signals with common sparse supports when they obey .*]{}

Note that there are other types of joint sparsity [@BaronDCStech] that fit into the MMV framework. For example, an MMV model with signal vectors that have slowly changing supports is discussed in Ziniel and Schniter [@ZinielSchniter2013MMV]. Since this paper only focuses on the MMV problem with signals sharing common sparse supports , we refer to joint sparsity with common sparse supports as joint sparsity for brevity.

[**Measurement models**]{}: Each signal $\x^{(j)}$ is measured by a measurement matrix $\A^{(j)}\in\mathbb{R}^{M\times N}$ before being corrupted by a random measurement channel, $$\label{eq:linearMixing}
\begin{split}
\w^{(j)}&=\A^{(j)}\x^{(j)},\ y_m^{(j)}=\mathcal{Z}(w_m^{(j)}),\\
 &\quad m\in\{1,\ldots,M\},\ j\in\{1,\ldots,J\},
\end{split}$$ where $y_m^{(j)}, m\in \{1,\ldots,M\}$ are the entries of the measurements $\y^{(j)}$, and the measurement channel $\mathcal{Z}(\cdot)$ is characterized by the pdf $f(y_m^{(j)}|w_m^{(j)})$. In this paper, we only focus on independent and identically distributed (i.i.d.) parallel measurement channels, i.e., the pdf’s $f(y_m^{(j)}|w_m^{(j)}), \forall m,j$, are identical and there is no cross-talk among different channels; our proposed algorithm is readily extended to parallel channels with different $f(y_m^{(j)}|w_m^{(j)}), \forall m,j$. When the number of signal vectors $J=1$, we call this MMV model an SMV problem .

\[def:largeSystemLimit\] [*The signal length $N$ scales to infinity, and the number of measurements 