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Maxeto0910 (talk | contribs) Added short description. Tags: Mobile edit Mobile web edit Advanced mobile edit |
Brom20110101 (talk | contribs) m Frank Lauren Hitchcock introduced tensor rank decompositions in 1927 in physics and mathematics. Ledyard Tucker extended the idea in 1966 in psychometrics. This article did not mention Hitchcock's work, so I added a sentence mentioning it. |
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* linear tensor models such as CANDECOMP/Parafac, or
* multilinear tensor models, such as multilinear principal component analysis (MPCA), or multilinear independent component analysis (MICA), etc.
The origin of MPCA can be traced back to the [[
| author = F. L. Hitchcock
| author-link = F. L. Hitchcock
| title = The expression of a tensor or a polyadic as a sum of products
| journal = [[Journal of Mathematics and Physics]]
| volume = 6
| pages = 164–189
| year = 1927
| issue = 1–4
| doi = 10.1002/sapm192761164
}}</ref> to the [[Tucker decomposition]];<ref>{{Cite journal|last1=Tucker| first1=Ledyard R
| authorlink1 = Ledyard R Tucker
| title = Some mathematical notes on three-mode factor analysis
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|date=September 1966
| doi = 10.1007/BF02289464 | pmid = 5221127
}}</ref> and to Peter Kroonenberg's "3-mode PCA" work.<ref name="Kroonenberg1980">P. M. Kroonenberg and J. de Leeuw, [https://doi.org/10.1007%2FBF02293599 Principal component analysis of three-mode data by means of alternating least squares algorithms], Psychometrika, 45 (1980), pp. 69–97.</ref> In 2000, De Lathauwer et al. restated Tucker and Kroonenberg's work in clear and concise numerical computational terms in their SIAM paper entitled "[[Multilinear Singular Value Decomposition]]",<ref name="DeLathauwer2000a">{{cite journal | last1 = Lathauwer | first1 = L.D. | last2 = Moor | first2 = B.D. | last3 = Vandewalle | first3 = J. | year = 2000 | title = A multilinear singular value decomposition | url = http://portal.acm.org/citation.cfm?id=354398 | journal = SIAM Journal on Matrix Analysis and Applications | volume = 21 | issue = 4| pages = 1253–1278 | doi = 10.1137/s0895479896305696 }}</ref> (HOSVD) and in their paper "On the Best Rank-1 and Rank-(R<sub>1</sub>, R<sub>2</sub>, ..., R<sub>N</sub> ) Approximation of Higher-order Tensors".<ref name=DeLathauwer2000b>{{cite journal | last1 = Lathauwer | first1 = L. D. | last2 = Moor | first2 = B. D. | last3 = Vandewalle | first3 = J. | year = 2000 | title = On the best rank-1 and rank-(R1, R2, ..., RN ) approximation of higher-order tensors | url = http://portal.acm.org/citation.cfm?id=354405 | journal = SIAM Journal on Matrix Analysis and Applications | volume = 21 | issue = 4| pages = 1324–1342 | doi = 10.1137/s0895479898346995 }}</ref>
Circa 2001, Vasilescu and Terzopoulos reframed the data analysis, recognition and synthesis problems as multilinear tensor problems. Tensor factor analysis is the compositional consequence of several causal factors of data formation, and are well suited for multi-modal data tensor analysis. The power of the tensor framework was showcased by analyzing human motion joint angles, facial images or textures in terms of their causal factors of data formation in the following works: Human Motion Signatures<ref name="Vasilescu2002b">M.A.O. Vasilescu (2002) [http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf "Human Motion Signatures: Analysis, Synthesis, Recognition," Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460.]</ref>
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