Some Super-approximation Rates of ReLU Neural Networks for Korobov Functions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918370417311744 |
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| author | Li, Yuwen Zhang, Guozhi |
| author_facet | Li, Yuwen Zhang, Guozhi |
| contents | This paper examines the $L_p$ and $W^1_p$ norm approximation errors of ReLU neural networks for Korobov functions. In terms of network width and depth, we derive nearly optimal super-approximation error bounds of order $2m$ in the $L_p$ norm and order $2m-2$ in the $W^1_p$ norm, for target functions with $L_p$ mixed derivative of order $m$ in each direction. The analysis leverages sparse grid finite elements and the bit extraction technique. Our results improve upon classical lowest order $L_\infty$ and $H^1$ norm error bounds and demonstrate that the expressivity of neural networks is largely unaffected by the curse of dimensionality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some Super-approximation Rates of ReLU Neural Networks for Korobov Functions Li, Yuwen Zhang, Guozhi Machine Learning This paper examines the $L_p$ and $W^1_p$ norm approximation errors of ReLU neural networks for Korobov functions. In terms of network width and depth, we derive nearly optimal super-approximation error bounds of order $2m$ in the $L_p$ norm and order $2m-2$ in the $W^1_p$ norm, for target functions with $L_p$ mixed derivative of order $m$ in each direction. The analysis leverages sparse grid finite elements and the bit extraction technique. Our results improve upon classical lowest order $L_\infty$ and $H^1$ norm error bounds and demonstrate that the expressivity of neural networks is largely unaffected by the curse of dimensionality. |
| title | Some Super-approximation Rates of ReLU Neural Networks for Korobov Functions |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2507.10345 |