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Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem

arXiv 1906.11945 ↗First posted 27 Jun 2019math.NA cs.LG cs.NAcited by 49

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Abstract (arXiv)

We prove a theorem concerning the approximation of multivariate functions by deep ReLU networks, for which the curse of the dimensionality is lessened. Our theorem is based on a constructive proof of the Kolmogorov--Arnold superposition theorem, and on a subset of multivariate continuous functions whose outer superposition functions can be efficiently approximated by deep ReLU networks.

Key facts. Cited by 49 papers in the map (data through 7 Oct 2026). First posted on arXiv 27 Jun 2019.

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Citing papers per year: 2021: 11; 2022: 5; 2023: 8; 2024: 8; 2025: 9; 2026: 8.

49 papers.

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  1. 2024KAN: Kolmogorov-Arnold Networkscited by 634
  2. 2023Specformer: Spectral Graph Neural Networks Meet Transformerscited by 102
  3. 2021On the approximation of functions by tanh neural networkscited by 81
  4. 2024A Survey on Kolmogorov-Arnold Networkcited by 67
  5. 2022ExSpliNet: An interpretable and expressive spline-based neural networkcited by 33
  6. 2023Deep Network Approximation: Beyond ReLU to Diverse Activation Functionscited by 33
  7. 2021Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networkscited by 32
  8. 2024Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KANcited by 25
  9. 2021Reproducing Activation Function for Deep Learningcited by 19
  10. 2023On the Kolmogorov neural networkscited by 19
  11. 2022Neural Network Architecture Beyond Width and Depthcited by 17
  12. 2021The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimationcited by 14
37 more
  1. 2021Convergence of Deep ReLU Networkscited by 14
  2. 2021Convergence of Deep Convolutional Neural Networkscited by 13
  3. 2023On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theoremcited by 13
  4. 2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
  5. 2021Deep Network Approximation in Terms of Intrinsic Parameterscited by 12
  6. 2024KANICE: Kolmogorov-Arnold Networks with Interactive Convolutional Elementscited by 12
  7. 2023Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustnesscited by 10
  8. 2021Stationary Density Estimation of Ito Diffusions Using Deep Learningcited by 9
  9. 2021Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on H"older Classcited by 7
  10. 2022Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraintcited by 6
  11. 2023On Enhancing Expressive Power via Compositions of Single Fixed-Size ReLU Networkcited by 6
  12. 2023Uniform Convergence of Deep Neural Networks with Lipschitz Continuous Activation Functions and Variable Widthscited by 5
  13. 2024KAN we improve on HEP classification tasks? Kolmogorov-Arnold Networks applied to an LHC physics examplecited by 4
  14. 2025Fourier Multi-Component and Multi-Layer Neural Networks: Unlocking High-Frequency Potentialcited by 4
  15. 2025KAC: Kolmogorov-Arnold Classifier for Continual Learningcited by 4
  16. 2024Structured and Balanced Multi-Component and Multi-Layer Neural Networkscited by 3
  17. 2021Analysis of Structured Deep Kernel Networkscited by 2
  18. 2023Approximating Langevin Monte Carlo with ResNet-like Neural Network architecturescited by 2
  19. 2025Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertaintycited by 2
  20. 2024Generative Modeling: A Reviewcited by 2
  21. 2025Concurrent training methods for Kolmogorov-Arnold networks: Disjoint datasets and FPGA implementationcited by 2
  22. 2022Convergence Analysis of Deep Residual Networkscited by 1
  23. 2025Kolmogorov GAM Networks are all you need!cited by 1
  24. 2025A Kolmogorov-Arnold Neural Model for Cascading Extremescited by 1
  25. 2026Multigrade Neural Network Approximationcited by 1
  26. 2026Layer-wise Lipschitz-Product Control for Deep Kolmogorov--Arnold Network Representations of Compositionally Structured Functionscited by 1
  27. 2022Convergence of Deep Neural Networks with General Activation Functions and Poolingcited by 0
  28. 2024Learnability of high-dimensional targets by two-parameter models and gradient flowcited by 0
  29. 2025Identifying Unknown Stochastic Dynamics via Finite expression methodscited by 0
  30. 2025Worst-Case Distance-Aware Error Bounds for Neural Networkscited by 0
  31. 2025Kolmogorov-Arnold graph neural networks for chemically informed prediction tasks on inorganic nanomaterialscited by 0
  32. 2026Fully Kolmogorov-Arnold Deep Model in Medical Image Segmentationcited by 0
  33. 2026Geometric Kolmogorov--Arnold Network (GeoKAN)cited by 0
  34. 2026Adaptive RBF-KAN: A Comparative Evaluation of Dynamic Shape Parameters in Kolmogorov-Arnold Networkscited by 0
  35. 2026Deep Learning for Reflected BSDEs: Regularization and Error Analysiscited by 0
  36. 2026Neural Approximation by Function Composition: Rigidity and Doubly Exponential Convergencecited by 0
  37. 2026SW-KAN: Kolmogorov-Arnold Networks with Stieltjes-Wigert q-Orthogonal Polynomialscited by 0