# Kolmogorov--Arnold against bounded translations

arXiv: https://arxiv.org/abs/2608.30710 · HTML: https://mlsanity.com/paper/2608.30710/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 28 references that are in the map.
- First posted on arXiv 31 Aug 2026.

## Builds on (most cited first, 4 of 4)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [Addressing common misinterpretations of KART and UAT in neural network literature](https://mlsanity.com/paper/2408.16389/) (2024)
- [Kolmogorov--Arnold stability](https://mlsanity.com/paper/2504.05255/) (2025)

## Abstract (arXiv)

Historically originating from Hilbert's 13th problem, the Kolmogorov-Arnold representation theorem (KART) has recently experienced a major revitalisation through its applications to neural networks, specifically Kolmogorov-Arnold Networks (KANs). While the exact representation is well established, its stability under continuous adversarial perturbations of the hidden layer remains a critical open question. In this paper, we investigate the robustness of KART against bounded adversarial translations. We provide an explicit, self-contained, and constructive proof of an approximate representation using fixed, piecewise linear inner functions. Crucially, our construction employs a single outer function that remains invariant for all summands and is independent of the specific adversarial translation, provided its maximum bound is known a priori.