Convergence of Hessian estimator from random samples on a manifold with boundary
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914027996708864 |
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| author | Chen, Chih-Wei Wu, Hau-Tieng |
| author_facet | Chen, Chih-Wei Wu, Hau-Tieng |
| contents | A common method for estimating the Hessian operator from random samples on a low-dimensional manifold involves locally fitting a quadratic polynomial. Although widely used, it is unclear if this estimator introduces bias, especially in complex manifolds with boundaries and nonuniform sampling. Rigorous theoretical guarantees of its asymptotic behavior have been lacking. We show that, under mild conditions, this estimator asymptotically converges to the Hessian operator, with nonuniform sampling and curvature effects proving negligible, even near boundaries. Our analysis framework simplifies the intensive computations required for direct analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_12547 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convergence of Hessian estimator from random samples on a manifold with boundary Chen, Chih-Wei Wu, Hau-Tieng Statistics Theory Differential Geometry 62-07 A common method for estimating the Hessian operator from random samples on a low-dimensional manifold involves locally fitting a quadratic polynomial. Although widely used, it is unclear if this estimator introduces bias, especially in complex manifolds with boundaries and nonuniform sampling. Rigorous theoretical guarantees of its asymptotic behavior have been lacking. We show that, under mild conditions, this estimator asymptotically converges to the Hessian operator, with nonuniform sampling and curvature effects proving negligible, even near boundaries. Our analysis framework simplifies the intensive computations required for direct analysis. |
| title | Convergence of Hessian estimator from random samples on a manifold with boundary |
| topic | Statistics Theory Differential Geometry 62-07 |
| url | https://arxiv.org/abs/2303.12547 |