Derivation of the AMP equations from belief propagation for the $\ell_2$ minimisation problem
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918342602784768 |
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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 |