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 $K\subset\mathbb{R}^n$ 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 $A_σ$ such that deep KANs with edge functions in $A_σ\cup\{σ\}$ 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.

翻译：我们从边缘函数的角度分析了Kolmogorov-Arnold网络（KANs）的普适逼近性质。若所有边缘函数均为仿射函数，则普适性显然不成立。在仿射函数之外还需多少非仿射函数才能保证普适性？我们证明：单个非仿射函数即已足够。更精确地说，我们证明：对于每个紧集$K\subset\mathbb{R}^n$，当且仅当$σ$是非仿射函数时，所有边缘函数均为仿射函数或等于固定连续函数$σ$的深层KAN在$C(K)$中稠密。相比之下，对于恰好具有两个隐藏层的KAN，普适性成立当且仅当$σ$是非多项式函数。我们进一步证明，并不需要完整的仿射函数类；它可被有限集替代而不影响普适性。特别地，在非多项式情形下，当深度任意时，固定的五个仿射函数族即已足够。更一般地，对于每个连续非仿射函数$σ$，存在有限仿射族$A_σ$，使得边缘函数属于$A_σ\cup\{σ\}$的深层KAN仍具有普适性。我们还证明，采用Liu等人~\cite{Liu2024}引入的基于样条的边缘参数化的KAN在经典意义下是普适逼近器，即使样条次数和节点序列是预先固定的。