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[2301.00201] Exploring Singularities in point clouds with the graph Laplacian: An explicit approach
Summary
This paper presents a novel approach using the graph Laplacian to analyze singularities in point clouds, offering theoretical guarantees and explicit bounds for estimating geometric properties.
Why It Matters
Understanding singularities in point clouds is crucial for improving data analysis techniques in machine learning and computer vision. This research provides a foundation for developing more accurate models and methods in these fields, enhancing the interpretation of complex datasets.
Key Takeaways
- Introduces a method for analyzing singularities using the graph Laplacian.
- Provides theoretical guarantees and explicit bounds for practical applications.
- Offers tests for identifying singularities in datasets.
- Suggests methods for estimating geometric properties related to singularities.
- Enhances understanding of the geometry underlying complex datasets.
Statistics > Machine Learning arXiv:2301.00201 (stat) [Submitted on 31 Dec 2022 (v1), last revised 23 Feb 2026 (this version, v4)] Title:Exploring Singularities in point clouds with the graph Laplacian: An explicit approach Authors:Martin Andersson, Benny Avelin View a PDF of the paper titled Exploring Singularities in point clouds with the graph Laplacian: An explicit approach, by Martin Andersson and Benny Avelin View PDF Abstract:We develop theory and methods that use the graph Laplacian to analyze the geometry of the underlying manifold of datasets. Our theory provides theoretical guarantees and explicit bounds on the functional forms of the graph Laplacian when it acts on functions defined close to singularities of the underlying manifold. We use these explicit bounds to develop tests for singularities and propose methods that can be used to estimate geometric properties of singularities in the datasets. Comments: Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Differential Geometry (math.DG) MSC classes: 58K99 (Primary), 68R99, 60B99, 62G10 (Secondary) Cite as: arXiv:2301.00201 [stat.ML] (or arXiv:2301.00201v4 [stat.ML] for this version) https://doi.org/10.48550/arXiv.2301.00201 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Journal of Computational Mathematics and Data Science 14 (2025) 100113 Related DOI: https://doi.org/10.1016/j.jcmds.2025.100113 Focus to learn more DOI(s) linking to related resources Submission history F...
