Summary: JACOBI ALGORITHM FOR THE BEST LOW MULTILINEAR RANK
APPROXIMATION OF SYMMETRIC TENSORS
MARIYA ISHTEVA, P.-A. ABSIL, AND PAUL VAN DOOREN
Abstract. The problem discussed in this paper is the symmetric best low multilinear rank
approximation of third-order symmetric tensors. We propose an algorithm based on Jacobi rotations,
for which symmetry is preserved at each iteration. Two numerical examples are provided indicating
the need of such algorithms. An important part of the paper consists of proving that our algorithm
converges to stationary points of the objective function. This can be considered an advantage of the
proposed algorithm over existing symmetry-preserving algorithms in the literature.
Key words. multilinear algebra, higher-order tensor, rank reduction, singular value decompo-
sition, Jacobi rotation.
AMS subject classifications. 15A69, 65F99
1. Introduction. Higher-order tensors (three-way arrays) have been used as
a tool in higher-order statistics (HOS) [35, 32, 41, 34] and independent component
analysis (ICA) [12, 13, 18, 8] for several decades already. Other application areas
include chemometrics, scientific computing, biomedical signal processing, image pro-
cessing and telecommunications. For an exhaustive list and references we refer to
[40, 30, 29, 7, 11].
Let us first consider the general low multilinear rank approximation of third-
order tensors. The problem consists of finding the best approximation of a given