Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning

2026-03-12

AI Summary

This work addresses the lack of systematic statistical and algorithmic support for hypercomplex algebras—particularly quaternions—in current machine learning methods, which hinders effective modeling of high-dimensional data such as 3D rotations. The study presents the first comprehensive statistical framework for quaternion-valued stochastic processes and introduces generalized linear models grounded in quaternion calculus and mean-square estimation theory, thereby establishing a unified foundation for quaternion-based machine learning. By bridging hypercomplex algebra with modern machine learning, this research provides directly applicable theoretical tools and algorithmic paradigms for modeling 3D rotations and processing high-dimensional signals, significantly advancing the practical adoption of quaternions in machine learning applications.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• quaternion
• machine learning
• widely linear processing
• hypercomplex
• augmented statistics

Innovation

• quaternion machine learning
• widely linear processing
• augmented statistics
• hypercomplex algebra
• mean square estimation