Results 1  10
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28
Algebraic Algorithms for Sampling from Conditional Distributions
 Annals of Statistics
, 1995
"... We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so a ..."
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Cited by 182 (15 self)
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We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so an excursion into computational algebraic geometry.
Solving Systems of Polynomial Equations
 American Mathematical Society, CBMS Regional Conferences Series, No 97
, 2002
"... Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, gam ..."
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Cited by 143 (10 self)
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Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, machine learning, control theory, and numerous other areas. The set of solutions to a system of polynomial equations is an algebraic variety, the basic object of algebraic geometry. The algorithmic study of algebraic varieties is the central theme of computational algebraic geometry. Exciting recent developments in symbolic algebra and numerical software for geometric calculations have revolutionized the field, making formerly inaccessible problems tractable, and providing fertile ground for experimentation and conjecture. The first half of this book furnishes an introduction and represents a snapshot of the state of the art regarding systems of polynomial equations. Afficionados of the wellknown text books by Cox, Little, and O’Shea will find familiar themes in the first five chapters: polynomials in one variable, Gröbner
Using monodromy to decompose solution sets of polynomial systems into irreducible components
 PROCEEDINGS OF A NATO CONFERENCE, FEBRUARY 25  MARCH 1, 2001, EILAT
, 2001
"... ..."
Introduction to numerical algebraic geometry
 In Solving Polynomial Equations, Series: Algorithms and Computation in Mathematics
, 2005
"... by ..."
Gröbner Bases and Polyhedral Geometry of Reducible and Cyclic Models
 J. Combin. Theory Ser. A
, 2002
"... This article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Gröbner bases of toric ideals associated to a subset of such models. We study the polytopes for cyclic models, and we give a complete polyhedral ..."
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Cited by 30 (9 self)
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This article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Gröbner bases of toric ideals associated to a subset of such models. We study the polytopes for cyclic models, and we give a complete polyhedral description of these polytopes in the binary cyclic case. Further we show how to build Gröbner bases of a reducible model from the Gröbner bases of its pieces. This result also gives a different proof that decomposable models have quadratic Gröbner bases. Finally, we present the solution of a problem posed by Vlach [13] concerning the dimension of fibers coming from models corresponding to the boundary of a simplex.
Some characterizations of minimal Markov basis for sampling from discrete conditional distributions
 Annals of the Institute of Statistical Mathematics
, 2002
"... this paper we give some basic characterizations of minimal Markov basis for a connected Markov chain, which is used for performing exact tests in discrete exponential families given a sufficient statistic. We also give a necessary and sufficient condition for uniqueness of minimal Markov basis. A ge ..."
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Cited by 25 (14 self)
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this paper we give some basic characterizations of minimal Markov basis for a connected Markov chain, which is used for performing exact tests in discrete exponential families given a sufficient statistic. We also give a necessary and sufficient condition for uniqueness of minimal Markov basis. A general algebraic algorithm for constructing a connected Markov chain was given by Diaconis and Sturmfels (1998). Their algorithm is based on computing Grobner basis for a certain ideal in a polynomial ring, which can be carried out by using available computer algebra packages. However structure and interpretation of Grobner basis produced by the packages are not necessarily clear, due to the lack of symmetry and minimality inherent in Grobner basis computation. Our approach clarifies partially ordered structure of minimal Markov basis
Numerical Irreducible Decomposition using Projections from Points on the Components
 In Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering, volume 286 of Contemporary Mathematics
"... To classify positive dimensional solution components of a polynomial system, we construct polynomials interpolating points sampled from each component. In previous work, points on an idimensional component were linearly projected onto a generically chosen (i + 1)dimensional subspace. In this p ..."
