Derivation of the AMP equations from belief propagation for the $\ell_2$ minimisation problem

We consider the $\ell_p$-minimisation, which consists of finding the vector $x\in\mathbb{R}^N$ which minimises $\|x\|_p$ subject to the linear constraint $y=Ax$, where $y\in\mathbb{R}^m$ is given and $A$ is a $m\times N$ random matrix with i.i.d. sub-Gaussian centred entries ($m<N$). This can be viewed as the zero temperature version of a statistical mechanics problem, in which one introduces a suitable Gibbs measure on $\mathbb{R}^N$. To such a Gibbs measure there are associated belief propagation equations. We prove in the easiest case $p=2$ that the means of the distributions obtained by the belief propagation iteration satisfy asymptotically the approximate message passing equations.