On the Nuclear Norm and the Singular Value Decomposition of Tensors

Finding the rank of a tensor is a problem that has many applications. Unfortunately, it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of the rank of a tensor. We determine the nuclear norm of var...

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Bibliographic Details
Published in:Foundations of computational mathematics 2016-06, Vol.16 (3), p.779-811
Main Author: Derksen, Harm
Format: Article
Language:English
Subjects:
Applications of Mathematics
Computer Science
Decomposition (Mathematics)
Economics
Foundations
Heuristic
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematical analysis
Mathematical models
Mathematical research
Mathematics
Mathematics and Statistics
Matrix Theory
Norms
Numerical Analysis
Orthogonality
Singular value decomposition
Tensors
Tensors (Mathematics)
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Summary:Finding the rank of a tensor is a problem that has many applications. Unfortunately, it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of the rank of a tensor. We determine the nuclear norm of various tensors of interest. Along the way, we also do a systematic study various measures of orthogonality in tensor product spaces and we give a new generalization of the singular value decomposition to higher-order tensors.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-015-9264-x