Domínguez Eduardo, Lage-Castellanos Alejandro, Mulet Roberto, Ricci-Tersenghi Federico
Department of Theoretical Physics, Physics Faculty, University of Havana, La Habana, CP 10400, Cuba.
"Henri Poincaré" group of Complex Systems, University of Havana, Cuba.
Phys Rev E. 2017 Apr;95(4-1):043308. doi: 10.1103/PhysRevE.95.043308. Epub 2017 Apr 25.
We present an implementation of the cluster variational method (CVM) as a message passing algorithm. The kind of message passing algorithm used for CVM, usually named generalized belief propagation (GBP), is a generalization of the belief propagation algorithm in the same way that CVM is a generalization of the Bethe approximation for estimating the partition function. However, the connection between fixed points of GBP and the extremal points of the CVM free energy is usually not a one-to-one correspondence because of the existence of a gauge transformation involving the GBP messages. Our contribution is twofold. First, we propose a way of defining messages (fields) in a generic CVM approximation, such that messages arrive on a given region from all its ancestors, and not only from its direct parents, as in the standard parent-to-child GBP. We call this approach maximal messages. Second, we focus on the case of binary variables, reinterpreting the messages as fields enforcing the consistency between the moments of the local (marginal) probability distributions. We provide a precise rule to enforce all consistencies, avoiding any redundancy, that would otherwise lead to a gauge transformation on the messages. This moment matching method is gauge free, i.e., it guarantees that the resulting GBP is not gauge invariant. We apply our maximal messages and moment matching GBP to obtain an analytical expression for the critical temperature of the Ising model in general dimensions at the level of plaquette CVM. The values obtained outperform Bethe estimates, and are comparable with loop corrected belief propagation equations. The method allows for a straightforward generalization to disordered systems.
我们提出了一种将簇变分法（CVM）实现为消息传递算法的方法。用于CVM的那种消息传递算法，通常称为广义置信传播（GBP），它是置信传播算法的一种推广，就如同CVM是用于估计配分函数的贝叶斯近似的一种推广一样。然而，由于存在涉及GBP消息的规范变换，GBP的不动点与CVM自由能的极值点之间的联系通常不是一一对应的。我们的贡献有两方面。首先，我们提出了一种在一般CVM近似中定义消息（场）的方法，使得消息从给定区域的所有祖先到达该区域，而不像标准的从父到子的GBP那样仅从其直接父节点到达。我们将这种方法称为最大消息。其次，我们专注于二元变量的情况，将消息重新解释为强制局部（边缘）概率分布的矩之间一致性的场。我们提供了一个精确的规则来强制所有的一致性，避免任何冗余，否则会导致消息上的规范变换。这种矩匹配方法是无规范的，即它保证得到的GBP不是规范不变的。我们应用我们的最大消息和矩匹配GBP在面元CVM水平上获得了一般维度下伊辛模型临界温度的解析表达式。得到的值优于贝叶斯估计值，并且与循环校正的置信传播方程相当。该方法可以直接推广到无序系统。
