On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimators

• URL: http://arxiv.org/abs/2509.19830v1
• Date: Wed, 24 Sep 2025 07:22:03 GMT
• Title: On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimators
• Authors: Wei Liu, Eleni Chatzi, Zhilu Lai,
• Abstract summary: Kolmogorov-Arnold Networks (KANs) offer a structured and interpretable framework for multivariate function approximation.<n>We prove that both additive and hybrid additive-multiplicative KANs attain the minimax-optimal convergence rate.<n>We derive guidelines for selecting the optimal number of knots in the B-splines.
• Score: 4.595923896761076
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Kolmogorov-Arnold Networks (KANs) offer a structured and interpretable framework for multivariate function approximation by composing univariate transformations through additive or multiplicative aggregation. This paper establishes theoretical convergence guarantees for KANs when the univariate components are represented by B-splines. We prove that both additive and hybrid additive-multiplicative KANs attain the minimax-optimal convergence rate $O(n^{-2r/(2r+1)})$ for functions in Sobolev spaces of smoothness $r$. We further derive guidelines for selecting the optimal number of knots in the B-splines. The theory is supported by simulation studies that confirm the predicted convergence rates. These results provide a theoretical foundation for using KANs in nonparametric regression and highlight their potential as a structured alternative to existing methods.

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