Author:

Keywords:

SISTA, Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, multilinear algebra, higher-order tensor, rank reduction, singular value decomposition, trust-region scheme, Riemannian manifold, Grassmann manifold, PARALLEL FACTOR-ANALYSIS, GEOMETRIC NEWTON METHOD, DIMENSIONALITY REDUCTION, COMPONENT ANALYSIS, DECOMPOSITION, ALGEBRA, 0102 Applied Mathematics, 0103 Numerical and Computational Mathematics, Numerical & Computational Mathematics, 4901 Applied mathematics

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 multiway 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 a minimization of a cost function on a product of three Grassmann manifolds.We apply the Riemannian trust-region scheme, using the truncated conjugategradient 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. © 2011 Society for Industrial and Applied Mathematics.