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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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Cited by 705 (17 self)
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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.
A multilinear singular value decomposition
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higherorder tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, firstorder perturbation effects, etc., are ..."
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Cited by 467 (20 self)
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Abstract. We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higherorder tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, firstorder perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pairwise symmetric tensors.
TENSOR RANK AND THE ILLPOSEDNESS OF THE BEST LOWRANK APPROXIMATION PROBLEM
"... There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, te ..."
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Cited by 193 (13 self)
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There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rankr approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders and ranks, regardless of the choice of norm (or even Brègman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank3 tensor has an optimal rank2 approximation. The notable exceptions to this misbehavior are rank1 tensors and order2 tensors (i.e. matrices). In a more positive spirit, we propose a natural way of overcoming the illposedness of the lowrank approximation problem, by using weak solutions when true solutions do not exist. For this to work, it is necessary to characterize the set of weak solutions, and we do this in the case of rank 2, order 3 (in arbitrary dimensions). In our work we emphasize the importance of closely studying concrete lowdimensional examples as a first step towards more general results. To this end, we present a detailed analysis of equivalence classes of 2 × 2 × 2 tensors, and we develop methods for extending results upwards to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; in other words, can be determined by a system of polynomial inequalities. We study some of these polynomials in cases of interest to us; in particular we make extensive use of the hyperdeterminant ∆ on R 2×2×2.
Orthogonal Tensor Decompositions
 SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
, 2001
"... We explore the orthogonal decomposition of tensors (also known as multidimensional arrays or nway arrays) using two different definitions of orthogonality. We present numerous examples to illustrate the difficulties in understanding such decompositions. We conclude with a counterexample to a tensor ..."
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Cited by 122 (9 self)
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We explore the orthogonal decomposition of tensors (also known as multidimensional arrays or nway arrays) using two different definitions of orthogonality. We present numerous examples to illustrate the difficulties in understanding such decompositions. We conclude with a counterexample to a tensor extension of the EckartYoung SVD approximation theorem by Leibovici and Sabatier [Linear Algebra Appl., 269 (1998), pp. 307329].
Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Cited by 101 (22 self)
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Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
Efficient MATLAB computations with sparse and factored tensors
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 2007
"... In this paper, the term tensor refers simply to a multidimensional or $N$way array, and we consider how specially structured tensors allow for efficient storage and computation. First, we study sparse tensors, which have the property that the vast majority of the elements are zero. We propose stori ..."
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Cited by 80 (15 self)
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In this paper, the term tensor refers simply to a multidimensional or $N$way array, and we consider how specially structured tensors allow for efficient storage and computation. First, we study sparse tensors, which have the property that the vast majority of the elements are zero. We propose storing sparse tensors using coordinate format and describe the computational efficiency of this scheme for various mathematical operations, including those typical to tensor decomposition algorithms. Second, we study factored tensors, which have the property that they can be assembled from more basic components. We consider two specific types: A Tucker tensor can be expressed as the product of a core tensor (which itself may be dense, sparse, or factored) and a matrix along each mode, and a Kruskal tensor can be expressed as the sum of rank1 tensors. We are interested in the case where the storage of the components is less than the storage of the full tensor, and we demonstrate that many elementary operations can be computed using only the components. All of the efficiencies described in this paper are implemented in the Tensor Toolbox for MATLAB.
Unsupervised multiway data analysis: A literature survey
 IEEE Transactions on Knowledge and Data Engineering
, 2008
"... Multiway data analysis captures multilinear structures in higherorder datasets, where data have more than two modes. Standard twoway methods commonly applied on matrices often fail to find the underlying structures in multiway arrays. With increasing number of application areas, multiway data anal ..."
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Cited by 80 (10 self)
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Multiway data analysis captures multilinear structures in higherorder datasets, where data have more than two modes. Standard twoway methods commonly applied on matrices often fail to find the underlying structures in multiway arrays. With increasing number of application areas, multiway data analysis has become popular as an exploratory analysis tool. We provide a review of significant contributions in literature on multiway models, algorithms as well as their applications in diverse disciplines including chemometrics, neuroscience, computer vision, and social network analysis. 1.
