In this thesis, we leverage finite graph covers to analyze the SPA and the Bethe partition function for both S-FGs and DE-FGs. There are two main contributions in this thesis. The first main contribution concerns a special class of S-FGs where the partition function of each S-FG equals the permanent of a nonnegative square matrix. The Bethe partition function for such an S-FG is called the Bethe permanent. A combinatorial characterization of the Bethe permanent is given by the degree-$M$ Bethe permanent, which is defined based on the degree-$M$ graph covers of the underlying S-FG. In this thesis, we prove a degree-$M$-Bethe-permanent-based lower bound on the permanent of a non-negative square matrix, resolving 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 square matrix. The second main contribution is giving a combinatorial characterization of the Bethe partition function for DE-FGs in terms of finite graph covers. In general, approximating the partition function of a DE-FG is more challenging than for an S-FG because the partition function of the DE-FG is a sum of complex values and not just a sum of non-negative real values. Moreover, one cannot apply the method of types for proving the combinatorial characterization as in the case of S-FGs. We overcome this challenge by applying a suitable loop-calculus transform (LCT) for both S-FGs and DE-FGs. Currently, we provide a combinatorial characterization of the Bethe partition function in terms of finite graph covers for a class of DE-FGs satisfying an (easily checkable) condition.

翻译：本论文利用有限图覆盖来分析标准因子图（S-FG）与双边缘因子图（DE-FG）的SPA及Bethe配分函数。本论文主要有两项贡献。第一项贡献涉及一类特殊的S-FG，其中每个S-FG的配分函数等于一个非负方阵的积和式。此类S-FG的Bethe配分函数称为Bethe积和式。Bethe积和式可通过基于底层S-FG的度-$M$图覆盖所定义的度-$M$ Bethe积和式进行组合刻画。本论文中，我们证明了基于度-$M$ Bethe积和式的非负方阵积和式的下界，解决了Vontobel在[IEEE Trans. Inf. Theory, Mar. 2013]中提出的猜想。同时，我们也证明了基于度-$M$ Bethe积和式的非负矩阵积和式的上界。当$M \to \infty$时，这些下界与上界收敛于已知的基于Bethe积和式的非负方阵积和式的下界与上界。第二项贡献是借助有限图覆盖对DE-FG的Bethe配分函数给出了组合刻画。一般而言，近似DE-FG的配分函数比近似S-FG更具挑战性，因为DE-FG的配分函数是复数值之和，而不仅仅是非负实数值之和。此外，无法像处理S-FG那样应用类型方法证明其组合刻画。我们通过对S-FG和DE-FG同时应用合适的环演算变换（LCT）来克服这一挑战。目前，我们针对满足一个（易于验证的）条件的DE-FG类，基于有限图覆盖给出了其Bethe配分函数的组合刻画。