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Neural network approximation of continuous functionals and continuous functions on compactifications.

M B. Stinchcombe1

  • 1Department of Economics, The University of Texas at Austin, Austin, TX, USA

Neural Networks : the Official Journal of the International Neural Network Society
|March 29, 2003
PubMed
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This study identifies specific activation functions enabling feedforward neural networks to approximate continuous functions in R(1). The findings differ for higher dimensions, impacting neural network approximation capabilities.

Area of Science:

  • Computational theory
  • Artificial intelligence
  • Mathematical analysis

Background:

  • Feedforward neural networks are powerful function approximators.
  • Understanding the role of activation functions is crucial for network capabilities.
  • Compactification of Euclidean spaces R(n) is a key mathematical concept.

Purpose of the Study:

  • To characterize activation functions for feedforward network approximation on compactified R(1).
  • To investigate limitations and possibilities for approximation in higher dimensions R(n), n>=2.
  • To explore implications for multi-layer and infinite-dimensional neural networks.

Main Methods:

  • Analysis of activation functions (bounded and unbounded) in feedforward networks.
  • Examination of function approximation on classic compactifications of R(n).

Related Experiment Videos

  • Comparison of nonpolynomial, analytic, and sigmoidal activation functions with limited weight sets.

    • Main Results:

    • Characterization achieved for R(1) compactifications.
    • Approximation capabilities differ significantly for R(n), n>=2.
    • Nonpolynomial analytic functions with limited weights approximate continuous functions on compact sets in R(n).
    • Sigmoidal functions with limited weights fail to approximate on compactifications.

    Conclusions:

    • The structure of compactification is key to approximation possibilities.
    • Results provide insights into the theoretical limits of neural network function approximation.
    • Findings have implications for designing neural networks in various dimensions and infinite settings.