Derivation of the AMP equations from belief propagation for the ${\mathrm{\ell }}_2$ minimisation problem

We consider the ${\mathrm{\ell }}_p$-minimisation, which consists of finding the vector $x\in {\mathbb{R}}^N$ which minimises $\mathrm{\parallel }x{\mathrm{\parallel }}_p$ 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.