Sharp Lower Bounds for Linearized ReLU^k Approximation on the Sphere
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908625187897344 |
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| author | Mao, Tong Xu, Jinchao |
| author_facet | Mao, Tong Xu, Jinchao |
| contents | We prove a saturation theorem for linearized shallow ReLU$^k$ neural networks on the unit sphere $\mathbb S^d$. For any antipodally quasi-uniform set of centers, if the target function has smoothness $r>\tfrac{d+2k+1}{2}$, then the best $\mathcal{L}^2(\mathbb S^d)$ approximation cannot converge faster than order $n^{-\frac{d+2k+1}{2d}}$. This lower bound matches existing upper bounds, thereby establishing the exact saturation order $\tfrac{d+2k+1}{2d}$ for such networks. Our results place linearized neural-network approximation firmly within the classical saturation framework and show that, although ReLU$^k$ networks outperform finite elements under equal degrees $k$, this advantage is intrinsically limited. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04060 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Lower Bounds for Linearized ReLU^k Approximation on the Sphere Mao, Tong Xu, Jinchao Numerical Analysis Machine Learning We prove a saturation theorem for linearized shallow ReLU$^k$ neural networks on the unit sphere $\mathbb S^d$. For any antipodally quasi-uniform set of centers, if the target function has smoothness $r>\tfrac{d+2k+1}{2}$, then the best $\mathcal{L}^2(\mathbb S^d)$ approximation cannot converge faster than order $n^{-\frac{d+2k+1}{2d}}$. This lower bound matches existing upper bounds, thereby establishing the exact saturation order $\tfrac{d+2k+1}{2d}$ for such networks. Our results place linearized neural-network approximation firmly within the classical saturation framework and show that, although ReLU$^k$ networks outperform finite elements under equal degrees $k$, this advantage is intrinsically limited. |
| title | Sharp Lower Bounds for Linearized ReLU^k Approximation on the Sphere |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2510.04060 |