{"paper":{"title":"Non-Asymptotic Rates for Manifold, Tangent Space, and Curvature Estimation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Cl\\'ement Levrard (UPD7), Eddie Aamari (DATASHAPE, LM-Orsay), SELECT","submitted_at":"2017-05-02T14:15:03Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.00989","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}