higher-order singular value decomposition

For the purpose of this article, the abstract tensor $\backslash mathcal\{A\}$ is assumed to be given in coordinates with respect to some basis as a M-way array, also denoted by $\backslash mathcal\{A\}\backslash in\backslash mathbb\{C\}^\{I\_1\; \backslash times\; I\_2\; \backslash cdots\; \backslash times\; \backslash cdots\; I\_m\; \backslash cdots\backslash times\; I\_M\}$, where M is the number of modes and the order of the tensor. $\backslash mathbb\{C\}$ is the complex numbers and it includes both the real numbers $\backslash mathbb\{R\}$ and the pure imaginary numbers.

Let $\backslash mathcal\{A\}\_\{[m]\}\backslash in\backslash mathbb\{C\}^\{I\_m\; \backslash times\; (I\_1\; I\_2\; \backslash cdots\; I\_\{m-1\}\; I\_\{m+1\}\; \backslash cdots\; I\_M)\}$ denote the standard mode-m flattening of $\backslash mathcal\{A\}$, so that the left index of $\backslash mathcal\{A\}\_\{[m]\}$ corresponds to the $m$'th index $\backslash mathcal\{A\}$ and the right index of $\backslash mathcal\{A\}\_\{[m]\}$ corresponds to all other indices of $\backslash mathcal\{A\}$ combined. Let $\{\backslash bf\; U\}\_m\; \backslash in\; \backslash mathbb\{C\}^\{I\_m\; \backslash times\; I\_m\}$be a unitary matrix containing a basis of the left singular vectors of the $\backslash mathcal\{A\}\_\{[m]\}$ such that the jth column $\backslash mathbf\{u\}\_j$ of $\{\backslash bf\; U\}\_m$ corresponds to the jth largest singular value of $\backslash mathcal\{A\}\_\{[m]\}$. Observe that the mode/factor matrix $\{\backslash bf\; U\}\_m$ does not depend on the particular on the specific definition of the mode m flattening. By the properties of the multilinear multiplication, we have$$\backslash begin\{array\}\{rcl\}$$

\mathcal{A}

&=& \mathcal{A}\times ({\bf I}, {\bf I}, \ldots, {\bf I}) \\

&=& \mathcal{A} \times ({\bf U}_1 {\bf U}_1^H, {\bf U}_2 {\bf U}_2^H, \ldots, {\bf U}_M {\bf U}_M^H) \\

&=& \left(\mathcal{A} \times ({\bf U}_1^H, {\bf U}_2^H, \ldots, {\bf U}_M^H) \right) \times ({\bf U}_1, {\bf U}_2, \ldots, {\bf U}_M),

\end{array}where $\backslash cdot^H$ denotes the conjugate transpose. The second equality is because the $\{\backslash bf\; U\}\_m$'s are unitary matrices. Define now the core tensor$$\backslash mathcal\{S\}\; :=\; \backslash mathcal\{A\}\; \backslash times\; (\{\backslash bf\; U\}\_1^H,\; \{\backslash bf\; U\}\_2^H,\; \backslash ldots,\; \{\backslash bf\; U\}\_M^H).$$Then, the HOSVD of $\backslash mathcal\{A\}$ is the decomposition$$\backslash mathcal\{A\}\; =\; \backslash mathcal\{S\}\backslash times\; (\{\backslash bf\; U\}\_1,\; \{\backslash bf\; U\}\_2,\; \backslash ldots,\; \{\backslash bf\; U\}\_M).$$ The above construction shows that every tensor has a HOSVD.

As in the case of the compact singular value decomposition of a matrix, where the rows and columns corresponding to vanishing singular values are dropped, it is also possible to consider a compact HOSVD, which is very useful in applications.

