, Volume 76, Issue 1, pp 3–12 | Cite as

Simplicity and Typical Rank Results for Three-Way Arrays

  • Jos M. F. ten BergeEmail author


Matrices can be diagonalized by singular vectors or, when they are symmetric, by eigenvectors. Pairs of square matrices often admit simultaneous diagonalization, and always admit block wise simultaneous diagonalization. Generalizing these possibilities to more than two (non-square) matrices leads to methods of simplifying three-way arrays by nonsingular transformations. Such transformations have direct applications in Tucker PCA for three-way arrays, where transforming the core array to simplicity is allowed without loss of fit. Simplifying arrays also facilitates the study of array rank. The typical rank of a three-way array is the smallest number of rank-one arrays that have the array as their sum, when the array is generated by random sampling from a continuous distribution. In some applications, the core array of Tucker PCA is constrained to have a vast majority of zero elements. Both simplicity and typical rank results can be applied to distinguish constrained Tucker PCA models from tautologies. An update of typical rank results over the real number field is given in the form of two tables.


tensor decomposition tensor rank typical rank sparse arrays Candecomp Parafac Tucker component analysis 


  1. Bennani Dosse, M., & Ten Berge, J.M.F. (2008). The assumption of proportional components when Candecomp is applied to symmetric matrices in the context of Indscal. Psychometrika, 73, 303–307. CrossRefGoogle Scholar
  2. 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. CrossRefGoogle Scholar
  3. Choulakian, V. (2010). Some numerical results on the rank of generic three-way arrays over ℜ. SIAM Journal on Matrix Analysis and Applications, 31, 1541–1551. CrossRefGoogle Scholar
  4. Comon, P., Ten Berge, J.M.F., De Lathauwer, L., & Castaing, J. (2009). Generic and typical ranks of multiway arrays. Linear Algebra & Applications, 430, 2997–3007. CrossRefGoogle Scholar
  5. De Lathauwer, L. (2006). A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization. SIAM Journal on Matrix Analysis and Applications, 28, 642–666. CrossRefGoogle Scholar
  6. Gurden, S.P., Westerhuis, J.A., Bijlsma, S., & Smilde, A.K. (2001). Modeling of spectroscopic batch process data using grey models to incorporate external information. Journal of Chemometrics, 15, 101–121. CrossRefGoogle Scholar
  7. Friedland, S. (2010). On the generic and typical rank of 3-tensors. arXiv:0805.3777v4.
  8. Harshman, R.A. (1970). Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multi-mode factor analysis. UCLA Working Papers in Phonetics, 16, 1–84. Google Scholar
  9. Harshman, R.A. (1972). Determination and proof of minimum uniqueness conditions for PARAFAC1. UCLA Working Papers in Phonetics, 16, 1–84. Google Scholar
  10. Hitchcock, F.L. (1927a). The expression of a tensor or a polyadic as a sum of products. Journal of Mathematical Physics, 6, 164–189. Google Scholar
  11. Hitchcock, F.L. (1927b). Multiple invariants and generalized rank of a p-way matrix or tensor. Journal of Mathematical Physics, 7, 39–79. Google Scholar
  12. Jiang, T., & Sidiropoulos, N.D. (2004). Kruskal’s permutation lemma and the identification of Candecomp/Parafac and bilinear models with constant modulus constraints. IEEE Transactions on Signal Processing, 52, 2625–2636. CrossRefGoogle Scholar
  13. Kiers, H.A.L. (1998). Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core to simple structure. Computational Statistics & Data Analysis, 28, 307–324. CrossRefGoogle Scholar
  14. Kiers, H.A.L., Ten Berge, J.M.F., & Rocci, R. (1997). Uniqueness of three-mode factor models with sparse cores: The 3×3×3 case. Psychometrika, 62, 349–374. CrossRefGoogle Scholar
  15. Kolda, T.G., & Brader, B.W. (2009). Tensor decompositions and applications. SIAM Review, 51, 455–500. CrossRefGoogle Scholar
  16. Kroonenberg, P.M., & De Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least-squares. Psychometrika, 45, 69–97. CrossRefGoogle Scholar
  17. Kruskal, J.B. (1977). Three-way arrays: Rank and uniqueness of trilinear decompositions with applications to arithmetic complexity and statistics. Linear Algebra & Applications, 18, 95–138. CrossRefGoogle Scholar
