Universal Sparse Superposition Codes with Spatial Coupling and GAMP Decoding

TL;DR

This paper demonstrates that spatially coupled sparse superposition codes, decoded with GAMP, can universally achieve capacity over all memoryless channels, with error floors vanishing as code parameters grow large.

Contribution

It generalizes the analysis of sparse superposition codes to all memoryless channels and establishes their capacity-achieving property under GAMP decoding with spatial coupling.

Findings

Spatial coupling enables efficient GAMP decoding to reach the potential threshold.

Error floors vanish and capacity is approached as code parameters increase.

Provides a closed-form formula for the GAMP algorithmic threshold based on Fisher information.

Abstract

Sparse superposition codes, or sparse regression codes, constitute a new class of codes which was first introduced for communication over the additive white Gaussian noise (AWGN) channel. It has been shown that such codes are capacity-achieving over the AWGN channel under optimal maximum-likelihood decoding as well as under various efficient iterative decoding schemes equipped with power allocation or spatially coupled constructions. Here, we generalize the analysis of these codes to a much broader setting that includes all memoryless channels. We show, for a large class of memoryless channels, that spatial coupling allows an efficient decoder, based on the generalized approximate message-passing (GAMP) algorithm, to reach the potential (or Bayes optimal) threshold of the underlying (or uncoupled) code ensemble. Moreover, we argue that spatially coupled sparse superposition codes universally achieve capacity under GAMP decoding by showing, through analytical computations, that the error floor vanishes and the potential threshold tends to capacity as one of the code parameter goes to infinity. Furthermore, we provide a closed form formula for the algorithmic threshold of the underlying code ensemble in terms of a Fisher information. Relating an algorithmic threshold to a Fisher information has theoretical as well as practical importance. Our proof relies on the state evolution analysis and uses the potential method developed in the theory of low-density parity-check (LDPC) codes and compressed sensing.