error for arbitrary additive error metrics. Our algorithm first runs MP until it converges or reaches some stopping criteria, and we then denoise MP’s output using a denoiser that minimizes the given additive error metric. Second, we prove rigorously that our algorithm is optimal in a specific SMV case,[^3] and further conjecture the optimality of our algorithm in MMV. Third, as an example, we derive performance limits for MAE and mean weighted support set error (MWSE) by designing the corresponding optimal denoisers, based on the scalar channel noise variance (Section \[sec:MPA\]) derived from replica analysis (Appendix \[app:inverMMSE\]) [@ZhuBaronKrzakala2017IEEE]. Simulation results show the superiority and optimality of our algorithm. We note in passing that having more signal vectors in MMV helps reduce the estimation error. Finally, as an application of MMV and our algorithm, we propose a novel setup for active user detection in multi-user communications (details in Section \[sec:app\]) and demonstrate the promise of our proposed setup through simulation.

[**Organization:**]{} The remainder of the paper is organized as follows. We introduce our problem setting and MP algorithms in Section \[sec:background\]. Our algorithmic framework, which can minimize arbitrary additive error metrics, is proposed in Section \[sec:achievable\]; we rigorously prove the optimality 