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

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