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. 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^[1][2]. 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^[1][3].

MSL methods are higher-order generalizations of linear subspace learning methods such as principal component analysis (PCA) and linear discriminant analysis (LDA). In the literature, MSL is also referred to as tensor subspace learning or tensor subspace analysis^[2]. Research on MSL has progressed from heuristic exploration in 2000s to systematic investigation in 2010s.

Background

With the advances in data acquisition and storage technology, massive multidimensional data are being generated on a daily basis in a wide range of emerging applications. These massive, multidimensional data 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.^[1][2][4][5]

MSL is closely related to tensor decompositions^[6]. They both employ multilinear algebra tools. The difference is that tensor decomposition focuses on factor analysis, while MSL focuses on dimensionality reduction.

Multilinear projection

A multilinear subspace is defined through a multilinear projection that maps the input tensor data from one space to another (lower-dimensional) space. The original idea is due to Hitchcock in 1927.^[7]

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.^[8] This projection is an extension of the higher-order singular value decomposition^[8] (HOSVD) to subspace learning^[4]. Hence, its origin is traced back to the Tucker decomposition^[9] 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^[10], also known as the parallel factors (PARAFAC) decomposition ^[11].

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 ^[12] 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 ^[13].

Pros and cons

The advantages of MSL are^[1][2][4][5]:

• It operates on natural tensorial representation of multidimensional data so the structure and correlation in the original data are preserved.
• The number of parameters to be estimated is much smaller than in the linear case.
• It has fewer problems in the small sample size scenario.

The disadvantages of MSL are^[1][2][4][5]:

• 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
  • Multilinear Principal Component Analysis (MPCA) ^[4]
  • Uncorrelated Multilinear Principal Component Analysis (UMPCA) ^[14]
• Multilinear extension of LDA
  • Discriminant Analysis with Tensor Representation (DATER) ^[5]
  • General tensor discriminant analysis (GTDA) ^[15]
  • Uncorrelated Multilinear Discriminant Analysis (UMLDA) ^[16]

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).

Tensor data sets

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

See also

• CP decomposition

References

1. ^ ^{a b c d e} Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. (2011). "A Survey of Multilinear Subspace Learning for Tensor Data". Pattern Recognition 44 (7): 1540–1551. doi:10.1016/j.patcog.2011.01.004. http://www.dsp.utoronto.ca/~haiping/Publication/SurveyMSL_PR2011.pdf. 
2. ^ ^{a b c d e} X. He, D. Cai, P. Niyogi, Tensor subspace analysis, in: Advances in Neural Information Processing Systemsc 18 (NIPS), 2005.
3. ^ 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. 
4. ^ ^{a b c d e} 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, Jan. 2008.
5. ^ ^{a b c d} 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.
6. ^ T. G. Kolda, B. W. Bader, Tensor decompositions and applications, SIAM Review 51 (3) (2009) 455–500.
7. ^ 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. 
8. ^ ^{a b} 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
9. ^ Ledyard R. Tucker (September 1966). "Some mathematical notes on three-mode factor analysis". Psychometrika 31 (3): 279–311. doi:10.1007/BF02289464. 
10. ^ 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. 
11. ^ 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.
12. ^ 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.
13. ^ 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.
14. ^ 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, Nov. 2009.
15. ^ 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, Oct. 2007.
16. ^ 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, Jan. 2009.