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<p>The application of tensor generalized inverses provides a robust mathematical framework for solving complex multi-linear systems, enabling advanced computational approaches in optimization, statistical estimation, and high-dimensional data analysis. This study proposes two tensor-based iterative methods, (1) a rapid and powerful iterative method and (2) a strong approximate inverse, to efficiently compute the <i>t</i>-Moore-Penrose inverse under the <i>t</i>-product. The fast Fourier transform plays a significant role in the design of tensor-based iterative algorithms and the enhancement of computational efficiency. These methods are accompanied by convergence theorems and perturbation error analyses. Numerical experiments demonstrate the usefulness of the tensor-based methods, providing insights into their applicability and superiority in scenarios where data exhibit multidimensional structures. This study presents a comparison between tensor methods and matrix-based approaches in iterative computations for the <i>t</i>-Moore-Penrose inverse. The proposed methods are empirically evaluated through a comparative study of image deblurring and denoising problems. Furthermore, we apply these methods to verify ordinary least squares (OLS) and generalized least squares (GLS) based on the <i>t</i>-product in the third-order Gauss-Markov theorem.</p>

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Efficient iterative methods for computing generalized inverse of tensors based on t-product

  • 期刊论文

  • 发表日期：

    2025年7月25日

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  • Biswarup Karmakar,
  • Ratikanta Behera

摘要

The application of tensor generalized inverses provides a robust mathematical framework for solving complex multi-linear systems, enabling advanced computational approaches in optimization, statistical estimation, and high-dimensional data analysis. This study proposes two tensor-based iterative methods, (1) a rapid and powerful iterative method and (2) a strong approximate inverse, to efficiently compute the t-Moore-Penrose inverse under the t-product. The fast Fourier transform plays a significant role in the design of tensor-based iterative algorithms and the enhancement of computational efficiency. These methods are accompanied by convergence theorems and perturbation error analyses. Numerical experiments demonstrate the usefulness of the tensor-based methods, providing insights into their applicability and superiority in scenarios where data exhibit multidimensional structures. This study presents a comparison between tensor methods and matrix-based approaches in iterative computations for the t-Moore-Penrose inverse. The proposed methods are empirically evaluated through a comparative study of image deblurring and denoising problems. Furthermore, we apply these methods to verify ordinary least squares (OLS) and generalized least squares (GLS) based on the t-product in the third-order Gauss-Markov theorem.