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Tightening LP Relaxations for MAP using Message Passing

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by David Sontag , Talya Meltzer , Amir Globerson , Tommi Jaakkola , Yair Weiss
Citations:38 - 8 self
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BibTeX

@MISC{Sontag_tighteninglp,
    author = {David Sontag and Talya Meltzer and Amir Globerson and Tommi Jaakkola and Yair Weiss},
    title = {Tightening LP Relaxations for MAP using Message Passing},
    year = {}
}

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Abstract

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 message-passing 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 real-world problems, however, and this has lead to the use of higher order cluster-based 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 re-solving with the added clusters by reusing the existing solution. Our dual message-passing algorithm finds the MAP configuration in protein sidechain placement, protein design, and stereo problems, in cases where the standard LP relaxation fails. 1

Citations

708 Taxonomy and evaluation of dense two-frame stereo correspondence algorithms - Scharstein, Szeliski
279 Constructing free-energy approximations and generalized belief propagation algorithms - Yedidia, Freeman, et al.
268 Graphical models, exponential families, and variational - Wainwright, Jordan - 2008
180 BConvergent tree-reweighted message passing for energy minimization - Kolmogorov - 2006
82 Finding MAPs for belief networks is NP-hard - Shimony
76 MAP estimation via agreement on trees: messagepassing and linear programming. Information Theory - Wainwright, Jaakkola, et al. - 2005
63 Residual belief propagation: Informed scheduling for asynchronous message passing - Elidan, McGraw, et al. - 2006
45 Fixing maxproduct: Convergent message passing algorithms for MAP LP-relaxations - Globerson, Jaakkola - 2007
44 Iterative Join Graph propagation - Dechter, Mateescu - 2002
41 On the optimality of treereweighted max-product message passing - Kolmogorov, Wainwright - 2005
38 On the generation of alternative explanations with implications for belief revision - Santos - 1991
33 Linear programming relaxations and belief propagation - an empirical study - Yanover, Meltzer, et al. - 1907
32 MAP estimation, linear programming and belief propagation with convex free energies - Weiss, Yanover, et al.
31 BNew outer bounds on the marginal polytope - Sontag, Jaakkola - 2007
24 Solving and analyzing side-chain positioning problems using linear and integer programming - Kingsford, Bernard, et al. - 2005
21 Globally optimal solutions for energy minimization in stereo vision using reweighted belief propagation - Meltzer, Yanover, et al. - 2005
16 On the choice of regions for generalized belief propagation - Welling - 2004
14 Structured region graphs: Morphing EP into GBP - Welling, Minka, et al. - 2005
1 Protein side-chain placement through MAP estimation and problem-size reduction - Hong, Lozano-Pérez - 2006
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