Approximate message-passing decoder and capacity-achieving sparse superposition codes

TL;DR

This paper analyzes an approximate message-passing decoder for sparse superposition codes on Gaussian channels, identifying phase transition limitations and proposing power allocation and spatial coupling to achieve Shannon capacity efficiently.

Contribution

It introduces spatial coupling as a novel approach for these codes and demonstrates its effectiveness in reaching capacity and improving finite-length performance.

Findings

Superposition codes asymptotically reach Shannon capacity.

Spatial coupling improves robustness and finite-length reconstruction.

Fast Hadamard-based operators enable efficient large-scale decoding.

Abstract

We study the approximate message-passing decoder for sparse superposition coding on the additive white Gaussian noise channel and extend our preliminary work [1]. We use heuristic statistical-physics-based tools such as the cavity and the replica methods for the statistical analysis of the scheme. While superposition codes asymptotically reach the Shannon capacity, we show that our iterative decoder is limited by a phase transition similar to the one that happens in Low Density Parity check codes. We consider two solutions to this problem, that both allow to reach the Shannon capacity: i) a power allocation strategy and ii) the use of spatial coupling, a novelty for these codes that appears to be promising. We present in particular simulations suggesting that spatial coupling is more robust and allows for better reconstruction at finite code lengths. Finally, we show empirically that the use of a fast Hadamard-based operator allows for an efficient reconstruction, both in terms of computational time and memory, and the ability to deal with very large messages.