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11
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.
MACAULAY STYLE FORMULAS FOR SPARSE RESULTANTS
, 2002
"... We present formulas for computing the resultant of sparse polynomials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials. ..."
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Cited by 22 (0 self)
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We present formulas for computing the resultant of sparse polynomials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials.
The topology of the Voronoi diagram of three lines
 PROCEEDINGS OF SYMPOSIUM ON COMPUTATIONAL GEOMETRY, ACM PRESS, SOUTHKOREA
, 2007
"... We give a complete description of the Voronoi diagram, in R³, of three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. In particular, we show that the topology of the Voronoi diagram is invariant for three such lines. The trisector consists of four ..."
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Cited by 14 (3 self)
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We give a complete description of the Voronoi diagram, in R³, of three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. In particular, we show that the topology of the Voronoi diagram is invariant for three such lines. The trisector consists of four unbounded branches of either a nonsingular quartic or of a nonsingular cubic and a line that do not intersect in real space. Each cell of dimension two consists of two connected components on a hyperbolic paraboloid that are bounded, respectively, by three and one of the branches of the trisector. We introduce a proof technique, which relies heavily upon modern tools of computer algebra, and is of interest in its own right. This characterization yields some fundamental properties of the Voronoi diagram of three lines. In particular, we present linear semialgebraic tests for separating the two connected components of each twodimensional Voronoi cell and for separating the four connected components of the trisector. This enables us to answer queries of the form, given a point, determine in which connected component of which cell it lies. We also show that the arcs of the trisector are monotonic in some direction. These properties imply that points on the trisector of three lines can be sorted along each branch using only linear semialgebraic tests.
On the Topology of Planar Algebraic Curves
"... We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e cienc ..."
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Cited by 13 (2 self)
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We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e ciency of our algorithm.
The predicates for the Voronoi diagram of ellipses
 In Proc. 22th Annual ACM Symp. on Computational Geometry
, 2006
"... This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exter ..."
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Cited by 12 (6 self)
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This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exterior point; κ2 decides the position of a query ellipse relative to an external bitangent line of 2 given ellipses; κ3 decides the position of a query ellipse relative to a Voronoi circle of 3 given ellipses; κ4 determines the type of conflict between a Voronoi edge, defined by 4 given ellipses, and a query ellipse. The paper is restricted to nonintersecting ellipses, but the extension to arbitrary ones is straightforward. The ellipses are input in parametric representation or constructively. For κ1 and κ2 we derive optimal algebraic conditions, solve them exactly and provide efficient implementations in C++. For κ3 we compute a tight bound on the number of complex tritangent circles and use the parametric form of the ellipses in order to design an exact subdivisionbased algorithm, which is implemented on Maple. This approach essentially answers κ4 as well. We conclude with current work on optimizing κ3 and implementing it in C++.
Multihomogeneous polynomial decomposition using moment matrices
 International Symposium on Symbolic and Algebraic Computation
, 2011
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 7 (4 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
GENERAL TENSOR DECOMPOSITION, MOMENT MATRICES AND APPLICATIONS
, 2011
"... 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 exten ..."
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Cited by 3 (2 self)
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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.
Computational geometry for curved objects. Voronoi diagrams in the plane
, 2009
"... Abstract. We examine the problem of computing exactly the Delaunay graph (and the dual Voronoi diagram) of a set of, possibly intersecting, smooth convex pseudocircles in the Euclidean plane, given in parametric form. Pseudocircles are (convex) closed curves, every pair of which has at most two ..."
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Cited by 1 (1 self)
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Abstract. We examine the problem of computing exactly the Delaunay graph (and the dual Voronoi diagram) of a set of, possibly intersecting, smooth convex pseudocircles in the Euclidean plane, given in parametric form. Pseudocircles are (convex) closed curves, every pair of which has at most two intersection points. We propose robust end efficient algorithms for all required predicates under the exact computation paradigm, analyzing their algebraic complexity. To speed up the algebraic computations, we exploit geometric properties of the problem and provide a subdivisionbased algorithm that exhibits quadratic convergence, allowing for realtime evaluations. Finally, we present a cgalbased c++ implementation for the case of ellipses, which is, to the best of our knowledge, the first exact implementation in nonlinear computational geometry. Our code spends about 98 sec to construct the Delaunay graph of 128 nonintersecting ellipses, when no degeneracies occur. It is faster than the cgal segment Delaunay graph, when ellipses are approximated by kgons for k> 15. 1
Kingdom. pp.525529. <hal00435908>
, 2009
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
Abstract
 Add to MetaCart
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.