noisy [*pseudo data*]{}, and $\v$ is the equivalent scalar channel additive white Gaussian noise (AWGN) whose variance $\Delta_v$ can be approximated. After obtaining $\q$ and $\Delta_v$, each variable node $x_n$ updates the estimate $\widehat{x}_n$ by denoising $q_n$. If the signal entries are i.i.d. and the matrix is either i.i.d. or sparse and locally tree-like, then MP algorithms yield a density function $f(x_n|q_n)$ that is statistically equivalent to $f(x_n|\y)$ [@RanganGAMP2011ISIT].

[**Factor graph for MMV:**]{} An MMV problem with jointly sparse signals can be expressed as the factor graph shown in the right panel of Fig. \[fig:factorGraph\]. We can see that the MP in each channel is similar to an SMV problem. The only difference is that the $J$ variable nodes $x_n^{(j)},\ j\in\{1,\ldots,J\}$, for fixed $n$ are driven by one factor node $f(\x_n)$. By grouping the entries from different signal vectors together into super-symbols as in , we have i.i.d. super-symbols. The noisy super-symbol pseudo data $\q_n=[q_n^{(1)},\ldots,q_n^{(J)}]$ is denoised, in order to update the estimate for $\x_n=[x_n^{(1)},\ldots,x_n^{(J)}]$.

Main Results {#sec:main}
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We first present our metric-optimal algorithm in Section \[sec:achievable\] and then in Section \[sec:converse\] we rigorously prove that our metric-optimal algorithm is optimal in the SMV case under certain conditions.[^4] Based 