On approximate diagonalization of third order symmetric tensors by orthogonal transformations

Abstract

In this paper, we study the approximate orthogonal diagonalization problem of third order symmetric tensors. We define several classes of approximately diagonal tensors, including the ones corresponding to the stationary points of this problem. We study the relationships between these classes, and other well-known objects, such as tensor Zeigenvalue and Z-eigenvector. We also prove results on convergence of the cyclic Jacobi (or Jacobi CoM2) algorithm.

Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization

“…It is easy to check that if X is symmetric, then sym(X ) = X . Symmetric tensors were studied in details in [12] and [11] where authors also work with a Jacobi-type algorithm. In the structure-preserving symmetric SVD-like tensor decomposition orthogonal transformations are the same in each mode and one has…”

“…Since a general tensor cannot be diagonalized, we aim to achieve approximate diagonalization. Similar problem for symmetric tensors has been studied in a series of papers by Comon, Li and Usevich [14,11,12] where a Jacobi-type method is also a method of choice. In this paper we are mostly concerned with general tensors, that is, we do not assume any tensor structure, except in Section 5, and our convergence results are alongside those for the symmetric case.…”

Trace maximization algorithm for the approximate tensor diagonalization

“…[12,11,2,3]. The problem has been studied as the orthogonal [1,13,8,9,10] and non-orthogonal [12] tensor diagonalization, for structured and unstructured tensors. In this paper we are interested in the orthogonal tensor diagonalization of a tensor A ∈ R n×nו••×n of order d ≥ 3.…”

“…Therefore, a diagonal tensor S is not achievable, but we want to maximize its diagonal in a certain way. In papers [1,13,8,9] authors develop the Jacobi-type algorithms for maximizing the Frobenius norm of the diagonal of S, that is, maximizing the sum of the squares of the diagonal elements of S. Here, inspired by the algorithm of Moravitz Martin and Van Loan [10], we design a Jacobi-type algorithm that maximizes the trace of S. For a given tensor A ∈ R n×nו••×n we are looking for its decomposition of the form (1.1) such that tr(S) = Using the properties of the mode-m product one can express the core tensor S as…”

Strict inclusions of high rank loci

“…Tensor rank and symmetric tensor ranks have recently attracted a lot of attention, because of their natural appearance in several pure and applied contexts ( [19,20,5,2,15]). However, the notion of rank may be generalized to any projective variety.…”