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A kernel-based analysis of Laplacian Eigenmaps
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abstract
Given i.i.d. observations uniformly distributed on a closed manifold $\mathcal{M}\subseteq \mathbb{R}^p$, 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 $\mathcal{M}$. In our analysis, we connect the empirical graph Laplacian to kernel principal component analysis, and consider the heat kernel of $\mathcal{M}$ as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.
Forward citations
Cited by 2 Pith papers
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Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds
The minimax rate for estimating eigenpairs of weighted Laplace-Beltrami operators from n samples on a d-dimensional manifold is n^{-2/(d+4)}, and graph Laplacians achieve this rate up to logarithmic factors.
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On the convergence of graph Laplacians with a symmetric divergence
Graph Laplacians constructed from a smooth nondegenerate symmetric divergence D on a compact Riemannian manifold converge pointwise to the Laplace–Beltrami operator under a fourth-order closeness condition to squared ...
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