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Signal reconstruction in linear mixing systems with different error metrics

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.