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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2406.19510 |
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| _version_ | 1866929450455662592 |
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| author | Akwei, Bernard Atkins, Bobita Bailey, Rachel Dalal, Ashka Dinin, Natalie Kerby-White, Jonathan McGuinness, Tess Patricks, Tonya Rogers, Luke Romanelli, Genevieve Su, Yiheng Teplyaev, Alexander |
| author_facet | Akwei, Bernard Atkins, Bobita Bailey, Rachel Dalal, Ashka Dinin, Natalie Kerby-White, Jonathan McGuinness, Tess Patricks, Tonya Rogers, Luke Romanelli, Genevieve Su, Yiheng Teplyaev, Alexander |
| contents | Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $ε$. If $ε$ is too small or too large, then the approximation is inaccurate or completely breaks down. However, an analytic expression for the optimal $ε$ is out of reach. In our work, we use some explicitly solvable models and Monte Carlo simulations to find the approximately optimal range of $ε$ that gives, on average, relatively accurate approximation of the eigenmaps. Numerically we can consider several model situations where eigen-coordinates can be computed analytically, including intervals with uniform and weighted measures, squares, tori, spheres, and the Sierpinski gasket. In broader terms, we intend to study eigen-coordinates on weighted Riemannian manifolds, possibly with boundary, and on some metric measure spaces, such as fractals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19510 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence, optimization and stability of singular eigenmaps Akwei, Bernard Atkins, Bobita Bailey, Rachel Dalal, Ashka Dinin, Natalie Kerby-White, Jonathan McGuinness, Tess Patricks, Tonya Rogers, Luke Romanelli, Genevieve Su, Yiheng Teplyaev, Alexander Probability Metric Geometry Statistics Theory 60D05 28A80 62R07 65J20 Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $ε$. If $ε$ is too small or too large, then the approximation is inaccurate or completely breaks down. However, an analytic expression for the optimal $ε$ is out of reach. In our work, we use some explicitly solvable models and Monte Carlo simulations to find the approximately optimal range of $ε$ that gives, on average, relatively accurate approximation of the eigenmaps. Numerically we can consider several model situations where eigen-coordinates can be computed analytically, including intervals with uniform and weighted measures, squares, tori, spheres, and the Sierpinski gasket. In broader terms, we intend to study eigen-coordinates on weighted Riemannian manifolds, possibly with boundary, and on some metric measure spaces, such as fractals. |
| title | Convergence, optimization and stability of singular eigenmaps |
| topic | Probability Metric Geometry Statistics Theory 60D05 28A80 62R07 65J20 |
| url | https://arxiv.org/abs/2406.19510 |