a certain SMV case is provided below, followed by our conjecture that  is optimal in the MMV case.

\[lemma:optSMV\] Consider an SMV  in the large system limit (Definition \[def:largeSystemLimit\]) with an AWGN measurement channel, $f(y_m|w_m)=\frac{1}{\sqrt{2\pi\sigma_Z^2}}\exp\l[\frac{(y_m-w_m)^2}{2\sigma_Z^2}\r], \forall m$. The estimate $\widetilde{\x}_n$  is optimal in the sense that $$\label{eq:optSMV}
\lim_{N\rightarrow\infty} \frac{1}{N}\sum_{n=1}^N d_{\text{UD}}(\widetilde{x}_n,x_n) = \text{MUDE},$$ where MUDE denotes the minimum user-defined error, if all the conditions below hold.

1.  The entries of the measurement matrix are i.i.d. Gaussian, $A_{mn}\sim \mathcal{N}(0,\frac{1}{N})$,

2.  the signal entries are i.i.d. with bounded fourth moment $\mathbb{E}[X^4]<B$, where $B$ is some constant,

3.  the free energy given by replica analysis has one fixed point [@ZhuBaronCISS2013; @ZhuBaronKrzakala2017IEEE; @Krzakala2012probabilistic],[^6]

4.  the user-defined error metric $d_{\text{UD}}(\cdot,\cdot)$ is pseudo-Lipschitz [@Bayati2011],[^7]

5.  the optimal estimator  as a function of $q_n,\ \widetilde{x}_n(q_n): \mathbb{R}\rightarrow \mathbb{R}$ , is Lipschitz continuous, and

6.  Part 1 converges before entering Part 2 in Algorithm \[algo:metric\_opt\_MMV\].

According to Theorem 1 in Bayati and Montanari [@Bayati2011], we know that $\lim_{N\rightarrow\infty} \frac{1}{N}\sum_{n=1}^N d_{\text{UD}}(\widetilde{x}_n(q_n),x_n) = \mathbb{E}\l[d_{\text{UD}}(\widetilde{x}_{n}(X+\Delta_v Z),X)\r]$, where $X$ is a random variable following the same distribution as the signal entries, $Z\sim \mathcal{N}(0,1)$, and $\Delta_v$ is given in Line \[line:mean\_delta\] of Algorithm \[algo:AMP\_MMV\]. The question boils down to showing that $\text{MUDE} = \mathbb{E}\l[d_{\text{UD}}(\widetilde{x}_{n}(X+\Delta_v Z),X)\r]$. 