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

TL;DR

This paper develops theoretical tools and methods using the graph Laplacian to analyze and detect singularities in the geometry of datasets modeled as manifolds, providing explicit bounds and tests.

Contribution

It introduces explicit bounds on the graph Laplacian near singularities and develops new tests and methods for estimating geometric properties of these singularities.

Findings

Provided explicit bounds on the graph Laplacian near singularities

Developed tests for detecting singularities in datasets

Proposed methods for estimating geometric properties of singularities

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.