Capacity threshold for the Ising perceptron

TL;DR

This paper establishes an upper bound on the capacity of the Ising perceptron that aligns with a longstanding conjecture, contingent on a specific mathematical condition involving a two-variable function.

Contribution

It provides a conditional proof confirming the conjectured capacity threshold for the Ising perceptron by analyzing a key optimization condition.

Findings

Upper bound on Ising perceptron capacity approximately 0.833

Conditional proof of Krauth and Me9zard's conjecture

Alignment with previous lower bound results

Abstract

We show that the capacity of the Ising perceptron is with high probability upper bounded by the constant $\alpha_\star \approx 0.833$ conjectured by Krauth and M\'ezard, under the condition that an explicit two-variable function $\mathscr{S}_*(\lambda_1,\lambda_2)$ is maximized at $(1,0)$. The earlier work of Ding and Sun proves the matching lower bound subject to a similar numerical condition, and together these results give a conditional proof of the conjecture of Krauth and M\'ezard.