Cavity Method: Message Passing from a Physics Perspective

TL;DR

This paper introduces the cavity method from a physics perspective as a heuristic for solving probabilistic graphical models, covering both RS and 1RSB levels, with applications to spin glasses, rCSP, and K-SAT.

Contribution

It presents a comprehensive physics-based explanation of the cavity method, including derivations and applications to complex models like K-SAT and spin glasses.

Findings

Derivation of RS cavity solution for the Sherrington-Kirkpatrick model

Application of cavity method to diluted graphs at RS and 1RSB levels

Analysis of K-SAT using belief and survey propagation

Abstract

In this three-sections lecture cavity method is introduced as heuristic framework from a Physics perspective to solve probabilistic graphical models and it is presented both at the replica symmetric (RS) and 1-step replica symmetry breaking (1RSB) level. This technique has been applied with success on a wide range of models and problems such as spin glasses, random constrain satisfaction problems (rCSP), error correcting codes etc. Firstly, the RS cavity solution for Sherrington-Kirkpatrick model---a fully connected spin glass model---is derived and its equivalence to the RS solution obtained using replicas is discussed. Then, the general cavity method for diluted graphs is illustrated both at RS and 1RSB level. The latter was a significant breakthrough in the last decade and has direct applications to rCSP. Finally, as example of an actual problem, K-SAT is investigated using belief and survey propagation.