M. Chertkov, A. B. Yedidia
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 BP 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 (FE) functional is parameterized by a scalar parameter $\gamma\in[-1;1]$, where $\gamma=-1$ corresponds to the BP limit and $\gamma=1$ corresponds to the exclusion principle (but ignoring perfect matching constraints) Mean-Field (MF) limit, and shows monotonicity and continuity of the functional with $\gamma$. We observe that the special value of $\gamma$, where the $\gamma$-parameterized FE functional is equal to the exact FE (defined as the minus log of the permanent), lies in the $[-1;0]$ range, with the low and high values from the range producing provable lower and upper bounds for the permanent. Our experimental analysis suggests that the special $\gamma$ varies for different ensembles but that it always lies in the $[-1;-1/2]$ interval. Besides, for all ensembles considered the behavior of the special $\gamma$ is highly distinctive, offering a practical potential for estimating permanents of non-negative matrices via the fractional FE approach.
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http://arxiv.org/abs/1108.0065
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