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.

翻译：多维四元数阵列（常称为“四元数张量”）及其分解方法近年来在彩色与偏振成像、视频处理等领域受到日益广泛的关注。尽管关注度不断提升，四元数张量的理论发展仍相对有限。本文提出了一种新颖的四元数阵列多线性分析框架，将经典张量分析以严格的方式推广至多维四元数数据。具体而言，我们提出将四元数张量定义为 $\mathbb{H}\mathbb{R}$-多线性形式的新概念，以应对四元数乘法的非交换性带来的挑战。在此框架下，我们建立了四元数张量的Tucker分解，并发展了一种四元数规范多元分解方法（Q-CPD）。我们深入研究了Q-CPD的性质，包括平凡歧义性、复数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，并通过数值实验验证了其有效性。本研究为四元数张量分解的进一步探索奠定了坚实的理论基础，并为处理四元数多路数据的实践者提供了新的计算工具。