Bayesian signal reconstruction for 1-bit compressed sensing

TL;DR

This paper introduces a Bayesian method for reconstructing sparse signals from 1-bit compressed sensing data, demonstrating superior performance over traditional methods through theoretical analysis and numerical experiments.

Contribution

It presents a Bayesian reconstruction framework for 1-bit compressed sensing and analyzes its performance using statistical mechanics, showing improvements over L1-norm minimization.

Findings

Bayesian approach outperforms L1 minimization in reconstruction accuracy.

Performance saturates when non-zero entry positions are known.

Numerical experiments confirm theoretical predictions.

Abstract

The 1-bit compressed sensing framework enables the recovery of a sparse vector x from the sign information of each entry of its linear transformation. Discarding the amplitude information can significantly reduce the amount of data, which is highly beneficial in practical applications. In this paper, we present a Bayesian approach to signal reconstruction for 1-bit compressed sensing, and analyze its typical performance using statistical mechanics. Utilizing the replica method, we show that the Bayesian approach enables better reconstruction than the L1-norm minimization approach, asymptotically saturating the performance obtained when the non-zero entries positions of the signal are known. We also test a message passing algorithm for signal reconstruction on the basis of belief propagation. The results of numerical experiments are consistent with those of the theoretical analysis.