arXiv Machine Learning September 1, 2026 By Sviatoslav V. Dzhenzher

Kolmogorov--Arnold against bounded translations

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The paper revisits the Kolmogorov–Arnold representation theorem (KART), which has gained renewed interest through its use in neural networks such as Kolmogorov–Arnold Networks (KANs). It addresses the open question of KART’s stability when the hidden layer is subjected to continuous adversarial perturbations, specifically bounded translations. The authors present a constructive proof of an approximate representation that uses fixed, piecewise‑linear inner functions and a single outer function that remains invariant across all summands, provided the maximum translation bound is known in advance.

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