On the Generalized Belief Propagation and its dynamics

Abstract

Numerous inference problems in statistical physics, computer vision or error-correcting coding theory consist in approximating the marginal probability distributions on Markov Random Fields (MRF). The Belief Propagation (BP) is an accurate solution that is optimal if the MRF is loop free and suboptimal otherwise. In the context of error-correcting coding theory, any Low-Density Parity-Check (LDPC) code has a graphical representation, the Tanner graph, which is a particular MRF. It is used as a media for the BP algorithm to correct the bits, damaged by a noisy channel, by estimating their probability distributions. Though loops and combination thereof in the Tanner graph prevent the BP from being optimal, especially harmful topological structures called the trapping-sets. The BP has been extended to the Generalized Belief Propagation (GBP). This message-passing algorithm runs on a non unique mapping of the Tanner graph, namely the regiongraph, such that its nodes are gatherings of the Tanner graph nodes. Then it appears the possibility to decrease the loops effect, making the GBP more accurate than the BP. In this article, we expose a novel region graph construction suited to the Tanner code, an LDPC code whose Tanner graph is entirely covered by trapping-sets. Furthermore, we investigate the dynamic behavior of the GBP compared with that of the BP to understand its evolution in terms of the Signal-to-Noise Ratio (SNR). To this end we make use of classical estimators and we introduce a new one called the hyperspheres method.