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Geometric Range Searching and Its Relatives
 CONTEMPORARY MATHEMATICS
"... ... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems. ..."
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Cited by 256 (40 self)
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... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems.
TENSOR RANK AND THE ILLPOSEDNESS OF THE BEST LOWRANK APPROXIMATION PROBLEM
"... There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, te ..."
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Cited by 75 (10 self)
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There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rankr approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders and ranks, regardless of the choice of norm (or even Brègman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank3 tensor has an optimal rank2 approximation. The notable exceptions to this misbehavior are rank1 tensors and order2 tensors (i.e. matrices). In a more positive spirit, we propose a natural way of overcoming the illposedness of the lowrank approximation problem, by using weak solutions when true solutions do not exist. For this to work, it is necessary to characterize the set of weak solutions, and we do this in the case of rank 2, order 3 (in arbitrary dimensions). In our work we emphasize the importance of closely studying concrete lowdimensional examples as a first step towards more general results. To this end, we present a detailed analysis of equivalence classes of 2 × 2 × 2 tensors, and we develop methods for extending results upwards to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; in other words, can be determined by a system of polynomial inequalities. We study some of these polynomials in cases of interest to us; in particular we make extensive use of the hyperdeterminant ∆ on R 2×2×2.
Multiplying matrices faster than coppersmithwinograd
 In Proc. 44th ACM Symposium on Theory of Computation
, 2012
"... We develop new tools for analyzing matrix multiplication constructions similar to the CoppersmithWinograd construction, and obtain a new improved bound on ω < 2.3727. 1 ..."
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Cited by 39 (5 self)
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We develop new tools for analyzing matrix multiplication constructions similar to the CoppersmithWinograd construction, and obtain a new improved bound on ω < 2.3727. 1
Some applications of generalized FFTs
 In Proceedings of DIMACS Workshop in Groups and Computation
, 1997
"... . Generalized FFTs are efficient algorithms for computing a Fourier transform of a function defined on finite group, or a bandlimited function defined on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applicat ..."
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Cited by 30 (5 self)
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. Generalized FFTs are efficient algorithms for computing a Fourier transform of a function defined on finite group, or a bandlimited function defined on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applications. This paper will attempt to survey some of these applications. Appendices include some more detailed examples. 1. A brief history The now "classical" Fast Fourier Transform (FFT) has a long and interesting history. Originally discovered by Gauss, and later made famous after being rediscovered by Cooley and Tukey [21], it may be viewed as an algorithm which efficiently computes the discrete Fourier transform or DFT. In between Gauss and CooleyTukey others developed special cases of the algorithm, usually motivated by the need to make efficient data analysis of one sort or another. To cite but a few examples, Gauss was interested in efficiently interpolating the orbits of asteroids [43...
Generalized polar varieties: Geometry and algorithms
, 2004
"... Let V be a closed algebraic subvariety of the n–dimensional projective space over the complex or real numbers and suppose that V is non–empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motiva ..."
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Cited by 27 (7 self)
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Let V be a closed algebraic subvariety of the n–dimensional projective space over the complex or real numbers and suppose that V is non–empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motivated in [2]. As particular instances of this notion of a generalized polar variety one reobtains the classic one and an alternative type of a polar varietiy, called dual. As main result of the present paper we show that for a generic choice of their parameters the generalized polar varieties of V are empty or equidimensional and smooth in any regular point of V. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we indicate how this description may be used in order to design in
Linear recurrences with polynomial coefficients and computation of the CartierManin operator on hyperelliptic curves
 In International Conference on Finite Fields and Applications (Toulouse
, 2004
"... Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of h ..."
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Cited by 21 (8 self)
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Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of hyperelliptic curves.
RIGIDITY AND POLYNOMIAL INVARIANTS OF CONVEX POLYTOPES
, 2004
"... We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obt ..."
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Cited by 19 (6 self)
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We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obtain sharp upper and lower bounds on the degree of these polynomial relations. In a special case of regular bipyramid we obtain explicit formulae for some of these relations. We conclude with a proof of Robbins Conjecture [R2] on the degree of generalized Heron polynomials.
Arithmetic circuits and counting complexity classes
 In Complexity of Computations and Proofs,J.Krajíček, Ed. Quaderni di Matematica
"... Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in ..."
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Cited by 17 (3 self)
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Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in
Change of ordering for regular chains in positive dimension
 IN ILIAS S. KOTSIREAS, EDITOR, MAPLE CONFERENCE 2006
, 2006
"... We discuss changing the variable ordering for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to d ..."
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Cited by 16 (7 self)
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We discuss changing the variable ordering for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to dimension zero and using NewtonHensel lifting techniques. The problems raised by the choice of the specialization points, the lack of the (crucial) information of what are the free and algebraic variables for the new ordering, and the efficiency regarding the other methods are discussed. Strong hypotheses (but not unusual) for the initial regular chain are required. Change of ordering in dimension zero is taken as a subroutine.
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 15 (1 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.