The permanent of a non-negative square matrix can be well approximated by finding the minimum of the Bethe free energy functions associated with some suitably defined factor graph; the resulting approximation to the permanent is called the Bethe permanent. Vontobel gave a combinatorial characterization of the Bethe permanent via degree-$M$ Bethe permanents, which is based on degree-$M$ covers of the underlying factor graph. In this paper, we prove a degree-$M$-Bethe-permanent-based lower bound on the permanent of a non-negative matrix, which solves a conjecture proposed by Vontobel in [IEEE Trans. Inf. Theory, Mar. 2013]. We also prove a degree-$M$-Bethe-permanent-based upper bound on the permanent of a non-negative matrix. In the limit $M \to \infty$, these lower and upper bounds yield known Bethe-permanent-based lower and upper bounds on the permanent of a non-negative matrix. Moreover, we prove similar results for an approximation to the permanent known as the (scaled) Sinkhorn permanent.

翻译：非负方阵的永久可通过寻找某适当定义的因子图对应的Bethe自由能函数的极小值进行良好近似；由此得到的永久近似值称为Bethe永久。Vontobel通过$M$度Bethe永久给出了Bethe永久的组合刻画，该刻画基于底层因子图的$M$度覆盖。本文证明了基于$M$度Bethe永久的非负矩阵永久下界，解决了Vontobel在《IEEE信息论汇刊》（2013年3月）中提出的猜想。我们还证明了基于$M$度Bethe永久的非负矩阵永久上界。当$M \to \infty$时，这些下界与上界退化为已知的基于Bethe永久的非负矩阵永久下界与上界。此外，我们针对被称为（缩放）Sinkhorn永久的永久近似值证明了对偶结果。