Enhanced line search: A novel method to accelerate Parafac
 in Eusipco’05
, 2005
"... Abstract. Several modifications have been proposed to speed up the alternating least squares (ALS) method of fitting the PARAFAC model. The most widely used is line search, which extrapolates from linear trends in the parameter changes over prior iterations to estimate the parameter values that woul ..."
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Cited by 58 (11 self)
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Abstract. Several modifications have been proposed to speed up the alternating least squares (ALS) method of fitting the PARAFAC model. The most widely used is line search, which extrapolates from linear trends in the parameter changes over prior iterations to estimate the parameter values that would be obtained after many additional ALS iterations. We propose some extensions of this approach that incorporate a more sophisticated extrapolation, using information on nonlinear trends in the parameters and changing all the parameter sets simultaneously. The new method, called “enhanced line search (ELS), ” can be implemented at different levels of complexity, depending on how many different extrapolation parameters (for different modes) are jointly optimized during each iteration. We report some tests of the simplest parameter version, using simulated data. The performance of this lowestlevel of ELS depends on the nature of the convergence difficulty. It significantly outperforms standard LS when there is a “convergence bottleneck, ” a situation where some modes have almost collinear factors but others do not, but is somewhat less effective in classic “swamp ” situations where factors are highly collinear in all modes. This is illustrated by examples. To demonstrate how ELS can be adapted to different Nway decompositions, we also apply it to a fourway array to perform a blind identification of an underdetermined mixture (UDM). Since analysis of this dataset happens to involve a serious convergence “bottleneck ” (collinear factors in two of the four modes), it provides another example of a situation in which ELS dramatically outperforms standard line search. Key words. PARAFAC, alternating least squares (ALS), line search, enhanced line search (ELS), acceleration, swamps, bottlenecks, collinear factors, degeneracy AMS subject classifications. Authors must provide DOI. 10.1137/06065577 1. Introduction. PARAFAC
Computation of the canonical decomposition by means of a simultaneous generalized schur decomposition
 SIAM J. Matrix Anal. Appl
, 2004
"... Abstract. The canonical decomposition of higherorder tensors is a key tool in multilinear algebra. First we review the state of the art. Then we show that, under certain conditions, the problem can be rephrased as the simultaneous diagonalization, by equivalence or congruence, of a set of matrices. ..."
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Cited by 55 (10 self)
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Abstract. The canonical decomposition of higherorder tensors is a key tool in multilinear algebra. First we review the state of the art. Then we show that, under certain conditions, the problem can be rephrased as the simultaneous diagonalization, by equivalence or congruence, of a set of matrices. Necessary and sufficient conditions for the uniqueness of these simultaneous matrix decompositions are derived. In a next step, the problem can be translated into a simultaneous generalized Schur decomposition, with orthogonal unknowns [A.J. van der Veen and A. Paulraj, IEEE Trans. Signal Process., 44 (1996), pp. 1136–1155]. A firstorder perturbation analysis of the simultaneous generalized Schur decomposition is carried out. We discuss some computational techniques (including a new Jacobi algorithm) and illustrate their behavior by means of a number of numerical experiments.
Multilinear operators for higherorder decompositions
, 2006
"... We propose two new multilinear operators for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions. The ﬁrst operator,
which we call the Tucker operator, is shorthand for performing an nmode matrix multiplication for every mode of a given tensor and ..."
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Cited by 48 (10 self)
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We propose two new multilinear operators for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions. The ﬁrst operator,
which we call the Tucker operator, is shorthand for performing an nmode matrix multiplication for every mode of a given tensor and can be employed to consisely express the Tucker decomposition. The second operator, which we call the Kruskal operator, is shorthand for the sum of the outerproducts of the columns of N matrices and allows a divorce from a matricized representation and a very consise expression of the PARAFAC decomposition. We explore the
properties of the Tucker and Kruskal operators independently of the related decompositions.
Additionally, we provide a review of the matrix and tensor operations that are frequently used in the context of tensor decompositions.