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Summary: BEST LOW MULTILINEAR RANK APPROXIMATION OF
HIGHER-ORDER TENSORS, BASED ON THE RIEMANNIAN
TRUST-REGION SCHEME
MARIYA ISHTEVA, LIEVEN DE LATHAUWER,
P.-A. ABSIL§, AND SABINE VAN HUFFEL
Abstract. Higher-order tensors are used in many application fields, such as statistics, signal
processing and scientific computing. Efficient and reliable algorithms for manipulating these multi-
way arrays are thus required. In this paper, we focus on the best rank-(R1, R2, R3) approximation
of third-order tensors. We propose a new iterative algorithm based on the trust-region scheme. The
tensor approximation problem is expressed as minimization of a cost function on a product of three
Grassmann manifolds. We apply the Riemannian trust-region scheme, using the truncated conjugate-
gradient method for solving the trust-region subproblem. Making use of second order information of
the cost function, superlinear convergence is achieved. If the stopping criterion of the subproblem is
chosen adequately, the local convergence rate is quadratic. We compare this new method with the
well-known higher-order orthogonal iteration method and discuss the advantages over Newton-type
methods.
Key words. multilinear algebra, higher-order tensor, rank reduction, singular value decompo-
sition, trust-region scheme, Riemannian manifold, Grassmann manifold.
AMS subject classifications. 15A69, 65F99, 90C48, 49M37, 53B20
1. Introduction. Higher-order tensors are multi-way arrays of numbers, i.e.,
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