Belief propagation and loop calculus for the permanent of a non-negative matrix

We consider computation of the permanent of a positive (N × N) non-negative matrix, P = (P^j_i|i, j = 1, ⋯, N), or equivalently the problem of weighted counting of the perfect matchings over the complete bipartite graph K_{N, N}. The problem is known to be of likely exponential complexity. Stated as the partition function Z of a graphical model, the problem allows for exact loop calculus representation (Chertkov M and Chernyak V 2006 Phys. Rev. E 72 065102) in terms of an interior minimum of the Bethe free energy functional over non-integer doubly stochastic matrix of marginal beliefs, β = (β^j_i|i, j = 1, ⋯, N), also correspondent to a fixed point of the iterative message-passing algorithm of the belief propagation (BP) type. Our main result is an explicit expression of the exact partition function (permanent) in terms of the matrix of BP marginals, β, as Z = Perm(P) = Z_BP Perm(β ^j_i(1 - β^j_i))/∏ _{i, j}(1 - β^j_i), where Z_BP is the BP expression for the permanent stated explicitly in terms of β. We give two derivations of the formula, a direct one based on the Bethe free energy and an alternative one combining the Ihara graph-ζ function and the loop calculus approaches. Assuming that the matrix β of the BP marginals is calculated, we provide two lower bounds and one upper bound to estimate the multiplicative term. Two complementary lower bounds are based on the Gurvits-van der Waerden theorem and on a relation between the modified permanent and determinant, respectively.