Provable approximation properties for deep neural networks

Uri Shaham, Alexander Cloninger, Ronald R. Coifman

Research output: Contribution to journalArticlepeer-review

115 Scopus citations


We discuss approximation of functions using deep neural nets. Given a function f on a d-dimensional manifold Γ⊂Rm, we construct a sparsely-connected depth-4 neural network and bound its error in approximating f. The size of the network depends on dimension and curvature of the manifold Γ the complexity of f, in terms of its wavelet description, and only weakly on the ambient dimension m. Essentially, our network computes wavelet functions, which are computed from Rectified Linear Units (ReLU).

Original languageEnglish
Pages (from-to)537-557
Number of pages21
JournalApplied and Computational Harmonic Analysis
Issue number3
StatePublished - May 2018
Externally publishedYes

Bibliographical note

Funding Information:
The authors thank Stefan Steinerberger, Roy Lederman for their help, and to Andrew Barron, Ed Bosch, Mark Tygert and Yann LeCun for their comments. Alexander Cloninger is supported by NSF Award No. DMS-1402254 .

Publisher Copyright:
© 2016 Elsevier Inc.


  • Function approximation
  • Neural nets
  • Wavelets


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