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Randomized Quaternion QLP Decomposition for Low-Rank Approximation

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Abstract

The low-rank approximation of a quaternion matrix has attracted growing attention in many applications including color image processing and signal processing. In this paper, based on quaternion normal distribution random sampling, we propose a randomized quaternion QLP decomposition algorithm for computing a low-rank approximation to a quaternion data matrix. For the theoretical analysis, we first present convergence results of the quaternion QLP decomposition, which provides slightly tighter upper bounds than the existing ones for the real QLP decomposition. Then, for the randomized quaternion QLP decomposition, the matrix approximation error and the singular value approximation error analyses are also established to show the proposed randomized algorithm can track the singular values of the quaternion data matrix with high probability. Finally, we present some numerical examples to illustrate the effectiveness and reliablity of the proposed algorithm.

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Data Availability Statement

All data generated or analysed during this study are included in this manuscript.

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Acknowledgements

The authors are grateful to the handling editor and three anonymous referees for their useful comments and suggestions, which greatly improved the original presentation.

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 360015, People’s Republic of China

    Huan Ren

  2. School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, 215009, People’s Republic of China

    Ru-Ru Ma

  3. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Qiaohua Liu

  4. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Xiamen, 361005, People’s Republic of China

    Zheng-Jian Bai

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Ren, H., Ma, RR., Liu, Q. et al. Randomized Quaternion QLP Decomposition for Low-Rank Approximation. J Sci Comput 92, 80 (2022). https://doi.org/10.1007/s10915-022-01917-5

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