Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks

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 σ are dense in C(K) for every compact set Ksubsetℝn if and only if σ is non-affine. In contrast, for KANs with exactly two hidden layers, universality holds if and only if σ 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 σ, there exists a finite affine family Aσ such that deep KANs with edge functions in Aσcupσ remain universal. We also prove that KANs with the spline-based edge parameterization introduced by Liu et al.~citeLiu2024 are universal approximators in the classical sense, even when the spline degree and knot sequence are fixed in advance.