NONASYMPTOTIC RATES FOR MANIFOLD, TANGENT SPACE AND CURVATURE ESTIMATION

被引：40

作者：

Aamari, Eddie ^{[1
]}

Levrard, Clement ^{[2
]}

机构：

[1] Univ Calif San Diego, Dept Math, 9500 Gilman Dr, La Jolla, CA 92093 USA

[2] Batiment Sophie Germain Univ Paris Diderot, Lab Probabilites & Modeles Aleatoires, F-75013 Paris, France

来源：

ANNALS OF STATISTICS | 2019年 / 47卷 / 01期

基金：

欧洲研究理事会;

关键词：

Geometric inference; minimax; manifold learning; CONFIDENCE;

D O I：

10.1214/18-AOS1685

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

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.

引用

页码：177 / 204

页数：28

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