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Randomized Quaternion UTV Decomposition and Randomized Quaternion Tensor UTV Decomposition

Liqiao Yang, Jifei Miao, Tai-Xiang Jiang, Yanlin Zhang, Kit Ian Kou

TL;DR

The quaternion matrix UTV (QUTV) decomposition and quaternion tensor UTV (QTUTV) decomposition are proposed and the theory analysis offers a solid theoretical basis and the experiments show significant potential for the associated processing tasks of color images and color videos.

Abstract

In this paper, the quaternion matrix UTV (QUTV) decomposition and quaternion tensor UTV (QTUTV) decomposition are proposed. To begin, the terms QUTV and QTUTV are defined, followed by the algorithms. Subsequently, by employing random sampling from the quaternion normal distribution, randomized QUTV and randomized QTUTV are generated to provide enhanced algorithmic efficiency. These techniques produce decompositions that are straightforward 9 to understand and require minimal cost. Furthermore, theoretical analysis is discussed. Specifically, the upper bounds for approximating QUTV on the rank-K and QTUTV on the TQt-rank K errors are provided, followed by deterministic error bounds and average-case error bounds for the randomized situations, which demonstrate the correlation between the accuracy of the low-rank approximation and the singular values. Finally, numerous numerical experiments are presented to verify that the proposed algorithms work more efficiently and with similar relative errors compared to other comparable decomposition methods. For the novel decompositions, the theory analysis offers a solid theoretical basis and the experiments show significant potential for the associated processing tasks of color images and color videos.

Randomized Quaternion UTV Decomposition and Randomized Quaternion Tensor UTV Decomposition

TL;DR

The quaternion matrix UTV (QUTV) decomposition and quaternion tensor UTV (QTUTV) decomposition are proposed and the theory analysis offers a solid theoretical basis and the experiments show significant potential for the associated processing tasks of color images and color videos.

Abstract

In this paper, the quaternion matrix UTV (QUTV) decomposition and quaternion tensor UTV (QTUTV) decomposition are proposed. To begin, the terms QUTV and QTUTV are defined, followed by the algorithms. Subsequently, by employing random sampling from the quaternion normal distribution, randomized QUTV and randomized QTUTV are generated to provide enhanced algorithmic efficiency. These techniques produce decompositions that are straightforward 9 to understand and require minimal cost. Furthermore, theoretical analysis is discussed. Specifically, the upper bounds for approximating QUTV on the rank-K and QTUTV on the TQt-rank K errors are provided, followed by deterministic error bounds and average-case error bounds for the randomized situations, which demonstrate the correlation between the accuracy of the low-rank approximation and the singular values. Finally, numerous numerical experiments are presented to verify that the proposed algorithms work more efficiently and with similar relative errors compared to other comparable decomposition methods. For the novel decompositions, the theory analysis offers a solid theoretical basis and the experiments show significant potential for the associated processing tasks of color images and color videos.
Paper Structure (19 sections, 16 theorems, 54 equations, 13 figures, 5 algorithms)

This paper contains 19 sections, 16 theorems, 54 equations, 13 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Notations
  4. Preliminary of quaternion and quaternion matrix
  5. Preliminary of quaternion tensor
  6. Quaternion UTV decompositions
  7. Quaternion matrix UTV decomposition
  8. Quaternion tensor UTV decomposition
  9. Randomized quaternion UTV decompositions
  10. Randomized Algorithms for QURV
  11. Randomized Algorithms for QTURV
  12. Theoretical Analysis
  13. Experimental Results
  14. Settings
  15. Testing the proposed QURV and CoR-QURV algorithms
  16. ...and 4 more sections

Key Result

Theorem 2.2

(QSVDzhang1997quaternions): \newlabeltheorem10 Let a quaternion matrix $\dot{\mathbf{A}}\in\mathbb{H}^{M \times N}$ be of rank $r$. There exist two unitary quaternion matrices $\dot{\mathbf{U}}\in\mathbb{H}^{M \times M}$ and $\dot{\mathbf{V}}\in\mathbb{H}^{N \times N}$ such that where $\mathbf{\Sigma}_r=diag({\sigma_1,\cdots, \sigma_r})\in\mathbb{R}^{r\times r}$, and all singular values $\sigma

Figures (13)

  • Figure 1: Comparison of the RE and running time of implementing truncated-QURV truncated-QSVD, and truncated-QQRPCP to Example \ref{['eg1']} with $K=10, 20, \cdots, 100.$
  • Figure 2: Comparison of the RE and running time of implementing truncated-QURV truncated-QSVD, and truncated-QQRPCP to Example \ref{['eg2']} with $K=2, 4, \cdots, 20.$
  • Figure 3: The test images: "Flower" and "House".
  • Figure 4: Comparison of the RE and running time of implementing CoR-QURV and randQSVD to Example \ref{['eg3']} with $K=20, 40, \cdots, 200.$
  • Figure 5: Comparison of the RE and running time of implementing CoR-QURV and randQSVD to Example \ref{['eg3']} with $K=40, 80, \cdots, 400.$
  • ...and 8 more figures

Theorems & Definitions (37)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3: Quaternion tensorDBLP:journals/pr/MiaoKL20
  • Definition 2.4: $\star_{QT}$-product DBLP:journals/sigpro/MiaoK23
  • Remark 2.5
  • Theorem 2.6: TQt-SVD DBLP:journals/sigpro/MiaoK23
  • Definition 2.7: TQt-rank DBLP:journals/sigpro/MiaoK23
  • Theorem 3.1: QTUTV
  • Remark 3.2
  • Remark 4.1
  • ...and 27 more