Assume that $\{\backslash bf\; U\}\_m\; \backslash in\; \backslash mathbb\{C\}^\{I\_m\; \backslash times\; R\_m\}$ is a matrix with unitary columns containing a basis of the left singular vectors corresponding to the nonzero singular values of the standard factor-m flattening $\backslash mathcal\{A\}\_\{[m]\}$ of $\backslash mathcal\{A\}$. Let the columns of $\{\backslash bf\; U\}\_m$ be sorted such that the $r\_m$ th column $\{\backslash bf\; u\}\_\{r\_m\}$ of $\{\backslash bf\; U\}\_m$ corresponds to the $r\_m$th largest nonzero singular value of $\backslash mathcal\{A\}\_\{[m]\}$. Since the columns of $\{\backslash bf\; U\}\_m$ form a basis for the image of $\backslash mathcal\{A\}\_\{[m]\}$, we have$$\backslash mathcal\{A\}\_\{[m]\}\; =\; \{\backslash bf\; U\}\_m\; \{\backslash bf\; U\}\_m^H\; \backslash mathcal\{A\}\_\{[m]\}\; =\; \backslash bigl(\; \backslash mathcal\{A\}\; \backslash times\_m\; (\{\backslash bf\; U\}\_m\; \{\backslash bf\; U\}\_m^H)\; \backslash bigr)\_\{[m]\},$$where the first equality is due to the properties of orthogonal projections (in the Hermitian inner product) and the last equality is due to the properties of multilinear multiplication. As flattenings are bijective maps and the above formula is valid for all $m=1,2,\backslash ldots,m,\backslash ldots,M$, we find as before that$$\backslash begin\{array\}\{rcl\}$$

\mathcal{A}

&=& \mathcal{A} \times ({\bf U}_1 {\bf U}_1^H, {\bf U}_2 {\bf U}_2^H, \ldots, {\bf U}_M {\bf U}_M^H)\\

&=& \left(\mathcal{A} \times ({\bf U}_1^H, {\bf U}_2^H, \ldots, {\bf U}_M^H)\right) \times ({\bf U}_1, {\bf U}_2, \ldots, {\bf U}_M) \\

&=& \mathcal{S} \times ({\bf U}_1, {\bf U}_2, \ldots, {\bf U}_M),

\end{array}where the core tensor $\backslash mathcal\{S\}$ is now of size $R\_1\; \backslash times\; R\_2\; \backslash times\; \backslash cdots\; \backslash times\; R\_M$.

The multilinear rank of $\backslash mathcal\{A\}$ is denoted with rank-$(R\_1,\; R\_2,\; \backslash ldots,\; R\_M)$. The multilinear rank is a tuple in $\backslash mathbb\{N\}^M$ where $R\_m\; :=\; \backslash mathrm\{rank\}(\; \backslash mathcal\{A\}\_\{[m]\}\; )$. Not all tuples in $\backslash mathbb\{N\}^M$ are multilinear ranks.{{Cite journal|last1=Carlini|first1=Enrico|last2=Kleppe|first2=Johannes|title=Ranks derived from multilinear maps|journal=Journal of Pure and Applied Algebra|volume=215|issue=8|pages=1999–2004|doi=10.1016/j.jpaa.2010.11.010|year=2011|doi-access=free}} The multilinear ranks are bounded by $1\; \backslash le\; R\_m\; \backslash le\; I\_m$ and it satisfy the constraint $R\_m\; \backslash le\; \backslash prod\_\{i\; \backslash ne\; m\}\; R\_i$ must hold.

The compact HOSVD is a rank-revealing decomposition in the sense that the dimensions of its core tensor correspond with the components of the multilinear rank of the tensor.

The following geometric interpretation is valid for both the full and compact HOSVD. Let $(R\_1,\; R\_2,\; \backslash ldots,\; R\_M)$ be the multilinear rank of the tensor $\backslash mathcal\{A\}$. Since $\backslash mathcal\{S\}\; \backslash in\; \{\backslash mathbb\; C\}^\{R\_1\; \backslash times\; R\_2\; \backslash times\; \backslash cdots\; \backslash times\; R\_M\}$ is a multidimensional array, we can expand it as follows$$\backslash mathcal\{S\}\; =\; \backslash sum\_\{r\_1=1\}^\{R\_1\}\; \backslash sum\_\{r\_2=1\}^\{R\_2\}\; \backslash cdots\; \backslash sum\_\{r\_M=1\}^\{R\_M\}\; s\_\{r\_1,r\_2,\backslash ldots,r\_M\}\; \backslash mathbf\{e\}\_\{r\_1\}\; \backslash otimes\; \backslash mathbf\{e\}\_\{r\_2\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \backslash mathbf\{e\}\_\{r\_M\},$$where $\backslash mathbf\{e\}\_\{r\_m\}$ is the $r\_m$th standard basis vector of $\{\backslash mathbb\; C\}^\{I\_m\}$. By definition of the multilinear multiplication, it holds that$$\backslash mathcal\{A\}\; =\; \backslash sum\_\{r\_1=1\}^\{R\_1\}\; \backslash sum\_\{r\_2=1\}^\{R\_2\}\; \backslash cdots\; \backslash sum\_\{r\_M=1\}^\{R\_M\}\; s\_\{r\_1,r\_2,\backslash ldots,r\_M\}$$

