A Jacobi-Type Method for Computing Orthogonal Tensor Decompositions

Abstract

Suppose $\mathcal{A}=(a_{ijk})\in\mathbb{R}^{n\times n\times n}$ is a three-way array or third-order tensor. Many of the powerful tools of linear algebra such as the singular value decomposition (SVD) do not, unfortunately, extend in a straightforward way to tensors of order three or higher. In the two-dimensional case, the SVD is particularly illuminating, since it reduces a matrix to diagonal form. Although it is not possible in general to diagonalize a tensor (i.e., $a_{ijk}=0$ unless $i=j=k$), our goal is to “condense” a tensor in fewer nonzero entries using orthogonal transformations. We propose an algorithm for tensors of the form $\mathcal{A}\in\mathbb{R}^{n\times n\times n}$ that is an extension of the Jacobi SVD algorithm for matrices. The resulting tensor decomposition reduces $\mathcal{A}$ to a form such that the quantity $\sum_{i=1}^n a_{iii}^2$ or $\sum_{i=1}^n a_{iii}$ is maximized.