图上的费米子和圈。I.行列式的循环演算
Fermions and Loops on Graphs. I. Loop Calculus for Determinant
摘要：
本文是第一次在一系列致力于评估的配分函数在统计模型的图形循环的Berezin/费米子积分。本文主要研究方阵行列式的有限级数表示，其中每一项对应于图上的一个环。该表示是基于费米子版本的循环演算，以前介绍了作者的图形模型与有限的字母表。我们的建设包含两个层面。首先，我们表示的行列式在反交换格拉斯曼变量的积分，与一些reparametrization/规范的自由隐藏在制定。其次，我们表明，一个特殊的选择的规范，称为BP（Bethe-Peierls或信念传播）规范，产生所需的循环表示。规范固定BP条件的集合等价于高斯BP方程，其在过去被讨论为用于估计稀疏正矩阵的协方差的有效（线性缩放）算法。
英文摘要：
This paper is the first in the series devoted to evaluation of the partition function in statistical models on graphs with loops in terms of the Berezin/fermion integrals. The paper focuses on a representation of the determinant of a square matrix in terms of a finite series, where each term corresponds to a loop on the graph. The representation is based on a fermion version of the Loop Calculus, previously introduced by the authors for graphical models with finite alphabets. Our construction contains two levels. First, we represent the determinant in terms of an integral over anti-commuting Grassman variables, with some reparametrization/gauge freedom hidden in the formulation. Second, we show that a special choice of the gauge, called BP (Bethe-Peierls or Belief Propagation) gauge, yields the desired loop representation. The set of gauge-fixing BP conditions is equivalent to the Gaussian BP equations, discussed in the past as efficient (linear scaling) heuristics for estimating the covariance of a sparse positive matrix.
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