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

2025-06-19

AI Summary

The Bethe free energy in double-edge factor graphs (DE-FGs) with complex-valued local functions and positive semidefinite constraints lacks a combinatorial characterization due to analytical intractability. Method: We propose the first verifiable, finite-graph-cover-based combinatorial characterization framework for the Bethe partition function in DE-FGs. Our approach generalizes loop calculus transformation (LCT) to accommodate zero-message components, unifying treatment of DE-FGs and standard factor graphs; it integrates refined sum-product algorithm fixed-point theory with a method-of-types-adapted graph-cover construction. Contribution/Results: Under verifiable conditions on DE-FGs, we rigorously prove the validity of this characterization. Numerical experiments suggest its broader applicability. This work establishes the first theoretically grounded, combinatorially explicit tool for inference on complex-weighted graphical models—particularly enabling rigorous analysis in quantum information applications involving complex-valued graphical inference.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• Characterize Bethe partition function for double-edge factor graphs
• Develop sum-product algorithm for complex-valued partition functions
• Extend loop-calculus transform to handle zero-valued message components

Innovation

• Developed SPA fixed-point-based Bethe approximation
• Applied loop-calculus transform for DE-FGs
• Generalized LCT for zero-valued message components