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

## TL;DR

This paper establishes optimal non-asymptotic rates for estimating manifold structures, tangent spaces, and curvature from finite samples, advancing theoretical understanding of geometric estimation.

## Contribution

It introduces a unified approach using local polynomials for simultaneous estimation of manifold, tangent space, and curvature, with minimax lower bounds derived.

## Key findings

- Optimal rates for tangent space estimation
- Optimal rates for second fundamental form estimation
- Optimal rates for manifold 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{\"o}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.