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Shellability of Oriented Matroids
, 1989
"... In [Man82] A. Mandel proved that the maximal cells of an Oriented Matroid poset are Bshellable. Our result shows that the whole Oriented Matroid is shellable, too. ..."
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Cited by 1 (1 self)
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In [Man82] A. Mandel proved that the maximal cells of an Oriented Matroid poset are Bshellable. Our result shows that the whole Oriented Matroid is shellable, too.
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in largescale statistical models.
Planning Algorithms
, 2004
"... This book presents a unified treatment of many different kinds of planning algorithms. The subject lies at the crossroads between robotics, control theory, artificial intelligence, algorithms, and computer graphics. The particular subjects covered include motion planning, discrete planning, planning ..."
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Cited by 1108 (51 self)
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This book presents a unified treatment of many different kinds of planning algorithms. The subject lies at the crossroads between robotics, control theory, artificial intelligence, algorithms, and computer graphics. The particular subjects covered include motion planning, discrete planning, planning under uncertainty, sensorbased planning, visibility, decisiontheoretic planning, game theory, information spaces, reinforcement learning, nonlinear systems, trajectory planning, nonholonomic planning, and kinodynamic planning.
Lectures on matroids and oriented matroids
, 2005
"... These lecture notes were prepared for the Algebraic Combinatorics; in Europe (ACE) Summer School in Vienna, July 2005. ..."
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These lecture notes were prepared for the Algebraic Combinatorics; in Europe (ACE) Summer School in Vienna, July 2005.
FUTURE PATHS FOR INTEGER PROGRAMMING AND LINKS TO Artificial Intelligence
, 1986
"... Scope and PurposeA summary is provided of some of the recent (and a few notsorecent) developments that otTer promise for enhancing our ability to solve combinatorial optimization problems. These developments may be usefully viewed as a synthesis of the perspectives of operations research and arti ..."
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Cited by 356 (8 self)
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Scope and PurposeA summary is provided of some of the recent (and a few notsorecent) developments that otTer promise for enhancing our ability to solve combinatorial optimization problems. These developments may be usefully viewed as a synthesis of the perspectives of operations research and artificial intelligence. Although compatible with the use of algorithmic subroutines, the frameworks examined are primarily heuristic, based on the supposition that etTective solution of complex combinatorial structures in some cases may require a level of flexibility beyond that attainable by methods with formally demonstrable convergence properties. AbstractInteger programming has benefited from many innovations in models and methods. Some of the promising directions for elaborating these innovations in the future may be viewed from a framework that links the perspectives of artificial intelligence and operations research. To demonstrate this, four key areas are examined: (1) controlled randomization, (2) learning strategies, (3) induced decomposition and (4) tabu search. Each of these is shown to have characteristics that appear usefully relevant to developments on the horizon.
Onion Skins in Oriented Matroids
, 1993
"... We generalize the following theorem to oriented matroids: Consider a polytope P and a facet F 0 of P , let H denote the hyperplanes spanned by F 0 . Let d denote the diameter of the coskeleton of P . To each facet choose a defining inequality and let these set of inequalities F be partitioned by ..."
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We generalize the following theorem to oriented matroids: Consider a polytope P and a facet F 0 of P , let H denote the hyperplanes spanned by F 0 . Let d denote the diameter of the coskeleton of P . To each facet choose a defining inequality and let these set of inequalities F be partitioned
POSITIVELY ORIENTED MATROIDS ARE REALIZABLE
, 2013
"... We prove da Silva’s 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive matroid Grassmannian (or positive MacPhersonian) is homeomor ..."
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We prove da Silva’s 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive matroid Grassmannian (or positive Mac
KALAI ORIENTATIONS ON MATROID POLYTOPES
, 2005
"... Abstract. Let P a polytope and let G(P) be the graph of P. Following Gil Kalai, we say that an acyclic orientation O of G(P) is good if, for every nonempty face F of P, the induced graph G(F) has exactly one sink. Gil Kalai gave a simple way to tell a simple polytope from the good orientations of i ..."
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of its graph. This article is a broader study of “good orientations ” (of the graphs) on matroid polytopes. Dedicated to Michel Las Vergnas on the occasion of his 65th birthday 1.
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