Deep ReLU neural networks in high-dimensional approximation

Abstract

We study the computation complexity of deep ReLU (Rectified Linear Unit) neural networks for the approximation of functions from the Hölder–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 Hölder–Zygmund space of mixed smoothness, we explicitly construct a deep ReLU neural network having an output that approximates f with a prescribed accuracy ɛ, and prove tight dimension-dependent upper and lower bounds of the computation complexity of the approximation, characterized as the size and depth of this deep ReLU neural network, explicitly in d and ɛ. The proof of these results in particular, relies on the approximation by sparse-grid sampling recovery based on the Faber series.