The Bethe Free Energy Allows to Compute the Conditional Entropy of Graphical Code Instances. A Proof from the Polymer Expansion

The main objective of this paper is to explore the precise relationship between the Bethe free energy (or entropy) and the Shannon conditional entropy of graphical error correcting codes. The main result shows that the Bethe free energy associated with a low-density parity-check code used over a binary symmetric channel in a large noise regime is, with high probability, asymptotically exact as the block length grows. To arrive at this result we develop new techniques for rather general graphical models based on the loop sum as a starting point and the polymer expansion from statistical mechanics. The true free energy is computed as a series expansion containing the Bethe free energy as its zero-th order term plus a series of corrections. It is easily seen that convergence criteria for such expansions are satisfied for general high-temperature models. We apply these general results to ensembles of low-density generator-matrix and parity-check codes. While the application to generator-matrix codes follows standard high temperature methods, the case of parity-check codes requires non-trivial new ideas because the hard constraints correspond to a zero-temperature regime. Nevertheless one can combine the polymer expansion with expander and counting arguments to show that the difference between the true and Bethe free energies vanishes with high probability in the large block

Approaching the Rate-Distortion Limit with Spatial Coupling, Belief propagation and Decimation

We investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled Low-Density Generator-Matrix codes. The degree of the check nodes is regular and the one of code-bits is Poisson distributed with an average depending on the compression rate. The performance of a low complexity Belief Propagation Guided Decimation algorithm is excellent. The algorithmic rate-distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows. Moreover, as the check degree grows both curves approach the ultimate Shannon rate-distortion limit. The Belief Propagation Guided Decimation encoder is based on the posterior measure of a binary symmetric test-channel. This measure can be interpreted as a random Gibbs measure at a temperature directly related to the noise level of the test-channel. We investigate the links between the algorithmic performance of the Belief Propagation Guided Decimation encoder and the phase diagram of this Gibbs measure. The phase diagram is investigated thanks to the cavity method of spin glass theory which predicts a number of phase transition thresholds. In particular the dynamical and condensation phase transition temperatures (equivalently test-channel noise thresholds) are computed. We observe that: (i) the dynamical temperature of the spatially coupled construction saturates towards the condensation temperature; (ii) for large degrees the condensation temperature approaches the temperature (i.e. noise level) related to the information theoretic Shannon test-channel noise parameter of rate-distortion theory. This provides heuristic insight into the excellent performance of the Belief Propagation Guided Decimation algorithm. The paper contains an introduction to the cavity method.

Decay of Correlations for Sparse Graph Error Correcting Codes

The subject of this paper is transmission over a general class of binary-input memoryless symmetric channels using error correcting codes based on sparse graphs, namely low-density generator-matrix and low-density parity-check codes. The optimal (or ideal) decoder based on the posterior measure over the code bits, and its relationship to the sub-optimal belief propagation decoder, are investigated. We consider the correlation (or covariance) between two codebits, averaged over the noise realizations, as a function of the graph distance, for the optimal decoder. Our main result is that this correlation decays exponentially fast for fixed general low-density generator-matrix codes and high enough noise parameter, and also for fixed general low-density parity-check codes and low enough noise parameter. This has many consequences. Appropriate performance curves - called GEXIT functions - of the belief propagation and optimal decoders match in high/low noise regimes. This means that in high/low noise regimes the performance curves of the optimal decoder can be computed by density evolution. Another interpretation is that the replica predictions of spin-glass theory are exact. Our methods are rather general and use cluster expansions first developed in the context of mathematical statistical mechanics.

Decay of Correlations in Low Density Parity Check Codes: Low Noise Regime

Consider transmission over a binary additive white gaussian noise channel using a fixed low-density parity check code. We consider the posterior measure over the code bits and the corresponding correlation between two codebits, averaged over the noise realizations. We show that for low enough noise variance this average correlation decays exponentially fast with the graph distance between the code bits. One consequence of this result is that for low enough noise variance the GEXIT functions (further averaged over a standard code ensemble) of the belief propagation and optimal decoders are the same.