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

Fuente: arXiv
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Main Authors: Genovese, Giuseppe, Piana, Arianna
Format: Preprint
Published: 2026
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author Genovese, Giuseppe
Piana, Arianna
author_facet Genovese, Giuseppe
Piana, Arianna
contents 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.
format Preprint
id arxiv_https___arxiv_org_abs_2602_15191
institution arXiv
publishDate 2026
record_format arxiv
spellingShingle Derivation of the AMP equations from belief propagation for the $\ell_2$ minimisation problem
Genovese, Giuseppe
Piana, Arianna
Probability
Statistics Theory
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.
title Derivation of the AMP equations from belief propagation for the $\ell_2$ minimisation problem
topic Probability
Statistics Theory
url https://arxiv.org/abs/2602.15191