Vladimir Y Chernyak and Michael Chertkov J. Stat. Mech. (2008) P12011 doi:10.1088/1742-5468/2008/12/P12011
Vladimir Y Chernyak1,2 and Michael Chertkov2
Show affiliationsThis paper is the first in a 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 Grassmann variables, with some reparametrization/gauge freedom hidden in the formulation. Second, we show that a special choice of the gauge, called the 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.
E-print Number: 0809.3479
Cited: by |
Refers: to
05Cxx Graph theory (For applications of graphs, see 68R10, 90C35, 94C15)
15A15 Determinants, permanents, other special matrix functions (See also 19B10, 19B14)
Issue 12 (December 2008)
Received 23 September 2008, accepted for publication 25 November 2008
Published 17 December 2008
Vladimir Y Chernyak and Michael Chertkov J. Stat. Mech. (2008) P12011
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