Graph-Cover-based Characterization of the Bethe Partition Function of Double-Edge Factor Graphs

For standard factor graphs (S-FGs) with non-negative real-valued local functions, Vontobel provided a combinatorial characterization of the Bethe approximation of the partition function, also known as the Bethe partition function, using finite graph covers. The proof of this characterization, i.e., the graph-cover theorem for S-FGs, heavily relied on the method of types. In this paper, we study double-edge factor graphs (DE-FGs), a class of factor graphs where each local function takes complex values and satisfies some positive semi-definiteness constraints. DE-FGs and their partition functions are particularly relevant for quantum information processing. Approximating the partition function of a DE-FG is more difficult than for an S-FG, as it involves summing complex values instead of non-negative real values. We develop the sum-product algorithm (SPA) fixed-point-based Bethe approximation of the partition function. However, one cannot directly apply the method of types to prove a similar combinatorial characterization as in the case of S-FGs. We provide a combinatorial characterization of the Bethe partition function in terms of finite graph covers for a class of DE-FGs that satisfy a specific, easily checkable condition. Towards proving this characterization, we apply a suitable loop-calculus transform (LCT) to these graphs. Originally, the LCT was introduced by Chertkov and Chernyak as a special linear transform for S-FGs and later extended by Mori. Our proposed LCT is applicable for both DE-FGs and S-FGs and generalizes prior versions by handling zero-valued SPA fixed-point message components, which are common in DE-FGs. Supported by numerical results, we conjecture that this combinatorial characterization of the Bethe partition function in terms of finite graph covers holds more broadly for DE-FGs.