Abstract
Tensor data are becoming important recently in various application fields. In this paper, we consider the maximal rank problem of 3-tensors and extend Atkinson and Stephens’ and Atkinson and Lloyd’s results over the real number field. We also prove the assertion of Atkinson and Stephens: \({{\rm max.rank}_{\mathbb{R}}(m,n,p) \leq m+\lfloor p/2\rfloor n}\), \({{\rm max.rank}_{\mathbb{R}}(n,n,p) \leq (p+1)n/2}\) if p is even, \({{\rm max.rank}_{\mathbb{F}}(n,n,3)\leq 2n-1}\) if \({\mathbb{F}=\mathbb{C}}\) or n is odd, and \({{\rm max.rank}_{\mathbb{F}}(m,n,3)\leq m+n-1}\) if m < n where \({\mathbb{F}}\) stands for \({\mathbb{R}}\) or \({\mathbb{C}}\).
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References
Atkinson M.D., Lloyd S. (1980) Bounds on the ranks of some 3-tensors. Linear Algebra and its Applications 31: 19–31
Atkinson M.D., Stephens N.M. (1979) On the maximal multiplicative complexity of a family of bilinear forms. Linear Algebra and its Applications 27: 1–8
Bro R., Kiers H.A.L. (2003) A new efficient method for determining the number of components in PARAFAC models. Journal of Chemometrics 17: 274–286
Harshman, R. A. (1970). Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multi-modal factor analysis. UCLA Working Papers in Phonetics, 16, 1–84 (University Microfilms, Ann Arbor, No. 10,085).
Harshman R.A., Hong S., Lundy M.E. (2003) Shifted factor analysis—part I: Models and properties. Journal of Chemometrics 17: 363–378
Kolda T.G., Bader B.W. (2009) Tensor decompositions and applications. SIAM Review 51(3): 455–500
Kruskal J.B. (1977) Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra and its Applications 18: 95–138
Kruskal J.B. (1989) Rank, decomposition, and uniqueness for 3-way and N-way arrays. In: Coppi R., Bolasco S. (eds) Multiway data analysis. Elsevier, Amsterdam, pp 7–18
Malinowski E.R. (2002) Factor analysis in chemistry (3rd ed). Wiley, New York
Miwakeichi, F., Martinez-Montes, E., Valdés-Sosa, P. A., Nishiyama, N., Mizuhara, H., Yamaguchi, Y. (2004). Decomposing EEG data into space–time–frequency components using parallel factor analysis. NeuroImage, 22, 1035–1045.
Muti D., Bourennane S. (2007) Survey on tensor signal algebraic filtering. Signal Processing Archive 87(2): 237–249
Smilde A., Bro R., Geladi P. (2004) Multi-way analysis: Applications in the chemical sciences. Wiley, New York
Sumi T., Miyazaki M., Sakata T. (2009) Rank of 3-tensors with 2 slices and Kronecker canonical forms. Linear Algebra and its Applications 431: 1858–1868
Vasilescu, M., Terzopoulos, D. (2005). Multilinear independent components analysis. In IEEE computer society conference on computer vision and pattern recognition, 2005, CVPR 2005 (Vol. 1, pp. 547–553).
Wu H.-L., Nie J.-F., Yu Y.-J., Yu R.-Q. (2009) Multi-way chemometric methodologies and applications: A central summary of our research work. Analytica Chimica Acta 650: 131–142
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This work was supported in part by Grant-in-Aid for Scientific Research (B) (No. 20340021) of the Japan Society for the Promotion of Science.
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Sumi, T., Miyazaki, M. & Sakata, T. About the maximal rank of 3-tensors over the real and the complex number field. Ann Inst Stat Math 62, 807–822 (2010). https://doi.org/10.1007/s10463-010-0294-5
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DOI: https://doi.org/10.1007/s10463-010-0294-5