Capacity lower bound for the Ising perceptron

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 $\alpha_N \equiv M_N/N$ for the largest integer $M_N$ such that the intersection in nonempty. It is conjectured by Krauth and M\'ezard (1989) that the (random) ratio $\alpha_N$ 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 $\gamma \le \alpha_N \le 1-\gamma$ with high probability; see also Talagrand (1999). In this paper we show that the Krauth--M\'ezard conjecture $\alpha_\star$ is a lower bound with positive probability, under the condition that an explicit univariate function $S(\lambda)$ 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). For verifying the condition on $S(\lambda)$ we outline one approach, which is implemented in the current version using (nonrigorous) numerical integration packages. In a future version of this paper we intend to complete the verification by implementing a rigorous numerical method.