A kernel-based analysis of Laplacian Eigenmaps
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913244287860736 |
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| author | Wahl, Martin |
| author_facet | Wahl, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16481 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A kernel-based analysis of Laplacian Eigenmaps Wahl, Martin Statistics Theory Probability Spectral Theory Machine Learning 62H25, 60B20, 15A42, 47A55, 47D07, 35K08 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. |
| title | A kernel-based analysis of Laplacian Eigenmaps |
| topic | Statistics Theory Probability Spectral Theory Machine Learning 62H25, 60B20, 15A42, 47A55, 47D07, 35K08 |
| url | https://arxiv.org/abs/2402.16481 |