In the context of channel coding, the Generalized Belief Propagation (GBP) is an iterative algorithm used to recover the transmission bits sent through a noisy channel. To ensure a reliable transmission, we apply a map on the bits, that is called a code. This code induces artificial correlations between the bits to send, and it can be modeled by a graph whose nodes are the bits and the edges are the correlations. This graph, called Tanner graph, is used for most of the decoding algorithms like Belief Propagation or Gallager-B. The GBP is based on a non unic transformation of the Tanner graph into a so called region-graph. A clear advantage of the GBP over the other algorithms is the freedom in the construction of this graph. In this article, we explain a particular construction for specific graph topologies that involves relevant performance of the GBP. Moreover, we investigate the behavior of the GBP considered as a dynamic system in order to understand the way it evolves in terms of the time and in terms of the noise power of the channel. To this end we make use of classical measures and we introduce a new measure called the hyperspheres method that enables to know the size of the attractors.
In the context of channel coding, the Generalized Belief Propagation (GBP) is an iterative algorithm used to recover the transmission bits sent through a noisy channel. To ensure a reliable transmission, we apply a map on the bits, that is called a code. This code induces artificial correlations between the bits to send, and it can be modeled by a graph whose nodes are the bits and the edges are the correlations. This graph, called Tanner graph, is used for most of the decoding algorithms like Belief Propagation or Gallager-B. The GBP is based on a non unic transformation of the Tanner graph into a so called region-graph. A clear advantage of the GBP over the other algorithms is the freedom in the construction of this graph. In this article, we explain a particular construction for specific graph topologies that involves relevant performance of the GBP. Moreover, we investigate the behavior of the GBP considered as a dynamic system in order to understand the way it evolves in terms of the time and in terms of the noise power of the channel. To this end we make use of classical measures and we introduce a new measure called the hyperspheres method that enables to know the size of the attractors.
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.
This thesis addresses the problem of inference in factor graphs, especially the LDPC codes, almost solved by message-passing algorithms. In particular, the Belief Propagation algorithm (BP) is investigated as a particular message-passing algorithm whose suboptimality is discussed in the case where the factor graph has a loop-like topology. From the equivalence between the BP and the Bethe approximation in statistical physics that is generalized to the region-based approximation, is detailed the Generalized Belief Propagation algorithm (GBP), a message-passing algorithm between clusters of the factor graph. It is experimentally shown to surpass the BP in the cases where the clustering deals with the harmful topological structures that prevents the BP from rightly decoding any LDPC code, namely the trapping sets. We do not only confront the BP and the GBP algorithms according to their performance from the point of view of the channel coding with the error-rate, but also according to their dynamical behaviors for non-trivial error-events for which both algorithms can exhibit chaotic beahviors. By means of classical and original dynamical quantifiers, it is shown that the GBP algorithm can overcome the BP algorithm.
Abstract—In this paper, we focus on the Generalized Belief Propagation (GBP) algorithm to solve trapping sets in Low-Density Parity-Check (LDPC) codes. Trapping sets are topo-logical structures in Tanner graphs of LDPC codes that are not correctly decoded by Belief Propagation (BP), leading to exhibit an error-floor in the Bit-Error Rate (BER). Stemming from statistical physics of spin glasses, GBP consists in passing messages between clusters of Tanner graph nodes in another graph called the region-graph. Here, we introduce a specific clustering of nodes, based on a novel local loopfree principle, that breaks trapping sets such that the resulting region-graph is locally loopfree. We then construct a hybrid decoder made of BP and GBP that proves to be a powerful decoder as it clearly improves the BER and defeats the error-floor. Index Terms—LDPC codes, Generalized Belief Propagation, trapping sets, error-floor, local clustering
In this paper, we focus on the Generalized Belief Propagation (GBP) algorithm to solve trapping sets in Low-Density Parity-Check (LDPC) codes. Trapping sets are topological structures in Tanner graphs of LDPC codes that are not correctly decoded by Belief Propagation (BP), leading to exhibit an error-floor in the Bit-Error Rate (BER). Stemming from statistical physics of spin glasses, GBP consists in passing messages between clusters of Tanner graph nodes in another graph called the region-graph. Here, we introduce a specific clustering of nodes, based on a novel local loopfree principle, that breaks trapping sets such that the resulting region-graph is locally loopfree. We then construct a hybrid decoder made of BP and GBP that proves to be a powerful decoder as it clearly improves the BER and defeats the error-floor.
The Belief Propagation (BP) is an inference algorithm used to estimate marginal probability distributions for any Markov Random Field (MRF). In the realm of Low-Density Parity-Check (LDPC) codes that can be represented by MRF called Tanner graphs, the BP is used as a decoding algorithm to estimate the states of bits sent through a noisy channel. Known to be optimal when the Tanner graph is a tree, the BP suffers from suboptimality when the Tanner graph has a loop-like topology. Furthermore, combinations of loops, namely the trapping sets, are particularly harmful for the decoding. To circumvent this problem were proposed other algorithms, like the Generalized Belief Propagation (GBP) that comes from statistical physics. This algorithm allows to absorb topological structures inside new nodes called regions. An advantage is that the resulting graph, the region graph, is not unique then according to its construction this region graph is a media for the GBP that can provide more accurate estimates than the BP. In this paper, we propose novel constructions of the region graph for the famous Tanner code of length N = 155 by making use of the trapping sets as basis for the regions.
Abstract. In this paper, we investigate the behaviors of the Belief Propagation algorithm considered as a dynamic system. In the context of LDPC (Low Density Parity-Check) codes, we use the noise power of the transmission channel as a potentiometer to evaluate the different motions that the BP can follow. The computations of dynamic quantifiers as the bifurcation diagram, the Lyapunov exponent and the reconstructed trajectory enable to bring out four main behaviors. In addition, we propose a novel measure that is the hyperspheres method, which provides the knowledge of the time evolution of the attractor size. The information collected from these different quantifiers helps to better understand the BP evolution and to focus on the noise power values for which the BP suffers from chaos.