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

Michael Kohler, Adam Krzyzak, Sophie Langer

Research output: Contribution to journalArticleAcademicpeer-review


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
Pages (from-to)4032-4042
Number of pages11
JournalIEEE transactions on information theory
Issue number6
Early online date26 Jan 2022
Publication statusPublished - Jun 2022


  • Clouds
  • Convergence
  • Curse of dimensionality
  • Data models
  • Deep learning
  • Deep neural networks
  • Neural networks
  • Nonparametric regression
  • Piecewise partitioning
  • rate of convergence
  • Temperature
  • Temperature dependence


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