$M=M(N)$ depends on $N$ and also scales to infinity, where the ratio approaches a positive constant $R$,*]{} $$\lim_{N\rightarrow\infty} \frac{M(N)}{N} = R>0.$$

We call $R$ the measurement rate.

For MMV problems, we are given the matrices $\A^{(j)}$ and measurements $\y^{(j)},\ \forall j$, as well as knowledge of the measurement channel . Our task is to estimate the underlying signal vectors $\x^{(j)},\ \forall j$. Suppose that the estimate is $\widehat{\x}^{(j)}$. Define $\X=[\x^{(1)},\cdots,\x^{(J)}]$ and $\widehat{\X}=[\widehat{\x}^{(1)},\cdots,\widehat{\x}^{(J)}]$. Therefore, $\X=[\x_1^T,\cdots,\x_N^T]^T$, where $\{\cdot\}^T$ denotes transpose. The estimation quality is quantified by a user-defined error metric $D_{\text{UD}}(\X,\widehat{\X})$, where the subscript UD denotes “user-defined." We define this [*additive error metric $D_{\text{UD}}(\cdot,\cdot)$ as*]{} $$D_{\text{UD}}(\X, \widehat{\X})={\sum_{n=1}^{N}}d_{\text{UD}}(\x_n,\widehat{\x}_n),$$ where $d_{\text{UD}}(\cdot,\cdot): \mathbb{R}^J\times \mathbb{R}^J \rightarrow \mathbb{R}$ is an arbitrary user-defined error metric on each super-symbol. The smaller the $D_{\text{UD}}(\cdot,\cdot)$ is, the better the estimation is.

Message passing algorithms {#sec:MPA}
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Message passing (MP) algorithms consider a factor graph [@RanganGAMP2011ISIT; @Krzakala2012probabilistic; @BarbierKrzakala2017IT], which expresses the relation between the signals $\x$ and measurements $\y$. We begin by discussing the factor graph for SMV, followed by that of MMV.

![Factor graph for SMV (left) and MMV (right).[]{data-label="fig:factorGraph"}](factorGraph.png){width="8cm"}

[**Factor graph for SMV:**]{} The left panel of Fig. \[fig:factorGraph\] illustrates the factor graph concept for an SMV problem  with i.i.d. 