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

arXiv: https://arxiv.org/abs/2509.19830 · HTML: https://mlsanity.com/paper/2509.19830/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 42 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 16 of 16)

- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [Kolmogorov-Arnold Networks (KANs) for Time Series Analysis](https://mlsanity.com/paper/2405.08790/) (2024)
- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [Convolutional Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.13155/) (2024)
- [Kolmogorov-Arnold Networks for Time Series: Bridging Predictive Power and Interpretability](https://mlsanity.com/paper/2406.02496/) (2024)
- [Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.11045/) (2024)
- [KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics](https://mlsanity.com/paper/2407.04192/) (2024)
- [Deep Neural Networks Learn Non-Smooth Functions Effectively](https://mlsanity.com/paper/1802.04474/) (2018)
- [GKAN: Graph Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.06470/) (2024)
- [Deep learning is adaptive to intrinsic dimensionality of model smoothness in anisotropic Besov space](https://mlsanity.com/paper/1910.12799/) (2019)
- [rKAN: Rational Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.14495/) (2024)
- [Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression](https://mlsanity.com/paper/2311.00140/) (2023)

## Built on by (most cited first, 4 of 4)

- [Optimization, Generalization and Differential Privacy Bounds for Gradient Descent on Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2601.22409/) (2026)
- [Optimization Risk Bounds for Kolmogorov-Arnold Networks Trained by DP-SGD with Correlated Noise](https://mlsanity.com/paper/2605.12648/) (2026)
- [Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks](https://mlsanity.com/paper/2604.23765/) (2026)
- [KAPLAN: Kolmogorov-Arnold Prognostic Learnable Activation Networks for Survival Analysis](https://mlsanity.com/paper/2605.23082/) (2026)

## Abstract (arXiv)

Kolmogorov-Arnold Networks (KANs) approximate multivariate functions by composing univariate transformations through additive or multiplicative aggregation. We establish convergence guarantees for KANs whose univariate components are B-splines. The least-squares estimator over the KAN spline sieve attains the rate O((log n / n)2r/(2r+1)), uniformly over a ball of regression functions admitting a KAN representation with univariate components of Sobolev smoothness r; a matching lower bound of order n-2r/(2r+1) shows this is minimax optimal up to the logarithmic factor, which we trace to the nonlinearity of the sieve rather than to the architecture. The rate is free of the ambient dimension d; this dimension-free exponent reflects the assumed KAN structure of the target, not an escape from the minimax rate n-2r/(2r+d) on Sobolev classes over [0,1]d. We derive a knot-selection rule, show that penalized selection over a dyadic knot grid attains the rate adaptively in the unknown smoothness, and show that univariate components are not identifiable under centering alone, so consistency of the fit does not imply consistency of the components. On targets of exactly known smoothness the fitted risk exponent is at least as steep as the bound in every configuration, and the predicted knot scaling and k-r approximation decay are checked directly.