Results 1  10
of
27
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 ..."
Abstract

Cited by 585 (13 self)
 Add to MetaCart
(Show Context)
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.
A New Class of Upper Bounds on the Log Partition Function
 In Uncertainty in Artificial Intelligence
, 2002
"... Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis [11, 5, 4]. We introduce a new class of upper bounds on the log partition function, based on convex combinations of distribution ..."
Abstract

Cited by 225 (32 self)
 Add to MetaCart
Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis [11, 5, 4]. We introduce a new class of upper bounds on the log partition function, based on convex combinations of distributions in the exponential domain, that is applicable to an arbitrary undirected graphical model. In the special case of convex combinations of treestructured distributions, we obtain a family of variational problems, similar to the Bethe free energy, but distinguished by the following desirable properties: (i) they are convex, and have a unique global minimum; and (ii) the global minimum gives an upper bound on the log partition function. The global minimum is defined by stationary conditions very similar to those defining xed points of belief propagation (BP) or treebased reparameterization [see 13, 14]. As with BP fixed points, the elements of the minimizing argument can be used as approximations to the marginals of the original model. The analysis described here can be extended to structures of higher treewidth (e.g., hypertrees), thereby making connections with more advanced approximations (e.g., Kikuchi and variants [15, 10]).
Tightening LP Relaxations for MAP using Message Passing
, 2008
"... Linear Programming (LP) relaxations have become powerful tools for finding the most probable (MAP) configuration in graphical models. These relaxations can be solved efficiently using messagepassing algorithms such as belief propagation and, when the relaxation is tight, provably find the MAP confi ..."
Abstract

Cited by 112 (18 self)
 Add to MetaCart
(Show Context)
Linear Programming (LP) relaxations have become powerful tools for finding the most probable (MAP) configuration in graphical models. These relaxations can be solved efficiently using messagepassing algorithms such as belief propagation and, when the relaxation is tight, provably find the MAP configuration. The standard LP relaxation is not tight enough in many realworld problems, however, and this has lead to the use of higher order clusterbased LP relaxations. The computational cost increases exponentially with the size of the clusters and limits the number and type of clusters we can use. We propose to solve the cluster selection problem monotonically in the dual LP, iteratively selecting clusters with guaranteed improvement, and quickly resolving with the added clusters by reusing the existing solution. Our dual messagepassing algorithm finds the MAP configuration in protein sidechain placement, protein design, and stereo problems, in cases where the standard LP relaxation fails.
Approximate Inference in Graphical Models using LP Relaxations
, 2010
"... Graphical models such as Markov random fields have been successfully applied to a wide variety of fields, from computer vision and natural language processing, to computational biology. Exact probabilistic inference is generally intractable in complex models having many dependencies between the vari ..."
Abstract

