to be numerically approximated.

![Comparison of simulation results to theoretic MMWSE for weighted support set estimation under different number of channels $J$ and measurement rates $R$ ($\beta=0.2$, noise variances $\Delta_z=0.01$).[]{data-label="fig:SimWSE"}](WeightSuppEstSimVsTheory.pdf){width="8cm"}

Synthetic Simulations {#sec:numeric_synth}
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After deriving the minimum mean weighted support set error (MMWSE) and minimum mean absolute error (MMAE) estimators, this section provides numerical results for Algorithm \[algo:metric\_opt\_MMV\]. In the case of i.i.d. random matrices and AWGN channels , replica analysis yields the MMSE of the MMV problem [@ZhuBaronKrzakala2017IEEE]. By inverting the MMSE (details in Appendix \[app:inverMMSE\]), we obtain the scalar channel noise variance $\Delta_v$,[^9] which characterizes the posterior $f(\x_n|\q_n)$. Given $\Delta_v$, we characterize the MMWSE and MMAE theoretically. In the following simulations, we use i.i.d. Gaussian matrices with unit-norm rows, i.i.d. $J$-dimensional Bernoulli-Gaussian signals  with $J=1,3$, and 5, and sparsity rate $\rho=0.1$. The signal length is $N=10000$, and the measurement rate $R=\frac{M}{N}$ varies from $0.3$ to $0.7$. For each setting, the simulation results are averaged over 50 realizations of the problem.

**Mean weighted support set error in AWGN channels:** We simulate AWGN channels  in this case with the noise variance being $\Delta_z\in \{0.01,0.001\}$. Fig. \[fig:SimWSE\] shows the weighted support set estimation results using our metric-optimal algorithm compared to 