成对MRF中MAP推理对偶分解的部分最优性
Partial Optimality of Dual Decomposition for MAP Inference in Pairwise MRFs
摘要：
马尔可夫随机场（MRF）是一个强大的工具，用于建模的统计依赖性的一组随机变量使用图形表示。最大后验概率（MAP）推理是与MRF相关的一个重要计算问题，它是寻找具有最大概率的联合变量分配。众所周知，这一任务的两个流行的优化技术，线性规划（LP）松弛和对偶分解（DD），有很强的联系，都提供了一个最佳的解决方案，MAP问题时，相应的LP松弛是紧的。然而，在相反和更现实的情况下，对它们的关系知之甚少。在本文中，我们解释了如何通过DD获得的完全积分的配置部分同意的最佳分数的配置通过LP松弛时，后者是不紧的。特别是，对于二进制成对MRF的相应结果表明，这两种方法共享其解决方案的部分最优性。
英文摘要：
Markov random fields (MRFs) are a powerful tool for modelling statistical dependencies for a set of random variables using a graphical representation. An important computational problem related to MRFs, called maximum a posteriori (MAP) inference, is finding a joint variable assignment with the maximal probability. It is well known that the two popular optimisation techniques for this task, linear programming (LP) relaxation and dual decomposition (DD), have a strong connection both providing an optimal solution to the MAP problem when a corresponding LP relaxation is tight. However, less is known about their relationship in the opposite and more realistic case. In this paper, we explain how the fully integral assignments obtained via DD partially agree with the optimal fractional assignments via LP relaxation when the latter is not tight. In particular, for binary pairwise MRFs the corresponding result suggests that both methods share the partial optimality property of their solutions.
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