A Jacobi-Type Method for Computing Orthogonal Tensor Decompositions

Abstract. Suppose A = (aijk) ∈ Rn×n×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 twodimensional 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., aijk = 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 A ∈ Rn×n×n that is an extension of the Jacobi SVD algorithm for matrices. The resulting tensor decomposition reduces A to a form such that the quantity ∑n i=1 a 2 iii or ∑n i=1 aiii is maximized.