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12
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.
Monica: Computing symmetric rank for symmetric tensors
 J. Symbolic Comput
"... 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 9 (4 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. Key words: Symmetric tensor, tensor rank, secant variety. 1.
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 7 (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.
SYMMETRIC TENSOR DECOMPOSITION
"... 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 6 (0 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. 1.
Lathauwer, A tensorbased blind DSCDMA receiver using simultaneous matrix diagonalization
 Proceedings of the Eighth IEEE Workshop on Signal Processing Advances in Wireless Communications (SPAWC 2007
, 2007
"... In this paper, we consider the problem of blind separationequalization of DSCDMA signals, from convolutive mixtures received by an antenna array. We suppose that multipath reflections occur in the farfield of this array and that InterSymbolInterference is caused by large delay spread. Our receiv ..."
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Cited by 5 (2 self)
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In this paper, we consider the problem of blind separationequalization of DSCDMA signals, from convolutive mixtures received by an antenna array. We suppose that multipath reflections occur in the farfield of this array and that InterSymbolInterference is caused by large delay spread. Our receiver is deterministic and relies on a thirdorder tensor decomposition, called decomposition in rank(L,L,1) terms, which is a generalization of the wellknown Parallel Factor (PARAFAC) decomposition. The technique we propose to calculate this decomposition is based on simultaneous matrix diagonalization, which is more accurate than the standard Alternating Least Squares (ALS) algorithm and also allows to blindly identify more users than previously stated. 1.
Lathauwer: Block component modelbased blind DSCDMA receiver
 IEEE Trans. Signal Process
, 2008
"... Abstract—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 t ..."
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Cited by 4 (0 self)
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Abstract—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
Batch and Adaptive PARAFACBased Blind Separation of Convolutive Speech Mixtures
"... Abstract—We present a frequencydomain technique based on PARAllel FACtor (PARAFAC) analysis that performs multichannel blind source separation (BSS) of convolutive speech mixtures. PARAFAC algorithms are combined with a dimensionality reduction step to significantly reduce computational complexity. ..."
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Cited by 2 (1 self)
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Abstract—We present a frequencydomain technique based on PARAllel FACtor (PARAFAC) analysis that performs multichannel blind source separation (BSS) of convolutive speech mixtures. PARAFAC algorithms are combined with a dimensionality reduction step to significantly reduce computational complexity. The identifiability potential of PARAFAC is exploited to derive a BSS algorithm for the underdetermined case (more speakers than microphones), combining PARAFAC analysis with timevarying Capon beamforming. Finally, a lowcomplexity adaptive version of the BSS algorithm is proposed that can track changes in the mixing environment. Extensive experiments with realistic and measured data corroborate our claims, including the underdetermined case. Signaltointerference ratio improvements of up to 6 dB are shown compared to stateoftheart BSS algorithms, at an order of magnitude lower computational complexity. Index Terms—Adaptive separation, blind speech separation,, joint diagonalization, PARAllel FACtor (PARAFAC), permutation ambiguity, underdetermined case.
HIGHER SECANT VARIETIES OF Pn × Pm EMBEDDED IN BIDEGREE (1, d)
, 2011
"... Abstract. Let X (n,m) (1,d) denote the SegreVeronese embedding of Pn × Pm via the sections of the sheaf O(1, d). We study the dimensions of higher secant varieties of X (n,m) (1,d) and we prove that there is no defective sth secant variety, except possibly for n values of s. Moreover when ( m+d) is ..."
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Abstract. Let X (n,m) (1,d) denote the SegreVeronese embedding of Pn × Pm via the sections of the sheaf O(1, d). We study the dimensions of higher secant varieties of X (n,m) (1,d) and we prove that there is no defective sth secant variety, except possibly for n values of s. Moreover when ( m+d) is a multiple of d (m + n + 1), the sth secant variety of X (n,m) has the expected dimension for (1,d) every s.
Author manuscript, published in "Linear Algebra and Applications 433, 1112 (2010) 18511872" SYMMETRIC TENSOR DECOMPOSITION
, 2009
"... Abstract. We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variab ..."
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Abstract. We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring’s problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester’s approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasiHankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in nongeneric cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. 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 global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding
Special issue, 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 ..."
Abstract
 Add to MetaCart
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. 1