We consider the Ising perceptron with gaussian disorder, which is equivalent to the discrete cube \(\{-1,+1\}^N\) intersected by M random half-spaces. The perceptron’s capacity is the largest integer \(M_N\) for which the intersection is nonempty. It is conjectured by Krauth and Mézard (1989) that the (random) ratio \(M_N/N\) converges in probability to an explicit constant \(\alpha _\star \doteq 0.83\) . Kim and Roche (1998) proved the existence of a positive constant \(\gamma \) such that \(\gamma \leqslant M_N/N \leqslant 1-\gamma \) with high probability; see also Talagrand (1999). In this paper we show that the Krauth–Mézard conjecture \(\alpha _\star \) is a lower bound with positive probability, under the condition that an explicit univariate function \(\mathscr {S}_\star (\lambda )\) is maximized at \(\lambda =0\) . Our proof is an application of the second moment method to a certain slice of perceptron configurations, as selected by the so-called TAP (Thouless, Anderson, and Palmer, 1977) or AMP (approximate message passing) iteration, whose scaling limit has been characterized by Bayati and Montanari (2011) and Bolthausen (2012).
