为四元数机器学习提供数学基础，支持三维旋转建模与工程应用。
Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning
- 构建四元数随机过程的扩展统计模型
- 提出广义线性模型以利用高级统计特性
- 适合从事三维视觉、机器人与深度学习的研究者
许多研究尝试将复数代数在工程与科学中的成功经验推广到其他超复数域，如四元数、双四元数、八元数等。其中，四元数的应用最为广泛，其核心优势在于可建模三维旋转，在航空与计算机图形学等领域有重要工业应用。近年来，随着机器学习兴起，四元数迎来新发展。本文旨在为该新兴领域奠定基础：包括四元数随机过程的扩展统计建模、利用先进统计的广义线性模型、算法推导所需的四元数微积分与代数工具，以及实际应用中的最小均方估计方法。通过多个实例帮助读者理解并采用这一多维域的分析框架。
原文摘要 · Abstract (English)
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.
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