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

Eddie Aamari (1, 2) , Clément Levrard (1, 3)

1 DATASHAPE - Understanding the Shape of Data
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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 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.

Mots clés: Tangent Space Estimation; Curvature Estimation; Minimax rates; manifold learning

Domaines: Statistiques [math.ST]

Eddie Aamari, Clément Levrard. Non-Asymptotic Rates for Manifold, Tangent Space, and Curvature Estimation. Annals of Statistics, 2019, 47 (1), ⟨10.1214/18-AOS1685⟩. ⟨hal-01516032v3⟩