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{{short description|Multilinear extension of principal component analysis}}
'''Multilinear principal component analysis''' ('''MPCA''') is a [[Multilinear algebra|multilinear]] extension of [[principal component analysis]] (PCA) * '''linear tensor models''', such as [[Tensor rank|CANDECOMP/Parafac]], or by
* '''multilinear tensor models''', such as '''multilinear principal component analysis (MPCA)
}}</ref> and 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>▼
(CVPR 2001, ICPR 2002), face recognition – [[TensorFaces]],<ref name="Vasilescu2002a"/><ref name="Vasilescu2003"/>▼
(ECCV 2002, CVPR 2003, etc.) and computer graphics – [[TensorTextures]]<ref name="Vasilescu2004"/> (Siggraph 2004).▼
Multilinear PCA may be applied to compute the causal factors of data formation, or as signal processing tool on data tensors whose individual observation have either been vectorized,<ref name="Vasilescu2002b"/><ref name="Vasilescu2002a">M.A.O. Vasilescu, [[Demetri Terzopoulos|D. Terzopoulos]] (2002) [http://www.media.mit.edu/~maov/tensorfaces/eccv02_corrected.pdf "Multilinear Analysis of Image Ensembles: TensorFaces," Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002, in Computer Vision – ECCV 2002, Lecture Notes in Computer Science, Vol. 2350, A. Heyden et al. (Eds.), Springer-Verlag, Berlin, 2002, 447–460. ]</ref><ref name="Vasilescu2003">M.A.O. Vasilescu, D. Terzopoulos (2003) [http://www.media.mit.edu/~maov/tensorfaces/cvpr03.pdf "Multilinear Subspace Analysis for Image Ensembles,'' M. A. O. Vasilescu, D. Terzopoulos, Proc. Computer Vision and Pattern Recognition Conf. (CVPR '03), Vol.2, Madison, WI, June, 2003, 93–99.]</ref><ref name="Vasilescu2004">M.A.O. Vasilescu, D. Terzopoulos (2004) [http://www.media.mit.edu/~maov/tensortextures/Vasilescu_siggraph04.pdf "TensorTextures: Multilinear Image-Based Rendering", M. A. O. Vasilescu and D. Terzopoulos, Proc. ACM SIGGRAPH 2004 Conference Los Angeles, CA, August, 2004, in Computer Graphics Proceedings, Annual Conference Series, 2004, 336–342. ]</ref> or whose observations are treated as a collection of column/row observations, an "
▲Historically, MPCA has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg in 1980.<ref name="Kroonenberg1980"/> In 2005, Vasilescu and [[Demetri Terzopoulos|Terzopoulos]] introduced the Multilinear PCA<ref name="MPCA-MICA2005">M. A. O. Vasilescu, D. Terzopoulos (2005) [http://www.media.mit.edu/~maov/mica/mica05.pdf "Multilinear Independent Component Analysis"], "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’05), San Diego, CA, June 2005, vol.1, 547–553."</ref> terminology as a way to better differentiate between linear and multilinear tensor decomposition, as well as, to better differentiate between the work<ref name="Vasilescu2002b"/><ref name="Vasilescu2002a"/><ref name="Vasilescu2003"/><ref name="Vasilescu2004"/> that computed 2nd order statistics associated with each data tensor mode(axis), and subsequent work on Multilinear Independent Component Analysis<ref name="MPCA-MICA2005"/> that computed higher order statistics associated with each tensor mode/axis.
▲Multilinear PCA may be applied to compute the causal factors of data formation, or as signal processing tool on data tensors whose individual observation have either been vectorized,<ref name="Vasilescu2002b"/><ref name="Vasilescu2002a">M.A.O. Vasilescu, [[Demetri Terzopoulos|D. Terzopoulos]] (2002) [http://www.media.mit.edu/~maov/tensorfaces/eccv02_corrected.pdf "Multilinear Analysis of Image Ensembles: TensorFaces," Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002, in Computer Vision – ECCV 2002, Lecture Notes in Computer Science, Vol. 2350, A. Heyden et al. (Eds.), Springer-Verlag, Berlin, 2002, 447–460. ]</ref><ref name="Vasilescu2003">M.A.O. Vasilescu, D. Terzopoulos (2003) [http://www.media.mit.edu/~maov/tensorfaces/cvpr03.pdf "Multilinear Subspace Analysis for Image Ensembles,'' M. A. O. Vasilescu, D. Terzopoulos, Proc. Computer Vision and Pattern Recognition Conf. (CVPR '03), Vol.2, Madison, WI, June, 2003, 93–99.]</ref><ref name="Vasilescu2004">M.A.O. Vasilescu, D. Terzopoulos (2004) [http://www.media.mit.edu/~maov/tensortextures/Vasilescu_siggraph04.pdf "TensorTextures: Multilinear Image-Based Rendering", M. A. O. Vasilescu and D. Terzopoulos, Proc. ACM SIGGRAPH 2004 Conference Los Angeles, CA, August, 2004, in Computer Graphics Proceedings, Annual Conference Series, 2004, 336–342. ]</ref> or whose observations are treated as a collection of column/row observations, "data matrix" and concatenated into a data tensor. The main disadvantage of this approach is that rather than computing all possible combinations
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or the '''Tucker''' which employ the power method or gradient descent, respectively.
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▲(CVPR 2001, ICPR 2002), face recognition – [[TensorFaces]],<ref name="Vasilescu2002a"/><ref name="Vasilescu2003"/>
▲(ECCV 2002, CVPR 2003, etc.) and computer graphics – [[TensorTextures]]<ref name="Vasilescu2004"/> (Siggraph 2004).
== The algorithm ==
The MPCA solution follows the alternating least square (ALS) approach. It is iterative in nature. As in PCA, MPCA works on centered data. Centering is a little more complicated for tensors, and it is problem dependent.
== Feature selection ==
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