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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2512.23195 |
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| _version_ | 1866914223385214976 |
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| author | Nakajima, Shuta |
| author_facet | Nakajima, Shuta |
| contents | We study the replica-symmetric saddle point equations for the Ising perceptron with Gaussian disorder and margin $κ\ge 0$. We prove that for each $κ\ge 0$ there is a critical capacity $α_c(κ)=\frac{2}{π\,\mathbb E[(κ-Z)_+^2]}$, where $Z$ is a standard normal and $(x)_+=\max\{x,0\}$, such that the saddle point equation has a unique solution for $α\in(0,α_c(κ))$ and has no solution when $α\ge α_c(κ)$. When $α\uparrow α_c(κ)$ and $κ>0$, the replica-symmetric free energy at this solution diverges to $-\infty$. In the zero-margin case $κ=0$, Ding and Sun obtained a conditional uniqueness result, with one step verified numerically. Our argument gives a fully analytic proof without computer assistance. We used GPT-5 to help develop intermediate proof steps and to perform sanity-check computations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23195 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness of Replica-symmetric Saddle Point for Ising Perceptron Nakajima, Shuta Probability Statistical Mechanics We study the replica-symmetric saddle point equations for the Ising perceptron with Gaussian disorder and margin $κ\ge 0$. We prove that for each $κ\ge 0$ there is a critical capacity $α_c(κ)=\frac{2}{π\,\mathbb E[(κ-Z)_+^2]}$, where $Z$ is a standard normal and $(x)_+=\max\{x,0\}$, such that the saddle point equation has a unique solution for $α\in(0,α_c(κ))$ and has no solution when $α\ge α_c(κ)$. When $α\uparrow α_c(κ)$ and $κ>0$, the replica-symmetric free energy at this solution diverges to $-\infty$. In the zero-margin case $κ=0$, Ding and Sun obtained a conditional uniqueness result, with one step verified numerically. Our argument gives a fully analytic proof without computer assistance. We used GPT-5 to help develop intermediate proof steps and to perform sanity-check computations. |
| title | Uniqueness of Replica-symmetric Saddle Point for Ising Perceptron |
| topic | Probability Statistical Mechanics |
| url | https://arxiv.org/abs/2512.23195 |