# Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks

arXiv: https://arxiv.org/abs/2604.23765 · HTML: https://mlsanity.com/paper/2604.23765/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 26 references that are in the map.
- First posted on arXiv 26 Apr 2026.

## Builds on (most cited first, 6 of 6)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [On the expressiveness and spectral bias of KANs](https://mlsanity.com/paper/2410.01803/) (2024)
- [A Practitioner's Guide to Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2510.25781/) (2025)
- [Approximation Rates in Besov Norms and Sample-Complexity of Kolmogorov-Arnold Networks with Residual Connections](https://mlsanity.com/paper/2504.15110/) (2025)
- [On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimators](https://mlsanity.com/paper/2509.19830/) (2025)
- [Free-RBF-KAN: Kolmogorov-Arnold Networks with Adaptive Radial Basis Functions for Efficient Function Learning](https://mlsanity.com/paper/2601.07760/) (2026)

## Abstract (arXiv)

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.