摘要翻译：
我们利用拉格朗日松弛技术建立了一个通用框架，用于离散和高斯图形模型中的地图估计。其关键思想是将一个难以处理的估计问题重新表述为一个定义在更容易处理的图上的问题，但受到额外的约束。放宽这些约束条件就得到了一个易于处理的对偶问题，一个由薄图定义的对偶问题，然后通过迭代过程进行优化。当这种迭代优化导致一个一致的估计，一个也满足约束条件，那么它对应于一个最优的MAP估计原模型。否则存在对偶间隙，我们得到了最优解的一个界。因此，我们的方法结合了凸优化和适用于薄图的动态规划技术。流行的树重加权最大乘积(TRMP)方法可以被视为解决一类特殊的松弛问题，其中难以处理的图被松弛为一组生成树。我们还考虑了对一组小的诱导子图、薄的子图（例如，环）和通过“展开”圈得到的连通树的松弛。此外，我们提出了一种新的多尺度弛豫，它引入了“总结”变量。这种推广的潜在好处包括：减少或消除困难问题中的“对偶间隙”，减少对偶问题中的拉格朗日乘子数，加速迭代优化过程的收敛。
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英文标题：
《Lagrangian Relaxation for MAP Estimation in Graphical Models》
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作者：
Jason K. Johnson, Dmitry M. Malioutov, Alan S. Willsky
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最新提交年份：
2007
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分类信息：

一级分类：Computer Science        计算机科学
二级分类：Artificial Intelligence        人工智能
分类描述：Covers all areas of AI except Vision, Robotics, Machine Learning, Multiagent Systems, and Computation and Language (Natural Language Processing), which have separate subject areas. In particular, includes Expert Systems, Theorem Proving (although this may overlap with Logic in Computer Science), Knowledge Representation, Planning, and Uncertainty in AI. Roughly includes material in ACM Subject Classes I.2.0, I.2.1, I.2.3, I.2.4, I.2.8, and I.2.11.
涵盖了人工智能的所有领域，除了视觉、机器人、机器学习、多智能体系统以及计算和语言（自然语言处理），这些领域有独立的学科领域。特别地，包括专家系统，定理证明（尽管这可能与计算机科学中的逻辑重叠），知识表示，规划，和人工智能中的不确定性。大致包括ACM学科类I.2.0、I.2.1、I.2.3、I.2.4、I.2.8和I.2.11中的材料。
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英文摘要：
  We develop a general framework for MAP estimation in discrete and Gaussian graphical models using Lagrangian relaxation techniques. The key idea is to reformulate an intractable estimation problem as one defined on a more tractable graph, but subject to additional constraints. Relaxing these constraints gives a tractable dual problem, one defined by a thin graph, which is then optimized by an iterative procedure. When this iterative optimization leads to a consistent estimate, one which also satisfies the constraints, then it corresponds to an optimal MAP estimate of the original model. Otherwise there is a ``duality gap'', and we obtain a bound on the optimal solution. Thus, our approach combines convex optimization with dynamic programming techniques applicable for thin graphs. The popular tree-reweighted max-product (TRMP) method may be seen as solving a particular class of such relaxations, where the intractable graph is relaxed to a set of spanning trees. We also consider relaxations to a set of small induced subgraphs, thin subgraphs (e.g. loops), and a connected tree obtained by ``unwinding'' cycles. In addition, we propose a new class of multiscale relaxations that introduce ``summary'' variables. The potential benefits of such generalizations include: reducing or eliminating the ``duality gap'' in hard problems, reducing the number or Lagrange multipliers in the dual problem, and accelerating convergence of the iterative optimization procedure.
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PDF链接：
https://arxiv.org/pdf/0710.0013