Approximation bounds of three-layered neural networks-A theorem on an integral transform with ridge functions

Noboru Murata*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


Neural networks have attracted attention due to their capability to perform nonlinear function approximation. In this paper, in order to better understand this capability, a new theorem on an integral transform was derived by applying ridge functions to neural networks. From the theorem, it is possible to obtain approximation bounds which clarify the quantitative relationship between the function approximation accuracy and the number of nodes in the hidden layer. The theorem indicates that the approximation accuracy depends on the smoothness of the target function. Furthermore, the theorem also shows that this type of approximation method differs from usual methods and is able to escape the so-called "curse of dimensionality," in which the approximation accuracy depends strongly of the input dimension of the function and deteriorates exponentially.

Original languageEnglish
Pages (from-to)23-33
Number of pages11
JournalElectronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)
Issue number3
Publication statusPublished - 1996 Mar
Externally publishedYes


  • Curse of dimensionality
  • Integral transform
  • Neural networks
  • Nonlinear function approximation
  • Ridge functions

ASJC Scopus subject areas

  • Electrical and Electronic Engineering


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