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Volume 18, Issue 2
Frontal Slice Approaches for Tensor Linear Systems

Hengrui Luo & Anna Ma

DOI: 10.4208/nmtma.OA-2024-0096

Numer. Math. Theor. Meth. Appl., 18 (2025), pp. 353-394.

Published online: 2025-05

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  • Abstract

Inspired by the row and column action methods for solving large-scale linear systems, in this work, we explore the use of frontal slices for solving tensor linear systems. In particular, this paper presents a novel approach for using frontal slices of a tensor $\mathcal{A}$ to solve tensor linear systems $\mathcal{A} ∗\mathcal{X} = \mathcal{B}$ where ∗ denotes the $t$-product. In addition, we consider variations of this method, including cyclic, block, and randomized approaches, each designed to optimize performance in different operational contexts. Our primary contribution lies in the development and convergence analysis of these methods. Experimental results on synthetically generated and real-world data, including applications such as image and video deblurring, demonstrate the efficacy of our proposed approaches and validate our theoretical findings.

  • Keywords

Tensor linear system, t-product, iterative method, tensor sketching.

  • AMS Subject Headings

15A69, 15A72, 65F10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-18-353, author = {Luo , Hengrui and Ma , Anna}, title = {Frontal Slice Approaches for Tensor Linear Systems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2025}, volume = {18}, number = {2}, pages = {353--394}, abstract = {

Inspired by the row and column action methods for solving large-scale linear systems, in this work, we explore the use of frontal slices for solving tensor linear systems. In particular, this paper presents a novel approach for using frontal slices of a tensor $\mathcal{A}$ to solve tensor linear systems $\mathcal{A} ∗\mathcal{X} = \mathcal{B}$ where ∗ denotes the $t$-product. In addition, we consider variations of this method, including cyclic, block, and randomized approaches, each designed to optimize performance in different operational contexts. Our primary contribution lies in the development and convergence analysis of these methods. Experimental results on synthetically generated and real-world data, including applications such as image and video deblurring, demonstrate the efficacy of our proposed approaches and validate our theoretical findings.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2024-0096}, url = {http://global-sci.org/intro/article_detail/nmtma/24069.html} }
TY - JOUR T1 - Frontal Slice Approaches for Tensor Linear Systems AU - Luo , Hengrui AU - Ma , Anna JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 353 EP - 394 PY - 2025 DA - 2025/05 SN - 18 DO - http://doi.org/10.4208/nmtma.OA-2024-0096 UR - https://global-sci.org/intro/article_detail/nmtma/24069.html KW - Tensor linear system, t-product, iterative method, tensor sketching. AB -

Inspired by the row and column action methods for solving large-scale linear systems, in this work, we explore the use of frontal slices for solving tensor linear systems. In particular, this paper presents a novel approach for using frontal slices of a tensor $\mathcal{A}$ to solve tensor linear systems $\mathcal{A} ∗\mathcal{X} = \mathcal{B}$ where ∗ denotes the $t$-product. In addition, we consider variations of this method, including cyclic, block, and randomized approaches, each designed to optimize performance in different operational contexts. Our primary contribution lies in the development and convergence analysis of these methods. Experimental results on synthetically generated and real-world data, including applications such as image and video deblurring, demonstrate the efficacy of our proposed approaches and validate our theoretical findings.

Luo , Hengrui and Ma , Anna. (2025). Frontal Slice Approaches for Tensor Linear Systems. Numerical Mathematics: Theory, Methods and Applications. 18 (2). 353-394. doi:10.4208/nmtma.OA-2024-0096
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