Neural Network Approximation of Refinable Functions

Ingrid Daubechies, Ronald Devore, Nadav Dym, Shira Faigenbaum-Golovin, Shahar Z. Kovalsky, Kung Chin Lin, Josiah Park, Guergana Petrova, Barak Sober*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions and, on the other hand, very rough functions such as the Weierstrass function, which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU neural networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.

Original languageAmerican English
Article number1
Pages (from-to)482-495
Number of pages14
JournalIEEE Transactions on Information Theory
Issue number1
StatePublished - 1 Jan 2023

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  • Neural networks
  • cascade algorithm
  • exponential accuracy
  • neural network approximation
  • refinable functions


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