\mathbf{u}_{r_1} \otimes \mathbf{u}_{r_2} \otimes \cdots \otimes \mathbf{u}_{r_M},where the $\backslash mathbf\{u\}\_\{r\_m\}$ are the columns of $\{\backslash bf\; U\}\_m\; \backslash in\; \{\backslash mathbb\; C\}^\{I\_m\; \backslash times\; R\_m\}$. It is easy to verify that $B\; =\; \backslash \{\; \backslash mathbf\{u\}\_\{r\_1\}\; \backslash otimes\; \backslash mathbf\{u\}\_\{r\_2\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \backslash mathbf\{u\}\_\{r\_M\}\; \backslash \}\_\{r\_1,r\_2,\backslash ldots,r\_M\}$ is an orthonormal set of tensors. This means that the HOSVD can be interpreted as a way to express the tensor $\backslash mathcal\{A\}$ with respect to a specifically chosen orthonormal basis $B$ with the coefficients given as the multidimensional array $\backslash mathcal\{S\}$.

Let $\backslash mathcal\{A\}\; \backslash in\; \{\backslash mathbb\; C\}^\{I\_1\; \backslash times\; I\_2\; \backslash times\; \backslash cdots\; \backslash times\; I\_M\}$ be a tensor with a rank-$(R\_1,\; R\_2,\; \backslash ldots,\; R\_M)$, where $\backslash mathbb\; C$ contains the reals $\backslash mathbb\{R\}$ as a subset.

The strategy for computing the Multilinear SVD and M-mode SVD was first introduced by L. R. Tucker in the 1960s, and later extended by L. De Lathauwer et al. and by Vasilescu and Terzopoulos. However, the algorithm widely referred to in the literature as the Tucker or HOSVD decomposition was developed by Vasilescu and Terzopoulos under the name M-mode SVD. This algorithm employs the matrix SVD to compute orthonormal mode matrices and suitable for parallel computation, making it computationally efficient and suitable for analyzing the causal factors of data formation. In contrast, Tucker and De Lathauwer proposed sequential algorithms based on gradient descent and the power method, respectively.

What is commonly referred to as the HOSVD or Tucker was developed by Vasilescu and Terzopoulos under the name
M-mode SVD, and suitable for parallel computation, but misattributed to L. R. Tucker or Lieven De Lathauwer et al..

• For $m=1,\backslash ldots,M$, do the following:

1. Construct the mode-m flattening $\backslash mathcal\{A\}\_\{[m]\}$;
2. Compute the (compact) singular value decomposition
   $\backslash mathcal\{A\}\_\{[m]\}\; =\; \{\backslash bf\; U\}\_m\; \{\backslash bf\; \backslash Sigma\}\_m\; \{\backslash bf\; V\}^T\_m$, and store the left singular vectors $\{\backslash bf\; U\}\; \backslash in\; \backslash mathbb\{C\}^\{I\_m\; \backslash times\; R\_m\}$;

• Compute the core tensor $\backslash mathcal\{S\}$ via the mode-m product
  $\backslash mathcal\{S\}\; =\; \backslash mathcal\{A\}\; \backslash times\_1\; \{\backslash bf\; U\}\_1^H\; \backslash times\_2\; \{\backslash bf\; U\}\_2^H\; \backslash ldots\; \backslash times\_m\; \{\backslash bf\; U\}\_m^H\; \backslash ldots\; \backslash times\_M\; \{\backslash bf\; U\}\_M^H$

A strategy that is significantly faster when some or all $R\_m\; \backslash ll\; I\_m$ consists of interlacing the computation of the core tensor and the factor matrices, as follows:{{Cite journal|last1=Vannieuwenhoven|first1=N.|last2=Vandebril|first2=R.|last3=Meerbergen|first3=K.|date=2012-01-01|title=A New Truncation Strategy for the Higher-Order Singular Value Decomposition|journal=SIAM Journal on Scientific Computing|volume=34|issue=2|pages=A1027–A1052|doi=10.1137/110836067|bibcode=2012SJSC...34A1027V |s2cid=15318433 |issn=1064-8275|url=https://lirias.kuleuven.be/handle/123456789/337210}}{{Cite book|title=Tensor Spaces and Numerical Tensor Calculus {{!}} SpringerLink|volume = 42|last=Hackbusch|first=Wolfgang|language=en-gb|doi=10.1007/978-3-642-28027-6|series = Springer Series in Computational Mathematics|year = 2012|isbn = 978-3-642-28026-9| s2cid=117253621 }}{{Cite conference|last1=Cobb|first1=Benjamin|last2=Kolla|first2=Hemanth|last3=Phipps|first3=Eric|last4=Çatalyürek|first4=Ümit V.|date=2022|title=FIST-HOSVD: Fused in-Place Sequentially Truncated Higher Order Singular Value Decomposition|conference=Platform for Advanced Scientific Computing(PASC) |language=en|isbn=9781450394109|doi=10.1145/3539781.3539798|doi-access=free}}

