Efficient Approximation of High-Dimensional Functions With Neural Networks

IEEE Trans Neural Netw Learn Syst. 2022 Jul;33(7):3079-3093. doi: 10.1109/TNNLS.2021.3049719. Epub 2022 Jul 6.


In this article, 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 recitifed linear unit networks without the curse of dimensionality.

MeSH terms

  • Neural Networks, Computer*