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Multilinear analysis of quaternion arrays: theory and computation

Julien Flamant, Xavier Luciani, Sebastian Miron, Yassine Zniyed

TL;DR

This work addresses the lack of a rigorous multilinear framework for quaternion data by defining quaternion tensors as $\mathbb{H}\mathbb{R}$-multilinear forms, which resolves noncommutativity issues. It develops a complete quaternion tensor theory, including the Tucker model and the Quaternion Canonical Polyadic Decomposition (Q-CPD), and establishes uniqueness criteria via complex-adjoint and CONFAC representations. Two ALS-based algorithms, one in the quaternion domain ($\mathbb{H}$-Q-ALS) and one in the complex domain (C-ALS), are proposed and shown to be equivalent in updates, with numerical experiments confirming accurate factor recovery and practical runtimes. The results provide a solid theoretical foundation and actionable computational tools for quaternion multiway data, with potential impact on color/polarimetric imaging and video processing where quaternion representations are advantageous.

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.

Multilinear analysis of quaternion arrays: theory and computation

TL;DR

This work addresses the lack of a rigorous multilinear framework for quaternion data by defining quaternion tensors as -multilinear forms, which resolves noncommutativity issues. It develops a complete quaternion tensor theory, including the Tucker model and the Quaternion Canonical Polyadic Decomposition (Q-CPD), and establishes uniqueness criteria via complex-adjoint and CONFAC representations. Two ALS-based algorithms, one in the quaternion domain (-Q-ALS) and one in the complex domain (C-ALS), are proposed and shown to be equivalent in updates, with numerical experiments confirming accurate factor recovery and practical runtimes. The results provide a solid theoretical foundation and actionable computational tools for quaternion multiway data, with potential impact on color/polarimetric imaging and video processing where quaternion representations are advantageous.

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 -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.

Paper Structure

This paper contains 25 sections, 13 theorems, 58 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quaternions
  4. Quaternion linear algebra
  5. Quaternion matrix operations
  6. Complex adjoints for quaternion matrix products
  7. Quaternion vector spaces
  8. The rank and Kruskal rank of quaternion matrices
  9. Quaternion tensors: construction and properties
  10. The quaternion multilinearity issue
  11. Quaternion tensors as $\bbH\bbR$-multilinear forms
  12. Transformations rules for quaternion tensors
  13. Tucker format of quaternion tensors
  14. Quaternion Canonical Polyadic Decomposition
  15. Definition
  16. ...and 10 more sections

Key Result

Proposition 2.1

Let $\mathbf{A}$ and $\bfB$ be two quaternion matrices with arbitrary dimensions. The direct eq:left_complexAdjointMatrix and reverse eq:right_complexAdjointMatrix complex adjoint satisfy the following properties: \newlabelprop:complexAdjointProperties0

Figures (1)

  • Figure 1: Median value of the cost function and median NMSE on the factor matrices with respect to the SNR.

Theorems & Definitions (40)

  • Proposition 2.1: Properties of the complex adjoint
  • Remark 2.2
  • Definition 2.3: Left quaternion vector space jamison1970extension
  • Definition 2.4: Left and right columns spans
  • Definition 2.5: Left and right ranks of a quaternion matrix
  • Remark 2.6
  • Proposition 2.7: Left and right ranks from complex adjoints
  • Proof 1
  • Definition 2.8: Left and right Kruskal ranks of a quaternion matrix
  • Definition 2.9: $\mathop{\mathrm{k'}}\nolimits$-rank of a partitioned matrix de_lathauwer_decompositions_2008PartI
  • ...and 30 more