Fermions and Loops on Graphs. I. Loop Calculus for Determinant

TL;DR

This paper introduces a novel fermion-based loop calculus representation for the determinant of matrices, linking it to graphical models with loops and providing a new perspective on evaluating partition functions.

Contribution

It develops a fermionic loop calculus framework for determinants, connecting Grassman integrals, gauge fixing, and belief propagation in graphical models.

Findings

Determinant expressed as a finite series over graph loops.

BP gauge choice simplifies the loop representation.

Connection established between gauge fixing and Gaussian BP equations.

Abstract

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.