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Computation of tensors generalized inverses under $M$-product and applications

Jajati Keshari Sahoo, Saroja Kumar Panda, Ratikanta Behera, Predrag S. Stanimirović

TL;DR

This work extends core tensor algebra by defining Drazin and core-EP inverses for third-order tensors under the $M$-product and introducing composite inverses CMP, DMP, and MPD. It develops algorithms to compute these inverses via the transform-domain $\mathbf{mat}$ representation, and establishes key characterizations and additive properties. The authors then build higher-order Jacobi and Gauss-Seidel iterative methods for solving multilinear systems, and introduce a Tikhonov-type regularization to handle ill-posed problems, with a practical color image deblurring example demonstrating the efficacy of the approach. Overall, the paper provides a cohesive framework for tensor generalized inverses under $M$-product, combining theoretical results, algorithmic tools, and real-world applications in image processing.

Abstract

This paper introduces notions of the Drazin and the core-EP inverses on tensors via M-product. We propose a few properties of the Drazin and core-EP inverses of tensors, as well as effective tensor-based algorithms for calculating these inverses. In addition, definitions of composite generalized inverses are presented in the framework of the M-product, including CMP, DMP, and MPD inverse of tensors. Tensor-based higher-order Gauss-Seidel and Gauss-Jacobi iterative methods are designed. Algorithms for these two iterative methods to solve multilinear equations are developed. Certain multilinear systems are solved using the Drazin inverse, core-EP inverses, and composite generalized inverses such as CMP, DMP, and MPD inverse. A tensor M-product-based regularization technique is applied to solve the color image deblurring.

Computation of tensors generalized inverses under $M$-product and applications

TL;DR

This work extends core tensor algebra by defining Drazin and core-EP inverses for third-order tensors under the -product and introducing composite inverses CMP, DMP, and MPD. It develops algorithms to compute these inverses via the transform-domain representation, and establishes key characterizations and additive properties. The authors then build higher-order Jacobi and Gauss-Seidel iterative methods for solving multilinear systems, and introduce a Tikhonov-type regularization to handle ill-posed problems, with a practical color image deblurring example demonstrating the efficacy of the approach. Overall, the paper provides a cohesive framework for tensor generalized inverses under -product, combining theoretical results, algorithmic tools, and real-world applications in image processing.

Abstract

This paper introduces notions of the Drazin and the core-EP inverses on tensors via M-product. We propose a few properties of the Drazin and core-EP inverses of tensors, as well as effective tensor-based algorithms for calculating these inverses. In addition, definitions of composite generalized inverses are presented in the framework of the M-product, including CMP, DMP, and MPD inverse of tensors. Tensor-based higher-order Gauss-Seidel and Gauss-Jacobi iterative methods are designed. Algorithms for these two iterative methods to solve multilinear equations are developed. Certain multilinear systems are solved using the Drazin inverse, core-EP inverses, and composite generalized inverses such as CMP, DMP, and MPD inverse. A tensor M-product-based regularization technique is applied to solve the color image deblurring.
Paper Structure (11 sections, 29 theorems, 103 equations, 2 figures, 8 tables, 6 algorithms)

This paper contains 11 sections, 29 theorems, 103 equations, 2 figures, 8 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Basic terms and preliminaries
  3. Drazin inverse of tensors under $M$-product
  4. Characterizations of the Drazin inverse
  5. Core-EP inverse of a tensor under $M$-product
  6. Composite generalized inverses
  7. Solution of Multilinear System
  8. Iterative methods
  9. Tikhonov's regularization
  10. Image deblurring
  11. Conclusion

Key Result

Proposition 2.10

Let $M\in\mathbb{C}^{p\times p}$ and $\mathcal{A},~\mathcal{B} \in \mathbb{C}^{m\times n\times p}$. Then ${\mathscr{R}}_M(\mathcal{A})\subseteq{\mathscr{R}}_M(\mathcal{B})\Longleftrightarrow\mathcal{A}=\mathcal{B}\!*\!{\!_M}\mathcal{T}$ for some $\mathcal{T}\in \mathbb{C}^{n\times n\times p}$.

Figures (2)

  • Figure 1: (a) True image of size $256\times 256$. (b) Blurred noisy image. (c) Reconstructed image.
  • Figure 2: (a) True image of size $512\times 512$ (b) Blurred noisy image. (c) Reconstructed image.

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2: Definition 2.6 and Lemma 3.1, Kernfeldlinear
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 64 more