Degeneracy in Candecomp/Parafac explained for p x p x 2 arrays of rank p+1 or higher

Alwin Stegeman*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

62 Citations (Scopus)

Abstract

The Candecomp/Parafac (CP) model decomposes a three-way array into a prespecified number R of rank-1 arrays and a residual array, in which the sum of squares of the residual array is minimized. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate solutions, in which some components are highly correlated in all three modes and the elements of these components become arbitrarily large. We consider the real-valued CP model in which p x p x 2 arrays of rank p + 1 or higher are decomposed into p rank-1 arrays and a residual array. It is shown that the CP objective function does not have a minimum in these cases, but an infimum. Moreover, any sequence of CP approximations, of which the objective value approaches the infimum, will become degenerate. This result extends Ten Berge, Kiers, & De Leeuw (1988), who consider a particular 2 x 2 x 2 array of rank 3.

Original languageEnglish
Pages (from-to)483-501
Number of pages19
JournalPsychometrika
Volume71
Issue number3
DOIs
Publication statusPublished - Sept-2006

Keywords

  • Candecomp
  • Parafac
  • three-way arrays
  • degenerate solutions
  • 2-FACTOR DEGENERACIES
  • PARAFAC
  • UNIQUENESS

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