• Set $\backslash mathcal\{A\}^0\; =\; \backslash mathcal\{A\}$;
• For $m\; =\; 1,2\; \backslash ldots,\; M$ perform the following:
• # Construct the standard mode-m flattening $\backslash mathcal\{A\}\_\{[m]\}^\{m-1\}$;
• # Compute the (compact) singular value decomposition $\backslash mathcal\{A\}\_\{[m]\}^\{m-1\}\; =\; U\_m\; \backslash Sigma\_m\; V^T\_m$, and store the left singular vectors $U\_m\; \backslash in\; F^\{I\_m\; \backslash times\; R\_m\}$;
• # Set $\backslash mathcal\{A\}^m\; =\; U\_m^H\; \backslash times\_m\; \backslash mathcal\{A\}^\{m-1\}$, or, equivalently, $\backslash mathcal\{A\}^m\_\{[m]\}\; =\; \backslash Sigma\_m\; V\_m^T$.

The HOSVD can be computed in-place via the Fused In-place Sequentially Truncated Higher Order Singular Value Decomposition (FIST-HOSVD) algorithm by overwriting the original tensor by the HOSVD core tensor, significantly reducing the memory consumption of computing HOSVD.

In applications, such as those mentioned below, a common problem consists of approximating a given tensor $\backslash mathcal\{A\}\; \backslash in\; \backslash mathbb\{C\}^\{I\_1\; \backslash times\; I\_2\; \backslash times\; \backslash cdots\; \backslash times\; I\_m\; \backslash cdots\; \backslash times\; I\_M\}$ by one with a reduced multilinear rank. Formally, if the multilinear rank of $\backslash mathcal\{A\}$ is denoted by $\backslash mathrm\{rank-\}(R\_1,R\_2,\backslash ldots,R\_m,\backslash ldots,R\_M)$, then computing the optimal $\backslash mathcal\{\backslash bar\; A\}$ that approximates $\backslash mathcal\{A\}$ for a given reduced $\backslash mathrm\{rank-\}(\backslash bar\; R\_1,\backslash bar\; R\_2,\backslash ldots,\backslash bar\; R\_m,\backslash ldots,\backslash bar\; R\_M)$ is a nonlinear non-convex $\backslash ell\_2$-optimization problem

$$\backslash min\_\{\backslash mathcal\{\backslash bar\; A\}\backslash in\; \backslash mathbb\{C\}^\{I\_1\; \backslash times\; I\_2\; \backslash times\; \backslash cdots\; \backslash times\; I\_M\}\}\; \backslash frac\{1\}\{2\}\; \backslash |\; \backslash mathcal\{A\}\; -\; \backslash mathcal\{\backslash bar\; A\}\; \backslash |\_F^2$$

\quad\text{s.t.}\quad \mathrm{rank-}(\bar R_1, \bar R_2, \ldots, \bar R_M), where $(\backslash bar\; R\_1,\; \backslash bar\; R\_2,\; \backslash ldots,\; \backslash bar\; R\_M)\; \backslash in\; \backslash mathbb\{N\}^M$ is the reduced multilinear rank with , and the norm $\backslash |.\backslash |\_F$ is the Frobenius norm.

