Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916731337834496 |
|---|---|
| author | Liu, Xinliang Mao, Tong Xu, Jinchao |
| author_facet | Liu, Xinliang Mao, Tong Xu, Jinchao |
| contents | This paper presents two main theoretical results concerning shallow neural networks with ReLU$^k$ activation functions. We establish a novel integral representation for Sobolev spaces, showing that every function in $H^{\frac{d+2k+1}{2}}(Ω)$ can be expressed as an $L^2$-weighted integral of ReLU$^k$ ridge functions over the unit sphere. This result mirrors the known representation of Barron spaces and highlights a fundamental connection between Sobolev regularity and neural network representations. Moreover, we prove that linearized shallow networks -- constructed by fixed inner parameters and optimizing only the linear coefficients -- achieve optimal approximation rates $O(n^{-\frac{1}{2}-\frac{2k+1}{2d}})$ in Sobolev spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00351 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks Liu, Xinliang Mao, Tong Xu, Jinchao Numerical Analysis This paper presents two main theoretical results concerning shallow neural networks with ReLU$^k$ activation functions. We establish a novel integral representation for Sobolev spaces, showing that every function in $H^{\frac{d+2k+1}{2}}(Ω)$ can be expressed as an $L^2$-weighted integral of ReLU$^k$ ridge functions over the unit sphere. This result mirrors the known representation of Barron spaces and highlights a fundamental connection between Sobolev regularity and neural network representations. Moreover, we prove that linearized shallow networks -- constructed by fixed inner parameters and optimizing only the linear coefficients -- achieve optimal approximation rates $O(n^{-\frac{1}{2}-\frac{2k+1}{2d}})$ in Sobolev spaces. |
| title | Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2505.00351 |