Abstract
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.
Article PDF
Avoid common mistakes on your manuscript.
References
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.
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.
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.
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.
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.
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.
Friedland, S. (2010). On the generic and typical rank of 3-tensors. arXiv:0805.3777v4.
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.
Harshman, R.A. (1972). Determination and proof of minimum uniqueness conditions for PARAFAC1. UCLA Working Papers in Phonetics, 16, 1–84.
Hitchcock, F.L. (1927a). The expression of a tensor or a polyadic as a sum of products. Journal of Mathematical Physics, 6, 164–189.
Hitchcock, F.L. (1927b). Multiple invariants and generalized rank of a p-way matrix or tensor. Journal of Mathematical Physics, 7, 39–79.
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.
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.
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.
Kolda, T.G., & Brader, B.W. (2009). Tensor decompositions and applications. SIAM Review, 51, 455–500.
Kroonenberg, P.M., & De Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least-squares. Psychometrika, 45, 69–97.
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.
Kruskal, J.B. (1983, unpublished). Statement of some current results about three-way arrays.
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.
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.
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.
Rocci, R., & Ten Berge, J.M.F. (2002). Transforming three-way arrays to maximal simplicity. Psychometrika, 67, 351–365.
Sidiropoulos, N.D., & Bro, R. (2000). On the uniqueness of multilinear decomposition of N-way arrays. Journal of Chemometrics, 14, 229–239.
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.
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.
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.
Stegeman, A., & Sidiropoulos, N.D. (2007). On Kruskal’s uniqueness condition for the Candecomp/Parafac decomposition. Linear Algebra & Applications, 420, 540–552.
Sumi, T., Sakata, T., & Miyazaki, M. (2010). Typical ranks for m×n×(m−1)n tensors with m≤n. Preprint, retrieved from http://polygon.aid.design.kyushu-u.ac.jp/~sumi/myarticles.html, October 14, 2010.
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. (2000). The typical rank of tall three-way arrays. Psychometrika, 65, 525–532.
Ten Berge, J.M.F. (2004). Partial uniqueness in CANDECOMP/PARAFAC. Journal of Chemometrics, 18, 12–16.
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.
Ten Berge, J.M.F., & Sidiropoulos, N.D. (2002). Some new results on uniqueness in Candecomp/Parafac. Psychometrika, 67, 399–409.
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.
Ten Berge, J.M.F., & Smilde, A.K. (2002). Non-triviality and identification of a constrained Tucker3 analysis. Journal of Chemometrics, 16, 609–612.
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.
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.
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.
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.
Thijsse, G.P.A. (1994). Simultaneous diagonal forms for pairs of matrices (Report 9450/B). Econometric Institute. Erasmus University, Rotterdam.
Tucker, L.R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was done jointly with Henk Kiers, Roberto Rocci, Alwin Stegeman, and Jorge Tendeiro. The author is obliged to Henk Kiers and Mohammed Bennani Dosse for helpful comments.
Rights and permissions
About this article
Cite this article
ten Berge, J.M.F. Simplicity and Typical Rank Results for Three-Way Arrays. Psychometrika 76, 3–12 (2011). https://doi.org/10.1007/s11336-010-9193-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11336-010-9193-1