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On global, local and stationary solutions in three‐way data analysis
Author(s) -
Henrion René
Publication year - 2000
Publication title -
journal of chemometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.47
H-Index - 92
eISSN - 1099-128X
pISSN - 0886-9383
DOI - 10.1002/1099-128x(200005/06)14:3<261::aid-cem583>3.0.co;2-i
Subject(s) - context (archaeology) , iterated function , mathematical optimization , component (thermodynamics) , simple (philosophy) , core (optical fiber) , global optimization , mathematics , maximization , computer science , decomposition , chemistry , mathematical analysis , paleontology , philosophy , telecommunications , physics , organic chemistry , epistemology , biology , thermodynamics
Abstract The issue of global and local solutions to optimization problems is of much interest in the context of three‐way analysis, in particular when dealing with the PARAFAC and Tucker3 models or core transformations within the latter. For clarity of statements, it is useful to consider the most simple yet reasonable situation, namely one‐component PARAFAC decomposition or, closely related, maximization of the leading squared core entry in Tucker3. In the paper, necessary and sufficient conditions for global solutions are derived. Furthermore, it is shown that, in general, the usual cyclic co‐ordinate optimization scheme of three‐way methods does not converge towards a local minimum (or maximum) even if the iterates yield global solutions in each co‐ordinate direction. Finally, an example for a proper local minimum in one‐component PARAFAC is given. Copyright © 2000 John Wiley & Sons, Ltd.