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Universal approximation theorem for vector- and hypercomplex-valued neural networks.

Marcos Eduardo Valle1, Wington L Vital2, Guilherme Vieira1

  • 1Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil.

Neural Networks : the Official Journal of the International Neural Network Society
|August 22, 2024
PubMed
Summary
This summary is machine-generated.

The universal approximation theorem now applies to a broader class of vector-valued neural networks. This extension, based on non-degenerate algebras, broadens the applicability of neural network theory.

Keywords:
Hypercomplex algebrasNeural networksUniversal approximation theorem

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Area of Science:

  • Mathematics
  • Computer Science
  • Artificial Intelligence

Background:

  • The universal approximation theorem is fundamental to neural network theory.
  • Existing theorems cover real-valued and some hypercomplex-valued neural networks.
  • Hypercomplex-valued neural networks are vector-valued networks with specific algebraic properties.

Purpose of the Study:

  • To extend the universal approximation theorem.
  • To encompass a wider range of vector-valued neural networks.
  • To include hypercomplex-valued neural networks as specific cases.

Main Methods:

  • Introduction of the concept of non-degenerate algebra.
  • Formulation of the universal approximation theorem for neural networks defined on non-degenerate algebras.

Main Results:

  • The universal approximation theorem is generalized to neural networks defined on non-degenerate algebras.
  • This generalization includes hypercomplex-valued neural networks.

Conclusions:

  • The extended theorem broadens the theoretical foundation for various vector-valued neural networks.
  • This work enhances the understanding and application of neural networks in diverse mathematical and computational domains.