Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks

ArXi:2604.23765v1 Announce Type: new We analyze the universal approximation property of Kolmogoro-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. 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.