{"paper":{"title":"Implicit Manifold Reconstruction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Man-Kwun Chiu, Siu-Wing Cheng","submitted_at":"2019-04-07T22:45:15Z","abstract_excerpt":"Let ${\\cal M} \\subset \\mathbb{R}^d$ be a compact, smooth and boundaryless manifold with dimension $m$ and unit reach. We show how to construct a function $\\varphi: \\mathbb{R}^d \\rightarrow \\mathbb{R}^{d-m}$ from a uniform $(\\varepsilon,\\kappa)$-sample $P$ of $\\cal M$ that offers several guarantees. Let $Z_\\varphi$ denote the zero set of $\\varphi$. Let $\\widehat{{\\cal M}}$ denote the set of points at distance $\\varepsilon$ or less from $\\cal M$. There exists $\\varepsilon_0 \\in (0,1)$ that decreases as $d$ increases such that if $\\varepsilon \\leq \\varepsilon_0$, the following guarantees hold. Fi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03764","kind":"arxiv","version":1},"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"}