Exploring Singularities in point clouds with the graph Laplacian: An explicit approach
This work addresses the challenge of understanding complex data structures for researchers in machine learning and data analysis, though it appears incremental as it builds on existing graph Laplacian methods.
The authors tackled the problem of analyzing the geometry of underlying manifolds in datasets, particularly near singularities, by developing theory and methods using the graph Laplacian, resulting in explicit bounds and tests for singularity detection and geometric property estimation.
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.