Maximum a posteriori (MAP) inference is an important task for graphical models. Due to complex dependencies among variables in realistic model, finding an exact solution for MAP inference is often intractable. Thus, many approximation methods have been developed, among which the linear programming (LP) relaxation based methods show promising performance. However, one major drawback of LP relaxation is that it is possible to give fractional solutions. Instead of presenting a tighter relaxation, in this work we propose a continuous but equivalent reformulation of the original MAP inference problem, called LS-LP. We add the L2-sphere constraint onto the original LP relaxation, leading to an intersected space with the local marginal polytope that is equivalent to the space of all valid integer label configurations. Thus, LS-LP is equivalent to the original MAP inference problem. We propose a perturbed alternating direction method of multipliers (ADMM) algorithm to optimize the LS-LP problem, by adding a sufficiently small perturbation epsilon onto the objective function and constraints. We prove that the perturbed ADMM algorithm globally converges to the epsilon-Karush-Kuhn-Tucker (epsilon-KKT) point of the LS-LP problem. The convergence rate will also be analyzed. Experiments on several benchmark datasets from Probabilistic Inference Challenge (PIC 2011) and OpenGM 2 show competitive performance of our proposed method against state-of-the-art MAP inference methods.

中文翻译：

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通过 $$\ell _2$$ℓ2-Sphere 线性规划重构进行 MAP 推理

最大后验 (MAP) 推理是图形模型的一项重要任务。由于现实模型中变量之间的复杂依赖关系，为 MAP 推理找到一个精确的解决方案通常是棘手的。因此，已经开发了许多近似方法，其中基于线性规划（LP）松弛的方法显示出良好的性能。然而，LP 松弛的一个主要缺点是可以给出分数解。在这项工作中，我们没有提出更严格的松弛，而是提出了原始 MAP 推理问题的连续但等效的重新表述，称为 LS-LP。我们将 L2 球体约束添加到原始 LP 松弛，导致与局部边缘多面体相交的空间，该空间等效于所有有效整数标签配置的空间。因此，LS-LP 等价于原始 MAP 推理问题。我们提出了乘法器的扰动交替方向方法 (ADMM) 算法来优化 LS-LP 问题，方法是在目标函数和约束上添加足够小的扰动 epsilon。我们证明扰动 ADMM 算法全局收敛到 LS-LP 问题的 epsilon-Karush-Kuhn-Tucker (epsilon-KK​​T) 点。收敛速度也将被分析。来自概率推理挑战赛 (PIC 2011) 和 OpenGM 2 的几个基准数据集的实验表明，我们提出的方法与最先进的 MAP 推理方法相比具有竞争力。通过在目标函数和约束上添加足够小的扰动 epsilon。我们证明扰动 ADMM 算法全局收敛到 LS-LP 问题的 epsilon-Karush-Kuhn-Tucker (epsilon-KK​​T) 点。收敛速度也将被分析。来自概率推理挑战赛 (PIC 2011) 和 OpenGM 2 的几个基准数据集的实验表明，我们提出的方法与最先进的 MAP 推理方法相比具有竞争力。通过在目标函数和约束上添加足够小的扰动 epsilon。我们证明扰动 ADMM 算法全局收敛到 LS-LP 问题的 epsilon-Karush-Kuhn-Tucker (epsilon-KK​​T) 点。收敛速度也将被分析。来自概率推理挑战赛 (PIC 2011) 和 OpenGM 2 的几个基准数据集的实验表明，我们提出的方法与最先进的 MAP 推理方法相比具有竞争力。