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27
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
Matrices in Elimination Theory
, 1997
"... The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in ..."
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Cited by 54 (16 self)
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The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bézout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with an emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay's matrix and further conjecture the generalization of Macaulay's exact ratio...
Tensor decompositions, state of the art and applications
 MATHEMATICS IN SIGNAL PROCESSING V
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A complete parameterization of all positive rational extensions of a covariance sequence
 IEEE TRANS. AUTOMAT. CONTROL
, 1995
"... In this paper we formalize the observation that filtering and interpolation induce complementary, or ”dual” decompositions of the space of positive real rational functions of degree less than or equal to n. From this basic result about the geometry of the space of positive real functions, we are abl ..."
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Cited by 32 (21 self)
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In this paper we formalize the observation that filtering and interpolation induce complementary, or ”dual” decompositions of the space of positive real rational functions of degree less than or equal to n. From this basic result about the geometry of the space of positive real functions, we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence. On the one hand, by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space, we are able to develop estimates on the asymptotic behavior of the Schur parameters of positive rational extensions. On the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence. Indeed, motivated by its application to signal processing, speech processing and stochastic realization theory, this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by Georgiou. However, our basic result also enables us to analyze the robustness of this parameterization with respect to variations in the problem data. The methodology employed is a combination of complex analysis, geometry, linear systems and nonlinear dynamics.
Generalized Resultants Over Unirational Algebraic Varieties
, 1999
"... In this paper, we propose a new method, based on Bezoutian matrices, for computing a nontrivial multiple of the resultant over a projective variety X, which is described on an open subset by a parameterization. This construction, which generalizes the classical and toric one, also applies for instan ..."
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Cited by 30 (9 self)
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In this paper, we propose a new method, based on Bezoutian matrices, for computing a nontrivial multiple of the resultant over a projective variety X, which is described on an open subset by a parameterization. This construction, which generalizes the classical and toric one, also applies for instance to blowing up varieties and to residual intersection problems. We recall the classical notion of resultant over a variety X. Then we extend it to varieties which are parameterized on a dense open subset and give new conditions for the existence of the resultant over these varieties. We prove that any maximal nonzero minor of the corresponding Bezoutian matrix yields a nontrivial multiple of the resultant. We end with some experiments.
The 40 "generic" Positions of a Parallel Robot
, 1993
"... In this paper, we consider the direct kinematic problem of a parallel robot (called the Stewart platform or left hand). We want to show how the use of formal tools help us to guess the solution of this problem and then to establish it. We do not try here to give real time and numerical solutions to ..."
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Cited by 22 (4 self)
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In this paper, we consider the direct kinematic problem of a parallel robot (called the Stewart platform or left hand). We want to show how the use of formal tools help us to guess the solution of this problem and then to establish it. We do not try here to give real time and numerical solutions to the problem of inverse images but focus on tools of effective algebra, which can help us to know a little more about the geometric aspects of the question. In a first part, we describe experimentations done in order to obtain the number of "generic" positions of this robot, once the length of the arms are known. In a second part, we sketch the proof that the degree of the corresponding map is 40 (details will be given in another paper). We use explicit eliminations techniques, in order to get rid of the solution at infinity and we use Bezout's theorem on surfaces with circularity to conclude. The mechanism of the Stewart Platform (or left hand) is the following. Consider six fixed points (X ...
Resultant Over the Residual of a Complete Intersection
, 2001
"... In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual of a variety, defined here by a complete intersection G. We show that it corresponds to an irreducible divisor and give an explicit formula fo ..."
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Cited by 17 (5 self)
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In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual of a variety, defined here by a complete intersection G. We show that it corresponds to an irreducible divisor and give an explicit formula for its degree in the coefficients of each polynomial. Using the resolution of the ideal (F : G) and computing its regularity, we give a method for computing the residual resultant using a matrix which involves a Macaulay and a Bezout part. In particular, we show that this resultant is the gcd of all the maximal minors of this matrix. We illustrate our approach for the residual of points and end by some explicit examples.
Enumeration problems in Geometry, Robotics and Vision
, 1994
"... This paper presents different examples of enumerative geometry in robotics and vision. The aim is to obtain intersection formulas giving the number of solutions (on C ) of some classes of problems which appear in such applied fields. The first family of examples involves curves and surfaces dealing ..."
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Cited by 13 (7 self)
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This paper presents different examples of enumerative geometry in robotics and vision. The aim is to obtain intersection formulas giving the number of solutions (on C ) of some classes of problems which appear in such applied fields. The first family of examples involves curves and surfaces dealing with distances. In such problems, the varieties have a common part at infinity, called the umbilic, that we must not take into account in the enumeration. A new formula is obtained by means of simple algebraic manipulations and the approach is compared with the usual techniques of blowing up. In a second part, we consider the variety of displacements from an algebraic point of view. A structure, similar to what is called an algebra with straightening laws, is exhibited and allows us to compute the degree of the algebra representing the functions on the displacements. This approach has an immediate application in the direct kinematic problem of a parallel robot and in the problem of reconstru...
Genericity and Rank Deficiency of High Order Symmetric Tensors
 Proc. IEEE Int. Conference on Acoustics, Speech, and Signal Processing (ICASSP
, 2006
"... Blind Identification of UnderDetermined Mixtures (UDM) is involved in numerous applications, including MultiWay factor Analysis (MWA) and Signal Processing. In the latter case, the use of HighOrder Statistics (HOS) like Cumulants leads to the decomposition of symmetric tensors. Yet, little has be ..."
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Cited by 9 (6 self)
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Blind Identification of UnderDetermined Mixtures (UDM) is involved in numerous applications, including MultiWay factor Analysis (MWA) and Signal Processing. In the latter case, the use of HighOrder Statistics (HOS) like Cumulants leads to the decomposition of symmetric tensors. Yet, little has been published about rankrevealing decompositions of symmetric tensors. Definitions of rank are discussed, and useful results on Generic Rank are proved, with the help of tools borrowed from Algebraic Geometry. 1.