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

TL;DR

This paper demonstrates that spatially coupled Low-Density Generator-Matrix codes combined with belief propagation guided decimation can approach the Shannon rate-distortion limit for lossy compression of binary sources, with performance improving as code parameters grow.

Contribution

It introduces a novel encoding scheme using spatially coupled LDGM codes and analyzes its performance through spin glass theory, showing near-optimal rate-distortion results.

Findings

Algorithmic rate-distortion curve approaches the optimal as coupling width increases.

Increasing check degree brings the curve closer to the Shannon limit.

Phase transition analysis explains the algorithm's excellent performance.

Abstract

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.