mlsanityData to 7 Oct 2026

The Kolmogorov-Arnold representation theorem revisited

arXiv 2007.15884 ↗First posted 31 Jul 2020cs.LG cs.NE stat.MLcited by 89

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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.

Key facts. Cited by 89 papers in the map (data through 7 Oct 2026). First posted on arXiv 31 Jul 2020.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 6; 2023: 4; 2024: 21; 2025: 41; 2026: 15.

89 papers.

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  1. 2024KAN: Kolmogorov-Arnold Networkscited by 634
  2. 2024Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximationcited by 115
  3. 2024Kolmogorov-Arnold Networks (KANs) for Time Series Analysiscited by 97
  4. 2024A Survey on Kolmogorov-Arnold Networkcited by 67
  5. 2024fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functionscited by 60
  6. 2024BSRBF-KAN: A combination of B-splines and Radial Basis Functions in Kolmogorov-Arnold Networkscited by 43
  7. 2024Suitability of KANs for Computer Vision: A preliminary investigationcited by 33
  8. 2024rKAN: Rational Kolmogorov-Arnold Networkscited by 31
  9. 2021On the Existence of Universal Lottery Ticketscited by 30
  10. 2024Prompting a Pretrained Transformer Can Be a Universal Approximatorcited by 25
  11. 2022Approximation bounds for norm constrained neural networks with applications to regression and GANscited by 21
  12. 2024A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative Modelscited by 20
38 more
  1. 2022A Survey on Over-the-Air Computationcited by 19
  2. 2023On the Kolmogorov neural networkscited by 19
  3. 2024FC-KAN: Function Combinations in Kolmogorov-Arnold Networkscited by 18
  4. 2025Kolmogorov-Arnold Fourier Networkscited by 18
  5. 2023On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theoremcited by 13
  6. 2024KANICE: Kolmogorov-Arnold Networks with Interactive Convolutional Elementscited by 12
  7. 2025LeanKAN: A Parameter-Lean Kolmogorov-Arnold Network Layer with Improved Memory Efficiency and Convergence Behaviorcited by 12
  8. 2025Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)cited by 12
  9. 2025Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNscited by 9
  10. 2022On the inability of Gaussian process regression to optimally learn compositional functionscited by 7
  11. 2024Architectural Scaling Surpass Basis Complexity? Efficient KANs with Single-Parameter Designcited by 6
  12. 2025Free-Knots Kolmogorov-Arnold Network: On the Analysis of Spline Knots and Advancing Stabilitycited by 6
  13. 2024Exploring Kolmogorov-Arnold networks for realistic image sharpness assessmentcited by 5
  14. 2024Kolmogorov-Arnold Networks in Low-Data Regimes: A Comparative Study with Multilayer Perceptronscited by 5
  15. 2025Transformers Can Overcome the Curse of Dimensionality: A Theoretical Study from an Approximation Perspectivecited by 5
  16. 2022KASAM: Spline Additive Models for Function Approximationcited by 4
  17. 2023ENN: A Neural Network with DCT Adaptive Activation Functionscited by 4
  18. 2025Introducing the Short-Time Fourier Kolmogorov Arnold Network: A Dynamic Graph CNN Approach for Tree Species Classification in 3D Point Cloudscited by 4
  19. 2025Approximation Bounds for Transformer Networks with Application to Regressioncited by 4
  20. 2021A structured proof of Kolmogorov's Superposition Theoremcited by 3
  21. 2024Acceleration of Grokking in Learning Arithmetic Operations via Kolmogorov-Arnold Representationcited by 3
  22. 2025AF-KAN: Activation Function-Based Kolmogorov-Arnold Networks for Efficient Representation Learningcited by 3
  23. 2025QuKAN: A Quantum Circuit Born Machine approach to Quantum Kolmogorov Arnold Networkscited by 3
  24. 2025KANMixer: a minimal KAN-centered mixer for long-term time series forecastingcited by 3
  25. 2025Interpretable Clinical Classification with Kolmogorov-Arnold Networkscited by 3
  26. 2026Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learningcited by 3
  27. 2022Few-shot human motion prediction for heterogeneous sensorscited by 2
  28. 2025Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertaintycited by 2
  29. 2025Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networkscited by 2
  30. 2025Relating Piecewise Linear Kolmogorov Arnold Networks to ReLU Networkscited by 2
  31. 2025Pushing the Performance of Synthetic Speech Detection with Kolmogorov-Arnold Networks and Self-Supervised Learning Modelscited by 2
  32. 2025On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimatorscited by 2
  33. 2025Assessing the performance of correlation-based multi-fidelity neural emulatorscited by 2
  34. 2026Fixed Aggregation Features Can Rival GNNscited by 2
  35. 2024Image Classification using Fuzzy Pooling in Convolutional Kolmogorov-Arnold Networkscited by 1
  36. 2024KAT to KANs: A Review of Kolmogorov-Arnold Networks and the Neural Leap Forwardcited by 1
  37. 2025Kolmogorov GAM Networks are all you need!cited by 1
  38. 2025DeepOHeat-v1: Efficient Operator Learning for Fast and Trustworthy Thermal Simulation and Optimization in 3D-IC Designcited by 1

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