Nonasymptotic rates for manifold, tangent space and curvature estimation

Given a noisy sample from a submanifold M.RD, we derive optimal rates for the estimation of tangent spaces TXM, the second fundamental form IIMX and the submanifold M. After motivating their study, we introduce a quantitative class of Ck-submanifolds in analogy with Hölder 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.