Graphical models (GM) represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function (PF), is the main inference challenge relevant to multiple statistical and optimization applications. The problem is of an exponential complexity with respect to the number of variables. In this manuscript, aimed at approximating the PF, we consider Multi-Graph Models (MGMs) where binary variables and multivariable factors are associated with edges and nodes, respectively, of an undirected multi-graph. We suggest a new methodology for analysis and computations that combines the Gauge Function (GF) technique with the technique from the field of real stable polynomials. We show that the GF, representing a single-out term in a finite sum expression for the PF which achieves extremum at the so-called Belief-Propagation (BP) gauge, has a natural polynomial representation in terms of gauges/variables associated with edges of the multi-graph. Moreover, GF can be used to recover the PF through a sequence of transformations allowing appealing algebraic and graphical interpretations. Algebraically, one step in the sequence consists in application of a differential operator over gauges associated with an edge. Graphically, the sequence is interpreted as a repetitive elimination/contraction of edges resulting in MGMs on decreasing in size (number of edges) graphs with the same PF as in the original MGM. Even though complexity of computing factors in the sequence of derived MGMs and respective GFs grow exponentially with the number of eliminated edges, polynomials associated with the new factors remain bi-stable if the original factors have this property. Moreover, we show that BP estimations in the sequence do not decrease, each low-bounding the PF.

中文翻译：

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图模型分区函数的量规、循环和多项式

图形模型 (GM) 表示多元且通常未归一化的概率分布。计算称为分区函数 (PF) 的归一化因子是与多个统计和优化应用程序相关的主要推理挑战。该问题相对于变量的数量具有指数级的复杂性。在本手稿中，旨在逼近 PF，我们考虑多图模型 (MGM)，其中二元变量和多变量因子分别与无向多图的边和节点相关联。我们提出了一种新的分析和计算方法，它将规范函数 (GF) 技术与实数稳定多项式领域的技术相结合。我们证明 GF，表示在所谓的置信传播 (BP) 规范处达到极值的 PF 的有限和表达式中的单出项，在与多重图的边相关联的规范/变量方面具有自然多项式表示。此外，GF 可用于通过允许有吸引力的代数和图形解释的一系列转换来恢复 PF。从代数上讲，该序列中的一个步骤是在与边相关联的规范上应用微分算子。从图形上看，该序列被解释为边的重复消除/收缩，导致 MGM 在大小（边数）图上减小，并具有与原始 MGM 中相同的 PF。即使派生 MGM 和相应 GF 序列中计算因子的复杂性随着消除边的数量呈指数增长，如果原始因子具有此属性，则与新因子相关联的多项式仍保持双稳态。此外，我们表明序列中的 BP 估计不会减少，每个低边界 PF。