Before revision: We investigate approximations of functions from the H\"older-Zygmund space of mixed smoothness H^\alpha_\infty( \mathbb I ^d) defined on the d-dimensional unit cube \mathbb I ^d:= 0,1 ^d, by linear algorithms of sparse-grid sampling recovery and by deep ReLU (Rectified Linear Unit) neural networks when the dimension d may be very large.
After revision: 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 H^\alpha_\infty( \mathbb I ^d) defined on the d-dimensional unit cube \mathbb I ^d:= 0,1 ^d, by linear algorithms of sparse-grid sampling recovery and by deep ReLU (Rectified Linear Unit) neural networks when the dimension d may be very large.