Spectral Convergence of Symmetrized Graph Laplacian on manifolds with boundary
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866912381640114176 |
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| author | Peoples, J. Wilson Harlim, John |
| author_facet | Peoples, J. Wilson Harlim, John |
| contents | We study the spectral convergence of a symmetrized Graph Laplacian matrix induced by a Gaussian kernel evaluated on pairs of embedded data, sampled from a manifold with boundary, a sub-manifold of $\mathbb{R}^m$. Specifically, we deduce the convergence rates for eigenpairs of the discrete Graph-Laplacian matrix to the eigensolutions of the Laplace-Beltrami operator that are well-defined on manifolds with boundary, including the homogeneous Neumann and Dirichlet boundary conditions. For the Dirichlet problem, we deduce the convergence of the \emph{truncated Graph Laplacian}, which is recently numerically observed in applications, and provide a detailed numerical investigation on simple manifolds. Our method of proof relies on the min-max argument over a compact and symmetric integral operator, leveraging the RKHS theory for spectral convergence of integral operator and a recent pointwise asymptotic result of a Gaussian kernel integral operator on manifolds with boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_06988 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Spectral Convergence of Symmetrized Graph Laplacian on manifolds with boundary Peoples, J. Wilson Harlim, John Numerical Analysis 34L15, 34L16, 65N25, 60D05, 58J50, 05C50, 47N40 We study the spectral convergence of a symmetrized Graph Laplacian matrix induced by a Gaussian kernel evaluated on pairs of embedded data, sampled from a manifold with boundary, a sub-manifold of $\mathbb{R}^m$. Specifically, we deduce the convergence rates for eigenpairs of the discrete Graph-Laplacian matrix to the eigensolutions of the Laplace-Beltrami operator that are well-defined on manifolds with boundary, including the homogeneous Neumann and Dirichlet boundary conditions. For the Dirichlet problem, we deduce the convergence of the \emph{truncated Graph Laplacian}, which is recently numerically observed in applications, and provide a detailed numerical investigation on simple manifolds. Our method of proof relies on the min-max argument over a compact and symmetric integral operator, leveraging the RKHS theory for spectral convergence of integral operator and a recent pointwise asymptotic result of a Gaussian kernel integral operator on manifolds with boundary. |
| title | Spectral Convergence of Symmetrized Graph Laplacian on manifolds with boundary |
| topic | Numerical Analysis 34L15, 34L16, 65N25, 60D05, 58J50, 05C50, 47N40 |
| url | https://arxiv.org/abs/2110.06988 |