An in-depth look at approximation via deep and narrow neural networks
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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_ | 1866918156699697152 |
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| author | Dommel, Joris Wegner, Sven A. |
| author_facet | Dommel, Joris Wegner, Sven A. |
| contents | In 2017, Hanin and Sellke showed that the class of arbitrarily deep, real-valued, feed-forward and ReLU-activated networks of width w forms a dense subset of the space of continuous functions on R^n, with respect to the topology of uniform convergence on compact sets, if and only if w>n holds. To show the necessity, a concrete counterexample function f:R^n->R was used. In this note we actually approximate this very f by neural networks in the two cases w=n and w=n+1 around the aforementioned threshold. We study how the approximation quality behaves if we vary the depth and what effect (spoiler alert: dying neurons) cause that behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An in-depth look at approximation via deep and narrow neural networks Dommel, Joris Wegner, Sven A. Machine Learning 68T07, 41A30 I.2.6; I.5.1 In 2017, Hanin and Sellke showed that the class of arbitrarily deep, real-valued, feed-forward and ReLU-activated networks of width w forms a dense subset of the space of continuous functions on R^n, with respect to the topology of uniform convergence on compact sets, if and only if w>n holds. To show the necessity, a concrete counterexample function f:R^n->R was used. In this note we actually approximate this very f by neural networks in the two cases w=n and w=n+1 around the aforementioned threshold. We study how the approximation quality behaves if we vary the depth and what effect (spoiler alert: dying neurons) cause that behavior. |
| title | An in-depth look at approximation via deep and narrow neural networks |
| topic | Machine Learning 68T07, 41A30 I.2.6; I.5.1 |
| url | https://arxiv.org/abs/2510.07202 |