Tensor decompositions are powerful tools for analyzing multi-dimensional data in their original format. Besides tensor decompositions like Tucker and CP, Tensor SVD (t-SVD) which is based on the t-product of tensors is another extension of SVD to tensors that recently developed and has found numerous applications in analyzing high dimensional data. This paper offers a new insight into the t-Product and shows that this product is a block convolution of two tensors with periodic boundary conditions. Based on this viewpoint, we propose a new tensor-tensor product called the $\star_c{}\text{-Product}$ based on Block convolution with reflective boundary conditions. Using a tensor framework, this product can be easily extended to tensors of arbitrary order. Additionally, we introduce a tensor decomposition based on our $\star_c{}\text{-Product}$ for arbitrary order tensors. Compared to t-SVD, our new decomposition has lower complexity, and experiments show that it yields higher-quality results in applications such as classification and compression.

翻译：张量分解是在原始格式下分析多维数据的强大工具。除了Tucker和CP等张量分解外，基于张量t-乘积的张量SVD（t-SVD）是近年来发展起来的另一种SVD向张量的扩展，并在高维数据分析中获得了广泛应用。本文对t-乘积提出了新的见解，表明该乘积是具有周期边界条件的两个张量的块卷积。基于这一观点，我们提出了一种新的张量-张量乘积，称为基于反射边界条件块卷积的$\star_c{}\text{-乘积}$。利用张量框架，该乘积可轻松扩展到任意阶张量。此外，我们还基于我们的$\star_c{}\text{-乘积}$引入了一种适用于任意阶张量的张量分解。与t-SVD相比，我们的新分解具有更低的复杂度，实验表明，在分类和压缩等应用中，它能获得更高质量的结果。