Finding the homology of submanifolds with high confidence from random samples
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Niyogi, Smale, and Weinberger [52] provide sufficient hypotheses guaranteeing that reconstruction of Riemannian manifolds from data is possible.
“Niyogi, Smale, and Weinberger in [52] provide sufficient hypotheses which guarantee that this is possible for Riemannian manifolds.”
From Topology and data (2009) · cites this paper as [52]
Section 2 (Preliminaries; page unavailable) considers compact Riemannian submanifolds sampled as data. Section 3 (An Outline of Our Main Results; page unavailable) gives conditions under which the samples recover the manifold’s homology with high confidence. It does not guarantee reconstruction of the manifold itself, so the claim is broader than the result. (LLM, conf=0.88)
Experimental — accuracy not yet measuredNiyogi, Smale and Weinberger (2008) give conditions showing that with high confidence, a construction based on ε balls around a finite sample from a submanifold of Euclidean space computes homology of the submanifold accurately.
“Niyogi, Smale and Weinberger (2008) give conditions which show that with high confidence, a construction based on ϵ balls around a finite sample from a submanifold of Euclidean space computes homology of the submanifold accurately.”
From Topological pattern recognition for point cloud data (2014) · cites this paper as Niyogi, Smale and Weinberger (2008)
The method of Niyogi, Smale, and Weinberger proves that the Čech complex associated to a covering by balls of a fixed radius is homotopy equivalent to the underlying manifold.
“Their method is to prove that the Čech complex associated to a covering by balls of a fixed radius is homotopy equivalent to the underlying manifold.”
From Topology and data (2009) · cites this paper as [52]
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“Further, even if one could assume that the data lies along a manifold, one is usually not in a position to verify that the stringent hypotheses of [52] are satisfied.”
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- Authors
- Niyogi, P., Smale, S., & Weinberger, S.
- Affiliations
- University of Chicago (US)
- Toyota Technological Institute at Chicago (US)
- Corresponding author
- Niyogi, Partha
- ORCID
- Weinberger, Shmuel
- Venue
- Discrete & Computational Geometry
- Published
- 2008-03
- Volume
- 39
- Issue
- 1-3
- Pages
- 419-441
- ISSN
- 0179-5376 (ISSN-L)
- 1432-0444
- OpenAlex
- W2157878484
- Cited by
- 523
- References
- 24
- Citations by year
- 2026: 17 · 2025: 21 · 2024: 26 · 2023: 25 · 2022: 25 · 2021: 37 · 2020: 32 · 2019: 23 · 2018: 43 · 2017: 24 · 2016: 26 · 2015: 23 · 2014: 31 · 2013: 24 · 2012: 26
- FWCI
- 41.28
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Discrete & Computational Geometry (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Engineering › Computational Mechanics › 3D Shape Modeling and Analysis
- Keywords
- submanifolds, random samples, topological inference, condition number, curvature, self-intersection, noisy data, high-dimensional data analysis, topological data analysis
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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