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Tensor Decompositions and Applications
 SIAM REVIEW
, 2009
"... This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal proce ..."
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This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decompo
sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rankone tensors, and the Tucker decomposition is a higherorder form of principal components analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The Nway Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.
Canonical Tensor Decompositions
 ARCC WORKSHOP ON TENSOR DECOMPOSITION
, 2004
"... The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short pap ..."
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Cited by 43 (16 self)
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The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short paper, such as the maximal achievable rank of a tensor of given dimensions, or the computation of a CanD. Some questions are answered, and it turns out that the answers depend on the choice of the underlying field, and on tensor symmetry structure, which outlines a major difference compared to matrices.
Dimensionality reduction in higherorder signal processing and rank(R_1,R__2,...,R_N) reduction in multilinear algebra
, 2004
"... ..."
Fast multilinear Singular Value Decomposition for higherorder Hankel tensors
"... (HOSVD) is a possible generalization of the Singular Value Decomposition (SVD) to tensors, which have been successfully applied in various domains. Unfortunately, this decomposition is computationally demanding. Indeed, the HOSVD of a N thorder tensor involves the computation of the SVD of N matric ..."
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Cited by 16 (2 self)
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(HOSVD) is a possible generalization of the Singular Value Decomposition (SVD) to tensors, which have been successfully applied in various domains. Unfortunately, this decomposition is computationally demanding. Indeed, the HOSVD of a N thorder tensor involves the computation of the SVD of N matrices. Previous works have shown that it is possible to reduce the complexity of HOSVD for thirdorder structured tensors. These methods exploit the columns redundancy, which is present in the mode of structured tensors, especially in Hankel tensors. In this paper, we propose to extend these results to fourth order Hankel tensor. We propose two ways to extend Hankel structure to fourth order tensors. For these two types of tensors, a method to build a reordered mode is proposed, which highlights the column redundancy and we derive a fast algorithm to compute their HOSVD. Finally we show the benefit of our algorithms in terms of complexity. I.
Combining blind source extraction with joint approximate diagonalization: Thin algorithms for ICA
 in Proc. of the Fourth Symposium on Independent Component Analysis and Blind Signal Separation
, 2003
"... In this paper a multivariate contrast function is proposed for the blind signal extraction of a subset of the independent components from a linear mixture. This contrast combines the robustness of the joint approximate diagonalization techniques with the flexibility of the methods for blind signal e ..."
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Cited by 15 (5 self)
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In this paper a multivariate contrast function is proposed for the blind signal extraction of a subset of the independent components from a linear mixture. This contrast combines the robustness of the joint approximate diagonalization techniques with the flexibility of the methods for blind signal extraction. Its maximization leads to hierarchical and simultaneous ICA extraction algorithms which are respectively based on the thin QR and thin SVD factorizations. The interesting similarities and differences with other existing contrasts and algorithms are commented. 1.
Lowrank decomposition of multiway arrays: A signal processing perspective
 In IEEE SAM
, 2004
"... In many signal processing applications of linear algebra tools, the signal part of a postulated model lies in a socalled signal subspace, while the parameters of interest are in onetoone correspondence with a certain basis of this subspace. The signal subspace can often be reliably estimated from ..."
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Cited by 12 (0 self)
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In many signal processing applications of linear algebra tools, the signal part of a postulated model lies in a socalled signal subspace, while the parameters of interest are in onetoone correspondence with a certain basis of this subspace. The signal subspace can often be reliably estimated from measured data, but the particular basis of interest cannot be identified without additional problemspecific structure. This is a manifestation of rotational indeterminacy, i.e., nonuniqueness of lowrank matrix decomposition. The situation is very different for three or higherway arrays, i.e., arrays indexed by three or more independent variables, for which lowrank decomposition is unique under mild conditions. This has fundamental implications for DSP problems which deal with such data. This paper provides a brief tour of the basic elements of this theory, along with many examples of application in problems of current interest in the signal processing community. Keywords: Threeway analysis, lowrank decomposition, parallel factor analysis (PARAFAC), canonical decomposition (CAN
On the Blind Separation of Noncircular Sources
, 2002
"... In this paper we address the blind separation of an instantaneous complex mixture of statistically independent noncircular signals. We show that, if the sources are noncircular at order 2, by exploiting all the secondorder information, the mixing matrix can be estimated up to a real orthogonal ..."
