Randomized Quaternion UTV Decomposition and Randomized Quaternion Tensor UTV Decomposition

TL;DR

This paper introduces randomized quaternion matrix and tensor UTV decompositions, providing efficient algorithms with theoretical error bounds and demonstrating their effectiveness in processing color images and videos.

Contribution

The paper proposes novel randomized quaternion UTV and tensor UTV decompositions with theoretical analysis and practical algorithms, enhancing efficiency and accuracy.

Findings

Algorithms are simple to understand and computationally efficient.

Theoretical error bounds are established for low-rank approximations.

Numerical experiments confirm high accuracy and efficiency in image and video processing.

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.