提出了两个基于不同张量乘法的四阶张量分解. 首先, 在矩阵乘法的基础上, 定义第一种四阶张量乘法(F-乘), 基于F-乘提出了第一种四阶张量分解(F-TD). 其次, 基于三阶张量t-product给出了第二种四阶张量乘法(B-乘)和分解(FT-SVD). 同时, 利用两种分解方法, 分别给出两个张量逼近定理. 最后, 三个数值算例阐明提出的两种分解方法的准确性和可行性.

In this paper, two factorizations for fourth-order tensors based on different multiplications of the fourth-order tensors are investigated. One is, called as F-TD, based on the fourth-order tensor multiplication (F-product). Another is, the fourth-order tensor multiplication and decomposition are defined, called as B-product and FT-SVD, based on the t-product of the third-order tensor multiplication.  Meanwhile, two tensor approximation theorems are present using two  decomposition methods. Finally, three numerical examples are given to demonstrate the accuracy and the feasibility of our proposed methods.