Efficient Approximation to Analytic and $L^p$ functions by Height-Augmented ReLU Networks
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915855016656896 |
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| author | Li, ZeYu Fan, FengLei Zeng, TieYong |
| author_facet | Li, ZeYu Fan, FengLei Zeng, TieYong |
| contents | This work addresses two fundamental limitations in neural network approximation theory. We demonstrate that a three-dimensional network architecture enables a significantly more efficient representation of sawtooth functions, which serves as the cornerstone in the approximation of analytic and $L^p$ functions. First, we establish substantially improved exponential approximation rates for several important classes of analytic functions and offer a parameter-efficient network design. Second, for the first time, we derive a quantitative and non-asymptotic approximation of high orders for general $L^p$ functions. Our techniques advance the theoretical understanding of the neural network approximation in fundamental function spaces and offer a theoretically grounded pathway for designing more parameter-efficient networks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11128 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Efficient Approximation to Analytic and $L^p$ functions by Height-Augmented ReLU Networks Li, ZeYu Fan, FengLei Zeng, TieYong Machine Learning Neural and Evolutionary Computing 68T07, 41A25 This work addresses two fundamental limitations in neural network approximation theory. We demonstrate that a three-dimensional network architecture enables a significantly more efficient representation of sawtooth functions, which serves as the cornerstone in the approximation of analytic and $L^p$ functions. First, we establish substantially improved exponential approximation rates for several important classes of analytic functions and offer a parameter-efficient network design. Second, for the first time, we derive a quantitative and non-asymptotic approximation of high orders for general $L^p$ functions. Our techniques advance the theoretical understanding of the neural network approximation in fundamental function spaces and offer a theoretically grounded pathway for designing more parameter-efficient networks. |
| title | Efficient Approximation to Analytic and $L^p$ functions by Height-Augmented ReLU Networks |
| topic | Machine Learning Neural and Evolutionary Computing 68T07, 41A25 |
| url | https://arxiv.org/abs/2603.11128 |