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21
Algebraic factor analysis: tetrads, pentads and beyond
"... Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a multivariate normal distribution. The parameter space of a factor analysis model is a subset of the cone of po ..."
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Cited by 28 (12 self)
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Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a multivariate normal distribution. The parameter space of a factor analysis model is a subset of the cone of positive definite matrices. This parameter space is studied from the perspective of computational algebraic geometry. Gröbner bases and resultants are applied to compute the ideal of all polynomial functions that vanish on the parameter space. These polynomials, known as model invariants, arise from rank conditions on a symmetric matrix under elimination of the diagonal entries of the matrix. Besides revealing the geometry of the factor analysis model, the model invariants also furnish useful statistics for testing goodnessoffit. 1
A finiteness theorem for markov bases of hierarchical models
 J. COMB. THEORY SER. A
, 2007
"... We show that the complexity of the Markov bases of multidimensional tables stabilizes eventually if a single table dimension is allowed to vary. In particular, if this table dimension is greater than a computable bound, the Markov bases consist of elements from Markov bases of smaller tables. We giv ..."
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Cited by 22 (3 self)
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We show that the complexity of the Markov bases of multidimensional tables stabilizes eventually if a single table dimension is allowed to vary. In particular, if this table dimension is greater than a computable bound, the Markov bases consist of elements from Markov bases of smaller tables. We give an explicit formula for this bound in terms of Graver bases. We also compute these Markov and Graver complexities for all K × 2 × 2 × 2 tables.
R.: Nfold integer programming
 Disc. Optim
"... Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. Th ..."
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Cited by 13 (5 self)
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Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations
Polyhedral conditions for the nonexistence of the MLE for hierarchical loglinear models
, 2006
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The list of indispensable moves of the unique minimal Markov basis for 3 × 4 × K and 4 × 4 × 4 contingency tables with fixed twodimensional marginals
, 2003
"... In this paper we present indispensable moves of Markov bases for connected Markov chains over threeway contingency tables with fixed twodimensional marginals. In Aoki and Takemura (2003a) we proved that there exists a unique minimal basis for 3 contingency tables consisting of four types of i ..."
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Cited by 9 (3 self)
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In this paper we present indispensable moves of Markov bases for connected Markov chains over threeway contingency tables with fixed twodimensional marginals. In Aoki and Takemura (2003a) we proved that there exists a unique minimal basis for 3 contingency tables consisting of four types of indispensable moves. Generalizing this result, we present a list of indispensable moves of the unique minimal Markov basis for 3 4 contingency tables. This list allows us to actually perform exact tests of no threefactor interaction in threeway tables of these sizes. There are 21 types of indispensable moves for the 3 K case and 14 types of indispensable moves for the 444 case. A proof of the fact that these indispensable moves form the unique minimal basis along the lines of Aoki and Takemura (2003a) is unfortunately too long and omitted. In addition we give a (nonexhaustive) list of indispensable moves for larger threeway tables. In this paper we prove some results on constructing indispensable moves from other indispensable moves. Our indispensable moves for larger tables were found by using these results combined with some computer searches. Closely connected notions to indispensability are the notions of fundamental moves and circuits discussed in Ohsugi and Hibi (1999, 2003). We also indicate whether our indispensable moves are fundamental or circuits.
The Graver complexity of integer programming
 Annals Combin
"... In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver complexity of the incidence matrix of the complete bipartite ..."
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Cited by 5 (3 self)
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In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver complexity of the incidence matrix of the complete bipartite graph K3,m satisfies g(m) = Ω(2 m), with g(m) ≥ 17 ·2 m−3 −7 for every m> 3.
Convex Integer Maximization via Graver Bases
, 2008
"... We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R d and w1x,..., wdx are linear forms on R n, max {c(w1x,..., wdx) : Ax = b, x ∈ N n}. This method works for arbitrary input data A, b, d, w1,..., wd, c. ..."
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Cited by 4 (2 self)
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We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R d and w1x,..., wdx are linear forms on R n, max {c(w1x,..., wdx) : Ax = b, x ∈ N n}. This method works for arbitrary input data A, b, d, w1,..., wd, c. Moreover, for fixed d and several important classes of programs in variable dimension, we prove that our algorithm runs in polynomial time. As a consequence, we obtain polynomial time algorithms for various types of multiway transportation problems, packing problems, and partitioning problems in variable dimension.
Finiteness theorems in stochastic integer programming
 Foundations of Computational Mathematics
, 2003
"... Abstract. We study Graver test sets for families of linear multistage stochastic integer programs with varying number of scenarios. We show that these test sets can be decomposed into finitely many “building blocks”, independent of the number of scenarios, and we give an effective procedure to comp ..."
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Cited by 3 (0 self)
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Abstract. We study Graver test sets for families of linear multistage stochastic integer programs with varying number of scenarios. We show that these test sets can be decomposed into finitely many “building blocks”, independent of the number of scenarios, and we give an effective procedure to compute them. The paper includes an introduction to NashWilliams ’ theory of betterquasiorderings, which is used to show termination of our algorithm. We also apply this theory to finiteness results for Hilbert functions.
On the Gröbner complexity of matrices
, 2007
"... In this paper we show that if for an integer matrix A the universal Gröbner basis of the associated toric ideal IA coincides with the Graver basis of A, then the Gröbner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. We conclude that for the matrices A3×3 ..."
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Cited by 3 (0 self)
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In this paper we show that if for an integer matrix A the universal Gröbner basis of the associated toric ideal IA coincides with the Graver basis of A, then the Gröbner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. We conclude that for the matrices A3×3 and A3×4, defining the 3 × 3 and 3 × 4 transportation problems,
FINITE GRÖBNER BASES IN INFINITE DIMENSIONAL POLYNOMIAL RINGS AND APPLICATIONS
"... Abstract. We introduce the theory of monoidal Gröbner bases, a concept which generalizes the familiar notion in a polynomial ring and allows for a description of Gröbner bases of ideals that are stable under the action of a monoid. The main motivation for developing this theory is to prove finitenes ..."
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Cited by 3 (1 self)
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Abstract. We introduce the theory of monoidal Gröbner bases, a concept which generalizes the familiar notion in a polynomial ring and allows for a description of Gröbner bases of ideals that are stable under the action of a monoid. The main motivation for developing this theory is to prove finiteness results in commutative algebra and applications. A basic theorem of this type is that ideals in infinitely many indeterminates stable under the action of the symmetric group are finitely generated up to symmetry. Using this machinery, we give new streamlined proofs of some classical finiteness theorems in algebraic statistics as well as a proof of the independent set conjecture of Ho¸sten and the second author. 1.