The Best Rank-One Approximation Ratio of a Tensor Space

In this paper we define the best rank-one approximation ratio of a tensor space. It turns out that in the finite dimensional case this provides an upper bound for the quotient of the residual of the best rank-one approximation of any tensor in that tensor space and the norm of that tensor. This uppe...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications Vol. 32; no. 2; pp. 430 - 442
Main Author: Qi, Liqun
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.04.2011
Subjects:
Acceleration of convergence
Algebra
Algorithms
Applied mathematics
Approximation
Eigenvalues
Exact sciences and technology
Functional analysis
Linear and multilinear algebra, matrix theory
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
Lower bound
Symmetric tensor
Normed space
Rank
tensors
Ordered space
Upper bound
Numerical analysis
Symmetric space
Linear algebra
Greedy algorithm
Convergence rate
bounds
Best approximation
Quotient
best rank-one approximation ratio
ISSN:0895-4798, 1095-7162
Online Access:Get full text
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Summary:In this paper we define the best rank-one approximation ratio of a tensor space. It turns out that in the finite dimensional case this provides an upper bound for the quotient of the residual of the best rank-one approximation of any tensor in that tensor space and the norm of that tensor. This upper bound is strictly less than one, and it gives a convergence rate for the greedy rank-one update algorithm. For finite dimensional general tensor spaces, third order finite dimensional symmetric tensor spaces, and finite biquadratic tensor spaces, we give positive lower bounds for the best rank-one approximation ratio. For finite symmetric tensor spaces and finite dimensional biquadratic tensor spaces, we give upper bounds for this ratio. [PUBLICATION ABSTRACT]
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ISSN:0895-4798
1095-7162
DOI:10.1137/100795802