Uniqueness of Replica-symmetric Saddle Point for Ising Perceptron

TL;DR

This paper proves the uniqueness of the replica-symmetric saddle point for the Ising perceptron with Gaussian disorder across different capacities, providing a fully analytic proof that clarifies the phase transition behavior.

Contribution

It establishes a rigorous, analytic proof of the uniqueness of the saddle point for the Ising perceptron model, extending previous numerical results and eliminating the need for computer assistance.

Findings

Unique saddle point for capacities below critical value

Divergence of free energy at critical capacity

Analytic proof without computer aid

Abstract

We study the replica-symmetric saddle point equations for the Ising perceptron with Gaussian disorder and margin $\kappa\ge 0$. We prove that for each $\kappa\ge 0$ there is a critical capacity $\alpha_c(\kappa)=\frac{2}{\pi\,\mathbb E[(\kappa-Z)_+^2]}$, where $Z$ is a standard normal and $(x)_+=\max\{x,0\}$, such that the saddle point equation has a unique solution for $\alpha\in(0,\alpha_c(\kappa))$ and has no solution when $\alpha\ge \alpha_c(\kappa)$. When $\alpha\uparrow \alpha_c(\kappa)$ and $\kappa>0$, the replica-symmetric free energy at this solution diverges to $-\infty$. In the zero-margin case $\kappa=0$, 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.