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Two-step parameterized tensor-based iterative methods for solving $\mathcal{A}_{*M}\mathcal{X}_{*M}\mathcal{B}=\mathcal{C}$

Ratikanta Behera, Saroja Kumar Panda, Jajati Keshari Sahoo

TL;DR

The paper tackles solving high-dimensional tensor equations of the form $\mathcal{A}_{*M}\mathcal{X}_{*M}\mathcal{B}=\mathcal{C}$ by introducing a two-step parameterized iterative framework under the $M$-product. It develops coupled tensor recursions with splittings $\mathcal{A}=\mathcal{F}_1-\mathcal{G}_1$ and $\mathcal{B}=\mathcal{F}_2-\mathcal{G}_2$, augmented by parameters $\alpha$ and $\beta$, and further strengthens convergence through preconditioning with $\mathcal{P}_1$ and $\mathcal{P}_2$. The work provides convergence proofs based on spectral radius conditions, derives an explicit preconditioned variant (PTPSI), and demonstrates applications to Sylvester tensor equations and regularized image deblurring, including Tikhonov-type formulations and least-squares solutions. Overall, the methodology offers scalable, efficient tensor-based solvers with practical impact on multidimensional data problems and image restoration, while outlining avenues for improved parameter selection and broader tensor-equation applications.

Abstract

Iterative methods based on tensors have emerged as powerful tools for solving tensor equations, and have significantly advanced across multiple disciplines. In this study, we propose two-step tensor-based iterative methods to solve the tensor equations $\mathcal{A}_{*M}\mathcal{X}_{*M}\mathcal{B}=\mathcal{C}$ by incorporating preconditioning techniques and parametric optimization to enhance convergence properties. The theoretical results were complemented by comprehensive numerical experiments that demonstrated the computational efficiency of the proposed two-step parametrized iterative methods. The convergence criterion for parameter selection has been studied and a few numerical experiments have been conducted for optimal parameter selection. Effective algorithms were proposed to compute iterative methods based on two-step parameterized tensors, and the results are promising. In addition, we discuss the solution of the Sylvester equations and a regularized least-squares solution for image deblurring problems.

Two-step parameterized tensor-based iterative methods for solving $\mathcal{A}_{*M}\mathcal{X}_{*M}\mathcal{B}=\mathcal{C}$

TL;DR

The paper tackles solving high-dimensional tensor equations of the form by introducing a two-step parameterized iterative framework under the -product. It develops coupled tensor recursions with splittings and , augmented by parameters and , and further strengthens convergence through preconditioning with and . The work provides convergence proofs based on spectral radius conditions, derives an explicit preconditioned variant (PTPSI), and demonstrates applications to Sylvester tensor equations and regularized image deblurring, including Tikhonov-type formulations and least-squares solutions. Overall, the methodology offers scalable, efficient tensor-based solvers with practical impact on multidimensional data problems and image restoration, while outlining avenues for improved parameter selection and broader tensor-equation applications.

Abstract

Iterative methods based on tensors have emerged as powerful tools for solving tensor equations, and have significantly advanced across multiple disciplines. In this study, we propose two-step tensor-based iterative methods to solve the tensor equations by incorporating preconditioning techniques and parametric optimization to enhance convergence properties. The theoretical results were complemented by comprehensive numerical experiments that demonstrated the computational efficiency of the proposed two-step parametrized iterative methods. The convergence criterion for parameter selection has been studied and a few numerical experiments have been conducted for optimal parameter selection. Effective algorithms were proposed to compute iterative methods based on two-step parameterized tensors, and the results are promising. In addition, we discuss the solution of the Sylvester equations and a regularized least-squares solution for image deblurring problems.

Paper Structure

This paper contains 11 sections, 31 theorems, 86 equations, 5 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and Definitions
  4. Nonnegative matrices, splittings and regularization
  5. Block tensor
  6. Basic Properties of tensors under different products
  7. Two-step parameterized iterative method
  8. Preconditioned two-step parametrized iterative (PTSPI) method
  9. Application to Sylvester equations and image deblurring
  10. Image deblurring
  11. Conclusion

Key Result

Lemma 2.4

varga1varga Let $A=F-G$ be a regular splitting ( regular: if $F^{-1}\geq 0$ and $G\geq 0$ ) of a nonsingular matrix $A$. Then $A^{-1}\geq 0$ if and only if $\rho(F^{-1}G)<1$.

Figures (5)

  • Figure 1: A comparative analysis of the average CPU times for higher-order TSI and TPSI methods, focusing on various parameters such as $\alpha$ and $\beta$
  • Figure 2: Comparison analysis of mean CPU time of higher order TSI and TPSI methods for different tensor sizes $n\times n\times n$ ( where $\mathcal{A},~\mathcal{B},~\mathcal{C}\in\mathbb{C}^{n\times n\times n}$ ) by varying parameters $\alpha$ and $\beta$
  • Figure 3: A comparative analysis of the average CPU time for higher-order TSPI methods focusing on fixed tensor sizes of $n\times n\times n$ ( where $\mathcal{A},~\mathcal{B},~\mathcal{C}\in\mathbb{C}^{n\times n\times n}$ ) with varying both parameters $\alpha$ and $\beta$
  • Figure 4: A visual comparison showcasing (a) the original true image at $128 \times 128$ resolution, (b) the degraded version affected by blurring and noise, and (c) the result of the reconstruction process aimed at restoring the image quality.
  • Figure 5: A visual comparison showcasing (a) the original true image at $256 \times 256$ resolution, (b) the degraded version affected by blurring and noise, and (c) the result of the reconstruction process aimed at restoring the image quality.

Theorems & Definitions (49)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Proposition 2.9
  • ...and 39 more