Beyond the Bethe Free Energy of LDPC Codes via Polymer Expansions

TL;DR

This paper rigorously analyzes the corrections to the Bethe free energy in LDPC codes using polymer expansions, proving that the Bethe approximation becomes exact in the large-size limit for highly noisy channels.

Contribution

It introduces a rigorous method to control loop series expansions for LDPC codes, establishing the Bethe entropy as exact in the asymptotic limit under high noise conditions.

Findings

Bethe free energy is exact for large LDPC codes over noisy channels

Polymer expansion methods effectively control loop series corrections

Results extend to more general graphical models

Abstract

The loop series provides a formal way to write down corrections to the Bethe entropy (and/or free energy) of graphical models. We provide methods to rigorously control such expansions for low-density parity-check codes used over a highly noisy binary symmetric channel. We prove that in the asymptotic limit of large size, with high probability, the Bethe expression gives an exact formula for the entropy (per bit) of the input word conditioned on the output of the channel. Our methods also apply to more general models.