Coded Compressive Sensing: A Compute-and-Recover Approach

TL;DR

This paper introduces coded compressive sensing using lattice codes and a compute-and-recover decoding method to efficiently recover sparse signals from noisy, quantized measurements with fewer measurements than signal dimension.

Contribution

It proposes a novel coded compressive sensing framework with a two-stage decoding process leveraging lattice codes, improving recovery performance over existing methods.

Findings

Theoretical conditions for perfect recovery are derived.

Empirical results show outperforming existing greedy algorithms in 1-bit compressive sensing.

The method effectively handles noisy, quantized measurements with fewer samples.

Abstract

In this paper, we propose \textit{coded compressive sensing} that recovers an $n$-dimensional integer sparse signal vector from a noisy and quantized measurement vector whose dimension $m$ is far-fewer than $n$. The core idea of coded compressive sensing is to construct a linear sensing matrix whose columns consist of lattice codes. We present a two-stage decoding method named \textit{compute-and-recover} to detect the sparse signal from the noisy and quantized measurements. In the first stage, we transform such measurements into noiseless finite-field measurements using the linearity of lattice codewords. In the second stage, syndrome decoding is applied over the finite-field to reconstruct the sparse signal vector. A sufficient condition of a perfect recovery is derived. Our theoretical result demonstrates an interplay among the quantization level $p$, the sparsity level $k$, the signal dimension $n$, and the number of measurements $m$ for the perfect recovery. Considering 1-bit compressive sensing as a special case, we show that the proposed algorithm empirically outperforms an existing greedy recovery algorithm.