四元数数组的多线性分析：理论与计算
Multilinear analysis of quaternion arrays: theory and computation
AI总结：
本文提出四元数张量的多线性框架，定义$\mathbb{H}\mathbb{R}$-多线性形式，建立Tucker分解与Q-CPD，研究其性质并给出两种算法，为四元数张量分解提供理论与计算工具。
摘要：
多维四元数数组（通常称为“四元数张量”）及其分解近年来在彩色和偏振成像、视频处理等各个领域受到越来越多的关注。尽管兴趣日益增长，四元数张量的理论发展仍然有限。本文提出了一种新颖的四元数数组多线性框架，该框架以严谨的方式将经典张量分析扩展到多维四元数数据。具体而言，我们提出了四元数张量作为$\mathbb{H}\mathbb{R}$-多线性形式的新定义，解决了四元数乘法非交换性带来的挑战。在此框架内，我们建立了四元数张量的Tucker分解，并开发了四元数典型多adic分解（Q-CPD）。我们深入研究了Q-CPD的性质，包括平凡模糊性、复等价模型以及唯一性的充分条件。此外，我们提出了两种计算Q-CPD的算法，并通过数值实验证明了其有效性。我们的结果为四元数张量分解的进一步研究提供了坚实的理论基础，并为处理四元数多路数据的实践者提供了新的计算工具。
英文摘要：
Multidimensional quaternion arrays (often referred to as "quaternion tensors") and their decompositions have recently gained increasing attention in various fields such as color and polarimetric imaging or video processing. Despite this growing interest, the theoretical development of quaternion tensors remains limited. This paper introduces a novel multilinear framework for quaternion arrays, which extends the classical tensor analysis to multidimensional quaternion data in a rigorous manner. Specifically, we propose a new definition of quaternion tensors as $\mathbb{H}\mathbb{R}$-multilinear forms, addressing the challenges posed by the non-commutativity of quaternion multiplication. Within this framework, we establish the Tucker decomposition for quaternion tensors and develop a quaternion Canonical Polyadic Decomposition (Q-CPD). We thoroughly investigate the properties of the Q-CPD, including trivial ambiguities, complex equivalent models, and sufficient conditions for uniqueness. Additionally, we present two algorithms for computing the Q-CPD and demonstrate their effectiveness through numerical experiments. Our results provide a solid theoretical foundation for further research on quaternion tensor decompositions and offer new computational tools for practitioners working with quaternion multiway data.
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