It has recently been observed that the permanent of a nonnegative square matrix, i.e., of a square matrix containing only nonnegative real entries, can very well be approximated by solving a certain Bethe free energy function minimization problem with the help of the sum-product algorithm. We call the resulting approximation of the permanent the Bethe permanent. In this paper, we give reasons why this approach to approximating the permanent works well. Namely, we show that the Bethe free energy function is convex and that the sum-product algorithm finds its minimum efficiently. We then discuss the fact that the permanent is lower bounded by the Bethe permanent, and we comment on potential upper bounds on the permanent based on the Bethe permanent. We also present a combinatorial characterization of the Bethe permanent in terms of permanents of so-called lifted versions of the matrix under consideration. Moreover, we comment on possibilities to modify the Bethe permanent so that it approximates the permanent even better, and we conclude the paper with some observations and conjectures about permanent-based pseudocodewords and permanent-based kernels.
As mentioned in the abstract, the present paper discusses a combinatorial characterization of the Bethe permanent in terms of permanents of so-called lifted versions of the matrix under consideration. For this we use results from @cite_22 that give a combinatorial characterization of the Bethe partition function of a factor graph in terms of the partition function of graph covers of this factor graph. Interestingly, very similar objects were considered by Greenhill, Janson, and Ruci 'n ski @cite_68 ; we will comment on this connection in .
@cite_68 @cite_22
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It has recently been observed that the permanent of a nonnegative square matrix, i.e., of a square matrix containing only nonnegative real entries, can very well be approximated by solving a certain Bethe free energy function minimization problem with the help of the sum-product algorithm. We call the resulting approximation of the permanent the Bethe permanent. In this paper, we give reasons why this approach to approximating the permanent works well. Namely, we show that the Bethe free energy function is convex and that the sum-product algorithm finds its minimum efficiently. We then discuss the fact that the permanent is lower bounded by the Bethe permanent, and we comment on potential upper bounds on the permanent based on the Bethe permanent. We also present a combinatorial characterization of the Bethe permanent in terms of permanents of so-called lifted versions of the matrix under consideration. Moreover, we comment on possibilities to modify the Bethe permanent so that it approximates the permanent even better, and we conclude the paper with some observations and conjectures about permanent-based pseudocodewords and permanent-based kernels. We present a combinatorial characterization of the Bethe entropy function of a factor graph, such a characterization being in contrast to the original, analytical, definition of this function. We achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph. Analogously, we give a combinatorial characterization of the Bethe partition function, whose original definition was also of an analytical nature. As we point out, our approach has similarities to the replica method, but also stark differences. The above findings are a natural backdrop for introducing a decoder for graph-based codes that we will call symbolwise graph-cover decoding, a decoder that extends our earlier work on blockwise graph-cover decoding. Both graph-cover decoders are theoretical tools that help toward a better understanding of message-passing iterative decoding, namely blockwise graph-cover decoding links max-product (min-sum) algorithm decoding with linear programming decoding, and symbolwise graph-cover decoding links sum-product algorithm decoding with Bethe free energy function minimization at temperature one. In contrast to the Gibbs entropy function, which is a concave function, the Bethe entropy function is in general not concave everywhere. In particular, we show that every code picked from an ensemble of regular low-density parity-check codes with minimum Hamming distance growing (with high probability) linearly with the block length has a Bethe entropy function that is convex in certain regions of its domain. Sudderth, Wainwright, and Willsky have conjectured that the Bethe approximation corresponding to any fixed point of the belief propagation algorithm over an attractive, pairwise binary graphical model provides a lower bound on the true partition function. In this work, we resolve this conjecture in the affirmative by demonstrating that, for any graphical model with binary variables whose potential functions (not necessarily pairwise) are all log-supermodular, the Bethe partition function always lower bounds the true partition function. The proof of this result follows from a new variant of the "four functions" theorem that may be of independent interest. We consider computation of the permanent of a positive (N × N) non-negative matrix, P = (Pji|i, j = 1, ..., N), or equivalently the problem of weighted counting of the perfect matchings over the complete bipartite graph KN, 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 = (βji|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) = ZBP Perm(βji(1 − βji)) ∏i, j(1 − βji), where ZBP 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.