Non-Asymptotic Rates for Manifold, Tangent Space, and Curvature Estimation

Abstract

Given an $n$-sample drawn on a submanifold $M\subset {\mathbb{R}}^D$, we derive optimal rates for the estimation of tangent spaces $T\_XM$, 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{ö}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.