Threshold Saturation of Spatially Coupled Sparse Superposition Codes for All Memoryless Channels

TL;DR

This paper extends the threshold saturation phenomenon for spatially coupled sparse superposition codes to all memoryless channels, showing that coupling enables near-optimal decoding performance approaching Shannon capacity.

Contribution

It generalizes previous results to all memoryless channels and connects the GAMP decoding threshold to Fisher information, demonstrating capacity achievement in large alphabet limits.

Findings

GAMP decoding reaches the potential threshold for all memoryless channels.

In large alphabet limits, the GAMP threshold is expressed via Fisher information.

Potential threshold approaches Shannon capacity as alphabet size increases.

Abstract

We recently proved threshold saturation for spatially coupled sparse superposition codes on the additive white Gaussian noise channel. Here we generalize our analysis to a much broader setting. We show for any memoryless channel that spatial coupling allows generalized approximate message-passing (GAMP) decoding to reach the potential (or Bayes optimal) threshold of the code ensemble. Moreover in the large input alphabet size limit: i) the GAMP algorithmic threshold of the underlying (or uncoupled) code ensemble is simply expressed as a Fisher information; ii) the potential threshold tends to Shannon's capacity. Although we focus on coding for sake of coherence with our previous results, the framework and methods are very general and hold for a wide class of generalized estimation problems with random linear mixing.