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70
Tightening LP Relaxations for MAP using Message Passing
"... 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 ..."
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Cited by 65 (10 self)
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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. 1
On Dual Decomposition and Linear Programming Relaxations for Natural Language Processing
 In Proc. EMNLP
, 2010
"... This paper introduces dual decomposition as a framework for deriving inference algorithms for NLP problems. The approach relies on standard dynamicprogramming algorithms as oracle solvers for subproblems, together with a simple method for forcing agreement between the different oracles. The approa ..."
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Cited by 48 (2 self)
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This paper introduces dual decomposition as a framework for deriving inference algorithms for NLP problems. The approach relies on standard dynamicprogramming algorithms as oracle solvers for subproblems, together with a simple method for forcing agreement between the different oracles. The approach provably solves a linear programming (LP) relaxation of the global inference problem. It leads to algorithms that are simple, in that they use existing decoding algorithms; efficient, in that they avoid exact algorithms for the full model; and often exact, in that empirically they often recover the correct solution in spite of using an LP relaxation. We give experimental results on two problems: 1) the combination of two lexicalized parsing models; and 2) the combination of a lexicalized parsing model and a trigram partofspeech tagger. 1
Global connectivity potentials for random field models
 In CVPR
, 2008
"... Markov random field (MRF, CRF) models are popular in computer vision. However, in order to be computationally tractable they are limited to incorporate only local interactions and cannot model global properties, such as connectedness, which is a potentially useful highlevel prior for object segment ..."
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Cited by 34 (4 self)
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Markov random field (MRF, CRF) models are popular in computer vision. However, in order to be computationally tractable they are limited to incorporate only local interactions and cannot model global properties, such as connectedness, which is a potentially useful highlevel prior for object segmentation. In this work, we overcome this limitation by deriving a potential function that enforces the output labeling to be connected and that can naturally be used in the framework of recent MAPMRF LP relaxations. Using techniques from polyhedral combinatorics, we show that a provably tight approximation to the MAP solution of the resulting MRF can still be found efficiently by solving a sequence of maxflow problems. The efficiency of the inference procedure also allows us to learn the parameters of a MRF with global connectivity potentials by means of a cutting plane algorithm. We experimentally evaluate our algorithm on both synthetic data and on the challenging segmentation task of the PASCAL VOC 2008 data set. We show that in both cases the addition of a connectedness prior significantly reduces the segmentation error. 1.
MRF energy minimization and beyond via dual decomposition
 IN: IEEE PAMI. (2011
"... This paper introduces a new rigorous theoretical framework to address discrete MRFbased optimization in computer vision. Such a framework exploits the powerful technique of Dual Decomposition. It is based on a projected subgradient scheme that attempts to solve an MRF optimization problem by first ..."
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Cited by 33 (2 self)
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This paper introduces a new rigorous theoretical framework to address discrete MRFbased optimization in computer vision. Such a framework exploits the powerful technique of Dual Decomposition. It is based on a projected subgradient scheme that attempts to solve an MRF optimization problem by first decomposing it into a set of appropriately chosen subproblems and then combining their solutions in a principled way. In order to determine the limits of this method, we analyze the conditions that these subproblems have to satisfy and we demonstrate the extreme generality and flexibility of such an approach. We thus show that, by appropriately choosing what subproblems to use, one can design novel and very powerful MRF optimization algorithms. For instance, in this manner we are able to derive algorithms that: 1) generalize and extend stateoftheart messagepassing methods, 2) optimize very tight LPrelaxations to MRF optimization, 3) and take full advantage of the special structure that may exist in particular MRFs, allowing the use of efficient inference techniques such as, e.g, graphcut based methods. Theoretical analysis on the bounds related with the different algorithms derived from our framework and experimental results/comparisons using synthetic and real data for a variety of tasks in computer vision demonstrate the extreme potentials of our approach.
Messagepassing for graphstructured linear programs: Proximal methods and rounding schemes
, 2008
"... The problem of computing a maximum a posteriori (MAP) configuration is a central computational challenge associated with Markov random fields. A line of work has focused on “treebased ” linear programming (LP) relaxations for the MAP problem. This paper develops a family of superlinearly convergen ..."
