entries in the signal $\x$. The round circles are the variable nodes (representing the distribution of the signal), and the squares denote the factor nodes (representing the measurement channel). The variables $x_n,\ \forall n$, are driven by each factor node $f(x_n)$ individually, because the signal has i.i.d. entries $x_n$. There are two types of messages passed in the factor graph shown on the left panel of Fig. \[fig:factorGraph\]: the message passed by the variable node $x_n$ to the factor node $y_m$, $\mathcal{M}_{n\rightarrow m}(x_n)$, and the message passed by the factor node $y_m$ to the variable node $x_n$, $\widehat{\mathcal{M}}_{m\rightarrow n}(x_n)$. According to the literature on MP algorithms [@RanganGAMP2011ISIT; @Krzakala2012probabilistic; @BarbierKrzakala2017IT], we have the following relation: $$\begin{split}
\mathcal{M}_{n\rightarrow m}(x_n) &=\frac{1}{Z_{n\rightarrow m}}f(x_n) \prod_{\widehat{m} \neq m} \widehat{\mathcal{M}}_{\widehat{m}\rightarrow n}(x_n), \\
\widehat{\mathcal{M}}_{m\rightarrow n}(x_n) &=\frac{1}{Z_{m\rightarrow n}} \int f(y_m | \x) \prod_{\widehat{n}\neq n} \mathcal{M}_{\widehat{n}\rightarrow m}(x_{\widehat{n}})\prod_{\widehat{n}\neq n}dx_{\widehat{n}},
\end{split}$$ where we use a single integral sign to denote a multi-dimensional integration for brevity, and $Z_{n\rightarrow m}$ and $Z_{m\rightarrow n}$ are normalization factors. The aim of this paper is not the detailed derivation of MP algorithms. Instead, the key property of MP utilized in this paper is that MP converts  into the following equivalent scalar channel, $$\q=\x+\v,$$ where $\q$ is the 