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.

翻译：我们发展了利用图拉普拉斯算子分析数据集底层流形几何性质的理论与方法。该理论为图拉普拉斯算子作用于底层流形奇点附近定义函数时的泛函形式提供了理论保证与显式边界。我们利用这些显式边界构建了奇异性检验方法，并提出了可用于估计数据集中奇异性几何性质的算法。