Multilinear principal component analysis: Difference between revisions

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{{context|date=June 2012}}

'''Multilinear principal component analysis''' (MPCA) is a [[multilinear]] extension of [[principal component analysis]] (PCA). MPCA is employed in the analysis of n-way arrays, i.e. a cube or hyper-cube of numbers, also informally referred to as a "data tensor". N-way arrays may be decomposed, analyzed, or modeled by

* linear tensor models such as CANDECOMP/Parafac, or

* multilinear tensor models, such multilinear principal component analysis (MPCA), or multilinear independent component analysis (MICA), etc.

The origin of MPCA can be traced back to the [[Tucker decomposition]]<ref>{{Cite journal|last1=Tucker| first1=Ledyard R

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|date=September 1966

| doi = 10.1007/BF02289464

}}</ref> and Peter Kroonenberg's "M-mode PCA/3-mode PCA" work.<ref name="Kroonenberg1980">P. M. Kroonenberg and J. de Leeuw, [http://www.springerlink.com/content/c8551t1p31236776/ 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">L.D. Lathauwer, B.D. Moor, J. Vandewalle (2000) [http://portal.acm.org/citation.cfm?id=354398 "A multilinear singular value decomposition"], ''SIAM Journal of Matrix Analysis and Applications'', 21 (4), 1253–1278</ref> (HOSVD) and in their paper "On the Best Rank-1 and Rank-(R₁, R₂, ..., R_N ) Approximation of Higher-order Tensors".<ref name=DeLathauwer2000b>L. D. Lathauwer, B. D. Moor, J. Vandewalle (2000) [http://portal.acm.org/citation.cfm?id=354405 "On the best rank-1 and rank-(R1, R2, ..., RN ) approximation of higher-order tensors"], ''SIAM Journal of Matrix Analysis and Applications'' 21 (4), 1324–1342.</ref>

Circa 2001, Vasilescu reframed the data analysis, recognition and synthesis problems as multilinear tensor problems based on the insight that most observed data are 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

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|doi=10.1016/j.patcog.2011.01.004

}}</ref>

*Uncorrelated MPCA (UMPCA) <ref name="UMPCA">H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "[http://www.dsp.utoronto.ca/~haiping/Publication/UMPCA_TNN09.pdf Uncorrelated multilinear principal component analysis for unsupervised multilinear subspace learning]," IEEE Trans. Neural Netw., vol. 20, no. 11, pp. 1820–1836, Nov. 2009.</ref> In contrast, the uncorrelated MPCA (UMPCA) generates uncorrelated multilinear features.<ref name="UMPCA"/>

*[[Boosting (meta-algorithm)|Boosting]]+MPCA<ref>H. Lu, K. N. Plataniotis and A. N. Venetsanopoulos, "[http://www.hindawi.com/journals/ivp/2009/713183.html Boosting Discriminant Learners for Gait Recognition using MPCA Features]", EURASIP Journal on Image and Video Processing, Volume 2009, Article ID 713183, 11 pages, 2009. {{doi|10.1155/2009/713183}}.</ref>

*Non-negative MPCA (NMPCA) <ref>Y. Panagakis, C. Kotropoulos, G. R. Arce, "Non-negative multilinear principal component analysis of auditory temporal modulations for music genre classificationclassification", IEEE Trans. on Audio, Speech, and Language Processing, vol. 18, no. 3, pp. 576–588, 2010.</ref>

*Robust MPCA (RMPCA) <ref>K. Inoue, K. Hara, K. Urahama, "Robust multilinear principal component analysis", Proc. IEEE Conference on Computer Vision, 2009, pp. 591–597.</ref>

*Multi-Tensor Factorization, that also finds the number of components automatically (MTF) <ref>{{Cite journal|last=Khan|first=Suleiman A.|last2=Leppäaho|first2=Eemeli|last3=Kaski|first3=Samuel|date=2016-06-10|title=Bayesian multi-tensor factorization|url=https://link.springer.com/article/10.1007/s10994-016-5563-y|journal=Machine Learning|language=en|volume=105|issue=2|pages=233–253|doi=10.1007/s10994-016-5563-y|issn=0885-6125}}</ref>

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* ''Matlab code'': [http://www.mathworks.com/matlabcentral/fileexchange/35432 UMPCA (including data)].

* ''R code:'' [http://research.cs.aalto.fi/pml/software/mtf/ MTF]

[[Category:Dimension reduction]]