictors among a list of p predictors, with log p = o(n); (2) the true regression surface depends on O(n) predictors but is an additive function where each additive component is sparse but may contain two or more interacting predictors and may have a smoothness level different from other components. For either modeling assumption, a practicable extension of the widely used Bayesian Gaussian process regression method is shown to adaptively attain the optimal minimax rate (up to log n terms) asymptotically as both n, p --> infinity with log p = o(n).