Sharp threshold sequence and universality for Ising perceptron models
Abstract
We study a family of Ising perceptron models with -valued activation functions. This includes the classical half-space models, as well as some of the symmetric models considered in recent works. For each of these models we show that the free energy is self-averaging, there is a sharp threshold sequence, and the free energy is universal with respect to the disorder. A prior work of Xu (2019) used very different methods to show a sharp threshold sequence in the half-space Ising perceptron with Bernoulli disorder. Recent works of Perkins--Xu (2021) and Abbe--Li--Sly (2021) determined the sharp threshold and limiting free energy in a symmetric perceptron model. The results of this paper apply in more general settings, and are based on new "add one constraint" estimates extending Talagrand's estimates for the half-space model (1999, 2011).
Cite
@article{arxiv.2204.03469,
title = {Sharp threshold sequence and universality for Ising perceptron models},
author = {Shuta Nakajima and Nike Sun},
journal= {arXiv preprint arXiv:2204.03469},
year = {2022}
}