Rapidly Mixing Gibbs Sampling for a Class of Factor Graphs Using Hierarchy Width.

Area of Science:

• Computational statistics
• Machine learning
• Graph theory

Background:

• Gibbs sampling is a key inference method for factor graphs, but its theoretical performance guarantees are limited, with potential for exponential mixing times.
• Understanding the factors influencing Gibbs sampling's efficiency is crucial for improving its reliability in complex models.

Purpose of the Study:

• Introduce a novel graph property, hierarchy width, to provide theoretical guarantees for Gibbs sampling performance.
• Investigate the relationship between hierarchy width and mixing time in factor graphs.
• Explore applications of hierarchy width in hierarchical templates and natural language processing.

Main Methods:

• Defined and analyzed the concept of hierarchy width for (hyper)graphs.
• Established conditions under which bounded hierarchy width ensures polynomial mixing time for Gibbs sampling.
• Applied hierarchy width analysis to hierarchical templates and a natural language processing application.

Main Results:

• Bounded hierarchy width guarantees polynomial mixing time for Gibbs sampling under specific weight conditions.
• Hierarchical templates inherently possess bounded hierarchy width.
• A natural language processing application using this approach demonstrated provably rapid mixing and superior accuracy compared to human performance.

Conclusions:

• Hierarchy width offers a powerful theoretical tool for analyzing and guaranteeing the efficiency of Gibbs sampling.
• The developed framework has practical implications, particularly in natural language processing, where it enhances model performance and reliability.