Approximation in shift-invariant spaces with deep ReLU neural networks

TL;DR

This paper investigates how deep ReLU neural networks can effectively approximate functions in shift-invariant spaces, providing error bounds, optimality results, and applications to classical function spaces like Sobolev and Besov spaces.

Contribution

The paper introduces a method to approximate functions in shift-invariant spaces using deep ReLU networks, with explicit error bounds and asymptotic optimality analysis.

Findings

Derived approximation error bounds based on network width and depth.

Established the asymptotic optimality of the neural network construction.

Applied results to classical function spaces such as Sobolev and Besov spaces.

Abstract

We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are estimated with respect to the width and depth of neural networks. The network construction is based on the bit extraction and data-fitting capacity of deep neural networks. As applications of our main results, the approximation rates of classical function spaces such as Sobolev spaces and Besov spaces are obtained. We also give lower bounds of the $L^p (1\le p \le \infty)$ approximation error for Sobolev spaces, which show that our construction of neural network is asymptotically optimal up to a logarithmic factor.