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Cited by 30 (1 self)
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The problem of computing a maximum a posteriori (MAP) configuration is a central computational challenge associated with Markov random fields. A line of work has focused on “treebased ” linear programming (LP) relaxations for the MAP problem. This paper develops a family of superlinearly convergent algorithms for solving these LPs, based on proximal minimization schemes using Bregman divergences. As with standard messagepassing on graphs, the algorithms are distributed and exploit the underlying graphical structure, and so scale well to large problems. Our algorithms have a doubleloop character, with the outer loop corresponding to the proximal sequence, and an inner loop of cyclic Bregman divergences used to compute each proximal update. Different choices of the Bregman divergence lead to conceptually related but distinct LPsolving algorithms. We establish convergence guarantees for our algorithms, and illustrate their performance via some simulations. We also develop two classes of graphstructured rounding schemes, randomized and deterministic, for obtaining integral configurations from the LP solutions. Our deterministic rounding schemes use a “reparameterization ” property of our algorithms so that when the LP solution is integral, the MAP solution can be obtained even before the LPsolver converges to the optimum. We also propose a graphstructured randomized rounding scheme that applies to iterative LP solving algorithms in general. We analyze the performance of our rounding schemes, giving bounds on the number of iterations required, when the LP is integral, for the rounding schemes to obtain the MAP solution. These bounds are expressed in terms of the strength of the potential functions, and the energy gap, which measures how well the integral MAP solution is separated from other integral configurations. We also report simulations comparing these rounding schemes. 1
Tree Block Coordinate Descent for MAP in Graphical Models
"... A number of linear programming relaxations have been proposed for finding most likely settings of the variables (MAP) in large probabilistic models. The relaxations are often succinctly expressed in the dual and reduce to different types of reparameterizations of the original model. The dual objecti ..."
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Cited by 25 (3 self)
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A number of linear programming relaxations have been proposed for finding most likely settings of the variables (MAP) in large probabilistic models. The relaxations are often succinctly expressed in the dual and reduce to different types of reparameterizations of the original model. The dual objectives are typically solved by performing local block coordinate descent steps. In this work, we show how to perform block coordinate descent on spanning trees of the graphical model. We also show how all of the earlier dual algorithms are related to each other, giving transformations from one type of reparameterization to another while maintaining monotonicity relative to a common objective function. Finally, we quantify when the MAP solution can and cannot be decoded directly from the dual LP relaxation. 1
NormProduct Belief Propagation: PrimalDual MessagePassing for Approximate Inference
"... Abstract—Inference problems in graphical models can be represented as a constrained optimization of a free energy function. In this paper we treat both forms of probabilistic inference, estimating marginal probabilities of the joint distribution and finding the most probable assignment, through a un ..."
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Cited by 21 (7 self)
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Abstract—Inference problems in graphical models can be represented as a constrained optimization of a free energy function. In this paper we treat both forms of probabilistic inference, estimating marginal probabilities of the joint distribution and finding the most probable assignment, through a unified messagepassing algorithm architecture. In particular we generalize the Belief Propagation (BP) algorithms of sumproduct and maxproduct and treerewaighted (TRW) sum and max product algorithms (TRBP) and introduce a new set of convergent algorithms based on ”convexfreeenergy ” and LinearProgramming (LP) relaxation as a zerotemprature of a convexfreeenergy. The main idea of this work arises from taking a general perspective on the existing BP and TRBP algorithms while observing that they all are reductions from the basic optimization formula of f + ∑ i hi
Convergent message passing algorithms  a unifying view
 In Proc. Twentyeighth Conference on Uncertainty in Artificial Intelligence (UAI ’09
, 2009
"... Messagepassing algorithms have emerged as powerful techniques for approximate inference in graphical models. When these algorithms converge, they can be shown to find local (or sometimes even global) optima of variational formulations to the inference problem. But many of the most popular algorithm ..."
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Cited by 20 (0 self)
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Messagepassing algorithms have emerged as powerful techniques for approximate inference in graphical models. When these algorithms converge, they can be shown to find local (or sometimes even global) optima of variational formulations to the inference problem. But many of the most popular algorithms are not guaranteed to converge. This has lead to recent interest in convergent messagepassing algorithms. In this paper, we present a unified view of convergent messagepassing algorithms. We algorithm, treeconsistency bound optimization (TCBO) that is provably convergent in both its sum and max product forms. We then show that many of the existing convergent algorithms are instances of our TCBO algorithm, and obtain novel convergent algorithms “for free ” by exchanging maximizations and summations in existing algorithms. In particular, we show that Wainwright’s nonconvergent sumproduct algorithm for tree based variational bounds, is actually convergent with the right update order for the case where trees are monotonic chains. 1
An LP View of the Mbest MAP problem
"... We consider the problem of finding the M assignments with maximum probability in a probabilistic graphical model. We show how this problem can be formulated as a linear program (LP) on a particular polytope. We prove that, for tree graphs (and junction trees in general), this polytope has a particul ..."
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Cited by 15 (1 self)
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We consider the problem of finding the M assignments with maximum probability in a probabilistic graphical model. We show how this problem can be formulated as a linear program (LP) on a particular polytope. We prove that, for tree graphs (and junction trees in general), this polytope has a particularly simple form and differs from the marginal polytope in a single inequality constraint. We use this characterization to provide an approximation scheme for nontree graphs, by using the set of spanning trees over such graphs. The method we present puts the Mbest inference problem in the context of LP relaxations, which have recently received considerable attention and have proven useful in solving difficult inference problems. We show empirically that our method often finds the provably exact M best configurations for problems of high treewidth. A common task in probabilistic modeling is finding the assignment with maximum probability given a model. This is often referred to as the MAP (maximum aposteriori) problem.