mlsanityData to 14 Oct 2026

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

arXiv 2509.19830 ↗First posted 24 Sep 2025cs.LG cs.AI stat.MLcited by 4

Not on the map. Why?

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.

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

16 of 42 references.

References · 16

  1. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,238
  2. 2024KAN: Kolmogorov-Arnold Networkscited by 649
  3. 2017Nonparametric regression using deep neural networks with ReLU activation functioncited by 433
  4. 2024Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximationcited by 118
  5. 2024KAN 2.0: Kolmogorov-Arnold Networks Meet Sciencecited by 118
  6. 2024Kolmogorov-Arnold Networks (KANs) for Time Series Analysiscited by 98
  7. 2020The Kolmogorov-Arnold representation theorem revisitedcited by 89
  8. 2024Convolutional Kolmogorov-Arnold Networkscited by 77
  9. 2024Kolmogorov-Arnold Networks for Time Series: Bridging Predictive Power and Interpretabilitycited by 70
  10. 2024Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networkscited by 67
  11. 2024KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physicscited by 59
  12. 2018Deep Neural Networks Learn Non-Smooth Functions Effectivelycited by 57
4 more
  1. 2024GKAN: Graph Kolmogorov-Arnold Networkscited by 53
  2. 2019Deep learning is adaptive to intrinsic dimensionality of model smoothness in anisotropic Besov spacecited by 46
  3. 2024rKAN: Rational Kolmogorov-Arnold Networkscited by 35
  4. 2023Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regressioncited by 5

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 4.

4 papers.

  1. 2026Optimization, Generalization and Differential Privacy Bounds for Gradient Descent on Kolmogorov-Arnold Networkscited by 1
  2. 2026Optimization Risk Bounds for Kolmogorov-Arnold Networks Trained by DP-SGD with Correlated Noisecited by 1
  3. 2026Necessary and sufficient conditions for universality of Kolmogorov-Arnold networkscited by 0
  4. 2026KAPLAN: Kolmogorov-Arnold Prognostic Learnable Activation Networks for Survival Analysiscited by 0