A class of random fields on complete graphs with tractable partition function

Area of Science:

• Computational Statistical Physics
• Machine Learning Theory
• Graph Theory

Background:

• Computing partition functions and marginal probabilities is fundamental in statistical physics and machine learning.
• Exact computation is often intractable for large-scale random fields, necessitating approximation algorithms.
• Tractable models are essential for benchmarking and understanding the performance of these algorithms.

Purpose of the Study:

• To introduce a polynomial-time method for computing partition functions and marginal probabilities.
• To identify specific classes of random fields on complete and complete bipartite graphs that are computationally tractable.
• To provide a basis for evaluating the accuracy of approximation algorithms through exact error estimation.

Main Methods:

• Development of an efficient algorithm for calculating the partition function.
• Application of the method to Ising models with homogeneous pairwise and arbitrary unary potentials on complete graphs.
• Extension of the method to random fields on complete bipartite graphs with homogeneous pairwise potentials.

Main Results:

• Demonstration of polynomial-time computability for partition functions and marginal probabilities in the specified random field classes.
• Identification of tractable models suitable for rigorous analysis of approximation algorithms.
• Establishment of a foundation for generating exact error bounds.

Conclusions:

• The proposed method offers significant computational advantages for specific random field models.
• These tractable models serve as valuable benchmarks for assessing approximation algorithms.
• The ability to compute exact error estimates will advance the development and understanding of approximate methods.