Deep ReLU neural networks in high-dimensional approximation

TL;DR

This paper analyzes the computational complexity of deep ReLU neural networks for high-dimensional function approximation, providing explicit bounds on network size and depth in relation to dimension and accuracy.

Contribution

It introduces explicit constructions and tight bounds for deep ReLU networks approximating functions in high dimensions, based on sparse-grid sampling and Faber series.

Findings

Derived dimension-dependent upper bounds on network size and depth.

Established lower bounds matching the upper bounds, proving tightness.

Connected neural network approximation complexity with sparse-grid sampling methods.

Abstract

We study the computation complexity of deep ReLU (Rectified Linear Unit) neural networks for the approximation of functions from the H\"older-Zygmund space of mixed smoothness defined on the $d$-dimensional unit cube when the dimension $d$ may be very large. The approximation error is measured in the norm of isotropic Sobolev space. For every function $f$ from the H\"older-Zygmund space of mixed smoothness, we explicitly construct a deep ReLU neural network having an output that approximates $f$ with a prescribed accuracy $\varepsilon$, and prove tight dimension-dependent upper and lower bounds of the computation complexity of this approximation, characterized as the size and the depth of this deep ReLU neural network, explicitly in $d$ and $\varepsilon$. The proof of these results are in particular, relied on the approximation by sparse-grid sampling recovery based on the Faber series.