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. This paper also has implications for the theory of superpositions of real functions. In particular, we show that every continuous multivariate function can be approximated arbitrarily well using only the coordinate functions and two fixed univariate functions under repeated addition and composition.

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。除了仿射函数之外，还需要多少个非仿射函数才能确保普适性？我们证明单个非仿射函数足矣。更精确地，我们证明当且仅当固定连续函数$σ$为非仿射时，所有边缘函数或为仿射或等于$σ$的深度KANs在任意紧集$K\subset\mathbb{R}^n$上于$C(K)$中稠密。相比之下，对于恰好包含两个隐藏层的KANs，普适性成立当且仅当$σ$为非多项式函数。我们进一步证明，完整的仿射函数类并非必需——可用有限集合替代而不影响普适性。特别地，在非多项式情形下，对于任意深度，固定五个仿射函数组成的族即足以保证普适性。更一般地，对任意连续非仿射函数$σ$，存在有限仿射族$A_σ$，使得边缘函数取自$A_σ\cup\{σ\}$的深度KANs仍保持普适性。我们还证明，采用Liu等~\cite{Liu2024}提出的基于样条的边缘参数化的KANs在经典意义下是普适逼近器，即使样条次数与节点序列预先固定。本文对实函数叠加理论亦具启示意义，特别地，我们证明仅利用坐标函数与两个固定单变量函数经重复加法与复合运算即可任意逼近任意连续多元函数。