Loop Corrections in Spin Models through Density Consistency.

Area of Science:

• Computational physics
• Statistical mechanics
• Machine learning

Background:

• Computing marginal distributions of discrete Markov random fields (MRFs) is crucial across scientific disciplines but is generally intractable.
• Existing methods like belief propagation and adaptive Thouless-Anderson-Palmer have limitations in handling complex correlations.

Purpose of the Study:

• To introduce a novel family of computational schemes for approximating marginals of discrete MRFs.
• To enhance accuracy by incorporating loop corrections that account for cycles in factor graphs.

Main Methods:

• The proposed method shares properties with belief propagation, offering exact marginals on acyclic graphs.
• It differs by including loop corrections, considering correlations from all cycles in the factor graph.
• Consistency is based on density values, not just moments, distinguishing it from adaptive Thouless-Anderson-Palmer.

Main Results:

• Demonstrated significant improvement over the Bethe-Peierls (tree) approximation for Ising-like models.
• Outperformed the plaquette cluster variational method approximation in many cases.
• Achieved exactness up to the d^{-4} order for the critical inverse temperature expansion on hypercubic lattices, surpassing the d^{-2} order of other methods.

Conclusions:

• The new computational schemes offer a more accurate approximation for discrete MRF marginals.
• The inclusion of loop corrections is key to improving performance, especially in models with cyclic dependencies.
• This work provides a valuable tool for analyzing complex systems in various scientific fields.