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

arXiv: https://arxiv.org/abs/2507.10345 · HTML: https://mlsanity.com/paper/2507.10345/

## Key facts

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

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [ResNet with one-neuron hidden layers is a Universal Approximator](https://mlsanity.com/paper/1806.10909/) (2018)
- [Deep Learning via Dynamical Systems: An Approximation Perspective](https://mlsanity.com/paper/1912.10382/) (2019)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces](https://mlsanity.com/paper/2211.14400/) (2022)
- [Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networks](https://mlsanity.com/paper/1903.10047/) (2019)
- [Nonlinear Approximation via Compositions](https://mlsanity.com/paper/1902.10170/) (2019)
- [Deep Network Approximation: Beyond ReLU to Diverse Activation Functions](https://mlsanity.com/paper/2307.06555/) (2023)
- [Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives](https://mlsanity.com/paper/2305.08466/) (2023)
- [Approximation bounds for norm constrained neural networks with applications to regression and GANs](https://mlsanity.com/paper/2201.09418/) (2022)
- [Neural Network Architecture Beyond Width and Depth](https://mlsanity.com/paper/2205.09459/) (2022)
- [On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theorem](https://mlsanity.com/paper/2308.05509/) (2023)
- [On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networks](https://mlsanity.com/paper/2409.00901/) (2024)
- [ReLU Deep Neural Networks from the Hierarchical Basis Perspective](https://mlsanity.com/paper/2105.04156/) (2021)
- [Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces](https://mlsanity.com/paper/2501.11275/) (2025)
- [Two-Dimensional Deep ReLU CNN Approximation for Korobov Functions: A Constructive Approach](https://mlsanity.com/paper/2503.07976/) (2025)

## Built on by (most cited first, 3 of 3)

- [On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions](https://mlsanity.com/paper/2511.12398/) (2025)
- [Approximation of Analytic Functions by ReLU Neural Networks with Adjustable Depth and Width](https://mlsanity.com/paper/2607.10589/) (2026)
- [Shallow neural network approximation in mixed Sobolev spaces](https://mlsanity.com/paper/2609.05263/) (2026)

## 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.