Approaching the rate-distortion limit by spatial coupling with belief propagation and decimation

We investigate an encoding scheme for lossy compression based on spatially coupled Low-Density GeneratorMatrix codes. The degree distributions are regular, or are Poisson on the code-bit side and check-regular which allows use for any compression rate. The performance of a low complexity Belief Propagation Guided Decimation algorithm is excellent, and for large check degrees it gets close to Shannon´s rate-distortion limit. We investigate links between the algorithmic performance and the phase diagram of a relevant random Gibbs measure. The associated dynamical and condensation thresholds are computed within the framework of the cavity method. We observe that: (i) the dynamical threshold of the spatially coupled construction saturates towards the condensation threshold; (ii) for large degrees the condensation threshold approaches the information theoretic test-channel parameter of rate-distortion theory. This provides heuristic insight into the excellent performance of the BPGD algorithm.