index.\
  $\delta=\frac{1}{NJ}\sum_{l=1}^N\sum_{j=1}^J(\widehat{a}^j_l-a^j_l)^2$ // Change in estimate\
[**Outputs:**]{} Estimate $\a^j,\forall j$

We simulated the signals in  with $J=3$ signal vectors and sparsity rate $\rho=0.1$ measured by a channel  with measurement rate $R\in[0.11,0.24]$ and noise variance $\Delta\in[-20,-50]$ dB. For each setting, we generated 50 signals of length $N=5000$, and the resulting MSE compared to the predicted BP MSE is shown in Fig. \[fig:AMPoverMSE\].[^13]

![AMP simulation results ($\text{MSE}_{\text{AMP}}$) compared to the predicted BP MSE ($\text{MSE}_{\text{BP}}$) with $J=3$ jointly sparse signal vectors. The dashed curve, solid curve, and the curve comprised of little circles correspond to thresholds $R_c(\Delta),\ R_l(\Delta)$, and $R_{BP}(\Delta)$, respectively. Regions 1-4 are also marked. The shade denotes $\ln \l(\frac{\text{MSE}_{\text{AMP}}}{\text{MSE}_{\text{BP}}}\r)$, which we expect to be 0 (completely dark shades) in the entire $R$ versus $\Delta$ plane. The narrow bright band above the BP threshold indicates the mismatch of AMP simulated MSE to the BP predicted MSE.[]{data-label="fig:AMPoverMSE"}](simMSE_over_predMSE_J3_TSP_v2.pdf){width="8cm"}

The labels of the thresholds are omitted for brevity. We can see that AMP simulation results match with the predicted MSE of BP and BP phase transition from the replica analysis of Section \[sec:phaseTrans\]. Note that there is a narrow band of light shades above the BP threshold, $R_{BP}(\Delta)$ (the top threshold), meaning that the simulated 