Coupled Graphical Models and Their Thresholds

TL;DR

This paper demonstrates that coupling large graphical models into chains enhances their message passing thresholds, approaching the models' static thresholds, revealing a broad applicability of the convolutional coding phenomenon.

Contribution

It generalizes the threshold boosting effect of spatial coupling from coding theory to various graphical models like Curie-Weiss and K-satisfiability.

Findings

Coupled models' thresholds approach static thresholds of individual models.

The phenomenon is confirmed through analytical and numerical methods.

Threshold improvement is consistent across different types of graphical models.

Abstract

The excellent performance of convolutional low-density parity-check codes is the result of the spatial coupling of individual underlying codes across a window of growing size, but much smaller than the length of the individual codes. Remarkably, the belief-propagation threshold of the coupled ensemble is boosted to the maximum-a-posteriori one of the individual system. We investigate the generality of this phenomenon beyond coding theory: we couple general graphical models into a one-dimensional chain of large individual systems. For the later we take the Curie-Weiss, random field Curie-Weiss, $K$-satisfiability, and $Q$-coloring models. We always find, based on analytical as well as numerical calculations, that the message passing thresholds of the coupled systems come very close to the static ones of the individual models. The remarkable properties of convolutional low-density parity-check codes are a manifestation of this very general phenomenon.