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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.
SemiSupervised Learning Literature Survey
, 2006
"... We review the literature on semisupervised learning, which is an area in machine learning and more generally, artificial intelligence. There has been a whole
spectrum of interesting ideas on how to learn from both labeled and unlabeled data, i.e. semisupervised learning. This document is a chapter ..."
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Cited by 757 (8 self)
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chapter excerpt from the author’s
doctoral thesis (Zhu, 2005). However the author plans to update the online version frequently to incorporate the latest development in the field. Please obtain the latest
version at http://www.cs.wisc.edu/~jerryzhu/pub/ssl_survey.pdf
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
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Cited by 801 (1 self)
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such
An introduction to variational methods for graphical models
 TO APPEAR: M. I. JORDAN, (ED.), LEARNING IN GRAPHICAL MODELS
"... ..."
Fusion, Propagation, and Structuring in Belief Networks
 ARTIFICIAL INTELLIGENCE
, 1986
"... Belief networks are directed acyclic graphs in which the nodes represent propositions (or variables), the arcs signify direct dependencies between the linked propositions, and the strengths of these dependencies are quantified by conditional probabilities. A network of this sort can be used to repre ..."
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Cited by 482 (8 self)
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Belief networks are directed acyclic graphs in which the nodes represent propositions (or variables), the arcs signify direct dependencies between the linked propositions, and the strengths of these dependencies are quantified by conditional probabilities. A network of this sort can be used to represent the generic knowledge of a domain expert, and it turns into a computational architecture if the links are used not merely for storing factual knowledge but also for directing and activating the data flow in the computations which manipulate this knowledge. The first part of the paper deals with the task of fusing and propagating the impacts of new information through the networks in such a way that, when equilibrium is reached, each proposition will be assigned a measure of belief consistent with the axioms of probability theory. It is shown that if the network is singly connected (e.g. treestructured), then probabilities can be updated by local propagation in an isomorphic network of parallel and autonomous processors and that the impact of new information can be imparted to all propositions in time proportional to the longest path in the network. The second part of the paper deals with the problem of finding a treestructured representation for a collection of probabilistically coupled propositions using auxiliary (dummy) variables, colloquially called "hidden causes. " It is shown that if such a treestructured representation exists, then it is possible to uniquely uncover the topology of the tree by observing pairwise dependencies among the available propositions (i.e., the leaves of the tree). The entire tree structure, including the strengths of all internal relationships, can be reconstructed in time proportional to n log n, where n is the number of leaves.
On the Chromatic Thresholds of Hypergraphs
, 2011
"... Let F be a family of runiform hypergraphs. The chromatic threshold of F is the infimum of all nonnegative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least c () V (H) r−1 has bounded chromatic number. This parameter has a long history for graphs (r = 2), ..."
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Cited by 2 (1 self)
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Let F be a family of runiform hypergraphs. The chromatic threshold of F is the infimum of all nonnegative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least c () V (H) r−1 has bounded chromatic number. This parameter has a long history for graphs (r = 2
On the Upper Chromatic Number of a Hypergraph
, 1995
"... We introduce the notion of a coedge of a hypergraph, which is a subset of vertices to be colored so that at least two vertices are of the same color. Hypergraphs with both edges and coedges are called mixed hypergraphs. The maximal number of colors for which there exists a mixed hypergraph colorin ..."
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Cited by 27 (8 self)
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coloring using all the colors is called the upper chromatic number of a hypergraph H and is denoted by (H). An algorithm for computing the number of colorings of a mixed hypergraph is proposed. The properties of the upper chromatic number and the colorings of some classes of hypergraphs are discussed. A
The Chromatic Numbers of Random Hypergraphs
 Random Struct. Alg
, 1998
"... : For a pair of integers 1### r, the #chromatic number of an runiform Z. hypergraph H# V, E is the minimal k, for which there exists a partition of V into subsets ## T,...,T such that e#T ## for every e#E. In this paper we determine the asymptotic 1 ki Z. behavior of the #chromatic number of t ..."
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Cited by 2 (1 self)
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: For a pair of integers 1### r, the #chromatic number of an runiform Z. hypergraph H# V, E is the minimal k, for which there exists a partition of V into subsets ## T,...,T such that e#T ## for every e#E. In this paper we determine the asymptotic 1 ki Z. behavior of the #chromatic number
Results 1  10
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