The minimal width of neural networks and immersions of real projective space

arXiv:2610.20229v1 Announce Type: cross Abstract: We investigate the minimal width of deep neural networks that can approximate all continuous functions from R^n to R^m in the topology of uniform convergence on compact subsets. We show that whenever the real projective space RP^{n-1} admits a smooth immersion into R^n, the minimal width is at least n+min{n,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 1le m le n, extending results of Hanin-Sellke and Hwang. This suggests an intriguing connection between the approximation theory of neural networks and differential topology.

Source: arXiv cs.LG | 2026-10-15