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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 228 (14 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.
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 37 (7 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.
ComplexField Coding for OFDM Over Fading Wireless Channels
 IEEE Trans. Inform. Theory
, 2003
"... Orthogonal frequencydivision multiplexing (OFDM) converts a timedispersive channel into parallel subchannels, and thus facilitates equalization and (de)coding. But when the channel has nulls close to or on the fast Fourier transform (FFT) grid, uncoded OFDM faces serious symbol recovery problems. ..."
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Cited by 32 (1 self)
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Orthogonal frequencydivision multiplexing (OFDM) converts a timedispersive channel into parallel subchannels, and thus facilitates equalization and (de)coding. But when the channel has nulls close to or on the fast Fourier transform (FFT) grid, uncoded OFDM faces serious symbol recovery problems. As an alternative to various errorcontrol coding techniques that have been proposed to ameliorate the problem, we perform complexfield coding (CFC) before the symbols are multiplexed. We quantify the maximum achievable diversity order for independent and identically distributed (i.i.d.) or correlated Rayleighfading channels, and also provide design rules for achieving the maximum diversity order. The maximum coding gain is given, and the encoder enabling the maximum coding gain is also found. Simulated performance comparisons of CFCOFDM with existing block and convolutionally coded OFDM alternatives favor CFCOFDM for the code rates used in a HiperLAN2 experiment.
Kruskal’s permutation lemma and the identification of Candecomp/Parafac and bilinear models with constant modulus constraints
 IEEE Trans. Signal Process
"... Abstract—CANDECOMP/PARAFAC (CP) analysis is an extension of lowrank matrix decomposition to higherway arrays, which are also referred to as tensors. CP extends and unifies several array signal processing tools and has found applications ranging from multidimensional harmonic retrieval and angleca ..."
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Cited by 27 (5 self)
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Abstract—CANDECOMP/PARAFAC (CP) analysis is an extension of lowrank matrix decomposition to higherway arrays, which are also referred to as tensors. CP extends and unifies several array signal processing tools and has found applications ranging from multidimensional harmonic retrieval and anglecarrier estimation to blind multiuser detection. The uniqueness of CP decomposition is not fully understood yet, despite its theoretical and practical significance. Toward this end, we first revisit Kruskal’s Permutation Lemma, which is a cornerstone result in the area, using an accessible basic linear algebra and induction approach. The new proof highlights the nature and limits of the identification process. We then derive two equivalent necessary and sufficient uniqueness conditions for the case where one of the component matrices involved in the decomposition is full column rank. These new conditions explain a curious example provided recently in a previous paper by Sidiropoulos, who showed that Kruskal’s condition is in general sufficient but not necessary for uniqueness and that uniqueness depends on the particular joint pattern of zeros in the (possibly pretransformed) component matrices. As another interesting application of the Permutation Lemma, we derive a similar necessary and sufficient condition for unique bilinear factorization under constant modulus (CM) constraints, thus providing an interesting link to (and unification with) CP. Index Terms—CANDECOMP, constant modulus, identifiablity, PARAFAC, SVD, threeway array analysis, uniqueness. I.
Dimensionality reduction in higherorder signal processing and rank(R_1,R__2,...,R_N) reduction in multilinear algebra
, 2004
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Decompositions of a higherorder tensor in block terms— Part III: Alternating Least Squares algorithms
 SIAM J. Matrix Anal. Appl
"... Abstract. In this paper we introduce a new class of tensor decompositions. Intuitively, we decompose a given tensor block into blocks of smaller size, where the size is characterized by a set of moden ranks. We study different types of such decompositions. For each type we derive conditions under w ..."
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Cited by 21 (3 self)
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Abstract. In this paper we introduce a new class of tensor decompositions. Intuitively, we decompose a given tensor block into blocks of smaller size, where the size is characterized by a set of moden ranks. We study different types of such decompositions. For each type we derive conditions under which essential uniqueness is guaranteed. The parallel factor decomposition and Tucker’s decomposition can be considered as special cases in the new framework. The paper sheds new light on fundamental aspects of tensor algebra.
