Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning

Researchers Sayed Pouria Talebi and Clive Cheong Took have authored a foundational guide, 'Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning,' set for publication in Elsevier's Handbook of Statistics in 2026. The work provides a comprehensive mathematical toolkit for applying quaternion algebra—a four-dimensional number system—to modern machine learning. Quaternions excel at modeling three-dimensional rotations, a capability that has historically been valuable in fields like computer graphics and aerospace engineering. The chapter aims to equip data scientists and engineers with the augmented statistics, widely linear models, and calculus needed to build AI systems that natively understand 3D space and complex motion.

This resurgence of interest in quaternions is driven by the demands of contemporary AI applications. As machine learning moves beyond 2D images and text into robotics, autonomous navigation, and advanced simulation, representing and processing 3D transformations efficiently becomes critical. The authors' framework lays the groundwork for developing new neural network architectures and algorithms that can handle multidimensional signal processing more naturally than current real- or complex-valued approaches. By formalizing the statistics and estimation techniques for quaternion-valued data, this work could accelerate AI research in areas requiring sophisticated spatial reasoning.