mlsanityData to 7 Oct 2026

Some Super-approximation Rates of ReLU Neural Networks for Korobov Functions

arXiv 2507.10345 ↗First posted 14 Jul 2025cs.LGcited by 3

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Abstract (arXiv)

This paper examines the Lp and W1p norm approximation errors of ReLU neural networks for Korobov functions. In terms of network width and depth, we derive nearly optimal super-approximation error bounds of order 2m in the Lp norm and order 2m-2 in the W1p norm, for target functions with Lp mixed derivative of order m in each direction. The analysis leverages sparse grid finite elements and the bit extraction technique. Our results improve upon classical lowest order L_∞ and H1 norm error bounds and demonstrate that the expressivity of neural networks is largely unaffected by the curse of dimensionality.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 22 of its 42 references that are in the map. First posted on arXiv 14 Jul 2025.

Builds on

22 of 42 references.

References · 22

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2017Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networkscited by 304
  3. 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 165
  4. 2020Deep Network Approximation for Smooth Functionscited by 163
  5. 2019Deep Network Approximation Characterized by Number of Neuronscited by 120
  6. 2019The phase diagram of approximation rates for deep neural networkscited by 86
  7. 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
  8. 2018ResNet with one-neuron hidden layers is a Universal Approximatorcited by 78
  9. 2019Deep Learning via Dynamical Systems: An Approximation Perspectivecited by 78
  10. 2020Neural Network Approximation: Three Hidden Layers Are Enoughcited by 63
  11. 2022Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spacescited by 50
  12. 2019Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networkscited by 40
10 more
  1. 2019Nonlinear Approximation via Compositionscited by 37
  2. 2023Deep Network Approximation: Beyond ReLU to Diverse Activation Functionscited by 33
  3. 2023Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivativescited by 30
  4. 2022Approximation bounds for norm constrained neural networks with applications to regression and GANscited by 21
  5. 2022Neural Network Architecture Beyond Width and Depthcited by 17
  6. 2023On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theoremcited by 13
  7. 2024On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networkscited by 13
  8. 2021ReLU Deep Neural Networks from the Hierarchical Basis Perspectivecited by 12
  9. 2025Higher Order Approximation Rates for ReLU CNNs in Korobov Spacescited by 4
  10. 2025Two-Dimensional Deep ReLU CNN Approximation for Korobov Functions: A Constructive Approachcited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 1; 2026: 2.

3 papers.

  1. 2025On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functionscited by 0
  2. 2026Approximation of Analytic Functions by ReLU Neural Networks with Adjustable Depth and Widthcited by 0
  3. 2026Shallow neural network approximation in mixed Sobolev spacescited by 0