Tensor decompositions, alternating least squares and other tales

This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple ‘bottlenecks’, and on ‘swamps’. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity...

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Bibliographic Details
Published in:Journal of chemometrics Vol. 23; no. 7-8; pp. 393 - 405
Main Authors: Comon, P., Luciani, X., de Almeida, A. L. F.
Format: Journal Article
Language:English
Published: Chichester, UK John Wiley & Sons, Ltd 01.07.2009
Wiley
Wiley Subscription Services, Inc
Subjects:
Algorithms
canonical decomposition
Chemical Sciences
Cheminformatics
Chemistry
Chemometrics
Classification
computational complexity
Computer Science
Computer simulation
degeneracy
Engineering Sciences
Environmental Sciences
Exact sciences and technology
General and physical chemistry
General. Nomenclature, chemical documentation, computer chemistry
Least squares method
Mathematical analysis
Numerical analysis
PARAFAC
Signal and Image Processing
Swamps
tensor rank
Tensors
Theory of reactions, general kinetics. Catalysis. Nomenclature, chemical documentation, computer chemistry
three-way array
canonical decomposition
degeneracy
Computer simulation
Classification
PARAFAC
Decomposition
three-way array
Algorithm
Chemometrics
tensor rank
computational complexity
Performance measure
Levenberg-Marquardt
Parafac
Tensor rank
Canonical Decomposition
Three-way Array
Candecomp
greedy algorithm
Indeterminacy
compression
dimension reduction
ISSN:0886-9383, 1099-128X, 1099-128X
Online Access:Get full text
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Summary:This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple ‘bottlenecks’, and on ‘swamps’. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the enhanced line search (ELS) enhancement in these algorithms is discussed. Computer simulations feed this discussion. Copyright © 2009 John Wiley & Sons, Ltd. Various aspects of tensor decompositions are addressed: existence, uniqueness and computation. The state of the art is surveyed, by making the difference between conjectures and proved results. Some numerical algorithms are described in details, and their numerical complexity is evaluated. The slowness of numerical algorithms is often due to a form of ill‐conditioning of the tensor to be decomposed. In particular, Richard Harshman called ‘bottleneck’ the fact that two or more factors in a mode are almost collinear.
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ISSN:0886-9383
1099-128X
1099-128X
DOI:10.1002/cem.1236