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When is approximation by Gaussian networks necessarily a linear process?

H N Mhaskar1

  • 1Department of Mathematics, California State University, Los Angeles, CA 90032, USA. hmhaskar@calstatela.edu

Neural Networks : the Official Journal of the International Neural Network Society
|August 18, 2004
PubMed
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This study links approximation degrees of Gaussian networks to Hermite expansions. It shows that if a function is well-approximated by Gaussian networks, its Hermite expansion also achieves a similar approximation degree.

Area of Science:

  • Numerical Analysis
  • Approximation Theory
  • Machine Learning Theory

Background:

  • Gaussian networks are functions used in approximation theory and machine learning.
  • Minimal separation of centers is crucial for stability and approximation properties.
  • Understanding the relationship between different approximation methods is essential.

Purpose of the Study:

  • To establish a connection between the approximation degree of Gaussian networks and Hermite expansions.
  • To investigate the conditions under which approximation by Gaussian networks implies a certain approximation degree by Hermite expansions.
  • To explore the construction of Gaussian networks for achieving specific approximation degrees.

Main Methods:

  • Analysis of L2 (nonlinear) approximation degrees for functions using Gaussian networks (Gm).

Related Experiment Videos

  • Comparison with the approximation degrees obtained from rectangular partial sums of Hermite expansions.
  • Investigation of Lp norms (1 <= p <= infinity) with conditions on the number of neurons (N).

    • Main Results:

    • A function's approximation degree using Gaussian networks (Gm) directly correlates with its Hermite expansion's approximation degree.
    • Gaussian networks with fixed centers and linear functional coefficients can achieve the same approximation degree.
    • Similar results hold for Lp norms, provided logN is proportional to m2.

    Conclusions:

    • The study provides a theoretical link between Gaussian network approximation and Hermite series approximation.
    • This finding has implications for understanding the representational power and approximation capabilities of Gaussian networks.
    • The results offer insights into constructing efficient Gaussian network models for function approximation.