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Is your business frequently losing out to competitors?<br><br>
In [[multilinear algebra]], there does not exist a general decomposition method for multi-way arrays (also known as ''N-arrays'', ''higher-order arrays'', or ''data-tensors'') with all the properties of a matrix [[singular value decomposition]] (SVD).  A matrix SVD simultaneously computes


:(a) a rank-''R'' decomposition and
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:(b) the orthonormal row/column matrices.
 
These two properties can be captured separately by two different decompositions for multi-way [[matrix (mathematics)|arrays]].
 
Property (a) is extended to higher order by a class of closely related constructions known collectively as [[CP decomposition]] (named after the two most popular and general variants, CANDECOMP and PARAFAC). Such decompositions represent a tensor as the sum of the n-fold outer products of rank-1 tensors, where n is the dimension of the tensor indices. 
 
Property (b) is extended to higher order by a class of methods known variably as ''[[Tucker decomposition|Tucker3]]'', ''N-mode SVD'', and ''N-mode [[principal component analysis]]'' (PCA).  (This article will use the general term "Tucker decomposition".)  These methods compute the othonormal spaces associated with the different axes (or modes) of a tensor.  The Tucker decomposition is also used in [[multilinear subspace learning]] as [[multilinear principal component analysis]]. This terminology was coined by P. Kroonenberg in the 1980s, but it was later called ''multilinear SVD'' and ''[[HOSVD]]'' (higher-order SVD) by L. De Lathauwer.
 
Historically, much of the interest in higher-order SVDs was driven by the need to analyze empirical data, especial in [[psychometrics]] and [[chemometrics]].   As such, many of the methods have been independently invented several times, often with subtle variations, leading to a confusing literature. Abstract and general mathematical theorems are rare (though see Kruskal<ref>Kruskal, J. B. (1989). "Rank, decomposition, and uniqueness for 3-way and N-way arrays". In R. Coppi & S. Bolasco (Eds.), ''Multiway data analysis'' (pp. 7–18). Amsterdam: Elsevier.  [ [http://publish.uwo.ca/~harshman/jbkrank.pdf PDF] ].</ref> with regard to the CP decomposition); instead, the methods are often designed for analyzing specific data types.  The 2008 review article by Kolda and Bader<ref>{{cite journal | first1 = Tamara G. | last1 = Kolda | first2 = Brett W. | last2 = Bader | url = http://epubs.siam.org/sirev/resource/1/siread/v51/i3/p455_s1 | title = Tensor Decompositions and Applications | journal = SIAM Rev. | volume = 51 | pages = 455–500 (46 pages) | id = {{citeseerx|10.1.1.153.2059}} | doi=10.1137/07070111X}}</ref> provides a compact summary of the history of these decompositions, and many references for further reading.
 
The concept of HOSVD was carried over to functions by Baranyi and Yam  via the [[TP model transformation]] <ref name=Baranyi04>{{cite journal
|author =      P. Baranyi
|title =        TP model transformation as a way to LMI based controller design
|journal =      IEEE Transaction on Industrial Electronics
|year =        2004
|volume =      51
|number =      2
|month =        April
|pages =        387&ndash;400
}}</ref>
.<ref name=compind>{{cite journal
|author =      P. Baranyi and D. Tikk and Y. Yam and R. J. Patton
|title =        From Differential Equations to PDC Controller Design via Numerical Transformation
|journal =      Computers in Industry, Elsevier Science
|year =        2003
|volume =      51
|pages =        281&ndash;297
}}</ref> This extension led to the definition of the HOSVD based canonical form of tensor product functions and Linear Parameter Varying system models <ref name=canon1>{{cite book
|author =      P. Baranyi and L. Szeidl and P. Várlaki and Y. Yam
|title =        Definition of the HOSVD-based canonical form of polytopic dynamic models
|booktitle =    3rd International Conference on Mechatronics (ICM 2006)
|year =        2006
|pages =        660–665
|location =      Budapest, Hungary
|month =        July 3–5
}}</ref> and to convex hull manipulation based control optimization theory, see [[TP model transformation in control theories]].
 
