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Blind identification of overcomplete mixtures of sources
 BIOME)”, Lin. Algebra Appl
, 2004
"... The problem of Blind Identification of linear mixtures of independent random processes is known to be related to the diagonalization of some tensors. This problem is posed here in terms of a non conventional joint approximate diagonalization of several matrices. In fact, a congruent transform is app ..."
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Cited by 15 (12 self)
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The problem of Blind Identification of linear mixtures of independent random processes is known to be related to the diagonalization of some tensors. This problem is posed here in terms of a non conventional joint approximate diagonalization of several matrices. In fact, a congruent transform is applied to each of these matrices, the left transform being rectangular full rank, and the right one being unitary. The application in antenna signal processing is described, and suboptimal numerical algorithms are proposed.
ICA: A Potential Tool for BCI Systems
, 2008
"... Several studies dealing with independent component analysis (ICA)based braincomputer interface (BCI) systems have been reported. Most of them have only explored a limited number of ICA methods, mainly FastICA and INFOMAX. The aim of this article is to help the BCI community researchers, especiall ..."
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Cited by 9 (5 self)
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Several studies dealing with independent component analysis (ICA)based braincomputer interface (BCI) systems have been reported. Most of them have only explored a limited number of ICA methods, mainly FastICA and INFOMAX. The aim of this article is to help the BCI community researchers, especially those who are not familiar with ICA techniques, to choose an appropriate ICA method. For this purpose, the concept of ICA is reviewed and different measures of statistical independence are reported. Then, the application of these measures is illustrated through a brief description of the widely used algorithms in the ICA community, namely SOBI, COM2, JADE, ICAR, FastICA, and INFOMAX. The implementation of these techniques in the BCI field is also explained. Finally, a comparative study of these algorithms, conducted on simulated electroencephalography (EEG) data, shows that an appropriate selection of an ICA algorithm may significantly improve the capabilities of BCI systems.
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.
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.
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
HAL author manuscript IEEE Signal Processing Magazine 2008;25(1):5768 HAL author manuscript inserm00202706, version 1
"... This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyrig ..."
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This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder. 1 ICA: a potential tool for BCI systems
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.