Hybrid least squares for learning functions from highly noisy data
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913160690139136 |
|---|---|
| author | Adcock, Ben Hientzsch, Bernhard Narayan, Akil Xu, Yiming |
| author_facet | Adcock, Ben Hientzsch, Bernhard Narayan, Akil Xu, Yiming |
| contents | Motivated by the need for efficient estimation of conditional expectations, we consider a least-squares function approximation problem with heavily polluted data. Existing methods that are effective in the small-noise regime are suboptimal when large noise is present. To address this issue, we propose a hybrid approach that combines Christoffel sampling with optimal experimental design. We show that the proposed algorithm enjoys appropriate optimality properties for both sample point generation and noise mollification, leading to improved computational efficiency and sample complexity compared to existing methods. We also extend the algorithm to convexity-constrained settings with similar theoretical guarantees. When the target function is defined as the expectation of a random field, we further extend our approach to leverage adaptive random subspaces and establish results on the approximation capacity of the adaptive procedure. Our theoretical findings are supported by numerical studies on both synthetic data and on a more challenging stochastic simulation problem in computational finance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02215 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hybrid least squares for learning functions from highly noisy data Adcock, Ben Hientzsch, Bernhard Narayan, Akil Xu, Yiming Machine Learning Numerical Analysis Motivated by the need for efficient estimation of conditional expectations, we consider a least-squares function approximation problem with heavily polluted data. Existing methods that are effective in the small-noise regime are suboptimal when large noise is present. To address this issue, we propose a hybrid approach that combines Christoffel sampling with optimal experimental design. We show that the proposed algorithm enjoys appropriate optimality properties for both sample point generation and noise mollification, leading to improved computational efficiency and sample complexity compared to existing methods. We also extend the algorithm to convexity-constrained settings with similar theoretical guarantees. When the target function is defined as the expectation of a random field, we further extend our approach to leverage adaptive random subspaces and establish results on the approximation capacity of the adaptive procedure. Our theoretical findings are supported by numerical studies on both synthetic data and on a more challenging stochastic simulation problem in computational finance. |
| title | Hybrid least squares for learning functions from highly noisy data |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2507.02215 |