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Finding the Homology of Submanifolds with High Confidence from Random Samples
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“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]
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“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)
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“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.”
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Metadata
- 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 › Computer Science › Computer Graphics and Computer-Aided Design › 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.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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