Determine the minimal width for unresolved input-output dimensions and standard activations

Determine the minimal width required for deep neural networks with standard activation functions to approximate every continuous map from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) uniformly on compact subsets for the values of \(n\) and \(m\) whose exact minimal widths are not known.

Background

The paper studies the least width permitting arbitrary-depth neural networks to approximate all continuous functions from Rn\mathbb{R}^n to Rm\mathbb{R}^m in the topology of uniform convergence on compact subsets. Although upper and lower bounds are known for several settings, the exact minimal width remains unresolved for most parameter choices when standard activations such as ReLU are used. The paper establishes new lower bounds under a differential-topological immersion hypothesis, but does not resolve the general minimal-width problem.

References

Surprisingly, for most values of $n$ and $m$, and standard activations, this minimal width, which we denote by $w_{\text{min}(n,m)$, remains unknown.

— The minimal width of neural networks and immersions of real projective space  (2610.20229 - Chu et al., 14 Oct 2026) in Section 1, Introduction