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Given i. i. d. observations uniformly distributed on a closed manifold M Rᵖ, we study the spectral properties of the associated empirical graph Laplacian based on a Gaussian kernel. Our main results are non-asymptotic error bounds, showing that the eigenvalues and eigenspaces of the empirical graph Laplacian are close to the eigenvalues and eigenspaces of the Laplace-Beltrami operator of M. In our analysis, we connect the empirical graph Laplacian to kernel principal component analysis, and consider the heat kernel of M as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.
Martin Wahl (Mon,) studied this question.