Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning

TL;DR

This paper establishes foundational concepts for quaternion-based machine learning, including augmented statistics, widely linear models, and quaternion calculus, to facilitate the development of advanced algorithms in multidimensional data processing.

Contribution

It introduces a comprehensive framework for quaternion-valued machine learning, covering statistical modeling, algebra, and estimation techniques, which is novel in advancing hypercomplex domain applications.

Findings

Developed augmented statistics for quaternion processes

Formulated widely linear quaternion models

Provided examples to aid understanding and adoption

Abstract

Numerous attempts have been made to replicate the success of complex-valued algebra in engineering and science to other hypercomplex domains such as quaternions, tessarines, biquaternions, and octonions. Perhaps, none have matched the success of quaternions. The most useful feature of quaternions lies in their ability to model three-dimensional rotations which, in turn, have found various industrial applications such as in aeronautics and computergraphics. Recently, we have witnessed a renaissance of quaternions due to the rise of machine learning. To equip the reader to contribute to this emerging research area, this chapter lays down the foundation for: - augmented statistics for modelling quaternion-valued random processes, - widely linear models to exploit such advanced statistics, - quaternion calculus and algebra for algorithmic derivations, - mean square estimation for practical considerations. For ease of exposure, several examples are offered to facilitate the learning, understanding, and(hopefully) the adoption of this multidimensional domain.