AI 中文总结
该研究将张量列车分解（TTD）、分层塔克分解（HTD）融入基于变换的多线性代数，开发相关公式并推广至高阶张量-张量积，应用于多线性模型降阶，通过数值示例验证其有效性与效率。
AI 中文摘要
基于变换的张量积，包括T-积及其更通用形式即高阶张量-张量积，已成为图像处理、信号重构、机器人学等应用中多线性数据分析的基础工具。尽管可逆变换可使张量计算在变换域通过矩阵运算完成，但对于高维、高阶张量，其存储和计算成本仍过高。为应对这一挑战，我们将低秩张量分解技术，具体为张量列车分解（TTD）和分层塔克分解（HTD），整合到基于变换的多线性代数中，以提升计算与内存效率。特别地，我们针对T-积及其相关核心代数（如块对角化和张量奇异值分解）开发了基于TTD和HTD的公式，通过直接操作分解的因子矩阵或张量实现。该框架进一步推广至高阶张量-张量积，并应用于多线性模型降阶问题。我们通过数值示例验证了所提框架的有效性与效率。
英文摘要
Transform-based tensor products, including the T-product and its more general form, namely the higher-order tensor-tensor product, have become fundamental tools for multilinear data analysis in applications such as image processing, signal reconstruction, and robotics. While invertible transforms enable tensor computations to be carried out via matrix operations in the transform domain, the resulting storage and computational costs remain prohibitive for high-dimensional, higher-order tensors. To address this challenge, we integrate low-rank tensor decomposition techniques, specifically tensor train decomposition (TTD) and hierarchical Tucker decomposition (HTD), into transform-based multilinear algebra to improve computational and memory efficiency. In particular, we develop TTD- and HTD-based formulations for the T-product and its associated key algebra, such as block diagonalization and tensor singular value decomposition, by operating directly on the factor matrices or tensors of the decompositions. The framework is further generalized to the higher-order tensor-tensor product and applied to multilinear model order reduction problems. We demonstrate the effectiveness and efficiency of our framework with numerical examples.
Comments20 pages, 3 figures, 1 table
