The Effect of Spatial Coupling on Compressive Sensing

TL;DR

This paper demonstrates that spatial coupling improves the sparsity-to-sampling threshold in compressed sensing, similar to its known benefits in channel coding, though empirical gains are modest at accessible block lengths.

Contribution

It extends the concept of spatial coupling from channel coding to compressed sensing, showing improved thresholds for sparse signal reconstruction.

Findings

Spatially-coupled measurement matrices enhance sparsity-sampling thresholds.

Empirical tests show modest gains in reconstruction performance.

Threshold analysis suggests further potential for spatial coupling in compressed sensing.

Abstract

Recently, it was observed that spatially-coupled LDPC code ensembles approach the Shannon capacity for a class of binary-input memoryless symmetric (BMS) channels. The fundamental reason for this was attributed to a "threshold saturation" phenomena derived by Kudekar, Richardson and Urbanke. In particular, it was shown that the belief propagation (BP) threshold of the spatially coupled codes is equal to the maximum a posteriori (MAP) decoding threshold of the underlying constituent codes. In this sense, the BP threshold is saturated to its maximum value. Moreover, it has been empirically observed that the same phenomena also occurs when transmitting over more general classes of BMS channels. In this paper, we show that the effect of spatial coupling is not restricted to the realm of channel coding. The effect of coupling also manifests itself in compressed sensing. Specifically, we show that spatially-coupled measurement matrices have an improved sparsity to sampling threshold for reconstruction algorithms based on verification decoding. For BP-based reconstruction algorithms, this phenomenon is also tested empirically via simulation. At the block lengths accessible via simulation, the effect is quite small and it seems that spatial coupling is not providing the gains one might expect. Based on the threshold analysis, however, we believe this warrants further study.