Gauge-free cluster variational method by maximal messages and moment matching

Area of Science:

• Statistical Physics
• Computational Physics
• Machine Learning Algorithms

Background:

• The cluster variational method (CVM) is a powerful technique for approximating partition functions in statistical physics.
• Generalized belief propagation (GBP) is commonly used for CVM but suffers from gauge ambiguities.
• Existing methods often lack a one-to-one correspondence between GBP fixed points and CVM free energy extrema.

Purpose of the Study:

• To develop a gauge-free message passing algorithm for CVM.
• To improve the accuracy of CVM approximations by addressing gauge transformations.
• To provide a method applicable to binary variable systems and disordered systems.

Main Methods:

• Implementation of CVM as a message passing algorithm using generalized belief propagation (GBP).
• Introduction of 'maximal messages' where messages propagate from all ancestors, not just direct parents.
• Development of a 'moment matching' method for binary variables to enforce consistency and eliminate gauge invariance.

Main Results:

• An analytical expression for the critical temperature of the Ising model using plaquette CVM was derived.
• The proposed method yields results comparable to loop-corrected belief propagation and superior to Bethe estimates.
• The approach is demonstrated to be gauge-free, ensuring a unique solution.

Conclusions:

• The maximal messages and moment matching GBP provides an accurate and robust implementation of CVM.
• This gauge-free method offers a significant improvement over standard GBP for CVM approximations.
• The framework is readily generalizable to complex systems, including disordered statistical models.