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arXiv · 2604.23765

Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks

Vugar Ismailov
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Abstract

We analyze the universal approximation property of Kolmogorov-Arnold Networks (KANs) in terms of their edge functions. If these functions are all affine, then universality clearly fails. How many non-affine functions are needed, in addition to affine ones, to ensure universality? We show that a single one suffices. More precisely, we prove that deep KANs in which all edge functions are either affine or equal to a fixed continuous function $\sigma$ are dense in $C(K)$ for every compact set $K\subset\mathbb{R}^n$ if and only if $\sigma$ is non-affine. In contrast, for KANs with exactly two hidden layers, universality holds if and only if $\sigma$ is nonpolynomial. We further show that the full class of affine functions is not required; it can be replaced by a finite set without affecting universality. In particular, in the nonpolynomial case, a fixed family of five affine functions suffices when the depth is arbitrary. More generally, for every continuous non-affine function $\sigma$, there exists a finite affine family $A_\sigma$ such that deep KANs with edge functions in $A_\sigma\cup\{\sigma\}$ remain universal. We also prove that KANs with the spline-based edge parameterization introduced by Liu et al.~\cite{Liu2024} are universal approximators in the classical sense, even when the spline degree and knot sequence are fixed in advance.

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Vugar Ismailov. 2026-04-26. Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks. https://arxiv.org/abs/2604.23765

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