通过t-积的半张量积分析张量方程$\mathcal{A} \ltimes \mathcal{X} \ltimes \mathcal{B}= \mathcal{C}$
Solution Analysis of Tensor Equation $\mathcal{A} \ltimes \mathcal{X} \ltimes \mathcal{B}= \mathcal{C}$ via Semi Tensor Product with t-product
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中文总结 AI 辅助
本文通过半张量积与t-积分析张量方程，提出了解的存在条件及显式结构刻画，通过示例验证结果。
中文摘要 AI 辅助
本文聚焦于通过半张量积与t-积所形成的张量方程$\mathcal{A\ltimes X\ltimes B=C}$的分析。对于未知向量$\mathcal{X}$，我们建立了必要且充分的条件，提供了解存在的等价判据。对于矩阵值和高阶张量值的未知$\mathcal{X}$，可解性由相应的相容性要求决定。此外，$\mathcal{C}$的显式结构（Toeplitz和Circulant）被刻画。所得结果通过多个示例加以支持。
英文摘要
Tensor equations involving both left and right tensor operators arise naturally in applications where multidimensional data are coupled through transformations acting from both sides. Motivated by the need to analyze such equations, this manuscript investigates a detailed solution analysis of $\mathcal{A}\ltimes \mathcal{X}\ltimes \mathcal{B} = \mathcal{C}$ via the semi-tensor product with the $t$-product. The manuscript provides a more general framework that includes, as a special case, the tensor equation $\mathcal{A}\ltimes\mathcal{X}=\mathcal{B}$ considered in \cite{J.FathitensorequationAX=BunderSTP}. Necessary and sufficient conditions are derived for the existence of vector and matrix valued solutions. Explicit compatibility conditions are established, and constructive Moore-Penrose-inverse-based algorithms are provided for the vector and matrix valued cases. The computational complexity and execution times are compared for Discrete Fourier Transform(DFT) and Fast Fourier Transform(FFT) based \(t\)-product implementations. The practical relevance of the proposed framework is further demonstrated through a color image deblurring application. Examples throughout illustrate the theoretical results.
发表机构
- NIT Raipur（印度理工学院赖普尔分校）
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