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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Veröffentlicht in:SIAM journal on matrix analysis and applications Jg. 32; H. 1; S. 115 - 135
Hauptverfasser: Ishteva, Mariya, Absil, P.-A., Van Huffel, Sabine, De Lathauwer, Lieven
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
Schlagworte:
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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Zusammenfassung: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]
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ISSN:0895-4798
1095-7162
DOI:10.1137/090764827