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arXiv · 2608.30710

Kolmogorov--Arnold against bounded translations

Sviatoslav V. Dzhenzher
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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.

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Sviatoslav V. Dzhenzher. 2026-08-31. Kolmogorov--Arnold against bounded translations. https://arxiv.org/abs/2608.30710

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