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Multilinear subspace learning
Approach to dimensionality reduction

Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction. The Dimensionality reduction can be performed on a data tensor that contains a collection of observations that have been vectorized, or observations that are treated as matrices and concatenated into a data tensor. Here are some examples of data tensors whose observations are vectorized or whose observations are matrices concatenated into data tensor images (2D/3D), video sequences (3D/4D), and hyperspectral cubes (3D/4D).

The mapping from a high-dimensional vector space to a set of lower dimensional vector spaces is a multilinear projection. When observations are retained in the same organizational structure as matrices or higher order tensors, their representations are computed by performing linear projections into the column space, row space and fiber space.

Multilinear subspace learning algorithms are higher-order generalizations of linear subspace learning methods such as principal component analysis (PCA), independent component analysis (ICA), linear discriminant analysis (LDA) and canonical correlation analysis (CCA).

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Background

Multilinear methods may be causal in nature and perform causal inference, or they may be simple regression methods from which no causal conclusion are drawn.

Linear subspace learning algorithms are traditional dimensionality reduction techniques that are well suited for datasets that are the result of varying a single causal factor. Unfortunately, they often become inadequate when dealing with datasets that are the result of multiple causal factors. .

Multilinear subspace learning can be applied to observations whose measurements were vectorized and organized into a data tensor for causally aware dimensionality reduction.11 These methods may also be employed in reducing horizontal and vertical redundancies irrespective of the causal factors when the observations are treated as a "matrix" (ie. a collection of independent column/row observations) and concatenated into a tensor.1213

Algorithms

Multilinear principal component analysis

Historically, multilinear principal component analysis has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg.14 In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA15 terminology as a way to better differentiate between multilinear tensor decompositions that computed 2nd order statistics associated with each data tensor mode,1617181920 and subsequent work on Multilinear Independent Component Analysis21 that computed higher order statistics for each tensor mode. MPCA is an extension of PCA.

Multilinear independent component analysis

Multilinear independent component analysis22 is an extension of ICA.

Multilinear linear discriminant analysis

  • Multilinear extension of LDA
    • TTP-based: Discriminant Analysis with Tensor Representation (DATER)23
    • TTP-based: General tensor discriminant analysis (GTDA)24
    • TVP-based: Uncorrelated Multilinear Discriminant Analysis (UMLDA)25

Multilinear canonical correlation analysis

  • Multilinear extension of CCA
    • TTP-based: Tensor Canonical Correlation Analysis (TCCA)26
    • TVP-based: Multilinear Canonical Correlation Analysis (MCCA)27
    • TVP-based: Bayesian Multilinear Canonical Correlation Analysis (BMTF)28
  • A TTP is a direct projection of a high-dimensional tensor to a low-dimensional tensor of the same order, using N projection matrices for an Nth-order tensor. It can be performed in N steps with each step performing a tensor-matrix multiplication (product). The N steps are exchangeable.29 This projection is an extension of the higher-order singular value decomposition30 (HOSVD) to subspace learning.31 Hence, its origin is traced back to the Tucker decomposition32 in 1960s.
  • A TVP is a direct projection of a high-dimensional tensor to a low-dimensional vector, which is also referred to as the rank-one projections. As TVP projects a tensor to a vector, it can be viewed as multiple projections from a tensor to a scalar. Thus, the TVP of a tensor to a P-dimensional vector consists of P projections from the tensor to a scalar. The projection from a tensor to a scalar is an elementary multilinear projection (EMP). In EMP, a tensor is projected to a point through N unit projection vectors. It is the projection of a tensor on a single line (resulting a scalar), with one projection vector in each mode. Thus, the TVP of a tensor object to a vector in a P-dimensional vector space consists of P EMPs. This projection is an extension of the canonical decomposition,33 also known as the parallel factors (PARAFAC) decomposition.34

Typical approach in MSL

There are N sets of parameters to be solved, one in each mode. The solution to one set often depends on the other sets (except when N=1, the linear case). Therefore, the suboptimal iterative procedure in35 is followed.

  1. Initialization of the projections in each mode
  2. For each mode, fixing the projection in all the other mode, and solve for the projection in the current mode.
  3. Do the mode-wise optimization for a few iterations or until convergence.

