MSE*]{} for BP. Moving from Region 1 to Region 2 by decreasing the measurement rate $R$ with fixed noise variance $\Delta$, the number of local maxima increases from 1 to 2. Therefore, BP estimation performance experiences a sudden deterioration (increase in MSE) when the measurement rate $R$ drops such that the combination of the noise variance $\Delta$ and measurement rate $R$ moves from Region 1 to Region 2. The BP threshold, $R_{BP}(\Delta)$, is the boundary between Regions 1 and 2, and is where the BP phase transition happens. That is, BP achieves poor estimation performance below $R_{BP}(\Delta)$, and good performance above $R_{BP}(\Delta)$.

[**Remark 2**]{}: In Fig. \[fig:PerformanceRegions\], we see that increasing $J$ reduces the BP threshold $R_{BP}(\Delta)$. Since BP achieves the MMSE when $R>R_{BP}(\Delta)$, increasing $J$ is beneficial to applications that use BP as the estimation algorithm.

[**Remark 3**]{}: We further analyzed the low noise ($\Delta\rightarrow 0$) and zero noise ($\Delta=0$) cases. The critical threshold $R_c(\Delta)$ converges to $\rho$ as the noise variance $\Delta$ is decreased for $J=1,3$, and $5$. We believe that this numerical result holds for every $J$. Moreover, this result matches the theoretical robust threshold of Wu and Verd[ú]{} [@WuVerdu2012] for $J=1$ in the low noise limit. 