Abstract
The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank ofI × J × K arrays withK = 2 to a particular class of arrays withK ≥ 2. It is shown that the typical rank isI when the array is tall in the sense thatJK − J < I < JK. In addition, typical rank results are given for the case whereI equalsJK − J.
Similar content being viewed by others
References
Atkinson, M.D., & Stevens, N.M. (1979). On the multiplicative complexity of a family of bilinear forms.Linear Algebra and its Applications, 27, 1–8.
Carroll, J.D., & Chang, J.J. (1970). Analysis of individual differences in multidimensional scaling via an n-way generalization of Eckart-Young decomposition.Psychometrika, 35, 283–319.
Fisher, F.M. (1966).The identification problem in Econometrics. New York, NY: McGraw-Hill.
Franc, A. (1992).Etudes algebriques des multitableaux: Apports de l'algebre tensorielle. Unpublished doctoral dissertation, University of Montpellier II.
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.
Kruskal, J.B. (1977). Three-way arrays: Rank and uniqueness of trilinear decompositions with applications to arithmetic complexity and statistics.Linear Algebra and its Applications, 18, 95–138.
Kruskal, J.B. (1983).Statement of some current results about three-way arrays. Unpublished manuscript, AT&T Bell Laboratories, Murray Hill, NJ.
Kruskal, J.B. (1989). Rank, decomposition, and uniqueness for 3-way and N-way arrays. In R. Coppi & S. Bolasco (Eds.),Multiway data analysis (pp. 7–18). Amsterdam: North-Holland.
Murakami, T., ten Berge, J.M.F., & Kiers, H.A.L. (1998). A case of extreme simplicity of the core matrix in three-mode principal components analysis.Psychometrika, 63, 255–261.
Rocci, R., & ten Berge, J.M.F. (1994). A simplification of a result by Zellini on the maximal rank of symmetric three-way arrays.Psychometrika, 59, 377–380.
ten Berge, J.M.F. (1991). Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays.Psychometrika, 56, 631–636.
ten Berge, J.M.F., & Kiers, H.A.L. (1999). Simplicity of core arrays in three-way principal component analysis and the typical rank ofP ×Q × 2 arrays.Linear Algebra and its Applications, 294, 169–179.
Thijsse, G.P.A. (1994).Simultaneous diagonal forms for pairs of matrices (Report 9450/B). Erasmus University, Rotterdam: Econometric Institute.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is obliged to Henk Kiers, Tom Snijders, and Philip Thijsse for helpful comments.
Rights and permissions
About this article
Cite this article
ten Berge, J.M.F. The typical rank of tall three-way arrays. Psychometrika 65, 525–532 (2000). https://doi.org/10.1007/BF02296342
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02296342