Signal reconstruction in linear mixing systems with different error metrics

TL;DR

This paper introduces a versatile algorithm for signal reconstruction in linear systems that minimizes various error metrics, including non-additive ones like the infinity norm, leveraging relaxed belief propagation and Wiener filtering.

Contribution

It presents a general, fast reconstruction algorithm capable of minimizing arbitrary additive error metrics and extends to non-additive metrics like the infinity norm.

Findings

Algorithm effectively minimizes different error metrics.

Performance analysis confirms the algorithm's accuracy.

Wiener filter application enables infinity norm minimization.

Abstract

We consider the problem of reconstructing a signal from noisy measurements in linear mixing systems. The reconstruction performance is usually quantified by standard error metrics such as squared error, whereas we consider any additive error metric. Under the assumption that relaxed belief propagation (BP) can compute the posterior in the large system limit, we propose a simple, fast, and highly general algorithm that reconstructs the signal by minimizing the user-defined error metric. For two example metrics, we provide performance analysis and convincing numerical results. Finally, our algorithm can be adjusted to minimize the $\ell_\infty$ error, which is not additive. Interestingly, $\ell_{\infty}$ minimization only requires to apply a Wiener filter to the output of relaxed BP.