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.

Machine Learning (cs.LG); Functional Analysis (math.FA)

arXiv:2608.30710 [cs.LG]

(or arXiv:2608.30710v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.30710 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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