This is originated from the alternating least square method for multi-way data analysis.36

Code

Tensor data sets

See also

References

  1. 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" http://www.cs.toronto.edu/~maov/tensorfaces/cvpr03.pdf ↩

  2. M. A. O. Vasilescu, D. Terzopoulos (2002) "Multilinear Analysis of Image Ensembles: TensorFaces", Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002 http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf ↩

  3. M. A. O. Vasilescu,(2002) "Human Motion Signatures: Analysis, Synthesis, Recognition", "Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460." http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf ↩

  4. Vasilescu, M.A.O.; Terzopoulos, D. (2007). Multilinear Projection for Appearance-Based Recognition in the Tensor Framework. IEEE 11th International Conference on Computer Vision. pp. 1–8. doi:10.1109/ICCV.2007.4409067.. /wiki/International_Conference_on_Computer_Vision ↩

  5. Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. (2013). Multilinear Subspace Learning: Dimensionality Reduction of Multidimensional Data. Chapman & Hall/CRC Press Machine Learning and Pattern Recognition Series. Taylor and Francis. ISBN 978-1-4398572-4-3. 978-1-4398572-4-3 ↩

  6. 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" http://www.cs.toronto.edu/~maov/tensorfaces/cvpr03.pdf ↩

  7. Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. (2011). "A Survey of Multilinear Subspace Learning for Tensor Data" (PDF). Pattern Recognition. 44 (7): 1540–1551. Bibcode:2011PatRe..44.1540L. doi:10.1016/j.patcog.2011.01.004. http://www.dsp.utoronto.ca/~haiping/Publication/SurveyMSL_PR2011.pdf ↩

  8. X. He, D. Cai, P. Niyogi, Tensor subspace analysis, in: Advances in Neural Information Processing Systemsc 18 (NIPS), 2005. http://books.nips.cc/papers/files/nips18/NIPS2005_0249.pdf ↩

  9. Vasilescu, M.A.O.; Terzopoulos, D. (2007). Multilinear Projection for Appearance-Based Recognition in the Tensor Framework. IEEE 11th International Conference on Computer Vision. pp. 1–8. doi:10.1109/ICCV.2007.4409067.. /wiki/International_Conference_on_Computer_Vision ↩

  10. Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. (2011). "A Survey of Multilinear Subspace Learning for Tensor Data" (PDF). Pattern Recognition. 44 (7): 1540–1551. Bibcode:2011PatRe..44.1540L. doi:10.1016/j.patcog.2011.01.004. http://www.dsp.utoronto.ca/~haiping/Publication/SurveyMSL_PR2011.pdf ↩

  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" http://www.cs.toronto.edu/~maov/tensorfaces/cvpr03.pdf ↩

  12. "Future Directions in Tensor-Based Computation and Modeling" (PDF). May 2009. http://www.cs.cornell.edu/cv/tenwork/finalreport.pdf ↩

  13. S. Yan, D. Xu, Q. Yang, L. Zhang, X. Tang, and H.-J. Zhang, "Discriminant analysis with tensor representation," in Proc. IEEE Conference on Computer Vision and Pattern Recognition, vol. I, June 2005, pp. 526–532. http://portal.acm.org/citation.cfm?id=1068959 ↩

  14. 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. https://doi.org/10.1007%2FBF02293599 ↩

  15. 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." http://www.media.mit.edu/~maov/mica/mica05.pdf ↩

  16. 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" http://www.cs.toronto.edu/~maov/tensorfaces/cvpr03.pdf ↩

  17. M. A. O. Vasilescu, D. Terzopoulos (2002) "Multilinear Analysis of Image Ensembles: TensorFaces", Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002 http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf ↩

  18. M. A. O. Vasilescu,(2002) "Human Motion Signatures: Analysis, Synthesis, Recognition", "Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460." http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf ↩

  19. M.A.O. Vasilescu, D. Terzopoulos (2004) "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. http://www.media.mit.edu/~maov/tensortextures/Vasilescu_siggraph04.pdf ↩

  20. H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "MPCA: Multilinear principal component analysis of tensor objects," IEEE Trans. Neural Netw., vol. 19, no. 1, pp. 18–39, January 2008. https://dx.doi.org/10.1109/TNN.2007.901277 ↩

