NONASYMPTOTIC RATES FOR MANIFOLD, TANGENT SPACE AND CURVATURE ESTIMATION
NONASYMPTOTIC RATES FOR MANIFOLD, TANGENT SPACE AND CURVATURE ESTIMATION
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DOI：
10.1214/18-aos1685
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发表时间：
2019-02-01
影响因子：
4.5
通讯作者：
Levrard, Clement
中科院分区：
文献类型：
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作者：
Aamari, Eddie;Levrard, Clement
给定R-D的子流形M子集的噪声样本，我们推导出切空间TXM，第二基本形式IIXM和子流形M的估计的最优速率。在激励他们的研究之后，我们引入了与Holder类类似的c -k子流形的定量类。所提出的估计是基于局部多项式的，并允许同时处理这三个问题。当基点X是随机时，使用Assouad引理的条件版本导出了极大极小下界。
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.