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We develop theory and methods that use the graph Laplacian to analyze thegeometry of the underlying manifold of point clouds. Our theory providestheoretical guarantees and explicit bounds on the functional form of the graphLaplacian, in the case when it acts on functions defined close to singularitiesof the underlying manifold. We also propose methods that can be used toestimate these geometric properties of the point cloud, which are based on thetheoretical guarantees.

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