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

Given an $n$-sample drawn on a submanifold $M \\subset \\mathbb{R}^D$, we\nderive optimal rates for the estimation of tangent spaces $T\\_X M$, the second\nfundamental form $II\\_X^M$, and the submanifold $M$.After motivating their\nstudy, we introduce a quantitative class of $\\mathcal{C}^k$-submanifolds in\nanalogy with H{\\"o}lder classes.The proposed estimators are based on local\npolynomials and allow to deal simultaneously with the three problems at stake.\nMinimax lower bounds are derived using a conditional version of Assouad's lemma\nwhen the base point $X$ is random.\n