Title: 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 $T\_X M$, 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{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.