  21. 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." http://www.media.mit.edu/~maov/mica/mica05.pdf ↩

  22. 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." http://www.media.mit.edu/~maov/mica/mica05.pdf ↩

  23. S. Yan, D. Xu, Q. Yang, L. Zhang, X. Tang, and H.-J. Zhang, "Discriminant analysis with tensor representation," in Proc. IEEE Conference on Computer Vision and Pattern Recognition, vol. I, June 2005, pp. 526–532. http://portal.acm.org/citation.cfm?id=1068959 ↩

  24. D. Tao, X. Li, X. Wu, and S. J. Maybank, "General tensor discriminant analysis and gabor features for gait recognition," IEEE Trans. Pattern Anal. Mach. Intell., vol. 29, no. 10, pp. 1700–1715, October 2007. https://dx.doi.org/10.1109/TPAMI.2007.1096 ↩

  25. H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "Uncorrelated multilinear discriminant analysis with regularization and aggregation for tensor object recognition," IEEE Trans. Neural Netw., vol. 20, no. 1, pp. 103–123, January 2009. https://dx.doi.org/10.1109/TNN.2008.2004625 ↩

  26. T.-K. Kim and R. Cipolla. "Canonical correlation analysis of video volume tensors for action categorization and detection," IEEE Trans. Pattern Anal. Mach. Intell., vol. 31, no. 8, pp. 1415–1428, 2009. https://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4547427 ↩

  27. H. Lu, "Learning Canonical Correlations of Paired Tensor Sets via Tensor-to-Vector Projection," Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI 2013), Beijing, China, August 3–9, 2013. http://www.dsp.utoronto.ca/~haiping/Publication/MCCA_IJCAI2013.pdf ↩

  28. Khan, Suleiman A.; Kaski, Samuel (2014-09-15). "Bayesian Multi-view Tensor Factorization". In Calders, Toon; Esposito, Floriana; Hüllermeier, Eyke; Meo, Rosa (eds.). Machine Learning and Knowledge Discovery in Databases. Lecture Notes in Computer Science. Vol. 8724. Springer Berlin Heidelberg. pp. 656–671. doi:10.1007/978-3-662-44848-9_42. ISBN 9783662448472. 9783662448472 ↩

  29. L.D. Lathauwer, B.D. Moor, J. Vandewalle, A multilinear singular value decomposition, SIAM Journal of Matrix Analysis and Applications vol. 21, no. 4, pp. 1253–1278, 2000 http://portal.acm.org/citation.cfm?id=354398 ↩

  30. L.D. Lathauwer, B.D. Moor, J. Vandewalle, A multilinear singular value decomposition, SIAM Journal of Matrix Analysis and Applications vol. 21, no. 4, pp. 1253–1278, 2000 http://portal.acm.org/citation.cfm?id=354398 ↩

  31. H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "MPCA: Multilinear principal component analysis of tensor objects," IEEE Trans. Neural Netw., vol. 19, no. 1, pp. 18–39, January 2008. https://dx.doi.org/10.1109/TNN.2007.901277 ↩

  32. Ledyard R Tucker (September 1966). "Some mathematical notes on three-mode factor analysis". Psychometrika. 31 (3): 279–311. doi:10.1007/BF02289464. PMID 5221127. S2CID 44301099. /wiki/Ledyard_R_Tucker ↩

  33. J. D. Carroll & J. Chang (1970). "Analysis of individual differences in multidimensional scaling via an n-way generalization of 'Eckart–Young' decomposition". Psychometrika. 35 (3): 283–319. doi:10.1007/BF02310791. S2CID 50364581. /wiki/Psychometrika ↩

  34. R. A. Harshman, Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis Archived 2004-10-10 at the Wayback Machine. UCLA Working Papers in Phonetics, 16, pp. 1–84, 1970. http://publish.uwo.ca/~harshman/wpppfac0.pdf ↩

  35. L. D. Lathauwer, B. D. Moor, J. Vandewalle, On the best rank-1 and rank-(R1, R2, ..., RN ) approximation of higher-order tensors, SIAM Journal of Matrix Analysis and Applications 21 (4) (2000) 1324–1342. http://portal.acm.org/citation.cfm?id=354405 ↩

  36. 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. https://doi.org/10.1007%2FBF02293599 ↩