  In this paper, we develop a framework for showing that neural networks can
overcome the curse of dimensionality in different high-dimensional
approximation problems. Our approach is based on the notion of a catalog
network, which is a generalization of a standard neural network in which the
nonlinear activation functions can vary from layer to layer as long as they are
chosen from a predefined catalog of functions. As such, catalog networks
constitute a rich family of continuous functions. We show that under
appropriate conditions on the catalog, catalog networks can efficiently be
approximated with rectified linear unit-type networks and provide precise
estimates on the number of parameters needed for a given approximation
accuracy. As special cases of the general results, we obtain different classes
of functions that can be approximated with ReLU networks without the curse of
dimensionality.