Cited by 27 (1 self)
 Add to MetaCart
Graphical models such as Markov random fields have been successfully applied to a wide variety of fields, from computer vision and natural language processing, to computational biology. Exact probabilistic inference is generally intractable in complex models having many dependencies between the variables. We present new approaches to approximate inference based on linear programming (LP) relaxations. Our algorithms optimize over the cycle relaxation of the marginal polytope, which we show to be closely related to the first lifting of the SheraliAdams hierarchy, and is significantly tighter than the pairwise LP relaxation. We show how to efficiently optimize over the cycle relaxation using a cuttingplane algorithm that iteratively introduces constraints into the relaxation. We provide a criterion to determine which constraints would be most helpful in tightening the relaxation, and give efficient algorithms for solving the search problem of finding the best cycle constraint to add according to this criterion.
Structured region graphs: Morphing EP into GBP
 in UAI
, 2005
"... GBP and EP are two successful algorithms for approximate probabilistic inference, which are based on different approximation strategies. An open problem in both algorithms has been how to choose an appropriate approximation structure. We introduce “structured region graphs, ” a formalism which marri ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
(Show Context)
GBP and EP are two successful algorithms for approximate probabilistic inference, which are based on different approximation strategies. An open problem in both algorithms has been how to choose an appropriate approximation structure. We introduce “structured region graphs, ” a formalism which marries these two strategies, reveals a deep connection between them, and suggests how to choose good approximation structures. In this formalism, each region has an internal structure which defines an exponential family, whose sufficient statistics must be matched by the parent region. Reduction operators on these structures allow conversion between EP and GBP free energies. Thus it is revealed that all EP approximations on discrete variables are special cases of GBP, and conversely that some wellknown GBP approximations, such as overlapping squares, are special cases of EP. Furthermore, region graphs derived from EP have a number of good structural properties, including maxentnormality and overall counting number of one. The result is a convenient framework for producing highquality approximations with a useradjustable level of complexity. 1
An edge deletion semantics for belief propagation and its practical impact on approximation quality
 In AAAI
, 2006
"... We show in this paper that the influential algorithm of iterative belief propagation can be understood in terms of exact inference on a polytree, which results from deleting enough edges from the original network. We show that deleting edges implies adding new parameters into a network, and that the ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
We show in this paper that the influential algorithm of iterative belief propagation can be understood in terms of exact inference on a polytree, which results from deleting enough edges from the original network. We show that deleting edges implies adding new parameters into a network, and that the iterations of belief propagation are searching for values of these new parameters which satisfy intuitive conditions that we characterize. The new semantics lead to the following question: Can one improve the quality of approximations computed by belief propagation by recovering some of the deleted edges, while keeping the network easy enough for exact inference? We show in this paper that the answer is yes, leading to another question: How do we choose which edges to recover? To answer, we propose a specific method based on mutual information which is motivated by the edge deletion semantics. Empirically, we provide experimental results showing that the quality of approximations can be improved without incurring much additional computational cost. We also show that recovering certain edges with low mutual information may not be worthwhile as they increase the computational complexity, without necessarily improving the quality of approximations.
Convexifying the bethe free energy
 in Conference on Uncertainty in Artifical Intelligence (UAI
, 2009
"... The introduction of loopy belief propagation (LBP) revitalized the application of graphical models in many domains. Many recent works present improvements on the basic LBP algorithm in an attempt to overcome convergence and local optima problems. Notable among these are convexified free energy appro ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
(Show Context)
The introduction of loopy belief propagation (LBP) revitalized the application of graphical models in many domains. Many recent works present improvements on the basic LBP algorithm in an attempt to overcome convergence and local optima problems. Notable among these are convexified free energy approximations that lead to inference procedures with provable convergence and quality properties. However, empirically LBP still outperforms most of its convex variants in a variety of settings, as we also demonstrate here. Motivated by this fact we seek convexified free energies that directly approximate the Bethe free energy. We show that the proposed approximations compare favorably with stateofthe art convex free energy approximations. 1
Approximating marginals using discrete energy minimization
, 2012
"... classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specifi ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission.
Focusing Generalizations of Belief Propagation on Targeted Queries
"... A recent formalization of Iterative Belief Propagation (IBP) has shown that it can be understood as an exact inference algorithm on an approximate model that results from deleting every model edge. This formalization has led to (1) new realizations of Generalized Belief Propagation (GBP) in which ed ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
A recent formalization of Iterative Belief Propagation (IBP) has shown that it can be understood as an exact inference algorithm on an approximate model that results from deleting every model edge. This formalization has led to (1) new realizations of Generalized Belief Propagation (GBP) in which edges are recovered incrementally to improve approximation quality, and (2) edgerecovery heuristics that are motivated by improving the approximation quality of all node marginals in a graphical model. In this paper, we propose new edgerecovery heuristics, which are focused on improving the approximations of targeted node marginals. The new heuristics are based on newlyidentified properties of edge deletion, and in turn IBP, which guarantee the exactness of edge deletion in simple and idealized cases. These properties also suggest new improvements to IBP approximations which are based on performing edgebyedge corrections on targeted marginals, which are less costly than improvements based on edge recovery.
Generalized Belief Propagation for the Noiseless Capacity and Information Rates of RunLength Limited Constraints
 IEEE TRANSACTIONS ON COMMUNICATIONS
, 2011
"... The performance of the generalized belief propagation algorithm to compute the noiseless capacity and mutual information rates of finitesize twodimensional and threedimensional runlength limited constraints is investigated. In both cases, the problem is reduced to estimating the partition func ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
The performance of the generalized belief propagation algorithm to compute the noiseless capacity and mutual information rates of finitesize twodimensional and threedimensional runlength limited constraints is investigated. In both cases, the problem is reduced to estimating the partition function of graphical models with cycles. The partition function is then estimated using the regionbased free energy approximation technique. For each constraint, a method is proposed to choose the basic regions and to construct the region graph which provides the graphical framework to run the generalized belief propagation algorithm. Simulation results for the noiseless capacity of different constraints as a function of the size of the channel are reported. In the cases that tight lower and upper bounds on the Shannon capacity exist, convergence to the Shannon capacity is discussed. For noisy constrained channels, simulation results are reported for mutual information rates as a function of signaltonoise ratio.