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On the Skeleton of the Metric Polytope
, 2001
"... We consider convex polyhedra with applications to wellknown combinatorial optimization problems: the metric polytope mn and its relatives. For n # 6 the description of the metric polytope is easy as mn has at most 544 vertices partitioned into 3 orbits; m7  the largest previously known instan ..."
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Cited by 14 (1 self)
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We consider convex polyhedra with applications to wellknown combinatorial optimization problems: the metric polytope mn and its relatives. For n # 6 the description of the metric polytope is easy as mn has at most 544 vertices partitioned into 3 orbits; m7  the largest previously known
On the face lattice of the metric polytope
 in: Discrete and computational geometry, Lecture Notes in Comput. Sci. 2866
, 2003
"... Abstract. In this paper we study enumeration problems for polytopes arising from combinatorial optimization problems. While these polytopes turn out to be quickly intractable for enumeration algorithms designed for general polytopes, tailormade algorithms using their rich combinatorial features can ..."
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Cited by 8 (1 self)
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can exhibit strong performances. The main engine of these combinatorial algorithms is the use of the large symmetry group of combinatorial polytopes. Specifically we consider a polytope with applications to the wellknown maxcut and multicommodity flow problems: the metric polytope mn on n nodes. We
The Combinatorial Structure of Small Cut and Metric Polytopes
 COMBINATORICS AND GRAPH THEORY, WORLD SCIENTIFIC
, 1995
"... We study the combinatorial structure of the cut and metric polytopes on n nodes for n <= 5. Those two polytopes have a complicated geometrical structure, but using their large symmetry group, we can completely describe their face lattices. We present, for any n, some orbits of faces and give ne ..."
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Cited by 7 (5 self)
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We study the combinatorial structure of the cut and metric polytopes on n nodes for n <= 5. Those two polytopes have a complicated geometrical structure, but using their large symmetry group, we can completely describe their face lattices. We present, for any n, some orbits of faces and give
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.
A fast procedure for computing the distance between complex objects in three space
 in Proc. IEEE Int. Conf. on Robotics and Automation
, 1987
"... AbstractAn efficient and reliable algorithm for computing the Euclidean distance between a pair of convex sets in Rm is described. Extensive numerical experience with a broad family of polytopes in R3 shows that the computational cost is approximately linear in the total number of vertices specifyi ..."
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Cited by 348 (9 self)
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AbstractAn efficient and reliable algorithm for computing the Euclidean distance between a pair of convex sets in Rm is described. Extensive numerical experience with a broad family of polytopes in R3 shows that the computational cost is approximately linear in the total number of vertices
Existence of minimal models for varieties of log general type
 J. AMER. MATH. SOC
, 2008
"... We prove that the canonical ring of a smooth projective variety is finitely generated. ..."
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Cited by 386 (34 self)
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We prove that the canonical ring of a smooth projective variety is finitely generated.
ROC Graphs: Notes and Practical Considerations for Researchers
, 2004
"... Receiver Operating Characteristics (ROC) graphs are a useful technique for organizing classifiers and visualizing their performance. ROC graphs are commonly used in medical decision making, and in recent years have been increasingly adopted in the machine learning and data mining research communitie ..."
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Cited by 378 (1 self)
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Receiver Operating Characteristics (ROC) graphs are a useful technique for organizing classifiers and visualizing their performance. ROC graphs are commonly used in medical decision making, and in recent years have been increasingly adopted in the machine learning and data mining research communities. Although ROC graphs are apparently simple, there are some common misconceptions and pitfalls when using them in practice. This article serves both as a tutorial introduction to ROC graphs and as a practical guide for using them in research.
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