better trade-offs between true positives and false positives.

![Receiver operating characteristic curves for weighted support set estimation under different channel noise variances $\Delta_z$ and number of channels $J$ ($\beta=0.2$, measurement rate $R=0.3$).[]{data-label="fig:ROC"}](ROCweightSuppEst.pdf){width="8cm"}

**Mean absolute error in logistic channels:** We simulate i.i.d. logistic channels  with parameters $a=10$ and $30$; the smaller $a$ is, the noisier the channel becomes. Fig. \[fig:MAEvsMMAE\] plots the simulated MAE (crosses) and the theoretic MMAE  (curves) for various settings.[^10] Different colors and line shapes refer to different $a$’s and $J$’s, respectively. We can see that our simulation results match the theoretically optimal performance.

![Comparison of simulation results to theoretic MMAE under different logistic channels , number of channels $J$, and measurement rates $R$.[]{data-label="fig:MAEvsMMAE"}](logitMAE.pdf){width="8cm"}

**Remark:** Both simulations yield better performance for larger $J$. This is intuitive, because more signal vectors that share the same support should make the estimation process easier due to more information being available.

![Simulation results of OMP and Algorithm \[algo:metric\_opt\_MMV\] for active user detection in multi-user communication (SMV).[]{data-label="fig:OMP"}](OMP_vs_metric_opt.pdf){width="8.5cm"}

Application {#sec:app}
===========

In this section, we discuss active user detection (AUD) in a multi-user communication setting that can be viewed as compressed sensing [@DonohoCS; @CandesRUP; @BaraniukCS2007] (CS, which is closely related to SMV) in some 