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arXiv · 1912.04310

Efficient approximation of high-dimensional functions with neural networks

Patrick Cheridito·Arnulf Jentzen·Florian Rossmannek
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Abstract

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.

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Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek. 2019-12-09. Efficient approximation of high-dimensional functions with neural networks. https://doi.org/10.1109/tnnls.2021.3049719

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