Estimation of a function of low local dimensionality by deep neural networks

Michael Kohler, Adam Krzyzak, Sophie Langer

Research output: Working paper

Abstract

Deep neural networks (DNNs) achieve impressive results for complicated tasks like
object detection on images and speech recognition. Motivated by this practical success, there is now a strong interest in showing good theoretical properties of DNNs.
To describe for which tasks DNNs perform well and when they fail, it is a key challenge to understand their performance. The aim of this paper is to contribute to the
current statistical theory of DNNs.
We apply DNNs on high dimensional data and we show that the least squares regression estimates using DNNs are able to achieve dimensionality reduction in case
that the regression function has locally low dimensionality. Consequently, the rate of convergence of the estimate does not depend on its input dimension d, but on its
local dimension d∗ and the DNNs are able to circumvent the curse of dimensionality in case that d∗ is much smaller than d. In our simulation study we provide numerical
experiments to support our theoretical result and we compare our estimate with other conventional nonparametric regression estimates. The performance of our estimates
is also validated in experiments with real data.
Original languageEnglish
PublisherarXiv.org
Number of pages69
Publication statusSubmitted - 2021

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