Non-Asymptotic Rates for Manifold, Tangent Space, and Curvature Estimation
Eddie Aamari, Clément Levrard · arXiv (Cornell University) · 2017
Given an $n$-sample drawn on a submanifold $M \\subset \\mathbb{R}^D$, we derive optimal rates for the estimation of tangent spaces $T\\_X M$, the second fundamental form $II\\_X^M$, and the submanifold $M$.After motivating their study, we introduce a quantitative class of $\\mathcal{C}^k$-submanifolds in analogy with H{\\"o}lder 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.