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Hyperbolic Gradient Operator and Hyperbolic Back-Propagation Learning Algorithms.

Tohru Nitta, Yasuaki Kuroe

    IEEE Transactions on Neural Networks and Learning Systems
    |March 31, 2017
    PubMed
    Summary

    Researchers extended the Wirtinger derivative for hyperbolic functions, creating hyperbolic gradient operators and backpropagation algorithms for hyperbolic neural networks (NNs). This simplifies NN algorithm development and improves convergence rates.

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    Area of Science:

    • Hyperbolic geometry
    • Neural network algorithms
    • Complex analysis

    Background:

    • The Wirtinger derivative is a powerful tool for complex-valued functions.
    • Backpropagation is a standard algorithm for training neural networks.
    • Hyperbolic neural networks offer unique properties but lack efficient training methods.

    Purpose of the Study:

    • Extend the Wirtinger derivative to hyperbolic functions.
    • Develop hyperbolic gradient operators and backpropagation learning algorithms.
    • Analyze the efficiency and properties of the new hyperbolic learning algorithms.

    Main Methods:

    • Extension of the Wirtinger derivative to hyperbolic functions.
    • Derivation of the hyperbolic gradient operator.

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  • Development of hyperbolic backpropagation (Hyperbolic-BP) learning algorithms for multilayered hyperbolic neural networks.
  • Comparison of Hyperbolic-BP with complex-valued backpropagation (Complex-BP).

    • Main Results:

    • The use of the Wirtinger derivative simplifies the derivation and implementation of hyperbolic neural network learning algorithms.
    • Hyperbolic-BP algorithms demonstrate high convergence rates, even with fully activated functions.
    • A specific Hyperbolic-BP algorithm exhibits an inherent ability to learn hyperbolic rotations.

    Conclusions:

    • The extended Wirtinger derivative provides a more efficient approach to developing hyperbolic neural network training algorithms.
    • Hyperbolic-BP algorithms offer improved performance and unique learning capabilities compared to existing methods.
    • This work lays the foundation for further research into hyperbolic neural networks and their applications.