CramérRao Lower Bounds for LowRank Decomposition of Multidimensional Arrays
 IEEE Trans. on Signal Processing
, 2001
"... Unlike lowrank matrix decomposition, which is generically nonunique for rank greater than one, lowrank threeand higher dimensional array decomposition is unique, provided that the array rank is lower than a certain bound, and the correct number of components (equal to array rank) is sought in the ..."
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Cited by 16 (5 self)
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Unlike lowrank matrix decomposition, which is generically nonunique for rank greater than one, lowrank threeand higher dimensional array decomposition is unique, provided that the array rank is lower than a certain bound, and the correct number of components (equal to array rank) is sought in the decomposition. Parallel factor (PARAFAC) analysis is a common name for lowrank decomposition of higher dimensional arrays. This paper develops CramrRao Bound (CRB) results for lowrank decomposition of three and fourdimensional (3D and 4D) arrays, illustrates the behavior of the resulting bounds, and compares alternating least squares algorithms that are commonly used to compute such decompositions with the respective CRBs. Simpletocheck necessary conditions for a unique lowrank decomposition are also provided. Index TermsCramrRao bound, least squares method, matrix decomposition, multidimensional signal processing. I.
Blind identification of underdetermined mixtures by simultaneous matrix diagonalization
 IEEE Transactions on Signal Processing
"... Abstract—In this paper, we study simultaneous matrix diagonalizationbased techniques for the estimation of the mixing matrix in underdetermined independent component analysis (ICA). This includes a generalization to underdetermined mixtures of the wellknown SOBI algorithm. The problem is reformula ..."
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Cited by 12 (2 self)
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Abstract—In this paper, we study simultaneous matrix diagonalizationbased techniques for the estimation of the mixing matrix in underdetermined independent component analysis (ICA). This includes a generalization to underdetermined mixtures of the wellknown SOBI algorithm. The problem is reformulated in terms of the parallel factor decomposition (PARAFAC) of a higherorder tensor. We present conditions under which the mixing matrix is unique and discuss several algorithms for its computation. Index Terms—Canonical decomposition, higher order tensor, independent component analysis (ICA), parallel factor (PARAFAC) analysis, simultaneous diagonalization, underdetermined mixture. I.
Robust iterative fitting of multilinear models
 IEEE Transactions on Signal Processing
, 2005
"... Abstract—Parallel factor (PARAFAC) analysis is an extension of lowrank matrix decomposition to higher way arrays, also referred to as tensors. It decomposes a given array in a sum of multilinear terms, analogous to the familiar bilinear vector outer products that appear in matrix decomposition. PAR ..."
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Cited by 12 (0 self)
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Abstract—Parallel factor (PARAFAC) analysis is an extension of lowrank matrix decomposition to higher way arrays, also referred to as tensors. It decomposes a given array in a sum of multilinear terms, analogous to the familiar bilinear vector outer products that appear in matrix decomposition. PARAFAC analysis generalizes and unifies common array processing models, like joint diagonalization and ESPRIT; it has found numerous applications from blind multiuser detection and multidimensional harmonic retrieval, to clustering and nuclear magnetic resonance. The prevailing fitting algorithm in all these applications is based on (alternating) least squares, which is optimal for Gaussian noise. In many cases, however, measurement errors are far from being Gaussian. In this paper, we develop two iterative algorithms for the least absolute error fitting of general multilinear models. The first is based on efficient interior point methods for linear programming, employed in an alternating fashion. The second is based on a weighted median filtering iteration, which is particularly appealing from a simplicity viewpoint. Both are guaranteed to converge in terms of absolute error. Performance is illustrated by means of simulations, and compared to the pertinent Cramér–Rao bounds (CRBs). Index Terms—Array signal processing, nonGaussian noise, parallel factor analysis, robust model fitting. I.
Tensor Decompositions, Alternating Least Squares and Other Tales
 JOURNAL OF CHEMOMETRICS
, 2009
"... This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple “bottlenecks”, and on “swamps”. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity ..."
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Cited by 10 (3 self)
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This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple “bottlenecks”, and on “swamps”. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the ELS enhancement in these algorithms is discussed. Computer simulations feed this discussion.