# Two-Dimensional Deep ReLU CNN Approximation for Korobov Functions: A Constructive Approach

arXiv: https://arxiv.org/abs/2503.07976 · HTML: https://mlsanity.com/paper/2503.07976/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 31 references that are in the map.
- First posted on arXiv 11 Mar 2025.

## Builds on (most cited first, 14 of 14)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Object Detection with Deep Learning: A Review](https://mlsanity.com/paper/1807.05511/) (2018)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [MgNet: A Unified Framework of Multigrid and Convolutional Neural Network](https://mlsanity.com/paper/1901.10415/) (2019)
- [Equivalence of approximation by convolutional neural networks and fully-connected networks](https://mlsanity.com/paper/1809.00973/) (2018)
- [Deep Network Approximation: Beyond ReLU to Diverse Activation Functions](https://mlsanity.com/paper/2307.06555/) (2023)
- [Approximation Properties of Deep ReLU CNNs](https://mlsanity.com/paper/2109.00190/) (2021)
- [ReLU Deep Neural Networks from the Hierarchical Basis Perspective](https://mlsanity.com/paper/2105.04156/) (2021)
- [Learning Ability of Interpolating Deep Convolutional Neural Networks](https://mlsanity.com/paper/2210.14184/) (2022)
- [Deep Convolutional Neural Networks with Zero-Padding: Feature Extraction and Learning](https://mlsanity.com/paper/2307.16203/) (2023)

## Built on by (most cited first, 2 of 2)

- [Some Super-approximation Rates of ReLU Neural Networks for Korobov Functions](https://mlsanity.com/paper/2507.10345/) (2025)
- [Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations](https://mlsanity.com/paper/2605.22557/) (2026)

## Abstract (arXiv)

This paper investigates approximation capabilities of two-dimensional (2D) deep convolutional neural networks (CNNs), with Korobov functions serving as a benchmark. We focus on 2D CNNs, comprising multi-channel convolutional layers with zero-padding and ReLU activations, followed by a fully connected layer. We propose a fully constructive approach for building 2D CNNs to approximate Korobov functions and provide a rigorous analysis of the complexity of the constructed networks. Our results demonstrate that 2D CNNs achieve near-optimal approximation rates under the continuous weight selection model, significantly alleviating the curse of dimensionality. This work provides a solid theoretical foundation for 2D CNNs and illustrates their potential for broader applications in function approximation.