Multilinear subspace learning

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 (MSL) aims to learn a specific small part of a large space of multidimensional objects having a particular desired property. ^{[1] [2] [3] [4]} It is a dimensionality reduction approach for finding a low-dimensional representation with certain preferred characteristics of high-dimensional tensor data through direct mapping, without going through vectorization.^[5][6] The term tensor in MSL refers to multidimensional arrays. Examples of tensor data include images (2D/3D), video sequences (3D/4D), and hyperspectral cubes (3D/4D). The mapping from a high-dimensional tensor space to a low-dimensional tensor space or vector space is named as multilinear projection.^[5][7]

MSL methods are higher-order generalizations of linear subspace learning methods such as principal component analysis (PCA), linear discriminant analysis (LDA) and canonical correlation analysis (CCA). In the literature, MSL is also referred to as tensor subspace learning or tensor subspace analysis.^[6] Research on MSL has progressed from heuristic exploration in 2000s (decade) to systematic investigation in 2010s.

Background

With the advances in data acquisition and storage technology, big data (or massive data sets) are being generated on a daily basis in a wide range of emerging applications. Most of these big data are multidimensional. Moreover, they are usually very-high-dimensional, with a large amount of redundancy, and only occupying a part of the input space. Therefore, dimensionality reduction is frequently employed to map high-dimensional data to a low-dimensional space while retaining as much information as possible.

Linear subspace learning algorithms are traditional dimensionality reduction techniques that represent input data as vectors and solve for an optimal linear mapping to a lower-dimensional space. Unfortunately, they often become inadequate when dealing with massive multidimensional data. They result in very-high-dimensional vectors, lead to the estimation of a large number of parameters, and also break the natural structure and correlation in the original data.^[5][6][8][9]

MSL is closely related to tensor decompositions.^[10] They both employ multilinear algebra tools. The difference is that tensor decomposition focuses on factor analysis, while MSL focuses on dimensionality reduction. MSL belongs to tensor-based computation^[11] and it can be seen as a tensor-level computational thinking of machine learning.

Multilinear projection

Multilinear projection to transform a tensor to a low-dimensional representation for multilinear subspace learning: elementary multilinear projection (EMP), tensor-to-vector projection (TVP), and tensor-to-tensor projection (TTP).

A multilinear subspace is defined through a multilinear projection that maps the (high-dimensional) input tensor space to a (lower-dimensional) vector/tensor space (the subspace). The original idea is due to Hitchcock in 1927.^[12]

Tensor-to-tensor projection (TTP)

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.^[13] This projection is an extension of the higher-order singular value decomposition^[13] (HOSVD) to subspace learning.^[8] Hence, its origin is traced back to the Tucker decomposition^[14] in 1960s.

Tensor-to-vector projection (TVP)

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,^[15] also known as the parallel factors (PARAFAC) decomposition.^[16]

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 in ^[17] 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.^[18]

Pros and cons

This figure compares the number of parameters to be estimated for the same amount of dimension reduction by vector-to-vector projection (VVP), (i.e., linear projection,) tensor-to-vector projection (TVP), and tensor-to-tensor projection (TTP). Multilinear projections require much fewer parameters and the representations obtained are more compact. (This figure is produced based on Table 3 of the survey paper ^[5])

The advantages of MSL are:^[5][6][8][9]

• It preserves the structure and correlation in the original data before projection by operating on natural tensorial representation of multidimensional data.
• It can learn more compact representations than its linear counterpart. It needs to estimate a much smaller number of parameters and it has fewer problems in the small sample size scenario.
• It can handle big tensor data more efficiently with computations in much lower dimensions than linear methods. Thus, it leads to lower demand on computational resources.

The disadvantages of MSL are:^[5][6][8][9]

• Most MSL algorithm are iterative. They may be affected by initialization method and have convergence problem.
• The solution obtained is local optimum.

Algorithms

• Multilinear extension of PCA
  • TTP-based: Multilinear Principal Component Analysis (MPCA) ^[8]
  • TVP-based: Uncorrelated Multilinear Principal Component Analysis (UMPCA) ^[19]
• Multilinear extension of LDA
  • TTP-based: Discriminant Analysis with Tensor Representation (DATER) ^[9]
  • TTP-based: General tensor discriminant analysis (GTDA) ^[20]
  • TVP-based: Uncorrelated Multilinear Discriminant Analysis (UMLDA) ^[21]
• Multilinear extension of CCA
  • TTP-based: Tensor Canonical Correlation Analysis (TCCA) ^[22]
  • TVP-based: Multilinear Canonical Correlation Analysis (MCCA) ^[23]

Pedagogical resources

• Survey: A survey of multilinear subspace learning for tensor data (open access version).
• Lecture: Video lecture on UMPCA at the 25th International Conference on Machine Learning (ICML 2008).

Code

• MATLAB Tensor Toolbox by Sandia National Laboratories.
• The MPCA algorithm written in Matlab (MPCA+LDA included).
• The UMPCA algorithm written in Matlab (data included).
• The UMLDA algorithm written in Matlab (data included).

Tensor data sets

• 3D gait data (third-order tensors): 128x88x20(21.2M); 64x44x20(9.9M); 32x22x10(3.2M);

See also

• CP decomposition
• Dimension reduction
• Multilinear algebra
• Tensor
• Tensor decomposition
• Tensor software
• Tucker decomposition

References

1. 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
2. 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."
3. 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"
4. 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.
5. 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. doi:10.1016/j.patcog.2011.01.004.
6. X. He, D. Cai, P. Niyogi, Tensor subspace analysis, in: Advances in Neural Information Processing Systemsc 18 (NIPS), 2005.
7. Vasilescu, M.A.O.; Terzopoulos, D. (2007). Multilinear Projection for Appearance-Based Recognition in the Tensor Framework. IEEE 11th International Conference on Computer Visioncc. pp. 1–8. doi:10.1109/ICCV.2007.4409067.
8. 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.
9. 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.
10. T. G. Kolda, B. W. Bader, Tensor decompositions and applications, SIAM Review 51 (3) (2009) 455–500.
11. "Future Directions in Tensor-Based Computation and Modeling" (PDF). May 2009.
12. F. L. Hitchcock (1927). "The expression of a tensor or a polyadic as a sum of products". Journal of Mathematics and Physics 6: 164–189.
13. 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
14. Ledyard R Tucker (September 1966). "Some mathematical notes on three-mode factor analysis". Psychometrika 31 (3): 279–311. doi:10.1007/BF02289464.
15. J. D. Carroll & J. Chang (1970). "Analysis of individual differences in multidimensional scaling via an n-way generalization of 'Eckart–Young' decomposition". Psychometrika 35: 283–319. doi:10.1007/BF02310791.
16. R. A. Harshman, Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis. UCLA Working Papers in Phonetics, 16, pp. 1-84, 1970.
17. 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.
18. 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.
19. H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "Uncorrelated multilinear principal component analysis for unsupervised multilinear subspace learning," IEEE Trans. Neural Netw., vol. 20, no. 11, pp. 1820–1836, November 2009.
20. 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.
21. 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.
22. 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.
23. 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.