In fact, when Algorithm \[algo:AMP\_MMV\] converges, $\Delta_v$ corresponds to the scalar channel noise variance associated with the MMSE given by replica analysis [@Bayati2011; @RushVenkataramanan2016]. In addition, the MMSE provided by replica analysis is proved to be exact under the conditions asserted in Lemma \[lemma:optSMV\] [@ReevesPfister2016]. Therefore, $\mathbb{E}\l[d_{\text{UD}}(\widetilde{x}_{n}(X+\Delta_v Z),X)\r]$ is indeed the MUDE. Hence, we proved , and the estimator  is optimal.

**Remark 1:** The optimality of the metric-optimal estimator $\widetilde{\x}_n$  is stated in the sense that the ensemble mean user-defined error converges almost surely to the MUDE. It is possible that there exist other estimators that achieve the MUDE.

**Remark 2:** The proof is made possible by linking three rigorous proofs [@ReevesPfister2016; @Bayati2011; @RushVenkataramanan2016] from the prior art. That said, numerical examples in Section \[sec:numeric\_synth\] demonstrate that Algorithm \[algo:metric\_opt\_MMV\] yields promising results even if the conditions required by Lemma \[lemma:optSMV\] are not met.

After proving the optimality of our metric-optimal algorithm in the SMV scenario with certain conditions, we state the following conjecture.

\[conj:2\] In the large system limit, for the MMV model with the signal in  and the user-defined additive error metric $d_{\text{UD}}(\x_n,\widehat{\x}_n)$, the optimal estimate of the signal vectors is $$\widetilde{\x}_n=\arg\min_{\widehat{\x}_n} \mathbb{E}[d_{\text{UD}}(\x_n,\widehat{\x}_n)|\q_n].$$

As the reader can see from 