Near concavity of the growth rate for coupled LDPC chains

TL;DR

This paper demonstrates that the growth rate of coupled LDPC ensembles approaches the concave hull of the underlying ensemble's growth rate as the coupling strength increases, revealing a universal behavior in coupled mean field models.

Contribution

It introduces the concept that the growth rate of coupled LDPC ensembles becomes nearly concave, extending the understanding of threshold phenomena to growth rate behavior in coupled systems.

Findings

Growth rate approaches the concave hull with increased coupling

Supports findings with combinatorial and replica method calculations

Universal behavior observed in coupled mean field models

Abstract

Convolutional Low-Density-Parity-Check (LDPC) ensembles have excellent performance. Their iterative threshold increases with their average degree, or with the size of the coupling window in randomized constructions. In the later case, as the window size grows, the Belief Propagation (BP) threshold attains the maximum-a-posteriori (MAP) threshold of the underlying ensemble. In this contribution we show that a similar phenomenon happens for the growth rate of coupled ensembles. Loosely speaking, we observe that as the coupling strength grows, the growth rate of the coupled ensemble comes close to the concave hull of the underlying ensemble's growth rate. For ensembles randomly coupled across a window the growth rate actually tends to the concave hull of the underlying one as the window size increases. Our observations are supported by the calculations of the combinatorial growth rate, and that of the growth rate derived from the replica method. The observed concavity is a general feature of coupled mean field graphical models and is already present at the level of coupled Curie-Weiss models. There, the canonical free energy of the coupled system tends to the concave hull of the underlying one. As we explain, the behavior of the growth rate of coupled ensembles is exactly analogous.