# Kolmogorov--Arnold stability

arXiv: https://arxiv.org/abs/2504.05255 · HTML: https://mlsanity.com/paper/2504.05255/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 19 references that are in the map.
- First posted on arXiv 7 Apr 2025.

## Builds on (most cited first, 3 of 3)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [The Proof of Kolmogorov-Arnold May Illuminate Neural Network Learning](https://mlsanity.com/paper/2410.08451/) (2024)

## Built on by (most cited first, 3 of 3)

- [Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps](https://mlsanity.com/paper/2607.03187/) (2026)
- [Kolmogorov--Arnold against bounded translations](https://mlsanity.com/paper/2608.30710/) (2026)
- [Kolmogorov--Arnold stability for discontinuous functions](https://mlsanity.com/paper/2609.07240/) (2026)

## Abstract (arXiv)

Regarding the representation theorem of Kolmogorov and Arnold (KA) as an algorithm for representing or <<expressing>> functions, we test its robustness by analyzing its stability to withstand re-parameterizations of the hidden space. One may think of such re-parameterizations as the work of an adversary attempting to foil the construction of the KA outer function. We find KA to be stable under countable collections of continuous re-parameterizations, but unearth a question about the equi-continuity of the outer functions that, so far, obstructs taking limits and defeating continuous groups of re-parameterizations. This question on the regularity of the outer functions is relevant to the debate over the applicability of KA to the general theory of NNs.