A simple idea for trying to solve this optimization problem is to truncate the (compact) SVD in step 2 of either the classic or the interlaced computation. A classically truncated HOSVD is obtained by replacing step 2 in the classic computation by

• Compute a rank-$\backslash bar\; R\_m$ truncated SVD $\backslash mathcal\{A\}\_\{[m]\}\; \backslash approx\; U\_m\; \backslash Sigma\_m\; V^T\_m$, and store the top $\backslash bar\; R\_m$ left singular vectors $U\_m\; \backslash in\; F^\{I\_m\; \backslash times\; \backslash bar\; R\_m\}$;

while a sequentially truncated HOSVD (or successively truncated HOSVD) is obtained by replacing step 2 in the interlaced computation by

• Compute a rank-$\backslash bar\; R\_m$ truncated SVD $\backslash mathcal\{A\}\_\{[m]\}^\{m-1\}\; \backslash approx\; U\_m\; \backslash Sigma\_m\; V^T\_m$, and store the top $\backslash bar\; R\_m$ left singular vectors $U\_m\; \backslash in\; F^\{I\_m\; \backslash times\; \backslash bar\; R\_m\}$. Unfortunately, truncation does not result in an optimal solution for the best low multilinear rank optimization problem,. However, both the classically and interleaved truncated HOSVD result in a quasi-optimal solution:{{Cite journal|last=Grasedyck|first=L.|date=2010-01-01|title=Hierarchical Singular Value Decomposition of Tensors|journal=SIAM Journal on Matrix Analysis and Applications|volume=31|issue=4|pages=2029–2054|doi=10.1137/090764189|issn=0895-4798|citeseerx=10.1.1.660.8333}} if $\backslash mathcal\{\backslash bar\; A\}\_t$ denotes the classically or sequentially truncated HOSVD and $\backslash mathcal\{\backslash bar\; A\}^*$ denotes the optimal solution to the best low multilinear rank approximation problem, then$$\backslash |\; \backslash mathcal\{A\}\; -\; \backslash mathcal\{\backslash bar\; A\}\_t\; \backslash |\_F\; \backslash le\; \backslash sqrt\{M\}\; \backslash |\; \backslash mathcal\{A\}\; -\; \backslash mathcal\{\backslash bar\; A\}^*\; \backslash |\_F;$$in practice this means that if there exists an optimal solution with a small error, then a truncated HOSVD will for many intended purposes also yield a sufficiently good solution.

The HOSVD is most commonly applied to the extraction of relevant information from multi-way arrays.

Starting in the early 2000s, Vasilescu addressed causal questions by reframing the data analysis, recognition and synthesis problems as multilinear tensor problems. The power of the tensor framework was showcased by decomposing and representing an image in terms of its causal factors of data formation, in the context of Human Motion Signatures for gait recognition,M. 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.] face recognition—TensorFacesM.A.O. Vasilescu, D. Terzopoulos (2003) [http://www.media.mit.edu/~maov/tensorfaces/cvpr03.pdf "Multilinear Subspace Analysis for Image Ensembles, M. A. O. Vasilescu, D. Terzopoulos, Proc. Computer Vision and Pattern Recognition Conf. (CVPR '03), Vol.2, Madison, WI, June, 2003, 93–99.]M.A.O. Vasilescu, D. Terzopoulos (2002) [http://www.media.mit.edu/~maov/tensorfaces/eccv02_corrected.pdf "Multilinear Analysis of Image Ensembles: TensorFaces," Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002, in Computer Vision -- ECCV 2002, Lecture Notes in Computer Science, Vol. 2350, A. Heyden et al. (Eds.), Springer-Verlag, Berlin, 2002, 447–460.] and computer graphics—TensorTextures.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.]