== CP decomposition ==
{{Main|CP decomposition}}
 
===Definition===
A CP decomposition of an N-way array ''X'', with elements <math>x_{i_1 \cdots i_N}</math>, is
 
:<math>X = \sum_{r=1}^{R} D^{(r)} = \sum_{r=1}^{R} a^{(r)} \otimes \cdots \otimes z^{(r)}</math>
 
where <math>\otimes</math> denotes the [[tensor product]]. The ''R'' tensors <math>D^{(r)}</math> (known as ''simple tensors'', ''rank-1 tensors'', ''dyads'', or, in [[quantum mechanics]], ''product states'') are constructed from the ''rN'' vectors <math>a^{(r)}, \cdots, z^{(r)}</math>. With indices, this is
 
:<math>x_{i_1 \cdots i_N} = \sum_{r=1}^{R} a^{(r)}_{i_1} \cdots z^{(r)}_{i_N}</math>
 
where <math>a^{(r)}_{i}</math> is the ''i''-th element of the vector <math>a^{(r)}</math>, etc.
 
== Tucker decomposition ==
{{Main|Tucker decomposition}}
 
=== History ===
In 1966, [[L. Tucker]] proposed a decomposition method for three-way arrays (referred to as a 3-mode "[[tensor]]s") as a multidimensional extension of [[factor analysis]].<ref>{{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
}}</ref> This decomposition was further developed in the 1980s by P. Kroonenberg, who coined the terms Tucker3, Tucker3ALS (an alternating least squares dimensionality reduction algorithm), 3-Mode SVD, and 3-Mode PCA.<ref>{{Cite journal
| author = P. M. Kroonenberg
| title = Three-mode principal component analysis: Theory and applications
| journal = DSWO Press, Leiden
| year = 1983
| url = http://three-mode.leidenuniv.nl/bibliogr/kroonenbergpm_thesis/index.html
}}</ref> In the intervening years, several authors developed the decomposition for ''N''-way arrays.
Most recently, this work was treated in an elegant fashion and introduced to the SIAM community by L. De Lathauwer et al. who referred to the decomposition as an ''N''-way SVD, multilinear SVD and HOSVD.<ref name="Lieven De Lathauwer, Bart De Moor and Joos Vandewalle 2000">{{Cite journal
| author = Lieven De Lathauwer, Bart De Moor and Joos Vandewalle
| title = A multilinear Singular Value Decomposition
| journal = [[Society for Industrial and Applied Mathematics|SIAM]]
| volume = 21
| issue = 4
|date=April 2000
| url = http://perso-etis.ensea.fr/~lathauwer/ldl-94-31.pdf
}}</ref>
 
=== Definitions ===
Let the SVD of a real matrix be <math>A = U S V^T</math>, then it can be written in an elementwise form as
 
:<math>a_{i_1,i_2} = \sum_{j_1} \sum_{j_2} s_{j_1,j_2} u_{i_1,j_1} v_{i_2,j_2}.</math>
 
<math>U</math> and <math>V</math> give, in a certain sense optimal, orthonormal basis for the column and row space, <math>S</math> is diagonal with decreasing elements.
''N''-mode SVD can be defined by the multidimensional generalization of this concept:
 
:<math>a_{i_1,i_2,\dots,i_N} = \sum_{j_1} \sum_{j_2}\cdots \sum_{j_N} s_{j_1,j_2,\dots,j_N} u^{(1)}_{i_1,j_1} u^{(2)}_{i_2,j_2} \dots u^{(N)}_{i_N,j_N},</math>
 
where the <math>U^{(n)} = [u^{(n)}_{i,j}]_{I_n \times I_n}</math> matrices and the <math>\mathcal{S} = [s_{j_1,\dots,j_N}]_{I_1 \times I_2 \times \cdots \times I_N}</math> core tensor should satisfy certain requirements (similar ones to the matrix SVD), namely
 
* Each <math>U^{(n)}</math> is an [[orthogonal matrix]] (called ''n''-mode singular matrix).
* Two subtensors of the core tensor <math>\mathcal{S}</math> are orthogonal i.e., <math>\langle\mathcal{S}_{i_n = p}, \mathcal{S}_{i_n = q}\rangle = 0</math> if <math>p \neq q</math>.
* The subtensors in the core tensor <math>\mathcal{S}</math> are ordered according to their [[Frobenius norm]], i.e. <math>\|\mathcal{S}_{i_n = 1}\| \geq  \|\mathcal{S}_{i_n = 2}\| \geq \dots \geq \|\mathcal{S}_{i_n = I_n}\|</math> for ''n''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''N''.
 
