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19
Computing symmetric rank for symmetric tensors
 J. SYMBOLIC COMPUT
, 2011
"... We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for 2 × · · · × 2 tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetri ..."
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Cited by 44 (14 self)
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We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for 2 × · · · × 2 tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.
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 35 (9 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.
SYMMETRIC TENSOR DECOMPOSITION
, 2009
"... We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented ..."
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Cited by 29 (4 self)
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We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented by a homogeneous polynomial in n variables of total degree d. Thus the decomposition corresponds to a sum of powers of linear forms. The impact of this contribution is twofold. First it permits an efficient computation of the decomposition of any tensor of subgeneric rank, as opposed to widely used iterative algorithms with unproved convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the tensor rank.
SUBTRACTING A BEST RANK1 APPROXIMATION MAY INCREASE TENSOR RANK
"... Is has been shown that a best rankR approximation of an orderk tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using Candecomp/Parafac and related models. It has been observed numerically that, generally, this issue cannot be solved by consecutively computi ..."
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Cited by 17 (0 self)
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Is has been shown that a best rankR approximation of an orderk tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using Candecomp/Parafac and related models. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and substracting best rank1 approximations. The reason for this is that subtracting a best rank1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for realvalued 2 × 2 × 2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2 × 2 × 2 tensors (which have rank 2 or 3), subtracting a best rank1 approximation will result in a tensor that has rank 3 and lies on the boundary between the rank2 and rank3 sets. Hence, for a typical tensor of rank 2, subtracting a best rank1 approximation has increased the tensor rank.
Block component modelbased blind DSCDMA receiver
 IEEE TRANS. SIGNAL PROCESS
, 2008
"... In this paper, we consider the problem of blind multiuser separationequalization in the uplink of a wideband DSCDMA system, in a multipath propagation environment with intersymbolinterference (ISI). To solve this problem, we propose a multilinear algebraic receiver that relies on a new thirdord ..."
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Cited by 12 (0 self)
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In this paper, we consider the problem of blind multiuser separationequalization in the uplink of a wideband DSCDMA system, in a multipath propagation environment with intersymbolinterference (ISI). To solve this problem, we propose a multilinear algebraic receiver that relies on a new thirdorder tensor decomposition and generalizes the parallel factor (PARAFAC) model. Our method is deterministic and exploits the temporal, spatial and spectral diversities to collect the received data in a thirdorder tensor. The specific algebraic structure of this tensor is then used to decompose it in a sum of user’s contributions. The socalled Block Component Model (BCM) receiver does not require knowledge of the spreading codes, the propagation parameters, nor statistical independence of the sources but relies instead on a fundamental uniqueness condition of the decomposition that guarantees identifiability of every user’s contribution. The development of fast and reliable techniques to calculate this decomposition is important. We propose a blind receiver based either on an alternating least squares (ALS) algorithm or on a LevenbergMarquardt (LM) algorithm. Simulations illustrate the performance of the algorithms. Index Terms—Blind signal extraction, block component model
Stratification of the fourth secant variety of Veronese variety via the symmetric rank
 Adv. Pure Appl. Math
"... ar ..."
Batch and adaptive PARAFACbased blind separation of convolutive speech mixtures
 IEEE Audio, Speech, Language Process
, 2010
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Multihomogeneous polynomial decomposition using moment matrices
 INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND ALGEBRAIC COMPUTATION
, 2011
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Fast Alternating LS Algorithms for High Order CANDECOMP/PARAFAC Tensor Factorizations
, 2012
"... CANDECOMP/PARAFAC (CP) has found numerous applications in wide variety of areas such as in chemometrics, telecommunication, data mining, neuroscience, separated representations. For an orderN tensor, most CP algorithms can be computationally demanding due to computation of gradients which are rela ..."
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Cited by 5 (1 self)
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CANDECOMP/PARAFAC (CP) has found numerous applications in wide variety of areas such as in chemometrics, telecommunication, data mining, neuroscience, separated representations. For an orderN tensor, most CP algorithms can be computationally demanding due to computation of gradients which are related to products between tensor unfoldings and KhatriRao products of all factor matrices except one. These products have the largest workload in most CP algorithms. In this paper, we propose a fast method to deal with this issue. The method also reduces the extra memory requirements of CP algorithms. As a result, we can accelerate the standard alternating CP algorithms 2030 times for order5 and order6 tensors, and even higher ratios can be obtained for higher order tensors (e.g., N ≥ 10). The proposed method is more efficient than the stateoftheart ALS algorithm which operates two modes at a time (ALSo2) in the Eigenvector PLS toolbox, especially for tensors with order N ≥ 5 and high rank.
Jacobi algorithm for the best low multilinear rank approximation of symmetric tensors
 SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
, 2013
"... The problem discussed in this paper is the symmetric best low multilinear rank approximation of thirdorder symmetric tensors. We propose an algorithm based on Jacobi rotations, for which symmetry is preserved at each iteration. Two numerical examples are provided indicating the need for such algo ..."
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Cited by 5 (1 self)
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The problem discussed in this paper is the symmetric best low multilinear rank approximation of thirdorder symmetric tensors. We propose an algorithm based on Jacobi rotations, for which symmetry is preserved at each iteration. Two numerical examples are provided indicating the need for such algorithms. An important part of the paper consists of proving that our algorithm converges to stationary points of the objective function. This can be considered an advantage of the proposed algorithm over existing symmetrypreserving algorithms in the literature.