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 language | English |
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Pages (from-to) | 483-501 |
Number of pages | 19 |
Journal | Psychometrika |
Volume | 71 |
Issue number | 3 |
DOIs | |
Publication status | Published - Sept-2006 |
Keywords
- Candecomp
- Parafac
- three-way arrays
- degenerate solutions
- 2-FACTOR DEGENERACIES
- PARAFAC
- UNIQUENESS