用分数置信传播逼近积和式
Approximating the Permanent with Fractional Belief Propagation
摘要：
我们讨论计划的精确和近似计算的永久性，并相互比较。具体来说，我们分析了置信传播（BP）方法和分数置信传播（FBP）的推广计算永久的非负矩阵。在实验中验证了已知的界限和公式，并提出了一些新的理论关系、界限和公式。分数自由能（FFE）泛函由标量参数$γ\in[-1;1]$参数化，其中$γ=-1$对应于BP极限，并且$γ=1$对应于排除原理（但忽略完美匹配约束）平均场（MF）极限。FFE相对于$γ$表现出单调性和连续性。对于每个非负矩阵，我们定义其特殊值$γ_*\in[-1;0]$为$γ$，其中$γ$ -参数化FFE泛函的最小值等于矩阵的永久值，其中$γ$ -区间的下限和上限对应于永久值的各自边界。我们的实验分析表明，$γ_*$的分布不同的合奏，但$γ_*$总是位于$[-1;-1/2]$间隔。此外，对于所有的系综考虑的行为$γ_*$是非常独特的，提供了一个经验的实际指导，估计永久的非负矩阵通过FFE方法。
英文摘要：
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) functional is parameterized by a scalar parameter $γ\in[-1;1]$, 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 $γ_*\in[-1;0]$ to be the $γ$ for which the minimum of the $γ$-parameterized FFE functional 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 emprirical practical guidance for estimating permanents of non-negative matrices via the FFE approach.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
