Kolmogorov--Arnold against bounded translations
Abstract
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.
Related
- Kolmogorov--Arnold stability for discontinuous functions
- Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations
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- Quantum Bernstein--Kolmogorov--Arnold approximation
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BibTeX
@misc{dzhenzher2026kolmogorovarnold,
title = {{Kolmogorov--Arnold against bounded translations}},
author = {Sviatoslav V. Dzhenzher},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30710},
eprint = {2608.30710},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30710},
}
RIS
TY - UNPB
TI - Kolmogorov--Arnold against bounded translations
AU - Sviatoslav V. Dzhenzher
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30710
UR - https://arxiv.org/abs/2608.30710
AB - 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.
ER -