NONASYMPTOTIC RATES FOR MANIFOLD, TANGENT SPACE AND CURVATURE ESTIMATION

• 成果类型: Article
• 署名作者: Aamari, Eddie; Levrard, Clement
• 署名单位: University of California System; University of California San Diego; Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1685
• 发表日期: 2019
• 页码: 177-204
• 关键词: confidence

摘要

Given a noisy sample from a submanifold M subset of R-D, we derive optimal rates for the estimation of tangent spaces TXM, the second fundamental form IIXM and the submanifold M. After motivating their study, we introduce a quantitative class of C-k-submanifolds in analogy with Holder classes. The proposed estimators are based on local polynomials and allow to deal simultaneously with the three problems at stake. Minimax lower bounds are derived using a conditional version of Assouad's lemma when the base point X is random.