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29
Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Cited by 101 (22 self)
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Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
Decomposition of quantics in sums of powers of linear forms
 Signal Processing
, 1996
"... Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent complementary use of HighOrder Statistics (HOS). However, very few special purpose tools are at disposal for manipulating such objects in engineering prob ..."
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Cited by 91 (23 self)
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Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent complementary use of HighOrder Statistics (HOS). However, very few special purpose tools are at disposal for manipulating such objects in engineering problems. In this paper, the decomposition of a symmetric tensor into a sum of simpler ones is focused on, and links with the theory of homogeneous polynomials in several variables (i.e. quantics) are pointed out. This decomposition may be seen as a formal extension of the Eigen Value Decomposition (EVD), known for symmetric matrices. By reviewing the state of the art, quite surprising statements are emphasized, that explain why the problem is much more complicated in the tensor case than in the matrix case. Very few theoretical results can be applied in practice, even for cubics or quartics, because proofs are not constructive. Nevertheless in the binary case, we have more freedom to devise numerical algorithms. Keywords. Tensors, Polynomials, Diagonalization, EVD, HighOrder Statistics, Cumulants. 1
Geometric aspects of polynomial interpolation in more variables and of Waring’s problem
 EUROPEAN CONGRESS OF MATHEMATICS, VOL. I (BARCELONA
, 2001
"... In this paper I treat the problem of determining the dimension of the vector space of homogeneous polynomials in a given number of variables vanishing with some of their derivatives at a finite set of general points in projective space. I will illustrate the geometric meaning of this problem and th ..."
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Cited by 55 (7 self)
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In this paper I treat the problem of determining the dimension of the vector space of homogeneous polynomials in a given number of variables vanishing with some of their derivatives at a finite set of general points in projective space. I will illustrate the geometric meaning of this problem and the main results and conjectures about it. I will finally point out its connection with the socalled Waring’s problem for forms, of which I will also indicate the geometric meaning.
On the Alexander–Hirschowitz theorem
 J. Pure Appl. Algebra
, 2008
"... The AlexanderHirschowitz theorem says that a general collection of k double points in Pn imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, con ..."
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Cited by 39 (8 self)
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The AlexanderHirschowitz theorem says that a general collection of k double points in Pn imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a selfcontained proof based mainly on previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case d = 3, where our proof is shorter. We end with an account of the history of the work on this problem.
A tropical approach to secant dimensions
, 2006
"... Tropical geometry yields good lower bounds, in terms of certain combinatorialpolyhedral optimisation problems, on the dimensions of secant varieties. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for th ..."
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Cited by 38 (3 self)
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Tropical geometry yields good lower bounds, in terms of certain combinatorialpolyhedral optimisation problems, on the dimensions of secant varieties. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are nondefective; this proof might be generalisable to cover all Veronese embeddings, whose secant dimensions are known from the groundbreaking but difficult work of Alexander and Hirschowitz. Also, the nondefectiveness of certain Segre embeddings is proved, which cannot be proved with the rook covering argument already known in the literature. Short selfcontained introductions to secant varieties and the required tropical geometry are included.
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 33 (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 30 (5 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.
A brief proof of a maximal rank theorem for generic double points in projective space
, 2000
"... We give a simple proof of the following theorem of J. Alexander and A. Hirschowitz: Given a general set of points in projective space, the homogeneous ideal of polynomials that are singular at these points has the expected dimension in each degree of 4 and higher, except in 3 cases. ..."
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Cited by 28 (2 self)
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We give a simple proof of the following theorem of J. Alexander and A. Hirschowitz: Given a general set of points in projective space, the homogeneous ideal of polynomials that are singular at these points has the expected dimension in each degree of 4 and higher, except in 3 cases.
Topics on Interpolation Problems in Algebraic Geometry
, 2004
"... These are notes of the lectures given by the authors during the school/workshop “Polynomial Interpolation and Projective Embeddings”. We mainly focus our attention on the planar case and on the Segre and HarbourneHirschowitz Conjectures. We discuss the state of the art introducing several results ..."
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Cited by 11 (3 self)
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These are notes of the lectures given by the authors during the school/workshop “Polynomial Interpolation and Projective Embeddings”. We mainly focus our attention on the planar case and on the Segre and HarbourneHirschowitz Conjectures. We discuss the state of the art introducing several results and different techniques.