Title: Determining the Necessary and Sufficient Criteria for the Universality of Kolmogorov-Arnold Networks

Abstract

This study investigates the universal approximation capabilities of Kolmogorov-Arnold Networks (KANs) by examining the nature of their edge functions. It is evident that if all edge functions are affine, the network fails to achieve universality. The central question addressed is the minimum number of non-affine functions required, alongside affine ones, to guarantee this property. We demonstrate that a single non-affine function is sufficient.

Specifically, we establish that deep KANs, where every edge function is either affine or identical to a fixed continuous function $\sigma$, form a dense set in $C(K)$ for any compact subset $K\subset\mathbb{R}^n$ if and only if $\sigma$ is non-affine. However, for KANs structured with exactly two hidden layers, the condition for universality is stricter: it holds if and only if $\sigma$ is nonpolynomial.

Furthermore, our analysis reveals that the inclusion of the entire class of affine functions is not a prerequisite for universality; this requirement can be satisfied by a finite set without compromising the approximation capability. In the nonpolynomial scenario, a specific family of five affine functions is adequate, regardless of the network's depth. More broadly, for any continuous non-affine function $\sigma$, we identify a finite affine family $A_\sigma$ such that deep KANs utilizing edge functions from the union $A_\sigma\cup{\sigma}$ retain their universal approximation properties. Finally, we prove that KANs employing the spline-based edge parameterization proposed by Liu et al.~\cite{Liu2024} serve as universal approximators in the classical sense, even when the spline degree and knot sequence are predetermined.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC