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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.23220 |
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| _version_ | 1866909904274456576 |
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| author | Lo, Phillip Khoo, Yuehaw |
| author_facet | Lo, Phillip Khoo, Yuehaw |
| contents | In this paper, we study the problem of recovering a ground truth high dimensional piecewise linear curve $C^*(t):[0, 1]\to\mathbb{R}^d$ from a high noise Gaussian point cloud with covariance $σ^2I$ centered around the curve. We establish that the sample complexity of recovering $C^*$ from data scales with order at least $σ^6$. We then show that recovery of a piecewise linear curve from the third moment is locally well-posed, and hence $O(σ^6)$ samples is also sufficient for recovery. We propose methods to recover a curve from data based on a fitting to the third moment tensor with a careful initialization strategy and conduct some numerical experiments verifying the ability of our methods to recover curves. All code for our numerical experiments is publicly available on GitHub. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23220 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Method of Moments for Estimation of Noisy Curves Lo, Phillip Khoo, Yuehaw Statistics Theory Optimization and Control In this paper, we study the problem of recovering a ground truth high dimensional piecewise linear curve $C^*(t):[0, 1]\to\mathbb{R}^d$ from a high noise Gaussian point cloud with covariance $σ^2I$ centered around the curve. We establish that the sample complexity of recovering $C^*$ from data scales with order at least $σ^6$. We then show that recovery of a piecewise linear curve from the third moment is locally well-posed, and hence $O(σ^6)$ samples is also sufficient for recovery. We propose methods to recover a curve from data based on a fitting to the third moment tensor with a careful initialization strategy and conduct some numerical experiments verifying the ability of our methods to recover curves. All code for our numerical experiments is publicly available on GitHub. |
| title | Method of Moments for Estimation of Noisy Curves |
| topic | Statistics Theory Optimization and Control |
| url | https://arxiv.org/abs/2410.23220 |