Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915105719975936 |
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| author | Lyons, Andrew |
| author_facet | Lyons, Andrew |
| contents | We study the eigenvalues and eigenfunctions of a differential operator that governs the asymptotic behavior of the unsupervised learning algorithm known as Locally Linear Embedding when a large data set is sampled from an interval or disc. In particular, the differential operator is of second order, mixed-type, and degenerates near the boundary. We show that a natural regularity condition on the eigenfunctions imposes a consistent boundary condition and use the Frobenius method to estimate pointwise behavior. We then determine the limiting sequence of eigenvalues analytically and compare them to numerical predictions. Finally, we propose a variational framework for determining eigenvalues on other compact manifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_09572 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary Lyons, Andrew Analysis of PDEs Machine Learning 35J20, 35M12 We study the eigenvalues and eigenfunctions of a differential operator that governs the asymptotic behavior of the unsupervised learning algorithm known as Locally Linear Embedding when a large data set is sampled from an interval or disc. In particular, the differential operator is of second order, mixed-type, and degenerates near the boundary. We show that a natural regularity condition on the eigenfunctions imposes a consistent boundary condition and use the Frobenius method to estimate pointwise behavior. We then determine the limiting sequence of eigenvalues analytically and compare them to numerical predictions. Finally, we propose a variational framework for determining eigenvalues on other compact manifolds. |
| title | Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary |
| topic | Analysis of PDEs Machine Learning 35J20, 35M12 |
| url | https://arxiv.org/abs/2501.09572 |