multilinear subspace learning

File:Video represented as a third-order tensor.jpg

Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction.M. A. O. Vasilescu, D. Terzopoulos (2003) [http://www.cs.toronto.edu/~maov/tensorfaces/cvpr03.pdf "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) [http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf "Multilinear Analysis of Image Ensembles: TensorFaces"], Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002M. 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."{{cite book

|first1=Haiping |last1=Lu

|first2=K.N. |last2=Plataniotis

|first3=A.N. |last3=Venetsanopoulos

|url=http://www.crcpress.com/product/isbn/9781439857243

|title=Multilinear Subspace Learning: Dimensionality Reduction of Multidimensional Data

|series=Chapman & Hall/CRC Press Machine Learning and Pattern Recognition Series

|publisher=Taylor and Francis

|isbn=978-1-4398572-4-3

|year=2013

}}

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.{{cite journal

|first1=Haiping |last1=Lu

|first2=K.N. |last2=Plataniotis

|first3=A.N. |last3=Venetsanopoulos

|url=http://www.dsp.utoronto.ca/~haiping/Publication/SurveyMSL_PR2011.pdf

|title=A Survey of Multilinear Subspace Learning for Tensor Data

|journal=Pattern Recognition

|volume=44 |number=7 |pages=1540–1551 |year=2011

|doi=10.1016/j.patcog.2011.01.004

|bibcode=2011PatRe..44.1540L

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

|first1=M.A.O. |last1=Vasilescu

|first2=D. |last2=Terzopoulos

|title=Multilinear Projection for Appearance-Based Recognition in the Tensor Framework

|conference=IEEE 11th International Conference on Computer Vision

|pages=1–8 |year=2007

|doi=10.1109/ICCV.2007.4409067

}}.

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).

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. 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.{{cite web |title=Future Directions in Tensor-Based Computation and Modeling |date=May 2009|url=http://www.cs.cornell.edu/cv/tenwork/finalreport.pdf}}S. Yan, D. Xu, Q. Yang, L. Zhang, X. Tang, and H.-J. Zhang, "[http://portal.acm.org/citation.cfm?id=1068959 Discriminant analysis with tensor representation]," in Proc. IEEE Conference on Computer Vision and Pattern Recognition, vol. I, June 2005, pp. 526–532.

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.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.

In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCAM. 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." terminology as a way to better differentiate between multilinear tensor decompositions that computed 2nd order statistics associated with each data tensor mode,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. ]H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "[https://dx.doi.org/10.1109/TNN.2007.901277 MPCA: Multilinear principal component analysis of tensor objects]," IEEE Trans. Neural Netw., vol. 19, no. 1, pp. 18–39, January 2008. and subsequent work on Multilinear Independent Component Analysis that computed higher order statistics for each tensor mode. MPCA is an extension of PCA.

= Multilinear independent component analysis =

Multilinear independent component analysis is an extension of ICA.

= Multilinear linear discriminant analysis =

Multilinear extension of LDA
TTP-based: Discriminant Analysis with Tensor Representation (DATER)
TTP-based: General tensor discriminant analysis (GTDA)D. Tao, X. Li, X. Wu, and S. J. Maybank, "[https://dx.doi.org/10.1109/TPAMI.2007.1096 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.
TVP-based: Uncorrelated Multilinear Discriminant Analysis (UMLDA)H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "[https://dx.doi.org/10.1109/TNN.2008.2004625 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.

= Multilinear canonical correlation analysis =

Multilinear extension of CCA
TTP-based: Tensor Canonical Correlation Analysis (TCCA)

T.-K. Kim and R. Cipolla. "[https://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4547427 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.

TVP-based: Multilinear Canonical Correlation Analysis (MCCA)H. Lu, "[http://www.dsp.utoronto.ca/~haiping/Publication/MCCA_IJCAI2013.pdf 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.
TVP-based: Bayesian Multilinear Canonical Correlation Analysis (BMTF){{Cite book|title=Machine Learning and Knowledge Discovery in Databases|last1=Khan|first1=Suleiman A.|last2=Kaski|first2=Samuel|chapter=Bayesian Multi-view Tensor Factorization |date=2014-09-15|publisher=Springer Berlin Heidelberg|isbn=9783662448472|editor-last=Calders|editor-first=Toon|series=Lecture Notes in Computer Science|volume=8724 |pages=656–671|language=en|doi=10.1007/978-3-662-44848-9_42|editor-last2=Esposito|editor-first2=Floriana|editor2-link=Floriana Esposito|editor-last3=Hüllermeier|editor-first3=Eyke|editor-last4=Meo|editor-first4=Rosa}}
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.L.D. Lathauwer, B.D. Moor, J. Vandewalle, [http://portal.acm.org/citation.cfm?id=354398 A multilinear singular value decomposition], SIAM Journal of Matrix Analysis and Applications vol. 21, no. 4, pp. 1253–1278, 2000 This projection is an extension of the higher-order singular value decomposition (HOSVD) to subspace learning. Hence, its origin is traced back to the Tucker decomposition{{Cite journal

| author = Ledyard R Tucker

| title = Some mathematical notes on three-mode factor analysis

| journal = Psychometrika

| volume = 31

| issue = 3

|date=September 1966

| doi = 10.1007/BF02289464

| pages = 279–311

| pmid = 5221127

| s2cid = 44301099

| author-link = Ledyard R Tucker

}} 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,{{Cite journal

| author = J. D. Carroll & J. Chang

| title = Analysis of individual differences in multidimensional scaling via an n-way generalization of 'Eckart–Young' decomposition

| journal = Psychometrika

| volume = 35

| issue = 3

| pages = 283–319

| year = 1970

| doi = 10.1007/BF02310791

| s2cid = 50364581

}} also known as the parallel factors (PARAFAC) decomposition.R. A. Harshman, [http://publish.uwo.ca/~harshman/wpppfac0.pdf Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis] {{webarchive|url=https://web.archive.org/web/20041010092429/http://publish.uwo.ca/~harshman/wpppfac0.pdf |date=2004-10-10 }}. UCLA Working Papers in Phonetics, 16, pp. 1–84, 1970.

= 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 inL. D. Lathauwer, B. D. Moor, J. Vandewalle, [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) (2000) 1324–1342. is followed.

Initialization of the projections in each mode
For each mode, fixing the projection in all the other mode, and solve for the projection in the current mode.
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.

Code

[https://web.archive.org/web/20110717172720/http://csmr.ca.sandia.gov/~tgkolda/TensorToolbox/ MATLAB Tensor Toolbox] by Sandia National Laboratories.
[http://www.mathworks.com/matlabcentral/fileexchange/26168 The MPCA algorithm written in Matlab (MPCA+LDA included)].
[http://www.mathworks.com/matlabcentral/fileexchange/35432 The UMPCA algorithm written in Matlab (data included)].
[http://www.mathworks.fr/matlabcentral/fileexchange/35782 The UMLDA algorithm written in Matlab (data included)].

Tensor data sets

3D gait data (third-order tensors): [http://www.dsp.utoronto.ca/~haiping/CodeData/USFGait17_128x88x20.zip 128x88x20(21.2M)]; [http://www.dsp.utoronto.ca/~haiping/CodeData/USFGait17_64x44x20.zip 64x44x20(9.9M)]; [http://www.dsp.utoronto.ca/~haiping/CodeData/USFGait17_32x22x10.zip 32x22x10(3.2M)];