Capacity threshold for the Ising perceptron

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 \(\mathscr{S}_\star(\lambda_1,\lambda_2)\) 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.