Probabilistic reasoning in intelligent systems: networks of plausible inference
Probabilistic reasoning in intelligent systems: networks of plausible inference
Zero-suppressed BDDs for set manipulation in combinatorial problems
DAC '93 Proceedings of the 30th international Design Automation Conference
Counting 1-factors in regular bipartite graphs
Journal of Combinatorial Theory Series B
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
Journal of the ACM (JACM)
Fast approximation of the permanent for very dense problems
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
The Art of Computer Programming, Volume 4, Fascicle 1: Bitwise Tricks & Techniques; Binary Decision Diagrams
Properties of Bethe free energies and message passing in Gaussian models
Journal of Artificial Intelligence Research
Constructing free-energy approximations and generalized belief propagation algorithms
IEEE Transactions on Information Theory
Max-Product for Maximum Weight Matching: Convergence, Correctness, and LP Duality
IEEE Transactions on Information Theory
Belief Propagation and LP Relaxation for Weighted Matching in General Graphs
IEEE Transactions on Information Theory
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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.