Notation:
:<math>\mathcal{A} = \mathcal{S} \times_{n=1}^N U^{(n)}</math>
 
=== Algorithm ===
 
The HOSVD can be built from several SVDs, as follows:<ref name="Lieven De Lathauwer, Bart De Moor and Joos Vandewalle 2000"/>
 
Given a tensor, ''T'' of shape <math>d_1\times d_2\times\cdots\times d_n</math>, the ''i''<sup>th</sup> singular tensor is given by making a matrix,
: <math>T_{(i)}</math>, the <math>d_i\times (\prod _{j \neq i} d_j)</math> matrix corresponding to <math>T</math>.
taking its left singular vectors, <math>U_{(i)}</math>.
 
The core tensor is then the tensor product of those <math>U_{(i)}</math>s.
 
== Applications ==
Main applications are extracting relevant information from multi-way arrays. Used in factor analysis, face recognition ([[TensorFaces]]), human motion analysis and synthesis.  
 
The HOSVD has been successfully applied to signal processing and big data, e.g., in genomic signal processing.<ref>{{Cite journal
| author = L. Omberg, G. H. Golub and O. Alter
| title = A Tensor Higher-Order Singular Value Decomposition for Integrative Analysis of DNA Microarray Data From Different Studies
  | journal = PNAS
| volume = 104
| issue = 47
| pages = 18371–18376
| date = November 2007
| doi = 10.1073/pnas.0709146104
| url = http://dx.doi.org/10.1073/pnas.0709146104
}}</ref><ref>{{Cite journal
| author = L. Omberg, J. R. Meyerson, K. Kobayashi, L. S. Drury, J. F. X. Diffley and O. Alter
| title = Global Effects of DNA Replication and DNA Replication Origin Activity on Eukaryotic Gene Expression
| journal = Molecular Systems Biology
| volume = 5
| pages = 312
| date = October 2009
| doi = 10.1038/msb.2009.70
| url = http://dx.doi.org/10.1038/msb.2009.70
| id = [http://www.alterlab.org/research/highlights/msb.2009.70_Highlight.pdf Highlight]
| issue=1
}}</ref><ref>{{Cite journal
| author = C. Muralidhara, A. M. Gross, R. R. Gutell and O. Alter
| title = Tensor Decomposition Reveals Concurrent Evolutionary Convergences and Divergences and Correlations with Structural Motifs in Ribosomal RNA
| journal = PLoS ONE
| volume = 6
| issue = 4
| pages = e18768
| date = April 2011
| doi = 10.1371/journal.pone.0018768
| url = http://dx.doi.org/10.1371/journal.pone.0018768
| id = [http://www.alterlab.org/research/highlights/pone.0018768_Highlight.pdf Highlight]
}}</ref> These applications also inspired a higher-order generalized singular value decomposition (HO GSVD).<ref>{{Cite journal
| author = S. P. Ponnapalli, M. A. Saunders, C. F. Van Loan and O. Alter
| title = A Higher-Order Generalized Singular Value Decomposition for Comparison of Global mRNA Expression from Multiple Organisms
| journal = PLoS ONE
| volume = 6
| issue = 12
| pages = e28072
| date = December 2011
| doi = 10.1371/journal.pone.0028072
  | url = http://dx.doi.org/10.1371/journal.pone.0028072
| id = [http://www.alterlab.org/research/highlights/pone.0028072_Highlight.pdf Highlight]
}}</ref>
 
It is also used in [[tensor product model transformation]]-based controller design.<ref name=Baranyi04 /><ref name=compind /> In [[multilinear subspace learning]],<ref>Haiping Lu, K.N. Plataniotis and A.N. Venetsanopoulos, "[http://www.dsp.utoronto.ca/~haiping/Publication/SurveyMSL_PR2011.pdf A Survey of Multilinear Subspace Learning for Tensor Data]", Pattern Recognition, Vol. 44, No. 7, pp. 1540–1551, Jul. 2011.</ref> it is modified to [[multilinear principal component analysis]]<ref>H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "[http://www.dsp.utoronto.ca/~haiping/Publication/MPCA_TNN08_rev2010.pdf MPCA: Multilinear principal component analysis of tensor objects]," IEEE Trans. Neural Netw., vol. 19, no. 1, pp. 18–39, Jan. 2008.</ref> for gait recognition.
 
== References ==
{{Reflist}}
 
{{DEFAULTSORT:Higher Order Singular Value Decomposition}}
[[Category:Multilinear algebra]]

Latest revision as of 14:33, 21 July 2014

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