In contrast, in Regions 1 and 2, the optimal algorithm yields an MMSE that decreases with the noise variance $\Delta$. To summarize, the optimal algorithm yields poor estimation performance below the low noise threshold $R_{l}(\Delta)$, and good performance above $R_{l}(\Delta)$.

We further examine the MMSE as a function of the number of jointly sparse signal vectors $J$ and the measurement rate $R$. We plotted the MMSE in dB scale in Fig. \[fig:MMSE\_R\_J\]. The noise variance is -35 dB. We can see that the MMSE decreases with more signal vectors $J$ and greater measurement rate $R$. However, the MMSE depends less on $J$ as $J$ is increased. Note that the discontinuity in the MMSE surface in Fig. \[fig:MMSE\_R\_J\] is a result of the different performance regions that the various settings (different $J$ and $R$) lie in.

BP phase transition {#sec:phaseTrans}
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Belief propagation (BP) [@DMM2009; @CSBP2010; @Montanari2012; @Bayati2011; @Krzakala2012probabilistic; @krzakala2012statistical; @Barbier2015] is an algorithmic framework motivated by statistical physics, which can often achieve the optimal estimation performance (MMSE). When there are multiple local maxima $D_1<D_2$ in the free energy , BP converges to the local maximum with the larger MSE, $D_2$ [@DMM2009; @Montanari2012; @Bayati2011; @Krzakala2012probabilistic; @krzakala2012statistical]. Hence, $D_2$ characterizes the [*predicted 