On the Efficacy and High-Performance Implementation of Quaternion Matrix Multiplication

Abstract

Quaternion symmetry is ubiquitous in the physical sciences. As such, much work has been afforded over the years to the development of efficient schemes to exploit this symmetry using real and complex linear algebra. Recent years have also seen many advances in the formal theoretical development of explicitly quaternion linear algebra with promising applications in image processing and machine learning. Despite these advances, there do not currently exist optimized software implementations of quaternion linear algebra. The leverage of optimized linear algebra software is crucial in the achievement of high levels of performance on modern computing architectures, and thus provides a central tool in the development of high-performance scientific software. In this work, a case will be made for the efficacy of high-performance quaternion linear algebra software for appropriate problems. In this pursuit, an optimized software implementation of quaternion matrix multiplication will be presented and will be shown to outperform a vendor tuned implementation for the analogous complex matrix operation. The results of this work pave the path for further development of high-performance quaternion linear algebra software which will improve the performance of the next generation of applicable scientific applications.