Fermions and loops on graphs: I. Loop calculus for determinants

Abstract

This paper is the first in a series devoted to evaluation of the partition function instatistical models on graphs with loops in terms of the Berezin/fermion integrals. Thepaper focuses on a representation of the determinant of a square matrix in terms of a finiteseries, where each term corresponds to a loop on the graph. The representation is based ona fermion version of the loop calculus, previously introduced by the authors for graphicalmodels with finite alphabets. Our construction contains two levels. First, we represent thedeterminant in terms of an integral over anti-commuting Grassmann variables, withsome reparametrization/gauge freedom hidden in the formulation. Second, weshow that a special choice of the gauge, called the BP (Bethe–Peierls or beliefpropagation) gauge, yields the desired loop representation. The set of gauge fixing BPconditions is equivalent to the Gaussian BP equations, discussed in the past asefficient (linear scaling) heuristics for estimating the covariance of a sparse positivematrix.