Best Low Multilinear Rank Approximation of Higher-Order Tensors, Based on the Riemannian Trust-Region Scheme

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 t...

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Vydáno v:	SIAM journal on matrix analysis and applications Ročník 32; číslo 1; s. 115 - 135
Hlavní autoři:	Ishteva, Mariya, Absil, P.-A., Van Huffel, Sabine, De Lathauwer, Lieven
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
Témata:	Algebra Algebraic geometry Algorithms Approximation Arrays Convergence Cost function Exact sciences and technology Geometry Iterative methods Linear algebra Linear and multilinear algebra, matrix theory Manifolds and cell complexes Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Optimization Sciences and techniques of general use Statistics Studies Tensors Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds Cost minimization Second order Product Iteration multilinear algebra Riemann manifold Grassmann manifold Three manifold 65F99 Approximation order Convergence rate Cost function Singular value decomposition Conjugate gradient method 53B20 Approximation Tensor Solving Rank 90C48 15A69 Riemannian manifold Statistics rank reduction Statistical method Array Numerical analysis Scientific computation Function minimization Linear algebra Signal processing Best approximation trust-region scheme 49M37 higher-order tensor
ISSN:	0895-4798, 1095-7162
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Shrnutí:	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 a 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. [PUBLICATION ABSTRACT]
Bibliografie:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23
ISSN:	0895-4798 1095-7162
DOI:	10.1137/090764827