# Multilinear principal component analysis

Multilinear principal component analysis (MPCA) is a multilinear extension of principal component analysis (PCA) that is used to analyze M-way arrays, also informally referred to as "data tensors". M-way arrays may be modeled by linear tensor models, such as CANDECOMP/Parafac, or by multilinear tensor models, such as multilinear principal component analysis (MPCA) or multilinear independent component analysis (MICA). The origin of MPCA can be traced back to the tensor rank decomposition introduced by Frank Lauren Hitchcock in 1927;[1] to the Tucker decomposition;[2] and to Peter Kroonenberg's "3-mode PCA" work.[3] 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",[4] (HOSVD) and in their paper "On the Best Rank-1 and Rank-(R1, R2, ..., RN ) Approximation of Higher-order Tensors".[5]

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[6] (CVPR 2001, ICPR 2002), face recognition – TensorFaces,[7][8] (ECCV 2002, CVPR 2003, etc.) and computer graphics – TensorTextures[9] (Siggraph 2004).

Historically, MPCA has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg in 1980.[3] In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA[10] terminology as a way to better differentiate between linear and multilinear tensor decomposition, as well as, to better differentiate between the work[6][7][8][9] that computed 2nd order statistics associated with each data tensor mode(axis), and subsequent work on Multilinear Independent Component Analysis[10] 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,[6][7][8][9] 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

MPCA computes a set of orthonormal matrices associated with each mode of the data tensor which are analogous to the orthonormal row and column space of a matrix computed by the matrix SVD. This transformation aims to capture as high a variance as possible, accounting for as much of the variability in the data associated with each data tensor mode(axis).

## 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

MPCA features: Supervised MPCA is employed in causal factor analysis that facilitates object recognition[11] while a semi-supervised MPCA feature selection is employed in visualization tasks.[12]

## Extensions

Various extension of MPCA:

• Robust MPCA (RMPCA)[13]
• Multi-Tensor Factorization, that also finds the number of components automatically (MTF)[14]

## References

1. ^ F. L. Hitchcock (1927). "The expression of a tensor or a polyadic as a sum of products". Journal of Mathematics and Physics. 6 (1–4): 164–189. doi:10.1002/sapm192761164.
2. ^ Tucker, Ledyard R (September 1966). "Some mathematical notes on three-mode factor analysis". Psychometrika. 31 (3): 279–311. doi:10.1007/BF02289464. PMID 5221127.
3. ^ a b P. M. Kroonenberg and J. de Leeuw, Principal component analysis of three-mode data by means of alternating least squares algorithms, Psychometrika, 45 (1980), pp. 69–97.
4. ^ Lathauwer, L.D.; Moor, B.D.; Vandewalle, J. (2000). "A multilinear singular value decomposition". SIAM Journal on Matrix Analysis and Applications. 21 (4): 1253–1278. doi:10.1137/s0895479896305696.
5. ^ Lathauwer, L. D.; Moor, B. D.; Vandewalle, J. (2000). "On the best rank-1 and rank-(R1, R2, ..., RN ) approximation of higher-order tensors". SIAM Journal on Matrix Analysis and Applications. 21 (4): 1324–1342. doi:10.1137/s0895479898346995.
6. ^ a b c
7. ^ a b c
8. ^ a b c
9. ^ a b c
10. ^ a b M. A. O. Vasilescu, D. Terzopoulos (2005) "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."
11. ^ M. A. O. Vasilescu, D. Terzopoulos (2003) "Multilinear Subspace Analysis of Image Ensembles", "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI, June, 2003"
12. ^ H. Lu, H.-L. Eng, M. Thida, and K.N. Plataniotis, "Visualization and Clustering of Crowd Video Content in MPCA Subspace," in Proceedings of the 19th ACM Conference on Information and Knowledge Management (CIKM 2010), Toronto, ON, Canada, October, 2010.
13. ^ K. Inoue, K. Hara, K. Urahama, "Robust multilinear principal component analysis", Proc. IEEE Conference on Computer Vision, 2009, pp. 591–597.
14. ^ Khan, Suleiman A.; Leppäaho, Eemeli; Kaski, Samuel (2016-06-10). "Bayesian multi-tensor factorization". Machine Learning. 105 (2): 233–253. arXiv:1412.4679. doi:10.1007/s10994-016-5563-y. ISSN 0885-6125.