The minimal width of neural networks and immersions of real projective space
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Abstract (arXiv)
We investigate the minimal width of deep neural networks that can approximate all continuous functions from ℝn to ℝm in the topology of uniform convergence on compact subsets. We show that whenever the real projective space mathbbRPn-1 admits a smooth immersion into ℝn, the minimal width is at least n+minn,m, assuming that the activation function is continuous and monotone. As a consequence, the minimal width of ReLU networks is n+m whenever n is equal to 2, 3, 4, 7 or 8 and 1≤ m ≤ n, extending results of Hanin-Sellke and Hwang. This suggests an intriguing connection between the approximation theory of neural networks and differential topology.
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References · 11
- 2016Error bounds for approximations with deep ReLU networkscited by 574
- 2017The Expressive Power of Neural Networks: A View from the Widthcited by 320
- 2019Universal Approximation with Deep Narrow Networkscited by 200
- 2017Approximating Continuous Functions by ReLU Nets of Minimal Widthcited by 122
- 2020Minimum Width for Universal Approximationcited by 94
- 2015Representation Benefits of Deep Feedforward Networkscited by 86
- 2018Deep, Skinny Neural Networks are not Universal Approximatorscited by 38
- 2022Achieve the Minimum Width of Neural Networks for Universal Approximationcited by 26
- 2023Minimum width for universal approximation using ReLU networks on compact domaincited by 18
- 2024New advances in universal approximation with neural networks of minimal widthcited by 5
- 2025Minimum width for universal approximation using squashable activation functionscited by 3
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