基于t-product的张量总体最小二乘法

Tensor Total Least Squares Based on the t-product

总体最小二乘法(TLS)考虑了线性方程组Ax=b中A和b同时存在误差的问题。对于该问题已经有了很多成熟的研究,并广泛应用于多个领域。随着张量理论的发展,我们有必要开展张量总体最小二乘法的研究。本文基于一种新的张量乘积t-product,研究了总体最小二乘法的相关理论在张量上的推广。首先,根据张量奇异值分解(t-SVD)研究了张量奇异管之间的关系。然后分别从奇异管和左奇异侧面切片的角度,给出了张量系统存在唯一总体最小二乘解(t-TLS)的条件。最后,数值算例表明了上述方法的可行性和有效性。

The total least squares(TLS) considers that there are errors in both A and b in the linear equations Ax=b. There have been many mature studies on this problem, and it has been widely used in many fields. With the development of tensor theory, it is necessary to study the tensor total least square method. In this paper, we study the relevant theories of tensor total least squares(t-TLS) based on the t-product. Firstly, the relationship between singular tubes is considered in terms of tensor singular value decomposition(t-SVD). Then, the conditions for the existence of a unique t-TLS solution for the tensor system are given from the perspective of singular tubes and left singular lateral slices respectively. Finally, numerical examples show the feasibility and effectiveness of the proposed method.