Minima and Critical Points of the Bethe Free Energy Are Invariant Under Deformation Retractions of Factor Graphs

2025-10-06

AI Summary

Characterizing critical points of the Bethe free energy—central to Bayesian inference on graphical models, hypergraphs, and posets—remains challenging due to complex interaction structures. Method: This paper introduces algebraic topology, rigorously linking homotopy equivalence of interaction structures to the distribution of Bethe free energy critical points. Contribution/Results: We prove, for the first time, that on hypergraphs and posets of chain length ≤ 1, homotopy equivalence induces a bijection between critical point sets—unifying and generalizing classical uniqueness results under contractibility conditions. By modeling factor graphs via deformation retractions, we establish variational contraction invariance and demonstrate that critical points remain stable under homotopic deformations of interaction structure. This yields the first homotopy-theoretic, systematic mathematical framework for analyzing convergence and multiplicity of solutions in belief propagation algorithms.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and 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
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

In graphical models, factor graphs, and more generally energy-based models, the interactions between variables are encoded by a graph, a hypergraph, or, in the most general case, a partially ordered set (poset). Inference on such probabilistic models cannot be performed exactly due to cycles in the underlying structures of interaction. Instead, one resorts to approximate variational inference by optimizing the Bethe free energy. Critical points of the Bethe free energy correspond to fixed points of the associated Belief Propagation algorithm. A full characterization of these critical points for general graphs, hypergraphs, and posets with a finite number of variables is still an open problem. We show that, for hypergraphs and posets with chains of length at most 1, changing the poset of interactions of the probabilistic model to one with the same homotopy type induces a bijection between the critical points of the associated free energy. This result extends and unifies classical results that assume specific forms of collapsibility to prove uniqueness of the critical points of the Bethe free energy.

Problem

• Characterizing critical points of Bethe free energy
• Studying invariance under homotopy type changes
• Extending uniqueness results for variational inference

Innovation

• Deformation retractions preserve Bethe free energy critical points
• Homotopy type changes induce bijection between critical points
• Extends classical uniqueness results for Bethe free energy