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Cluster Algebras I: Foundations
 Journal of the American Mathematical Society
"... Abstract. In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras. Contents ..."
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Abstract. In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras. Contents
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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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
Tensor decompositions, state of the art and applications
 MATHEMATICS IN SIGNAL PROCESSING V
"... ..."
Canonical Tensor Decompositions
 ARCC WORKSHOP ON TENSOR DECOMPOSITION
, 2004
"... The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short pap ..."
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Cited by 42 (16 self)
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The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short paper, such as the maximal achievable rank of a tensor of given dimensions, or the computation of a CanD. Some questions are answered, and it turns out that the answers depend on the choice of the underlying field, and on tensor symmetry structure, which outlines a major difference compared to matrices.
Cluster algebras: Notes for the CDM03 conference
"... Abstract. This is an expanded version of the notes of our lectures ..."
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Cited by 39 (7 self)
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Abstract. This is an expanded version of the notes of our lectures
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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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.
Algebraic methods and arithmetic filtering for exact predicates on circle arcs
 Computational Geometry: Theory and Applications
"... The purpose of this paper is to present a new method to design exact geometric predicates in algorithms dealing with curved objects such as circular arcs. We focus on the comparison of the abscissae of two intersection points of circle arcs, which is known to be a difficult predicate involved in the ..."
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Cited by 38 (6 self)
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The purpose of this paper is to present a new method to design exact geometric predicates in algorithms dealing with curved objects such as circular arcs. We focus on the comparison of the abscissae of two intersection points of circle arcs, which is known to be a difficult predicate involved in the computation of arrangements of circle arcs. We present an algorithm for deciding the xorder of intersections from the signs of the coefficients of a polynomial, obtained by a general approach based on resultants. This method allows the use of efficient arithmetic and filtering techniques leading to fast implementation as shown by the experimental results. 1
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.