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Paige’s Algorithm for solving a class of tensor least squares problem

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Abstract

In this paper, we consider a class of tensor least squares problem with an invertible linear transform, which arises in image restoration. Based on the operator-bidiagonal procedure, two Paige’s algorithms are designed to solve it. The convergence theorems of the new methods are derived. Numerical experiments are performed to illustrate the feasibility and efficiency of the new methods, including when the algorithm is tested with the synthetic data and on some image restoration problems. Comparisons with some previous methods are also given.

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The data used to support the findings of this study are included within the article.

Notes

  1. When we use an algorithm to solve the ill-posed problem, the algorithm exhibits a gradual convergence behavior for the first k iterations. However, after the kth step, the error will increase and the convergence will disappear. This is called "semi-convergence".

  2. Set \({{\mathscr {X}}}_k\) be the kth iterative value of an algorithm. If the noise \({\mathscr {E}}\) satisfies \(\Vert {{\mathscr {E}}}\Vert _F \le \Vert {{\mathscr {A}}}{*_L}{{\mathscr {X}}}_k - {{{\mathscr {B}}}}\Vert _F\), it is called as a low noise level. Conversely, if \({\mathscr {E}}\) satisfies \( \Vert {{\mathscr {E}}}\Vert _F > \Vert {{\mathscr {A}}}{*_L}{{\mathscr {X}}}_k - {{{\mathscr {B}}}}\Vert _F \), it is called as a high noise level.

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Acknowledgements

The authors thank the editor and the reviewers for the constructive and helpful comments on the revision of this article.

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Authors and Affiliations

  1. College of Mathematics and Computational Science, Center for Applied Mathematics of Guangxi (GUET), Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin, 541004, Guangxi, People’s Republic of China

    Xue-Feng Duan, Yong-Shen Zhang & Chun-Mei Li

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

    Qing-Wen Wang

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  1. Xue-Feng Duan
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Correspondence to Xue-Feng Duan.

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This work was supported by the National Natural Science Foundation of China (Grant No. 12361079; 12201149; 12261026; 12371023), the Natural Science Foundation of Guangxi Province (Grant No. 2023GXNSFAA026067), and the Innovation Project of GUET Graduate Education (Grant No. 2022YCXS147; 2022YCXS140).

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Duan, XF., Zhang, YS., Wang, QW. et al. Paige’s Algorithm for solving a class of tensor least squares problem. Bit Numer Math 63, 48 (2023). https://doi.org/10.1007/s10543-023-00990-y

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