Hierarchical Singular Value Decomposition of Tensors

The authors define the hierarchical singular value decomposition (SVD) for tensors of order d ≥ 2. This hierarchical SVD has properties like the matrix SVD (and collapses to the SVD in d = 2), and they prove these. In particular, one can find low rank (almost) best approximations in a hierarchical f...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications Vol. 31; no. 4; pp. 2029 - 2054
Main Author: Grasedyck, Lars
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
Subjects:
Algebra
Algorithms
Approximation
Decomposition
Digits
Error analysis
Exact sciences and technology
Format
Geometry
Linear and multilinear algebra, matrix theory
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
Studies
Tensors
Collapse
SVD
Tucker
high-dimensional
Approximation
Error analysis
Tensor
low rank
90C06
Rank
15A69
Algorithm
N order
hierarchical Tucker
Complexity
Truncation
Numerical analysis
Linear algebra
Best approximation
65K10
Singular value decomposition
ISSN:0895-4798, 1095-7162
Online Access:Get full text
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Summary:The authors define the hierarchical singular value decomposition (SVD) for tensors of order d ≥ 2. This hierarchical SVD has properties like the matrix SVD (and collapses to the SVD in d = 2), and they prove these. In particular, one can find low rank (almost) best approximations in a hierarchical format (H-Tucker) which requires only ... parameters, where d is the order of the tensor, n the size of the modes, and k the (hierarchical) rank. The H-Tucker format is a specialization of the Tucker format and it contains as a special case all (canonical) rank k tensors. Based on this new concept of a hierarchical SVD the authors present algorithms for hierarchical tensor calculations allowing for a rigorous error analysis. The complexity of the truncation (finding lower rank approximations to hierarchical rank k tensors) is in ... and the attainable accuracy is just 2-3 digits less than machine precision.(ProQuest: ... denotes formulae/symbols omitted.)
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ISSN:0895-4798
1095-7162
DOI:10.1137/090764189