# The Kolmogorov-Arnold representation theorem revisited

arXiv: https://arxiv.org/abs/2007.15884 · HTML: https://mlsanity.com/paper/2007.15884/

## Key facts

- Cited by 89 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jul 2020.

## Built on by (most cited first, 30 of 89)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [Kolmogorov-Arnold Networks (KANs) for Time Series Analysis](https://mlsanity.com/paper/2405.08790/) (2024)
- [A Survey on Kolmogorov-Arnold Network](https://mlsanity.com/paper/2411.06078/) (2024)
- [fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions](https://mlsanity.com/paper/2406.07456/) (2024)
- [BSRBF-KAN: A combination of B-splines and Radial Basis Functions in Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.11173/) (2024)
- [rKAN: Rational Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.14495/) (2024)
- [Suitability of KANs for Computer Vision: A preliminary investigation](https://mlsanity.com/paper/2406.09087/) (2024)
- [On the Existence of Universal Lottery Tickets](https://mlsanity.com/paper/2111.11146/) (2021)
- [Prompting a Pretrained Transformer Can Be a Universal Approximator](https://mlsanity.com/paper/2402.14753/) (2024)
- [Approximation bounds for norm constrained neural networks with applications to regression and GANs](https://mlsanity.com/paper/2201.09418/) (2022)
- [A Survey on Over-the-Air Computation](https://mlsanity.com/paper/2210.11350/) (2022)
- [A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative Models](https://mlsanity.com/paper/2401.07187/) (2024)
- [On the Kolmogorov neural networks](https://mlsanity.com/paper/2311.00049/) (2023)
- [FC-KAN: Function Combinations in Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2409.01763/) (2024)
- [Kolmogorov-Arnold Fourier Networks](https://mlsanity.com/paper/2502.06018/) (2025)
- [On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theorem](https://mlsanity.com/paper/2308.05509/) (2023)
- [Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)](https://mlsanity.com/paper/2503.05617/) (2025)
- [KANICE: Kolmogorov-Arnold Networks with Interactive Convolutional Elements](https://mlsanity.com/paper/2410.17172/) (2024)
- [LeanKAN: A Parameter-Lean Kolmogorov-Arnold Network Layer with Improved Memory Efficiency and Convergence Behavior](https://mlsanity.com/paper/2502.17844/) (2025)
- [Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNs](https://mlsanity.com/paper/2506.07958/) (2025)
- [On the inability of Gaussian process regression to optimally learn compositional functions](https://mlsanity.com/paper/2205.07764/) (2022)
- [Architectural Scaling Surpass Basis Complexity? Efficient KANs with Single-Parameter Design](https://mlsanity.com/paper/2410.14951/) (2024)
- [Free-Knots Kolmogorov-Arnold Network: On the Analysis of Spline Knots and Advancing Stability](https://mlsanity.com/paper/2501.09283/) (2025)
- [KASAM: Spline Additive Models for Function Approximation](https://mlsanity.com/paper/2205.06376/) (2022)
- [Exploring Kolmogorov-Arnold networks for realistic image sharpness assessment](https://mlsanity.com/paper/2409.07762/) (2024)
- [Kolmogorov-Arnold Networks in Low-Data Regimes: A Comparative Study with Multilayer Perceptrons](https://mlsanity.com/paper/2409.10463/) (2024)
- [Transformers Can Overcome the Curse of Dimensionality: A Theoretical Study from an Approximation Perspective](https://mlsanity.com/paper/2504.13558/) (2025)
- [ENN: A Neural Network with DCT Adaptive Activation Functions](https://mlsanity.com/paper/2307.00673/) (2023)
- [Introducing the Short-Time Fourier Kolmogorov Arnold Network: A Dynamic Graph CNN Approach for Tree Species Classification in 3D Point Clouds](https://mlsanity.com/paper/2503.23647/) (2025)

## Abstract (arXiv)

There is a longstanding debate whether the Kolmogorov-Arnold representation theorem can explain the use of more than one hidden layer in neural networks. The Kolmogorov-Arnold representation decomposes a multivariate function into an interior and an outer function and therefore has indeed a similar structure as a neural network with two hidden layers. But there are distinctive differences. One of the main obstacles is that the outer function depends on the represented function and can be wildly varying even if the represented function is smooth. We derive modifications of the Kolmogorov-Arnold representation that transfer smoothness properties of the represented function to the outer function and can be well approximated by ReLU networks. It appears that instead of two hidden layers, a more natural interpretation of the Kolmogorov-Arnold representation is that of a deep neural network where most of the layers are required to approximate the interior function.