  18. Kruskal, J.B. (1983, unpublished). Statement of some current results about three-way arrays. Google Scholar
  19. Kruskal, J.B. (1989). Rank, decomposition, and uniqueness for 3-way and N-way arrays. In Coppi, R., & Bolasco, S. (Eds.) Multiway data analysis (pp. 7–18). Amsterdam: North-Holland. Google Scholar
  20. 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 component analysis. Psychometrika, 63, 255–261. CrossRefGoogle Scholar
  21. Rocci, R., & Ten Berge, J.M.F. (1994). A simplification of a result by Zellini on the maximal rank of a symmetric three-way array. Psychometrika, 59, 377–380. CrossRefGoogle Scholar
  22. Rocci, R., & Ten Berge, J.M.F. (2002). Transforming three-way arrays to maximal simplicity. Psychometrika, 67, 351–365. CrossRefGoogle Scholar
  23. Sidiropoulos, N.D., & Bro, R. (2000). On the uniqueness of multilinear decomposition of N-way arrays. Journal of Chemometrics, 14, 229–239. CrossRefGoogle Scholar
  24. Stegeman, A.W. (2009). On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode. Linear Algebra & Applications, 431, 211–227. CrossRefGoogle Scholar
  25. Stegeman, A.W., & Ten Berge, J.M.F. (2006). Kruskal’s condition for uniqueness in Candecomp/Parafac when ranks and k-ranks coincide. Computational Statistics & Data Analysis, 50, 210–220. CrossRefGoogle Scholar
  26. Stegeman, A., Ten Berge, J.M.F., & De Lathauwer, L. (2006). Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices. Psychometrika, 71, 219–229. CrossRefGoogle Scholar
  27. Stegeman, A., & Sidiropoulos, N.D. (2007). On Kruskal’s uniqueness condition for the Candecomp/Parafac decomposition. Linear Algebra & Applications, 420, 540–552. CrossRefGoogle Scholar
  28. Sumi, T., Sakata, T., & Miyazaki, M. (2010). Typical ranks for m×n×(m−1)n tensors with mn. Preprint, retrieved from, October 14, 2010.
  29. 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. CrossRefGoogle Scholar
  30. Ten Berge, J.M.F. (2000). The typical rank of tall three-way arrays. Psychometrika, 65, 525–532. CrossRefGoogle Scholar
  31. Ten Berge, J.M.F. (2004). Partial uniqueness in CANDECOMP/PARAFAC. Journal of Chemometrics, 18, 12–16. CrossRefGoogle Scholar
  32. Ten Berge, J.M.F., & Kiers, H.A.L. (1999). Simplicity of core arrays in three-way principal component analysis and the typical rank of P×Q×2 arrays. Linear Algebra & Applications, 294, 169–179. CrossRefGoogle Scholar
  33. Ten Berge, J.M.F., & Sidiropoulos, N.D. (2002). Some new results on uniqueness in Candecomp/Parafac. Psychometrika, 67, 399–409. CrossRefGoogle Scholar
  34. Ten Berge, J.M.F., Sidiropoulos, N.D., & Rocci, R. (2004). Typical rank and Indscal dimensionality for symmetric three-way arrays of order I×2×2 or I×3×3. Linear Algebra & Applications, 388, 363–377. CrossRefGoogle Scholar
  35. Ten Berge, J.M.F., & Smilde, A.K. (2002). Non-triviality and identification of a constrained Tucker3 analysis. Journal of Chemometrics, 16, 609–612. CrossRefGoogle Scholar
  36. Ten Berge, J.M.F., & Stegeman, A. (2006). Symmetry transformations for square sliced three-way arrays, with applications to their typical rank. Linear Algebra & Applications, 418, 215–224. CrossRefGoogle Scholar
  37. Ten Berge, J.M.F., Stegeman, A., & Bennani Dosse, M. (2009). The Carroll-Chang conjecture of equal Indscal components when Candecomp/Parafac gives perfect fit. Linear Algebra & Applications, 430, 818–829. CrossRefGoogle Scholar
  38. Ten Berge, J.M.F., & Tendeiro, J.N. (2009). The link between sufficient conditions by Harshman and by Kruskal for uniqueness in Candecomp/Parafac. Journal of Chemometrics, 23, 321–323. CrossRefGoogle Scholar
  39. Tendeiro, J.N., Ten Berge, J.M.F., & Kiers, H.A.L. (2009). Simplicity transformations for three-way arrays with symmetric slices, and applications to Tucker-3 models with sparse core arrays. Linear Algebra & Applications, 430, 924–940. CrossRefGoogle Scholar
  40. Thijsse, G.P.A. (1994). Simultaneous diagonal forms for pairs of matrices (Report 9450/B). Econometric Institute. Erasmus University, Rotterdam. Google Scholar
  41. Tucker, L.R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311. PubMedCrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2010

Authors and Affiliations

  1. 1.University of GroningenGroningenThe Netherlands

Personalised recommendations