## On the optimality of tree-reweighted max-product message passing (2005)

Venue: | In UAI |

Citations: | 54 - 4 self |

### BibTeX

@INPROCEEDINGS{Kolmogorov05onthe,

author = {Vladimir Kolmogorov},

title = {On the optimality of tree-reweighted max-product message passing},

booktitle = {In UAI},

year = {2005}

}

### Years of Citing Articles

### OpenURL

### Abstract

Tree-reweighted max-product (TRW) message passing [9] is a modified form of the ordinary max-product algorithm for attempting to find minimal energy configurations in Markov random field with cycles. For a TRW fixed point satisfying the strong tree agreement condition, the algorithm outputs a configuration that is provably optimal. In this paper, we focus on the case of binary variables with pairwise couplings, and establish stronger properties of TRW fixed points that satisfy only the milder condition of weak tree agreement (WTA). First, we demonstrate how it is possible to identify part of the optimal solution—i.e., a provably optimal solution for a subset of nodes — without knowing a complete solution. Second, we show that for submodular functions, a WTA fixed point always yields a globally optimal solution. We establish that for binary variables, any WTA fixed point always achieves the global maximum of the linear programming relaxation underlying the TRW method. 1

### Citations

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Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference
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Citation Context ...linear programming relaxation underlying the TRW method. 1 Introduction Markov random fields (MRFs) provide a powerful framework for capturing dependencies among large collections of random variables =-=[8]-=-. One problem is determining a most probable configuration, or equivalently, a configuration with minimal energy. For tree-structured MRFs, minimal energy configurations can be computed efficiently by... |

323 | Convergent tree-reweighted message passing for energy minimization
- Kolmogorov
(Show Context)
Citation Context ...algorithm outputs a provably optimal configuration. However, not all fixed points satisfy this requirement; in general, they are characterized by a milder condition known as weak tree agreement (WTA) =-=[6]-=-. The main contribution of this paper is to establish a number of optimality properties of TRW fixed points that satisfy only the WTA condition. Our work focuses in particular on the case of binary va... |

188 | On the optimality of solutions of the max-product beliefpropagation algorithm in arbitrary graphs
- Weiss, Freeman
- 2001
(Show Context)
Citation Context ...d, though convergent on trees, may fail to converge when applied to a graph with cycles. Moreover, although there are certain “local optimality” guarantees associated with any max-product fixed point =-=[3, 10]-=-, it is also straightforward to construct problems for which the max-product algorithm will yield a non-optimal configuration. In contrast to the ordinary max-product algorithm, the class of tree-rewe... |

163 | É.: Approximation algorithms for classification problems with pairwise relationships: Metric labeling and Markov random fields. In: FOCS
- Kleinberg, Tardos
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(Show Context)
Citation Context ...ugh it is known that submodular problems can be solved by flow-based [1] and other LP formulations [2], our work establishes that distributed TRW message-passing also solves suchsproblems. Other work =-=[2, 5]-=- has studied LP relaxations closely related to the TRW-LP, and established performance guarantees for certain MRFs (e.g., for metric labeling). The results of this paper differ qualitatively from this... |

114 | Pseudo-boolean optimization
- Boros, Hammer
(Show Context)
Citation Context ... fixed point is a dual TRW-LP solution for binary problems) allows us to relate our results to other work on LP formulations. Although it is known that submodular problems can be solved by flow-based =-=[1]-=- and other LP formulations [2], our work establishes that distributed TRW message-passing also solves suchsproblems. Other work [2, 5] has studied LP relaxations closely related to the TRW-LP, and est... |

114 | Map estimation via agreement on (hyper)trees: Message-passing and linear-programming approaches
- Wainwright, Jaakkola, et al.
- 2005
(Show Context)
Citation Context ...truct problems for which the max-product algorithm will yield a non-optimal configuration. In contrast to the ordinary max-product algorithm, the class of tree-reweighted (TRW) max-product algorithms =-=[9]-=- have certain correctness guarantees. In particular, for any TRW fixed point that satisfies strong tree agreement, the algorithm outputs a provably optimal configuration. However, not all fixed points... |

71 |
Roof duality, complementation and persistency in quadratic 0-1 optimization
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Citation Context ...tly specify (a subset of) variables in a globally optimal configuration. Our results also have connections with prior work on binary integer programming from the combinatorial optimization literature =-=[1, 4]-=-. In particular, previous work [4] has established a weak persistency property for the so-called Rhys relaxation. This result turns out to be related to our characterization of WTA fixed points, as we... |

65 | Approximation algorithms for the metric labeling problem via a new linear programming formulation - Chekuri, Khanna, et al. - 2001 |

58 | Tree consistency and bounds on the performance of the max-product algorithm and its generalizations - Wainwright, Willsky - 2004 |