A video or an image sequence represented as a third-order tensor of column x row x time for multilinear subspace learning.
Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction.[1][2][3][4][5]
The Dimensionality reduction can be performed on a data tensor that contains a collection of observations have been vectorized,[1] or observations that are treated as matrices and concatenated into a data tensor.[6][7] 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.[4] 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.[6]
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).
^ abM. 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"
^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
^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."
^ abVasilescu, 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..
^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.
^ abLu, 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.
^X. He, D. Cai, P. Niyogi, Tensor subspace analysis, in: Advances in Neural Information Processing Systemsc 18 (NIPS), 2005.
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