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Approximation of smooth functionals using deep ReLU networks.

Linhao Song1, Ying Liu2, Jun Fan3

  • 1School of Mathematical Science, Beihang University, Beijing, China; School of Data Science, City University of Hong Kong, Kowloon, Hong Kong.

Neural Networks : the Official Journal of the International Neural Network Society
|August 7, 2023
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Summary
This summary is machine-generated.

Deep neural networks, particularly functional deep ReLU networks, show improved approximation capabilities for nonlinear continuous functionals. This research provides new theoretical analysis for ReLU networks, offering better approximation rates for various function spaces.

Keywords:
Approximation theoryDeep learning theoryFréchet derivativePolynomial ratesReLUSmooth functionals

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Area of Science:

  • Applied Mathematics
  • Machine Learning Theory
  • Numerical Analysis

Background:

  • Deep neural networks are used to approximate nonlinear continuous functionals.
  • Existing theoretical analyses have limitations, especially for Rectified Linear Unit (ReLU) activation functions.
  • There is a need for improved theoretical understanding of functional deep ReLU networks' approximation power.

Purpose of the Study:

  • To investigate the approximation power of functional deep ReLU networks.
  • To analyze approximation capabilities for continuous functionals with restricted continuity moduli and those with higher-order Fréchet derivatives.
  • To develop novel network structures for feature extraction in functional approximation.

Main Methods:

  • Proposing a novel functional network architecture designed for higher-order smoothness feature extraction.
  • Deriving quantitative approximation rates based on network depth, width, and total number of weights.
  • Analyzing approximation errors on Hölder and analytic function spaces.

Main Results:

  • Achieved logarithmic approximation rates on the unit ball of Hölder spaces.
  • Established nearly polynomial approximation rates for analytic function spaces.
  • Demonstrated improved approximation results compared to existing literature, specifically for ReLU networks.

Conclusions:

  • Functional deep ReLU networks possess significant approximation power for nonlinear continuous functionals.
  • The proposed network structure and theoretical analysis provide a strong foundation for understanding deep learning in functional approximation.
  • This work advances the theoretical understanding of deep ReLU networks in approximating complex mathematical functions.