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Multilayer Perceptrons to Approximate Quaternion Valued Functions.

M G. Xibilia1, G Muscato, L Fortuna

  • 1University of Catania, Italy

Neural Networks : the Official Journal of the International Neural Network Society
|March 1, 1997
PubMed
Summary

A new hypercomplex multilayer perceptron (HMLP) uses quaternion algebra for efficient processing of quaternionic signals. This neural network structure reduces neuron count and computational complexity for complex function approximation.

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Area of Science:

  • Artificial Intelligence
  • Computational Neuroscience
  • Algebraic Topology

Background:

  • Multilayer perceptrons (MLPs) are foundational in machine learning.
  • Existing complex multilayer perceptrons (CMLPs) handle complex-valued inputs.
  • A need exists for neural networks capable of handling higher-dimensional or non-real number systems.

Purpose of the Study:

  • Introduce a novel neural network architecture, the hypercomplex multilayer perceptron (HMLP).
  • Develop an HMLP utilizing quaternion algebra for quaternionic signal processing.
  • Establish the HMLP as a universal approximator for quaternion-valued functions.

Main Methods:

  • Development of a feedforward neural network based on quaternion algebra.
  • Formulation and proof of a new density theorem for HMLPs.

Related Experiment Videos

  • Adaptation of the density theorem proof for complex space (CMLP).

    • Main Results:

    • The HMLP processes quaternionic inputs/outputs with fewer neurons than real MLPs.
    • HMLPs demonstrate universal approximation capabilities for continuous quaternion-valued functions.
    • The structure efficiently approximates multidimensional real-valued functions, reducing local minima issues.

    Conclusions:

    • The HMLP offers reduced computational complexity and improved efficiency for specific signal processing tasks.
    • The HMLP generalizes CMLPs and provides a powerful tool for approximating complex, multidimensional functions.
    • The proposed structure shows promise in reducing training complexities and enhancing neural network performance.