ReLU neural network approximation to piecewise constant functions
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866929553982619648 |
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| author | Cai, Zhiqiang Choi, Junpyo Liu, Min |
| author_facet | Cai, Zhiqiang Choi, Junpyo Liu, Min |
| contents | This paper studies the approximation property of ReLU neural networks (NNs) to piecewise constant functions with unknown interfaces in bounded regions in $\mathbb{R}^d$. Under the assumption that the discontinuity interface $Γ$ may be approximated by a connected series of hyperplanes with a prescribed accuracy $\varepsilon >0$, we show that a three-layer ReLU NN is sufficient to accurately approximate any piecewise constant function and establish its error bound. Moreover, if the discontinuity interface is convex, an analytical formula of the ReLU NN approximation with exact weights and biases is provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16506 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | ReLU neural network approximation to piecewise constant functions Cai, Zhiqiang Choi, Junpyo Liu, Min Functional Analysis Machine Learning Numerical Analysis 68T07, 41A25, 41A46 This paper studies the approximation property of ReLU neural networks (NNs) to piecewise constant functions with unknown interfaces in bounded regions in $\mathbb{R}^d$. Under the assumption that the discontinuity interface $Γ$ may be approximated by a connected series of hyperplanes with a prescribed accuracy $\varepsilon >0$, we show that a three-layer ReLU NN is sufficient to accurately approximate any piecewise constant function and establish its error bound. Moreover, if the discontinuity interface is convex, an analytical formula of the ReLU NN approximation with exact weights and biases is provided. |
| title | ReLU neural network approximation to piecewise constant functions |
| topic | Functional Analysis Machine Learning Numerical Analysis 68T07, 41A25, 41A46 |
| url | https://arxiv.org/abs/2410.16506 |