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690
Constructing Free Energy Approximations and Generalized Belief Propagation Algorithms
 IEEE Transactions on Information Theory
, 2005
"... Important inference problems in statistical physics, computer vision, errorcorrecting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation (BP) algorithm is an efficient way to solve these problems t ..."
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Cited by 412 (12 self)
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Important inference problems in statistical physics, computer vision, errorcorrecting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation (BP) algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree, but only approximate when the factor graph has cycles. We show that BP fixed points correspond to the stationary points of the Bethe approximation of the free energy for a factor graph. We explain how to obtain regionbased free energy approximations that improve the Bethe approximation, and corresponding generalized belief propagation (GBP) algorithms. We emphasize the conditions a free energy approximation must satisfy in order to be a “valid ” or “maxentnormal ” approximation. We describe the relationship between four different methods that can be used to generate valid approximations: the “Bethe method, ” the “junction graph method, ” the “cluster variation method, ” and the “region graph method.” Finally, we explain how to tell whether a regionbased approximation, and its corresponding GBP algorithm, is likely to be accurate, and describe empirical results showing that GBP can significantly outperform BP.
Hierarchical Cooperation Achieves Optimal Capacity Scaling in Ad Hoc Networks
, 2007
"... n source and destination pairs randomly located in an area want to communicate with each other. Signals transmitted from one user to another at distance r apart are subject to a power loss of r −α as well as a random phase. We identify the scaling laws of the information theoretic capacity of the ne ..."
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Cited by 153 (15 self)
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n source and destination pairs randomly located in an area want to communicate with each other. Signals transmitted from one user to another at distance r apart are subject to a power loss of r −α as well as a random phase. We identify the scaling laws of the information theoretic capacity of the network. In the case of dense networks, where the area is fixed and the density of nodes increasing, we show that the total capacity of the network scales linearly with n. This improves on the best known achievability result of n 2/3 of [1]. In the case of extended networks, where the density of nodes is fixed and the area increasing linearly with n, we show that this capacity scales as n 2−α/2 for 2 ≤ α < 3 and n for α ≥ 3. The best known earlier result [2] identified the scaling law for α> 4. Thus, much better scaling than multihop can be achieved in dense networks, as well as in extended networks with low attenuation. The performance gain is achieved by intelligent node cooperation and distributed MIMO communication. The key ingredient is a hierarchical and digital architecture for nodal exchange of information for realizing the cooperation.
Some Combinatorial Properties of Schubert Polynomials
, 1993
"... Schubert polynomials were introduced by Bernstein Gelfand Gelfand and De mazure, and were extensively developed by Lascoux, Schiitzenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polyno mial in terms of the reduced decompositions of the permutation ..."
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Cited by 123 (10 self)
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Schubert polynomials were introduced by Bernstein Gelfand Gelfand and De mazure, and were extensively developed by Lascoux, Schiitzenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polyno mial in terms of the reduced decompositions of the permutation w. Using this Supported by the National Physical Science Consortium.
TreeBased Reparameterization Framework for Analysis of Belief Propagation and Related Algorithms
, 2001
"... We present a treebased reparameterization framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles. This class includes the belief propagation or sumproduct algorithm [39, 36], as well as a rich set of variations and ext ..."
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Cited by 101 (21 self)
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We present a treebased reparameterization framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles. This class includes the belief propagation or sumproduct algorithm [39, 36], as well as a rich set of variations and extensions of belief propagation. Algorithms in this class can be formulated as a sequence of reparameterization updates, each of which entails refactorizing a portion of the distribution corresponding to an acyclic subgraph (i.e., a tree). The ultimate goal is to obtain an alternative but equivalent factorization using functions that represent (exact or approximate) marginal distributions on cliques of the graph. Our framework highlights an important property of BP and the entire class of reparameterization algorithms: the distribution on the full graph is not changed. The perspective of treebased updates gives rise to a simple and intuitive characterization of the fixed points in terms of tree consistency. We develop interpretations of these results in terms of information geometry. The invariance of the distribution, in conjunction with the fixed point characterization, enables us to derive an exact relation between the exact marginals on an arbitrary graph with cycles, and the approximations provided by belief propagation, and more broadly, any algorithm that minimizes the Bethe free energy. We also develop bounds on this approximation error, which illuminate the conditions that govern their accuracy. Finally, we show how the reparameterization perspective extends naturally to more structured approximations (e.g., Kikuchi and variants [52, 37]) that operate over higher order cliques.
An Algorithmic Theory of Lattice Points in Polyhedra
, 1999
"... We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higherdimensional Dedekind sums, complexity of the Presburger arithmetic, efficien ..."
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Cited by 91 (6 self)
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We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higherdimensional Dedekind sums, complexity of the Presburger arithmetic, efficient computations with rational functions, and others. Although the main slant is algorithmic, structural results are discussed, such as relations to the general theory of valuations on polyhedra and connections with the theory of toric varieties. The paper surveys known results and presents some new results and connections.
Exact Enumeration Of 1342Avoiding Permutations A Close Link With Labeled Trees And Planar Maps
, 1997
"... Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of inde ..."
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Cited by 84 (7 self)
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Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of indecomposable 1342avoiding permutations of length n equals that of labeled plane trees of a certain type on n vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, H(x) turns out to be algebraic, proving the first nonmonotonic, longerthanthree instance of a conjecture of Zeilberger and Noonan. We also prove that n p Sn (1342) converges to 8, so in particular, limn!1 (Sn (1342)=Sn (1234)) = 0.
Geometry of the Space of Phylogenetic Trees
 Adv. in Appl. Math
, 1999
"... ields to graphically represent various types of hierarchical relationships, including evolutionary relationships between species, divergent patterns between subpopulations and evolutionary relationships between genes. These trees are generally rooted and semilabeled, i.e., they descend from a singl ..."
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Cited by 74 (1 self)
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ields to graphically represent various types of hierarchical relationships, including evolutionary relationships between species, divergent patterns between subpopulations and evolutionary relationships between genes. These trees are generally rooted and semilabeled, i.e., they descend from a single node called the root, bifurcate at lower nodes and end at terminal nodes, called tips or leaves; the leaves are labeled by the names of the species, subpopulations or genes being studied. In biological studies the latter are called operational taxonomic units (OTU's). Traditionally, trees were inferred form morphological similarities among the OTU's. To build an evolutionary species tree, or phylogenetic tree, two species which shared the most characteristics were classified as `siblings' and assumed to share a common ancestor which is not the ancestor of any other species. Such `siblings' are said to be homologous, and it is this basic homo
Comparing clusterings: an axiomatic view
 In ICML ’05: Proceedings of the 22nd international conference on Machine learning
, 2005
"... This paper views clusterings as elements of a lattice. Distances between clusterings are analyzed in their relationship to the lattice. From this vantage point, we first give an axiomatic characterization of some criteria for comparing clusterings, including the variation of information and the unad ..."
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Cited by 69 (3 self)
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This paper views clusterings as elements of a lattice. Distances between clusterings are analyzed in their relationship to the lattice. From this vantage point, we first give an axiomatic characterization of some criteria for comparing clusterings, including the variation of information and the unadjusted Rand index. Then we study other distances between partitions w.r.t these axioms and prove an impossibility result: there is no “sensible” criterion for comparing clusterings that is simultaneously (1) aligned with the lattice of partitions, (2) convexely additive, and (3) bounded. 1.
Generating Functions for Generating Trees
 PROCEEDINGS OF 11TH FORMAL POWER SERIES AND ALGEBRAIC COMBINATORICS
, 1999
"... Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumerati ..."
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Cited by 68 (18 self)
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Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.