One of the basic issues in the analysis of three-way arrays by CANDECOMP/PARAFAC (CP) has been the question of uniqueness of the decomposition. Kruskal (1977) has proved that uniqueness is guaranteed when the sum of thek-ranks of the three component matrices involved is at least twice the rank of the solution plus 2. Since then, little has been achieved that might further qualify Kruskal's sufficient condition. Attempts to prove that it is also necessary for uniqueness (except for rank 1 or 2) have failed, but counterexamples to necessity have not been detected. The present paper gives a method for generating the class of all solutions (or at least a subset of that class), given a CP solution that satisfies certain conditions. This offers the possibility to examine uniqueness for a great variety of specific CP solutions. It will be shown that Kruskal's condition is necessary and sufficient when the rank of the solution is three, but that uniqueness may hold even if the condition is not satisfied, when the rank is four or higher.
Key wordsCandecomp Parafac uniqueness three-way arrays
Unable to display preview. Download preview PDF.
- Carroll, J.D., & Chang, J.J. (1970). Analysis of individual differences in multidimensional scaling via ann-way general-isation of “Eckart-Young” decomposition.Psychometrika, 35, 283–319.Google Scholar
- Harshman, R.L. (1970). Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multimodal factor analysis.UCLA Working Papers in Phonetics, 16, 1–84. (http://publish.uwo.ca/~harshman/)Google Scholar
- Harshman, R.L. (1972). Determination and proof of minimum uniqueness conditions for PARAFAC1.UCLA Working Papers in Phonetics, 22, 111–117. (http://publish.uwo.ca/~harshman/)Google Scholar
- Krijnen, W.P. (1993).The analysis of three-way arrays by constrained PARAFAC methods. Leiden, Netherlands: DSWO Press.Google Scholar
- Kruskal, J.B. (1989). Rank, decomposition, and uniqueness for 3-way andN-way arrays. In R. Coppi & S. Bolasco (Eds.),Multiway data analysis (pp. 7–18). Amsterdam, Netherlands: North-Holland.Google Scholar
- Liu, X., & Sidiropoulos, N.D. (2001). Cramer-Rao lower bounds for low-rank decomposition of multidimensional arrays.IEEE Transactions on Signal Processing, 49, 2074–2086.Google Scholar
- Paatero, P. (1999). The multilinear engine—A table-driven least squares program for solving multilinear programs, including then-way parallel factor analysis model.Journal of Computational and Graphical Statistics, 8, 854–888.Google Scholar