The HOSVD has been successfully applied to signal processing and big data, e.g., in genomic signal processing.{{Cite journal|author1=L. Omberg|author2=G. H. Golub|author3=O. Alter|date=November 2007|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|doi=10.1073/pnas.0709146104|pmc=2147680|pmid=18003902|bibcode=2007PNAS..10418371O|doi-access=free}}{{Cite journal|author1=L. Omberg|author2=J. R. Meyerson|author3=K. Kobayashi|author4=L. S. Drury|author5=J. F. X. Diffley|author6=O. Alter|date=October 2009|title=Global Effects of DNA Replication and DNA Replication Origin Activity on Eukaryotic Gene Expression|journal=Molecular Systems Biology|volume=5|pages=312|doi=10.1038/msb.2009.70|pmc=2779084|pmid=19888207|id=[http://www.alterlab.org/research/highlights/msb.2009.70_Highlight.pdf Highlight]}}{{Cite journal|author1=C. Muralidhara|author2=A. M. Gross|author3=R. R. Gutell|author4=O. Alter|date=April 2011|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|doi=10.1371/journal.pone.0018768|pmc=3094155|pmid=21625625|id=[http://www.alterlab.org/research/highlights/pone.0018768_Highlight.pdf Highlight]|bibcode=2011PLoSO...618768M|doi-access=free}} These applications also inspired a higher-order GSVD (HO GSVD){{Cite journal|author1=S. P. Ponnapalli|author2=M. A. Saunders|author3=C. F. Van Loan|author4=O. Alter|date=December 2011|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|doi=10.1371/journal.pone.0028072|pmc=3245232|pmid=22216090|id=[http://www.alterlab.org/research/highlights/pone.0028072_Highlight.pdf Highlight]|bibcode=2011PLoSO...628072P|doi-access=free}} and a tensor GSVD.{{Cite journal|author1=P. Sankaranarayanan|author2=T. E. Schomay|author3=K. A. Aiello|author4=O. Alter|date=April 2015|title=Tensor GSVD of Patient- and Platform-Matched Tumor and Normal DNA Copy-Number Profiles Uncovers Chromosome Arm-Wide Patterns of Tumor-Exclusive Platform-Consistent Alterations Encoding for Cell Transformation and Predicting Ovarian Cancer Survival|journal=PLOS ONE|volume=10|issue=4|pages=e0121396|doi=10.1371/journal.pone.0121396|pmc=4398562|pmid=25875127|id=[http://www.eurekalert.org/pub_releases/2015-04/uouh-nmi040915.php AAAS EurekAlert! Press Release] and [https://www.nae.edu/Projects/20730/wtop/134897.aspx NAE Podcast Feature]|bibcode=2015PLoSO..1021396S|doi-access=free}}

A combination of HOSVD and SVD also has been applied for real-time event detection from complex data streams (multivariate data with space and time dimensions) in disease surveillance.{{Cite journal|author1=Hadi Fanaee-T|author2=João Gama|date=May 2015|title=EigenEvent: An algorithm for event detection from complex data streams in Syndromic surveillance|journal=Intelligent Data Analysis|volume=19|issue=3|pages=597–616|arxiv=1406.3496|bibcode=2014arXiv1406.3496F|doi=10.3233/IDA-150734|s2cid=17966555}}

It is also used in tensor product model transformation-based controller design.{{cite journal|author=P. Baranyi|date=April 2004|title=TP model transformation as a way to LMI based controller design|journal=IEEE Transactions on Industrial Electronics|volume=51|pages=387–400|doi=10.1109/tie.2003.822037|number=2|s2cid=7957799}}{{cite journal|author1=P. Baranyi|author2=D. Tikk|author3=Y. Yam|author4=R. J. Patton|year=2003|title=From Differential Equations to PDC Controller Design via Numerical Transformation|journal=Computers in Industry|volume=51|issue=3|pages=281–297|doi=10.1016/s0166-3615(03)00058-7}}

The concept of HOSVD was carried over to functions by Baranyi and Yam via the TP model transformation. This extension led to the definition of the HOSVD-based canonical form of tensor product functions and Linear Parameter Varying system models{{cite conference|title=Definition of the HOSVD-based canonical form of polytopic dynamic models|author1=P. Baranyi|author2=L. Szeidl|author3=P. Várlaki|author4=Y. Yam|date=July 3–5, 2006|location=Budapest, Hungary|pages=660–665|conference=3rd International Conference on Mechatronics (ICM 2006)}} and to convex hull manipulation based control optimization theory, see TP model transformation in control theories.

HOSVD was proposed to be applied to multi-view data analysis{{Cite journal|author1=Y-h. Taguchi|date=August 2017|title=Tensor decomposition-based unsupervised feature extraction applied to matrix products for multi-view data processing|journal=PLOS ONE|volume=12|issue=8|pages=e0183933|doi=10.1371/journal.pone.0183933|pmc=5571984|pmid=28841719|bibcode=2017PLoSO..1283933T|doi-access=free}} and was successfully applied to in silico drug discovery from gene expression.{{Cite journal|author1=Y-h. Taguchi|date=October 2017|title=Identification of candidate drugs using tensor-decomposition-based unsupervised feature extraction in integrated analysis of gene expression between diseases and DrugMatrix datasets|journal=Scientific Reports|volume=7|issue=1|pages=13733|doi=10.1038/s41598-017-13003-0|pmc=5653784|pmid=29062063|bibcode=2017NatSR...713733T}}

L1-Tucker is the L1-norm-based, robust variant of Tucker decomposition. L1-HOSVD is the analogous of HOSVD for the solution to L1-Tucker.

{{DEFAULTSORT:Higher Order Singular Value Decomposition}}

Category:Multilinear algebra

Category:Tensors