On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimators
This provides a theoretical foundation for using KANs in nonparametric regression, offering a structured alternative to existing methods for researchers and practitioners in machine learning and statistics.
The paper tackled the problem of establishing theoretical convergence guarantees for Kolmogorov-Arnold Networks (KANs) in multivariate function approximation, proving that additive and hybrid KANs achieve the minimax-optimal convergence rate O(n^{-2r/(2r+1)}) for functions in Sobolev spaces of smoothness r, with simulation studies confirming these rates.
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.