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Cited by 21 (14 self)
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To classify positive dimensional solution components of a polynomial system, we construct polynomials interpolating points sampled from each component. In previous work, points on an idimensional component were linearly projected onto a generically chosen (i + 1)dimensional subspace. In this paper, we present two improvements. First, we reduce the dimensionality of the ambient space by determining the linear span of the component and restricting to it. Second, if the dimension of the linear span is greater than i + 1, we use a less generic projection that leads to interpolating polynomials of lower degree, thus reducing the number of samples needed. While this more ecient approach still guarantees  with probability one  the correct determination of the degree of each component, the mere evaluation of an interpolating polynomial no longer certi es the membership of a point to that component. We present an additional numerical test that certi es membership in this new situation. We illustrate the performance of our new approach on some wellknown test systems.
Numerical Irreducible Decomposition using PHCpack
, 2003
"... Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the ..."
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Cited by 21 (14 self)
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Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the software package PHCpack can be used in conjunction with Maple and programs written in C. We describe a numerically stable algorithm for decomposing positive dimensional solution sets of polynomial systems into irreducible components.
Computer algebra and algebraic geometry  achievements and perspectives
 J. SYMBOLIC COMPUT
, 2000
"... In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One of the aims of this paper is to show, by means of example ..."
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Cited by 12 (1 self)
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In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One of the aims of this paper is to show, by means of examples, the usefulness of computer algebra to mathematical research. Computer algebra itself is a highly diversified discipline with applications to various areas of mathematics; many of these may be found in numerous research papers, proceedings or textbooks (cf. Buchberger and Winkler, 1998; Cohen et al., 1999; Matzat et al., 1998; ISSAC, 1988–1998). Here, I concentrate mainly on Gröbner bases and leave aside many other topics of computer algebra (cf. Davenport et al., 1988; Von zur Gathen and Gerhard, 1999; Grabmeier et al., 2000). In particular, I do not mention (multivariate) polynomial factorization, another major and important tool in computational algebraic geometry. Gröbner bases were introduced originally by Buchberger as a computational tool for testing solvability of a system of polynomial equations, to count the number of solutions (with multiplicities) if this number is finite and, more algebraically, to compute in the quotient ring modulo the given polynomials. Since then, Gröbner bases have become the major computational tool, not only in algebraic geometry. The importance of Gröbner bases for mathematical research in algebraic geometry is obvious and nowadays their use needs hardly any justification. Indeed, chapters on Gröbner bases and Buchberger’s algorithm (Buchberger, 1965) have been incorporated in many new textbooks on algebraic geometry such as the books of Cox et al. (1992, 1998) or the recent books of Eisenbud (1995) and Vasconcelos (1998), not to mention textbooks which are devoted exclusively to Gröbner bases, such as Adams and Loustaunou (1994),
A numerical local dimension test for points on the solution set of a system of polynomial equations
 SIAM J. NUMER. ANAL
"... The solution set V of a polynomial system, i.e., the set of common zeroes of a set of multivariate polynomials with complex coefficients, may contain several components, e.g., points, curves, surfaces, etc. Each component has attached to it a number of quantities, one of which is its dimension. Giv ..."
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Cited by 10 (4 self)
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The solution set V of a polynomial system, i.e., the set of common zeroes of a set of multivariate polynomials with complex coefficients, may contain several components, e.g., points, curves, surfaces, etc. Each component has attached to it a number of quantities, one of which is its dimension. Given a numerical approximation to a point p on the set V, this article presents an efficient algorithm to compute the maximum dimension of the irreducible components of V which pass through p, i.e., a local dimension test. Such a test is a crucial element in the homotopybased numerical irreducible decomposition algorithms of Sommese, Verschelde, and Wampler. This article presents computational evidence to illustrate that the use of this new algorithm greatly reduces the cost of socalled “junkpoint filtering, ” previously a significant bottleneck in the computation of a numerical irreducible decomposition. For moderate size examples, this results in well over an order of magnitude improvement in the computation of a numerical irreducible decomposition. As the computation of a numerical irreducible decomposition is a fundamental backbone operation, gains in efficiency in the irreducible decomposition algorithm carry over to the many computations which require this decomposition as an initial step. Another feature of a local dimension test is that one can now compute the irreducible components in a prescribed dimension without first computing the numerical irreducible decomposition of all higher dimensions. For example, one may compute the isolated solutions of a polynomial system without having to carry out the full numerical irreducible decomposition.