Springer-Verlag
New York
11336
0033-3123
1860-0980
Psychometrika
Psychometrika
Psychology
Psychometrics
Assessment, Testing and Evaluation
Statistics for Social Science, Behavorial Science, Education, Public Policy, and Law
Statistical Theory and Methods
73
73
4
3
3
12
2008
9
19
2008
9
26
2008
9
2008
The Psychometric Society
2008
9056
10.1007/s11336-008-9056-1
5
On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model
Theory and Methods
431
439
2008
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3
2007
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27
2007
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The Author(s)
2008
Wim
P.
Krijnen
Theo
K.
Dijkstra
Alwin
Stegeman
+31-50-3636193
+31-50-3636304
a.w.stegeman@rug.nl
Hanze University Groningen
P.O. Box 3037
9701
DA
Groningen
The Netherlands
Department of Econometrics
University of Groningen
P.O. Box 800
9700
AV
Groningen
The Netherlands
Heymans Institute for Psychological Research
University of Groningen
Grote Kruisstraat 2/1
9712
TS
Groningen
The Netherlands
Abstract
The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called “degeneracy”. That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal.
Keywords
Candecomp
Parafac
level sets
bounded sequences
factor analysis
A. Stegeman is supported by the Dutch Organisation for Scientific Research (NWO), Veni grant 451-04-102.
ftp_PUB_09-12-12_06-07-18.zip11336-2008-Article-9056.pdfPDF1.3