Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks

arXiv:2604.23765v1 Announce Type: new 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 KsubsetR^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_sigmaup{sigma} remain universal. We also prove that KANs with the spline-based edge parameterization introduced by Liu et al.~ite{Liu2024} are universal approximators in the classical sense, even when the spline degree and knot sequence are fixed in advance.

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Source: arXiv cs.LG | 2026-04-28