outliers in the estimated signal, using the mean absolute error (MAE) metric might be more appropriate. Seeing that there is no prior work discussing the optimal performance with user-defined error metrics, we study the optimal performance with user-defined additive error metrics in MMV problems where the signals share common sparse supports, and each entry of the measurements is contaminated by parallel measurement channels (i.e., the channel in  satisfies $f(\y|\w)=\prod_{m=1}^M f(y_m|w_m)$). Note that a specific error metric, the MSE, has been studied in Zhu et al. [@ZhuBaronKrzakala2017IEEE], which focuses on the MSE performance limits (i.e., the MMSE) of MMV signal estimation. In contrast, this work explores performance limits and designs an algorithm that can minimize [*arbitrary additive error metrics*]{} beyond MSE.

[**Contributions:**]{} This paper combines insights from Zhu et al. [@ZhuBaronKrzakala2017IEEE] and Tan and coauthors [@Tan2012SSP; @Tan2014], thus yielding a stronger understanding of the MMV problem, which could be extended in future work to other distributed signal acquisition settings, beyond MMV. To be more specific, we make several contributions in this paper. First, by extending Tan and coauthors [@Tan2012SSP; @Tan2014] we provide an algorithm based on a message passing (MP) framework [@RanganGAMP2011ISIT; @Krzakala2012probabilistic] that can be adapted to minimize the expected 