It was recently conjectured that the permanent of a @math -lifting @math of a matrix @math of degree @math is less than or equal to the @math th power of the permanent perm @math , i.e., perm @math and, consequently, that the degree- @math Bethe permanent @math of a matrix @math is less than or equal to the permanent perm @math of @math , i.e., perm @math . In this paper, we prove these related conjectures and show in addition a few properties of the permanent of block matrices that are lifts of a matrix. As a corollary, we obtain an alternative proof of the inequality perm @math on the Bethe permanent of the base matrix @math that uses only the combinatorial definition of the Bethe permanent.
The literature on permanents and on adjacent areas (of counting perfect matchings, counting 0-1 matrices with specified row and column sums, etc.) is vast. Apart from the previously mentioned papers, the most relevant papers to our work are the one by Chertkov & Yedidia @cite_12 that studies the so-called fractional free energy functionals and resulting lower and upper bounds on the permanent of a non-negative matrix, the papers @cite_7 (on counting perfect matchings in random graph covers), @cite_1 (on counting matchings in graphs with the help of the sum-product algorithm Computing the permanent is related to counting perfect matchings. ), and @cite_18 @cite_32 @cite_21 (on max-product min-sum algorithms based approaches to the maximum weight perfect matching problem). Relevant is also the line of work on approximating the permanent of a non-negative matrix using Markov-chain-Monte-Carlo-based methods @cite_6 , polynomial-time randomized approximation schemes @cite_0 , and Bethe-approximation based methods or sum-product-algorithm (SPA) based method @cite_18 @cite_22 . See @cite_27 for a more detailed account of these and other related papers.
@cite_18 @cite_22 @cite_7 @cite_21 @cite_1 @cite_32 @cite_6 @cite_0 @cite_27 @cite_12
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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 discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze and generalize the Belief Propagation (BP) approach to computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. We introduce a fractional free energy functional parameterized by a scalar parameter @math , where @math corresponds to the BP limit and @math corresponds to the exclusion principle Mean-Field (MF) limit, and show monotonicity and continuity of the functional with @math . We observe that the optimal value of @math , where the @math -parameterized functional is equal to the exact free energy (defined as the minus log of the permanent), lies in the @math range, with the low and high values from the range producing provable low and upper bounds for the permanent. Our experimental analysis suggests that the optimal @math varies for different ensembles considered but it always lies in the @math interval. Besides, for all ensembles considered the behavior of the optimal @math is highly distinctive, thus offering a lot of practical potential for estimating permanents of non-negative matrices via the fractional free energy functional approach. 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. We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze the belief propagation (BP) approach and its fractional belief propagation (FBP) generalization for computing the permanent of a non-negative matrix. Known bounds and Conjectures are verified in experiments, and some new theoretical relations, bounds and Conjectures are proposed. The fractional free energy (FFE) function is parameterized by a scalar parameter γ ∈ [-1;1], where γ = -1 corresponds to the BP limit and γ = 1 corresponds to the exclusion principle (but ignoring perfect matching constraints) mean-field (MF) limit. FFE shows monotonicity and continuity with respect to γ. For every non-negative matrix, we define its special value γ* ∈ [-1;0] to be the γ for which the minimum of the γ-parameterized FFE function is equal to the permanent of the matrix, where the lower and upper bounds of the γ-interval corresponds to respective bounds for the permanent. Our experimental analysis suggests that the distribution of γ* varies for different ensembles but γ* always lies within the [-1;-1 2] interval. Moreover, for all ensembles considered, the behavior of γ* is highly distinctive, offering an empirical practical guidance for estimating permanents of non-negative matrices via the FFE approach.