Kolmogorov--Arnold against bounded translations
2026-08-31 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study a mathematical theorem called the Kolmogorov-Arnold representation theorem (KART), which has applications in neural networks. They focus on how stable this theorem's representation is when small, controlled changes (adversarial translations) are made to the network's hidden parts. They prove that an approximate version of this representation can remain stable using a fixed approach that doesn't change even when these small perturbations happen, as long as the size of the changes is known beforehand. Their work helps understand how certain neural network models can resist small, targeted disruptions.
Kolmogorov-Arnold representation theoremadversarial perturbationsneural networksKolmogorov-Arnold Networksrobustnessapproximate representationpiecewise linear functionshidden layerbounded translationsstability
Authors
Sviatoslav V. Dzhenzher
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.