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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.
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.
PARAFAC analysis of the three dimensional tongue shape
 JASA
, 2003
"... this paper is to demonstrate that PARAFAC successfully represents the threedimensional shape of the tongue surface extracted from coronal magnetic resonance (MR) image stacks. Two types of measurement vectors are analyzed: a vector of 3D pseudofleshpoint coordinates extracted uniformly from the le ..."
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this paper is to demonstrate that PARAFAC successfully represents the threedimensional shape of the tongue surface extracted from coronal magnetic resonance (MR) image stacks. Two types of measurement vectors are analyzed: a vector of 3D pseudofleshpoint coordinates extracted uniformly from the length and width of the tongue surface, and a vector of 2D pseudofleshpoint coordinates extracted from a curve along the tongue surface close to the midsagittal plane. The 2D pseudofleshpoint coordinates are structurally similar to the type of data analyzed by Nix et al. (1996). Measurement data are indexed by speaker identity, phonemic vowel identity, and twodimensional measurement position. A variety of data preprocessing strategies were attempted; the method that yields the best results is similar but not identical to the preprocessing methods of Nix et al. (1996)
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
GENERAL TENSOR DECOMPOSITION, MOMENT MATRICES AND APPLICATIONS
, 2011
"... Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for fl ..."
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Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of QuasiHankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.