Compressed Sensing under Matrix Uncertainty: Optimum Thresholds and Robust Approximate Message Passing

TL;DR

This paper analyzes compressed sensing with uncertain measurement matrices, deriving optimal error thresholds and proposing a robust algorithm that achieves near-optimal reconstruction performance in large systems.

Contribution

It introduces a replica-based analysis for optimal reconstruction error under matrix uncertainty and develops a robust approximate message passing algorithm that performs near optimally.

Findings

Replica method determines mean-squared error of Bayes-optimal reconstruction.

Robust AMP algorithm matches optimal performance in large systems.

Algorithm performs well across a wide parameter range.

Abstract

In compressed sensing one measures sparse signals directly in a compressed form via a linear transform and then reconstructs the original signal. However, it is often the case that the linear transform itself is known only approximately, a situation called matrix uncertainty, and that the measurement process is noisy. Here we present two contributions to this problem: first, we use the replica method to determine the mean-squared error of the Bayes-optimal reconstruction of sparse signals under matrix uncertainty. Second, we consider a robust variant of the approximate message passing algorithm and demonstrate numerically that in the limit of large systems, this algorithm matches the optimal performance in a large region of parameters.