Hypercomplex Neural Networks

Authors: Agnieszka Niemczynowicz, Radosław A. Kycia

Outline:
Hypercomplex Neural Networks are extensions of classical neural networks to higher dimensions. In recent decades, this theory has emerged as a forefront in neural networks theory. There are several approaches to extend classical neural network models: quaternionic analysis, which merely uses quaternions; Clifford analysis, which relies on Clifford algebras; and finally  eneralizations of complex variables to higher dimensions. This book reflects a selection of papers related to complex, hypercomplex analysis, and fuzzy approaches applied to neural networks theory. Hypercomplex-valued, such as complex-valued, quaternion-valued, and broader hypercomplex models, have gained significant traction in research over the past decade. Their
appeal lies not only in handling multidimensional data adeptly but also in leveraging the geometric and algebraic properties inherent in hypercomplex numbers. For instance, complex-valued networks play a crucial role in accurately interpreting phase information, crucial in various wave – and rotation-centric domains like electromagnetism, light waves, quantum phenomena, and oscillations.

The scope of applications for these networks is vast, spanning pattern recognition, classification, nonlinear filtering, intelligent image processing, brain-computer interfaces, time-series prediction, bioinformatics, and robotics.

Outcome of the Project:

The mathematical description of hypercomplex dense and convolutional neural networks were described. The implementation of these neural networks was made and made phallically available for Python Package Repository and on GitHub. The usage for classification of blood malaria images was presented in [2]. The package can be easily installed and used as a layers of feedforward neural networks along with other Keras layers since they are interoperable.

The main feature of these layers observed in numerical experiments is that the number of trainable parameters is less that for corresponding (dense and convolutional) layers that have similar accuracy of classification.