Exploring Singularities in point clouds with the graph Laplacian: An explicit approach

2022-12-31

arXiv.org

AI Summary

This work addresses the challenge of accurately characterizing geometric singularities—such as corners, edges, and self-intersections—in the underlying manifold of point cloud data. We propose a singularity modeling and estimation framework grounded in the graph Laplacian operator. For the first time, we establish explicit functional bounds for the graph Laplacian within singular neighborhoods, thereby bridging spectral methods with local geometric inference. Based on this theoretical foundation, we derive a principled existence test for singularities and develop robust, nonparametric estimators for intrinsic geometric quantities—including manifold dimension and curvature. The method provides rigorous theoretical guarantees while maintaining interpretability. Extensive experiments on both synthetic and real-world point clouds demonstrate high estimation accuracy and strong consistency with theoretical predictions. Our approach establishes a novel paradigm for singularity-aware manifold learning, enabling geometrically faithful analysis of complex, nonsmooth structures in point cloud data.

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.

Problem

• Analyze manifold geometry using graph Laplacian
• Provide theoretical guarantees for singularity analysis
• Develop methods to estimate geometric properties

Innovation

• Graph Laplacian singularity analysis
• Explicit bounds on functional forms
• Geometric property estimation methods