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

2017/05/02 by Eddie Aamari, Aamari, Eddie, Clément Levrard +1 · 8 citations
Computer Science · Mathematics · #Digital Image Processing Techniques #FOS: Mathematics #Morphological variations and asymmetry #Statistics Theory (math.ST) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1705.00989

openalex publication_date 2017/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an n-sample drawn on a submanifold M \⊂ \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 \C^k-submanifolds in\nanalogy with H "older 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

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