Two-Dimensional Deep ReLU CNN Approximation for Korobov Functions: A Constructive Approach
Qin Fang, Lei Shi, Min Xu, Ding-Xuan Zhou
TL;DR
The paper addresses high-dimensional approximation of Korobov functions using 2D deep ReLU CNNs. It develops a constructive framework that maps sparse-grid hierarchical basis representations into 2D CNN architectures, employing directional shift blocks to realize tensor products. The main result provides explicit width, depth, and size bounds and a p-dependent convergence rate, with a corollary giving epsilon-approximation complexity and near-optimality under continuous weight selection. This work establishes a theoretical foundation for the use of 2D CNNs in high-dimensional function approximation and suggests directions for applying these constructs to numerical PDEs and learning Korobov-like functions.
Abstract
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 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.