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Cited by 11 (3 self)
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In this paper we address the blind separation of an instantaneous complex mixture of statistically independent noncircular signals. We show that, if the sources are noncircular at order 2, by exploiting all the secondorder information, the mixing matrix can be estimated up to a real orthogonal factor. This is based on a link with the Takagi factorization, for the computation of which we derive a Jacobitype algorithm. We prove that, if the sources are noncircular at order 4, after a classical prewhitening, the remaining unitary factor can be found via a simultaneous Takagi factorization / Hermitian Eigenvalue Decomposition (EVD). We also describe a variant in which no hard prewhitening is carried out. In addition, we pay some attention to the issue of dimensionality reduction, in the case where there are fewer sources than sensors.
Tensors: a Brief Introduction
, 2014
"... Tensor decompositions are at the core of many Blind Source Separation (BSS) algorithms, either explicitly or implicitly. In particular, the Canonical Polyadic (CP) tensor ..."
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Cited by 11 (3 self)
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Tensor decompositions are at the core of many Blind Source Separation (BSS) algorithms, either explicitly or implicitly. In particular, the Canonical Polyadic (CP) tensor
A Jacobitype method for computing orthogonal tensor decompositions
 SIAM J. Matrix Anal. Appl
, 2006
"... Abstract. Suppose A =(aijk) ∈ Rn×n×n is a threeway array or thirdorder tensor. Many of the powerful tools of linear algebra such as the singular value decomposition (SVD) do not, unfortunately, extend in a straightforward way to tensors of order three or higher. In the twodimensional case, the SV ..."
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Cited by 9 (1 self)
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Abstract. Suppose A =(aijk) ∈ Rn×n×n is a threeway array or thirdorder tensor. Many of the powerful tools of linear algebra such as the singular value decomposition (SVD) do not, unfortunately, extend in a straightforward way to tensors of order three or higher. In the twodimensional case, the SVD is particularly illuminating, since it reduces a matrix to diagonal form. Although it is not possible in general to diagonalize a tensor (i.e., aijk = 0 unless i = j = k), our goal is to “condense ” a tensor in fewer nonzero entries using orthogonal transformations. We propose an algorithm for tensors of the form A∈Rn×n×n that is an extension of the Jacobi SVD algorithm for matrices. The resulting tensor decomposition reduces A to a form such that the quantity ∑n i=1 a2 iii or ∑n i=1 aiii is maximized. Key words. multilinear algebra, tensor decomposition, singular value decomposition, multidimensional arrays
Bioinformatics: Organisms from Venus, Technology from Jupiter, Algorithms from Mars
 JUPITER, ALGORITHMS FROM MARS, EUROPEAN JOURNAL OF CONTROL
, 2003
"... In this paper, we discuss datasets that are being generated by microarray technology, which makes it possible to measure in parallel the activity or expression of thousands of genes simultaneously. We discuss the basics of the technology, how to preprocess the data, and how classical and newly dev ..."
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Cited by 5 (0 self)
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In this paper, we discuss datasets that are being generated by microarray technology, which makes it possible to measure in parallel the activity or expression of thousands of genes simultaneously. We discuss the basics of the technology, how to preprocess the data, and how classical and newly developed algorithms can be used to generate insight in the biological processes that have generated the data. Algorithms we discuss are Principal Component Analysis, clustering techniques such as hierarchical clustering and Adaptive Quality Based Clustering and statistical sampling methods, such as Monte Carlo Markov Chains and Gibbs sampling. We illustrate these algorithms with several reallife cases from diagnostics and class discovery in leukemia, functional genomics research on the mitotic cell cycle of yeast, and motif detection in Arabidopsis thaliana using DNA background models. We also discuss some bioinformatics software platforms. In the final part of the manuscript, we present some future perspectives on the development of bioinformatics, including some visionary discussions on technology, algorithms, systems biology and computational biomedicine.