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

Abstract

For a general third-order tensor A ∈ R n×n×n the paper studies two closely related problems, the SVD-like tensor decomposition and the (approximate) tensor diagonalization. We develop the alternating least squares Jacobi-type algorithm that maximizes the squares of the diagonal entries of A. The algorithm works on 2 × 2 × 2 subtensors such that in each iteration the sum of the squares of two diagonal entries is maximized. We show how the rotation angles are calculated and prove the convergence of the algorithm. Different initializations of the algorithm are discussed, as well as the special cases of symmetric and antisymmetric tensors. The algorithm can be generalized to work on the higher-order tensors.

Jacobi-type algorithms for homogeneous polynomial optimization on Stiefel manifolds with applications to tensor approximations

“…, where D is a diagonal tensor with D jjjj = √ j + ji for 1 ≤ j ≤ 7, U (p) ∈ C np×np is a unitary matrix for p = 1, 2, 3, 4 and E = randn (7,7,8,8). The ratio Per = 8.1327e − 04 on this tensor.…”

“…If r = n in problem (1), or r p = n p for 1 ≤ p ≤ d in problem (2), then these problems have orthogonal constraints on unitary groups (or orthogonal groups in real case). In this case, besides the above optimization methods, one important approach is to use the Jacobi-type rotations [7,11,16,21,23,46,38,40,41,56]. From the computational point of view, a key step of Jacobi-type methods is to solve a subproblem.…”