Multilinear analysis of quaternion arrays: theory and computation

TL;DR

This paper develops a rigorous multilinear framework for quaternion arrays, extending tensor analysis to quaternion data, and introduces new decompositions and algorithms with theoretical and practical significance.

Contribution

It proposes a novel quaternion tensor framework, defines Tucker and CPD decompositions, and provides algorithms with theoretical analysis for quaternion multiway data.

Findings

Established the quaternion Tucker decomposition.

Developed the quaternion CPD and analyzed its properties.

Demonstrated the effectiveness of algorithms through numerical experiments.

Abstract

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.