Deep ReLU neural networks in high-dimensional approximation

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