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：非负方阵的积和式可通过求解适当定义因子图对应的Bethe自由能函数最小值得到良好近似；由此得到的积和式近似称为Bethe积和式。Vontobel通过度数-$M$ Bethe积和式给出了Bethe积和式的组合刻画，该刻画基于底层因子图的度数-$M$覆盖。本文证明了基于度数-$M$ Bethe积和式的非负矩阵积和式下界，解决了Vontobel在[IEEE Trans. Inf. Theory, Mar. 2013]中提出的猜想。同时，本文还证明了基于度数-$M$ Bethe积和式的非负矩阵积和式上界。当$M \to \infty$时，这些上下界退化为已知的基于Bethe积和式的非负矩阵积和式上下界。此外，我们针对称为（缩放）Sinkhorn积和式的积和式近似证明了类似结论。