IJPAM: Volume 42, No. 3 (2008)


Mariya Ishteva$^1$, Lieven De Lathauwer$^2$, P.-A. Absil$^3$, Sabine Van Huffel$^4$
$^{1,2,4}$Departement of Electrical Engineering
Faculty of Engineering
Katholieke Universiteit Leuven
Kasteelpark Arenberg 10, Leuven, B-3001, BELGIUM
$^1$e-mail: mariya.ishteva@esat.kuleuven.be
$^2$e-mail: lieven.delathauwer@esat.kuleuven.be
$^4$e-mail: sabine.vanhuffel@esat.kuleuven.be
$^2$Subfaculty Sciences, Campus Kortrijk
Katholieke Universiteit Leuven
Kortrijk, BELGIUM
$^3$Department of Numerical Analysis
Katholieke Universiteit Leuven
Av. Georges Lemaître 4, Louvain-la-Neuve, B-1348, BELGIUM
web: https://www.inma.ucl.ac.be/$\sim$absil/

Abstract.Higher-order tensors have applications in many areas such as biomedical engineering, image processing, and signal processing. For example, dimensionality reduction of a multi-way problem can be achieved by the best rank- $(\!R_1,\!R_2,\!\ldots,\!R_N\!)$ approximation of tensors. Contrary to the matrix case, the tensor best rank- $(R_1,R_2,\ldots,R_N)$ approximation cannot be computed in a straightforward way. In this paper, we present the higher-order orthogonal iterations and outline two new algorithms, based on the trust-region and conjugate gradient methods on manifolds. We touch on some of the applications.

Received: August 17, 2007

AMS Subject Classification: 15A69, 15A18

Key Words and Phrases: multilinear algebra, higher-order tensor, rank reduction, trust-region, conjugate gradients, Grassmann manifold

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 42
Issue: 3