Multilinear subspace learning is an approach to dimensionality reduction.
[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"[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."
Dimensionality reduction
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally ...
can be performed on a data
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
that contains a collection of observations have been vectorized,
or observations that are treated as matrices and concatenated into a data tensor.
[X. He, D. Cai, P. Niyogi]
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
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
s (2D/3D),
video
Video is an electronic medium for the recording, copying, playback, broadcasting, and display of moving visual media. Video was first developed for mechanical television systems, which were quickly replaced by cathode-ray tube (CRT) sy ...
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
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
learning methods such as
principal component analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
(PCA),
independent component analysis
In signal processing, independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents. This is done by assuming that at most one subcomponent is Gaussian and that the subcomponents ar ...
(ICA),
linear discriminant analysis
Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features ...
(LDA) and
canonical correlation analysis
In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors ''X'' = (''X''1, ..., ''X'n'') and ''Y' ...
(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
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
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.
[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.
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]
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 PCA
[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." 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]
"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.
/ref>[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.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)
**TVP-based: Uncorrelated Multilinear Discriminant Analysis (UMLDA)
Multilinear canonical correlation analysis
*Multilinear extension of CCA
**TTP-based: Tensor Canonical Correlation Analysis (TCCA)
**TVP-based: Multilinear Canonical Correlation Analysis (MCCA)
**TVP-based: Bayesian Multilinear Canonical Correlation Analysis (BMTF)
*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 ''N''th-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,]
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
In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one generalization of the matrix singular value decomposition. It has applications in ...
(HOSVD) to subspace learning. Hence, its origin is traced back to the Tucker decomposition
In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker
although it goes back to Hitchcock in 1927.
Initially described as a three-mode extension of factor an ...
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, also known as the parallel factors (PARAFAC) decomposition.
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[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. 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
MATLAB Tensor Toolbox
by Sandia National Laboratories
Sandia National Laboratories (SNL), also known as Sandia, is one of three research and development laboratories of the United States Department of Energy's National Nuclear Security Administration (NNSA). Headquartered in Kirtland Air Force Bas ...
.
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
* Multilinear Principal Component Analysis
*Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
*Tensor decomposition
In multilinear algebra, a tensor decomposition is any scheme for expressing a "data tensor" (M-way array) as a sequence of elementary operations acting on other, often simpler tensors. Many tensor decompositions generalize some matrix decompositi ...
*Tensor software
Tensor software is a class of mathematical software designed for manipulation and calculation with tensors.
Standalone software
* SPLATT is an open source software package for high-performance sparse tensor factorization. SPLATT ships a stand-alo ...
*Tucker decomposition
In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker
although it goes back to Hitchcock in 1927.
Initially described as a three-mode extension of factor an ...
References
{{Reflist, 2
Dimension reduction
Multilinear algebra
Tensors