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arXiv · 2510.07202

An in-depth look at approximation via deep and narrow neural networks

Joris Dommel·Sven A. Wegner
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Abstract

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.

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Joris Dommel, Sven A. Wegner. 2025-10-08. An in-depth look at approximation via deep and narrow neural networks. https://arxiv.org/abs/2510.07202

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