MINIMAX OPTIMAL RATES OF ESTIMATION IN HIGH DIMENSIONAL ADDITIVE MODELS

• 成果类型: Article
• 署名作者: Yuan, Ming; Zhou, Ding-Xuan
• 署名单位: University of Wisconsin System; University of Wisconsin Madison; City University of Hong Kong
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1422
• 发表日期: 2016
• 页码: 2564-2593
• 关键词: component selection; variable selection; DANTZIG SELECTOR; regression; Lasso; CONVERGENCE; sparsity

摘要

We establish minimax optimal rates of convergence for estimation in a high dimensional additive model assuming that it is approximately sparse. Our results reveal a behavior universal to this class of high dimensional problems. In the sparse regime when the components are sufficiently smooth or the dimensionality is sufficiently large, the optimal rates are identical to those for high dimensional linear regression and, therefore, there is no additional cost to entertain a nonparametric model. Otherwise, in the so-called smooth regime, the rates coincide with the optimal rates for estimating a univariate function and, therefore, they are immune to the curse of dimensionality.